window-integrated low concentration planar light guide solar concentrators by daniel james

Window-Integrated Low Concentration Planar Light Guide Solar Concentrators

by

Daniel James Lawler Williams

Submitted in Partial Fulfillment of the

Requirements for the Degree

Doctor of Philosophy

Supervised by Professor Duncan T. Moore

The Institute of Optics

Arts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of Rochester

Rochester, NY

2016

ii

Biographical Sketch

Daniel Williams was born in Newark, Delaware. He attended the University of Nebraska –

Lincoln and graduated with distinction, receiving a Bachelor of Science degree in both Physics

and Mathematics. He was awarded the Summer Undergraduate Research Fellowship by the

National Institute of Standards and Technology in 2006. He continued his education at The

Institute of Optics, University of Rochester. During graduate studies he was awarded the

Integrative Graduate Education and Research Traineeship by the National Science Foundation in

2010. He received a Master of Technical Entrepreneurship and Management degree, a joint

degree of the Simon School of Business and Hajim School of Engineering and Applied Science in

2012 and a Master of Science degree in Optics from The Institute of Optics in 2014. He pursued

his research in Optics under the direction of Professor Duncan T. Moore.

The following publications were a result of the work conducted during this doctoral study:

Yang Zhao, Daniel J. L. Williams, Peter McCarthy, et al., "Chromatic correction for a VIS-SWIR zoom lens using optical glasses", Proceedings of SPIE Vol. 9580, 95800E (2015)

Anthony J. Yee, Daniel J. L. Williams, Gustavo A. Gandara-Montano, et al., "New tools for finding first-order zoom lens solutions and the analysis of zoom lenses during the design process", Proceedings of SPIE Vol. 9580, 958006 (2015)

Peter McCarthy, Rebecca Berman, Daniel J. L. Williams, et al., "Optical design study in the 1-5μm spectral band with gradient-index materials", Proceedings of SPIE Vol. 9293, 92930X (2015)

Berman, Rebecca, et al. "Optical design study of a VIS-SWIR 3X zoom lens." SPIE Optical Engineering+ Applications. International Society for Optics and Photonics, 2015.

iii

Acknowledgements

I would like to thank my advisor, Duncan Moore, for his academic guidance and

foresight in preparing me for post-graduate life.

I would like to thank Blair Unger, Eric Christensen, and Mick Brown for their

indispensable mentorship and their work in the Moore group that made my research possible.

I would like to thank Greg Schmidt for his selflessness and absolute willingness to help

his colleagues and share his broad range of knowledge.

I would like to thank my colleagues, Rebecca Berman, Anthony Yee, and Yang Zhao for

making our office a great and friendly place to work.

I would like to thank Per Adamson for his help on numerous occasions and his incredible

drive to help the students of The Institute of Optics.

I would like to thank Lynn Doescher and Evelyn Sheffer for their support of the Moore

group and for bringing a caring atmosphere to our offices.

I would like to thank my parents, John and Kathleen, for their consistent support and

interest in my work both during and prior to my graduate studies.

I would like to thank my parents-in-law for supporting me and my wife and welcoming

me into their family.

I would like to thank my friends, especially Zack Lapin, Richard Smith, Nicole Kendrot,

and Greg Schmidt for making life outside of research so much more enjoyable.

Finally, I would like to thank my amazing wife and best friend, Jennifer Muniak, for her

incredible support and encouragement during my graduate studies, and for bringing our

wonderful daughter, Penelope, into this world.

iv

Abstract

Several novel low concentration solar concentrator photovoltaic designs are presented,

based on the planar light guide architecture pioneered by the University of Rochester. These

systems are designed for integration into windows, requiring them to be stationary and to have

a large acceptance angle since they cannot move to track the sun. The application goal is to use

solar generated electricity to offset the energy lost through the window during cold times of the

year. The systems are evaluated for their effective insulation properties given the calculated net

energy lost. Without moving parts, they optimize to have acceptance angles of about 20° to 35°

in the vertical direction and ±90° in the horizontal direction, but have peak optical efficiencies of

less than 50%. By including internally moving parts, the acceptance angle is increased to nearly

a full π steradians (the full sky from the point of view of the window) and the average optical

efficiency increases to over 50%. Systems in certain locations are not viable due to low solar

irradiance in the wintertime, e.g., Rochester, NY. Others, however, reduce net energy loss to

zero for much of the year. A prototype of one of the systems is fabricated, measured, and

modeled. The simulated and measured performance data are compared and are in close

agreement, validating the model and the evaluation methods used during system optimization.

v

Contributors and Funding Sources

This work was supervised by a dissertation committee consisting of Professors Duncan

Moore, James Zavislan, and Julie Bentley of the Institute of Optics and Matthew Yates of the

Chemical Engineering Department at the University of Rochester. The initial design work was

funded by Rambus International Ltd. under Award Number – 056105-002. Additional funding

was provided by the National Science Foundation under the Integrative Graduate Education

Research and Traineeship (IGERT) program and L-3 Communications.

vi

Table of Contents

Biographical Sketch ......................................................................................................................................... ii

Acknowledgements ........................................................................................................................................ iii

Abstract .......................................................................................................................................................... iv

Contributors and Funding Sources .................................................................................................................. v

List of Tables ................................................................................................................................................. viii

List of Figures .................................................................................................................................................. ix

Chapter 1: Background ................................................................................................................................ 1

1.1 Solar Power ...................................................................................................................... 1

1.2 Concentrating Photovoltaics (CPV) .................................................................................. 2

1.3 Concentrator Design Theory .......................................................................................... 13

1.4 Low Concentration Photovoltaics .................................................................................. 18

1.5 Examples of Existing Low Concentration Systems ......................................................... 24

1.6 Examples of Non-Concentrating BIPV ............................................................................ 29

1.7 Window Integrated LCPV Planar Light Guide Solar Concentrators ................................ 30

Chapter 2: Design ...................................................................................................................................... 31

2.1 Application ..................................................................................................................... 31

2.2 The Louver Concept ....................................................................................................... 35

2.3 Optimization and Results ............................................................................................... 39

2.4 Analysis .......................................................................................................................... 49

2.5 Conclusion ...................................................................................................................... 63

Chapter 3: Prototype and Metrology ........................................................................................................ 65

3.1 Prototype ....................................................................................................................... 65

vii

3.2 Fabrication Methods ...................................................................................................... 69

3.3 Metrology Tools ............................................................................................................. 80

Chapter 4: Measurements and Computer Model ................................................................................... 100

4.1 Component Measurements ......................................................................................... 100

4.2 Performance ................................................................................................................ 121

4.3 Conclusion .................................................................................................................... 137

Chapter 5: Conclusion ............................................................................................................................. 138

5.1 Summary ...................................................................................................................... 138

5.2 Future Work ................................................................................................................. 139

References ................................................................................................................................................... 141

Appendix A. Design Specifications ......................................................................................................... 148

Appendix B. Deflectometer MATLAB Code ........................................................................................... 156

Appendix C. Optimization Routine MATLAB Code ................................................................................ 261

viii

List of Tables

Table 2-1: Thermal conductivities for various materials [26]. ...................................................................... 54

Table 2-2: Vertical angle field of view summary. The left data column indicates the vertical angle field of

view range for each design, the center column shows the average optical efficiency over that range,

and the right column gives the peak optical efficiency. ...................................................................... 63

Table 4-1: Guide layer designed vs. measured dimensions. ...................................................................... 100

Table 4-2: Injection prism designed vs. measured dimensions. ................................................................. 106

Table 4-3: Light I-V curve parameters for both PV cells. ............................................................................ 121

Table 4-4: Summary of simulated manufacturing flaws. The flaws are added to the model in the order

shown. ............................................................................................................................................... 124

ix

List of Figures

Figure 1.1: Residential silicon flat panel PV solar installation (top) and the Solucar PS10 solar thermal

power plant near Seville, Spain (bottom, [55], originally posted to Flickr by afloresm at

http://flickr.com/photos/74424373@N00/1448540890). .................................................................... 1

Figure 1.2: Fresnel lens based CPV array on a large tracking mechanism. "Amonix7700" by Mbudzi - Own

work - https://commons.wikimedia.org/wiki/File:Amonix7700.jpg#/media/File:Amonix7700.jpg. .... 2

Figure 1.3: Spectral response of a silicon solar cell. Efficiency on the vertical axis is in units of current (A)

output per light power (W) input. The red curve is an ideal cell considering only the band

gap/photon energy differential and the blue is a measured real-world cell. [5] .................................. 5

Figure 1.4: Example IV curves. The orange curve is measured data from a silicon PV cell, and the blue is

derived from the diode law with I0 and IL adjusted to match the measured data. The areas above the

dashed lines indicate the peak power output for each curve. The fill factors for the measured and

ideal curves are 0.52 and 0.82, respectively. ......................................................................................... 7

Figure 1.5: Electrical grid demand during a hot day in California [57]. .......................................................... 9

Figure 1.6: Fresnel lens CPV system diagram with secondary concentrator. Image source: Light

Prescriptions Innovators, [59]. ............................................................................................................. 11

Figure 1.7: Planar light guide CPV form factor. Sunlight enters the large face and is guided toward one

edge (in red) where the PV chip is placed. .......................................................................................... 12

Figure 1.8: Refractive planar light guide CPV components. ......................................................................... 12

Figure 1.9: Schematic of polar plane orientation with respect to the earth (blue) and the sun on the

equinox at solar noon (yellow). ........................................................................................................... 18

Figure 1.10: Recreated from Figure 2 of ref [15]. The bold black circle is the unit circle, the green lines

represent the sun’s position throughout the day of the winter (left) and summer (right) solstices, the

x

solid vertical lines represent the sun’s path for the first day of each month from Jan 1 to June 1 from

left to right, and the dashed lines do the same for July 1 to Dec 1 from right to left, and the blue

ellipse represents the eight hours per day boundary. ......................................................................... 19

Figure 1.11: Compound parabolic concentrator profile illustrating the edge ray principle. The blue rays

come from the edge of the source and focus to a point on the edge of the dark red target surface.

The red rays come from inside the edges of the source. .................................................................... 22

Figure 1.12: Direction cosine representation of the sun's position for eight hours per day, year round (grey

shaded area) and the fields of view of trough CPCs with an index of 1 (red ellipse) and 1.5 (blue

ellipse) that are just wide enough to capture sunlight for eight hours every day without tracking. .. 23

Figure 1.13: tenKsolar's RAIS™ PV system. The deep blue colored rows are silicon solar panels and the

translucent plates are cold mirrors that concentrate convertible sunlight onto the PV cells. [16] .... 25

Figure 1.14: See-through window CPV system schematic (top) and picture of prototype (bottom, [17]). . 26

Figure 1.15: Reproduced rendering of Stellaris' window CPV module showing extruded CPC troughs [18].

............................................................................................................................................................. 26

Figure 1.16: Reproduction from [21]. A schematic for a LSC showing the incoming sunlight, a luminescent

particle (orange circle), and re-emitted light that is lost (1) and trapped in the light guide (2). ......... 27

Figure 1.17: Luminescent concentrator plates. Image courtesy Fraunhofer ISE. ........................................ 28

Figure 1.18: Prism solar BIPV modules integrated into balcony railings [22]. .............................................. 29

Figure 2.1: LCPV field of view coordinate system definition. ....................................................................... 33

Figure 2.2: Map of sun's position in azimuth and elevation for Rochester, NY (latitude = 43.12°). Yellow

dots indicate sun position at various times. The black dashed lines indicate constant vertical angle.

............................................................................................................................................................. 33

Figure 2.3: Weighting functions for three cities in the U.S. ......................................................................... 34

Figure 2.4: Shadowing advantage of the louver design space, where part of the louver (a) is designed for

greater vertical angles and other parts (b) are optimized for lesser ones. ......................................... 36

xi

Figure 2.5: Illustration of a full window LCPV system consisting of several modules stacked top to bottom.

............................................................................................................................................................. 37

Figure 2.6: Cross-section view of a suspended louver design. The horizontal grey lines on the injection

features indicate a reflective surface. ................................................................................................. 37

Figure 2.7: Cross-section of an immersed louver design. ............................................................................. 38

Figure 2.8: Cross-section of an extruded segment lens design. Grey lines indicate mirrored surfaces. ..... 39

Figure 2.9: Optimization variables for suspended (and immersed) louver designs indicated in red. .......... 41

Figure 2.10: Solutions from the suspended louver optimization for the three chosen locations. Geometry

of the reflectors and injection features are displayed across the top, and the efficiency as a function

of vertical angle is shown below. ......................................................................................................... 42

Figure 2.11: Solutions from the immersed louver optimization for the three chosen locations. Geometry

of the reflectors and injection features are displayed across the top, and the efficiency as a function

of vertical angle is shown below. ......................................................................................................... 43

Figure 2.12: Optimization variables for extruded segment lens design indicated in red lettering. Traced

rays show the additional angled fold mirror in the ESL design (ray colors indicate angle after injection

into the guide layer)............................................................................................................................. 44

Figure 2.13: Solutions from the extruded segment lens optimization for the three chosen locations.

Geometry of the reflectors and injection features are displayed across the top, and the efficiency as

a function of vertical angle is shown below. ....................................................................................... 45

Figure 2.14: Tracking louver optimization variables shown in red. .............................................................. 47

Figure 2.15: Solutions from the tracking louver optimization for the three chosen locations. Geometry of

the reflector and injection features for the Rochester design are displayed at the top for several

vertical angles, and the efficiency as a function of vertical angle is shown below for all three cities. 48

Figure 2.16: Suspended louver Rochester design performance showing injection efficiency and weighting

function. ............................................................................................................................................... 49

xii

Figure 2.17: Rochester suspended louver design normalized efficiency components as a function of

position on the system's front aperture. The total is the sum of the top and bottom cells, while the

average is the mean of the total over all positions. ............................................................................ 50

Figure 2.18: Rochester SL design average efficiency and effective concentration as a function of guide

length, both normalized at the designed guide length of 300 mm. .................................................... 51

Figure 2.19: Effective U-factor calculations for the SL Rochester design. The bold blue line represents a

sunny year and the thin green line is from “typical meteorological year” data. The red line of the U-

factor plot represents the passive U-factor of the window. ................................................................ 54

Figure 2.20: Effective U-factor calculations for the SL Rochester design. Battery storage is included and its

charge level is shown in the bottom plot. ........................................................................................... 55

Figure 2.21: Effective U-factor calculations for the immersed louver (IL) El Paso design including battery

storage. ................................................................................................................................................ 57

Figure 2.22: Effective U-factor calculations for the tracking louver Fairbanks design including battery

storage. ................................................................................................................................................ 58

Figure 2.23: Effect of backside retroreflective texture on static louver designs: rejecting direct sunlight

during summer. Left image shows high vertical angle rays striking back of louver and continuing

through the window. Right image shows rays being turned back by a retroreflective texture on the

adjacent louver rear surface. ............................................................................................................... 59

Figure 2.24: Photorealistic renderings of views through the three static designs: SL (top), IL (middle), and

ESL (bottom). ....................................................................................................................................... 61

Figure 2.25: Schematic of building with passive solar heating. Shows absorbing indoor surfaces, thermal

mass storage, and shading control eve. .............................................................................................. 62

Figure 3.1: Dimensioned cross section of prototype design (lengths in mm). Shows nomenclature for faces

of prism and guide. The dashed red line shows that the prism has minimal interference with light

xiii

that moves past the baffle and is headed for reflector. The blue dotted lines show that the prism

back can be extended without obscuring the intended optical path through the device. .................. 66

Figure 3.2: Tolerance study on prototype prism dimensions. Solid blue line shows designed shape, dashed

black lines show envelope of the tolerance study which is achieved by varying α, β, and the lengths

of the front and back edges. ................................................................................................................ 68

Figure 3.3: Error envelope of tolerance study. The blue line shows the performance of the as-designed

prototype, the orange line shows the minimum performance from the compound geometry errors,

while the grey line shows the maximum. ............................................................................................ 69

Figure 3.4: Cross-section of prism shape modified for the milling fabrication process. The red dashed line

indicates the unmodified shape. ......................................................................................................... 71

Figure 3.5: Modeled (top) and 3D printed (bottom) reflector with reflective Mylar film adhered to surface.

Printed part is a first full scale manufacturing attempt with visible flaws in the surface. .................. 72

Figure 3.6: Mylar adhesive trial. "SC" and "LT" represent glues that harden as they set, and cause the

Mylar to become diffuse as a result. "RC" (rubber cement) and "GE" (silicone sealant) both stay

somewhat flexible as they set, allowing the Mylar surfaces to remain smooth. ................................ 73

Figure 3.7: 3D printed reflector structure secured to an optical bench (top), and a roller with Mylar sheet

and protective paper layer (bottom). .................................................................................................. 74

Figure 3.8: A silicon wafer with conductive gridlines. Red box indicates ideal dicing lines and the green,

the necessary dicing lines due to the crystal lattice orientation. Dashed lines indicate extra diced cell

area that does not contact the guide layer edge in the prototype assembly, leaving room for solder

connections.......................................................................................................................................... 75

Figure 3.9: Exploded view of prototype assembly with (a) prism and guide in green, (b) reflector in grey,

(c) baffle in black, (d) structural braces in brown, (e) PV cell clamp components in yellow, (f) wire

management hanger in black, and (g) and (h) indicating the top and bottom PV cell locations,

respectively. ......................................................................................................................................... 78

xiv

Figure 3.10: PV cell optical contact trial (top) and in the prototype looking through the length of the guide

layer (bottom). Top: the darker regions of the cell are area of good optical contact with the PMMA

cover plate and the lighter areas have poor contact. Bottom: good optical contact in the prototype

assembly. ............................................................................................................................................. 79

Figure 3.11: Magnified view of PV cell surface showing a ridged texture running perpendicular to the small

gridlines. .............................................................................................................................................. 79

Figure 3.12: Deflectometer reference flat mirror. ....................................................................................... 84

Figure 3.13: Fringe Reflection deflectometer schematic showing relative positions of the monitor display,

camera, and reflective part. ................................................................................................................ 84

Figure 3.14: Camera response to monitor input values for several viewing angles using default display

settings. ............................................................................................................................................... 86

Figure 3.15: Camera and monitor response curves with modifications (top), and the calculated phase error

based on the nonlinear response (bottom). The phase error is based on a four step phase shifting

algorithm and the shaded region estimate the error based on the noise in the measured data. ...... 88

Figure 3.16: Deflectometer calibration during monitor reflection positioning step. Shows (a) drawn

fiducials on monitor, (b) reflection of drawn white fiducials and black mirror fiducials on calibration

flat, and (c) camera. ............................................................................................................................. 90

Figure 3.17: Reflection of horizontal fringes from prototype reflector as recorded by camera. From phase

shift of the fringes from left to right is 0, π/2, π, and 3π/2. ................................................................ 91

Figure 3.18: Pixel line data for the calculation of absolute position. Horizontal (left) and vertical (right)

lines are shown. These images are superimposed with another photo of the part so the part outline

is visible. ............................................................................................................................................... 92

Figure 3.19: Results of PUMA phase unwrapping algorithm performed on the horizontal fringe data for

prototype reflector. Wrapped phase [-π, π] (left) and unwrapped phase (right). ............................. 93

xv

Figure 3.20: Illustration of multiple solutions to surface position problem. The solid red line is the known

ray coming from the camera, the green patches are possible surface positions with the black arrows

indicating the surface normals. ........................................................................................................... 94

Figure 3.21: Illustration of two different slope calculations. S1 and S2 are the slopes calculated from the

monitor position reflected in the part, and S12 is the slope calculated from the positions of the data

points. In this example, to get the average of S1 and S2 to equal S12, σ2 will need to increase relative

to σ1. .................................................................................................................................................... 95

Figure 3.22: Calibration flat measured in a Zygo laser interferometer. A cylindrical bend can be seen over

the aperture of the part with a peak to valley departure of about 1 µm. The bottom graph shows the

slice chart indicated by the line in the top graph. ............................................................................... 97

Figure 3.23: Comparison of the spectral power distribution of sunlight after passing through Earth's

atmosphere and a xenon arc lamp. ..................................................................................................... 98

Figure 3.24: Schematic of solar simulator with labeled components. Path of light though system shown in

yellow. .................................................................................................................................................. 98

Figure 4.1: Laser interferometer data of the guide front face. Dark blue indicates invalid data. ............. 101

Figure 4.2: Guide layer immersed in bed of corn syrup to suppress back surface reflection. ................... 102

Figure 4.3: Top: Deflectometer data from guide layer front surface. Bottom: magnified region as marked

in top graphic with surface figure subtracted out. A scratch and tool marks become visible. Vertical

and horizontal slice charts shown, all units are mm. ......................................................................... 103

Figure 4.4: Portion of guide front surface from the raw deflectometer data. Scratch and tool marks are

visible. ................................................................................................................................................ 104

Figure 4.5: Zygo laser interferometer data of the injection prism back surface. ....................................... 107

Figure 4.6: Representation of prism front surface slope in the (top) horizontal and (bottom) vertical

directions. .......................................................................................................................................... 108

xvi

Figure 4.7: White light interferometer data of prism top tool marks. Top graph shows registered data sets

and position of slice, bottom shows surface height values along the slice (blue line) and the

smoothed data (red line). .................................................................................................................. 109

Figure 4.8: Deflectometer data of prism top near corner showing tool marks. All dimensions are in mm.

........................................................................................................................................................... 110

Figure 4.9: Looking through the prism at the back surface mirror. ............................................................ 110

Figure 4.10: Schematic of prism mirror reflectance measurement. “Fresnel Reflection” and “Output” rays

measured with power meters. .......................................................................................................... 111

Figure 4.11: Reflectance data from the prism mirror as a function of position in the extrusion direction.

The blue line is the calculated reflectance with grey error bars and the orange is the stepped

function used in the prototype model. .............................................................................................. 112

Figure 4.12: Geometry of the modeled reflectivity zones for the prism mirror. Blue regions are uncoated,

and the remaining regions have uniform reflectances as indicated. ................................................. 113

Figure 4.13: Reflector surface shape measured via CMM (top) before Mylar film is applied and via the

deflectometer (middle) with the Mylar film in place. Error values are mm of departure from the

designed shape. The bottom image shows difference of the two data sets. ................................... 114

Figure 4.14: Principal curvature analysis of reflector surface as measured by the deflectometer. ........... 116

Figure 4.15: Stock Mylar film showing wavy topology. .............................................................................. 116

Figure 4.16: Magnified reflector surface data with the principal curvature, k2, on the left and a set of raw

vertical fringe data on the right. ........................................................................................................ 117

Figure 4.17: White light interferogram of the reflector's Mylar surface. A 2nd order 2D polynomial is

removed to reveal surface roughness features. ................................................................................ 117

Figure 4.18: Schematic of PV cell slit scan experiment. ............................................................................. 118

Figure 4.19: PV chip geometry data and results. Photograph of back of PV cell over ruled background

(top), photograph of front of cell (middle), and generated binary apodization mask (bottom). ...... 118

xvii

Figure 4.20: Measured and modeled efficiency profiles for prototype photovoltaic cells. ....................... 119

Figure 4.21: Dark I-V curves of prototype PV cells before and after assembly. ......................................... 120

Figure 4.22: Light I-V curve of each PV cell before assembly. .................................................................... 121

Figure 4.23: Performance measurement schematic of the prototype system. Shows the three positional

variables in the system. The Vertical Angle axis of rotation remains constant with respect to the

prototype structure. .......................................................................................................................... 122

Figure 4.24: Slit scan simulation data for ideal model (red), added prism dimensions flaw (green), then

added reflector shape flaw (blue). Simulation is performed with wide source to average out

translational variations in the measured reflector shape. Vertical axis is in relative units and are

equivalent for all three plots. ............................................................................................................ 125

Figure 4.25: Raytraces of the "Reflector Shape" model at three different vertical angles as indicated.

There are peaks in the top cell output at 24° and 34° with a valley in between. .............................. 126

Figure 4.26: Results of Field of View (FoV) simulation for ideal prototype model. .................................... 127

Figure 4.27: Results of slit scan simulations for a 4.88 mm wide slit fixed at the translational origin. The

reflector shape data set includes the prism dimensions flaw. The remaining data sets are for and

added back prism fillet then front fillet. ............................................................................................ 128

Figure 4.28: Effect of front fillet on bottom cell output near 30° vertical angle and 9 mm translational

position. ............................................................................................................................................. 129

Figure 4.29: Slit Scan simulation results for final three modeled manufacturing flaws. Scales are constant

with each row to show attenuation effects of each flaw. ................................................................. 130

Figure 4.30: Field of View (FoV) simulation results for final three modeled manufacturing flaws. Scales are

constant across each row to show attenuation effects of each flaw. ............................................... 132

Figure 4.31: Slit scan performance comparison. Measured results in the left column, simulated results in

the right column. Scale is consistent across each row. ..................................................................... 133

xviii

Figure 4.32: Field of view performance comparison. Measured results in the left column, simulated

results in the right column. Scale is consistent across each row. ..................................................... 134

Figure 4.33: Photo peering through back face of guide layer at the bottom of the reflector structure. Resin

from the bottom cell seeping into the gap between the reflector and guide appears as irregularly

shaped darker regions. ...................................................................................................................... 135

Figure 4.34: Simulation of final prototype model at a vertical angle of 47° showing many rays interacting

with the area containing the unwanted resin. .................................................................................. 136

Figure B.1: Shows vertical and horizontal wrapped phase data and asks user to outline part leaving a

buffer of background pixels. .............................................................................................................. 167

Figure B.2: PUMA algorithm unwraps data within region of interest. ....................................................... 168

Figure B.3: Results of noise amplitude analysis. ......................................................................................... 169

Figure B.4: Histogram of noise amplitudes. Red line shows cutoff amplitude for what is considered noise.

........................................................................................................................................................... 169

Figure B.5: Left column: noisy data deleted. Right column: noise mask edges smoothed and rouge data

pixels mended. ................................................................................................................................... 170

Figure B.6: Calculated X and Y positions on the monitor. Shading relates loosely to surface slope in each

direction. ............................................................................................................................................ 171

Figure B.7: Sample points from ideal surface shape are overlayed over data from the deflectometer

camera. The green and cyan dots on the edge denote the surface points used as fiducial lines

pertaining to the part edges. If a caliper tip fiducial is used, the user must pick the location of the

caliper tips in the image instead. ....................................................................................................... 173

Figure B.8: Optimization iterations to find calculated XYZ points of the part surface. The value is the

average distance from the camera origin. In this case, the initial guess is rather accurate. ............ 174

Figure B.9: Initial guess vs. optimization solution. ..................................................................................... 175

Figure B.10: XYZ data for calculated surface. ............................................................................................. 175

xix

Figure B.11: Principal curvature maps of calculated data. Principal curvature k1 (left), principal curvature

k2 (center), Gaussian curvature (right). ............................................................................................. 176

Figure B.12: Registered data showing error from ideal surface. ................................................................ 177

1

Chapter 1: Background

1.1 Solar Power

The recent push for alternative energy solutions stems from the multifaceted problem

caused by the overuse of fossil fuels in the U.S. and worldwide. The current U.S. federal

government administration supports the move towards alternative energy production and

energy efficiency as a way of “building a new, clean energy economy, ending our dependence on

foreign oil, and limiting the dangerous pollutants that threaten our health and the health of our

planet.” [1] One focus of alternative energy electricity production is solar energy.

By far, the most common method of converting solar radiation into electricity is through

the use of large areas of silicon photovoltaic (PV) cells, whose prices have been falling rapidly

since 2009. While utility companies have been harvesting more solar energy in the past several

years, the majority of the explosive growth in

U.S. solar energy production has come from

the residential and commercial sectors [2],

both of which primarily utilize silicon PV cells.

The next most common form of solar

energy harvesting is via solar thermal

systems [2]. For residential applications,

these are usually water heating systems

which avoid converting the energy into

electricity. At a utility scale, electricity is

produced by focusing sunlight using

thousands of movable mirrors to heat a fluid,

Figure 1.1: Residential silicon flat panel PV solar installation (top) and the Solucar PS10 solar thermal power plant near Seville, Spain (bottom, [55], originally posted to Flickr by afloresm at http://flickr.com/photos/74424373@N00/1448540890).

https://en.wikipedia.org/wiki/Flickr

http://flickr.com/photos/74424373@N00/1448540890

2

often molten salt, and using it to run steam turbines and create electricity. These systems come

in the form of “solar power towers” or parabolic trough concentrators and are only effective on

a large scale, thus not suitable for residential and commercial applications. Finally, a still

emerging technology uses PV cells for conversion to electricity, but unlike traditional silicon

panels, optical components are used to concentrate the sunlight onto a small area of PV cells.

This concept is known as concentrating photovoltaics (CPV).

1.2 Concentrating Photovoltaics (CPV)

The purpose of CPV systems is to

optically relay an area of solar flux to a

photovoltaic cell of smaller area. This

effectively replaces photovoltaic cell area with

relatively inexpensive optical materials,

potentially lowering the levelized cost of

electricity (LCOE) produced by the system,

especially when utilizing highly efficient PV cells that carry a high monetary cost per unit area.

Additionally, CPV modules often have much higher conversion efficiencies than even the best

silicon flat panel PV technologies. A more recent area of interest in CPV is for building

integrated photovoltaics (BIPV, or BICPV) where PV modules exist as part of the architecture of

buildings. CPV offers great flexibility in terms of the aesthetics of such systems.

1.2.1 Concentration Ratio

CPV systems are classified by their concentration which is the ratio of the area of the

entrance aperture of a system to the area of its target surface, i.e. the photovoltaic chip. More

Figure 1.2: Fresnel lens based CPV array on a large tracking mechanism. "Amonix7700" by Mbudzi - Own work - https://commons.wikimedia.org/wiki/File:Amonix7700.jpg#/media/File:Amonix7700.jpg.

https://commons.wikimedia.org/wiki/File:Amonix7700.jpg%23/media/File:Amonix7700.jpg

https://commons.wikimedia.org/wiki/File:Amonix7700.jpg%23/media/File:Amonix7700.jpg

3

specifically, this is called the geometric concentration ratio given the spatial nature of its

definition. Classifications of CPV systems vary widely and have yet to be standardized. High

concentration (HCPV) systems have been defined as those with geometric concentration ratios

that range from 300X up to 1000X or higher. Low concentration (LCPV) system concentration

classifications vary greatly: anywhere from 3X and lower [3] to 100X and lower [4]. Some even

define a medium concentration class (MCPV) that exists between LCPV and HCPV, between

100X and 300X. Even given the variation of the definitions, HCPV systems almost exclusively

employ high-efficiency, high-cost, multi-junction PV cells and two-axis tracking systems to follow

the sun as it moves across the sky. Systems meant to maximize cost effectiveness often fall into

the HCPV category in order to minimize the area of expensive photovoltaic material and

increase efficiency of energy production, making this class of CPV a favorite of utility companies

for centralized solar plants.

LCPV system designs vary widely due to the numerous and highly diverse applications.

Some LCPV is meant to replace flat plate silicon while decreasing cell area while others are more

concerned with aesthetics, like in BIPV applications. Some LCPV requires two-axis tracking much

like HCPV, while others need only on-axis tracking or none at all. The PV material for LCPV is

most often inexpensive silicon in one of its several forms (single crystal, polycrystalline, etc.)

There is no dominant technology in the LCPV space at this time given the wide array of

applications and the fact that the market is still in a very early stage.

1.2.2 Tracking

The “field of view” (FoV) or “acceptance angle” of a CPV system refers to the angular

region (relative to the optical axis) from which the system accepts incident light. Due to

4

fundamental limits on concentration ratio, high concentration systems have smaller fields of

view than lower concentration systems. A small field of view of, say, a ±1° cone requires the

module to be pointed fairly precisely at the sun in order to function properly. Since the sun

moves approximately 1° across the sky every four minutes, a mechanical tracking system is

required.

Including a tracker in a CPV system has the disadvantages of an increased cost and

increased complexity with moving parts. One advantage of the tracker besides enabling the

functioning of high concentration systems is that it can keep the projected area of the CPV

system at a maximum throughout the day, most notably in the evening during the peak

residential electricity usage. For stationary solar panels, the projected area can become very

small, especially in the mornings and evenings, so very little sunlight strikes the cells during

those times. Another concern for solar trackers is their accuracy. Due to wind loading and

other pointing errors, even the most advanced trackers on the market have an accuracy of

around ±0.5° and the sun is approximately a ±0.25° source, so a practical high concentration CPV

design must have a field of view of at least ±0.75°.

Lower concentration systems can be designed to have an oblong field of view, accepting

light from large angles in one direction, but only a small range of angles in the orthogonal

direction. Such a system has the advantage of requiring tracking along a single axis, decreasing

the mechanism’s complexity and cost while maintaining a respectable concentration ratio.

1.2.3 Photovoltaics

Clearly, a key component of any CPV device is the photovoltaic cell, which converts the

concentrated light energy into electricity.

5

1.2.3.1 Photovoltaic Effect

As the name implies, photovoltaics function on the physical principal known as the

photovoltaic effect, which is related to the famous photoelectric effect. The photovoltaic effect

describes the process of photons being absorbed by a material and forming a mobile electron-

hole pair. If the electron-hole pair is separated and extracted from the material, the consequent

voltage potential can be used to perform work. On a molecular level, the absorbed photons are

exciting bound electrons to a higher discreet energy state, and the minimum energy required to

cause that excitation is referred to as the band gap of the material. In the case of a silicon PV

device, the band gap is 1.12 electron volts, which is equivalent to the energy contained in a

photon with a wavelength of approximately 1100 nm according to the following equation:

𝜆 =ℎ𝑐

𝐸 ,

where ℎ is Plank’s constant, 𝑐 is the speed of

light in vacuum, and 𝐸 is the photon energy.

Excluding effects such as two-photon

absorption and those of indirect bandgaps,

this means that photons of lower energy than

the band gap, or longer wavelength, cannot

be converted to electron hole pairs.

Another important fact is that, under

normal conditions, absorption of one photon results in one and only one electron-hole pair. In

the scope of a PV device, this means that a photon at the band gap energy and a photon with

twice that energy produce the same useable output current. Thus, the higher energy photons

Figure 1.3: Spectral response of a silicon solar cell. Efficiency on the vertical axis is in units of current (A) output per light power (W) input. The red curve is an ideal cell considering only the band gap/photon energy differential and the blue is a measured real-world cell. [5]

0

0.2

0.4

0.6

0.8

1

0 0.4 0.8 1.2

Spe

ctra

l Re

spo

nse

(A

/W)

Wavelength (µm)

Ideal Cell

Measured Cell

6

are not converted as efficiently as the photons just above the band gap energy. Figure 1.3

shows the ideal spectral response of a silicon PV cell as well as a measured device. The short

wavelength depression of efficiency in the measured data is due to absorption of a protective

glass cover layer, and the extension of the spectral response past the band gap wavelength is

due to effects associated with the indirect band gap nature of the device [5].

Knowledge of the spectral response of PV materials is important in CPV design. If the

concentration optics change the spectral distribution of the light incident on the cell, the

electrical output can be expected to change in a way that is not linear with the total power

striking the cell. Additionally, in order to convert sunlight with maximum efficiency, it is possible

to optically split the solar spectrum onto several different PV materials with various band gap

energies so that each one receives wavelengths of light that it can convert most efficiently. This

is a concept known as spectral splitting. Multi-junction cells can be thought of as PV devices

with integrated spectral splitting since different spectral bands are absorbed by each PV

material. The most efficient concentrating systems to date use these types of cells [6].

1.2.3.2 IV Curve

Typical inorganic PV devices consist of semiconductor p-n junctions and function as

diodes with the added photovoltaic effect as described previously. As such, PV devices follow a

similar law to diodes with a forward voltage bias:

𝐼 = 𝐼0 [𝑒𝑞𝑉

𝜂𝑘𝑇 − 1] − 𝐼𝐿 ,

where 𝐼 is the net current through the diode, 𝐼0 is the dark saturation current, 𝑉 is the applied

forward bias voltage, 𝑞 is the absolute value of electron charge, 𝑘 is Boltzmann’s constant, and

𝑇 is absolute temperature in Kelvin. 𝜂 is called the ideality factor which encompasses certain

7

effects that cause real diodes to function less efficiently than the theoretical models. Finally, 𝐼𝐿

is the light generated current, a term that does not exist in the diode law [5].

An IV curve of a PV device can

be directly measured by

simultaneously applying voltage and

reading the output current. Some of

the important metrics of PV cells are

the short circuit current (𝐼𝑆𝐶), open

circuit voltage (𝑉𝑂𝐶), the fill factor

(𝐹𝐹), and the peak power (𝑃𝑃). These

values are visualized in Figure 1.4. 𝐼𝑆𝐶

is the current produced by the cell under illumination with no voltage bias. This is the quantity

measured to assess performance of prototype systems in the lab since it increases linearly with

incident power assuming the spectrum of the incident light does not change. 𝑉𝑂𝐶 is the forward

bias voltage at which the net current is zero. For high quality cells, the open circuit voltage is

nearly constant with small changes in incident power, and devices of the same PV material will

have roughly the same open circuit voltages. The peak power is the largest absolute value of

the product of the applied voltage and output current along the IV curve and represents the

maximum power that can be extracted from the cell. Finally, the fill factor is a measure of how

square the IV curve is, and it is calculated by

𝐹𝐹 =𝑃𝑃

𝐼𝑆𝐶𝑉𝑂𝐶 .

-0.1

-0.08

-0.06

-0.04

-0.02

00 0.2 0.4 0.6

Net

Cu

rren

t (A

)

Voltage (V)

Figure 1.4: Example IV curves. The orange curve is measured data from a silicon PV cell, and the blue is derived from the diode law with I0 and IL adjusted to match the measured data. The areas above the dashed lines indicate the peak power output for each curve. The fill factors for the measured and ideal curves are 0.52 and 0.82, respectively.

VOC

ISC

8

A higher fill factor often indicates a cell of greater quality that is closer to functioning as an ideal

diode. As seen in Figure 1.4, the measured silicon cell is far from ideal. This is likely largely due

to parasitic resistances within the device [5].

In practice, measuring IV curves in both dark and illuminated conditions can give insights

into the quality of the cell. More importantly for the research presented here, comparing the

dark IV curves of a cell before and after it is adhered to an optical component can indicate if it

was damaged during assembly.

1.2.4 Viability of Solar Power

One of the major hurdles with all forms of solar power generation is its intermittency.

No solar technology can produce significant amounts of electrical power at night or under heavy

cloud cover. If solar is to satisfy a large portion of electricity demand, this issue must be

resolved. Perhaps the most direct solution is to use electrical energy storage systems which

exist in a variety of forms from chemical batteries to flywheels to pumped hydro-power [7],

though these would increase the total cost of the systems. There are two major benchmarks in

solar power storage: three and twenty hours. With three hours’ worth of energy storage

capacity, a non-tracking solar harvesting system could shift its output to match the peak

demand on the electrical grid, effectively producing electrical energy when it is needed most.

With twenty hours of storage, solar could effectively deliver electricity around the clock, making

it a viable primary energy source for utilities [8].

Even without extensive storage capabilities, solar has proven to be a valuable ancillary

energy source through what is called “peak shaving.” This concept refers to providing electricity

during times of peak demand on the electrical grid in order to reduce the maximum capacity

9

required of the primary energy generation methods. Peak demand for commercial

establishments tends to occur in the early afternoon (see Figure 1.5), which is also near the peak

production of most solar energy systems. Including solar as part of the energy infrastructure

(distributed or centralized) can help utility companies avoid having to build new fossil fuel

power plants that would only be needed for a few hours each day.

Peak demand for the residential sector, however, typically occurs in the early evening

(Figure 1.5) when the sun is lower in the sky. Non-tracking systems like most flat plate silicon

installations produce a fraction of their peak output at these times since the projected area of

the panels from the perspective of the sun is greatly reduced. This is one area where a tracking

system has an advantage, since the projected area is always kept at a maximum. Therefore,

solar installations with tracking capabilities, e.g. HCPV, can perform peak shaving for both major

demand peaks on the electrical grid, making them an attractive ancillary energy source, even

without large electrical storage capacity.

Figure 1.5: Electrical grid demand during a hot day in California [57].

10

1.2.5 Current Market

The principal metric used by utility companies is the aforementioned levelized cost of

electricity, or LCOE. This value has units of $/kWh and includes all the costs of building,

financing, and maintaining an energy production facility as well as how much electricity it is

predicted to generate over its lifetime. In the utility-scale energy production market, solar PV

must compete with traditional fossil fuels such as coal and natural gas on the LCOE metric which

intrinsically involves government policies such as renewable energy subsidies and greenhouse

gas emission penalties (e.g. “carbon tax” or “cap and trade”). Silicon-based PV is currently

competitive with coal and natural gas in some regions of the U.S. with an LCOE as low as $0.09

up to $0.18/kWh including the as-of-now permanent 10% investment tax credit provided by the

U.S. federal government. For comparison, the LCOE for a new coal power plant ranges from

$0.09 to $0.16/kWh depending on the geographical region and technology employed. Similarly,

natural gas ranges from $0.07 to $0.16/kWh [9]. As a subset of solar PV, CPV technologies are

often compared to flat plate silicon PV to evaluate their competitiveness in the energy market as

a whole.

A recent joint study by the Fraunhofer Institute for Solar Energy Systems (Fraunhofer

ISE; Freiburg, Germany) and the National Renewable Energy Laboratory (NREL; Golden, CO, USA)

reviewed the current state of the CPV market and technology [4]. While the market is still small

and young, it is growing quickly, mostly due to utility scale high concentration CPV (HCPV)

installations in the last few years. HCPV is becoming more attractive as commercially available

system efficiencies improve because it directly reduces area-related costs (e.g. land and number

of tracking mechanisms), lowering the LCOE. In fact, high efficiency is one of the key drivers to

make HCPV competitive with flat plate PV [4]. The record laboratory CPV module efficiencies

11

have now exceeded 36% and are projected to continue improving [10]. In comparison, the far

more mature silicon technologies have champion efficiencies around 22%. Despite these

impressive numbers, HCPV is not yet competitive with silicon on a LCOE basis due to higher

system costs and the relative difficulty of manufacturing and scalability [4].

1.2.6 Fresnel Lens CPV

The dominant technology in the CPV arena

consists of Fresnel lenses point-focusing direct sunlight

onto secondary concentration elements that are adhered

to PV chips, as in the Figure 1.6 schematic. These systems

typically have high geometric concentration ratios in the

500X to 1000X range, fitting squarely into the HCPV

classification. Due to their long focal lengths, on the order

of tens of centimeters, they have the disadvantage of

being bulky, requiring a large volume that needs to be

environmentally isolated. The main reason Fresnel lens

systems make up the majority of the CPV market is that they are relatively simple and

inexpensive to manufacture, and it is a time tested, proven technology.

Figure 1.6: Fresnel lens CPV system diagram with secondary concentrator. Image source: Light Prescriptions Innovators, [59].

12

1.2.7 Planar Light Guide CPV

Another class of CPV system that utilizes a planar light

guide has been extensively studied at the University of

Rochester [11, 12]. These systems eliminate the large air gap of

the Fresnel lens systems, allowing for a more manageable, flat

form factor while mitigating the requirements for environmental

isolation. Additionally, the solid layered construction decreases

risk for misalignment of the different components after assembly. The systems are in the shape

of flat rectangular plates, where direct sunlight enters through

one of the large faces and exits one of the edges (Figure 1.7).

The designs created at the University of Rochester vary from

about 50X to 500X concentration for the application of utility-

scale electricity production [11, 12].

The three key components of these designs are the

primary optic, injection feature, and the light guide. An

example of one planar light guide CPV design is shown in

Figure 1.8 with the components labeled. The role of the

primary optic is to decrease the spatial extent of the light so it

can interact with a small injection feature. In the figure, the

primary optics are an array of “lenslets.” The injection features

then redirect the light to travel parallel to the guide layer such

that it is trapped by total internal reflection (TIR). The purpose of the guide layer is to transport

light from the injection features to the output edge of the system.

Figure 1.7: Planar light guide CPV form factor. Sunlight enters the large face and is guided toward one edge (in red) where the PV chip is placed.

Figure 1.8: Refractive planar light guide CPV components.

13

The challenge with these designs is to keep light trapped in the guide layer long enough

for it to reach the output surface. The problem arises from the interactions of the injected light

with downstream injection features, which can cause some of the light to couple out of the

guide prematurely. Many of the designs try to reduce the severity of these interactions with

novel injection feature geometries [11, 12].

1.3 Concentrator Design Theory

The design of solar concentrator optical systems primarily uses concepts from the

broader field of non-imaging optics, which also encompasses illumination design (e.g.

automobile headlights, display illumination, street lighting, etc.). One of the core concepts of

non-imaging theory is a quantity called étendue, a French term that translates to “extent”.

1.3.1 Étendue and Maximum Concentration Ratio

Étendue relates to the angular and spatial extent of light captured and propagated

through an optical system and can be loosely defined as the product of the index of refraction,

an area, and the solid angle taken up by the light incident on that area. The differential étendue

is defined by

𝑑휀 = 𝑛2 ∙ (𝑑𝑥 ∙ 𝑑𝑦) ∙ (𝑑𝐿 ∙ 𝑑𝑀),

where 𝑛 is the index of refraction, 𝑑𝑥 ∙ 𝑑𝑦 is the differential area of the target surface, and 𝑑𝐿 ∙

𝑑𝑀 is the compensated differential solid angle taken up by the incident light, L and M being the

x and y direction cosines, respectively. This definition assumes that the surface lies in the XY

plane. Étendue is then defined by the following integral:

휀 = ∫ 𝑑휀 = 𝑛2 ∬ ∬ 𝑑𝑥𝑑𝑦𝑑𝐿𝑑𝑀,

or equivalently,

14

휀 = 𝑛2 ∭ cos 𝜃 𝑑𝑥𝑑𝑦𝑑Ω = 𝑛2 ∬ ∬ cos 𝜃 sin 𝜃 𝑑𝑥𝑑𝑦𝑑𝜃𝑑𝜑 [13]

in polar coordinates with 𝑑Ω being the differential solid angle, and with the assumption of a

spatially and directionally constant (non-birefringent) index of refraction. The integrand

contains a cosine term where 𝜃 is the angle between the surface normal and the direction of the

incident light, compensating for the solid angle in polar coordinates [13]. In a lossless system

limited to reflective and refractive interfaces and gradient index materials, étendue is conserved

and the invariant quantity is 𝑛2 𝑑𝑥 𝑑𝑦 𝑑𝐿 𝑑𝑀 [14], so we have

𝑛′2 𝑑𝑥′ 𝑑𝑦′ 𝑑𝐿′ 𝑑𝑀′ = 𝑛2 𝑑𝑥 𝑑𝑦 𝑑𝐿 𝑑𝑀.

This ignores effects such as diffraction and Fresnel reflection losses and does not account for

components such as scatterers. The consequence of étendue conservation is that within an

optical system, light can be incident on a small area from a large angular range or it can be

incident on a larger area from a smaller angular range.

In the extreme case, the conservation of étendue predicts that light from a perfectly

collimated source can be concentrated to an infinitesimally small point (again ignoring

diffractive effects). On the other hand, if the light entering a finite entrance aperture comes

from a non-zero solid angle, the light cannot possibly be concentrated to a single point given

that the maximum possible solid angle on the target is 4π if light is incident from every direction

in three dimensional space. The minimum area which the light can be concentrated onto is

finite. In this way, étendue defines an upper limit to the concentration ratio of a CPV system

given its acceptance angle.

If we assume a spatially invariant Lambertian angular distribution and make the

following simplifications of

15

∬ 𝑑𝑥𝑑𝑦 = 𝐴

and

∬ cos 𝜃 sin 𝜃 𝑑𝜃𝑑𝜑 = 𝜋 sin2(𝜃𝑜),

where A is the area of the surface in question and 𝜃𝑜 is the half angle of a cone with its axis of

symmetry along the surface normal, then the integrated étendue becomes

휀 = 𝑛2 ∙ 𝐴 ∙ 𝜋 sin2(𝜃𝑜).

For a lossless system (i.e. an optical efficiency of unity), the geometric concentration, C, is

𝐶 = 𝐴

𝐴′= (

𝑛′ ∙ sin(𝜃𝑜′ )

𝑛 ∙ sin(𝜃𝑜))

2

,

since étendue is conserved (ε = ε’). Assuming that the ambient index of refraction is unity and

𝜃𝑜′ is maximized to π/2 (for incident light from the 2π steradian hemisphere accessible to one

face of a flat target surface), we get a maximum possible concentration, Cmax, of

𝐶𝑚𝑎𝑥 = (𝑛′

sin(𝜃𝑜))

2 [14].

Notice that the maximum concentration can be increased n’2 fold by immersing the target solar

cell in a material with a refractive index of n’.

Now consider a CPV system that only captures direct sunlight, making the acceptance

angle range (or field of view) of the system equal to the angular size of the sun from Earth, a

±0.27 degree cone. Also assume that the PV cell is immersed in a material of index n’ = 1.5 and

can accept light from 2π steradians, then the maximum theoretical concentration of this system

is

𝐶𝑚𝑎𝑥 = (1.5

sin(0.27°))

2

≈ 101000

16

This concentration is impractically high as any system that could achieve this would have to be

made of extremely resilient materials since the PV cell would be receiving about as much

radiation as it would at the surface of the sun. CPV systems in production do not often have

concentrations higher than about 1000X, and these systems must employ advanced cooling

systems to avoid destroying the PV cells while keeping them at peak efficiency. Also note that

given a 1000X system with a target accepting light from a ±40° cone while immersed in an index

of 𝑛′=1.5 has a maximum acceptance cone of approximately ±1.75°. This is why HCPV systems

necessitate a two-axis tracking system to remain effective throughout the day.

All of the above discussion assumes that the system is comprised of only lossless,

smooth surfaces and non-diffuse materials. If the system includes a scattering component, for

example, the étendue increases when the beam interacts with it. To illustrate this, imagine a

perfectly collimated beam (étendue = 0) with a finite cross section hitting a diffuse surface.

Immediately after that interaction, the beam has the same finite cross section, but now has a

finite angular range in which it propagates, resulting in a positive, non-zero étendue.

It is also possible for étendue to decrease, and the most common way for this to happen

is through optical loss in a system. For example, if a small aperture is placed relatively far away

from a finite extended Lambertian source, the light that passes through the aperture looks more

and more collimated the further from the source it is. The large étendue of the initial extended

source becomes very small in the form of a nearly collimated beam, however, only a small

fraction of the light is successfully passing through the aperture, so the optical loss is high. This

decrease in étendue would also be evident in a CPV system which has a geometric

concentration ratio which is greater than the theoretical maximum concentration ratio: there is

guaranteed to be some optical loss. While there will always be some amount of loss in real

17

systems, the étendue-derived theoretical maximum concentration is still a useful concept in CPV

design, providing a fundamental limit to what is possible.

1.3.2 Optical Efficiency and Effective Concentration

Since the concentration ratio is a purely geometrical quantity, a CPV system could

technically obtain a higher concentration by simply shrinking the size of the PV cell, allowing

much of the light to miss its target. Of course, this would be a pointless exercise because much

of the light is wasted, which decreases the system’s efficiency. For this reason, another

important quantity for CPV systems is the optical efficiency, which is a metric describing the

fraction of light entering the system that eventually strikes the target PV cell. Note that this

quantity is independent of the PV cell efficiency.

In common vernacular, “module efficiency” describes the ratio of output power in the

form of DC electricity to the input power in the form of solar radiation, and “system efficiency”

is the same metric but for an array of modules instead of a single one. Both of these are usually

used in the context of real-world measurements, hence the differentiation of module and

system efficiencies. This document uses the module efficiency term as described here, but not

constrained to real-world data, and ignores this definition of the system efficiency entirely, using

the term more generally, e.g. as the efficiency of an optical system.

Additionally in this document, a new term, effective concentration, is defined as the

product of the optical efficiency and the geometric concentration. This quantity is especially

useful for systems with low optical efficiency, when the geometric concentration is less

meaningful in isolation. Effective concentration is meant to describe the ratio of the average

solar irradiance striking the cell to the solar irradiance entering the CPV system, or in other

words, how many sun’s worth of energy is striking the target PV cell.

18

1.4 Low Concentration Photovoltaics

The aptly named low concentration photovoltaic (LCPV) systems have low concentration

ratios of significantly less than 100X [4]. The low concentration ratio is often paired with a large

field of view, meaning that a tracker is not required, or a simpler system that tracks in only one

dimension may be sufficient. Lesser tracking requirements can make these systems attractive,

especially when high concentration is not necessary. Additionally, LCPV systems allow for some

interesting form factors and applications.

1.4.1 Concentration Limits on Stationary Concentrators

Having a stationary concentrator requires it to

have a large field of view in order to capture sunlight for

a significant amount of time throughout the year. The

necessary field of view can be calculated by following the

elegant description from Winston and Zhang [15]. First,

define a coordinate system for a given location on Earth’s

surface where the X axis is parallel to Earth’s rotational

axis and the Y axis points due west (Figure 1.9). Then,

the Z axis points towards the sun during solar noon on

equinox. The Z axis can be found by imagining a line pointing straight up, normal to the earth’s

surface, then tilting it by the latitude angle of the current position on Earth towards the equator

(due South in the northern hemisphere). This coordinate system is stationary with respect to

Earth, so it spins with Earth’s rotation and moves with Earth’s orbit around the sun. The XY

plane is defined as the polar plane.

Figure 1.9: Schematic of polar plane orientation with respect to the earth (blue) and the sun on the equinox at solar noon (yellow).

19

Recalling that the tilt of the Earth’s rotational axis is 23.45° with respect to its orbital

axis, the unit vector of the sun’s angular position can be written approximately as

𝑛𝑠 = (sin 𝛿 , cos 𝛿 sin 𝜔 , cos 𝛿 cos 𝜔),

where the solar declination, δ, is defined as

sin 𝛿 = − sin 23.45° cos (360°(𝑛 + 10)

365.25),

for the nth day of the year, and the hour angle, ω, is defined as

𝜔 =360°

24𝑡,

for the tth hour of the day starting from

solar noon (t<0 in the morning) [15]. If L

and M are the direction cosines for X and

Y, respectively, the sun’s position can be

visualized in convenient way as in Figure

1.10, with L = sinδ and M = cosδ∙sinω. The

black unit circle represents the positions in

the sky that lie on the polar plane as

defined above. Located on the equator,

this circle marks the horizon. If the sun’s

position is tracked from the bottom of this circle to the top on any given day, the time between

the two endpoints will be twelve hours (𝑡 ∈ [−6, +6]). Similarly, the blue ellipse marks the

boundaries for which it takes the sun eight hours to traverse. The various vertical lines trace the

sun’s position for single days throughout the year, with the green lines representing the days of

the solstices when the sun is at its highest and lowest points in the sky. On the equinoxes, the

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

L

M

Figure 1.10: Recreated from Figure 2 of ref [15]. The bold black circle is the unit circle, the green lines represent the sun’s position throughout the day of the winter (left) and summer (right) solstices, the solid vertical lines represent the sun’s path for the first day of each month from Jan 1 to June 1 from left to right, and the dashed lines do the same for July 1 to Dec 1 from right to left, and the blue ellipse represents the eight hours per day boundary.

20

sun’s path is represented by a vertical line at L=0. The region inside the closed loop formed by

the green lines and the solid sections of the blue ellipse would be the minimum field of view

required for a stationary concentrator to capture sunlight for eight hours per day, year round.

Likewise, the central region formed by the green lines and black circle would be the necessary

field of view for capturing sunlight twelve hours every day for the entire year, assuming the sun

is visible. Note that sunrise and sunset times throughout the year varies if not located on the

equator. At a latitude of 43° (Rochester, NY), the shortest day of the year is roughly nine hours

long, so if a concentrator were designed to capture exactly eight hours of sunlight every day, it

would never be attempting to capture sunlight before dawn or after dusk in Rochester. The

blue eight hour ellipse of Figure 1.10 is calculated by using the equation

sin2 𝛿 + cos2 𝛿 = sin2 𝛿 +cos2 𝛿 ∙ sin2 𝜔

sin2 𝜔= 𝐿2 +

𝑀2

sin2 𝜔= 1 ,

and holding the hour angle, ω, constant such that t = 4 (four hours before noon and four hours

after noon for a total of eight hours). This defines an ellipse with a major radius of 1 in the L

direction and a minor radius of sinω in the M direction.

Now that the sun’s position is calculated in direction cosine space, consider the

following definition of concentration, directly following the above section on étendue,

𝐶 =∫ 𝑑𝑥𝑑𝑦

∫ 𝑑𝑥′𝑑𝑦′=

𝑛′2 ∫ 𝑑𝐿′𝑑𝑀′

𝑛2 ∫ 𝑑𝐿𝑑𝑀 [15].

Assuming the light can hit the target surface from a full hemisphere, ∫ 𝑑𝐿′𝑑𝑀′ is equal to the

area of the unit circle, which is simply π. The area of the region for capturing eight hours of

sunlight per day can be calculated using the following equation:

∫ 𝑑𝐿𝑑𝑀 = sin(𝜔′) [2𝛿𝑠′ + sin(2𝛿𝑠

′)] ≈ 1.56 sin(𝜔′) [15] ,

21

where 𝛿𝑠′ is the solar declination on solstice plus the angular radius of the sun (0.27°), equaling

23.72° or 0.414 radians, and ω’ is the hour angle modified in a similar way according to the

equation,

𝜔′ =360°

24∙

𝐻

2+ 0.27° ,

where H is the number of hours per day captured. In this way, the maximum achievable

concentration for capturing eight hours of sunlight per day is calculated to be about 2.3X for n’ =

1. If the concentrator immerses the target surface in n’ = 1.5, the maximum concentration is

increased 1.52 = 2.25 times to around 5.2X.

1.4.2 Stationary Compound Parabolic Concentrator

The compound parabolic concentrator, or CPC, is a simple and well-known ideal form of

concentrator in nonimaging optics, which is a prime example of a design using the edge-ray

principle [14]. This principle essentially describes a nonimaging design method where the edge

of the source is mapped to the edge of the target, often in a point-to-point manner which is

reminiscent of imaging optical design. However, light incident from the source away from the

edges need not form a point on the target surface, but all rays will land somewhere within the

boundaries of the target surface. This principle is illustrated in the cross section of the CPC in

Figure 1.11. The blue rays come from the edge of the source, or equivalently the edge of the

concentrator’s field of view, and they converge on the edge of the target surface. However, the

red rays which form less of an angle with respect to the optical axis do not converge in any

discernable way.

The CPC is formed by two mirrored parabolic profiles. The axis of each parabola is tilted

at the maximum field of view angle of the system so that rays coming from that direction are

22

focused to a point. The focus of each parabola

is positioned at the bottom point of the

opposite reflector, which is where the edge of

the target surface will be. This is pictured by

the blue rays of Figure 1.11. Theoretically, this

concentrator is 100% efficient in getting light

within the designed field of view from the large

input aperture to the smaller target surface.

This assumes that the mirrors are 100%

reflective. Additionally, for some three dimensional manifestations of these concentrators such

as a rotationally symmetric version, rays that do not share a common plane with the optical axis,

called skew rays, can sometimes be ejected before reaching the target surface, slightly lowering

the concentrator’s efficiency.

As an example of what kind of CPC would be necessary to capture direct sunlight for

eight hours per day, year round, consider a hollow (n=1) extruded, or “trough,” CPC. The system

would be aligned so the input aperture exists in the polar plane, and the extruded dimension

runs in an east-west direction. The end caps of the extrusion are flat mirrors, perpendicular to

the input aperture. This CPC would capture 100% of all rays whose projected angle on the XZ

plane is within its designed acceptance angle, θ1. In equation form, this inequality is

𝐿2

sin2(𝜃1)+ 𝑀2 ≤ 1 [15].

If θ1 is adjusted so that it is just large enough to capture sunlight for eight hours per day, then θ1

≈ 41.5°. The field of view for such a system is shown in Figure 1.12 as a red ellipse. One can see

Figure 1.11: Compound parabolic concentrator profile illustrating the edge ray principle. The blue rays come from the edge of the source and focus to a point on the edge of the dark red target surface. The red rays come from inside the edges of the source.

23

that the corners of the shaded eight hours per day region intersect the field of view (FoV)

ellipse. Also notice that there is much angular

area in the FoV where the sun will never be

located. This is essentially wasted field of view

since the overly large angular range limits how

small the target surface can be and, therefore,

how high the system’s concentration is. For

this system, the area inside the ellipse is

∫ 𝑑𝐿𝑑𝑀 = 𝜋 ∙ sin 41.5° ,

making the concentration

𝐶 =𝜋

𝜋 ∙ sin 41.5°=

1

sin 41.5°≈ 1.5 .

This is about 65% of the theoretical maximum

concentration of 2.3X. This reduction is

entirely due to the extra regions in the field of view that the sun will never occupy.

To significantly increase the concentration, the same basic design can be filled with a

dielectric material of index greater than one, n = 1.5 for this example. Now, another factor to

consider is the refraction of rays at the input surface of the CPC. This is easy to implement with

the direction cosine form of Snell’s Law:

𝐿 = 𝑛𝐿𝑛, 𝑀 = 𝑛𝑀𝑛 [15],

where L and M are the direction cosines before the light enters the concentrator, and Ln and Mn

are the direction cosines just after the light enters the dielectric material of index n. For the

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

L

M

Figure 1.12: Direction cosine representation of the sun's position for eight hours per day, year round (grey shaded area) and the fields of view of trough CPCs with an index of 1 (red ellipse) and 1.5 (blue ellipse) that are just wide enough to capture sunlight for eight hours every day without tracking.

24

field of view plots, this essentially means the ellipse shape will be scaled by a factor of n = 1.5.

In equation form, the field of view of the dielectric-filled CPC is

𝐿2

sin2 𝜃1+

𝑀2

𝑛2 ≤ 1 [15],

where θ1 is the acceptance angle in air and n is the index of the dielectric. The equation shows

that the field of view ellipse is simply scaled by n in the M direction. This change of ellipse shape

allows the acceptance angle to be smaller (≈28.3°) and still capture direct sunlight eight hours

per day. The blue ellipse in Figure 1.12 represents the dielectric CPC field of view. The dashed

portions of the ellipse are not physical since they assume rays incident upon the input aperture

at greater than 90° from the surface normal, so only portions of the ellipse inside the unit circle

are physically realizable. That being said, the non-physical regions are still considered during the

calculation of concentration ratio. Compared to the hollow CPC from earlier, the dielectric CPC

has much less extraneous area in its field of view. As such, a higher concentration is expected.

In fact, the concentration is calculated to be

𝐶 =𝑛2 ∙ 𝜋

𝑛 ∙ 𝜋 ∙ sin 28.3°=

𝑛

sin 28.3°≈ 3.2 .

This is about 2.1 times higher than the concentration of the hollow CPC, a significant

improvement. Immersing a CPV system, especially one with a reflective concentrating

component, in a dielectric material can often have beneficial effects and is implemented in

some of the designs presented as part of this thesis.

1.5 Examples of Existing Low Concentration Systems

This section explores low concentration solar photovoltaic systems that exist today.

There are a wide variety of different design types, some that simply modify existing flat plate

solar arrays, and other designs that have interesting niche applications. With the exception of

25

the first device presented here, these designs fall into the building-integrated concentrating

photovoltaics (BICPV) sector which is the focus of this thesis.

1.5.1 RAIS™ PV from tenKsolar

A company called tenKsolar sells a

silicon PV system that has a concentration

ratio of roughly 1.5X which is pictured in

Figure 1.13 [16]. It is a simple but effective

design that is created by modifying a typical

flat plate solar array, placing dielectric

mirrors between the rows of tilted PV

panels. Much of the sunlight that would otherwise fall between the panel rows is reflected onto

the PV cells, boosting energy output. Additionally, the mirrors preferentially reflect photons

with energies greater than the band gap of silicon so light that the cells cannot convert is not

concentrated onto them, helping to temperature control the system.

Figure 1.13: tenKsolar's RAIS™ PV system. The deep blue colored rows are silicon solar panels and the translucent plates are cold mirrors that concentrate convertible sunlight onto the PV cells. [16]

26

1.5.2 Nagaoka University of Technology

The application on which this thesis focuses is

the integration of LCPV into architectural windows. One

such design has been developed, prototyped, and tested

by a research group at the Nagaoka University of

Technology [17]. The basic concept is similar to

tenKsolar’s PV module with alternating rows of PV

material and mirrors, but this system is mounted

vertically and the mirrors work solely on total internal

reflection so they are see-through from the observer’s

perspective. Figure 1.14 shows the operation principle

as well as a picture of the prototype. Notice that an

image of the building can be seen through the CPV

module, though it is significantly obstructed and shifted due to the prismatic nature of the

design. As tested, this module generates 1.15 times the electricity of a standard flat plate PV

module with just 37% of the cell area [17], or a geometric concentration of about 2.7X.

1.5.3 ClearPower PV Window Tile from Stellaris

Stellaris has introduced a “photovoltaic window

tile” that integrates optical components and silicon PV

cells into windows such that a feint image can still be

seen through the module [18]. A similar design

integrates the modules into roofing shingles [19]. This

Figure 1.14: See-through window CPV system schematic (top) and picture of prototype (bottom, [17]).

Figure 1.15: Reproduced rendering of Stellaris' window CPV module showing extruded CPC troughs [18].

27

technology appears to use a well-known shape in illumination design called a compound

parabolic concentrator, or CPC [14], introduced earlier in this chapter. In Stellaris’ design, the

CPC shape is employed in one dimension and linearly extruded in the other (Figure 1.15). It is

constructed of a solid dielectric that uses total internal reflection (TIR) to create a reflection off

of the CPC surfaces [20]. Since the surfaces are transparent, some light outside of the

acceptance angle of the CPC can still pass through the device, though heavily obstructed. While

this design is likely very efficient in terms of capturing solar energy, it loses much of its utility as

a window.

1.5.4 Luminescent Solar Concentrators

A class of concentrators on its own,

luminescent solar concentrators (LSC) utilize

luminescent particles, e.g. quantum dots or

organic dyes, to absorb and re-emit light towards

the PV cells [21]. These systems are shaped like

flat plates that act as light guides with the

luminescent particles throughout, accepting

sunlight from any angle on the large faces and sending light towards the relatively small target

edges. A significant fraction of the light is able to pass through the plate, thus causing the

devices to appear transparent. Re-emission from the luminescent particles sends photons in

random directions, so much of light is prematurely ejected from the system through one of the

large faces. The rest of the light is trapped in the plate by virtue of TIR and travels towards the

edges of the device (Figure 1.16). A major challenge in these systems is avoiding the re-

Figure 1.16: Reproduction from [21]. A schematic for a LSC showing the incoming sunlight, a luminescent particle (orange circle), and re-emitted light that is lost (1) and trapped in the light guide (2).

28

absorption of light that has been successfully trapped in the light guide. Many tricks have been

played such as wavelength conversion to decrease absorption after re-emission, and multiple

layers that specialize in either absorption or efficient light transfer. However, these devices

continue to have low conversion efficiencies of well under 10%, even with highly efficient and

expensive gallium-based PV cells. The geometric concentration ratios can be rather high at 10X

or greater, but the optical efficiencies are often extremely poor. This can be independently

deduced by noticing that the system can accept light incident on the large faces from a full 2π

steradians and the light incident on the smaller target surfaces is incident only from angles that

satisfy TIR conditions in the light guide. Note that these systems also do not conserve étendue

since they are effectively comprised of a diffuse material.

As these devices are highly inefficient at

converting sunlight, applications are often artistic.

Since the luminescent particles re-emit light in a

narrow band of wavelengths, the output light is

colored and the systems can be used, for example, to

accentuate a product display in a big box store. The

light from the overhead illumination is captured and

sent towards the edges of the guide making them glow somewhat brightly as seen in Figure

1.17.

Figure 1.17: Luminescent concentrator plates. Image courtesy Fraunhofer ISE.

29

1.6 Examples of Non-Concentrating BIPV

The building-integrated photovoltaic (BIPV) examples shown in this section do not

explicitly use concentrating optics, but are included since they are integrated into buildings on

vertical surfaces and allow a partially unobstructed view through the modules.

1.6.1 Prism Solar Technologies

Prism Solar leverages bi-facial PV cells and the

natural reflectance of surrounding surfaces in the

environment to create a product that will “outperform any

traditional module in a vertical application” [22]. Bi-facial

PV cells use a transparent conductor on the rear face of

the cell, allowing light to be absorbed in the PV material

from virtually any direction. Prism Solar advertises its

modules as being used in a vertical orientation, which will

severely impact how much sunlight is incident on the cells

due to the effects of projected area. However, some light energy is regained through diffuse

reflections from the ground or from surfaces behind the modules. The modules are designed

for integration into fencing, railings, balconies, and building facades, but would not perform well

as windows due to the high degree of obscuration.

1.6.2 Solaria’s BIPV Laminates

A similar system is available through Solaria, but these modules are meant for

integration into windows [23]. The individual PV cells are very narrow, and hundreds of them

are strung together leaving gaps in between adjacent cells. The individual cells are about 156

Figure 1.18: Prism solar BIPV modules integrated into balcony railings [22].

30

mm by 3 mm with about 6 mm between centers of adjacent cells, resulting in about 50%

transparency. The overall effect of the module looks similar to a tinted window.

1.7 Window Integrated LCPV Planar Light Guide Solar Concentrators

This thesis introduces novel low concentration planar light guide CPV designs for

integration into windows. The architecture of the University of Rochester planar light guide

systems [11, 12] are modified to greatly expand the field of view for use without a tracking

system. The resulting systems and target application are discussed in detail in Chapter 2. The

above examples of commercial BIPV systems show that there exists interest in such a

technology.

31

Chapter 2: Design

The LCPV systems presented in this thesis are designed to meet the criteria of a specific

application. They are integrated into South-facing architectural windows and the electricity

produced is used to offset the thermal energy lost through the windows during colder times of

the year. The application drives the specifications and figures of merit during optimization.

2.1 Application

The application imposes the constraints of being stationary, planar in shape, and

translucent. The stationary requirement implies that the module’s field of view must be very

wide in order to capture sunlight during a significant portion of the day since traditional tracking

is undesirable. However, moving parts are not prohibited, so there can be tracking via moving

components within the stationary module. The design spaces of completely static concentrators

and those with moving parts for one dimensional tracking are considered.

The planar constraint exists to keep the system in a shape that is expected of a window,

and it rules out systems with large air gaps such as Fresnel lens-based designs. The UR planar

light guide design [11, 12] is a good starting point since it has an appropriate form factor and

consists of only refractive surfaces.

Finally, in order to remain a window, a translucency constraint is imposed. Here,

“translucent” means that some outdoor light must have a path through the module so it can

illuminate the indoor area. If the specific application calls for integration into a picture window,

the image of the outdoor environment cannot be heavily distorted nor obstructed, thus

changing the constraint to “transparency.” In the application of a privacy window, the image

32

coming through the module is required to be heavily distorted, thus translucent but not

transparent. Both of these spaces are investigated.

2.1.1 Design Goals

This application does not focus on maximizing power generation. The specific use of the

electricity influences the optical design. The presented designs have the purpose of heating the

indoor air during the winter time in order to effectively improve the insulation properties of the

window. This use is chosen based on the poor insulation properties of architectural windows,

especially during winter when the temperature difference between the indoor and outdoor air

is greatest. Much energy in the form of heat is lost through windows as compared to typical

building outer walls. Thus, windows with the built-in capacity to offset that lost energy,

effectively having an increased R-value, could be a useful and marketable concept.

This use for the produced electricity affects the optical design mainly in the field of view

requirements. Heat should be produced only during the times of year when the outside air is

colder than a comfortable room temperature of 20°C. Thus, the field of view of the system must

only capture direct sunlight when the sun is lower in the sky during winter. The location where

the system is operating has a large effect on when electricity should be produced and where the

sun is located during those times.

2.1.2 Field of View

The sun’s position is often given in terms of an elevation angle (degrees above the

horizon) and an azimuthal angle (for the purposes of this paper, degrees to the west of due

south). From the perspective of the concentrator, the field of view (a.k.a. acceptance angle) is

33

more conveniently considered in the

horizontal and vertical angle coordinate

system defined in Figure 2.1. If the

window exists in the YZ plane and the

positive Y axis points straight up

relative to flat ground, then the

direction vector is parameterized into a vertical angle and horizontal angle. The vertical angle is

the angle between the Z axis and the projection of the direction vector onto the YZ plane, while

the horizontal angle is the angle between the Z axis and the projection of the direction vector on

the XZ plane.

All designs have geometry that is extruded in the horizontal direction. This means that

the optical performance of the design is constant with respect to horizontal angle if certain

effects (e.g. Fresnel losses) are ignored. Thus, the optimization’s figure of merit is weighted as a

function of vertical angle only. This weighting function is location specific, derived from the

sun’s position and the

typical high and low

ambient temperatures

throughout the year.

First consider

the sun’s position

throughout the year.

Figure 2.2 shows the

sun’s position in

vertical angle

horizontal angle

Figure 2.1: LCPV field of view coordinate system definition.

Figure 2.2: Map of sun's position in azimuth and elevation for Rochester, NY (latitude = 43.12°). Yellow dots indicate sun position at various times. The black dashed lines indicate constant vertical angle.

20°

40°

60°

80°

34

Rochester, NY every ten days and every 20 minutes during those select days. Lines of constant

vertical angle are shown as dashed black curves. Based on sun position alone, some vertical

angles are weighted lower since the sun rarely appears at those coordinates. For example, the

80° vertical angle line only coincides with the sun’s position in the sky for absolute azimuthal

angles of about 75° to 90°, near dawn and dusk, and only in the summertime.

The projected area of the window from the sun’s point of view must be considered since

it affects the solar irradiance on the window’s surface. This value is indicated in Figure 2.2 by

the size of the yellow dots. The closer to an azimuth and elevation of 0°, the greater the

projected area, and the greater the weighting in those regions.

Finally, the temperature differential between the inside and outside air must be

included in the vertical angle weighting function. There are two days of the year during which

the sun passes through each point within its range: one in the spring semester and another in

the fall. The outside temperature is estimated from average temperature in the specified

location as a function of calendar day. Further, the estimated temperatures for the two days

corresponding to each position of the sun are averaged. This value is then subtracted from

room temperature (20°C) and factored

into the weighting function. If the

average temperature is greater than

room temperature, its weighting is

negative.

The sun’s position, window’s

projected area, and temperature

differential are all integrated along each Figure 2.3: Weighting functions for three cities in the U.S.

35

line of constant vertical angle. The resulting values describe the final weighting function with

respect to vertical angle. The weighting functions for El Paso, TX, Rochester, NY, and Fairbanks

AK are shown in Figure 2.3. El Paso is relatively warm and at a lower latitude of 32°N. Rochester

is temperate with a latitude of 43°N. Lastly, Fairbanks is cold and at a high latitude of 65°N.

Each weighting function has a sharp peak that corresponds with the vertical angle of the

sun at solar noon on the winter solstice. This is the smallest vertical angle that exists completely

within the region occupied by the sun at some point during the year. Additionally, lower vertical

angles also correspond to a larger projected area of the window and a larger temperature

differential. The peak location can be calculated simply by subtracting the tilt of the earth’s

rotational axis and the latitude from 90°. For example, Fairbanks, located just south of the arctic

circle, has a peak in its weighting function at a vertical angle of nearly zero:

90° − 23.45° − 𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 = 90° − 23.45° − 64.82° = 1.73°

Another feature of note is the negative weighting function values for El Paso at vertical

angles above 60°. This occurs because the integrated average temperature is above 20°C for

those vertical angles.

2.2 The Louver Concept

For most CPV designs, the field of view is centered about the optical axis which is

normal to the input aperture plane. For example, a typical HCPV FoV is a ±1° cone centered on

the optical axis. In the window LCPV application, the ideal field of view is not only extremely

large, but it is far off center and oddly shaped. To accommodate this field of view, a novel

optical design is created.

36

Just like the planar light guide CPV introduced in

Chapter 1, there are three optical components to the

designs presented in this thesis: primary focusing optics,

injection features, and a guide layer. The novel primary

optics come in the form of curved reflective louvers that

direct sunlight into the guide layer via injection features.

The louvers are curved blades reminiscent of metal

window blinds. This design space takes advantage of the

high, non-normal incidence angles of the incoming

sunlight.

Each louver casts a shadow on the louver directly below, and the amount that the

adjacent louver is shadowed depends on the vertical position of the sun (Figure 2.4). This allows

part of the louver surface to be optimized for a range of vertical angles at which that section is

not shadowed. The horizontal extrusion geometry of this design creates no concentration in the

horizontal direction, thus keeping the theoretical horizontal field of view at ±90°.

The application calls for a large area of CPV. However, the further light travels through

the guide layer, the greater the chance it is coupled out of the guide prematurely, resulting in

optical efficiency loss. Splitting the window area into a number of CPV modules mitigates this

inefficiency. The louver designs send light towards both the top and bottom guide edges, so PV

cells must be located on each. If the modules are stacked top to bottom as illustrated in Figure

2.5, adjacent modules can share a single, bi-facial PV cell. The extruded architecture allows the

CPV to be extended to any length without impacting its optical efficiency.

Figure 2.4: Shadowing advantage of the louver design space, where part of the louver (a) is designed for greater vertical angles and other parts (b) are optimized for lesser ones.

37

Sending light towards both the

top and bottom edges of the guide

provides the opportunity to intrinsically

decrease guiding losses by reducing the

footprint of the injection features on the

guide surface. This is done by

shortening the injection feature

subsystem by essentially adding a fold

mirror. This approach is used in all

louver designs and appears in the form

of small mirrored surface located on the

upper extreme of the injection feature, perpendicular to the window plane.

For a single module, sending light towards two target edges instead of one decreases

the system’s concentration ratio by a factor of two. However, if the modules are stacked and

bifacial PV cells are used to accept light from two adjacent modules, it can be argued that the

concentration ratio is unaffected.

Four distinct designs

based on the louver concept

are presented here. The

“suspended louver” (SL) has

louvers suspended in air with

extruded dielectric prisms as

injection features. The

Thre

e st

acke

d L

CP

V m

od

ule

s

Bif

acia

l PV

cel

ls b

etw

een

mo

du

les

Figure 2.5: Illustration of a full window LCPV system consisting of several modules stacked top to bottom.

Figure 2.6: Cross-section view of a suspended louver design. The horizontal grey lines on the injection features indicate a reflective surface.

38

extruded injection features use a flat refractive surface to bend the light to angles where it is

subject to TIR in the guide layer. The top side of the injection features are mirrored to reduce

the footprint of the features on the guide surface while sending the light towards two edges of

the guide as opposed to a single edge as in the UR light guide module of Chapter 1. An example

of this design is shown in Figure 2.6 between the glass of a double-paned window. Hashed lines

indicate reflective surfaces which include the louver and the topmost injection feature edge.

The “immersed louver”

(IL) immerses the louvers in a

dielectric material. This has the

unique advantage of narrowing

the angular range of the light

that strikes the louver surface,

allowing them to be smaller and

flatter. In addition, if the

immersion material contacts the outer window pane, number of air-dielectric interfaces in the

system decreases. This means fewer Fresnel losses and greater optical efficiency. The injection

features in this design are areas of optical contact between the immersion material and the

guide layer which act as “holes” that allow light to pass through. The cross-section of an IL

design can be seen in Figure 2.7. Notice the fold mirror associated with each injection feature is

present.

The “extruded segment lens” (ESL) deviates from the original concept by having a flat

reflective surface and using a curved refractive interface to add focusing power. This design

loses much of the shadowing advantage around which the louver concept is based. The cross-

Figure 2.7: Cross-section of an immersed louver design.

Injection Feature

Fold Mirror

39

section of one such design is

displayed in Figure 2.8. Since the

window area is filled with curved

refractive surfaces, this design

heavily distorts the image of the

landscape on the opposite side of

the window. Thus the ESL is not

applicable to picture windows,

but has promise in the form of a privacy window.

Finally, the “tracking louver” once again suspends louvers in air, but allows them to

move and track the sun in a vertical direction. The cross-section looks qualitatively similar to the

suspended louver in Figure 2.6, but the louver’s position and orientation vary as a function of

the vertical angle of incoming sunlight. The injection features and guide layer remain static, but

the injection feature utilizes a cylindrical refractive surface. Though the moving parts increase

the complexity of the system, they allow greater optical efficiency and a broader vertical angle

field of view.

2.3 Optimization and Results

Initially, the primary optics of the static designs were optimized in segments, in a

sequential manner. The surface was constrained to being continuous, but not necessarily

smooth. In this scheme, each segment is described by a quadratic function with three degrees

of freedom. Other degrees of freedom include the starting position of the first segment relative

to the injection feature, the injection feature size, and the weighting function. Unlike those

Figure 2.8: Cross-section of an extruded segment lens design. Grey lines indicate mirrored surfaces.

40

presented in Section 2.1.2, the weighting function for each segment is dynamically calculated

according to the fraction of light from each vertical angle (within a predefined range) captured

by that segment.

While this optimization method has a large number of variables and its sequential

architecture is slow and cumbersome, it provides insight by explicitly incorporating the

shadowing effect central to the louver concept. After numerous results for each static design, it

was clear that the best performing designs exhibited relatively smooth surfaces that did not

appear segmented. The optimization routines were then rewritten to describe the louver (and

refractive lens) surface shape as a single higher order polynomial while implementing the

vertical angle weighting functions of Section 2.1.2. The new optimization method greatly

reduces the number of system variables and eliminates the need for a sequential process.

Though the performance values for each optimization method are comparable, the single

segment method does produce slightly better results since it can evaluate a greater number of

variable values in a shorter amount of time.

For comparison purposes, all designs are optimized for a guide layer length of 300 mm

and thickness of 13 mm. All dielectrics are simulated as PMMA with a refractive index of about

1.49 and do not include internal absorption. The reflectance of all reflective surfaces is a

constant 95% and the spacing between adjacent louvers is 2 mm. The size of the louvers is

proportional to their spacing (larger size adds to the thickness of the window as a whole), so 2

mm is chosen in order to not add too much thickness to the system since the guide already adds

13 mm. It is advantageous to the designs to have a thick guide layer rather than large primary

optics. Of course, the dimensions may be adjusted to improve manufacturability.

41

2.3.1 Suspended and Immersed Louver

The optimization parameters for the

suspended and immersed louver designs are similar.

The variables consist of the height of the injection

feature, the distance from the guide to the extreme

endpoint of the louver, and the coefficients of a

seventh order polynomial describing the shape of the

louver reflector (Figure 2.9). During optimization,

solutions in which the reflectors collide with adjacent

injection features are invalidated.

For the suspended louver (SL), the injection prism shape is calculated so the refractive

surface of the prism is oriented in a way that allows rays from any point on the reflector to be

refracted at an angle compatible with TIR in the guide layer. The size of the injection feature

fold mirrors is calculated for both SL and immersed louver (IL) so that all rays from the extreme

endpoint of the louver that strike said mirror will subsequently pass through the injection

feature/guide layer interface.

Figure 2.10 shows the final SL solutions for each location. The geometry of each

solution is shown across the top with the same scale for each. The bold lines in the geometry

plots represent one iteration of the arrayed optics, and the thin curve is the adjacent louver that

provides the shadowing effect. The optical efficiency as a function of vertical angle is presented

in the bottom graph. Notice that the efficiency peaks for the three cities appear in the same

order as the peaks in their respective weighting functions seen in Figure 2.3.

x0

h

Figure 2.9: Optimization variables for suspended (and immersed) louver designs indicated in red.

42

The efficiency values represent the optical efficiency of the concentrator, so the effect

of projected area for off-axis incidence angles is not included. This value may be broken down

into two components:

the “injection

efficiency” and “guiding

efficiency”. Injection

efficiency refers to

fraction of light energy

entering the system’s

front aperture that is

successfully injected

into the guide layer

within its TIR limits. The

guiding efficiency is the

fraction of successfully

injected light that

travels in the guide until

it strikes a PV cell surface. For these designs, the injection efficiency is the same for every louver

in the CPV module. The guiding efficiency, however, varies depending on distance that light

must travel in the guide layer before reaching the output edges. This is explored in greater

detail in Section 2.4.1.

As indicated by the geometry plots, the Rochester and Fairbanks design have injection

prisms that occupy about half of the adjacent guide surface. This means that the guiding losses

Figure 2.10: Solutions from the suspended louver optimization for the three chosen locations. Geometry of the reflectors and injection features are displayed across the top, and the efficiency as a function of vertical angle is shown below.

43

are large since there is a high chance that a ray traveling along the guide couples out via a

downstream injection feature. The efficiency plots take this into account since they simulate

the full 300 mm tall system.

Similar

efficiency trends are

found in the immersed

louver solutions as

shown in Figure 2.11.

The geometry of the

injection features are

displayed as a thick

horizontal line

representing the fold

mirrors and a narrow

vertical line indicating

the extent of the

injection feature.

Recall that the IL

injection features are

areas of optical

contact between the immersion material and the guide layer. The IL solutions typically have

better performance than their SL counterparts. Part of the reason for this is the reduced size of

the injection features, allowing for a greater guiding efficiency.

Figure 2.11: Solutions from the immersed louver optimization for the three chosen locations. Geometry of the reflectors and injection features are displayed across the top, and the efficiency as a function of vertical angle is shown below.

44

There exists a tradeoff between the width of the efficiency peaks and their height. For

example, comparing the Fairbanks solutions from the SL to the IL architecture reveals that the

efficiency peak has become narrower and taller. This tradeoff is also evident in the multiple

solutions found for a constant location and architecture. The peaks tend to be fairly narrow and

tall since the weighting functions have relatively narrow peaks.

2.3.2 Extruded Segment Lens

The extruded segment lens

(ESL) design was originally named for

its segmented curved surface and its

horizontally extruded geometry,

though the surface presented here is

defined by a single polynomial function.

ESL designs have analogous

optimization variables to the SL and IL

architectures (Figure 2.12). Instead of a

polynomial describing the louver shape,

there is a polynomial describing the

refractive surface shape. The width of

the refractive surface and size of the

injection feature are also variable as were the louver widths and injection features in the

previous designs. The calculated quantities include the injection feature fold mirror (horizontal

h y0

Figure 2.12: Optimization variables for extruded segment lens design indicated in red lettering. Traced rays show the additional angled fold mirror in the ESL design (ray colors indicate angle after injection into the guide layer).

Fold Mirror

← G

uid

e →

45

lines in Figure 2.12) height, and the angle of an additional fold mirror that makes up the bottom-

most face of the extruded lens structure. This added fold mirror essentially serves the same

purpose as the refractive face of the injection prism feature in the suspended louver

architecture.

Figure 2.12 also shows rays traced for a given incidence angle, displaying three ways

light can enter the guide layer. With the louver designs, all light reflects off of the louver and

possibly the fold mirror before successful injection into the guide layer. Those two paths remain

true for the ESL design, replacing the louver with the angled fold mirror. There is an additional

optical path that interacts with neither mirror, injecting into the guide immediately after passing

through the refractive lens surface.

Another note for this design

is the effect of the dielectric

material that fills the shape drawn

in Figure 2.12. It is modeled as a

dispersive material (PMMA) and

rays of several wavelengths are

traced during optimization. This is

done because the chromatic effects

are more significant than in the

reflective designs due to the highly

curved refractive surface as the first

interface in the ESL system. Figure 2.13: Solutions from the extruded segment lens optimization for the three chosen locations. Geometry of the reflectors and injection features are displayed across the top, and the efficiency as a function of vertical angle is shown below.

46

However, for the presented designs, the performance does not depend strongly on wavelength.

Optimization results for the three locations are shown in Figure 2.13. These refractive

designs have a uniquely shaped field of view compared to the two static reflective designs, but

the same trends are observed among the different locations. The El Paso optimization yields a

taller, narrower field of view than the Rochester and Fairbanks locations. The peaks still follow

the same order as the two louver designs since the weighting functions are independent of

system architecture.

While the ESL performance is roughly as good as the IL, it does have significant

disadvantage in this application. Since the curved refractive surface fills the majority of the

window’s aperture, the image of the outside world is heavily distorted.

2.3.3 Tracking Louver

The final design space explored is the “tracking louver” (TL). As the name implies, it

utilizes a reflective louver primary optic that moves to track the sun. The louver is allowed to

translate in the cross-sectional plane and rotate about an axis parallel to the extrusion direction.

The louver shape is defined as a section of a parabola to achieve a tight line focus. The injection

feature changes shape as well: the refractive surface is cylindrical instead of flat. The focus of

the reflector is shifted to a part of the injection feature such that the light rays refract into an

angle supported by TIR in the guide layer.

Since the reflector’s focus spot size is infinitesimally small for incident rays parallel to

the parabola axis, an extended source with the angular size of the sun is considered during

optimization to prevent an unrealistically small injection feature.

47

Optimization variables consist of the radius of the

injection feature refractive surface, the size of the injection

feature’s footprint on the guide surface, the focal length of

the reflector’s parabola, and the extent of the reflector

(Figure 2.14). For each vertical angle (at sub-1° intervals),

the reflector is moved to a position such that its axis is

aligned with the sun and its focus is positioned on the face

of the injection feature. Then the alignment and lateral

position is refined through an optimization routine for each

vertical angle. Thus the tracking motion is determined.

Occasionally, the reflector collides with the guide

layer after initial alignment. This is a physically disallowed

state and its occurrence depends primarily on the

reflector’s length and shape. In this case, the optimization

procedure brings the bottom of the reflector in contact with

the guide surface and rotates it out of alignment. The new

alignment causes the sharp focus to be lost, but a relatively

concentrated caustic is formed and aimed at the injection feature. This reflector-guide collision

occurs in the lower range of vertical angles as in the 0° position of Figure 2.15 (guide surface at x

= 0 mm). The lower the vertical angle, the more misalignment is required to prevent collision,

and the less well defined the caustic becomes. This reduces the optical efficiency at the vertical

angles in question.

Injection Feature

Guide

Figure 2.14: Tracking louver optimization variables shown in red.

Fold Mirror

48

Another concern

regarding the mobile

louvers is light blocked by

an adjacent louver after

reflection but before

striking the injection

feature. This essentially

constrains the maximum

reflector focal distance to

be approximately the

same as the louver

periodicity. This way, the

adjacent louver does not

interfere with the light

headed towards the

injection feature.

The optimization

solutions in Figure 2.15 show that the performance of the tracking louver design changes little

among the three cities investigated. Note, the weighting function for El Paso is modified to

ignore the negative values for this optimization procedure. Further investigation shows that the

optimized louver shapes are similar for all three locations as well. This means that a single

tracking louver may be used effectively at all three locations without modification.

Figure 2.15: Solutions from the tracking louver optimization for the three chosen locations. Geometry of the reflector and injection features for the Rochester design are displayed at the top for several vertical angles, and the efficiency as a function of vertical angle is shown below for all three cities.

Injection Feature

49

The magnitude of the efficiency for the TL designs is far greater than that of the static

designs, and the field of view is much broader. This vastly improved performance comes at the

cost of increased system complexity due to moving parts. The small size of the optimized

injection features mean that the reflector has a high positional tolerance making manufacture

difficult. Looser tolerances require a larger injection feature, therefore reducing guiding

efficiency.

2.4 Analysis

Insights regarding the loss mechanisms in the presented systems are achieved through a

more detailed analysis of the ray trace data recorded during optimization. Additionally, the data

are analyzed within the scope of the application: in terms of effective R-value gains.

2.4.1 Efficiency

Recall that the efficiency for the CPV systems can be broken into components of

injection efficiency (front aperture through injection feature) and guiding efficiency (injection

feature to PV cell). The suspended louver Rochester design is used as an example of this

analysis in Figure 2.16. The full

system field of view is the

same as that found in Figure

2.10, and the thin red line

shows the injection efficiency

as a function of vertical angle.

The guiding efficiency

(averaged over the system Figure 2.16: Suspended louver Rochester design performance showing injection efficiency and weighting function.

50

aperture) is the ratio of FoV to injection efficiency at each vertical angle. The guiding efficiency

is low for the SL designs, on the order of 40%.

The full length for the optimized

systems is 300 mm from the top edge of

the module to the bottom, where the two

PV cells are located. Given the louver

periodicity of 2 mm, each module contains

about 150 louvers and injection features,

giving ample opportunity for light to couple

out of the guide prematurely. To illustrate

the guiding efficiency in another way, the

performance data is averaged over vertical

angle and presented as a function of

aperture position instead. Additionally, the

output is broken into the top and bottom PV cells to show the effect of the low guiding

efficiency. A guide position of 0 mm in Figure 2.17 represents the location of the bottom PV cell,

and 300 mm is the location of the top cell.

Each cell has high efficiency in positions near their respective locations. In fact, the first

several millimeters near each guide edge reveal the injection efficiencies for rays travelling

towards the top and bottom cells. Guiding losses do not occur for approximately the first 26

mm of travel along the guide since the injected light does not contact the lossy guide surface

immediately. At a ray angle matching the critical angle of the guide material, about 45°, it

travels a distance roughly equal to twice the guide thickness (13 mm) before interacting with the

Figure 2.17: Rochester suspended louver design normalized efficiency components as a function of position on the system's front aperture. The total is the sum of the top and bottom cells, while the average is the mean of the total over all positions.

51

guide surface containing injection features. In contrast, very little light travels along the entire

length of the guide to strike the opposing PV cell since that requires multiple interactions with

the lossy guide surface. This is indicated by the low efficiencies of each cell at aperture positions

opposite their respective locations.

The sum of the top and bottom cells (“Total” in Figure 2.17) shows the relative efficiency

with which light is captured by either PV cell as a function of aperture position. The efficiency

for this design in the center of the aperture is about half of its maximum value found near the

edges. This efficiency depression is less pronounced in systems with greater guiding efficiencies,

e.g. tracking louver.

If the system remains as is except for the

total length of the guide, the efficiency is expected to

increase as the guide becomes shorter. A short

guide means less distance for light to travel before

encountering a cell and therefore greater guiding

efficiencies. Thinking graphically, the two curves of

Figure 2.17 for the top and bottom cells maintain the

same shape, but are shifted closer to each other if

the guide is shortened. The efficiency for the bottom

cell remains as is at the position of 0 mm and the

same is true for the top cell, but at the maximum

aperture position. The results of this analysis are

summarized in Figure 2.18. As expected, the

efficiency increases as the guide length decreases.

Figure 2.18: Rochester SL design average efficiency and effective concentration as a function of guide length, both normalized at the designed guide length of 300 mm.

52

However, the geometric concentration ratio is directly proportional to the guide length, so the

effective concentration ratio (geometric concentration x optical efficiency) actually decreases

with shorter guide lengths. For example, if the guide is shortened from 300 to 100 mm, the

optical efficiency for the system doubles, but the geometric concentration ratio is only a third of

its initial value. Thus, the resulting effective concentration at a 100 mm guide length is 66% of

its value for a 300 mm long guide.

Recall that this design is optimized for a guide length of 300 mm and thickness of 13

mm. If the system were re-optimized for a shorter guide, the results would not coincide with

those in Figure 2.18. For example, at a 100 mm guide length it is advantageous to achieve

higher injection efficiency at the expense of guiding efficiency since the average distance light

must travel along the guide layer decreases. This changes the shape of the optical components,

i.e. larger injection features, and achieves a higher efficiency (and effective concentration) than

seen in Figure 2.18 which assumes the optical components remain as they are for a 300 mm

long guide.

2.4.2 R-Value

The final analysis attempts to predict the CPV system’s effect on the insulatory

properties of a window. The R-value is a measure of thermal insulation efficiency. The higher

the R-value, the better the insulator. A well-insulated window has an R-value of R-3.0 to R-5.0

while a typical outer wall of a house is about R-13. R-value is defined by the expression, Q =∆T

R,

where Q is the energy flux lost through a barrier, ΔT is the temperature differential from one

side of the barrier to the other, and R is the R-value of the barrier. The US units for R-value are

ft2∙°F∙hr/Btu and the SI units are m2∙°K/W [24]. The U-factor, or U-value, is simply the inverse of

53

the R-value. When the energy produced by the window LCPV modules is equivalent to the

energy lost through the window, the U-factor is zero while the R-value is infinite. For this

reason, the analysis results are presented in terms of U-factor.

The “effective U-factor” is hereby defined as the passive U-factor above with a modified

net energy flux, Q, incorporating the energy recovered by the integrated CPV. This quantity is

highly dependent on weather conditions since Q varies with the temperature differential across

the window and the CPV electricity production depends on the amount of solar irradiance

incident on the window. As such, two weather conditions are considered in the effective U-

factor analysis: a year that is sunny at all times with temperatures based on monthly averages,

and a “typical meteorological year,” or TMY, that includes solar irradiance and temperatures

from historical data. The TMY database is managed by the U.S. National Renewable Energy

Laboratory (NREL) and includes data for hundreds of cities and towns across the United States

[25]. TMY data include ambient temperatures, direct and indirect solar irradiance, and several

other atmospheric conditions during every day of a single (synthetic) year per location.

The effective U-factor is calculated year-round and the results for the SL Rochester

design are displayed in Figure 2.19. The bold blue lines are the data for the sunny year and the

thin green lines correspond to the TMY data. The passive heat flux is the energy lost in the form

of heat per window area from the inside of the window to the outside based on the passive R-

value. Rochester does not often have average ambient temperatures (average of daytime highs

and nighttime lows) that exceed room temperature (20°C). The temperature gradient changes

sign mostly during the months of July and August.

54

The passive R-value for

this data is assumed to be 5.0

equivalent to a U-factor of 0.2.

This corresponds to a very well-

insulated, triple pane window

filled with a noble gas such as

krypton [26]. The passive R-value

is affected by the inclusion of the

CPV structure. The R-value is

inversely proportional to the

thermal conductivity of a bulk

material at a constant thickness,

the latter being an intrinsic

material property. Therefore, the

lower the thermal conductivity,

the more effective a material is for

use as insulation. Krypton, for instance, is sometimes used

between window panes since it has a low thermal conductivity

of 0.009 W/m∙K, less than half that of air. PMMA, a candidate

material for the CPV guide layer, has a relatively high

conductivity of 0.2 W/m∙K, but is still five times more effective

than glass (Table 2-1). If the construction of a CPV integrated

Material

Thermal

Conductivity

( W/m∙K )

Air 0.024

Argon 0.016

Krypton 0.009

PMMA 0.200

Glass 0.960

Copper 401

Table 2-1: Thermal conductivities for various materials [26].

Figure 2.19: Effective U-factor calculations for the SL Rochester design. The bold blue line represents a sunny year and the thin green line is from “typical meteorological year” data. The red line of the U-factor plot represents the passive U-factor of the window.

Passive Heat Flux (Indoors to Outdoors) W

h/m

2 /day

W

h/m

2 /day

55

window requires PMMA to replace part of the gas volume between the glass panes, the passive

R-value will decrease. If the window is allowed to become thicker to accommodate the

additional CPV structure, however, the R-value can be recovered. One can imagine the CPV

guide layer replacing the central pane of a triple paned window.

Referring once again to Figure 2.19, the CPV electricity production is the energy

converted to electricity per window area assuming a PV cell efficiency of 18%, typical of silicon.

The calculations account for both direct and indirect solar irradiance. When this value equals

that of the passive heat flux, the CPV is generating as much energy as is lost through the

window. One effect that is not included here is the heating of the cells which accounts for the

majority of the energy striking the cell that is not converted to electricity (100 – 18 = 82%). This

heats the inside of the window and is mostly lost to the outside, but would bolster the system’s

performance overall.

Finally, the effective U-

factor is plotted for both simulated

weather conditions along with the

passive U-factor of 0.2. The sunny

weather conditions yield a

significant decrease in the effective

U-factor in the winter. In February,

the U-factor is as little as half the

passive value alone. This equates to

an R-value of about 10, almost as

well insulated as the surrounding

Figure 2.20: Effective U-factor calculations for the SL Rochester design. Battery storage is included and its charge level is shown in the bottom plot.

56

outer walls of the building. Most of the month of October has an even more impressive U-factor

of around 0. This means there is no net loss of energy through the window, an effective R-value

of infinity. The TMY weather conditions, however, show significantly less optimistic

performance. Unfortunately, Rochester is extraordinarily cloudy during the winter months so

the solar irradiance is much reduced. This effect is seen clearly in the CPV electricity production

plot.

The U-factor data becomes negative and greater than the passive value at some times

during the year. A negative effective U-factor corresponds to the electricity production being

greater than the passive heat loss while the ambient temperature is below room temperature.

The U-factor wraps around through infinity when the ambient temperature becomes greater

than room temperature. These positive values greater than the passive U-factor indicate that

electricity is being produced when the indoor air is being cooled during warmer times of the

year.

To mitigate this unwanted effect, especially for TMY conditions, some battery storage

could be included with the window in order to save and use the electricity when it is needed

most. The U-factor is recalculated assuming an electrical storage capacity of 100 Wh (equivalent

to a large laptop battery) per square meter of window area. The battery is modeled as having

some current leakage that heats the indoor air when discharging. The U-factor in TMY

conditions is considerably improved during the summer and early fall months. This is because

any excess energy can be stored and used on a cooler or cloudier day.

57

Even with a large storage

capacity, the SL Rochester design

cannot provide a significant

benefit. The other static designs

fall similarly short for the

Rochester location due to the

cloudy winters. For the static

designs, the best case scenario for

effective U-factor is not

surprisingly a warm, sunny location

with a more efficient CPV

architecture. As such, the

immersed louver design is

analyzed for the El Paso location.

The results are presented in Figure

2.21. Indeed, the U-factor is

improved considerably and it is

often the case that the windows

have a net negative energy loss, as

is the case in February, October,

and November. Figure 2.21: Effective U-factor calculations for the immersed louver (IL) El Paso design including battery storage.

Wh

/m2 /d

ay

Wh

/m2 /d

ay

58

The final analysis is for

Fairbanks, a cold location with

little solar radiation in the

winters. The only truly effective

CPV for Fairbanks is the tracking

louver with its high conversion

efficiencies during the summer

and winter months. The results

are shown in Figure 2.22. The

highly efficient tracking louver

design shows promise even in the

demanding conditions of

Fairbanks. The advantages of the

CPV are especially noticeable

during the summer months,

which have average daily

temperatures below 20°C and a

fair amount of solar radiation.

The U-factor from April through

September is well below zero, so

the window has a net positive

effect for about half of the year.

Figure 2.22: Effective U-factor calculations for the tracking louver Fairbanks design including battery storage.

Wh

/m2/d

ay

Wh

/m2 /d

ay

59

2.4.3 Cooling Effect

One final effect that can be engineered into the reflective louver designs is a way to

reject direct sunlight at higher vertical angles associated with warmer months. If sunlight passes

into a building and strikes indoor surfaces, it will warm the indoor air. This is undesirable during

the summer when outdoor temperatures are high. At vertical angles above the range of the FoV

for the static reflective louver designs, direct sunlight tends to strike the back of the adjacent

louver before making its way indoors. If the back of the louvers are textured to be

retroreflective, the light could be redirected back outside where it will not have negative effects.

An immersed louver design is depicted in Figure 2.23. The left image illustrates solar

rays incident from a large vertical angle outside the FoV of the system. The smooth back of the

louver redirects the rays that miss the injection feature towards the indoor space. This is

undesirable when trying to keep the indoor air cool. The image on the right side of Figure 2.23

Figure 2.23: Effect of backside retroreflective texture on static louver designs: rejecting direct sunlight during summer. Left image shows high vertical angle rays striking back of louver and continuing through the window. Right image shows rays being turned back by a retroreflective texture on the adjacent louver rear surface.

60

shows the same system under the same illumination with the only difference being a

retroreflective texture on the back side of the louvers. Instead of sunlight being redirected

indoors, it is reflected back toward the primary louver and subsequently outdoors.

This system generally works for both the suspended and immersed louvers, but is not

obviously applicable to the refractive design space. The tracking louver can reject sunlight

without a retroreflective texture by simply moving the louvers into a positon that directs

sunlight back outside after a single reflection.

2.4.4 Aesthetics

For the application as specified, the aesthetics of the CPV are important. For picture

windows, an observer must be able to see a partially unobstructed view of the scene on the

opposite side of the glass. On the other hand, a privacy window application requires severe

distortion of the scene while still allowing light through to illuminate the room.

Three photorealistic renderings are shown in Figure 2.24. All ESL designs nearly fill the

system aperture with curved surfaces extruded in the horizontal direction. This blurs the scene

on the opposite side of the window in a vertical direction so no details can be distinguished,

thus its potential as a privacy window. This blurring is especially evident when comparing the

background of the bottom and middle images in Figure 2.24. The suspended louvers exhibit a

high degree of obstruction due to the louver shape and some obscuration from the large

prismatic injection features.

The IL designs offer the least amount of obstruction of any static design given their

relatively flat louver shapes and injection feature architecture. The only obstructive portion of

the injection features are the fold mirrors. If the injection feature is located close to the

61

adjacent louver (e.g. the Fairbanks design, Figure

2.11), the scene can be minimally obstructed

when peering through the window at certain

angles. The middle image in Figure 2.24

simulates such an IL design at the optimal

viewing angle for low obstruction ratio, about

25% of the total aperture is obstructed.

The tracking designs have even flatter

louver shapes and very small injection features,

resulting in obstruction ratios as low as 10%. Of

course, windows containing any of the louver

designs can have high obstruction ratios if

viewed from certain directions, for example, a

steep upward angle. Traditional window blinds

have the same, familiar effect.

IL No CPV

No CPV ESL

No CPV SL

Figure 2.24: Photorealistic renderings of views through the three static designs: SL (top), IL (middle), and ESL (bottom).

62

2.4.5 Passive Solar Heating

Passive solar heating is a

technology that has a nearly identical

application to the presented LCPV window

designs. Instead of using photovoltaic

materials, however, this approach relies on

indoor surfaces absorbing sunlight energy

passing through windows, converting the

energy directly to heat. Thoughtfully placed thermal masses embedded into a building’s

structure can offer storage of that thermal energy with the ability to continue heating the

indoor space when sunlight is not available, albeit in an uncontrolled manner. A passive control

mechanism may be included in the form of an eve to shade the windows during the summer, a

similar concept to that presented in Section 2.4.3 [27].

Effective passive solar heating design requires a disproportionate number of south-

facing windows (in the Northern Hemisphere) compared windows of other orientations. In a

typical home, about one quarter of the window glass is south-facing, amounting to about 3% of

the building’s floor area. A house with basic passive solar heating (without additional thermal

masses) increases this to between 5% and 7% the total floor area. Once the area of south-facing

window glass exceeds about 12% of the floor area (depending on climate), gains from solar

heating begin to wane since it is difficult to include enough thermal mass into the structure.

Additionally, problems such as glare from the large glass area and fading of indoor fabrics

become significant [27]. A major advantage of passive solar heating is that it is essentially free if

incorporated prior to construction.

Figure 2.25: Schematic of building with passive solar heating. Shows absorbing indoor surfaces, thermal mass storage, and shading control eve.

63

For some climates, passive solar heating can surpass the performance of the presented

LCPV designs for this application. For example, aggressive passive solar design can reduce the

amount of energy needed for heating by as much as 75% in favorable climates [27]. It is

important to note, however, that the electricity produced by photovoltaic systems is highly

versatile in its end use and the ability to transport and store the energy. Additionally, an effect

that is not included in the earlier analysis is the heating of the PV cells. During operation, the

cells convert roughly 18% of the energy that strikes them. Much of the remaining energy heats

the PV material, some of which will ultimately heat the indoor air, constituting a passive heating

effect.

2.5 Conclusion

A summary of the optical efficiency with respect to vertical angle is presented in Table

2-2. One notable trend is the highest peak efficiency in the El Paso designs compared to the

other two locations for each static design. This occurs because the El Paso weighting function

Table 2-2: Vertical angle field of view summary. The left data column indicates the vertical angle field of view range for each design, the center column shows the average optical efficiency over that range, and the right column gives the peak optical efficiency.

64

has the narrowest peak. The tradeoff between vertical angle range and average efficiency is

evident as well. For each static design, it is nearly always the case that a smaller FoV range

corresponds to a higher average optical efficiency.

In order to achieve large fields of view, efficiency must be sacrificed. In fact, several of

the presented designs have peak optical efficiencies of less than 40%. The ideal balance

between efficiency and field of view depends largely on the application, and integration into a

window demands light collection from a wide range of angles. All designs improve the effective

insulation properties of the windows to varying degrees. The immersed louver design in

Rochester has a small impact and is not a viable solution. Conversely, the immersed louver in El

Paso offers a net energy loss of zero or better during much of the year, which may be a viable

product in today’s energy conscious market.

65

Chapter 3: Prototype and Metrology

3.1 Prototype

The primary goal of the prototype is to serve as a proof of concept. If the performance

measurements taken on the prototype unit match the performance of a computer model of the

prototype, the modelling techniques are validated. Further study of the model can give insights

into the loss mechanisms present in the as-built prototype.

3.1.1 Prototype Design Choice

For ease of fabrication and alignment, the prototype unit contains a single primary

optical component and injection feature at roughly a 10X scale. A baffle is also included to

incorporate the shadowing effect of the adjacent primary reflector.

The choice for which of the aforementioned designs to attempt to prototype was made

through a process of elimination based on the availability and difficulty of the required

fabrication methods. The tracking louver, for the purposes of this thesis, is prohibitively

complex to build and align due to its moving parts. The refractive design requires the very

difficult manufacture of optically clear extruded freeform surfaces. The immersed louver design

is a strong candidate since it has the best non-tracking performance and provides the least

obstructed view. However, immersing a large scale prototype unit in a dielectric material while

maintaining a small air gap adjacent to the front surface of the guide layer is a difficult task given

the available fabrication techniques. The only remaining option is the suspended louver design,

and though it does not have the best performance nor the least obstructed view, it serves as an

acceptable proof of concept and is able to lend credence to the optical modeling techniques.

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The chosen location for the model is Rochester since it has a wider field of view than the

El Paso design and the efficiency as a function of vertical angle is less disjointed than the

Fairbanks design. Due to these features, the Rochester version is thought to be the best design

for diagnosing fabrication errors in the prototype based on performance measurements, but

likely any location choice is suited to achieving the goals of the prototype. The specific choice of

which Rochester design is based on similar merits.

An additional benefit to the chosen design is that the dimension of the prism back

surface can be extended to help make fabrication easier without interfering with the adjacent

reflector, or baffle in the case of the prototype (Figure 3.1).

3.1.2 Guide Layer Dimensions

In the full suspended louver design, the thickness of the guide layer is many times larger

than any dimension on the reflector or the injection prism, but this is impractical and

unnecessary for the large scale prototype. For the prototype, the guide need only be thick

enough so that light entering through the prism will not TIR off the back surface of the guide and

reenter the prism. With a critical angle of about 45° for the guide layer material, this means a

Back

Bo

tto

m

Top

Bottom Fr

on

t

Bac

k

Gu

ide

Prism

Baffle

Reflector

Figure 3.1: Dimensioned cross section of prototype design (lengths in mm). Shows nomenclature for faces of prism and guide. The dashed red line shows that the prism has minimal interference with light that moves past the baffle and is headed for reflector. The blue dotted lines show that the prism back can be extended without obscuring the intended optical path through the device.

PV Cell

PV Cell

67

minimum guide layer thickness of half the width of the prism bottom. Additionally, the guide

needs to be long enough so that light entering the guide via the prism at angles that do not TIR

off the guide surfaces cannot strike the PV cell surfaces before interacting with at least one

guide surface. The PV cells are located on the top and bottom faces of the guide compontent.

3.1.3 Extruded Dimension

The dimension of the prototype in the extruded direction is a balance between the

measurable azimuthal field of view and the difficulty of fabrication. A longer extrusion

dimension means that light entering the system from further off-axis in azimuthal angle can

traverse through the system without interacting with the side walls of the guide or prism.

However, manufacturing a longer prism and reflector becomes difficult to do accurately. The

extrusion dimension for the prototype is 100 mm, largely due to limitations of the fabrication

method chosen for the reflector.

3.1.4 Injection Prism Tolerancing

The prototype consists of three separate optical components: the guide layer, the

injection prism, and the reflector. The guide layer is the simplest to manufacture since it is a flat

slab of dielectric material, and it is constrained primarily by the minimum dimensions described

above. The prism and reflector present more difficulties, but the fabrication method of the

reflector is extremely difficult to control and adjust. For these reasons, tolerance analysis is

focused on the prism since there is more freedom in the choice of fabrication methods for an

optical component comprised entirely of flat surfaces.

Note that during the tolerance analysis, some simulated errors increase the expected

efficiency, however, this does not imply that the performance of the full system is improved if

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the same error is introduced. The prototype and the full design must be thought of as disparate

systems with the prototype functioning as a proof of concept of the reflector-prism geometry

only.

In cross-section, the injection prism is a four-sided polygon with two sides, the “bottom”

and the “back”, perpendicular to each other (see Figure 3.1). The tolerance analysis performed

perturbs the two adjacent vertices that form the “top” edge of the prism. The perturbations are

parameterized into the angle between the front (α) and back (β) edges relative to the bottom

edge as well as the length of the front and back edges. The values of the dimensions as

designed can be found in Figure 3.1. Each of α and β are perturbed by ±1 degree and the

lengths of both front and

back edges are each

perturbed by ±1 millimeter

(Figure 3.2). Each parameter

is varied individually then all

four are varied

simultaneously to find the

maximum compound error in the performance of the prototype model.

The figure of merit in this study is the deviation of total irradiance on the top and

bottom guide faces as compared to the ideal geometry. This is equivalent to the relative total

power incident on the PV cells. The results show that both edge length perturbations had a

negligible effect on the prototype performance, and so they were specified to be within the

maximum range of the study, ±1 mm. The back edge is allowed to be extended further in order

to make it easier to manufacture, i.e. to create a right angle with the top edge. The angle

0

5

10

15

20

25

-40 -30 -20 -10 0

Ideal Shape

Front Vertex Variation

Back Vertex Variation

Envelopeα

β

back

Figure 3.2: Tolerance study on prototype prism dimensions. Solid blue line shows designed shape, dashed black lines show envelope of the tolerance study which is achieved by varying α, β, and the lengths of the front and back edges.

69

perturbations had a more profound effect. To limit the change in performance, the maximum

error specifications for angle α are ±0.75°. The angle β is more sensitive still and has a specified

maximum error of ±0.25°, though it only affects the performance of the bottom PV cell. The

effect of the compound errors given the above specifications are shown in Figure 3.3.

3.2 Fabrication Methods

Given the varied specifications and applications, the fabrication and metrology methods

for each component have different requirements. This section discusses fabrication methods

for the components as well as assembly of the prototype.

3.2.1 Injection Prism and Guide Layer

A primary restriction on the injection prism and the guide layer is the material. Since

both components are used in transmission, absolute refractive index and its uniformity are

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25 30 35 40 45 50 55

Rel

ativ

e To

tal P

ow

er O

utp

ut

Vertical Angle ( ° )

Compound Error Tolerance Study

As Designed Minimum Error Maximum Error

Figure 3.3: Error envelope of tolerance study. The blue line shows the performance of the as-designed prototype, the orange line shows the minimum performance from the compound geometry errors, while the grey line shows the maximum.

70

major concerns. Anticipating the need to fabricate the system, the material used in the

computer model is poly(methyl methacrylate), and is indeed the choice for the real world

prototype. Poly(methyl methacrylate), or PMMA, is more commonly known as acrylic or by one

of its trade names, e.g. Plexiglas [28]. PMMA is easily and inexpensively sourced in bulk, and is

machined and polished with relative ease. Like many of the physical properties, the refractive

index can vary depending on the plastic’s specific formulation, but it is typically around 1.49 in

the visible spectrum. This has been verified for the fabricated prism and guide layer using a

refractometer (see Section 3.3.1.3) which measured an index of about 1.489 at a wavelength of

632.8 nm, matching published data [29]. Amongst typical optical glasses, PMMA would be

considered a crown, i.e. a low index and low dispersion material.

The fabrication method used for both the prism and guide is a milling procedure,

common for creating structures out of metals and plastics. The faces are then buffed to remove

surface roughness caused by the milling bit. This is not a common fabrication method for

optical components due to the relatively imprecise reproduction of the surface geometry and

high surface roughness. However, due to the fact that all surfaces were designed to be flat, the

loose tolerances on the component dimensions, and the non-imaging nature of the prototype

device, this fabrication method proved to be adequate.

Stock PMMA material is often in the form of flat plates, and the factory-produced faces

are usually optically smooth and fairly flat. In order to increase the chances of a successful

prototype, all of the faces of the prototype components that are used in reflection (back of the

prism, and front and back of the guide) are chosen to be factory faces of the original bulk

material. The reason for this is that surfaces used in reflection are more sensitive to slope errors

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than refractive surfaces, and subjugating them to the milling process would have created a

much rougher finish.

To further ease the fabrication process for the prism, its “top”

face is rotated to be perpendicular to the “back” face (Figure 3.4). This

adds material that is not part of the optical path of the device, but does

not interfere with intended path of the light. The three perpendicular

faces are more conducive to the milling process with only one face at a

non-right angle to the others.

In order to create a mirrored surface on the prism, aluminum is used in a physical vapor

deposition (PVD) process. The PVD process used does not have precise control over the layer

thickness or uniformity, so some variation in thickness across the surface is expected.

3.2.2 Reflector

The reflector requires unorthodox fabrication methods to remain cost effective due to

its unusual shape. Precision methods such as diamond turning, injection molding, and slump

molding are cost prohibitive. Though not yet widely used in the field of optics, an additive

manufacturing method was chosen to create the shape of the reflector. Specifically, a Form1+

desktop 3D printer from FormLabs [30] is used to create the freeform surface along with its

supportive structure. This printer is based on selective laser sintering technology where each

layer of the part to be printed is created out of a liquid polymer resin that is selectively cured

using a UV laser and two galvanometer mirrors. It is capable of reproducing geometrical

features as small as 300 µm, with layer thicknesses down to 25 µm. While these specifications

are top of the line in desktop 3D printing, they are far from good enough to recreate an optical

Figure 3.4: Cross-section of prism shape modified for the milling fabrication process. The red dashed line indicates the unmodified shape.

72

surface for use with visible light. The printed surfaces are

very rough on the microscopic level and not suitable as a

reflector without post-processing.

In order to create a smooth, reflective surface,

several methods were considered. The chosen method

adheres a reflective Mylar film to the printed surface

(Figure 3.5, bottom). Mylar is a thin polyester film that is

metallized with aluminum using a vapor deposition

process [31]. The process of drawing the polyester into a

sheet creates surfaces that are smooth enough to be used

in optical applications. The reflectivity of the aluminum

layer depends on its thickness, but is often similar to bulk

aluminum, around 90%, since the skin depth for visible

wavelengths is less than 5 nm [32].

The thickness of the plastic film is another important factor for this research. If too thin,

the film picks up the higher spatial frequency roughness of the printed surface to which it is

adhered. If too thick, it is more difficult to flatten the film onto a surface that may be curved in

two directions, e.g. if the printed part is not perfectly flat in the extrusion direction. Mapping a

flat surface to a curved one is a well-known problem in the fields of projective geometry and

cartography [33]. If a surface is curved in a single direction, it can be mapped to a flat surface

without distortion. However, if the surface is curved in both directions, distortion is a necessary

byproduct of mapping that surface to a flat plane. This is the same problem encountered when

attempting to draw a world map on a flat surface as opposed to a globe. A thicker piece of

Figure 3.5: Modeled (top) and 3D printed (bottom) reflector with reflective Mylar film adhered to surface. Printed part is a first full scale manufacturing attempt with visible flaws in the surface.

73

Mylar is more resistant to stretching and conforming to the 3D printed reflector surface if is not

completely flat in the extrusion direction.

The Mylar film sourced for this prototype has a thickness of about 40 µm and is coated

with aluminum on a single side. It is used as a first surface mirror with the aluminum coating

opposite the reflector structure since the film appears slightly more diffuse when used as a

second surface mirror.

The final component of the reflector is

the adhesive used to join the 3D printed

structure and Mylar film. Several different

adhesives were investigated, and it was quickly

found that adhesives that hardened and shrank

as they set are not plausible options. These

adhesives cause the Mylar surface to become

rough and diffuse as it shrinks and stretches with

the adhesive as it did for the “SC” and “LT”

adhesives in Figure 3.6. Adhesives that remain

flexible are more appropriate for this application as the surface distortion is much reduced.

Rubber cement (“RC”) fairs well, but after a long dry time it creates dimples in the Mylar

surface. The best option found is a silicon-based adhesive, as the Mylar remains visibly

undistorted even after long dry times. “GE” of Figure 3.6 is a silicone sealant meant for

construction applications. The adhesive used in the prototype is a silicon elastomer from NuSil

meant for encapsulating electronic components [34].

Figure 3.6: Mylar adhesive trial. "SC" and "LT" represent glues that harden as they set, and cause the Mylar to become diffuse as a result. "RC" (rubber cement) and "GE" (silicone sealant) both stay somewhat flexible as they set, allowing the Mylar surfaces to remain smooth.

74

The process of adhering the Mylar film is done

manually using a roller to create a thin, relatively even layer

of adhesive. First, the bare reflector surface is wetted with

the uncured silicone elastomer. Care is taken to remove as

many air bubbles as possible from the adhesive before

proceeding. An oversized piece of Mylar film is wrapped

around a roller, in this case a large opto-mechanical post,

with a piece of paper in between to protect the reflective

surface during assembly. The roller is placed at one edge of

the curved surface and rolled towards the opposite edge

while laying down the Mylar and squeezing out excess

adhesive. The roller is then moved back and forth over the

part several times to ensure good lamination of the

component layers. After the silicone is set, the excess Mylar film is trimmed back to the edge of

the reflector structure using a razor blade.

The assembly becomes problematic when the extrusion direction is curved since the

straight roller cannot press evenly over the entire surface. This often results in pockets of

silicone or air in the low areas, creating large bubble-like features in the laminated surface. This

is the case for the reflector shown in Figure 3.5, where the part is slightly concave along the

extrusion direction and the Mylar has visibly deformed, especially in the center portion of the

part.

Figure 3.7: 3D printed reflector structure secured to an optical bench (top), and a roller with Mylar sheet and protective paper layer (bottom).

75

The printed reflector structures tend to warp and become concave in the extrusion

direction as they dry. To combat this effect, the final prototype reflector was printed as a

slightly convex surface in the extrusion direction so it would flatten as it dried.

3.2.3 Photovoltaic Cells

The PV material best suited to this design is silicon due to the low concentration ratio.

For the prototype, mono-crystalline silicon wafers are diced to an appropriate size to cover the

edge of the guide layer component while leaving some extra area for solder joints on the active

face of the cells. The conductor on the front face of the cells is a sliver-based screen printed grid

of lines as seen in Figure 3.8. The

thicker horizontal gridlines create low

resistance pathways for electrical

current extraction, but also inhibit

incident light from reaching the silicon

PV material. The numerous, thin,

vertical gridlines create a more

uniform functioning of the cell, but

individually have higher electrical

resistance.

The ideal dicing geometry

avoids the thick gridlines on the area

of the cell that attaches to the guide edge, but has a thick gridline running parallel to the length

of the cell adjacent to the active area (pictured as a red outline in Figure 3.8). Unfortunately,

Figure 3.8: A silicon wafer with conductive gridlines. Red box indicates ideal dicing lines and the green, the necessary dicing lines due to the crystal lattice orientation. Dashed lines indicate extra diced cell area that does not contact the guide layer edge in the prototype assembly, leaving room for solder connections.

76

the wafers sourced for this project have a crystal lattice orientation such that the material

preferentially breaks at a roughly 45° angle to the gridlines. As such, the best possible dicing

geometry leaves an area free of thick gridlines near the middle of the cell, pictured as a green

outline in Figure 3.8. In order to ensure the best possible extraction of electrical current, leads

are soldered to both thick gridlines on the face of the cell and in similar positions on the rear of

the cell.

3.2.4 Assembly

The prototype assembly is a three step process consisting of joining the prism and guide

layer, mounting that subassembly along with the reflector, and attaching the PV cells.

3.2.4.1 Joining Prism and Guide Layer

The primary consideration for joining of the two refractive components is how to

optically couple the adjacent surfaces. Since neither adjoining surface is completely flat, an

intermediate liquid is used to fill in the gaps. Ideally this medium is of the same refractive index

as the parts it is joining so that light passing through this interface does not experience any

change in refractive index. Finding an index matching fluid with precisely the correct properties

is often impossible, especially for multiple wavelengths. However, a fluid with a nearly matched

index is sufficient for this application. Additionally, the prism and guide layer are attached with

machine screws for mechanical stability, so the index matching fluid need not serve as a

structural adhesive.

The first attempt at joining the prism and guide used a UV curable resin formulated

specifically for matching PMMA once set. Before curing, this resin has an index of

approximately 1.46 in the visible wavelengths and has a low viscosity similar to cooking oil. The

77

curing process increases the index to 1.49, matching PMMA. Unfortunately, the curing process

is made difficult by the large thickness of the PMMA parts that absorb UV wavelengths before

they can reach the resin. Before the resin could set, the UV source heats the PMMA parts

enough to warp them and create air pockets in the resin layer. If left uncured, the resin escapes

the interface due to its low viscosity in addition to maintaining a low index of 1.46. The resin

was rejected as a possible solution.

During the cleaning process to remove the resin, crazing of the guide layer was revealed

where it had been in contact with the uncured resin for a prolonged period. Crazing is a

phenomenon in plastics where local stresses are able to overcome the forces bonding the

molecules in the material, creating narrow fissures [35]. Some light that interacts with these

features is scattered, making them visible under certain illumination conditions. It is sometimes

possible to remove the surface crazing from thermoplastics by heating the material with a

blowtorch to reduce the internal stresses that prevent the fissures from closing. However, this

was not attempted due to the risk of damage to the part.

Ideally, a silicone gel would be used in this interface for its known stability in under solar

illumination, but it is not available with an index near that of PMMA. The final choice of index

matching fluid is Karo corn syrup. While not a conventional material in optical systems, it

satisfies the requirements of this application well. The high viscosity helps prevent the material

from flowing out of the interface, and its refractive index closely matches that of PMMA. The

corn syrup is measured with an Abbe refractometer (Section 3.3.1.4) and determined to have a

refractive index of 1.4832 ± 0.0005 at the Helium “d” Fraunhofer line, 587.6 nm.

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3.2.4.2 Reflector Mounting

The next assembly step mounts the

reflector and refractive subassembly together

while providing the hardware necessary to

mechanically mount the finished prototype to a

rotation stage. Two custom braces are designed

to fit on the end caps of the prototype on either

side of the extruded dimension (Figure 3.9d,

pictured in red). The braces have features that

position the guide and prism subassembly relative

to the reflector surface and a baffle that simulates

the shadowing of an adjacent reflector. Set screws hold all the components in place. The two

braces are connected with a threaded rod and essentially create a clamp to form a rigid

prototype structure.

A hole in the brace is positioned at the vertical center of the prototype’s entrance

aperture and provides a place to mount the prototype to a rotation stage on axis. This way, as

the rotation stage moves during the measurement process the entrance aperture center

position remains nearly constant. The mounting structure also provides mechanical clamps to

keep the PV cells positioned after they are placed during assembly and a wire management

structure to handle the eight wire leads attached to the PV cells.

(a)

(b)

(d)

(e)

(c)

(f)

Figure 3.9: Exploded view of prototype assembly with (a) prism and guide in green, (b) reflector in grey, (c) baffle in black, (d) structural braces in brown, (e) PV cell clamp components in yellow, (f) wire management hanger in black, and (g) and (h) indicating the top and bottom PV cell locations, respectively.

(g)

(h)

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3.2.4.3 Attaching Photovoltaic Cells

There are two PV cells, one for each

output edge of the guide layer, and each cell

has four wire leads as previously described.

The cells are weak and brittle, and they tend

to warp after being diced. The solution for

mounting the cells is to include two clamps

that contact the back of the opposing cells to

sandwich the top cell and the bottom cell

with the guide layer in between. The UV

curing resin that was utilized in the first prism-guide assembly attempt is placed between the

cell and guide layer edge to achieve good optical contact. The prototype is placed under a UV

lamp in a final attempt to cure the resin and secure the cells, but most of the resin remains fluid.

The clamp structure is enough to keep the cells in the correct positions.

During assembly, good optical contact can be easily assessed with the unaided eye, as

bubbles in the liquid layer appear with great contrast since TIR selectively occurs in those areas

(Figure 3.10). The liquid layers allow light to couple out of the guide and interact with the cells

with maximum efficiency. If there were an air interface

between the cell and the guide, Fresnel losses would occur at

the guide-air surface as well as the air-cell interface. The

Fresnel losses would be severe given the high index of silicon,

around 4.0 in the visible [36]. For this reason, the sourced

Figure 3.10: PV cell optical contact trial (top) and in the prototype looking through the length of the guide layer (bottom). Top: the darker regions of the cell are area of good optical contact with the PMMA cover plate and the lighter areas have poor contact. Bottom: good optical contact in the prototype assembly.

Figure 3.11: Magnified view of PV cell surface showing a ridged texture running perpendicular to the small gridlines.

1 mm

80

silicon cells have a textured front surface that helps combat Fresnel losses as seen in Figure

3.11.

3.3 Metrology Tools

This section introduces the metrology instruments used to measure the prototype

components as well as the full assembly. Special attention is given to the fringe reflection

deflectometer since it is a one-of-a-kind apparatus, built specifically for this project.

3.3.1 Commercial Instruments

This section briefly describes the commercially available metrology instruments used

during for the prototype component measurements. Details about each instrument can be

obtained from the manufacturers, and is thus not presented here.

3.3.1.1 Zygo Verifire PE Interferometer

The Zygo Verifire PE is a versatile laser interferometer system capable of measuring flat

and spherical surfaces in reflection [37]. The source has a high spatial and temporal coherence,

so it is most useful for measuring large, smooth parts. It is used to measure the flatness of the

calibration mirror for the deflectometer apparatus discussed later. This interferometer could

not obtain good data from most of the prototype component surfaces since they are too rough,

or have too large a departure from flat.

3.3.1.2 Zygo NewView 600 White Light Interferometer

Unlike the laser interferometer, the Zygo NewView 600 [38] white light interferometer

uses a source that has a small temporal coherence and a precision scanning stage. Due to the

low coherence, fringes are only visible over a small range along the optical axis, and as the

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scanning stage moves, the fringes are visible at different depths on the test surface. The end

result is a system with high depth resolution, even on rough surfaces. Unfortunately, the active

measurement area is several orders of magnitude smaller than the laser interferometer, on the

order of 0.1 mm2.

This instrument is most useful for gaining information about surface roughness as well

as small topical features. It is used to measure the surface quality of the PMMA components

and the Mylar.

3.3.1.3 Metricon Model 2010/M Refractometer

The Metricon Model 2010/M [39] is a refractometer that measures the critical angle of

an interface comprised of a high index prism and the sample material. The prism’s index is well

known and the critical angle is found by measuring the power in a reflected beam as a function

of incidence angle. Once the critical angle is found, the index of refraction of the unknown

sample is calculated. This instrument is used to verify the index of the PMMA prototype

components.

3.3.1.4 Abbe Refractometer

Abbe refractometers measure the index of refraction of an unknown material by

measuring the critical angle at an interface between a known high index prism and the unknown

material. This instrument is very similar to the aforementioned Metricon, but is more

appropriate for measuring fluid media.

3.3.1.5 Optical Gaging Products SmartScope Quest 300

The SmartScope Quest is a line of coordinate measurement machines (CMMs) that use

one of a variety of sensing methods to detect discreet points on a surface [40]. The method

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used for this research employs optical interference to find the position of a surface without

physically contacting the part. High precision stages move the part in three dimensional space

and the point data is recorded. The SmartScope is used in the measurements of the reflector

surface shape prior to applying the Mylar film for comparison with the data from the fringe

reflection deflectometer discussed next.

3.3.2 Fringe Reflection Deflectometer

The primary reflector prototype component is a highly specular freeform surface that

has significant figure errors, no common focal point, and deforms when small pressure is

applied to certain areas. Traditional optical metrology is not capable of measuring such a

surface with a large departure from spherical and no focal point. A contact profilometer risks

deforming the surface if it passes over an air pocket under the Mylar film, and the stylus could

damage the first surface coating on the Mylar.

After a review of measurement techniques for freeform surfaces [41] a structured light

system was chosen as an appropriate technology for this application. Structured light systems

come in two major categories: projection and reflection. The former projects a patterned light

beam onto the part to be measured and a camera images the light scattered from the part. The

images are analyzed to reveal the shape of the part’s surface in three dimensions.

Unfortunately, this system is not compatible with specular surfaces, but the other category of

structured light system uses reflections from a surface instead of scattered light. While this is a

less direct system for measuring points on a surface, it is far more sensitive to surface slopes.

One study shows that such systems are capable of measuring feature depths to an accuracy of 1

nm [42]!

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Current applications of this technology include qualitative fabrication flaw detection for

parts such as automobile body panels [43]. Applications that involve quantitative reconstruction

are less common and are very much still in the research stage. For example, the Software

Configurable Optical Test System (SCOTS) from the University of Arizona focuses on the accurate

reconstruction of optical surfaces, often ones with large apertures [44].

3.3.2.1 Hardware

The hardware needed for a basic reflective structured light system is widely and

inexpensively available: a camera, a flat digital display (e.g. LCD computer monitor), and a

computer for control, data acquisition, and analysis. Additionally, a mirrored calibration flat

with at least five fiducial points is needed to build the system presented here.

The display is a Dell UltraSharp 2009W with a resolution of 1680 by 1050 pixels, a pixel

periodicity of 0.258 mm in both the horizontal and vertical directions, and has a 20 inch diagonal

[45]. This display is large enough to capture all reflections off of the prototype reflector from

the point of view of the camera. For maximum accuracy, it is necessary to have a linear

relationship between the output of the monitor and the pixel brightness as perceived by the

camera. However, the chosen display has a nonlinear response which changes depending on

the angle from which each pixel is viewed. The steps taken to mitigate the nonlinearity are

outlined in section 3.3.2.3.

The deflectometer uses an Allied Vision Marlin F-201B monochrome machine vision

camera with a sensor resolution of 1628 by 1236 pixels and a pixel pitch of 4.4 µm. The lens is

an off-the-shelf TV lens with a nominal focal length of 25 mm for which the aperture stop is set

to the minimum diameter so as to increase the depth of focus during operation. Ideally, the

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lens would be telecentric so that the image does not shift depending on object’s distance from

the focal plane in object space. The lens used is not telecentric, which introduces error into the

final data since the entire part surface does not exist in a single focal plane.

The computer includes MathWorks® MATLAB computational software for which the

deflectometer code is written. It controls the monitor display and

the camera for data acquisition and analyzes the resulting data.

Finally, the calibration flat is a first surface mirror with nine

disk-shaped fiducial marks arranged in a 3 by 3 grid (Figure 3.12).

The relative positions of the fiducials are obtained via a measuring

microscope. The fiducials were created by ablating the surface with a CO2 laser writing system.

In order to avoid rigid mounting of the calibration flat and the part to be measured, the

monitor is suspended above the

tabletop facing down, and the camera

is looking down at a portion of the

tabletop that is underneath the

monitor, as pictured in Figure 3.13.

This way, the calibration flat and

measured parts may be set on the table

surface and held in place by gravity.

The monitor mount is vertically

adjustable so the distance from the

part to the display is variable.

Camera

Monitor

Part to measure

Monitor stand

Figure 3.13: Fringe Reflection deflectometer schematic showing relative positions of the monitor display, camera, and reflective part.

Figure 3.12: Deflectometer reference flat mirror.

85

3.3.2.2 Operation Summary

The operation of the fringe reflection deflectometer consists of three stages: calibration,

data acquisition, and data processing. The details for each stage can be found in the following

sections.

The primary goal of calibration is to find the relative position and orientation of the

camera and monitor in three dimensional space. This is done using the camera to image a

minimum of five coplanar fiducial points on the calibration flat. Given the relative fiducial

positions, the best fit rotational and translational transformations are found describing the

relative position of the camera and fiducial plane [46]. Additionally, the focal length and the

radial distortion of the camera lens are calculated. This process is used once to locate the

calibration flat and again to locate the reflection of the monitor off of the calibration flat’s

mirrored surface. The real position of the monitor is then calculated.

The data acquisition stage replaces the calibration flat with the part to be measured.

The fundamental goal is to map each pixel of the camera that is viewing the part surface to a

position on the monitor. This is done by displaying a short series of patterns on the monitor and

recording images of the part surface illuminated by each pattern.

The data analysis stage begins by calculating the coordinates on the monitor seen by

each camera pixel using knowledge of the patterns displayed during data acquisition. Then the

shape of the part surface is found using geometrical arguments and some justified assumptions

about the surface.

86

3.3.2.3 Monitor Nonlinearity Mitigation

To illustrate the magnitude of the nonlinear response of the monitor, the

deflectometer’s camera is pointed directly at the monitor from five different viewing angles:

normal to the screen, and from an angle of about 45 degrees looking down from above, up from

below, and horizontally from the left and right. Data are taken at each position while increasing

the input to the monitor and recording the brightness as perceived by the camera. These data

describe the response function between the camera and the value output by the computer.

Both the monitor and the camera run in 8-bit mode so the displayed and recorded values are

discretized to whole numbers between 0 and 255.

The data from this study

are seen in Figure 3.14. Notice

that the data at normal incidence

as well as that from the left and

right perspectives have similar

shapes, but all are nonlinear.

Much of this nonlinearity is an

intentional effect implemented

by the computer’s display driver.

Given the logarithmic response of the human eye, displays often implement what is called

gamma encoding to the display signal so that the output intensity appears more linear to human

viewers. Gamma is simply an exponent on the value to be displayed (𝐼𝑜𝑢𝑡 = 𝐼𝑖𝑛𝛾), and is often

chosen to be around 2.2 for input values between 0 and 1 [47].

Figure 3.14: Camera response to monitor input values for several viewing angles using default display settings.

Mea

sure

d In

ten

sity

Input Value

87

More problematically, the data from the above and below viewpoints are significantly

different from each other and from the other three perspectives. Further still, the slope of the

data from below actually changes sign around an input value of 100, making it impossible to

create a one to one mapping between input and output. In other words, looking from below,

the pixels actually appear to become dimmer as the input value is increased over a portion of

the full range. This is an effect that is also detectable by the unaided eye. The source of this

error is unknown and not intuitive, but may have to do with the color balance among display’s

red, green, and blue subpixels while a white pixel is being displayed and its value is increased.

Several measures are taken to create a more linear response between the camera and

monitor at different viewing angles. First, the display driver gamma values are readjusted so the

camera perceives a nearly linear increase in brightness from normal incidence. Second, two

films, the vertical brightness enhancement film and a textured light recycler film, were removed

from the LCD display stack which modified the angular output of the device. Third, the monitor

is operated with an increased minimum and reduced maximum pixel value in order to utilize

only the most linear region of the response curves. Finally, the monitor, camera, and part to be

tested are oriented in such a way that the monitor is rarely seen “from below” in order to avoid

the most egregiously nonlinear viewing angles.

The results of all the above adjustments are shown in Figure 3.15. The greyed out

ranges in the top graph indicate unused pixel value regions. Notice the normal, left, and right

viewing angles are very nearly linear within the chosen range. The phase error in the bottom

graph of Figure 3.15 is calculated for a system that uses a four step phase shifting algorithm as

does this deflectometer. The phase error is related to a lateral position error on the monitor

and is discussed further in a later section. Notice that the phase error has a period that is one

88

quarter that of the actual phase, an

effect that is observable in the data

generated by this apparatus.

The color-coded shaded

regions on the bottom graph

indicate an estimated range of the

phase error based on the noise in

the measured data. The viewing

angle from the right is omitted since

it is very similar to that from the

left. The data from below are of

much greater amplitude and

somewhat nonsensical due to the

highly nonlinear and inverse nature

of the response curve, and are also

omitted since that viewing angle is

avoided during operation of the

deflectometer.

Recall that the non-normal viewing angles are at a fairly severe 45 degrees. For most

measurements, the apparatus can be configured to have lower incidence angles during data

acquisition, further reducing the severity of the phase error.

Figure 3.15: Camera and monitor response curves with modifications (top), and the calculated phase error based on the nonlinear response (bottom). The phase error is based on a four step phase shifting algorithm and the shaded region estimate the error based on the noise in the measured data.

89

3.3.2.4 Calibration

The calibration procedure follows a 1987 publication by R. Y. Tsai for the case of

coplanar fiducial marks and a single captured frame [46]. This method finds a three dimensional

rotation matrix and a three element translation vector to map the coordinate system of the

fiducials to the camera’s coordinate system. This mapping is defined as

[𝑥𝑦𝑧

] = 𝑅 [

𝑥𝑤

𝑦𝑤

𝑧𝑤

] + 𝑇 ,

where (xw, yw, zw) is a point in the fiducial coordinate system, (x, y, z) is a point in the camera

coordinate system, R is a 3 by 3 rotation matrix, and T is a three element translation vector. The

camera is simplified to a pinhole geometry with the addition of a single radial distortion term

(r2). The focal length of the lens, f (i.e. the distance from the pin hole to the image plane), and

the distortion coefficient, κ1, are calculated along with R and T.

First, the camera records an image with the fiducials visible, and the fiducials are

identified in the software with the help of user input. This defines the pixel positions of each

fiducial from which a physical position on the sensor is calculated given the sensor array

geometry and dimensions. The lens distortion is assumed to be zero until the final step of the

process.

From this data, the entire orthonormal rotation matrix is calculated along with the first

two components of the translation vector which control position perpendicular to the camera’s

optical axis. This calculation requires there be at least five fiducial marks, and uses a least

squares solution for greater than five fiducials.

90

Next, a linear relationship between the lens focal length and the third component of the

translation vector (Tz , distance along the camera’s optical axis) is leveraged to find an initial

guess for these two quantities. The lens distortion is still assumed to be zero at this step.

Finally, f, Tz ,and κ1 are varied in an optimization scheme with a merit function that

compares the actual pixel positions of the imaged fiducials to the calculated positions using R, T

(including Tz), f, and κ1.

This method is first applied to the reference flat pictured in Figure 3.12 to find its

position and orientation relative to the camera along with the lens focal length and distortion

coefficient. The monitor displays a solid white image to obtain the highest contrast of the

optically absorbing fiducials.

Then, the monitor displays

solid black and the user is asked to

draw white dots on the display so

that they can be seen in the reflective

surface of the reference flat (Figure

3.16). These white dots act as

fiducials on the monitor and their

position is calculated through the

known pixel positions and pixel pitch

of the monitor. The above method is

applied to the new set of fiducials to

find the position of the monitor’s reflection relative to the camera. The lens focal length and

distortion are taken from the previous step and are not recalculated.

Figure 3.16: Deflectometer calibration during monitor reflection positioning step. Shows (a) drawn fiducials on monitor, (b) reflection of drawn white fiducials and black mirror fiducials on calibration flat, and (c) camera.

(a)

(b)

(c)

91

Now that the position of the reference mirror and the monitor’s reflection are known,

the real position of the monitor is calculated. To maintain a calibrated state, the positions of the

camera and monitor must remain fixed and the focus position of the lens must not be adjusted,

but the calibration flat may be removed.

3.3.2.5 Data Acquisition

The calibration flat is replaced with the part to be measured. In order to find the

position on the monitor that each camera pixel sees reflected off the part surface, a four step

algorithm is borrowed from phase shifting interferometry [48]. For this method, sinusoidal

fringes are displayed on the monitor and four images are recorded, shifting the fringes by a

phase of π/2 between frames. Example data of the prototype reflector is shown in Figure 3.17.

If I1, I2, I3, and I4 are the values of a single pixel for the images with a fringe phase shift of 0, π/2,

π, and 3π/2, respectively, the modulo 2π phase, φ, of that pixel is calculated as follows:

𝜙 = 𝑡𝑎𝑛−1 [𝐼4 − 𝐼2

𝐼1 − 𝐼3] .

Figure 3.17: Reflection of horizontal fringes from prototype reflector as recorded by camera. From phase shift of the fringes from left to right is 0, π/2, π, and 3π/2.

92

This calculation is done independently for each camera pixel, meaning the fringe apexes do not

need to be located to obtain the phase. Additionally, a constant background signal and constant

scaling do not affect the calculated phase, so if the reflections from one area are not as bright as

they are from of another area, the calculation is still valid [48].

During the data acquisition procedure, the best lateral resolution is gained by focusing

on the part surface, so the pattern on the monitor is out of focus [43]. However, given the

theoretically singular spatial frequency composition of sinusoidal fringes, an out of focus image

of this pattern is still sinusoidal with the same frequency and phase [49, 43]. The main concern

then becomes the signal to noise ratio, which will decrease with more defocus. The defocus is

advantageous by effectively eliminating the possibility of resolving individual pixels on the

display, producing a smooth, continuous sine pattern.

The period of the fringes is chosen to be as short as possible while retaining fringe

contrast, so a given error in the detected phase (e.g. Section 3.3.2.3) is less significant when

converted to spatial coordinates on the monitor. Once again, loss of signal to noise ratio limits

how short the period may be.

The phase calculated by the four step

algorithm in this case relates to a position on the

monitor’s screen. To gain knowledge of the two

dimensional position on the monitor, two sets of

fringes must be recorded: one for the horizontal

position (vertically oriented fringes), and one for

the vertical position (horizontally oriented

fringes). Since the calculation only gives the

Figure 3.18: Pixel line data for the calculation of absolute position. Horizontal (left) and vertical (right) lines are shown. These images are superimposed with another photo of the part so the part outline is visible.

93

relative phase of each camera pixel, two more images are acquired that reveal the absolute

position on the monitor. This is done by displaying a single line of pixels at a known position

with a vertical line orientation in one image, and a horizontal orientation in the other (Figure

3.18).

Finally, an optional frame is captured where calipers are set on the part’s surface and

the distance reading is recorded. This may serve as a reference distance during data analysis.

Thus, a total of eleven frames are captured during this stage.

3.3.2.6 Data Analysis

The wrapped phase (modulo

2π) on the part surface found via the

aforementioned four step algorithm

[48] is unwrapped using publicly

available “PUMA Algorithm”

software [50, 51]. Figure 3.19

shows the wrapped and unwrapped

phase for the prototype reflector. This phase is translated into absolute position on the monitor

using the information from the frames that captured the single horizontal and vertical lines

displayed on the monitor.

At this point, an inverse propagating ray can be traced for each camera pixel from the

camera origin towards the part, and the ray position on the monitor after reflecting off of the

part surface is now known. To calculate the part’s surface geometry, the distance the ray travels

between the camera and the surface must be calculated. Unfortunately, this is a

Figure 3.19: Results of PUMA phase unwrapping algorithm performed on the horizontal fringe data for prototype reflector. Wrapped phase [-π, π] (left) and unwrapped phase (right).

94

mathematically ill-posed problem with infinite

solutions for each pixel (Figure 3.20), and is an active

research topic in machine vision [42]. The ambiguity

comes from the fact that the ray directions at the

monitor surface are unknown.

To remove the relative ambiguity amongst all

the data points on the surface, it is assumed to be

continuous and smooth over the distance separating

adjacent data points. These assumptions are

integrated using a slope equalization technique

borrowed from the field of wave-front sensing [52].

First, an initial guess is made for the distance (σ) between the surface and the camera origin for

each camera pixel. This guess is based on the designed geometry of the surface and the position

of the reference mirror during the calibration stage. The slope of the surface is calculated at

each data point given the knowledge of where the ray strikes the monitor. The slope is broken

into components in the vertical and horizontal directions with respect to the camera pixel array

dimensions. Slopes of horizontally and vertically adjacent data points are averaged to calculate

the slope between them. The slope between horizontally and vertically adjacent data points is

separately calculated based on their σ values and the distance between them. The solution

matches these two slope quantities as closely as possible over the entire surface by adjusting

the σ values of each data point. A simplified illustration of the slope equalization calculation is

shown in Figure 3.21. This solution finds the relative σ of all data points, and additional

information is needed to determine absolute position.

Figure 3.20: Illustration of multiple solutions to surface position problem. The solid red line is the known ray coming from the camera, the green patches are possible surface positions with the black arrows indicating the surface normals.

95

Note that there exist two distinct coordinate

systems: 3D Cartesian coordinates and a spherical

coordinate system defined by σ and the pixel position

on the camera sensor. A surface with constant σ is a

spherical shell centered at the camera origin, and the

two dimensional pixel position determines the

position on that shell. Care must be taken to

transform all slopes into a single coordinate system

before solving the least squares problem. The

camera’s coordinate system is chosen for this apparatus since it provides a more direct solution

using the mathematical methods in [52].

To find an absolute distance from the camera, a constant is added to all data point

distances so that a physical measurement of the part (e.g. from the picture of the calipers) is

maintained in the reconstructed data. In the wave-front reconstruction analogy, this would be

equivalent to adding a piston term, which does not affect the wave-front shape.

The solution depends highly on the initial guess of the σ values, so the process is

iterated using the previous iteration’s calculated values as the new initial guess. In practice, the

solution converges very quickly, within 3 to 5 iterations.

Finally, once the surface shape is reconstructed, it is registered to a position that is

consistent with the part as it was originally modeled and the departure from the ideal surface

shape is calculated.

S1

S2

S12

σ1

σ2

Figure 3.21: Illustration of two different slope calculations. S1 and S2 are the slopes calculated from the monitor position reflected in the part, and S12 is the slope calculated from the positions of the data points. In this example, to get the average of S1 and S2 to equal S12, σ2 will need to increase relative to σ1.

96

3.3.2.7 Sources of Error

The primary sources of error in the fringe reflection deflectometer are calibration error,

variable nonlinearity of the display, and noise in the captured image data. As previously

discussed in section 3.3.2.3, the monitor nonlinearity varies depending on the viewing angle

causing an error in the measured phase. The phase is also adversely affected by noise in the

captured images. The signal to noise ratio is maximized by adjusting the exposure time of the

camera and the fringe period on the display for high contrast. Additional noise is inherent in the

discreet, 8-bit nature of the monitor pixel brightness and the camera’s digital sensor.

Calibration error refers to an inaccurate knowledge of the position and orientation of

the display surface relative to the camera. This will affect the geometrical calculations in the

data analysis stage. This error is very difficult to quantify, partially due to the virtual nature of

the camera origin. The calibration error also includes uncertainty in the calculated focal length

and distortion coefficient of the camera lens, which directly impact the ray directions associated

with each camera pixel. Calibration error can stem from a variety of sources including

inaccurate measurement of the calibration flat fiducials, a curved calibration flat, a curved

display surface or camera sensor, tilt or decenter of the camera sensor relative to the lens, or

inconsistent spacing of pixels on the display or camera sensor. The flatness of the calibration

mirror is simple to measure using the Zygo laser interferometer introduced earlier. The results

are seen in Figure 3.22, and they reveal a cylindrical bend in the mirror, but the amplitude is a

mere 1 µm over a distance of 100 mm. The effects of such a defect pale in comparison with the

nonlinear response of the monitor, for example.

97

The cumulative error in this system is

noticeable in the final measurements, but the

result is accurate enough to yield a robust

computer model of the large scale prototype. The

subsequent chapter presents data from the

deflectometer and compares it with similar

measurements performed via established, highly

accurate methods.

3.3.3 Solar Simulator

Given the varying properties of sunlight

and the difficulty of accurately tracking the sun as

it moves across the sky, the assembled prototype is tested in the laboratory under controlled

illumination meant to simulate sunlight. The apparatus is referred to as a solar simulator which

ideally mimics the sun in angular extent, spectral power distribution, and absolute irradiance.

However, it is difficult to simultaneously meet all three of these requirements with currently

available light sources. The solar simulator employed for this research satisfies the angular

extent and spectrum specifications fairly well, but has an irradiance that is approximately 20%

that of the sun on a clear day. This is a suitable device for relative power measurements.

The solar simulator uses a xenon short arc lamp with an integrated parabolic mirror.

This is a well-suited source since it is bright and the spectrum closely matches that of the sun

Figure 3.22: Calibration flat measured in a Zygo laser interferometer. A cylindrical bend can be seen over the aperture of the part with a peak to valley departure of about 1 µm. The bottom graph shows the slice chart indicated by the line in the top graph.

98

with the exception of excess power in

the near infrared (Figure 3.23), a defect

which can be filtered out. The

integrated parabolic reflector creates a

roughly collimated output from the

bulb. This beam is focused by a singlet

lens into a straight integrating rod with

a hexagonal cross section. The

integrating rod mixes the beam to

create a spatially and angularly uniform output. The output face of the integrating rod is placed

at the focus of a large off-axis parabolic mirror which collimates the light with a finite angular

extent. Angular extent of the final beam is determined by the focal length of the mirror and the

size and shape of the rod’s output face. The size of the mirror determines the size of the output

beam, and the F-number of the aforementioned focusing lens is chosen so that the mirror’s

aperture is not significantly over- or under-filled.

0

1

2

3

4

5

6

7

8

9

10

0 500 1000 1500 2000 2500

Re

lati

ve S

pec

tral

Po

wer

Dis

trib

uti

on

Wavelength (nm)

Xenon Lamp Sunlight on Earth's Surface

Figure 3.23: Comparison of the spectral power distribution of sunlight after passing through Earth's atmosphere and a xenon arc lamp.

Focusing Lens

Integrating Rod

Off-Axis Parabolic

Mirror

Test Area

Figure 3.24: Schematic of solar simulator with labeled components. Path of light though system shown in yellow.

99

A schematic of the solar simulator is shown in Figure 3.24, where “Test Area” signifies

position of the prototype during testing. The solar simulator is combined with computer

controlled translation and rotation stages and electronic source-meters to create a versatile test

bed for solar PV devices. Automated measurements utilizing this apparatus include I-V curves

for PV cells and 2D fields of view and sub-aperture mask scans for CPV prototypes.

100

Chapter 4: Measurements and Computer Model

This chapter presents metrology data from the assembled prototype as well as the

individual components. Measurement results are used to create an accurate, as-built computer

model of the prototype. The performance of the simulated system is compared to that of the

real-world prototype, and insights about the measured performance data are derived from the

validated model.

4.1 Component Measurements

Measurements taken on the individual components allow an accurate computer model

to be created using a bottom-up approach. The geometry of each component is found along

with the topology and optical properties of critical surfaces. The relative alignment of the

system’s components is also important for a high fidelity model.

4.1.1 Guide Layer

The guide layer has the simplest geometry of any component in the prototype. It is a

rectangular slab with designed dimensions of ≥11 x ≥60 x 100 mm. The dimensions of the

fabricated component are measured using calipers,

assuming the angles between all faces are close to 90

degrees. It was found to have dimensions of 11.7,

61.0, and 100.1 mm, with an accuracy of ± 0.05 mm.

There is no discernable wedge in the part based on caliper data, so the part is assumed to have

all perpendicular faces for the purposes of the computer model.

(mm) Designed Measured

Thickness ≥ 11 11.7 ± 0.05

Length ≥ 60 61.0 ± 0.05

Width 100 100.1 ± 0.05

Table 4-1: Guide layer designed vs. measured dimensions.

101

The topology of the guide surfaces is only of great importance on the large faces.

During performance testing, the prototype is oriented in such a way that the faces

perpendicular to the extrusion direction are not part of the system’s optical path. The faces

adjacent to the PV cells are immersed in a resin with an index of refraction near that of guide

material, so any refractive effects due to surface topology on those faces are reduced to the

point of insignificance. In fact, there are tool marks on all four of the narrow guide faces, but

they become invisible when immersed in the index matching resin.

The large faces of the guide layer, however, are critical to the system’s operation. If the

surface roughness is too great, they will not support total internal reflection, which is the

primary function of the guide. If these surfaces are smooth, but not perfectly flat, they would

have little impact on the prototype’s performance.

The guide front face is

measured with the Zygo laser

interferometer described in Chapter 3,

and the results are shown in Figure 4.1.

Only about half the surface area can be

resolved by the interferometer in a

single measurement, but it shows that

much of the surface is smooth enough to be measurable by the interferometer using a

wavelength of 632.8 nm. The missing data is either due to a high surface roughness that

produces a diffuse reflection or smooth surfaces with high local surface slopes that produce high

fringe densities that are unresolvable by the interferometer’s sensor. In this case, the primary

contribution to the invalid data is the latter. This is evident when the invalid regions of Figure

Figure 4.1: Laser interferometer data of the guide front face. Dark blue indicates invalid data.

102

4.1 become resolvable as the part is tilted. This is a strong indication that the surface is smooth

enough to support TIR, though not extremely flat. However, the peak to valley departure from

flat for the resolved data is a mere 3.5 µm over the extent of the 100 mm x 61 mm face. Even a

peak to valley surface figure error of 10 times this value does not produce detectable changes in

the prototype performance. For these reasons, the surface is adequately modeled as being flat

and smooth.

The guide layer surfaces are measured with the fringe reflection deflectometer as well.

Measuring data on refractive parts is more difficult in the deflectometer than opaque reflective

surfaces such as the prototype primary reflector. This is due to a reduced signal from the tested

surface relying solely upon Fresnel reflection and unwanted signals from the part’s other

surfaces.

The deflectometer is configured in such

a way that the surface reflections occur at a

more grazing incidence and the part is oriented

to reduce or eliminate signals from other

surfaces. In the case of the guide layer, the

reflection of the monitor from the opposing

guide surface is unavoidable through orientation

change alone. To attenuate the competing signal, the offending surface is immersed in index

matching fluid (corn syrup) and coupled to absorptive black fabric as shown in Figure 4.2. The

reflection from the back surface becomes imperceptible.

Results from the deflectometer are shown in Figure 4.3. There is a large discrepancy in

the data between the deflectometer and interferometer results of Figure 4.1. The departure

Figure 4.2: Guide layer immersed in bed of corn syrup to suppress back surface reflection.

103

from flat is more than a magnitude

greater in the deflectometer data over

the breadth of the surface, and the

shape does not correlate well. The

reconstructed surface shape from the

deflectometer data is a smooth saddle

shape that has a high positive error

magnitude in two opposing corners,

and a high negative error magnitude in

the other two corners. In contrast, the

valid interferometer data shows a finer

structure to the surface. The

discrepancy is primarily due to two

effects: the nonlinear response of the

monitor (especially considering the

deflectometer’s high incidence angle

configuration) and the cumulative error

of integrating the surface slopes during 3D reconstruction. For this measurement, there are

over 1 million data points scattered across the surface, so some error is expected when

reconstructing the data in three dimensions based on measured slope data. That being said, the

data has a peak to valley value of just 0.1 mm over a 60 x 100 mm part. While the surface figure

is not entirely accurate, the slopes of the surface are more trustworthy since they are measured

more directly.

Figure 4.3: Top: Deflectometer data from guide layer front surface. Bottom: magnified region as marked in top graphic with surface figure subtracted out. A scratch and tool marks become visible. Vertical and horizontal slice charts shown, all units are mm.

x 10 -4

104

The close-up data in Figure 4.3 shows a scratch

and some shallow vertical markings, probably from the

buffing process used to help remove the aged

protective tape that was covering the surface of the

bulk PMMA material. These features are visible in the

raw data of the same region as illuminated by a set of

vertical fringes, shown in Figure 4.4 (note the raw data

is from a perspective view which slightly changes the

directionality of the features).

The close-up data illustrates well the strengths and weaknesses of the deflectometer

apparatus. As mentioned previously, commercial applications of this technology use it to find

qualitative flaws in reflective surfaces, for example, the scratch and the tool marks in the

presented data. However, surface reconstruction is difficult due to cumulative errors in the

slope integration. This effect is clearly seen by looking at either side of the scratch in Figure 4.3.

There is a jump in the data values from one side of the scratch to the other. This is not a real

feature of the surface and is primarily a result of slope integration error. The scratch acts as a

region of invalid data that the path of integration must avoid. Essentially, the path of

integration is routed around the scratch, staying in the smoother regions. The longer that path,

the more error accumulates from one end to the other. Further down in the frame, the scratch

has gaps where the slope integration calculation has a much shorter path between the two

sides. The discrepancy in topology across the scratch disappears in this region.

The reconstructed surface is more trustworthy in smooth regions over short distances

since the slope integration error is less severe. For this reason, it is still worthwhile to analyze

Figure 4.4: Portion of guide front surface from the raw deflectometer data. Scratch and tool marks are visible.

105

the shape of the tool marks from the deflectometer data. According to the data, they have a

period of about 1 mm and a depth of about 0.1 µm in the selected region, which will have little

effect on the direction of reflected light.

Even with the numerous defects detected by the deflectometer and interferometer, the

guide front (and back) surface can be accurately modeled as smooth and flat in the scope of the

prototype system.

4.1.2 Injection Prism

4.1.2.1 Geometry

The injection prism geometry is measured in a similar manner to the guide using

calipers. The non-right angles in combination with the filleted edges make this a less precise

measurement than for the right-angled guide component. This is indicated by greater

uncertainties in the measured data (Table 4-2).

To corroborate the caliper measurements, the angles between adjoining faces are

measured using a precision rotation stage (±5 arcminutes) and the reflections of a laser off of

adjacent faces. The prism is placed on the stage such that the rotational axis is aligned with the

extrusion direction of the part and is located near the edge corresponding to the angle being

measured. A screen is placed at a distance and the center of the laser spot is marked. Finally,

the rotation stage is turned so the laser strikes the adjacent face and its reflection hits the

previously marked spot on the screen. The difference in the start and end positions read on the

rotation stage is the internal angle on the part. Note that this measurement is only indicative of

the internal angle between the two points on the surfaces that the laser interacts with, and it

may vary depending on position if the surfaces are not flat.

106

A comparison of the designed and

measured dimensions are listed in Table 4-2. To

further validate the data, the distance

measurements from the bottom, back, and top are

used to create the prism geometry assuming right

angles between those edges. The resulting

geometry has calculated values of 35.58 mm for

the front edge, 75.85° for α, and 104.15° for δ.

Recall that the tolerance specification for α is

0.75°. These data correspond fairly well with the

other measured values and are the measurements

used in the computer model.

The largest discrepancy found in Table 4-2 is the length of the top and bottom edges,

both of which are about 3.5 mm longer than specified. The manufactured prism is thicker front

to back than intended, which has the effect of widening the field of view of the prototype.

Though this may appear advantageous to the design, the same flaw introduced into the full

system design has the effect of increasing guiding losses, thus reducing the efficiency of the

system at all points in the field of view.

The thicker prism also creates a potential issue in the prototype system. Recall that the

guide thickness is chosen based on the length of the bottom of the prism so that light trapped in

the guide by TIR has no chance of reentering the prism. The larger bottom dimension of the

fabricated prism increases the specification of the guide thickness to ≥13.0 mm, but the

manufactured guide is 11.7 mm. A similar discrepancy is observed for the guide length

Table 4-2: Injection prism designed vs. measured dimensions.

α

β γ

δ

Back

Bo

tto

m

Top

Edges (mm) Designed Measured

Front 35.50 35.9 ±1

Bottom 20.00 23.6 +0.5/-0.05

Back 34.38 34.5 ±0.05

Top 11.17 14.9 ±0.5

Angles ( ° ) Designed Measured

α 75.60 75.67 ±0.08

β 90.00 90.00 ±0.08

γ 90.00 90.00 ±0.08

δ 104.40 104.42 ±0.08

107

specification which would need to increase by about 3.5 mm. The guide satisfies the original

specifications, but not those modified to accommodate the thicker prism, so a fraction of the

light that is injected into the guide within its TIR limits reenters the prism and does not

contribute to the output of either PV cell.

4.1.2.2 Surface Finish

The surfaces of the prism

that are milled (top, bottom, and

front) have areas of tools marks

visible with the unaided eye. This

is an indication of a fairly rough

surface even in the areas where the marks have been more completely buffed out. As a result,

the Zygo laser interferometer is unable to resolve any fringes on these surfaces. The back

surface, however, is a smooth factory-finished surface that is easily measured in the laser

interferometer. The data are shown in Figure 4.5. The solid dark blue regions of invalid data are

areas outside of the interferometer’s maximum aperture and do not indicate flaws in the prism

surface. Notice the surface is slightly curved across the extrusion dimension, with a peak to

valley of about 8 µm over a distance of 100 mm. This is not of great concern with respect to the

prototype performance since its primary effect is a slight deviation of the rays in the extrusion

direction. This surface is modeled as being smooth and flat.

Lacking interferometer data on the remaining prism surfaces, they are measured in the

deflectometer under similar configurations to that used in the guide measurement. The data

are recorded and analyzed with the deflectometer’s limitations in mind. The prism front surface

Figure 4.5: Zygo laser interferometer data of the injection prism back surface.

108

reveals buffing tool marks much like those found on the guide layer. The marks have a depth of

about 100 to 200 nm and run parallel to the extrusion direction. Additionally, towards the

edges of the surface, milling marks become apparent in the form of more dramatic curved lines

that run somewhat perpendicular to the buffing marks. The more severe milling marks have

depths of around 500 nm.

The various markings can be seen most clearly by viewing the component slopes of the

measured surface data as shown in Figure 4.6. It appears that the central region was buffed

most agressively given the diminished milling marks and appearance of the finer buffing marks

in that region. Note that these emphasized features are not primarily due to phase error

stemming from the monitor nonlinearity. This is evidenced by the fact that the marks do not

run parallel to the fringes in the deflectometer’s raw data images.

The bottom surface of the prism has a similar finish to the other two milled sides, but

the surface topology is less significant since it will be immersed in an index matching fluid. The

top surface is even less important since it is not in the optical path of the system. However, in

Figure 4.6: Representation of prism front surface slope in the (top) horizontal and (bottom) vertical directions.

109

order to test the deflectometer, some

outstanding tool marks on the top

surface were investigated further by

both the deflectometer and the Zygo

white light interferometer. The two

data sets are compared to show the

merit of the deflectometer apparatus.

Data from the white light

interferometer is limited to a lateral

range of 0.35 x 0.26 mm, and the

milling marks in question have a

period of roughly 0.4 mm, so several

data sets must be stitched together.

The results are shown in Figure 4.7.

On the slice chart, the large

amplitude, large period undulations are the mill marks. There are some smaller features as well

that appear to be finer scratches left by the milling bit. These are seen most easily in the top

graph of Figure 4.7. On the prism top face, regions near the surface corners are visibly hazy.

This is due to a high concentration of the smaller scratch marks. The buffing procedure

evidently removes the scratches very quickly compared to the deeper milling marks.

The average depth of the milling marks in the tested region is approximately 0.2 µm

with a period of roughly 400 µm. This region is near the edge of what is resolvable by the

deflectometer. The very corner of the surface becomes too diffuse for the deflectometer to

Figure 4.7: White light interferometer data of prism top tool marks. Top graph shows registered data sets and position of slice, bottom shows surface height values along the slice (blue line) and the smoothed data (red line).

110

detect a usable specular signal. A portion of

the deflectometer data that is as close as

possible to this region is shown in Figure 4.8.

The vertical slice chart shows a period

between 400 and 500 µm, and an amplitude

of between 0.15 and 0.2 µm. This is

consistent with the white light interferometer

data (period and amplitude of 400 and 0.2 µm,

respectively).

4.1.2.3 Back Surface Mirror

The final aspect of the injection prism

metrology is the reflectivity of the back

surface mirror coating. Upon visual

inspection, as in Figure 4.9, the coating is clearly far from ideal. The edges appear blackened

and the coating does not extend all the way to the edges of the surface. This fabrication error is

due to issues during the vapor deposition process where the aluminum material pellet

completely evaporated during the first run. The process was run a second time to create a

thicker layer of material. Fortunately,

the center portion has a relatively

uniform reflectance, though much of

the light is absorbed. The process

was not repeated since cleaning off

Figure 4.9: Looking through the prism at the back surface mirror.

x 10-4

Figure 4.8: Deflectometer data of prism top near corner showing tool marks. All dimensions are in mm.

-12 -

111

the coating risks damaging the prism component. Additionally, the center portion of the mirror

can be selectively used during performance measurements and it only affects one of the two

cells, making it far easier to isolate in the final performance measurements.

For a robust computer model, the prism mirror must have a spatially varying

reflectance, so the reflectance as a function of position along the extrusion dimension is

measured. This is done using a green (543 nm) helium-neon laser, aiming it through the front

prism face, reflecting off the back face, and returning through the front face. A schematic of the

measurement process is pictured in Figure 4.10.

The prism is oriented in such a way that

the beam interacts with the front surface at the

same angle on the way in as on the way out.

The beam strikes each interface at nearly

normal incidence and is positioned to hit the

mirrored surface near its vertical center. The

prism is mounted on a translation stage and

moved in the extrusion direction after each

measurement.

Each data point consists of the power in

the Fresnel Reflection (beam A) and the Output

beam (beam B), a la Figure 4.10. The unattenuated power of the laser output is measured

separately. Beam A is the Fresnel reflection off of a single uncoated prism surface, so the

Fresnel transmission is easily calculated by

Figure 4.10: Schematic of prism mirror reflectance measurement. “Fresnel Reflection” and “Output” rays measured with power meters.

Fresnel Loss

Fresnel Loss

Reflection from prism

mirror

112

𝐹𝑡 = 1 −𝐼𝐴

𝐼0 ,

where I0 is the laser output power, IA is the power in beam A, and Ft is the Fresnel transmission.

To find an estimate of the internal transmittance of the prism material, the prism is moved so

the beam strikes an uncoated portion of the prism back surface. The output beam is subjected

to two Fresnel transmissions, two internal transmittance legs, and a Fresnel reflection off of the

back surface which is assumed to be 1 - Ft . The internal transmittance, τ, value is thus

calculated by

𝜏 = √𝐼𝐵

𝐼0(1 − 𝐹𝑡)𝐹𝑡2 ,

where IB is the power in beam

B. After Ft and τ are known,

the reflection, R, from the

coated portion of the back

surface is calculated by

𝑅 =𝐼𝐵

𝐼0𝐹𝑡2𝜏2

.

Primarily due to

estimated scattering from the surfaces, the uncertainty in the measurments is significant. The

results are shown in Figure 4.11. As expected, there is great variation in the reflectance with

especially poor performance near the edges. Even the best region in the center has just 36%

reflectance. The mirror surface is modeled as a series of uniform regions. The values of the

reflectance in each region along with the region widths are indicated by the orange line in the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 20 40 60 80 100

Ref

lect

ance

of

Seco

nd

Su

rfac

e

Position (mm)

Reflectance

Figure 4.11: Reflectance data from the prism mirror as a function of position in the extrusion direction. The blue line is the calculated reflectance with grey error bars and the orange is the stepped function used in the prototype model.

113

presented data. Figure 4.12 shows the geometry of the

uniform regions based on the reflectivity data and

computer analysis of prism photographs, such as that in

Figure 4.9.

4.1.2.4 Prism Model

Given all the above data, the prism is modeled

as having the measured dimensions and the mirror reflectivity is modeled as presented in the

previous section. All surfaces, however, are modeled as being flat and smooth. One feature not

previously mentioned is the fillets between the faces of the prism. The fillets on both edges

associated with the bottom prism face are important to model in order to coordinate the

measured and simulated prototype performances. The fillet on the front bottom edge is

modeled with a radius of 0.9 mm, tapering to 0.4 mm near the ends. The back bottom edge is

modeled as a scattering surface that extends 0.55 mm up the back face. This narrow strip does

not apply the reflective properties of the prism mirror since the coating does not extend around

the filleted edge. The effect of both of these fillets is inseparable from the effect of the index

matching fluid between the prism and guide layer near these two edges. Details of the modeled

prototype are found in Section 4.2.2.

4.1.3 Reflector

Given the reflector’s fabrication method, the primary concern regarding its

characterization is the surface shape. The surface has significant flaws and is expected to be a

major source of error in the final performance results. As previously indicated, measuring a

23% 11%

Figure 4.12: Geometry of the modeled reflectivity zones for the prism mirror. Blue regions are uncoated, and the remaining regions have uniform reflectances as indicated.

2% 2%

36% 31%

9% 22%

114

highly flawed, afocal, freeform optical surface is a difficult task, and is the motivation behind the

fringe reflection deflectometer apparatus.

Before the reflective Mylar is applied, the bare part is measured in the coordinate

measurement machine (CMM) introduced in Chapter 3. The data is shown in Figure 4.13

alongside the deflectometer data

after the Mylar film is applied.

These two data sets are compared

even though the surfaces being

measured are fundamentally

different: one is the bare 3D

printed reflector structure, and

the other is the surface of the

Mylar film that is adhered to that

surface. The macroscopic shape of

the surface is not expected to

change appreciably after applying

the Mylar film, so the data are

compared to assess the accuracy

of the deflectometer apparatus. A

direct comparison of the surface

shape is difficult since the CMM

optical probe experiences

Figure 4.13: Reflector surface shape measured via CMM (top) before Mylar film is applied and via the deflectometer (middle) with the Mylar film in place. Error values are mm of departure from the designed shape. The bottom image shows difference of the two data sets.

115

difficulty with the highly specular surfaces, and the deflectometer is incapable of measuring the

diffuse 3D printed surface.

From measurements of other components, the deflectometer is known to impart a

systematic surface figure error to the measured data as seen in Figure 4.3. This is thought to be

the primary reason for the large difference in the data sets of Figure 4.13. Note, however, the

CMM data does not extend to the very edges of the surface where the deflectometer predicts

the largest departure from the designed shape. The black outline in the top graph of Figure 4.13

shows the extent of the ideal surface geometry. The values shown in the CMM data also

depend on how the data are registered to the ideal surface.

From a qualitative standpoint, both the CMM and deflectometer data predict the

manufactured reflector to be slightly more curved along the X direction than designed. The

CMM shows a low amplitude convex bump in the extrusion (Y) direction which is absent in the

deflectometer data. It is possible that this is due to actual surface figure differences between

the bare and Mylar surfaces, though the demonstrated surface figure error in the deflectometer

reconstructions makes it impossible to conclude definitively. The effect of this error on

prototype performance is insignificant compared to the more egregious errors detected by the

deflectometer discussed below.

The CMM data shows the topology from the layered construction of the 3D printing

process which appears as striations in the extrusion direction. These marks are nonexistent in

the deflectometer data. This indicates that the Mylar film does not pick up the higher spatial

frequency topology present in the printed surface: an intended and beneficial effect.

Analyzing the principal curvatures of the deflectometer data reveals additional features

in the Mylar surface. The principal curvatures, k1 and k2, are the maximum and minimum

116

curvatures at each data point, respectively. For the chosen surface normal direction, a positive

curvature is convex and a negative, concave. The Gaussian curvature is the product of k1 and k2,

meaning it is negative for a saddle point and zero if either one of the principal curvatures is zero

[53].

The principal curvature data for the reflector surface based on the deflectometer results

are shown in Figure 4.14. Two notable features are the horizontal banding and the bumps near

the left edge of the data. The banding is seen exclusively in the k1 data. This is a real effect and

is present in the Mylar film even before it is applied to the reflector surface as seen in Figure

4.15. Since the waviness is evident when little external force is applied (e.g. freely hanging), it is

concluded to be caused by internal stresses in the film. Possible

origins of the internal stress include the stretching of the film or

metal deposition process during manufacture. The bumps near the

left edge of the data are predicted to be related to the adhesive

layer containing air bubbles. This conclusion is evidenced by the

Figure 4.15: Stock Mylar film showing wavy topology.

Figure 4.14: Principal curvature analysis of reflector surface as measured by the deflectometer.

117

observation of air bubbles in that region just prior to the Mylar lamination process.

A fine fringe-like structure is

observed in the curvature data as

shown in Figure 4.16 alongside the raw

fringe data of the same region. The fine

structure has the same directionality as

the raw fringe data and has a period

one fourth that of the imaged fringes.

For this reason, the fine structure is an

artifact of the nonlinearity of the deflectometer’s display as discussed in Chapter 3. Recall that

phase errors for the four step algorithm occur with a period that is one quarter that of the

displayed fringes. This fine error structure does not significantly affect the small scale topology

of the reconstructed surface, but contributes to cumulative integration error. However, due to

the comparatively small scale of the phase error, no attempt is made to remove it from the

reconstructed data.

The final test on the reflector

evaluates the surface roughness via

white light interferometry. The results

are shown in Figure 4.17 with a second

order 2D polynomial removed to reveal

the fine structure. Apart from an

occasional dust particle or scratch, the

Mylar surface is quite smooth with a

Figure 4.16: Magnified reflector surface data with the principal curvature, k2, on the left and a set of raw vertical fringe data on the right.

Figure 4.17: White light interferogram of the reflector's Mylar surface. A 2nd order 2D polynomial is removed to reveal surface roughness features.

118

peak to valley roughness of about 40 nm and an RMS roughness of about 120 Å. For reference,

a typical finish on a glass optical component is 50 Å RMS [54].

Given all the above data, the reflector surface is modeled according to the shape

measured by the deflectometer that imparts a very narrow (nearly specular) scattering angle

distribution to reflected rays, on the order of 0.1°.

4.1.4 Photovoltaic Cells

The geometry of the

PV cells is measured through

the computer analysis of

photographs of the cells over

a ruled background. The cell’s

size, shape, and gridline

pattern are measured and

modeled. The gridlines are a

necessary inclusion since they prevent light from interacting with the underlying PV material.

The photographs are used to create a binary apodization mask that is applied to the chip faces in

the model. The mask and raw photographs of one of

the prototype cells is shown in Figure 4.19.

To verify the mask geometries, each PV cell is

placed in the solar simulator beam behind a 4.88 mm

wide slit aperture. The output current of the cell is

Figure 4.19: PV chip geometry data and results. Photograph of back of PV cell over ruled background (top), photograph of front of cell (middle), and generated binary apodization mask (bottom).

Figure 4.18: Schematic of PV cell slit scan experiment.

119

measured as it is translated

along its long dimension with

the slit aperture remaining

stationary (Figure 4.18). The

same experiment is simulated

using the generated

apodization mask while

recording the incident

irradiance on the PV material.

The measured and simulated

results are compared in Figure

4.20, and they show excellent

agreement. The only simulated

effect is the geometry of the

exposed PV material, and the

simulated data is modified by

adjusting only the Y scale and X

and Y offsets. This data demonstrates that the output current of the cells is proportional to the

incident irradiance in the computer model, a fact that is used to compare the full prototype

measured and modeled performances. Additionally, this close agreement shows that there is no

appreciable drop in performance of the cells near the center which is furthest away from the

wire leads used to extract electrical current.

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

-80 -60 -40 -20 0 20 40 60

Translation Stage Position

Cell 1 Efficiency Profile

Measured Simulated

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

-60 -40 -20 0 20 40 60 80

Translation Stage Position

Cell 2 Efficiency Profile

Measured Simulated

Figure 4.20: Measured and modeled efficiency profiles for prototype photovoltaic cells.

Cu

rren

t (A

) C

urr

ent

(A)

120

The cell efficiency profile data show dips in output near the both ends of each cell. This

is due entirely to the presence of the thick gridlines at those positions. A more subtle feature is

the slight slope of the output data from cell 1 as a function of translation stage position. This is

caused by the geometry of the cell which is slightly wider at one end than the other, a

characteristic captured in the apodization mask.

The cells are also characterized

by their I-V curves as introduced in

Chapter 1. For this research, the primary

goal of these measurements is to

determine if the cells are damaged

during assembly. All that is needed for

this purpose is the dark I-V curves before

and after assembly. Prior to assembly,

the light I-V curve of each cell

illuminated by the solar simulator is

recorded for completeness.

The dark I-V curves shown in

Figure 4.21 reveal an obvious

discrepancy in cell 2 before and after assembly. These results indicate that cell 2 may see a

lower peak power and open circuit voltage than it did prior to assembly. However, the cell still

performs acceptably well, so it is not replaced. Cell 1 shows very little difference in performance

before and after assembly.

Figure 4.21: Dark I-V curves of prototype PV cells before and after assembly.

-0.01

0.01

0.03

0.05

0.07

0 0.1 0.2 0.3 0.4 0.5 0.6

Ou

tpu

t C

urr

ent

(A)

Forward Bias Voltage (V)

Cell 1 Dark I-V Curves

Pre-Assembly Post-Assembly

-0.01

0.01

0.03

0.05

0.07

0 0.1 0.2 0.3 0.4 0.5 0.6

Ou

tpu

t C

urr

ent

(A)


Cell 2 Dark I-V Curves

Pre-Assembly Post-Assembly

121

The light I-V curve of each cell

prior to assembly is presented in Figure

4.22. The effect of parasitic resistances

can be seen in the data, manifesting in

the form of a significantly sloped IV

curve at lower bias voltages. The two

cells have similar performance

parameters as indicated in Table 4-3.

The ratio of short circuit currents

between cells 1 and 2 is 0.991 which

approximately matches the ratio in

exposed PV material area between the

cells: 0.986. The light I-V curves are not taken after the prototype is assembled since it is

difficult to illuminate the cells in a controlled fashion. This means that comparisons between

the pre- and post-assembly cells as well as comparisons between both post-assembly cells

would be inconclusive.

4.2 Performance

4.2.1 Measurement Methods

The performance measurements are chosen to assist with a full characterization of the

prototype system and verify the computer model. The first performance metric is called the

“slit scan,” in which a vertically oriented, 4.88 mm wide, stationary slit aperture is placed in the

beam before the prototype. The prototype is then translated horizontally so the beam samples

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Ou

tpu

t C

urr

ent

(A)


Light I-V Curves

Cell 1 Cell 2

Figure 4.22: Light I-V curve of each PV cell before assembly.

Cell 1 Cell 2

Short Circuit Current 95.2 mA 96.1 mA

Open Circuit Voltage 0.574 V 0.576 V

Peak Power 28.5 mW 30.3 mW

Fill Factor 52% 55%

Table 4-3: Light I-V curve parameters for both PV cells.

122

different parts of the prototype aperture in the extrusion direction. At each translational

position, the prototype is swept through a range of elevation, or vertical, angles producing a 2D

data set. This measurement is especially useful for assessing the performance as a function of

variations in the extrusion dimension such as the reflector shape, prism mirror reflectivity, and

cell gridline geometry.

The second measurement, called the “field of view” (FoV) is a 2D data set sweeping

through vertical and azimuthal angles. The slit aperture remains as it is in the previous

measurement, and the translational position is linked to the azimuthal angle in an attempt to

prevent the beam from interacting heavily with the poor reflectance regions of the prism mirror

or the thick gridlines on the PV cells. The FoV measurement is less for the purpose of verifying

Slit

Ap

ertu

re

Vertical Angle

Azimuthal Angle

Figure 4.23: Performance measurement schematic of the prototype system. Shows the three positional variables in the system. The Vertical Angle axis of rotation remains constant with respect to the prototype structure.

Bottom cell location

123

the model and more for assessing the prototype performance relating to the illumination

conditions expected for a full system. It maps the relative efficiency of the system for different

positions of the sun in the sky. The prototype dimensions and mounting structure limit the FoV

to azimuth angles between about +5° and -30°.

Each data point consists of two measured values, one is the output of the “top PV cell”

located on the guide top surface, closest to the prism back, and the other is the output of the

“bottom cell” on the opposing guide face (see Figure 4.23). Note that the prism mirror reflects

light towards the bottom cell while the top cell output is nearly, if not completely, independent

of the prism mirror quality. The top and bottom cell outputs are simply added together to

obtain the total performance metric.

All the data from the simulations is shifted in translational position, vertical angle, and

azimuthal angle to account for the offsets present in the mounted prototype. The data values

are also offset (background signal) and scaled to fit the measured data. These adjustments are

applied uniformly to all simulated results over both the slit scan and FoV data sets. For example,

if the reflectance value of the prism mirror is lower than modeled, the simulated data will

appear too high since the top and bottom cell values are not scaled independently.

As mentioned previously, the measured value in the real-world prototype is the

electrical current output from each cell, while the simulation records only the incident

irradiance on the cells’ active areas. These two values are proportional given the highly linear

current response of silicon PV cells [5]. This is supported by the slit scan data in Section 4.1.4

with the close agreement of the measured current and simulated incident irradiance.

124

4.2.2 Simulated Performance

To illustrate the effects of each modeled manufacturing flaw, the system performance is

first simulated without any flaws. Then the flaws are added to the model one at a time and the

performance is reevaluated. All of the flaws have already been discussed in detail and are

summarized in Table 4-4.

The ideal model has no variation in the translation direction, and is thus measured as a

function of vertical angle only for the slit scan experiment. The FoV measurement retains both

data dimensions.

Name Ideal Modeled Flaw

Prism Dimensions

Prism dimensions as specified for

manufacter (right angle between top and

back faces), 75.6° between front and

bottom faces, and a 20 mm wide bottom

face

75.847° angle between front and bottom

faces, and a 23.6 mm wide bottom face

Reflector ShapeAs-designed reflector shape with no

variation in the extrusion direction

Reflector shape as measured by the

deflectometer

Back Prism FilletSharp edge between back and bottom

prism faces

0.55 mm tall strip at the bottom of the

prism mirror face is modeled as a diffuse

scatterer since the edge is filleted and

the mirror coating does not extend to the

very edge of the surface

Front Prism FilletSharp edge between front and bottom

prism faces

Edge is filleted with a radius of 0.9 mm,

tapering to 0.4 mm near the ends of the

prism

Cell Gridlines No cell gridlinesCell gridlines with thick lines near either

end of each cell

Prism Mirror Spatially invariant reflectance of 90%Spatially varying reflectance with a peak

reflectance of 36%

Table 4-4: Summary of simulated manufacturing flaws. The flaws are added to the model in the order shown.

125

The results of several slit scan simulations are presented in Figure 4.24. The ideal model

is shown with none of the listed manufacturing flaws. Then the measured prism dimensions are

added followed by the measured reflector shape. A wide source filling most of the prototype

aperture is used for the simulations in order to average out the translational variability of the

reflector for these data. Notice that the totaled output does not appear to match the data

presented in the tolerance analysis of Chapter 3. While there are slight differences in the

models, the main cause of the discrepancy is that the previous data does not take into account

the cosine falloff from the projected area of the prototype aperture.

Including Flaws in:

0

0.2

0.4

0.6

0 20 40 60

Top Cell

Figure 4.24: Slit scan simulation data for ideal model (red), added prism dimensions flaw (green), then added reflector shape flaw (blue). Simulation is performed with wide source to average out translational variations in the measured reflector shape. Vertical axis is in relative units and are equivalent for all three plots.

0

0.2

0.4

0.6

0 20 40 60

Bottom Cell

0

0.2

0.4

0.6

0 20 40 60


Total

0

0.2

0.4

0.6

10

Ideal

Prism Dimensions

Reflector Shape

None (Ideal Model)

Prism Dimensions

Reflector Shape

126

When the measured prism dimensions are included in the model, the field of view is

widened as predicted. A much more profound difference is seen when the measured reflector

shape is incorporated. This is the single most significant source of error in the prototype, as

expected. The measured reflector shape causes two peaks to appear in the output of the top

cell around vertical angles of 24° and 34° with a valley in between at 29°. Figure 4.25 illustrates

the cause of this phenomenon by visualizing the traced rays. At the 24° peak, the ray bundle

just fills the extent of the prism-guide interface, but as the vertical angle increases, those rays

begin to strike the prism mirror and are sent towards the bottom cell instead. Increasing the

vertical angle further begins to utilize the leftmost portion of the reflector whose rays are sent

towards the top cell via TIR inside the guide layer.

The ideal reflector shape has a more definitive caustic and an overall flatter shape. For

that model, the top cell output peaks when the prism-guide interface is just filled, and sends

more rays toward the bottom cell as the vertical angle begins to increase. The difference comes

in the lack of a second peak since the leftmost portion of the reflector is at a different angle and

does not send rays toward the top cell with increasing vertical angle.

Figure 4.25: Raytraces of the "Reflector Shape" model at three different vertical angles as indicated. There are peaks in the top cell output at 24° and 34° with a valley in between.

127

The FoV results for the ideal model are displayed

in Figure 4.26. This simulation reverts to a narrow slit

aperture, so there is no cosine falloff effect in the

azimuthal direction. The performance is nearly constant

as a function of azimuth except for a slight depression in

the output at the more extreme azimuthal angles. This is

primarily due to increased Fresnel losses for the more

extreme incidence angles at all refractive interfaces.

The FoV data for the ideal model is symmetric

about the azimuthal origin given the lack of variation in

the extrusion direction. However, this is no longer true

once certain flaws are introduced. Note that the FoV

data at an azimuth of 0° is equivalent to the slit scan data

at the zero translational position.

The next flaws to be introduced are the front and

back fillets on the edges adjacent to the injection prism’s

bottom face. The effect of these flaws depends greatly

on the translational position, so the slit aperture is

reduced once again to 4.88 mm wide. The results with a

fixed translational position of 0 are shown in Figure 4.27.

The legend lists the most recently added flaw, but the

model also includes all previously added flaws in the

order listed in Table 4-4. For example, a data set labeled

Figure 4.26: Results of Field of View (FoV) simulation for ideal prototype model.

128

“Back Fillet” includes the back fillet flaw along with all previous flaws: prism dimensions and

reflector shape in this case.

As indicated by the data, the primary effect of the front fillet is a depression of the top

cell output for lower vertical angles. Likewise, the back fillet depresses the output of the

bottom cell for lower vertical angles. Note the small bump in the bottom cell output near the

30° vertical angle. For some translational positions, this bump is significantly suppressed by the

inclusion of the front fillet. This is because some rays are reflecting off the prism mirror and

passing very close the front edge of the prism before continuing into the guide layer. The

0

0.2

0.4

0 20 40 60

Top Cell

0

0.2

0.4

0 20 40 60

Bottom Cell

0

0.2

0.4

0.6

0 20 40 60


Total

0

0.2

0.4

0

Reflector Shape

Back Fillet

Front Fillet

Figure 4.27: Results of slit scan simulations for a 4.88 mm wide slit fixed at the translational origin. The reflector shape data set includes the prism dimensions flaw. The remaining data sets are for and added back prism fillet then front fillet.

Including Flaws from:

129

inclusion of the fillet deviates those rays so they do not reach the bottom cell with the same

efficiency. Figure 4.28 demonstrates this phenomenon at a translational position of 9 mm for

the most affected range of vertical angles. Rays are traced over the entire aperture height, but

only a subset are drawn for illustration purposes.

0

0.02

0.04

0.06

0.08

0.1

0.12

23 25 27 29 31 33

Translational Position = 9 mm

Back Fillet Front Fillet

Figure 4.28: Effect of front fillet on bottom cell output near 30° vertical angle and 9 mm translational position.

Front fillet

location

130

Fig

ure

4.2

9: S

lit S

can

sim

ula

tio

n r

esu

lts

for

fin

al t

hre

e m

od

ele

d m

anu

fact

uri

ng

flaw

s. S

cale

s ar

e co

nst

ant

wit

h e

ach

ro

w t

o s

ho

w a

tten

uat

ion

ef

fect

s o

f ea

ch f

law

. Fro

nt

Fille

t C

ell

Gri

dlin

es

Pri

sm M

irro

r

131

For the remainder of the data, all simulations are extended into two dimensions to show

the full effect of the included manufacturing flaws. Figure 4.29 shows the slit scan simulations

for the added flaws of the front fillet, cell gridlines, and prism mirror, which is the final modeled

flaw. The addition of the cell gridlines decreases the output of both cells since it is preventing

some of the light from reaching the PV material. In particular, there is a large decrease in

output near both extremes of the translation position for both cells. This is due to the thick

gridlines which are located on the corresponding parts of the cells. The top and bottom cells see

the decreased output at slightly different translational positions since the large gridlines are not

located on precisely the same parts of both cells.

Including the measured prism mirror reflectance has a substantial effect on the

bottom cell output, but the top cell experiences no distinguishable change. This is expected

since the purpose of the prism mirror is to redirect light rays towards the bottom cell. The low

reflectance of the mirror decreases the total output of the bottom cell at all data points. There

is an especially large depression at translational positions of less than -20 mm and greater than

about 30 mm. This is due to the reflectance falloff of the mirror near the translational extremes.

Figure 4.30 shows the same manufacturing flaws for the field of view measurement.

The addition of the cell gridlines appears to simply uniformly decrease output of both top and

bottom cells, while the poor reflectance of the prism mirror uniformly effects the bottom cell

only. The lack of added structure is achieved with the linked translational and azimuthal axes

during the simulation. The relationship between these two parameters is chosen so the beam

strikes the relatively uniform center portion of the prism mirror while avoiding the thick

gridlines on both top and bottom cells. As such, the uniformly decreasing output is due to the

thin cell gridlines and the ~36% reflectance center portion of the prism mirror only.

132

Figure 4.30: Field of View (FoV) simulation results for final three modeled manufacturing flaws. Scales are constant across each row to show attenuation effects of each flaw.

133

4.2.3 Measured Performance Comparison

The validation of the above simulation results is achieved by comparing them to

measured results from the real-world prototype. Figure 4.31 shows a comparison of the slit

scan performance and Figure 4.32 presents the field of view data. The data are processed only

in its relative output scale and offset, and the relative zero positions of the vertical angle,

Slit Scan Performance Comparison

Figure 4.31: Slit scan performance comparison. Measured results in the left column, simulated results in the right column. Scale is consistent across each row.

134

azimuth angle, and translation

position. The same post-processing

values are used ubiquitously for all

data presented in this section.

Many features match very well

between the measured and simulated

results. For example, the relative

position of peaks between the top and

bottom cells and the general envelope

of non-zero output values. Even the

fine translationally variant output of

the top cell at the 23° vertical angle

matches extraordinarily well.

However, there are certain structures

that the computer model did not

capture.

For instance, the measured slit

scan top cell data drops to zero at the

translational extremes while the

simulation predicts output to increase.

This is easily explained by the

prototype’s support structures blocking absorbing the incident rays at either end of the

extrusion. The support structure was purposely ignored during the simulation since its effects

Field of View Performance Comparison

Figure 4.32: Field of view performance comparison. Measured results in the left column, simulated results in the right column. Scale is consistent across each row.

135

would be easily discernable in the measured data. The simulation predicts a rise in output in

these regions because the beam would have passed over the thick gridlines as the translational

position nears ±50 mm. The measured results show the interaction with the thick gridlines, but

the output continues to decrease as the beam begins to be attenuated by the support structure.

A more mysterious discrepancy occurs near the bottom cell peak in both the slit scan

and field of view experiments. The simulated data shows a single peak at a vertical angle of

about 47° while the measured data shows the peak shifted to about 51° in the case of the slit

scan, and two separate peaks at about 47° and 52° in the FoV data. The FoV data at an azimuth

of 0° are the same exact measurement as the slit scan data at a translation position of 0 mm.

This is definitely the case for the simulated data and the measured top cell data seems to exhibit

this property. The change in this data subset in the bottom cell implies that part of the

prototype changed between the two experiments in a way that only significantly effects the

bottom cell output. A fabrication error that is not included in the computer model is some resin

trapped between the guide and the reflector structure.

This error occurred during the attachment of the bottom cell to the guide layer. At that

stage, the reflector was already in place and there existed a small air gap between that structure

and the front face of the guide layer. Some excess liquid resin from the bottom cell filled in part

of that gap via capillary action, against the force of gravity. Due to the construction of the

Bottom Cell

Guide Bottom of Reflector

Unwanted Resin

Figure 4.33: Photo peering through back face of guide layer at the bottom of the reflector structure. Resin from the bottom cell seeping into the gap between the reflector and guide appears as irregularly shaped darker regions.

136

braces, the offending resin cannot be cleaned off without disassembling the prototype.

Unfortunately, this resin allows light to couple out of the guide layer just prior to reaching the

bottom cell. The resin shape also changes unpredictably depending on the stresses in the

prototype structure, making it temporally variant as the prototype is rotated and moved during

the experiments. A snapshot of the resin’s irregular shape is shown in Figure 4.33 by looking

through the back surface of the guide layer.

The effect this has on the performance is

highly dependent upon the shape of the resin layer.

Investigating the ray paths near the discrepant

region in the data sets, say a vertical angle of 47°,

many rays strike the region of front guide layer

adjacent to the reflector structure where the excess

resin is located. By changing the shape of the resin

layer and assuming it is partially absorptive, one can

selectively and significantly change the bottom cell

output between vertical angles of about 44° and 51°

according to the as-built model. In combination

with adjusting the reflectivity values of the prism

mirror, this degree of freedom could appear to shift the simulated peak’s vertical angle position

or even split it in two. In a similar way, it can also explain the decreased output in the bottom

cell near an azimuth of -7° and a vertical angle of 47°. Since the resin layer shape is temporally

inconsistent and cannot feasibly be measured in real time, its effect is omitted from the

Unwanted Resin Region

Figure 4.34: Simulation of final prototype model at a vertical angle of 47° showing many rays interacting with the area containing the unwanted resin.

137

computer model. However, it offers a credible explanation for some inconsistencies in the data

that fits all observations.

Another more subtle inconsistency is the decreased top cell output in the simulation

data near the 24° vertical angle peak. This peak is slightly lower in both the simulated slit scan

and field of view studies. While no definite explanation is offered here, one possible cause is

the crazing of the guide surface adjacent to the prism as described in Chapter 3. This crazing is

not included in the model since its effect is difficult to measure. Ideally, the bidirectional

scattering distribution function (BSDF) of the surface would be measured as it is implemented in

the prototype: immersed in corn syrup. The BSDF is a notoriously difficult and data-intensive

measurement and the equipment needed was not readily available. The crazing is suggested as

the source of the data inconsistency since its degree of scattering depends on incident angle and

the 24° vertical angle peak has a significant portion of rays traversing the prism-guide interface

at grazing incidence angles (refer back to Figure 4.25).

4.3 Conclusion

Apart from the mentioned discrepancies, the data coincides very well, especially given

the major performance changes incurred by adding manufacturing flaws presented throughout

Section 4.2.2. This is a strong indication that the model is accurate to a degree such that

features in the data can be investigated and explained by visualizing light rays traced in the

computer model.

138

Chapter 5: Conclusion

5.1 Summary

Current research in CPV is focused mainly on high concentration systems with high

conversion efficiency. Low concentration systems are not as well explored since they cannot

offer the efficiencies of HCPV, but niche applications can take advantage of their flexible form

factor. Interest in building-integrated photovoltaics has led to several commercialized

technologies that integrate solar photovoltaics into windows [17, 18, 22, 23].

Several novel window-integrated low concentration solar concentrator designs have

been presented with the goal of reducing the net energy lost through the windows in the

wintertime. They are based on previous planar light guide concentrators from the University of

Rochester [11, 12]. The systems are evaluated regarding their effective insulation properties

given the calculated net energy lost. Without moving parts, they optimize to have acceptance

angles of about 20° to 35° in the vertical direction and ±90° in the horizontal direction, but have

peak optical efficiencies of less than 50%. By including internally moving parts, the acceptance

angle is increased to nearly a full π steradians (the full sky from the point of view of the window)

and the average optical efficiency increases to over 50%. Systems in certain locations are not

viable due to low solar irradiance in the wintertime. Others, however, reduce net energy loss to

zero or better for much of the year.

The “suspended louver” design optimized for Rochester NY is prototyped as a proof of

concept. The prototype is at a 10X scale and includes one primary optical element and its

associated injection feature. Its performance is measured using a solar simulator and is

compared to the performance of the computer model. Measurements of the prototype system

139

and its individual components help create an accurate model of the as-built system. The

measured and simulated performance results are shown to coincide closely.

The well-matched performance results indicate an accurate computer model of the

prototype system. This validates the simulation methods used during the optimization of the

designs and shows that they have the potential to reach the predicted performance values if

manufactured accurately.

The static designs must be optimized for a specific location while a single tracking design

operates efficiently for a wide range of latitudes. Another location specific consideration is the

variability of available solar irradiance throughout the year. Rochester, NY is a challenging

location for this application due to its frequent cloud cover during the cold times of year when

the CPV system is optimized to operate at peak efficiency. Designs for other locations, however,

show great promise in recovering energy lost through the windows.

5.2 Future Work

The next logical step to validate this LCPV design space is to create a prototype with an

array of optics at a smaller scale than done in this thesis. This requires sophisticated fabrication

methods and introduces more error largely due to the difficult task of aligning an array of micro-

optics. However, a similar undertaking has been achieved with some success at the University

of Rochester for HCPV systems [11, 12]. Fabrication methods that could be used for mass

manufacture are of particular interest. For example, the suspended louvers could be strips of

reflective Mylar film that are glued to plates of glass at either endpoint. The angle of the film

surface at the endpoints could be fixed during the glue setting process and the curvature may be

140

controlled by varying the film’s thickness (Dr. James Zavislan, personal communication, January,

2014).

The window-integrated light guide LCPV design space has not been thoroughly explored.

The designs presented in this thesis offer novel solutions in this space, but there is still much to

be discovered. In particular, the tracking design presented in Chapter 2 is based off of the static

louver designs and essentially tracks with a horizontal rotational axis. However, the more

sensible tracking direction uses a vertical (possibly tilted) rotational axis given the wider

acceptance angle requirements in the horizontal direction. This results in a drastically different

design form than the louvers presented here.

Finally, the presented static designs would be more marketable if they operated

efficiently for a wide range of latitudes rather than being designed for a series of singular

locations. A more broadly applicable design form seems to be possible given the similar shapes

of individual components for designs presented in this thesis. It may be possible to fabricate the

same components for a variety of locations, but assemble them with a different alignment

depending on latitude and typical annual weather conditions.

141

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148

Appendix A. Design Specifications

Suspended Louver

149

Immersed Louver

150

Extruded Segment Lens

Curve

151

Static Designs: Departures from Best Fit Circles of Primary Optic

152

Tracking Louver

156

Appendix B. Deflectometer MATLAB Code

DeflectometerWizard.m

This code is the high level function for data acquisition from the deflectometer. The

dependency functions written for this research are included after the high level code. All other

functions are included in MATLAB 2012a or are publically available on the MATLAB Central File

Exchange website.

function [cutup,cutdn,v1234lin,h1234lin,linmm,permm,CalibrationData] =

DeflectometerWizard(varargin)

p = inputParser;

p.StructExpand = true;

addParamValue(p,'exposure',0,@isnumeric);

addParamValue(p,'f',1,@(x) (isnumeric(x) && numel(x)==1));

addParamValue(p,'k1',0,@(x) (isnumeric(x) && numel(x)==1));

addParamValue(p,'R',zeros(3),@(x) (isnumeric(x) && all(size(x)==[3,3])));

addParamValue(p,'T',zeros(3),@(x) (isnumeric(x) && all(size(x)==[3,1])));

addParamValue(p,'Rmir',zeros(3),@(x) (isnumeric(x) && all(size(x)==[3,3])));

addParamValue(p,'Tmir',zeros(3),@(x) (isnumeric(x) && all(size(x)==[3,1])));

addParamValue(p,'cxcy',[0 0],@(x) (isnumeric(x) && numel(x)==2));

addParamValue(p,'dxy',[0 0],@(x) (isnumeric(x) && numel(x)==2));

parse(p,varargin{:});

close all;

global vid src fs;

cutup = 200; % upper 8-bit cutoff for fringe display

cutdn = 30; % lower 8-bit cutoff for fringe display

monpp = [0.258,0.258]; % monitor pixel pitch [x,y]

% dummy values in case of early termination

v1234lin=0; h1234lin=0; linmm=0; permm=0;

if any(cellfun(@(str)

any(strcmp(str,p.UsingDefaults)),{'f','k1','R','T','Rmir','Tmir','cxcy','dxy'}))

CalibrationData=0;

else

f = p.Results.f;

k1 = p.Results.k1;

R = p.Results.R;

T = p.Results.T;

Rmir = p.Results.Rmir;

157

Tmir = p.Results.Tmir;

cxcy = p.Results.cxcy;

dxy = p.Results.dxy;

CalibrationData =

struct('f',f,'k1',k1,'R',R,'T',T,'Rmir',Rmir,'Tmir',Tmir,'cxcy',cxcy,'dxy',dxy);

end

%% create persistent objects and figures

vid = videoinput('AVTMatlabAdaptor_R2009b', 1, 'F7M0_Mono8_1628x1236'); % create

VideoInput object

% vid = videoinput('winvideo');

set(vid,'Timeout',70);

src = getselectedsource(vid);

[fsfig,fsax,fsimg,fssiz] = FullscreenFig(2); % open full screen figure on monitor 2

caxis(fsax,[0,255]);

caxis('manual');

set(fsimg,'cdata',255.*ones(fssiz));

fs = struct('FigureHandle', fsfig, 'AxisHandle', fsax, 'ImageHandle', fsimg, 'Size',

fssiz);

% wizard control and instruction figure

figwiz = pptfig('Deflectometer Wizard',1);

ss = get(0,'screensize');

fwsiz = [500,250];

set(figwiz,'outerposition',[1+2, ss(4)-fwsiz(2)+2,

fwsiz],'resize','off','CloseRequestFcn',@cancel);

numbut = 4;

but = zeros(numbut,1);

spac = 0.1/numbut;

for ij = 1:numbut

but(ij) = uicontrol('style','pushbutton','units','normal','position',[(ij-1)*(1-

spac)/numbut+spac, spac*fwsiz(1)/fwsiz(2), (1-spac*(numbut+1))/numbut, 0.2]);

end

txwiz = uicontrol(figwiz,'style','text','units','normal','position',[0.05,

spac*fwsiz(1)/fwsiz(2)*2+0.2, 0.4, 1-(spac*fwsiz(1)/fwsiz(2)*2+0.2)-

0.1],'backgroundcolor','w','HorizontalAlignment','left','fontsize',12);

axwiz = axes('parent',figwiz,'units','normal','position',[0.5,

spac*fwsiz(1)/fwsiz(2)*2+0.2, 0.5, 1-(spac*fwsiz(1)/fwsiz(2)*2+0.2)],

'visible','off','ylim',[-0.2 1.8]);

awpos = getpixelposition(axwiz);

set(axwiz,'xlim',awpos(3)./awpos(4).*[-1,1]);

% graphics objects

opengl software;

ang = 30; % graphical camera angle from vertical

raydir = [tand(ang),1];

raydirc = raydir.*[1 -1];

camsca = 0.3;

camverts = -1.6.*raydirc; % camera origin

camverts = [camverts;camverts+camsca.*raydirc+camsca./4.*raydirc([2,1]).*[-1

1];camverts+camsca.*raydirc-camsca./4.*raydirc([2,1]).*[-1 1]];

158

gcam = patch('vertices',camverts,'faces',1:3,'edgecolor',[0 .7

0],'linewidth',1.5,'facecolor','g','facealpha',0.5,'parent',axwiz,'linesmoothing','on');

trapx0 = bsxfun(@plus,repmat(-1:1,3,1),0.5.*(-1:1).');

trapy0 = repmat(-0.5.*(-1:1).',1,3);

mirsca = 0.38;

gmir(1) = patch('vertices',mirsca.*[trapx0(:),trapy0(:)],'faces',[1 3 9

7],'edgecolor','k','facecolor','none','linewidth',1.5,'parent',axwiz,'linesmoothing','on'

);

gmir(2) =

line('xdata',0.5.*mirsca.*trapx0(:),'ydata',0.7.*mirsca.*trapy0(:),'color','k','marker','

.','linestyle','none','parent',axwiz);

monsca = 0.7;

gmon =

patch('vertices',[monsca.*trapx0(:)+raydir(1),monsca.*trapy0(:)+raydir(2)],'faces',[1 3 9

7],'edgecolor','b','facecolor','b','facealpha',0.5,'linewidth',1.5,'parent',axwiz,'linesm

oothing','on');

parsca = 0.15;

traparx = parsca.*(bsxfun(@plus,trapx0,[0.3;0;0]));

trapary = parsca.*(bsxfun(@plus,trapy0,[0.5;0;0]));

gpar = patch('vertices',[traparx(:),trapary(:)],'faces',[1 2 3 6 9 8 7

4],'edgecolor','r','facecolor','r','facealpha',0.5,'linewidth',1.5,'parent',axwiz,'linesm

oothing','on');

grai =

line('xdata',[camverts(1,1);0;raydir(1)],'ydata',[camverts(1,2);0;raydir(2)],'linestyle',

'--','color','k','linesmoothing','on');

% preview window with slice chart

figprev = pptfig('Camera Preview',10);

vidres = get(vid,'VideoResolution');

set(figprev,'position',[4, ss(4)-fwsiz(2)-round(vidres(2)/2)-100,

round(vidres./2)],'CloseRequestFcn',@hideprev);

axprev =

axes('parent',figprev,'units','normal','position',[0,0,1,1],'visible','off','xlimmode','m

anual','xlim',[0.5,vidres(1)+0.5],'ylimmode','manual','ylim',[0.5,vidres(2)+0.5]);

imgprev = imshow(zeros(vidres([2,1])),[0,255],'parent',axprev);

preview(vid,imgprev);

linroi = imline(axprev,[100,100;300,100]);

setColor(linroi,'b');

boxfcn = makeConstrainToRectFcn('imline',[1 vidres(1)],[1 vidres(2)]); % confine line ROI

to image boundaries

setPositionConstraintFcn(linroi,boxfcn);

slicax(1) = line('xdata',[100;100],'ydata',100+[-10,10],'color','b','HitTest','off');

slicax(2) = line('xdata',[300;300],'ydata',100+[-10,10],'color','b','HitTest','off');

slicplot = line('xdata',[100;300],'ydata',100+[-10,10],'color','r','HitTest','off');

hideprev();

setappdata(imgprev,'UpdatePreviewWindowFcn',@slicechart);

%% Focus and Exposure adjustments

set(txwiz,'string','Adjust Focus and/or Exposure?');

buttonset({'Exit','Focus','Exposure','Next'},{@cancel,@focadj,@expadj,@OK},{'off','on','o

n','off'});

159

set([gcam,gmir,gmon,gpar,grai],'visible','off');

exitflag = false;

figure(figwiz);

uiwait(figwiz);

if exitflag

return;

end

%% Deflectometer Calibration

buttonset({'Exit','','',''},{@cancel,0,0,0});

set(txwiz,'string','Checking for existing calibration data...');


if any(cellfun(@(str)

any(strcmp(str,p.UsingDefaults)),{'f','k1','R','T','Rmir','Tmir','cxcy','dxy'}))

set(txwiz,'string','Position MIRROR in camera''s field of view for calibration.');

set([gcam,gmir,gmon,grai],'visible','on');

buttonset({'Cancel','','','Continue'},{{@cancel,figwiz},0,0,@(~,~)

uiresume(figwiz)});


drawnow expose;

% pfig = prevlink(@(~,~) uiresume(figwiz));


uiwait(figwiz);

% if ishandle(pfig)

% delete(pfig);

% end

hideprev();

if exitflag

exitflag = false;

return;

end


set(txwiz,'string','Deflectometer Calibration');

[f,k1,R,T,Rmir,Tmir,cxcy,dxy,exitflag] = DW_DeflectometerCalibration(txwiz,but(1));

if exitflag

return;

end

else

f = p.Results.f;

k1 = p.Results.k1;

R = p.Results.R;

T = p.Results.T;

Rmir = p.Results.Rmir;

Tmir = p.Results.Tmir;

cxcy = p.Results.cxcy;

dxy = p.Results.dxy;

end

CalibrationData =

struct('f',f,'k1',k1,'R',R,'T',T,'Rmir',Rmir,'Tmir',Tmir,'cxcy',cxcy,'dxy',dxy);

assignin('base','CalibrationData',CalibrationData);

160

%% Focus and Exposure adjustment for part only

wizFE();

exitflag = false;

figure(figwiz);

uiwait(figwiz);

if exitflag

return;

end

%% Capture Phase Shift Data

set(txwiz,'string','Position PART in camera''s field of view.');

set([gcam,gpar,gmon,grai],'visible','on');

buttonset({'Exit','','','Continue'},{@cancel,0,0,@(~,~) uiresume(figwiz)});


drawnow expose;

% pfig = prevlink(@(~,~) uiresume(figwiz));


uiwait(figwiz);

permm = 100.*monpp;

set(txwiz,'string','Adjust VERTICAL fringe period. Aim for low value while maintaining

contrast.');


slid = uicontrol('style','slider','units','normal','position',[0.05,

spac*fwsiz(1)/fwsiz(2)+0.25, 0.9,

0.2],'value',round(permm(1)/monpp(1)),'callback',{@peradj,1},'min',20,'max',500);

peradj(slid,0,1);

uiwait(figwiz);

permm(1) = get(slid,'value').*monpp(1);

set(txwiz,'string','Adjust HORIZONTAL fringe period. Aim for low value while maintaining

contrast.');

set(slid,'value',round(permm(2)/monpp(2)),'callback',{@peradj,2},'min',20,'max',500);

peradj(slid,0,2);

uiwait(figwiz);

permm(2) = get(slid,'value').*monpp(2);

hideprev();

delete(slid);

if exitflag

exitflag = false;

return;

end


set(txwiz,'string','');

[v1234lin,h1234lin,linmm,permm,exitflag] =

DW_PhaseShiftCapture(txwiz,cutdn,cutup,monpp,permm,but(1));

%% Fiducial picture

set(txwiz,'string','Place caliper on surface for fiducial definition.');

set([gcam,gpar,gmon,grai],'visible','on');

buttonset({'Exit','','','Take Pic'},{@cancel,0,0,@(~,~) uiresume(figwiz)});

set(fsimg,'cdata',cutup.*ones(fssiz));

161

drawnow expose;


uiwait(figwiz);

fidpic = capic(cutup.*ones(fssiz));

fidfig = pptfig('Fiducial Picture',8);

fidimg = imagesc(fidpic);

buttonset({'Exit','','Re-take','Continue'},{@cancel,0,@retake,@(~,~) uiresume(figwiz)});

uiwait(figwiz);

fidist0 = inputdlg('Enter caliper distance in inches:','Fiducial spacing');

if isempty(fidist0)

fidist0 = {0};

end

fidist = str2double(fidist0{1})*25.4; % fiducial spacing in mm

if isnan(fidist)

fidist = 0;

end

%% Prepare output data

[curpath,~,~] = fileparts(mfilename('fullpath'));

pn = [curpath '\Data\Part.mat'];

set(txwiz,'string','Save data.');

set([gcam,gpar,gmon,grai,gmir],'visible','off');

buttonset({'Exit','','','Save Data'},{@cancel,0,0,{@savdat,pn}});

savdat(0,0,pn);

if ~exitflag

uiwait(figwiz);

end

%% Cleanup

cancel(0,0);

%% nested functions

function retake(~,~)

fidpic = capic(cutup.*ones(fssiz));

set(fidimg,'cdata',fidpic);

drawnow expose;

end

function pic = capic(cdat)

set(fsimg,'cdata',cdat);

drawnow expose;

pause(0.5);

start(vid);

pic = uint8(squeeze(mean(getdata(vid),4)));

end

function slicechart(~,event,~)

cdat = mean(event.Data,3); % if image is in color

slic = getPosition(linroi);

tvec = diff(slic,1,1);

tdist = sqrt(sum(tvec.^2));

162

if tdist==0

return;

end

tval = linspace(0,1,ceil(tdist)).';

tpxy = round(bsxfun(@plus,slic(1,:),tval*tvec));

tpid = sub2ind(size(cdat),tpxy(:,2),tpxy(:,1));

tdat0 = cdat(tpid);

tdat = (tdat0-min(tdat0))./(max(tdat0)-min(tdat0)).*2-1; % normalize to interval

[-1,1]

aspec = 10; % plot aspect ratio Width/halfHeight

if round(slic(1,1))==round(slic(2,1))

zvec = [tdist/aspec,0];

else

zvec = [-1,1].*tvec([2,1])./aspec;

zvec = -sign(zvec(2)).*zvec;

end

xy = bsxfun(@plus,slic(1,:),tval*tvec+tdat*zvec);

set(slicplot,'xdata',xy(:,1),'ydata',xy(:,2));

set(slicax(1),'xdata',zvec(1).*[-1;1]+slic(1,1),'ydata',zvec(2).*[-

1;1]+slic(1,2));

set(slicax(2),'xdata',zvec(1).*[-1;1]+slic(2,1),'ydata',zvec(2).*[-

1;1]+slic(2,2));

set(imgprev,'cdata',cdat);

drawnow expose;

end

function peradj(slid,~,vh)

val0 = get(slid,'value');

val = round(val0);

set(slid,'value',val);

switch vh

case 1

dir = [1,0];

case 2

dir = [0,1];

end

apod = DW_apodizationgen(dir,val*monpp(vh),0,'cos',[cutdn,cutup],monpp);


set(fsimg,'cdata',apod);

end

function savdat(~,~,pn)

uisave({'cutup','cutdn','v1234lin','h1234lin','linmm','permm','CalibrationData','fidpic',

'fidist'},pn);

end

function wizFE()

set(txwiz,'string','Adjust Focus and/or Exposure?');

163

buttonset({'Exit','Focus','Exposure','Next'},{@cancel,@focadj,@expadj,@OK},{'off','on','o

n','off'});


uiwait(figwiz);

end

function expadj(~,~)

set(txwiz,'string','Position MIRROR and PART in camera''s field of view.');

set([gcam,gmir,gmon,gpar,grai],'visible','on');


uiresume(figwiz)});


set(fsimg,'cdata',cutup.*ones(fssiz));

drawnow expose;


uiwait(figwiz);

hideprev();

if exitflag

exitflag = false;

wizFE();

return;

end

set(txwiz,'string','Adjust slider until pixels are bright but do not saturate.');

figcon = DW_ExposureAdjust(cutup);

buttonset({'','','','Done'},{0,0,0,{@OK,figcon}});

uiwait(figcon);

if ishandle(figcon)

close(figcon);

end

wizFE(); % reset wizard window

end

function focadj(~,~)

set(txwiz,'string','Position mirror in camera''s field of view.');

set([gcam,gmir,gmon,grai],'visible','on');


uiresume(figwiz)});



uiwait(figwiz);

hideprev();

if exitflag

exitflag = false;

wizFE();

return;

end

set(txwiz,'string','');

figimg = pptfig('Focus Helper',7);

set(figimg,'visible','off');

buttonset({'','','',''},{0,0,0,''});

164

DW_FocusAdjust(figimg,txwiz,but(4));

wizFE(); % reset wizard window

end

% function fig = prevlink(xfcn)

% fig = preview(vid);

% % if the object is a figure or figure descendent, return the figure. Otherwise

return [].

% while ~isempty(fig) && ~strcmp('figure', get(fig,'type'))

% fig = get(fig,'parent');

% end

% set(fig,'CloseRequestFcn',xfcn);

% end

function hideprev(~,~)

if strcmp(get(figprev,'visible'),'on');

closepreview(vid);

set(figprev,'visible','off');

end

end

function buttonset(strings,callbacks,varargin)

% strings is a cell array of strings corresponding to the buttons in the

% wizard figure.

% an empty string will hide the associated button

% callbacks is a cell array of callback function handles

exind = find(~strcmpi(strings,'')); % find indices of non-empty strings

set(but,'visible','off');

if nargin>2

intrup = varargin{1};

else

intrup = {'off','off','off','off'};

end

for ji = exind(:).'

set(but(ji),'string',strings{ji},'visible','on','callback',callbacks{ji},'interruptible',

intrup{ji});

end

drawnow expose;

end

function OK(~,~,varargin)

if nargin>2

uiresume(varargin{1});

else

uiresume(figwiz);

end

end

function cancel(~,~,varargin)

if nargin>2

165

exitflag = true;

uiresume(varargin{1});

else

qans = questdlg('Exit the Deflectometer Wizard?','Quit?','No','Yes','Yes');

if strcmp(qans,'Yes');

exitflag = true;

figHandles = findobj('Type','figure');

set(figHandles,'CloseRequestFcn','');

uiresume(figwiz);

delete(figHandles);

close all;

end

end

end

end

%% accessory functions

% function [tax,txt] = axtext(fhan,string,varargin)

% p = inputParser;

% p.KeepUnmatched = true;

% addRequired(p,'fhan',@ishandle);

% addRequired(p,'string',@ischar);

% addParamValue(p,'units','normal',@(x)

any(validatestring(x,{'normalized','inches','centimeters','points','pixels','characters'}

)));

% addParamValue(p,'position',[0.1,0.1,0.8,0.8],@(x) (isnumeric(x) &&

all(size(x)==[1,4])));

% addParamValue(p,'HorizontalAlignment','left',@(x)

any(validatestring(x,{'left','center','right'})));

% addParamValue(p,'VerticalAlignment','bottom',@(x)

any(validatestring(x,{'bottom','middle','top'})));

% parse(p,fhan,string,varargin{:});

%

% textinputs = transpose(fieldnames(p.Unmatched));

% for ij = 1:numel(textinputs)

% textinputs{2,ij} = p.Unmatched.(textinputs{1,ij});

% end

%

% switch p.Results.HorizontalAlignment

% case 'left'

% x = -1;

% case 'center'

% x = 0;

% case 'right'

% x = 1;

% end

% switch p.Results.VerticalAlignment

% case 'bottom'

% y = -1;

% case 'middle'

166

% y = 0;

% case 'top'

% y = 1;

% end

% tax =

axes('parent',p.Results.fhan,'units',p.Results.units,'position',p.Results.position,'xtick

',[],'ytick',[],'xlim',[-1,1],'ylim',[-1,1],'visible','off');

% txt =

text(x,y,p.Results.string,'parent',tax,'HorizontalAlignment',p.Results.HorizontalAlignmen

t,'VerticalAlignment',p.Results.VerticalAlignment,textinputs{:});

% end

DataProcessor.m

This section shows the high level MATLAB code for the fringe reflection deflectometer’s

data processing function. Images of the figures that appear during its execution are

interspersed. The purpose of this section is to illustrate the user interface of the data processing

code. The queried input requires a data file produced by DeflectometerWizard.m.

Get data file from data acquisition

pn0 = 'C:\Users\Dan\Google Drive\Thesis\Fringe Reflection Metrology\Deflectometer

Wizard\Data';

[fn, pn] = uigetfile('*.mat', 'Select a data file', pn0);

if isequal(fn,0)

disp('User selected Cancel')

return;

end

load('-mat', [pn fn]);

str = {'R3mini'; 'R3wrong'; 'R3fixed';'flat'};

[s,v] = listdlg('PromptString','Select a

shape:','SelectionMode','single','ListString',str);

if isequal(v,0)

disp('User selected Cancel')

return;

elseif s==4

shapename_cell = inputdlg({'Dimension in guiding direction (mm)','Dimension in

extrusion direction (mm)'});

shapename = reshape(cellfun(@str2double, shapename_cell),1,2);

else

shapename = str{s};

end

167

Unwrap phase

[unwpx,unwpy,croprect,xyoffset,nmask,wpx,wpy] =

DP_PhaseUnwrapper(v1234lin,h1234lin,linmm,permm);

Figure B.1: Shows vertical and horizontal wrapped phase data and asks user to outline part leaving a buffer of background pixels.

168

Figure B.2: PUMA algorithm unwraps data within region of interest.

169

Figure B.3: Results of noise amplitude analysis.

Figure B.4: Histogram of noise amplitudes. Red line shows cutoff amplitude for what is considered noise.

170

Figure B.5: Left column: noisy data deleted. Right column: noise mask edges smoothed and rouge data pixels mended.

171

Figure B.6: Calculated X and Y positions on the monitor. Shading relates loosely to surface slope in each direction.

172

Edit mask if necessary

Gives opportunity to manually edit noise mask.

v1234lincrop = zeros([size(nmask),5]);

h1234lincrop = v1234lincrop;

for ij = 1:5

v1234lincrop(:,:,ij) = imcrop(v1234lin(:,:,ij),croprect);

h1234lincrop(:,:,ij) = imcrop(h1234lin(:,:,ij),croprect);

end

fidpicrop = imcrop(fidpic,croprect);

nmask = DP_MaskEditor(nmask,v1234lincrop,h1234lincrop,wpx,wpy,unwpx,unwpy,fidpicrop);

unwpxc = unwpx;

unwpxc(~nmask) = nan;

unwpyc = unwpy;

unwpyc(~nmask) = nan;

Find approximate location of data points relative to camera based on ideal shape

[xs,ys,zs,xc,yc,Ropt,Topt,lindist,lmask1,lmask2,shapename] =

DP_placepart3d_edge(CalibrationData.f,CalibrationData.k1,CalibrationData.cxcy,Calibration

Data.dxy,CalibrationData.Rmir,CalibrationData.Tmir,nmask,xyoffset,shapename,imcrop(fidpic

,croprect),fidist);

173

Figure B.7: Sample points from ideal surface shape are overlayed over data from the deflectometer camera. The green and cyan dots on the edge denote the surface points used as fiducial lines pertaining to the part edges. If a caliper tip fiducial is used, the user must pick the location of the caliper tips in the image instead.

Find xyz position of each pixel of data

[surfptsx,surfptsy,surfptsz] =

DP_lsqiter(nmask,unwpx,unwpy,CalibrationData.R,CalibrationData.T,xyoffset,CalibrationData

.cxcy,CalibrationData.dxy,CalibrationData.f,CalibrationData.k1,xs,ys,zs,xc,yc,Ropt,Topt,l

indist,lmask1,lmask2,CalibrationData.Rmir,CalibrationData.Tmir);

174

Figure B.8: Optimization iterations to find calculated XYZ points of the part surface. The value is the average distance from the camera origin. In this case, the initial guess is rather accurate.

175

Figure B.9: Initial guess vs. optimization solution.

Figure B.10: XYZ data for calculated surface.

176

Figure B.11: Principal curvature maps of calculated data. Principal curvature k1 (left), principal curvature k2 (center), Gaussian curvature (right).

Register data to model shape and position to relative to assembled system origin

The user is asked to provide input to position the data with respect to the ideal surface

coordinate system by adjusting translational and rotational variables in a semi-automated

fashion. The surface error from ideal is calculated from the final registered position.

if isnumeric(shapename)

spacing = 0.5;

sampnum = ceil(shapename./spacing);

else

sampnum = [29,200];%[114,200]; % number of samples [along curve, in extrusion

direction]

end

edthresh = 1; % mm, longest allowed edge of triangulation simplex (to get rid of

"webbing")

[xyzreg,xyzfin,Mxyzfin,err,errvec] =

DP_DataRegister3D(surfptsx,surfptsy,surfptsz,shapename,sampnum,edthresh);

177

Figure B.12: Registered data showing error from ideal surface.

Dependency: DW_apodizationgen.m

Generates fringe images at native monitor resolution for accurate display in

deflectometer apparatus.

function [apod,X,Y] = DW_apodizationgen(dir,per,phas,shape,lvls,varargin)

% apodizationgen(siz,dir,per,varargin)

%

% INPUTS

% dir = direction vector of modulation [x,y]

%

% per = period of modulation in mm

%

% phas = phase shift (radians)

%

% shape = modulation shape ('sine'|'sin', 'cosine'|'cos', 'step')

%

% lvls = [low,high] displayed level cuttoff between 0 and 255 for uint8 display (e.g.

[cutdn,cutup])

%

178

% varargin{1} = pixel pitch in mm (scalar or 1x2 vector [x,y])

% or

% varargin{1,2} = physical size of display in mm (x,y)

% need one of these options

%

% OUTPUTS

% X = x spatial coordinates in mm

%

% Y = y spatial coordinates in mm

%

% apod = apodization function

global fs

siz = fs.Size([2,1]); % matrix size [x,y] of apodization file

switch nargin

case 6

% sizmm = siz.*varargin{1};

pp = [1,1].*varargin{1};

case 7

% sizmm = [varargin{1},varargin{2}];

pp = [varargin{1},varargin{2}]./siz;

otherwise

error('Invalid pixel pitch or screen size inputs');

end

% umin = -sizmm(1)/2;

% vmin = -sizmm(2)/2;

% umax = -umin;

% vmax = -vmin;

% x = linspace(umin,umax,siz(1))./siz(1).*(siz(1)-1);

% y = linspace(vmin,vmax,siz(2))./siz(2).*(siz(2)-1);

x = (1:1:siz(1)).*pp(1);

y = (1:1:siz(2)).*pp(2);

[X,Y] = meshgrid(x,y);

dirnorm = dir./norm(dir);

dist = X.*dirnorm(1) + Y.*dirnorm(2); % distance from origin in direction of 'dir'

vector

switch shape

case {'sine','sin'}

apod = sin(2.*pi.*(dist)./per + phas);

apod = uint8((apod+1)./2.*diff(lvls)+lvls(1));

case {'cosine','cos'}

apod = cos(2.*pi.*(dist)./per + phas);

apod = uint8((apod+1)./2.*diff(lvls)+lvls(1));

case 'step'

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apod = uint8((mod(dist+phas/(2*pi)*per,per) > (per/2)).*diff(lvls)+lvls(1));

otherwise

error('Invalid shape input');

end

% % write to LT apodization file

% fn = 'C:\Users\Dan\Documents\Thesis\Fringe Reflection Metrology\TempApod.txt';

% fid = fopen(fn,'w');

% % header = ['MESH: ' num2str(n) ' ' num2str(m) ' ' num2str(umin) ' ' num2str(vmin) ' '

num2str(umax) ' ' num2str(vmax)];

% fprintf(fid,'MESH: %u\t%u\t%0.3f\t%0.3f\t%0.3f\t%0.3f\r\n',n,m,umin,vmin,umax,vmax);

% fclose(fid);

% dlmwrite(fn,apod,'-append','delimiter','\t');

%

% % plot

% figure(1);

% imagesc(x,y,apod);

% set(gca,'YDir','normal');

% hold on;

% plot(0,0,'dr');

% hold off;

% colormap('gray');

% axis equal;

Dependency: DW_CamCalibration.m

Camera calibration routine that finds camera position and orientation relative to

calibration flat coordinate system and intrinsic camera properties (lens focal length and radial

distortion).

function [R,T,f,k1,mert,cxcy] = DW_CamCalibration(CamInfo,TargetInfo,img0,varargin)

%

% Based on paper:

% "A Versatile Camera Calibration Technique for High-Accuracy 3D Machine

% Vision Metrology Using Off-the-Shelf TV Cameras and Lenses", Roger Y.

% Tsai, IEEE Journal of Robotics and Automation, Vol. RA-3, NO.4, August

% 1987

% Note paper confuses units of variables Xd, Yd, Xu, Yu, so they don't line

% up with how they are used in the following code

%

% [R,T,f,k1,mert] = CamCalibration(CamInfo,TargetInfo,Image);

% [R,T,f,k1,mert] = CamCalibration(___,'ParameterName',ParameterValue);

%

% INPUT:

% CamInfo = structure containing camera data, fields: 'dx','Ncx','dy'

% TargetInfo = structure containing target data, fields: 'ptx','pty','names'

% Image = input image of calibration points

180

% INPUT PARAMETERS:

% 'fk1' = [f,k1] if known

% 'cpts' = image coordinates of control points if known

% 'invert' = logical flag of whether to invert image before processing (want black

background)

% 'opense' = imopen structing element i.e. strel('disk',7). if empty then no opening

will be performed (default)

%

% OUTPUT:

% R = Rotation matrix to go from target object plane to image plane

% T = Translation matrix to go from target object plane to image plane

% where [ximg;yimg;zimg] = R*[xtarget;ytarget;ztarget] + T

% f = lens focal length

% k1 = lens distortion coefficient ( X_real = X_distorted*(1+k1*r^2), where r^2 =

X_distorted^2 + Y_distorted^2 )

% mert = merit function value of fit (greatest deviation from control point, in

pixels)

% parse input arguments

p = inputParser;

p.addRequired('CameraInformation',@(x) all(isfield(x,{'dx','Ncx','dy'})));

p.addRequired('TargetInformation',@(x) all(isfield(x,{'ptx','pty','names'})));

p.addRequired('Image',@isnumeric);

p.addParamValue('fk1',[0,0],@(x) isnumeric(x) & numel(x)==2);

p.addParamValue('cpts',[0,0],@(x) isnumeric(x) & size(x,2)==2);

p.addParamValue('invert',true,@(x) numel(x)==1);

p.addParamValue('opense',[],@(x) numel(x)<2); % imopen structing element i.e.

strel('disk',7), if = [], then no opening will be performed

p.parse(CamInfo,TargetInfo,img0,varargin{:});

fk1 = p.Results.fk1;

cpts = p.Results.cpts;

invert = p.Results.invert;

opense = p.Results.opense;

fk1_unknown = all(fk1==0); % true if f and k1 are not provided

cpts_unknown = all(cpts==0); % true if image coordinates are not provided

%%% 1)a)ii) Obtain Ncx, Nfx, dxp, dy according to (6c)-(6h) using information of camera

and frame memory supplied by manufacturer.

dx = CamInfo.dx; % x direction pixel pitch in mm

Ncx = CamInfo.Ncx; % number of pixels in x direction on camera sensor

dy = CamInfo.dy; % y direction pixel pitch in mm

sx = 1; % uncertainty scale factor for x, due to TV camera scanning error and CCD

acquisition timing error

%%% 1)a)i) Grab a frame into the computer frame memory. Detect the row and column number

of each calibration point i. Call it (Xfi, Yfi).

if size(img0,3) == 3;

img = rgb2gray(img0);

181

else

img = img0;

end

imgsiz = size(img);

Nfx = imgsiz(2); % number of pixels in raw captured image

dxp = dx*Ncx/Nfx; % Ncx/Nfx will probably always be unity for the cameras used (not so

for TV cameras from 1987)

%%% 1)a)iii)

Cx = ceil(imgsiz(2)/2);

Cy = ceil(imgsiz(1)/2);

cxcy = [Cx,Cy];

xw = TargetInfo.ptx;

yw = TargetInfo.pty;

names = TargetInfo.names;

gsiz = size(xw);

zw = zeros(gsiz);

numpts = numel(xw);

% plot original image

figimg = pptfig('Original Image', 1); % set(gcf,'name','Original Image');

imshow(img);

aximg = gca;

origxlim = xlim;

origylim = ylim;

% might need some extra steps here before converting to binary (e.g. imopen)

if ~isempty(opense)

if invert

img = imclose(img,opense);

else

img = imopen(img,opense);

end

figopen = pptfig('Morphologically Opened Image', 2);

imshow(img);

figure(figimg);

end

imgbw = im2bw(img, graythresh(img)); % convert to binary image using automatic threshold

value

if invert

imgbw = ~imgbw; % invert image

end

imgcc = bwconncomp(imgbw, 8); % find connected components

ccnum0 = imgcc.NumObjects;

bndsiz = [1,1,imgsiz([2,1])];

if ccnum0 > 2*numpts

mbo = msgbox('Too many objects found. Select a region of interest.');

uiwait(mbo);

figure(figimg);

set(figimg,'name','Choose ROI');

roimask = 0;

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okbut = uicontrol(figimg,'style','pushbutton','units','normal','position',[.45 .01 .1

.08],'string','Continue','callback',@roisel,'fontsize',12,'fontweight','bold');

roih = imrect(aximg);

uiwait(figimg);

delete(okbut);

bndsiz = getPosition(roih);

set(roih,'visible','off');

set(figimg,'name','Original Image');

drawnow;

imgbw = imgbw.*roimask;

imgcc = bwconncomp(imgbw, 8); % find connected components

ccnum0 = imgcc.NumObjects;

end

imgrp = regionprops(imgcc,img,'all'); % find connected component properties (Area,

Centroid, BoundingBox)

% areas = [imgrp.Area]; % get vector of CC areas

% [~, backind] = max(areas); % find large foreground object (originally black background

beyond mirror edges)

bbs = reshape([imgrp.BoundingBox],4,[]).';

bbs(:,3) = bbs(:,1)+bbs(:,3);

bbs(:,4) = bbs(:,2)+bbs(:,4);

bndsiz(3) = bndsiz(1)+bndsiz(3);

bndsiz(4) = bndsiz(2)+bndsiz(4);

backind = find( bbs(:,1)<=(bndsiz(1)+0.5) | bbs(:,3)>=(bndsiz(3)-0.5) |

bbs(:,2)<=(bndsiz(2)+0.5) | bbs(:,4)>=(bndsiz(4)-0.5) ); % find foreground objects that

touch edges of frame

% % display binary image

% imgbw(imgcc.PixelIdxList{backind}) = false; % delete "backind" object

% figure(5); set(gcf,'name','Dot Find');

% imshow(imgbw); % show new binary image

% axbw = gca;

% update connected component and region properties lists

ccnum = ccnum0-numel(backind);

newcc = setdiff(1:ccnum0,backind);

imgcc.NumObjects = ccnum;

imgcc.PixelIdxList = imgcc.PixelIdxList(newcc);

imgrp = imgrp(newcc);

centroids = [imgrp.Centroid];

centroids = reshape(centroids,2,ccnum).';

diams = [imgrp.EquivDiameter];

% plot found points

for ri = 1:ccnum

line(imgrp(ri).ConvexHull(:,1),imgrp(ri).ConvexHull(:,2),'color','g','parent',aximg,'line

width',2);

end

% get input for camera coordinates of calibration points

ccind = zeros(gsiz);

183

cplab = ccind;

txtoff = mean(diams);

Xf = zeros(gsiz);

Yf = Xf;

if cpts_unknown

% poind = [1;gsiz(1);numpts-

gsiz(1)+1;numpts;sub2ind(gsiz,ceil(gsiz(1)/2),ceil(gsiz(2)/2))]; % indicies of initial

colibration points: [top left; bottom left; top right; bottom right; center]

poind = [1;3;7;9;4];

figure(figimg);

for ci = poind(:).'

title(['Click near "' names{ci} '"']);

[Xf0,Yf0] = ginput(1);

[~,ccind(ci)] = min( (centroids(:,1)-Xf0).^2 + (centroids(:,2)-Yf0).^2 );

cplab(ci) =

text(centroids(ccind(ci),1)+txtoff,centroids(ccind(ci),2)+txtoff,names{ci},'parent',aximg

,'color','r');

xlim(origxlim);

ylim(origylim);

end

title('Calibrataion Points');

Xf(poind) = centroids(ccind(poind),1);

Yf(poind) = centroids(ccind(poind),2);

scootcoord = 100.*max([max(abs(Xf(:))),max(abs(Yf(:)))]); % scalar, out-of-the-way

coordinate value (max of x and y) to put invalid centroids so they are not picked later

gooddiam = mean(diams(ccind(poind)));

badlist = unique([find(diams>1.5*gooddiam);find(diams<0.5*gooddiam)]);

centroids(badlist,:) = scootcoord;

% initial calibration with 5 points

Xui = Xf-Cx; % image plane coordinates in px

Yui = Yf-Cy;

%%% 1)a)iv) Compute (Xdi, Ydi) using (6a) and (6b)

Xdi = sx^(-1) .* dxp .* (Xui); % image plane coordinates in mm

Ydi = dy .* (Yui);

%%% 1)b) Compute the five unknowns r1/Ty, r2/Ty, Tx/Ty, r4/Ty, r5/Ty

initlinsysLi = [Ydi(poind).*xw(poind), Ydi(poind).*yw(poind), Ydi(poind), -

Xdi(poind).*xw(poind), -Xdi(poind).*yw(poind)]; % A

initlinsysRi = Xdi(poind); % b

fui = initlinsysLi\initlinsysRi; % solution to: A * x = b where x = fui (see Matlab

"systems of linear equations" help)

%%% 1)c)1) Compute abs(Ty) from r1/Ty, r2/Ty, Tx/Ty, r4/Ty, r5/Ty

Ci = fui([1,2;4,5]);

if any(sum(Ci.^2,1)==0) || any(sum(Ci.^2,2)==0)

Ty2i = 1/sum(Ci(:).^2);

else

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Sri = sum(Ci(:).^2);

Ty2i = (Sri-sqrt(Sri.^2-4*det(Ci)^2))/(2*det(Ci)^2);

end

%%% 1)c)2)i) Pick an object point i whose computer image coordinate (Xfi, Yfi) is

away from the image center (Cx, Cy)

[~,Tytestind0] = max(Ydi(poind).^2+Xdi(poind).^2);

Tytestind = poind(Tytestind0);

%%% 1)c)2)ii) Pick the sign of Ty to be +1

Ty0i = sqrt(Ty2i);

%%% 1)c)2)iii) Compute r1, r2, r4, r5, Tx, x, and y

Ri = zeros(3); % 3D rotation matrix, Ri

Ri(1:2,1:2) = Ci.*Ty0i;

Txi = fui(3)*Ty0i;

xi = Ri(1).*xw(Tytestind) + Ri(4).*yw(Tytestind) + Txi;

yi = Ri(2).*xw(Tytestind) + Ri(5).*yw(Tytestind) + Ty0i;

%%% 1)c)2)iv) Calculate sign of Ty

if sign(xi)==sign(Xdi(Tytestind)) && sign(yi)==sign(Ydi(Tytestind));

Tyi = Ty0i;

else

%%% 1)c)3)i) Adjust calculated values from 1)c)2)iii) if Ty<0

Tyi = -Ty0i;

Ri = -Ri;

Txi = -Txi;

end

%%% 1)c)3)ii) Compute R

si = -sign(Ri(1)*Ri(2)+Ri(4)*Ri(5));

Ri(7) = sqrt(1-Ri(1)^2-Ri(4)^2);

Ri(8) = si*sqrt(1-Ri(2)^2-Ri(5)^2);

Ri(3,:) = cross(Ri(1,:),Ri(2,:));

Ri(3,:) = Ri(3,:)./norm(Ri(3,:));

%%% 2)d) Compute an approximation of f and Tz by ignoring lens distortion

yi = Ri(2).*xw(poind) + Ri(5).*yw(poind) + Tyi;

wi = Ri(3).*xw(poind) + Ri(6).*yw(poind);

if fk1_unknown

soliniti = [yi(:), -dy.*Yui(poind)] \ (wi(:).*dy.*Yui(poind));

finiti = soliniti(1);

Tziniti = soliniti(2);

else

soliniti0 = [yi(:), -dy.*Yui(poind)] \ (wi(:).*dy.*Yui(poind));

finiti = sign(soliniti0(1))*fk1(1);

soliniti = (dy.*Yui(poind)) \ (yi(:).*finiti-wi(:).*dy.*Yui(poind));

Tziniti = soliniti;

end

%%% 1)c)3)iii) IF (f < 0) from 2)d), THEN do the following sign switches

if finiti<0

Ri = Ri.*[1 1 -1;1 1 -1;-1 -1 1];

finiti = -finiti;

185

Tziniti = -Tziniti;

end

%% guess where the rest of calibration points are located on the image plane

% disp(Ri);

[Xftst,Yftst,~] =

world2cam(xw,yw,zw,Ri,[Txi;Tyi;Tziniti],finiti,0,[Cx,Cy],[dxp/sx,dy]);

poindi = poind; % indicies of initial calibration points

poind = (1:numpts).'; % indices of all calibration points

poindn = setdiff(poind,poindi); % indices of just the new calibration points

else

poindn = (1:numpts).';

Xftst = reshape(cpts(:,1),gsiz);

Yftst = reshape(cpts(:,2),gsiz);

end

Xf(poindn) = Xftst(poindn);

Yf(poindn) = Yftst(poindn);

line(Xftst(:),Yftst(:),'linestyle','none','marker','x','color','r','parent',aximg);

% find closest centroid to test points

for ci = poindn(:).'

[~,ccind(ci)] = min( (centroids(:,1)-Xf(ci)).^2 + (centroids(:,2)-Yf(ci)).^2 );

cplab(ci) =

text(centroids(ccind(ci),1)+txtoff,centroids(ccind(ci),2)+txtoff,names{ci},'parent',aximg

,'color','r');

end

Xf(poindn) = centroids(ccind(poindn),1);

Yf(poindn) = centroids(ccind(poindn),2);

line(Xf(:),Yf(:),'linestyle','none','marker','+','color','g','parent',aximg);

%% calibration using all calibration points

% method finding all possible solutions, then averaging results

Xu = Xf-Cx; % focal plane coordinates in px

Yu = Yf-Cy;

%%% 1)a)iv) Compute (Xdi, Ydi) using (6a) and (6b)

Xd = sx^(-1) .* dxp .* (Xu);

Yd = dy .* (Yu);

%%% 1)b) Compute the five unknowns r1/Ty, r2/Ty, Tx/Ty, r4/Ty, r5/Ty

initlinsysL = [Yd(:).*xw(:), Yd(:).*yw(:), Yd(:), -Xd(:).*xw(:), -Xd(:).*yw(:)]; % A

initlinsysR = Xd(:); % b

fu = initlinsysL\initlinsysR; % solution to: A * x = b where x = fu (see 'Matlab systems

of linear equations' help)

%%% 1)c) Compute (rl, ..., r9, Tx, Ty) from (r1/Ty, r2/Ty, Tx/Ty, r4/Ty, r5/Ty):

%%% 1)c)1) Compute abs(Ty) from r1/Ty, r2/Ty, Tx/Ty, r4/Ty, r5/Ty

C = fu([1,2;4,5]);

if any(sum(C.^2,1)==0) || any(sum(C.^2,2)==0)

186

Ty2 = 1/sum(C(:).^2);

else

Sr = sum(C(:).^2);

Ty2 = (Sr-sqrt(Sr.^2-4*det(C)^2))/(2*det(C)^2);

end

%%% 1)c)2)i) Pick an object point i whose computer image coordinate (Xfi, Yfi) is away

from the image center (Cx, Cy)

[~,Tytestind] = max(Yd(:).^2+Xd(:).^2);

%%% 1)c)2)ii) Pick the sign of Ty to be +1

Ty0 = sqrt(Ty2); % col vec

%%% 1)c)2)iii) Compute r1, r2, r4, r5, Tx, x, and y

R = zeros(3); % 3D rotation matrix, Ri

R(1:2,1:2) = C.*Ty0;

Tx = fu(3)*Ty0;

x = R(1).*xw(Tytestind) + R(4).*yw(Tytestind) + Tx;

y = R(2).*xw(Tytestind) + R(5).*yw(Tytestind) + Ty0;

%%% 1)c)2)iv) Calculate sign of Ty

if sign(x)==sign(Xd(Tytestind)) && sign(y)==sign(Yd(Tytestind));

Ty = Ty0;

else

%%% 1)c)3)i) Adjust calculated values from 1)c)2)iii) if Ty<0

Ty = -Ty0;

R = -R;

Tx = -Tx;

end

%%% 1)c)3)ii) Compute R

s = -sign(R(1)*R(2)+R(4)*R(5));

R(7) = sqrt(1-R(1)^2-R(4)^2);

R(8) = s*sqrt(1-R(2)^2-R(5)^2);

R(3,:) = cross(R(1,:),R(2,:));

R(3,:) = R(3,:)./norm(R(3,:));

%%% 2)d) Compute an approximation of f and Tz by ignoring lens distortion

y_i = R(2).*xw(:) + R(5).*yw(:) + Ty;

w_i = R(3).*xw(:) + R(6).*yw(:);

if fk1_unknown

solinit = [y_i(:), -dy.*Yu(:)] \ (w_i(:).*dy.*Yu(:));

finit = solinit(1);

Tzinit = solinit(2);

else

solinit0 = [y_i(:), -dy.*Yu(:)] \ (w_i(:).*dy.*Yu(:));

finit = sign(solinit0(1))*fk1(1);

solinit = (dy.*Yu(:)) \ (y_i(:).*finit-w_i(:).*dy.*Yu(:));

Tzinit = solinit;

end

%%% 1)c)3)iii) IF (f < 0) from 2)d), THEN do the following sign switches

if finit<0

R = R.*[1 1 -1;1 1 -1;-1 -1 1];

187

finit = -finit;

Tzinit = -Tzinit;

end

% calculate euler angles from rotation matrix for each case

theta = atan2(-R(7),sqrt(R(1).^2+R(4).^2)); % pitch (rotation about y)

psi = atan2(R(4),R(1)); % roll (rotation about x)

phi = atan2(R(8),R(9)); % yaw (rotation about z)

%%% 2)e) Compute the exact solution for f, Tz, k1

fTzk10 = [finit,Tzinit,0];

if fk1_unknown % f and k1 are NOT provided, optimize over variables f, Tz, and k1

% options = optimset('plotfcns',{@optimplotx,@optimplotfval,@optimplotstepsize});

% flim = [0, finit*2];

% Tzlim = [0, Tzinit*2];

% fTzk1opt = lsqnonlin(@genmert,fTzk10,[min(flim),min(Tzlim),-

0.1],[max(flim),max(Tzlim),0.1]);

% fTzk1opt = lsqnonlin(@custmert,fTzk10,[-7 -800 -0.5],[-3 -200 0.5],options);

% fTzk1opt = fTzk10;

fTzk1opt = fminsearch(@genmert,fTzk10);

else % f and k1 are provided, optimize over variable Tz only

Tzopt = fminsearch(@(x) genmert([fTzk10(1),x,fTzk10(3)]), fTzk10(2));

fTzk1opt = fTzk10;

fTzk1opt(2) = Tzopt;

end

f = fTzk1opt(1);

Tz = fTzk1opt(2);

k1 = fTzk1opt(3);

% mert = eq8b(fTzk1opt);

[mert0,mert0i] = genmert(fTzk10);

[mert,merti] = genmert(fTzk1opt);

T = [Tx;Ty;Tz];

[Xfin,Yfin,xyzfin] = world2cam(xw,yw,zw,R,T,f,k1,[Cx,Cy],[dxp/sx,dy]);

% % plot camera and calibration points in 3D

% fname = '3D Plot';

% fogm = findobj('-regexp','Name',fname,'type','figure');

% if isempty(fogm);

% fig3d = figure(); % create figure

% else

% figure(fogm(1));

% fig3d = clf('reset'); % clear previous figure

% end

% set(fig3d,'Name',fname);

% cpts3 = plot3(xyzfin(:,1),xyzfin(:,2),xyzfin(:,3),'ob');

% camlin3 = line([0;0],[0;0],[0;mean(xyzfin(:,3))],'linestyle','-

','color','k','linewidth',2);

% xlabel('X');ylabel('Y');zlabel('Z');

% buff = [0.5 0.5 0.2];

188

% limt = zeros(3,2);

% limt(1,:) = get(gca,'xlim');

% limt(2,:) = get(gca,'ylim');

% limt(3,:) = get(gca,'zlim');

% line([limt(1,1)-buff(1)*diff(limt(1,:));limt(1,2)+buff(1)*diff(limt(1,:))],...

% [limt(2,1)-buff(2)*diff(limt(2,:));limt(2,2)+buff(2)*diff(limt(2,:))],...

% [limt(3,1)-buff(3)*diff(limt(3,:));limt(3,2)+buff(3)*diff(limt(3,:))],...

% 'linestyle','none','marker','.','color','w');

% axis equal;

% plot final points

figure(figimg);

blucirc = line(Xfin(:),Yfin(:),'linestyle','none','marker','o','color','b');

errsca = 100;

errhi = zeros(size(Xfin));

errh = errhi;

for cp = 1:numpts

set(cplab(cp),'color','b','position',[Xfin(cp),Yfin(cp)]+txtoff);

errhi(cp) = line([Xf(cp);Xf(cp)+errsca*(Xftst(cp)-

Xf(cp))],[Yf(cp);Yf(cp)+errsca*(Yftst(cp)-Yf(cp))],'color','r','linewidth',2);

errh(cp) = line([Xf(cp);Xf(cp)+errsca*(Xfin(cp)-

Xf(cp))],[Yf(cp);Yf(cp)+errsca*(Yfin(cp)-Yf(cp))],'color','b','linewidth',2);

end

errlabi = text(Xf(mert0i)+errsca*(Xftst(mert0i)-

Xf(mert0i)),Yf(mert0i)+errsca*(Yftst(mert0i)-

Yf(mert0i)),num2str(mert0,3),'fontweight','bold','color','r');

errlab = text(Xf(merti)+errsca*(Xfin(merti)-Xf(merti)),Yf(merti)+errsca*(Yfin(merti)-

Yf(merti)),num2str(mert,3),'fontweight','bold','color','b');

if true

else

% create manual adjustment control figure

fname = 'Manual Controls';

fpos = [200 200 800 400];

fogm = findobj('-regexp','Name',fname,'type','figure');

if isempty(fogm);

fhan = figure(); % create figure

else

figure(fogm(1));

fhan = clf('reset'); % clear previous figure

end

set(fhan,'Position',fpos,'color','w','units','pixels','Name',fname);

% create 3 UIpanels for T (Tx,Ty,Tz), R (roll, pitch, yaw), and (f,k1)

pan(1) = uipanel(fhan,'units','normal','position',[0.05 0.2 0.2875

0.75],'title','Translation','backgroundcolor','w');


0.75],'title','Rotation','backgroundcolor','w');


0.75],'title','Focal Length and Distortion','backgroundcolor','w');

% create sliders, edit boxes, and titles

slidenames = {{'Tx','Ty','Tz'},{'Roll','Pitch','Yaw'},{'f','Distortion (k1)'}};

189

def = {[Tx,Ty,Tz],[psi,theta,phi].*180/pi,[f,k1]}; % default starting positions

rnghi = cell(size(def));

rnglo = rnghi;

rngt = 20; % plus or minus range for T

rngf = 1; % plus or minus range for f

rngk = 0.02; % plus or minus range for k1

% rnghi{1} = (1+sign(def{1}).*rngfactor).*def{1};

% rnglo{1} = (1-sign(def{1}).*rngfactor).*def{1};

rnghi{1} = def{1}+rngt;

rnglo{1} = def{1}-rngt;

rnghi{2} = 180.*ones(size(def{2}));

rnglo{2} = -rnghi{2};

% rnghi{3} = (1+sign(def{3}).*rngfactor).*def{3};

% rnglo{3} = (1-sign(def{3}).*rngfactor).*def{3};

rnghi{3} = def{3}+[rngf,rngk];

rnglo{3} = def{3}-[rngf,rngk];

numc = cellfun(@numel,def); % number of control groups in each panel

numcc = cumsum(numc);

numcc = [0,numcc(1:end-1)];

slidehan = zeros(1,sum(numc));

edithan = slidehan;

for ip = 1:numel(numc)

ppos = getpixelposition(pan(ip));

hpos = ppos(3).*linspace(1/numc(ip)/2,1-1/numc(ip)/2,numc(ip));

wid = ppos(3)/numc(ip);

for ic = 1:numc(ip)

slidehan(numcc(ip)+ic) =

uicontrol(pan(ip),'style','slider','position',[hpos(ic)-10 10 20 ppos(4)-

85],'min',rnglo{ip}(ic),'max',rnghi{ip}(ic),'value',def{ip}(ic),'callback',@slidecall);

edithan(numcc(ip)+ic) =

uicontrol(pan(ip),'style','edit','position',[hpos(ic)-(wid-10)/2, ppos(4)-65, wid-10,

20],'string',num2str(def{ip}(ic)),'callback',@editcall);

uicontrol(pan(ip),'style','text','position',[hpos(ic)-(wid-10)/2, ppos(4)-40,

wid-10, 20],'string',slidenames{ip}{ic},'backgroundcolor','w');

end

end

set(slidehan(4:6),'sliderstep',[1/360, 1/60]);

axmert = axes('parent',fhan,'units','normal','position',[0.05 0.01 0.7

0.18],'xtick',[],'ygrid','on');

merthist = ones(30,1).*mert;

mertlin = line('xdata',(0:(numel(merthist)-

1)).','ydata',merthist,'color','b','linewidth',2,'parent',axmert);

mertxt = uicontrol(fhan,'style','text','units','normal','position',[0.76 0.01 0.13

0.18],'backgroundcolor','w','horizontalalignment','left','string',num2str(mert),'fontsize

',20,'fontweight','bold');

updateplot();

end

% save_to_base(1);

%% accessory functions

190

function [delt] = eq8b(fTzk1)

f0 = fTzk1(1);

Tz0 = fTzk1(2);

k10 = fTzk1(3);

r = sqrt( (dxp.*Xu./sx).^2 + (dy.*Yu).^2 );

LHS = dy.*Yu + dy.*Yu.*k10.*r.^2;

RHS = f0 .* (R(2).*xw+R(5).*yw+R(8).*zw+Ty) ./ (R(3).*xw+R(6).*yw+R(9).*zw+Tz0);

delt = LHS(:)-RHS(:);

end

function [delt] = custmert(fTzk1)

f0 = fTzk1(1);

Tz0 = fTzk1(2);

k10 = fTzk1(3);

r = sqrt( (dxp.*Xu./sx).^2 + (dy.*Yu).^2 );

LHSy = dy.*Yu + dy.*Yu.*k10.*r.^2;

RHSy = f0 .* (R(2).*xw+R(5).*yw+R(8).*zw+Ty) ./ (R(3).*xw+R(6).*yw+R(9).*zw+Tz0);

LHSx = sx^(-1).*dxp.*Xu + sx^(-1).*dxp.*Xu.*k10.*r.^2;

RHSx = f0 .* (R(1).*xw+R(4).*yw+R(7).*zw+Tx) ./ (R(3).*xw+R(6).*yw+R(9).*zw+Tz0);

delt = sqrt((LHSy(:)-RHSy(:)).^2+(LHSx(:)-RHSx(:)).^2);

end

function [delt,delti] = genmert(fTzk1,varargin)

f0 = fTzk1(1);

Tz0 = fTzk1(2);

k10 = fTzk1(3);

if nargin>1

T0 = [varargin{1}(:);Tz0];

else

T0 = [Tx;Ty;Tz0];

end

[Xup,Yup,~] = world2cam(xw,yw,zw,R,T0,f0,k10,[Cx,Cy],[dxp/sx,dy]);

[delt,delti] = max(sqrt((centroids(ccind(:),1)-Xup(:)).^2 +

(centroids(ccind(:),2)-Yup(:)).^2));

end

function slidecall(~,~)

vals0 = get(slidehan,'value');

vals = cellfun(@num2str,vals0,'UniformOutput',false);

set(edithan,{'string'},vals);

T = cell2mat(vals0(1:3));

angs = cell2mat(vals0(4:6)).*pi./180;

R = [ cos(angs(1))*cos(angs(2))

sin(angs(1))*cos(angs(2)) -sin(angs(2)) ;...

-sin(angs(1))*cos(angs(3))+cos(angs(1))*sin(angs(2))*sin(angs(3))

cos(angs(1))*cos(angs(3))+sin(angs(1))*sin(angs(2))*sin(angs(3))

cos(angs(2))*sin(angs(3));...

sin(angs(1))*sin(angs(3))+cos(angs(1))*sin(angs(2))*cos(angs(3)) -

cos(angs(1))*sin(angs(3))+sin(angs(1))*sin(angs(2))*cos(angs(3))

cos(angs(2))*cos(angs(3))];

191

f = vals0{7};

k1 = vals0{8};

updateplot();

end

function editcall(~,~)

vals0 = get(edithan,'string');

vals = cellfun(@str2num,vals0);

mins = [vals(1:3)-rngt;-180.*ones(3,1);vals(7)-rngf;vals(8)-rngk];

maxs = [vals(1:3)+rngt;180.*ones(3,1);vals(7)+rngf;vals(8)+rngk];

for ij = 1:numel(slidehan)

set(slidehan(ij),'value',vals(ij),'min',mins(ij),'max',maxs(ij));

end

T = vals(1:3);

angs = vals(4:6).*pi./180;

R = [ cos(angs(1))*cos(angs(2))

sin(angs(1))*cos(angs(2)) -sin(angs(2)) ;...

-sin(angs(1))*cos(angs(3))+cos(angs(1))*sin(angs(2))*sin(angs(3))

cos(angs(1))*cos(angs(3))+sin(angs(1))*sin(angs(2))*sin(angs(3))

cos(angs(2))*sin(angs(3));...

sin(angs(1))*sin(angs(3))+cos(angs(1))*sin(angs(2))*cos(angs(3)) -

cos(angs(1))*sin(angs(3))+sin(angs(1))*sin(angs(2))*cos(angs(3))

cos(angs(2))*cos(angs(3))];

f = vals(7);

k1 = vals(8);

updateplot();

end

function updateplot()

[Xup,Yup,xyzup] = world2cam(xw,yw,zw,R,T,f,k1,[Cx,Cy],[dxp/sx,dy]);

set(blucirc,'xdata',Xup(:),'ydata',Yup(:));

for ck = 1:numpts

set(cplab(ck),'position',[Xup(ck),Yup(ck)]+txtoff);

end

mert = genmert([f,T(3),k1],T(1:2));

merthist = [merthist(2:end);mert];

set(mertlin,'ydata',merthist);

set(mertxt,'string',num2str(mert));

set(cpts3,'xdata',xyzup(:,1),'ydata',xyzup(:,2),'zdata',xyzup(:,3));

% set(camlin3,'zdata',[0;mean(xyzup(:,3))]);

end

function [Xw2c,Yw2c,xyzw2c] = world2cam(xw,yw,zw,R,T,f,k1,Cxy,dxy)

% INPUT:

% xw,yw,zw are world coordinates (zw(:) = 0 for planar coordinate points)

% R is 3x3 rotation matrix (world to camera)

% T is three element translation vector [Tx;Ty;Tz] (world to camera)

% f is lens focal length

% k1 is r^2 coefficient of distortion

% Cxy is two element vector [x,y] of pixel position of optical center in computer

frame memory

192

% dxy is two element vector [x,y] of pixel pitch (units of length)

% OUTPUT:

% Xw2c is an array of X pixel coordinates on camera sensor of real world points,

the same size as xw input

% Yw2c is an array of Y pixel coordinates on camera sensor of real world points,

the same size as xw input

% xyzw2c is an array of [x1,y1,z1;x2,y2,z2;...] 3D camera coordinates of world

points

camcoordsw2c = R*[xw(:).'; yw(:).'; zw(:).'] + repmat(T(:),1,numel(xw));

xyzw2c = camcoordsw2c.';

xw2c = reshape(camcoordsw2c(1,:),size(xw));

yw2c = reshape(camcoordsw2c(2,:),size(xw));

zw2c = reshape(camcoordsw2c(3,:),size(xw));

Xdw2c0 = f.*xw2c./zw2c;

Ydw2c0 = f.*yw2c./zw2c;

rsqw2c = Xdw2c0.^2 + Ydw2c0.^2;

Xdw2c = Xdw2c0./(1+k1.*rsqw2c);

Ydw2c = Ydw2c0./(1+k1.*rsqw2c);

Xw2c = dxy(1)^(-1).*Xdw2c + Cxy(1);

Yw2c = dxy(2)^(-1).*Ydw2c + Cxy(2);

end

function [out] = inbounds(in,dir) % dir = 'x' or 'y'

switch dir

case 'x'

dfd = 2; %dimension from 'dir' input

case 'y'

dfd = 1;

end

out = in + max(1-min(in(:)),0);

out = out + min(imgsiz(dfd)-max(in(:)),0);

end

function [] = continuebutton(~,~)

% set(prevnextbut,'enable','off');

set(contbut,'enable','off');

set(prevnextbut,'visible','off');

set(contbut,'visible','off');

end

function roisel(~,~)

% set([okbut,roih],'visible','off');

% drawnow;

roimask = createMask(roih);

uiresume;

end

end

193

Dependency: DW_DeflectometerCalibration.m

Deflectometer calibration routine that finds relative position and orientation of camera

and display.

function [f,k1,R,T,Rmir,Tmir,cxcy,dxy,xflag] = DW_DeflectometerCalibration(txwiz,xbut)

global vid fs;

fsfig = fs.FigureHandle;

fsax = fs.AxisHandle;

fsimg = fs.ImageHandle;

fssiz = fs.Size;

% set output default values

f=0; k1=0; R=0; T=0; Rmir=0; Tmir=0; cxcy=0; dxy=0;

xflag = false;

set(xbut,'callback',@exitnow);

% Capture image of mirror calibration points


drawnow expose;

mirrorimg = getsnapshot(vid);

% Calibrate camera to mirror calibration points

sensor = camerasensordatabase('Marlin');

dxy = [sensor.dx,sensor.dy];

target = targetcoorddatabase('mirror');

set(txwiz,'string','Finding mirror position and lens specs. Point out mirror control

points on image.');

[Rmir,Tmir,f,k1,~,cxcy] =

DW_CamCalibration(sensor,target,mirrorimg,'opense',strel('disk',7));

if xflag

return;

end

% Interactively create control points on monitor

discpoint = nan(16);

[Xpointergen,Ypointergen]=meshgrid(linspace(-7.5,7.5,16),linspace(-7.5,7.5,16));

discpoint(sqrt(Xpointergen.^2+Ypointergen.^2)<=7.6) = 2; % disk pointer icon

pointercenter = [1,1];

[yfill,xfill] = find(~isnan(discpoint));

yfill = yfill - pointercenter(1)+1;

xfill = xfill - pointercenter(2)+1;

pointercenterf = pointercenter.*[-1,1]+[17,0]; % flip y direction of pointer center since

picture axis y direction is un-reversed

194

set(fsfig,'pointer','custom','pointershapecdata',discpoint,'pointershapehotspot',pointerc

enterf);

% set(fsimg,'hittest','off');

% set(fsax,'hittest','on');

% capture initial comparison frame

blackscreen = zeros(fssiz);

blackscreen(1) = 1;

% %%%

% blackscreen(105.5+[-8.5,8.5],:) = 1;

% blackscreen(:,200.5+[-8.5,8.5]) = 1;

% %%%

fsdisp = blackscreen; % CData to display on monitor

set(fsimg,'cdata',fsdisp); % turn screen to black (with one white pixel for contrast)

initimg = getsnapshot(vid);

drawnow expose;

if xflag

return;

end

% set up point coordinate databases and control figure

moncoords = [];

imgcoords = [];

figcon = pptfig('Control Point Selector',1);

set(figcon,'position',[100,100,fliplr(round(size(initimg)./[2,4/3]))]); % rightmost 2/3

of figure is for image preview

set(figcon,'DefaultUicontrolFontSize',14,'DefaultUicontrolFontWeight','bold');

prevax = axes('parent',figcon,'units','normal','position',[1/3,0,2/3,1],'visible','off');

pancon = uipanel(figcon,'units','normal','position',[0,0,1/3,1],'title','Control Panel');

preimg = image(ones(size(initimg)),'parent',prevax);

monptlist =

uicontrol(pancon,'style','listbox','units','normal','position',[0.05,0.4,0.425,0.5],'min'

,0,'max',10,'value',[],'callback',@listpoint);

imgptlist =

uicontrol(pancon,'style','listbox','units','normal','position',[0.525,0.4,0.425,0.5],'min

',0,'max',10,'value',[],'callback',@listpoint);

delbut =

uicontrol(pancon,'style','pushbutton','units','normal','position',[0.1,0.05,0.35,0.1],'st

ring','Delete Point','callback',@deletepoint);

contbut =

uicontrol(pancon,'style','pushbutton','units','normal','position',[0.55,0.05,0.35,0.1],'s

tring','Continue (5)','enable','on','callback',@contin);

checkx =

uicontrol(pancon,'style','checkbox','units','normal','position',[0.1,0.20,0.8,0.05],'stri

ng','Reverse X Axis','min',0,'max',1,'value',0,'callback',@(~,~) updatecontrols([]));

checky =

uicontrol(pancon,'style','checkbox','units','normal','position',[0.1,0.25,0.8,0.05],'stri

ng','Reverse Y Axis','min',0,'max',1,'value',1,'callback',@(~,~) updatecontrols([]));

selpt =

line('xdata',0,'ydata',0,'parent',prevax,'marker','x','linestyle','none','color','r','vis

ible','off');

195

updatecontrols([]);

if xflag

return;

end

% pick points with mouse

preview(vid,preimg);

set(fsimg,'ButtonDownFcn',@addpoint,'interruptible','on');

set(txwiz,'string','Draw a minimum of five(5) control points on monitor where camera can

view them.');

uiwait(figcon);

stoppreview(vid);

close(figcon);

if xflag

return;

end

monimg = getsnapshot(vid);

set(txwiz,'string','Calculating monitor position...');

% Calibrate camera to monitor (reflection) control points for extrinsic parameters (R,T)

only

monpp = [0.258,0.258]; % monitor pixel pitch, [x,y] mm/pixel

monptnames = arrayfun(@(x) num2str(x), (1:size(moncoords,1)).', 'UniformOutput', false);

monitarget =

struct('ptx',moncoords(:,1).*monpp(1),'pty',moncoords(:,2).*monpp(2),'names',[]);

monitarget.names = monptnames;

[Rmon,Tmon,~,~,~] =

DW_CamCalibration(sensor,monitarget,monimg,'fk1',[f,k1],'cpts',imgcoords,'invert',false);

%,'opense',strel('disk',7));

if xflag

return;

end

% Find monitor position relative to camera using mirror position and monitor reflected

image postion

invz = [1 0 0; 0 1 0; 0 0 -1];

Rmon = Rmon*invz; % invz is to create left handed coordinate system for the monitor

reflection

R = Rmir*invz*Rmir.'*Rmon;

T = Rmir*invz*Rmir.'*(Tmon-Tmir) + Tmir;

if xflag

return;

end

% plot setup in 3D

fig3d = pptfig('3D Render',1);

196

siz = size(initimg);

ax3d = axes;

axis equal;

% camera graphic

mx = sensor.dx*siz(2)/2/f;

my = sensor.dy*siz(1)/2/f;

camsca = 100;

camverts0 = [[-mx -my 1;... % bottom left corner

mx -my 1;... % bottom right corner

mx my 1;... % top right corner

-mx my 1].*camsca;... % top left corner, scaled

0 0 0;... % origin

eye(3) ]; % unit vectors

% take camera vertices from camera reference frame to mirror reference frame

camverts = (Rmir.'*(camverts0.'-repmat(Tmir,1,size(camverts0,1)))).';

camfaces = [1 2 5;...

2 3 5;...

3 4 5;...

4 1 5];

patch('vertices',camverts,'faces',camfaces,'facecolor','k','facealpha',0.3,'edgecolor','k

');

cosysplot(camverts(5:8,:),50);

% monitor graphic

mx = fssiz(2).*monpp(1);

my = fssiz(1).*monpp(2);

monverts0 = [0 0 0;... % corners

0 my 0;...

mx my 0;...

mx 0 0;...

0 0 0;... % origin

eye(3) ]; % unit vectors

% take monitor vertices from monitor reference frame to camera reference frame

monverts1 = R*monverts0.'+repmat(T,1,size(monverts0,1));

% take monitor vertices from camera reference frame to mirror reference frame

monverts = (Rmir.'*(monverts1-repmat(Tmir,1,size(monverts0,1)))).';

patch('vertices',monverts,'faces',[1 2 3

4],'facecolor','b','facealpha',0.5,'edgecolor','k');

xyzlin = cosysplot(monverts(5:8,:),50);

legend(xyzlin,{'X','Y','Z'});

% reflected monitor graphic

% take monitor vertices from monitor reference frame to camera reference frame

monvertsr1 = Rmon*monverts0.'+repmat(Tmon,1,size(monverts0,1));

% take monitor vertices from camera reference frame to mirror reference frame

monvertsr = (Rmir.'*(monvertsr1-repmat(Tmir,1,size(monverts0,1)))).';

patch('vertices',monvertsr,'faces',[1 2 3

4],'facecolor','b','facealpha',0.3,'edgecolor','k','linestyle','--');

cosysplot(monvertsr(5:8,:),50);

197

% mirror graphic

ang = linspace(0,2*pi,5*4+1).';

ang = ang(1:end-1);

dotx = cos(ang);

doty = sin(ang);

dotz = zeros(size(ang));

for di = 1:numel(target.ptx)

patch(dotx+target.ptx(di),doty+target.pty(di),dotz,dotz,'facecolor','k');

end

cosysplot([zeros(1,3);eye(3)],50);

% viewing area graphic

[xv,yv] =

atz0(camverts(5,1),camverts(5,2),camverts(5,3),camverts(1:4,1),camverts(1:4,2),camverts(1

:4,3));

line('xdata',xv([1;2;3;4;1]),'ydata',yv([1;2;3;4;1]),'zdata',zeros(5,1),'linestyle','--

','color','k');

set(ax3d,'zdir','reverse','ydir','reverse');

%% Nested Functions/Callbacks

function exitnow(~,~)

xflag = true;

end

function addpoint(~,~)

cp = get(fsax,'currentpoint'); % get the position of the mouse click

set(fsax,'hittest','off'); % make axis inactive

set(fsfig,'pointershapecdata',nan(16)); % hide cursor

cpx = round(cp(1,1));

cpy = round(cp(1,2));

% %%%

% disp(cp);

% disp([cpx,cpy]);

% save_to_base(1); pause(1);

% %%%

fillinds = sub2ind(fssiz,yfill+cpy-1,xfill+cpx-1); % indices to "paint" the area

where the cursor was when clicked

fsdisp(fillinds) = 255;

set(fsimg,'cdata',fsdisp); % update display data

drawnow expose; pause(0.5); % update graphics

initimg0 = initimg; % store previous image

initimg = getsnapshot(vid); % capture new image

imgdiff = initimg-initimg0; % image difference

xsignal = sum(imgdiff,1); % 1-D image difference in X direction

xsignal = xsignal./max(xsignal); % normalized

ysignal = sum(imgdiff,2); % 1-D image difference in Y direction

ysignal = ysignal./max(ysignal); % normalized

[~,xloc] = findpeaks(xsignal,'SortStr','descend','NPeaks',1); % finds tallest

peak location in X signal

198

[~,yloc] = findpeaks(ysignal,'SortStr','descend','NPeaks',1); % finds tallest

peak location in Y signal

imgcoords(end+1,:) = [xloc,yloc]; % approximate coordinates of new spot in camera

frame

moncoords(end+1,:) = [cpx,cpy]-pointercenter([2,1])+8.5; % exact coordinates of

control point on monitor

updatecontrols(size(moncoords,1));

set(fsax,'hittest','on'); % make axis active

set(fsfig,'pointershapecdata',discpoint); % show cursor

if xflag

uiresume(figcon);

end

end

function updatecontrols(selind) % input is logical flag whether to change the

selected values of the list boxes

if isempty(moncoords)

monptstr = {''};

imgptstr = {''};

else

monptstr = arrayfun(@(x,y) sprintf('(%0.1f, %0.1f)',x,y), moncoords(:,1),

moncoords(:,2), 'UniformOutput', false);

imgptstr = arrayfun(@(x,y) sprintf('(%0.1f, %0.1f)',x,y), imgcoords(:,1),

imgcoords(:,2), 'UniformOutput', false);

end

set(monptlist, 'string', monptstr);

set(imgptlist, 'string', imgptstr);

if size(moncoords,1) < 5

set(contbut,'string',['Continue (' int2str(5-size(moncoords,1))

')'],'enable','off');

else

set(contbut,'string','Continue','enable','on');

end

% selind = get(monptlist,'value');

set([monptlist,imgptlist], 'value', selind);

if isempty(selind) || all(selind==0)

set(selpt,'visible','off');

else

set(selpt,'visible','on','xdata',imgcoords(selind,1),'ydata',imgcoords(selind,2));

end

if get(checkx,'value')>0

set(prevax,'xdir','reverse');

else

set(prevax,'xdir','normal');

end

if get(checky,'value')>0

set(prevax,'ydir','normal');

else

set(prevax,'ydir','reverse');

end

199

end

function deletepoint(~,~)

delind = get(monptlist,'value');

moncoords(delind,:) = [];

imgcoords(delind,:) = [];

set(monptlist,'value',[]);

updatecontrols([]);

end

function listpoint(src,~)

selind = get(src,'value');

updatecontrols(selind);

end

function contin(~,~)

uiresume(figcon);

end

function linh = cosysplot(pts,sca) % pts must contain 4 row unit vectors

[origin;X_hat;Y_hat;Z_hat], sca is coordinate glyph scale

diffs = pts(2:4,:)-repmat(pts(1,:),3,1);

pts(2:4,:) = (diffs.*sca)+repmat(pts(1,:),3,1);

linh = zeros(3,1);

linh(1) =

line('xdata',pts([1,2],1),'ydata',pts([1,2],2),'zdata',pts([1,2],3),'color','r','linewidt

h',3);

linh(2) =

line('xdata',pts([1,3],1),'ydata',pts([1,3],2),'zdata',pts([1,3],3),'color','g','linewidt

h',3);

linh(3) =

line('xdata',pts([1,4],1),'ydata',pts([1,4],2),'zdata',pts([1,4],3),'color','b','linewidt

h',3);

end

function [x,y] = atz0(x1,y1,z1,x2,y2,z2)

% parameterized equation of line in 3D

% x = (x2-x1)*t + x1

% y = (y2-y1)*t + y1

% z = (z2-z1)*t + z1

t = (0-z1)./(z2-z1); % t at z=0

x = (x2-x1).*t + x1;

y = (y2-y1).*t + y1;

end

end

Dependency: DW_ExposureAdjust.m

Camera exposure adjustment routine.

200

function [figcon] = DW_ExposureAdjust(maxval)

global vid src fs

% Create full screen display figure

assignin('base','fs_in_wxpadj',fs);


fssiz = fs.Size;


caxis(fsax,[0 255]); % 8-bit color data

cdat = maxval.*ones(size(fssiz));


drawnow expose;

src = getselectedsource(vid);

% Marlin

expvar = 'Shutter'; % video source setting name to change

explims = [1,4095]; % slider limits

expsteps = [10,100]./diff(explims); % slider step sizes

extvar = 'ExtendedShutter'; % video source setting name to change

extlims = [0 67108863]; % slider limits

extsteps = [0.01, 0.1]; % slider step sizes

% % Webcam

% set(src,'ExposureMode','Manual');

% set(vid,'FramesPerTrigger',1);

% expvar = 'Exposure';

% explims = [-9,-1];

% steps = [1,3]./diff(explims);

set(vid,'FramesPerTrigger',5);

shutter = get(src,expvar);

set(src,expvar,shutter);

extshutter = get(src,extvar);

set(src,extvar,extshutter);

% create control figure

figcon = pptfig('Exposure Control',1);

set(figcon,'position',[100,100,round(get(vid,'VideoResolution')./[4/1.25,4/2])]); %

leftmost 4/5 of figure is for image preview and histogram

set(figcon,'DefaultUicontrolFontSize',14,'DefaultUicontrolFontWeight','bold');

imgpan = uipanel(figcon,'units','normal','position',[0,0.5,0.8,0.5],'title','Image

Preview');

histpan = uipanel(figcon,'units','normal','position',[0,0,0.8,0.5],'title','Histogram');

conpan = uipanel(figcon,'units','normal','position',[0.8,0,0.2,1],'title','Shutter');

imgax = axes('parent',imgpan,'units','normal','position',[0,0,1,1],'visible','off');

histax = axes('parent',histpan);

grayred = gray(256);

grayred(end,:) = [1 0 0]; % make saturated pixels red

201

colormap(grayred);

shlider =

uicontrol(conpan,'style','slider','units','normal','position',[0.1,0.1,0.3,0.7],...

'min',explims(1),'max',explims(2),'value',shutter,'sliderstep',expsteps,...

'callback',@shlidercall);

shedit =

uicontrol(conpan,'style','edit','units','normal','position',[0,0.85,0.5,0.05],'string',in

t2str(shutter),'callback',@sheditcall,'fontsize',11);

extshlider =

uicontrol(conpan,'style','slider','units','normal','position',[0.6,0.1,0.3,0.7],...

'min',extlims(1),'max',extlims(2),'value',extshutter,'sliderstep',extsteps,...

'callback',@extshlidercall);

extshedit =

uicontrol(conpan,'style','edit','units','normal','position',[0.5,0.85,0.5,0.05],'string',

int2str(extshutter),'callback',@extsheditcall,'fontsize',11);

imgh = imagesc(ones(fliplr(get(vid,'VideoResolution'))),'parent',imgax);

set(imgax,'visible','off');

updatepic();

function updatepic()

set([shlider,shedit,extshlider,extshedit],'enable','off');

set(shlider,'value',shutter);

set(shedit,'string',int2str(shutter));

set(extshlider,'value',extshutter);

set(extshedit,'string',int2str(extshutter));

drawnow expose;

set(src,expvar,shutter);

set(src,extvar,extshutter);

start(vid);

pic = mean(getdata(vid),4);

pic = mean(pic,3); % in case image is in color

pic = pic./255; % 8-bit image

caxis(imgax,[0,1]);

set(imgh,'cdata',pic);

hist(histax,pic(:),256);

set([shlider,shedit,extshlider,extshedit],'enable','on');

drawnow expose;

end

function shlidercall(~,~)

shutter = round(get(shlider,'value'));

shutter = min([shutter,explims(2)]);

shutter = max([shutter,explims(1)]);

updatepic();

end

function sheditcall(~,~)

202

shutter = round(str2double(get(shedit,'string')));

shutter = min([shutter,explims(2)]);

shutter = max([shutter,explims(1)]);

updatepic();

end

function extshlidercall(~,~)

extshutter = round(get(extshlider,'value'));

extshutter = min([extshutter,extlims(2)]);

extshutter = max([extshutter,extlims(1)]);

updatepic();

end

function extsheditcall(~,~)

extshutter = round(str2double(get(extshedit,'string')));

extshutter = min([extshutter,extlims(2)]);

extshutter = max([extshutter,extlims(1)]);

updatepic();

end

end

Dependency: DW_FocusAdjust.m

Manual camera focus adjustment routine based on user defined regions of interest.

function DW_FocusAdjust(figimg,txwiz,donebut)

% Provides focus merit functions in real time based on selected ROIs for

% manual focusing of camera lens

global vid fs

% get frame

img = getsnapshot(vid);

imgsiz = size(img);

imgsiz = imgsiz(1:2); % if video input is rgb

% display frame

figure(figimg);

set(figimg,'Renderer','opengl')

imgprev = imshow(img,gray(256));

colorbar;

aximg = gca;


% choose ROI(s)

set(txwiz,'string','Choose region(s) of interest.');

roicolors = repmat(get(0,'DefaultAxesColorOrder'),[5,1]);

figure(figimg);

roih = {};

203

okbut = uicontrol(figimg,'style','pushbutton','units','normal','position',[.51 .01 .19

.06],'string','Finish','callback',@roifinish,'fontsize',12,'fontweight','bold');

addbut = uicontrol(figimg,'style','pushbutton','units','normal','position',[.30 .01 .19

.06],'string','Another ROI','callback',@roisel,'fontsize',12,'fontweight','bold');

roisel(0,0);

set([okbut,addbut],'visible','off');

ROI(numel(roih)) = struct;

for ri = 1:numel(roih)

% ROI(ri).handle = roih(ri);

ROI(ri).position = getPosition(roih{ri});

ROI(ri).mask = createMask(roih{ri});

% [riX,riY] = meshgrid(linspace(-1,1,ROI(ri).position(3))./sqrt(2),linspace(-

1,1,ROI(ri).position(4))./sqrt(2));

[riX,riY] = meshgrid(linspace(-1,1,ROI(ri).position(3)),linspace(-

1,1,ROI(ri).position(4)));

ridist = sqrt(riX.^2+riY.^2);

[ridistunique,~,revorder] = unique(ridist(:));

ROI(ri).centerdist = ridistunique;

pixind = arrayfun(@(ind) find(revorder==ind), 1:numel(ridistunique),

'UniformOutput',false);

ROI(ri).pixelindices = pixind;

ROI(ri).size = size(ridist);

end

drawnow;

% create focus feedback figure

figfoc = pptfig('Focus Merit Function',2);

set(figfoc,'units','pixels','position',[100,100,500,800]);

subwid = 3;

margins = [0.07,0.02];

axmert = subplot_tight(numel(ROI)+1,subwid,1:subwid,[margins(1),.1]);

% fftthresh = 0.95;

results = evalfocfft(img,ROI);%,fftthresh);

assignin('base', 'results', results);

set(axmert,'xlim',[0,1],'xtick',[],'box','on');

maxavgmert = max(0,mean([results.merit]));

maxavglin = line('xdata',[0;1],'ydata',maxavgmert.*[1;1],'color','r','linewidth',3);

avglin =

line('xdata',linspace(0,1,50).','ydata',[0;ones(49,1).*mean([results.merit])],'linewidth'

,2);

avgtit = title(['Average Focus Index = ' num2str(mean([results.merit]),2) ' (max. '

num2str(maxavgmert,2) ')'],'fontweight','bold');

soundfreqs = (261.63/20).*[20;25;30;36]*(1:3); % list of sound frequencies in Hz

axroi = zeros(numel(ROI),1);

imroi = axroi;

axfft = axroi;

linefft = axroi;

linemert = axroi;

fftit = axroi;

maxmert = axroi;

linemaxmert = axroi;

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aph = cell(size(axroi)); % audio player objects

for rj = 1:numel(axroi)

axroi(rj) = subplot_tight(numel(ROI)+1,subwid,subwid*rj+1,margins);

imroi(rj) = imshow(reshape(img(ROI(rj).mask),ROI(rj).size),'parent',axroi(rj));

set(axroi(rj),'visible','on','box','on','xtick',[],'ytick',[],'xcolor',roicolors(rj,:),'y

color',roicolors(rj,:));

axfft(rj) = subplot_tight(numel(ROI)+1,subwid,subwid*rj+(2:subwid),margins);

set(axfft(rj),'xlim',[0,1],'ylim',[0,1],'xtick',0:0.25:1,'ytick',[],'box','on','xcolor',r

oicolors(rj,:),'ycolor',roicolors(rj,:));

maxmert(rj) = max(maxmert(rj),results(rj).merit);

linemaxmert(rj) =

line('xdata',ones(2,1).*maxmert(rj),'ydata',[0;1],'color',roicolors(rj,:),'linewidth',3);

linefft(rj) =

line('xdata',results(rj).fftx(:),'ydata',results(rj).ffty(:),'color','k');

linemert(rj) =

line('xdata',ones(2,1).*results(rj).merit,'ydata',[0;1],'color',0.4.*[1 1

1],'linewidth',2);

fftit(rj) =

title(num2str(results(rj).merit,2),'color',roicolors(rj,:),'fontweight','bold');

[svec,fs0] = VibratoGen(10,soundfreqs(rj));

aph{rj} = audioplayer(svec,fs0);

end

% threshlide = uicontrol(figfoc,'style','slider','units','normal','position',[0.875-

0.025,0.1,0.05,1-1/(numel(ROI)+1)-

0.2],'min',0,'max',1,'value',fftthresh,'sliderstep',[0.001,0.05],'callback',@threshcall);

% threshtxt = uicontrol(figfoc,'style','text','units','normal','position',[0.875-0.075,1-

1/(numel(ROI)+1)-0.1,0.15,0.05],'string',['Threshold' char(10)

num2str(fftthresh,3)],'backgroundcolor','w','fontweight','bold');

updatemertfig();

%% Focus Evaluation Loop

set(txwiz,'string','Adjust lens focus. Tone indicates proximity to historically best

focus for each ROI');

set(donebut,'callback',@(~,~)

set([figimg,figfoc],'visible','off'),'visible','on','string','Done');

set([figimg,figfoc],'CloseRequestFcn',@(x,y) set([figimg,figfoc],'visible','off'));

runLoop = true;

frameCount = 0;

fochist = zeros(numel(ROI),10);

while runLoop && frameCount < 4000

% get the next frame

img = getsnapshot(vid);

frameCount = frameCount + 1;

% evaluate focus

results = evalfocfft(img,ROI);

instfoc = [results.merit];

fochist = [instfoc(:),fochist(:,1:(end-1))];

newmert = mean(fochist,2);

for nf = 1:numel(ROI)

205

results(nf).merit = newmert(nf);

end

updatemertfig();

% check whether the video preview window has been "closed"

runLoop = all(strcmp('on',{get(figimg,'visible'),get(figfoc,'visible')}));

end

% clean up

if ishandle(figimg)

delete(figimg);

end

if ishandle(figfoc)

delete(figfoc);

end

%% Nested Functions

function roisel(~,~)

roitemp = imrect(aximg,[round(imgsiz([2,1])./2)-50.5, 101, 101]);

resume(roitemp);

roinum = numel(roih)+1;

roih{roinum} = roitemp;

setColor(roitemp,roicolors(roinum,:));

setFixedAspectRatioMode(roitemp,true);

uiwait(figimg);

end

function squareroi(newpos,roi)

centerxy = round(newpos(1:2)+newpos(3:4)./2);

hafsiz = min(round(max(newpos(3:4))/2) + 0.5, ceil(min(imgsiz)/2)-0.5);

if centerxy(1)-hafsiz<0.5

centerxy(1) = 0.5+hafsiz;

elseif centerxy(1)+hafsiz>imgsiz(2)+0.5

centerxy(1) = imgsiz(2)+0.5-hafsiz;

end

if centerxy(2)-hafsiz<0.5

centerxy(2) = 0.5+hafsiz;

elseif centerxy(2)+hafsiz>imgsiz(1)+0.5

centerxy(2) = imgsiz(1)+0.5-hafsiz;

end

squarepos = [centerxy-hafsiz, ones(1,2).*2.*hafsiz];

setPosition(roi,squarepos);

end

function roifinish(~,~)

for rl = 1:numel(roih)

squareroi(getPosition(roih{rl}),roih{rl});

setResizable(roih{rl},false);

end

uiresume(figimg);

end

function outstruct = evalfocfft(frame,ROI)%,thresh)

206

outstruct(numel(ROI)) = struct;

sampnum = 100;

for rk = 1:numel(ROI)

roiimg = reshape(frame(ROI(rk).mask),ROI(rk).size);

roifft = abs(fftshift(fft2(roiimg))); % in this case, 'abs' returns the

complex amplitude

fftx0 = ROI(rk).centerdist;

ffty0 = cellfun(@(ind) mean(roifft(ind)), ROI(rk).pixelindices);

eefft = cumsum(ffty0); % encircled energy of fft

fftx = linspace(min(fftx0),max(fftx0),sampnum).';

eefftsamp = interp1(fftx0,eefft,fftx,'pchip'); % sampled using cubic

interpolation

eefftsampn = eefftsamp./max(eefftsamp);

ffty = max(0,diff([0;eefftsamp]));

% ffty(2:end+1) = ffty;

fftyn = ffty./max(ffty(2:end));

% mert = interp1(eefft,fftx0,thresh*max(eefft(:))); % encircled energy

threshold method

mert = sum(fftx(:).*ffty(:)) / sum(ffty(:)); % weighted mean method

% avglen = 3;

% movavg = conv(ffty,ones(avglen,1),'same');

% mert = sum(fftx(:).*movavg(:)) / sum(movavg(:)); % smoothed weighted mean

method

outstruct(rk).fftx = fftx;

outstruct(rk).ffty = fftyn;

outstruct(rk).eefft = eefftsampn;

outstruct(rk).merit = mert;

outstruct(rk).fft2d = roifft;

end

end

function updatemertfig()

ytemp = get(avglin,'ydata');

set(avglin,'ydata',[ytemp(2:end),mean([results.merit])]);

set(avgtit,'string',['Average Focus Index = ' num2str(mean([results.merit]),2) '

(max. ' num2str(maxavgmert,2) ')']);

maxavgmert = max(maxavgmert,mean([results.merit]));

set(maxavglin,'ydata',maxavgmert.*[1;1]);

for rm = 1:numel(ROI)

set(imroi(rm),'cdata',reshape(img(ROI(rm).mask),ROI(rm).size));

% set(imroi(rm),'cdata',reshape((results(rm).fft2d).^(1/2),ROI(rm).size));

set(linefft(rm),'ydata',results(rm).ffty(:));

set(linemert(rm),'xdata',ones(2,1).*results(rm).merit);

set(fftit(rm),'string',num2str(results(rm).merit,2));

maxmert(rm) = max(maxmert(rm),results(rm).merit);

set(linemaxmert(rm),'xdata',ones(2,1).*maxmert(rm));

if results(rm).merit>=0.9*maxmert(rm)

play(aph{rm});

set(axroi(rm),'linewidth',4);

else

pause(aph{rm});

207

set(axroi(rm),'linewidth',0.5);

end

end

drawnow;

end

% function threshcall(src,~)

% fftthresh = max(0,min(1,get(src,'value')));

% set(threshtxt,'string',['Threshold' char(10) num2str(fftthresh,3)]);

% maxavgmert = 0;

% maxmert(:) = 0;

% results = evalfocfft(img,ROI,fftthresh);

% updatemertfig();

% end

end

Dependency: DW_PhaseShiftCapture.m

Function that iterates phase images, triggering the camera after each pattern change.

function [v1234lin,h1234lin,linmm,permm,xflag] =

DW_PhaseShiftCapture(txwiz,cutdn,cutup,monpp,permm,xbut)

global vid fs;

% set output default values

v1234lin=0; h1234lin=0; linmm=0;

xflag = false;

set(xbut,'callback',@exitnow);

% Create full screen display figure

fsfig = fs.FigureHandle;



fssiz = fs.Size;

blkscrn = zeros(fssiz);

set(fsimg,'cdata',blkscrn);

% Take pics of horizontal and vertical lines

% prevsize = round(get(vid,'VideoResolution')./2);

% fig = figure('name','Choose Line

Positions','units','pixels','position',[100,100,prevsize+100]);

% prevax = axes('parent',fig,'units','pixels','position',[50,50,prevsize]);

% previmg = imshow(ones(prevsize),'parent',prevax);

% set(prevax,'xdir','reverse');

set(txwiz,'string','Place crosshair so both vertical and horizontal lines are reflected

off of test part.');

fig = figure('name','Choose Line Positions');

previmg = imshow(ones(fliplr(get(vid,'VideoResolution'))));

208

set(gca,'xdir','normal','ydir','normal');

preview(vid,previmg);

axes(fsax);

linx = line([100 100],get(fsax,'ylim'),'linewidth',3,'color','w');

liny = line(get(fsax,'xlim'),[100 100],'linewidth',3,'color','w');

set(fsfig,'WindowButtonMotionFcn',@moti,'WindowButtonDownFcn',@sel);

x = 0;

y = 0;

set(fsfig, 'PointerShapeCData', nan(16));

set(fsfig, 'Pointer', 'custom'); % hide mouse cursor

uiwait(fsfig);

% [x,y] = ginput(1); % pick position of line

x = round(x);

y = round(y);

linmm = [x,y].*monpp;

v1234lin = uint8(zeros([fliplr(get(vid,'VideoResolution')),5]));

h1234lin = v1234lin;

stoppreview(vid);

close(fig)

if xflag

return;

end

% background pic

bkgnd = capic(blkscrn);

if xflag

return;

end

% vertical line pic

vscrn = zeros(fssiz);

vscrn(:,(x-0):(x+0)) = 255; % 1 pixel(s) wide

v1234lin(:,:,5) = capic(vscrn);

if xflag

return;

end

% horizontal line pic

hscrn = zeros(fssiz);

hscrn((y-0):(y+0),:) = 255; % 1 pixel(s) wide

h1234lin(:,:,5) = capic(hscrn);

if xflag

return;

end

209

% line phase shifting pics

phaseshift = (0:0.25:0.75).*2.*pi;

% permm = monpp.*100.1; % period in mm

for ij = 1:numel(phaseshift)

apodv = DW_apodizationgen([1,0],permm(1),phaseshift(ij),'cos',[cutdn,cutup],monpp);

v1234lin(:,:,ij) = capic(apodv);

if xflag

return;

end

apodh = DW_apodizationgen([0,1],permm(2),phaseshift(ij),'cos',[cutdn,cutup],monpp);

h1234lin(:,:,ij) = capic(apodh);

if xflag

return;

end

end

v1234lin = bsxfun(@minus,v1234lin,bkgnd);

h1234lin = bsxfun(@minus,h1234lin,bkgnd);

set(fsfig, 'Pointer', 'arrow');

function exitnow(~,~)

xflag = true;

end

function pic = capic(cdat)


drawnow expose;

pause(0.5);

start(vid);

pic = uint8(squeeze(mean(getdata(vid),4)));

end

function moti(~,~)

cp = get(fsax,'CurrentPoint');

xdat = [1,1].*cp(1,1);

ydat = [1,1].*cp(1,2);

set(linx,'XData',xdat);

set(liny,'YData',ydat);

drawnow;

end

function sel(~,~)

cp = get(fsax,'CurrentPoint');

x = cp(1,1);

y = cp(1,2);

set([linx,liny],'visible','off');

set(fsfig,'WindowButtonMotionFcn',[]);

set(fsax,'ButtonDownFcn',[]);

uiresume(fsfig);

210

end

end

Dependency: DP_DataRegister3D.m

Registers reconstructed surface data to shape and position of ideal part.

function [xyzreg,xyzfin,Mxyzfin,err,errvec] =

DP_DataRegister3D(surfptsx,surfptsy,surfptsz,shapename,samples,edthresh)

% pn = 'C:\Users\Dan\Google Drive\Thesis\Reflector Profile from QVI\';

% fn =

{'Roch3_20mm_InverseFlex1.0_FullScan_1.DAT','Roch3_20mm_InverseFlex1.0_FullScan_2.DAT','R

och3_20mm_InverseFlex1.0_PartialHighDensityScan.DAT'};

% % scandir1 = [1,1,1];

% % scandir2 = [2,2,2];

% % edthresh = [3,3,1];

% angxyz = [pi(),0,-pi()/2;...

% pi(),0,-pi()/2;...

% 0,pi(),pi()];

figure(2);

figure(3);

set([2,3],'CloseRequestFcn',@finish);

errthresh = 0.98;

% ij = 1; % for loop implemetation

% raw = importdata([pn,fn{ij}],',',2);

%% Iterative Closest Point (ICP) calculations

shapedata = reflectorshapedatabase(shapename);

shape = shapedata.shape;

% xbnd = shapedata.xbnd;

length = shapedata.length; % extruded length of part

tilt = shapedata.tilt; % degrees

% thik = shapedata.thik; % perpendicular distance from xbnd(1) point on curve to plane

describing the rear face of the part

[Mx,My,Mz] = DP_SLSurfacePoints(shapename,samples);

Mz = -Mz;

M = [Mx(:).';My(:).';Mz(:).'];

D0 = [surfptsx(:),surfptsy(:),surfptsz(:)].';

D0(:,isnan(sum(D0,1))) = [];

R = eye(3);

D1 = R*D0;

T = mean(M,2)-mean(D1,2);

D = bsxfun(@plus,D1,T);

211

mplot = [0;0];

ICPplot(0,0,false);

%%

x = D(1,:).';

y = D(2,:).';

z = D(3,:).';

% dt = delaunay(x,y);

dt0temp = DelaunayTri(x,y);

% ed = edges(dt0);

% edlen = sqrt( diff(x(ed),1,2).^2 + diff(y(ed),1,2).^2 );

% edel = edlen>edthresh(ij);

% si0 = edgeAttachments(dt0,ed(edel,:));

% si = cell2mat(transpose(si0)).';

% sikeep = ~ismember((1:size(dt0,1)).',si);

% dt = dt0(sikeep,:);

% tri = TriRep(dt,x,y,z);

% fe = unique(freeBoundary(dt));

% fehi = fe(y(fe)>0);

% felo = fe(y(fe)<0);

% figure(1);

% clf;

% trisurf(tri);

% shading interp;

% xlabel('X'); ylabel('Y'); zlabel('Z');

% tsi = TriScatteredInterp(dt0,z);

% zs = tsi(Mx(:),My(:));

% err = Mz(:)-zs;

% zs(isnan(zs)) = 0;

%

line('xdata',Mx(:),'ydata',My(:),'zdata',zs,'color','k','linestyle','none','marker','d');

% axis equal;

tsitemp = TriScatteredInterp(dt0temp,z);

zstemp = tsitemp(Mx(:),My(:));

err = -Mz(:)+zstemp;

figure(3);

clf;

chlen = 32; % colormap half length

grad = linspace(0,1,chlen)';

cmp = [grad,grad,ones(chlen,1);

ones(chlen,1),flipud(grad),flipud(grad)];

colormap(3,cmp);

erax(1) = subplot_tight(1,2,1,0.05);

errmap = surf(Mx,My,Mz,reshape(err,size(Mx)));

if all(isnan(err))

caxis([-1,1]);

212

else

caxis(max(abs(err)).*[-1,1]);

end

xlabel('X'); ylabel('Y'); zlabel('Z');

axis equal;

colorbar;

erax(2) = subplot_tight(1,2,2,0.05);

errsurf = surf(Mx,My,reshape(err,size(Mx)));

if all(isnan(err))

caxis([-1,1]);

else

caxis(max(abs(err)).*[-1,1]);

end

xlabel('X'); ylabel('Y'); zlabel('Z');

% figure(2);

% plot3(x,y,z);

% axis equal;

% xlabel('X');

% ylabel('Y');

% zlabel('Z');

% manual adjustment figure

figure(4);

clf;

set(4,'color','w','position',[766 309 300 420],'CloseRequestFcn',@finish);

lab = {'X','Y','Z'};

lft = [0.05,0.25,0.65];

bot = [0.6,0.35,0.1];

bw = 0.3;

bh = 0.08;

for ik = 1:3

uicontrol(4,'style','text','string',lab{ik},'units','normal','position',[lft(1),bot(ik),0

.1,0.12],'backgroundcolor','w','fontweight','bold','fontsize',12);

uicontrol(4,'style','pushbutton','string','-

','units','normal','position',[lft(2),bot(ik),bw,bh],'UserData',-

ik,'callback',@rotclick,'fontweight','bold','fontsize',16,'backgroundcolor',[1 .5 .5]);

uicontrol(4,'style','pushbutton','string','+','units','normal','position',[lft(2),bot(ik)

+0.1,bw,bh],'UserData',ik,'callback',@rotclick,'fontweight','bold','fontsize',12,'backgro

undcolor',[1 .5 .5]);

uicontrol(4,'style','pushbutton','string','-

','units','normal','position',[lft(3),bot(ik),bw,bh],'UserData',-

ik,'callback',@transclick,'fontweight','bold','fontsize',16,'backgroundcolor',[.5 .5 1]);

uicontrol(4,'style','pushbutton','string','+','units','normal','position',[lft(3),bot(ik)

+0.1,bw,bh],'UserData',ik,'callback',@transclick,'fontweight','bold','fontsize',12,'backg

roundcolor',[.5 .5 1]);

end

213

rotinc =

uicontrol(4,'style','edit','string','0.1','units','normal','position',[lft(2),0.8,bw,0.05

],'UserData',0.1,'callback',@numinput);

transinc =

uicontrol(4,'style','edit','string','0.5','units','normal','position',[lft(3),0.8,bw,0.05

],'UserData',0.5);

uicontrol(4,'style','text','string','Increment','units','normal','position',[0.02,0.8,0.1

8,0.05],'backgroundcolor','w');

uicontrol(4,'style','text','string',sprintf('Rotation\ndegrees'),'units','normal','positi

on',[lft(2),0.87,bw,0.12],'backgroundcolor',[1 .5 .5],'fontweight','bold','fontsize',12);

uicontrol(4,'style','text','string',sprintf('Translation\nmm'),'units','normal','position

',[lft(3),0.87,bw,0.12],'backgroundcolor',[.5 .5 1],'fontweight','bold','fontsize',12);

uicontrol(4,'style','pushbutton','string','Rerun Iterative Closest

Point','units','normal','position',[0.05,0.01,0.9,0.07],'callback',{@ICPplot,true},'fonts

ize',12);

uiwait(4);

% perform final spatial transformations on original measured data (just in case)

Dreg = bsxfun(@plus,R*D0,T);

xreg = Dreg(1,:).';

yreg = Dreg(2,:).';

zreg = Dreg(3,:).';

xyzreg = [xreg,yreg,zreg]; % unsampled data for output

% get registered point locations (regular grid in x,y)

xsamp = Mx;

ysamp = My;

zsamp = updaterr();

zsamp = reshape(zsamp,size(xsamp));

% replace z=NaN points with value of nearest neighbor and change x and y values to be

near edge of valid data

nans = isnan(zsamp);

nnind = knnsearch([xreg,yreg],[xsamp(nans),ysamp(nans)]);

zsamp(nans) = zreg(nnind);

xsamp(nans) = xsamp(nans)+(xreg(nnind)-xsamp(nans)).*0.99;

ysamp(nans) = ysamp(nans)+(yreg(nnind)-ysamp(nans)).*0.99;

% position points relative to CPV origin

T1 = [-Mx(end);-length/2;-Mz(end)];

R1 = [cosd(-tilt),0,sind(-tilt);0,1,0;-sind(-tilt),0,cosd(-tilt)];

T2 = [0;0;shape(end)];

% Rfin = R1*R;

% Tfin = R1*(T+T1) + T2;

xyzfin = bsxfun(@plus,R1*[xsamp(:),ysamp(:),zsamp(:)].',R1*T1+T2);

xyzfin = reshape(xyzfin.',[size(xsamp),3]);

Mxyzfin = bsxfun(@plus,R1*[Mx(:),My(:),Mz(:)].',R1*T1+T2);

Mxyzfin = reshape(Mxyzfin.',[size(Mx),3]);

xyzreg = bsxfun(@plus,R1*xyzreg.',R1*T1+T2).';

% xfin = reshape(xyzfin(1,:),size(xsamp));

214

% yfin = reshape(xyzfin(2,:),size(ysamp));

% zfin = reshape(xyzfin(3,:),size(zsamp));

errvec = (R1*[0;0;1]).';

delete(2,3,4);

pause(0.1);

figure(2);

surf(xyzfin(:,:,1),xyzfin(:,:,2),xyzfin(:,:,3),reshape(err,size(xyzfin(:,:,1))));

errlistfin = sort(err);

cmaxfin = [errlistfin(round(sum(~isnan(err))*(1-

errthresh))),errlistfin(round(sum(~isnan(err))*errthresh))];

caxis(cmaxfin);

shading interp;

colorbar;

hold on;

modsurf = surf(Mxyzfin(:,:,1),Mxyzfin(:,:,2),Mxyzfin(:,:,3));

hold off;

axis equal;

surfchek =

uicontrol('style','checkbox','units','pixels','position',[10,10,200,15],'string','Fill

ideal surface','min',0,'max',1,'callback',@filltoggle);

filltoggle(surfchek,0);

%% Nested Functions

function filltoggle(src,~)

switch get(src,'value')

case 0

set(modsurf,'facecolor','none');

case 1

set(modsurf,'facecolor','k');

end

end

function zs = updaterr()

x = D(1,:).';

y = D(2,:).';

z = D(3,:).';

dt0 = DelaunayTri(x,y);

ed = edges(dt0);

edlen = sqrt( diff(x(ed),1,2).^2 + diff(y(ed),1,2).^2 );

edel = edlen>edthresh;

si0 = edgeAttachments(dt0,ed(edel,:));

si = cell2mat(transpose(si0)).';

% sikeep = ~ismember((1:size(dt0,1)).',si);

% dt1 = TriRep(dt0(sikeep,:),x,y);

% fe = unique(freeBoundary(dt1));

tsi = TriScatteredInterp(dt0,z);

215

zs = tsi(Mx(:),My(:));

pls = pointLocation(dt0,Mx(:),My(:));

zs(ismember(pls,si)) = NaN; % values in non-valid simplexes are set to NaN

err = -Mz(:)+zs;

set(errmap,'cdata',reshape(err,size(Mx)));

set(errsurf,'zdata',reshape(err,size(Mx)));

if all(isnan(err))

cmax = 1;

else

errlist = sort(abs(err));

cmax = errlist(round(sum(~isnan(err))*errthresh));

end

caxis(erax(1),cmax.*[-1,1]);

caxis(erax(2),cmax.*[-1,1]);

set(erax(2),'zlim',cmax.*1.1.*[-1,1]);

set(mplot(2),'xdata',x,'ydata',y,'zdata',z); % update ICP Result plot data

end

function ICPplot(~,~,errup)

[Ricp Ticp ER t] = icp(M, D, 25, 'Matching', 'kDtree', 'WorstRejection', 1-

errthresh);

R = Ricp*R;

T = T+Ticp;

% Plot model points blue and transformed points red

figure(2);

clf;

sp(1) = subplot(2,2,1);

plot3(M(1,:),M(2,:),M(3,:),'bo',D(1,:),D(2,:),D(3,:),'r.');

axis equal;

xlabel('x'); ylabel('y'); zlabel('z');

title('Red: measured, blue: modeled');

% Transform data-matrix using ICP result

D = bsxfun(@plus,R*D0,T);

% Plot the results

sp(2) = subplot(2,2,2);

mplot = plot3(M(1,:),M(2,:),M(3,:),'bo',D(1,:),D(2,:),D(3,:),'r.'); % modeled and

measure results

axis equal;

xlabel('x'); ylabel('y'); zlabel('z');

title('ICP result');

linkprop(sp, 'CameraPosition');

% Plot RMS curve

subplot(2,2,[3 4]);

plot((1:numel(ER))-1,ER,'--x');

xlabel('iteration#');

ylabel('d_{RMS}');

216

title(['Total elapsed time: ' num2str(t(end),2) ' s']);

if errup

updaterr();

end

end

function rotclick(bhan,~)

rax = abs(get(bhan,'UserData'));

dir = sign(get(bhan,'UserData'));

inc = abs(str2double(get(rotinc,'string')));

Rinc = [1,0,0;0,cosd(dir*inc),-sind(dir*inc);0,sind(dir*inc),cosd(dir*inc)];

switch rax

case 1 % rotation about x axis

case 2 % rotation about y axis

Rinc = Rinc([3,1,2],[3,1,2]);

case 3 % rotation about z axis

Rinc = Rinc([2,3,1],[2,3,1]);

end

Tinc = mean(D,2);

R = Rinc*R;

T = Rinc*(T-Tinc)+Tinc; % rotates about the data centroid

D = bsxfun(@plus,R*D0,T); % update D

updaterr();

end

function transclick(bhan,~)

tax = abs(get(bhan,'UserData'));

dir = sign(get(bhan,'UserData'));

inc = abs(str2double(get(transinc,'string')));

Tinc = zeros(3,1);

Tinc(tax) = dir*inc;

T = T+Tinc;

D = bsxfun(@plus,R*D0,T); % update D

updaterr();

end

function numinput(han,~)

val = str2double(get(han,'string'));

if isnan(val)

set(han,'string',num2str(get(han,'UserData'))); % if input is not numeric,

set to default

else

set(han,'string',num2str(val));

end

end

function finish(~,~)

uiresume(4);

end

end

217

Dependency: DP_datazoom.m

Resamples surface data in a user input region of interest and subtracts a two

dimensional polynomial of user-specified order.

function [x,y,z,Ds,A,Ps,polyv,Aexp] = DP_datazoom(xyzorig,xsamp,ysamp,polyord,varargin)

% xyzorig = [x,y,z] containing pointcloud data, each row defining a point

% xsamp = vector of x positions for sampling

% ysamp = vector of y positions for sampling

% sample points will be a mesh of xsamp and ysamp, size of numel(ysamp)

% by numel(xsamp)

% polyord = order of 2D polynomial for subtracting out surface figure

% % create interpolated data set

% F = TriScatteredInterp(xyzorig(:,1),xyzorig(:,2),xyzorig(:,3));

% [x,y] = meshgrid(xsamp,ysamp);

% z = F(x,y); % resampled raw data

% create inrerpolated data set while omitting gaps in data

edthresh = max([mean(diff(xsamp)),mean(diff(ysamp))]).*3;

dt0 = DelaunayTri(xyzorig(:,1),xyzorig(:,2));

ed = edges(dt0);

edlen = sqrt( (xyzorig(ed(:,2),1)-xyzorig(ed(:,1),1)).^2 + (xyzorig(ed(:,2),2)-

xyzorig(ed(:,1),2)).^2 );

edel = edlen>edthresh;

si0 = edgeAttachments(dt0,ed(edel,:));

si = cell2mat(transpose(si0)).';

F = TriScatteredInterp(dt0,xyzorig(:,3));

[x,y] = meshgrid(xsamp,ysamp);

z = F(x(:),y(:));

pls = pointLocation(dt0,x(:),y(:));

z(ismember(pls,si)) = NaN; % values in non-valid simplexes are set to NaN

z = reshape(z,size(x));

% % calcs for best fit plane analysis

% Xs = x-mean(x(:));

% Ys = y-mean(y(:));

% Zs = z-mean(z(:));

% XYZs = [Xs(:),Ys(:),Zs(:)];

% [V,D] = eig(XYZs.'*XYZs); % Find eigenvectors, columns of V [Z,Y,X] axes, where Z is

normal to plane. D contains the eigenvalues.

% dists = XYZs*V(:,1);

% Ds = reshape(dists,size(Zs)); % data point perpendicular distances from best fit plane

Ds = 0; % placeholder value

% calcs for best fit 2D nth order polynomial subtraction

nnind = ~isnan(x) & ~isnan(y) & ~isnan(z);

xZ = x(nnind);

218

yZ = y(nnind);

zZ = z(nnind);

polylen = sum(0:(polyord+1));

Aexp = zeros(2,polylen);

for ord = 0:polyord

posoff = sum(0:ord);

Aexp(1:2,(1:(ord+1))+posoff) = [ord:-1:0;0:1:ord];

end

A = bsxfun(@power,xZ(:),Aexp(1,:)) .* bsxfun(@power,yZ(:),Aexp(2,:)) \ zZ(:);

% A = [ones(size(xZ(:))) xZ(:) yZ(:) xZ(:).^2 xZ(:).*yZ(:) yZ(:).^2] \ zZ(:);

Ps = z-polyvaln(A,Aexp,x,y);

fsiz = fliplr(size(x));

scrn = [1600,800];

if fsiz(1)/fsiz(2) >= scrn(1)/scrn(2)

wh = 1;

else

wh = 2;

end

fsiz = fsiz./fsiz(wh).*scrn(wh);

fig = figure('color','w','position',[40,1050-(100+fsiz(2)),fsiz]);

ax = subplot(4,4,[1:3,5:7,9:11]);

axv = subplot(4,4,[4,8,12]);

axh = subplot(4,4,13:15);

flipy = false;

% contourf(x,y,Ps,50,'parent',ax);

% surf(x,y,Ps,'parent',ax,'edgecolor','none');

if nargin>4

cutofffrac = varargin{1};

else

cutofffrac = 0.01;

end

PsSort = sort(Ps(~isnan(Ps)));

cutlim = PsSort(round([numel(PsSort)*cutofffrac+1,numel(PsSort)*(1-cutofffrac)]));

imagesc(xsamp,ysamp,Ps,'parent',ax);

caxis(ax,cutlim);

title(ax, ['Subtracted 2D Polynomial of order ' int2str(polyord)]);

if flipy

set(axv,'ydir','reverse');

elseif ~flipy

set(ax,'ydir','normal');

end

set(ax,'dataaspectratio',[1,1,1],'dataaspectratiomode','manual');

axis(ax,'fill');

slicept = [mean(get(ax,'xlim')),mean(get(ax,'ylim'))];

colh = [0,.5,0];

colv = [0,0,1];

linh = line('xdata',[0,1],'ydata',[0,0],'parent',ax,'linewidth',2,'color',colh);

linv = line('xdata',[0,0],'ydata',[0,1],'parent',ax,'linewidth',2,'color',colv);

slich = line('xdata',[0,1],'ydata',[0,1],'parent',axh,'linewidth',2,'color',colh);

slicv = line('xdata',[0,1],'ydata',[0,1],'parent',axv,'linewidth',2,'color',colv);

219

pth = line('xdata',0,'ydata',0,'marker','o','markerfacecolor',colv,'parent',axh);

ptv = line('xdata',0,'ydata',0,'marker','o','markerfacecolor',colh,'parent',axv);

upslice(0,0);

set(fig,'WindowButtonDownFcn',@(dum1,dum2)

set(fig,'WindowButtonMotionFcn',@sliclick),'WindowButtonUpFcn',@(dum1,dum2)

set(fig,'WindowButtonMotionFcn',[]));

set(axv,'xgrid','on','YAxisLocation','right');

set(axh,'ygrid','on');

% zoh = zoom;

% setAllowAxesZoom(zoh,ax,true);

% setAllowAxesZoom(zoh,axh,false);

% setAllowAxesZoom(zoh,axv,false);

% set(zoh,'ActionPostCallback',@upslice);

axc = subplot(4,4,16);

colorvec = linspace(cutlim(1),cutlim(2),100).';

imagesc([0,abs(diff(cutlim))/10],colorvec,repmat(colorvec,1,2));

axis image;

set(axc,'YAxisLocation','right','XTick',[],'YDir','normal');

polyv = @(x,y) polyvaln(A,Aexp,x,y);

function sliclick(~,~)

cp = get(ax,'currentpoint');

slicept = cp(1,1:2);

upslice(0,0);

end

function upslice(~,~)

[~,slicexind] = min(abs(x(1,:)-slicept(1)));

[~,sliceyind] = min(abs(y(:,1)-slicept(2)));

slicexval = x(1,slicexind);

sliceyval = y(sliceyind,1);

slicezval = Ps(sliceyind,slicexind);

hdata = [x(sliceyind,:).',Ps(sliceyind,:).'];

vdata = [y(:,slicexind),Ps(:,slicexind)];

set(linh,'xdata',[x(1,1);x(1,end)],'ydata',sliceyval.*[1;1]);

set(linv,'xdata',slicexval.*[1;1],'ydata',[y(1,1);y(end,1)]);

set(slich,'xdata',hdata(:,1),'ydata',hdata(:,2));

set(slicv,'xdata',vdata(:,2),'ydata',vdata(:,1));

set(pth,'xdata',slicexval,'ydata',slicezval);

set(ptv,'xdata',slicezval,'ydata',sliceyval);

limx = get(ax,'xlim');

limy = get(ax,'ylim');

set(axh,'xlim',limx);

set(axv,'ylim',limy);

set(axh,'ylimmode','auto');

set(axv,'xlimmode','auto');

sliclim0 = [get(axh,'ylim');get(axv,'xlim')];

sliclim1 = [min(sliclim0(:,1)),max(sliclim0(:,2))];

sliclim = [max(sliclim1(1),cutlim(1)), min(sliclim1(2),cutlim(2))];

220

set(axh,'ylim',sliclim);

set(axv,'xlim',sliclim);

end

end

function z = polyvaln(P,Pexp,x,y)

zc = sum(bsxfun(@times, bsxfun(@power,x(:),Pexp(1,:)) .* bsxfun(@power,y(:),Pexp(2,:)),

P(:).'),2);

z = reshape(zc, size(x));

% z = P(1) + P(2).*x + P(3).*y + P(4).*x.^2 + P(5).*x.*y + P(6).*y.^2;

end

Dependency: DP_DualNoiseDeleteFilter.m

Detects white noise in two coincident images and creates a mask to isolate larger

smooth data regions.

function nmask = DP_DualNoiseDeleteFilter(nimgs,mask,aximg)

% nimgs = stack of 2D images to filter on;

% masks = ROI mask to apply to all input images

merts = nan(size(nimgs));

mask([1,end],:) = 0;

mask(:,[1,end]) = 0;

for im = 1:size(nimgs,3)

nimg = nimgs(:,:,im);

intinds = find(mask);

merts0 = nan(size(mask));

% tic;

for ij = intinds(:).'

merts0(ij) = stdnn(ij,nimg);

end

% toc;

% normalize merit values from each image before combining

merts0 = merts0-nanmin(merts0(:));

merts0 = merts0./nanmax(merts0(:));

merts(:,:,im) = merts0;

end

mertsum = nansum(merts,3); % sum merit values pixel by pixel across all images

mertsum(sum(~isnan(merts),3)==0) = nan;

mertsum = mertsum-nanmin(mertsum(:)); % re-normalize

mertsum = mertsum./nanmax(mertsum(:));

numbins = 100;

[yh,xh] = hist(mertsum(:),numbins);

[~,plocs] = findpeaks(yh,'sortstr','descend','minpeakdistance',round(numbins/10));

thresh = mean(xh(plocs(1:2)));

timg = nimgs; % thresholded image

nanfilt = mertsum>thresh | isnan(mertsum);

221

timg(repmat(nanfilt,[1,1,size(nimgs,3)])) = nan;

imgbw = imclose(~nanfilt,strel('square',3));

% tic;

stats = regionprops(imgbw,{'FilledArea','FilledImage','BoundingBox'});

% toc;

[~,ccind] = max([stats.FilledArea]);

nmask = stats(ccind).FilledImage;

bb = stats(ccind).BoundingBox;

nmask = padarray(nmask,[bb(2)-0.5,bb(1)-0.5],0,'pre');

nmask = padarray(nmask,[size(nimgs,1)-(bb(2)+0.5+bb(4)),size(nimgs,2)-

(bb(1)+0.5+bb(3))]+1,0,'post');

% gimg = imcrop(nimg,bb+[.5 .5 -1 -1]);

% gimg(~ccmask) = nan;

% gimg = padarray(gimg,[bb(2)-0.5,bb(1)-0.5],nan,'pre');

% gimg = padarray(gimg,[size(nimg,1)-(bb(2)+0.5+bb(4)),size(nimg,2)-

(bb(1)+0.5+bb(3))],nan,'post');

gimg = nimgs;

gimg(repmat(~nmask,[1,1,size(nimgs,3)])) = nan;

figure();

imagesc(mertsum);

colormap(jet);

colorbar;

axmer = gca;

figure();

subplot(2,1,1);

imagesc(nimgs(:,:,1));

colormap(jet);

aximg(1) = gca;

subplot(2,1,2);

imagesc(nimgs(:,:,2));

colormap(jet);

aximg(2) = gca;

figure();

bar(xh,yh);

line('xdata',[1;1].*thresh,'ydata',[0;max(yh)],'color','r');

figure();

subplot(2,1,1);

imagesc(timg(:,:,1));

colormap(jet);

axnew(1) = gca;

subplot(2,1,2);

imagesc(timg(:,:,2));

colormap(jet);

axnew(2) = gca;

figure();

222

subplot(2,1,1);

imagesc(gimg(:,:,1));

colormap(jet);

axfin(1) = gca;

subplot(2,1,2);

imagesc(gimg(:,:,2));

colormap(jet);

axfin(2) = gca;

linkaxes([axmer,axnew,axfin,aximg]);

% works well, but takes a while to run

function out = stdnn(ind,nimg) % std dev of nearest neighbors' distance from best fit

plane

inds = bsxfun(@plus,ind+(-1:1).',(-1:1).*size(nimg,1));

inds = inds(:);

[Xfit,Yfit] = meshgrid(-1:1,-1:1);

ptsc = [Xfit(:),Yfit(:),nimg(inds)-mean(nimg(inds))]; % points centered around

the mean in each dimension

[V,D] = eig(ptsc.'*ptsc); % Find eigenvectors, columns of V [Z,Y,X] axes, where Z

is normal to plane. D contains the eigenvalues.

dists = ptsc*V(:,1);

out = std(dists);

end

% toc = 17.0

% % bad filter for nearly flat data with a small amount of noise

% % doesn't pick up severe noise reliably

% function out = pathwalk(ind) % walk path around pixel and count number of

inflection points

% inds0 = nebind(ind,size(nimg,1),1);

% inds = [inds0;inds0(1)];

% out = nnz(diff(sign(diff(nimg(inds))))); % number of inflection points while

walking around pixel (max = 7);

% end

% % toc = 4.6

end

% function inds = nebind(ind,siz1,ord) % gives pixel indices of nearest neighbors of

order 'ord' around pixel with index 'ind' in image with 'siz1' number of rows

% top = ind + (-ord:ord).'.*siz1 - ord;

% bottom = ind + (-ord:ord).'.*siz1 + ord;

% left = ind + (-ord:ord).' - ord*siz1;

% right = ind + (-ord:ord).' + ord*siz1;

% inds = [top(2:1:end);right(2:1:end);bottom(end-1:-1:1);left(end-1:-1:1)];

% end

223

Dependency: DP_puma_ho.m

This code is identical to puma_ho.m provided in [50], except it points to

DP_energy_ho.m (below) instead of the original energy_ho.m function.

Dependency: DP_energy_ho.m

Function modified from [50] to completely ignore specified regions. The vast majority of

this code is unmodified from its original, publicly available source.

function erg = DP_energy_ho(kappa,psi,base,p,cliques,disc_bar,th,quant)

%energy_ho Energy from kappa labeling and psi phase measurements.

% erg = energy_ho(kappa,psi,base,p,cliques,disc_bar,p,th,quant) returns the energy of

kappa labeling given the

% psi measurements image, the base ROI image (having ones in the region of interest

(psi) and a passe-partout

% made of zeros), the exponent p, the cliques matrix (each row indicating a

displacement vector corresponding

% to each clique), the disc_bar (complement to one of the quality maps), a threshold th

defining a region for

% which the potential (before a possible quantization) is quadratic, and quant which is

a flag defining whether

% the potential is or is not quantized.

% (see J. Bioucas-Dias and G. Valadão, "Phase Unwrapping via Graph Cuts"

% submitted to IEEE Transactions Image Processing, October, 2005).

% SITE: www.lx.it.pt/~bioucas/

[m,n] = size(psi);

[cliquesm,cliquesn] = size(cliques); % Size of input cliques

maxdesl = max(max(abs(cliques))); % This is the maximum clique length used

% Here we put a passe-partout (constant length = maxdesl+1) in the images kappa and psi

base_kappa = zeros(2*maxdesl+2+m,2*maxdesl+2+n); base_kappa(maxdesl+2:maxdesl+2+m-

1,maxdesl+2:maxdesl+2+n-1) = kappa;

psi_base = zeros(2*maxdesl+2+m,2*maxdesl+2+n); psi_base(maxdesl+2:maxdesl+2+m-

1,maxdesl+2:maxdesl+2+n-1) = psi;

z = size(disc_bar,3);

base_disc_bar = repmat(zeros(2*maxdesl+2+m,2*maxdesl+2+n),[1 1 z]);

base_disc_bar(maxdesl+2:maxdesl+2+m-1,maxdesl+2:maxdesl+2+n-1,:) = disc_bar;

for t = 1:cliquesm

% The allowed start and end pixels of the "interpixel" directed edge

base_start(:,:,t) = circshift(base,[-cliques(t,1),-cliques(t,2)]).*base;

base_end(:,:,t) = circshift(base,[cliques(t,1),cliques(t,2)]).*base;

% By convention the difference images have the same size as the

224

% original ones; the difference information is retrieved in the

% pixel of the image that is subtracted (end of the diff vector)

auxili = circshift(base_kappa,[cliques(t,1),cliques(t,2)]);

t_dkappa(:,:,t) = (base_kappa-auxili);

auxili2 = circshift(psi_base,[cliques(t,1),cliques(t,2)]);

dpsi = auxili2 - psi_base;

% Beyond base, we must multiply by

% circshift(base,[cliques(t,1),cliques(t,2)]) in order to

% account for frontier pixels that can't have links outside ROI

a(:,:,t) = (2*pi*t_dkappa(:,:,t)-

dpsi).*base.*circshift(base,[cliques(t,1),cliques(t,2)])...

.*base_disc_bar(:,:,t);

end

erg = sum(sum(sum((clique_energy_ho(a,p,th,quant)))));

Dependency: DP_lsqiter.m

This code implements the iterative slope matching process described in Chapter 3 [52].

function [surfptsx,surfptsy,surfptsz] =

DP_lsqiter(nmask,unwpx,unwpy,R,T,xyoffset,cxcy,dxy,f,k1,xs,ys,zs,xc,yc,Ropt,Topt,lindist,

lmask1,lmask2,Rmir,Tmir)

% Least Squares Iteration Method

% find l vector for each data pixel

l = bsxfun(@plus,R*[unwpx(nmask),unwpy(nmask),zeros(nnz(nmask),1)].',T).';

% lmir = (Rmir.'*bsxfun(@minus,lcam.',Tmir)).';

% find if pixels in the up, down, left, and/or right directions are valid

imask = zeros(size(nmask));

numpix = nnz(nmask);

imask(nmask) = 1:numpix;

up = padarray(imask(1:(end-1),:),[1,0],0,'pre');

up = up(nmask);

uplog = logical(up);

down = padarray(imask(2:end,:),[1,0],0,'post');

down = down(nmask);

downlog = logical(down);

left = padarray(imask(:,1:(end-1)),[0,1],0,'pre');

left = left(nmask);

leftlog = logical(left);

right = padarray(imask(:,2:end),[0,1],0,'post');

right = right(nmask);

rightlog = logical(right);

% form sparse matrix D for slope averaging (slope vector will have u

% component of slopes followed by the v component of slopes

pind = repmat((1:numpix).',2,1); % pixel index (repeated once to have a copy for u and v)

225

nextp = [right;down]; % adjacent pixel index, u then v (adjacent pixel in the u direction

is to the right, v is down)

Diu0 = find(right); % row position for elements in D (u quadrant)

Dju = [Diu0; right(rightlog)]; % column position for elements in D (u quadrant)

Diu = [Diu0; Diu0];

Div0 = find(down); % row position for elements in D (v quadrant)

Djv = [Div0; down(downlog)]; % column position for elements in D (v quadrant)

Div = [Div0; Div0];

D = sparse([Diu;Div+numpix],[Dju;Djv+numpix],0.5,2*numpix,2*numpix); % v subscripts

shifted by numpix to put values in lower right quadrant

% find 3D ray direction vectors and their partial derivatives in u and v for all data

pixels

xo = xyoffset(1)-cxcy(1); % xyoffset(1): match indicies to uncropped camera frame,

cxcy(1): move origin to optical center

yo = xyoffset(2)-cxcy(2);

dx = dxy(1);

dy = dxy(2);

[v,u] = find(nmask);

sx =

dx.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx

.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*

(v+yo).^2)+1.0).^2.*(v+yo).^2);

sy =

dy.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx

.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*

(v+yo).^2)+1.0).^2.*(v+yo).^2);

sz =

f.*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^

2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(v+yo).^2);

s = [sx(:),sy(:),sz(:)];

sxdu =

dx.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(u+

xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^

2)+1.0).^2.*(v+yo).^2)+dx.^3.*k1.*(u.*2.0+xo.*2.0).*(u+xo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(d

x.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.

*(v+yo).^2)+1.0).^2.*(v+yo).^2)-

dx.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).*(dx.^2.*(u.*2.0+xo.*2.0).*(k1.

*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dx.^4.*k1.*(u.*2.0+xo.*2.0).*(k1.*(dx.^2.*(u

+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0+dx.^2.*dy.^2.*k1.*(u.*2.0+xo.*2.0).*(k1.*(

dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0).*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*

(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo

).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(1.0./2.0);

sydu =

dy.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).*(dx.^2.*(u.*2.0+xo.*2.0).*(k1.

*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dx.^4.*k1.*(u.*2.0+xo.*2.0).*(k1.*(dx.^2.*(u

+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0+dx.^2.*dy.^2.*k1.*(u.*2.0+xo.*2.0).*(k1.*(

dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0).*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*


226

).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(-

1.0./2.0)+dx.^2.*dy.*k1.*(u.*2.0+xo.*2.0).*(v+yo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(u+


2)+1.0).^2.*(v+yo).^2);

szdu =

f.*(dx.^2.*(u.*2.0+xo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dx.^4.*k1.*

(u.*2.0+xo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0+dx.^2.*dy

.^2.*k1.*(u.*2.0+xo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0)

.*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k

1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(-1.0./2.0);

sdu = [sxdu(:),sydu(:),szdu(:)];

sdu = bsxfun(@rdivide,sdu,sqrt(sum(sdu.^2,2)));

sxdv =

dx.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).*(dy.^2.*(v.*2.0+yo.*2.0).*(k1.

*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dy.^4.*k1.*(v.*2.0+yo.*2.0).*(k1.*(dx.^2.*(u

+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0+dx.^2.*dy.^2.*k1.*(v.*2.0+yo.*2.0).*(k1.*(

dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0).*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*


).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(-

1.0./2.0)+dx.*dy.^2.*k1.*(v.*2.0+yo.*2.0).*(u+xo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(u+


2)+1.0).^2.*(v+yo).^2);

sydv =

dy.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(u+


2)+1.0).^2.*(v+yo).^2)+dy.^3.*k1.*(v.*2.0+yo.*2.0).*(v+yo).*1.0./sqrt(f.^2+dx.^2.*(k1.*(d

x.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.

*(v+yo).^2)+1.0).^2.*(v+yo).^2)-

dy.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).*(dy.^2.*(v.*2.0+yo.*2.0).*(k1.

*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dy.^4.*k1.*(v.*2.0+yo.*2.0).*(k1.*(dx.^2.*(u

+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0+dx.^2.*dy.^2.*k1.*(v.*2.0+yo.*2.0).*(k1.*(

dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0).*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*


).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(1.0./2.0);

szdv =

f.*(dy.^2.*(v.*2.0+yo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2+dy.^4.*k1.*

(v.*2.0+yo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(v+yo).^2.*2.0+dx.^2.*dy

.^2.*k1.*(v.*2.0+yo.*2.0).*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).*(u+xo).^2.*2.0)

.*1.0./(f.^2+dx.^2.*(k1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(u+xo).^2+dy.^2.*(k

1.*(dx.^2.*(u+xo).^2+dy.^2.*(v+yo).^2)+1.0).^2.*(v+yo).^2).^(3.0./2.0).*(-1.0./2.0);

sdv = [sxdv(:),sydv(:),szdv(:)];

sdv = bsxfun(@rdivide,sdv,sqrt(sum(sdv.^2,2)));

% create sparse matrix, 'H', used to calculate 'h', applied to previous iteration's sigma

vector

hustat = acos(dot(s(rightlog,:),s(right(rightlog),:),2));

hvstat = acos(dot(s(downlog,:),s(down(downlog),:),2));

H = sparse([Diu;Div+numpix],[Dju;Djv],[hustat;hustat;hvstat;hvstat]./2,2*numpix,numpix);

% apply to old sigma vector to get new 'h' values

227

% Aones = sparse([Diu;Div+numpix],[Dju;Djv],[-ones(size(Diu0));ones(size(Diu0));-

ones(size(Div0));ones(size(Div0))],2*numpix,numpix);

% to create matrix A = bsxfun(@rdivide,Aones,H*sigma_old);

% % give initial values for sigma

% sigold = 420.*ones(nnz(nmask),1);

% initial control point values

XinitFcn = TriScatteredInterp(xc(:),yc(:),xs(:)); % using instead of 'griddata' function

for speed

YinitFcn = TriScatteredInterp(xc(:),yc(:),ys(:));

xinit = XinitFcn(u,v);

yinit = YinitFcn(u,v);

zinit = interp2(xs,ys,zs,xinit,yinit);

initmir = bsxfun(@plus,Ropt*[xinit(:),yinit(:),zinit(:)].',Topt);

initcam = bsxfun(@plus,Rmir*initmir,Tmir);

sig = reshape(sqrt(sum(initcam.^2,1)),size(u));

siguess = sig;

% create sparse matrices to find discrete slope vectors (u and v) at each point

ucenter = find(rightlog & leftlog);

uleftedge = find(rightlog & ~leftlog);

urightedge = find(~rightlog & leftlog);

vcenter = find(downlog & uplog);

vtopedge = find(downlog & ~uplog);

vbottomedge = find(~downlog & uplog);

unextpix = [right(ucenter);right(uleftedge);urightedge];

uprevpix = [left(ucenter);uleftedge;left(urightedge)];

vnextpix = [down(vcenter);down(vtopedge);vbottomedge];

vprevpix = [up(vcenter);vtopedge;up(vbottomedge)];

Bu =

sparse(repmat([ucenter;uleftedge;urightedge],2,1),[unextpix;uprevpix],[ones(size(unextpix

));-ones(size(uprevpix))],numpix,numpix);

Bv =

sparse(repmat([vcenter;vtopedge;vbottomedge],2,1),[vnextpix;vprevpix],[ones(size(vnextpix

));-ones(size(vprevpix))],numpix,numpix);

% prepare edge pixel data

edge1 = imask(logical(lmask1)); % valid pixel indices of edge 1

edge2 = imask(logical(lmask2)); % valid pixel indices of edge 2

fidflag = false; % flag for fiducial registration

if any(lmask1(:)>0 & lmask1(:)<1) || any(lmask2(:)>0 & lmask2(:)<1)

fidflag = true;

wedge1 = lmask1(logical(lmask1));

wedge2 = lmask2(logical(lmask2));

end

count = 1;

if fidflag

numit = 20;

else

numit = 5;

228

end

sigiter = nan(numit,1);

sigmeans = sigiter;

sigiter(1) = 0;

sigmeans(1) = mean(sig);

% plot

close all;

figure(7);

siglin = plot(sigiter,sigmeans,'-b','linewidth',2);

set(gca,'ygrid','on');

drawnow

% sub sampling indicies for plots

% sampvec = 1:30:length(nmask);

% [ssu,ssv] = meshgrid(sampvec,sampvec);

% subsamp = [];

% for ti = 1:numel(ssu)

% subsamp = [subsamp;find(u==ssu(ti) & v==ssv(ti))];

% end

xcc = round(xc(2:(end-1),2:(end-1)));

ycc = round(yc(2:(end-1),2:(end-1)));

offscrind = find( xcc<1 | xcc>size(imask,2) | ycc<1 | ycc>size(imask,1) );

xcc(offscrind) = 1; % false x subscript for sample points outside the ROI where there is

no image data

ycc(offscrind) = 1; % false y subscript for sample points outside the ROI where there is

no image data

ic = sub2ind(size(imask),ycc,xcc);

ic(offscrind) = []; % get rid of indices from false subscripts

subsamp0 = imask(ic);

subsamp = subsamp0(:);

subsamp(subsamp<1) = [];

% corner indicies for mirror plane

ucorner = [1;1;size(nmask,2);size(nmask,2)];

vcorner = [1;size(nmask,1);size(nmask,1);1];

sxcorner =

dx.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).*(ucorner+xo).*1.0./sqrt(f.

^2+dx.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(ucorner+xo).^2+d

y.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(vcorner+yo).^2);

sycorner =

dy.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).*(vcorner+yo).*1.0./sqrt(f.

^2+dx.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(ucorner+xo).^2+d

y.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(vcorner+yo).^2);

szcorner =

f.*1.0./sqrt(f.^2+dx.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(u

corner+xo).^2+dy.^2.*(k1.*(dx.^2.*(ucorner+xo).^2+dy.^2.*(vcorner+yo).^2)+1.0).^2.*(vcorn

er+yo).^2);

scorner0 = [sxcorner(:),sycorner(:),szcorner(:)];

scorner1 = [scorner0;0,0,0].';

mircorner0 = Rmir.'*bsxfun(@minus,scorner1,Tmir);

229

mircorner1 = bsxfun(@minus,mircorner0(:,1:4),mircorner0(:,5));

sigcorner = -mircorner0(3,5)./mircorner1(3,1:4);

mircorner = bsxfun(@plus,mircorner0(:,5),bsxfun(@times,sigcorner,mircorner1));

scorner = bsxfun(@plus,Rmir*mircorner,Tmir).';

figure(1);

ho =

plot3(s(subsamp(:),1).*siguess(subsamp(:)),s(subsamp(:),2).*siguess(subsamp(:)),s(subsamp

(:),3).*siguess(subsamp(:)),'or');

ax1 = gca;

hold on;

hn =

plot3(s(subsamp(:),1).*sig(subsamp(:)),s(subsamp(:),2).*sig(subsamp(:)),s(subsamp(:),3).*

sig(subsamp(:)),'+b');

% hl = 30;

% plot3([lindat.pt1(1)+hl.*lindat.dir1(1).*[-1;1],lindat.pt2(1)+hl.*lindat.dir2(1).*[-

1;1]],...

% [lindat.pt1(2)+hl.*lindat.dir1(2).*[-1;1],lindat.pt2(2)+hl.*lindat.dir2(2).*[-

1;1]],...

% [lindat.pt1(3)+hl.*lindat.dir1(3).*[-1;1],lindat.pt2(3)+hl.*lindat.dir2(3).*[-

1;1]],'-k','linewidth',2);

%

plot3([lindat.allpts1(:,1);lindat.allpts2(:,1)],[lindat.allpts1(:,2);lindat.allpts2(:,2)]

,[lindat.allpts1(:,3);lindat.allpts2(:,3)],'.g');

patch(scorner(:,1),scorner(:,2),scorner(:,3),zeros(4,1),'facecolor','k','facealpha',0.5);

hold off;

axis equal;

set(gca,'ydir','reverse','zdir','reverse');

legend([ho,hn],{'Start','Final'});

for it = 1:numit

%%%%%%%%%%% iteration start

tic;

sigold = sig;

[~,~,Su,Sv] = slpuv(l,sigold,s,sdu,sdv);

DS = D*[Su;Sv];

h = H*sigold;

hu = h((1:numpix).');

hv = h(numpix+(1:numpix).');

A = sparse([Diu;Div+numpix],[Dju;Djv],[-1./hu(rightlog);1./hu(rightlog);-

1./hv(downlog);1./hv(downlog)],2*numpix,numpix);

sig0 = A\DS;

sigshift0 = mean(sigold)-mean(sig0);

% find piston error (sigshift)

options = optimset('TolX',0.1,'PlotFcns',{@optimplotfval,@optimplotx});

if fidflag

% sigshift = fminsearch(@(ss) mertfid(ss,sig0), sigshift0, options);

% [~,surfptsx,surfptsy,surfptsz] = mertfid(sigshift,sig0);

[sigshift,fidold,fidnew] = fidsig(sigshift0,sig0);

line(fidold(:,1),fidold(:,2),fidold(:,3),'color','g','marker','d','parent',ax1);

% line connecting fiducial points on each iteration

230

line(fidnew(:,1),fidnew(:,2),fidnew(:,3),'color','c','marker','d','parent',ax1);

% line connecting fiducial points on each iteration

else

sigshift = fminsearch(@(ss) mertedge(ss,sig0), sigshift0, options);

% [~,surfptsx,surfptsy,surfptsz,lindat] = mertedge(sigshift,sig0);

end

% [~,~,~,~,C1,C2] = mertcurv(sigshift,sig0);

sig = sig0+sigshift;

sigiter(it+1) = it+1;

sigmeans(it+1) = mean(sig);

set(siglin,'xdata',sigiter,'ydata',sigmeans);

[~,surfptsx,surfptsy,surfptsz] = mertedge(0,sig);

set(hn,'xdata',s(subsamp(:),1).*sig(subsamp(:)));

set(hn,'ydata',s(subsamp(:),2).*sig(subsamp(:)));

set(hn,'zdata',s(subsamp(:),3).*sig(subsamp(:)));

drawnow

if all(sig==sigold)

break

end

% sigshifts = linspace(sigshift0-200, sigshift0+600, 81);

% merts = zeros(size(sigshifts));

% merts0 = merts;

% merts1 = merts;

% for ij = 1:numel(sigshifts)

% merts(ij) = mertedge(sigshifts(ij),sig0);

% [merts0(ij),merts1(ij)] = mertslpmtch(sigshifts(ij),sig0);

% end

% [~,surfptsx,surfptsy,surfptsz] = mertedge(sigshift0,sig0);

% sig = sig0+sigshift0;

toc;

%%%%%%%%%%% iteration end

end

[~,~,~,~,C1,C2] = mertcurv(0,sig);

% % incident, reflected, and normal unit vectors for old sigma values

% figure(2);

% nquiver =

quiver3(s(subsamp(:),1).*sigold(subsamp(:)),s(subsamp(:),2).*sigold(subsamp(:)),s(subsamp

(:),3).*sigold(subsamp(:)),n(subsamp(:),1),n(subsamp(:),2),n(subsamp(:),3),0);

% axis equal;

% set(gca,'ydir','reverse','zdir','reverse');

% hold on;

% squiver = quiver3(s(subsamp(:),1).*(sigold(subsamp(:))-

1),s(subsamp(:),2).*(sigold(subsamp(:))-1),s(subsamp(:),3).*(sigold(subsamp(:))-

1),s(subsamp(:),1),s(subsamp(:),2),s(subsamp(:),3),0,'color','r');

% rquiver =

quiver3(s(subsamp(:),1).*sigold(subsamp(:)),s(subsamp(:),2).*sigold(subsamp(:)),s(subsamp

(:),3).*sigold(subsamp(:)),r(subsamp(:),1),r(subsamp(:),2),r(subsamp(:),3),0,'color','g')

;

231

% % ulins =

permute(cumsum(cat(3,bsxfun(@times,s(subsamp(:),:),sigold(subsamp(:))),sdu(subsamp(:),:),

bsxfun(@times,s(subsamp(:),:),Su(subsamp(:)))),3),[3,1,2]);

% % plot3(ulins(:,:,1),ulins(:,:,2),ulins(:,:,3),'-k');

% hold off;

% new surface in mirror reference frame

figure(3);

surfl(surfptsx,surfptsy,surfptsz);

shading interp;

colormap(gray);

axis equal;


% % ray error plot with sigshift slider

% figure(4);

% panh = uipanel('units','normal','position',[0 0 .9 1]);

% axes('parent',panh);

% rays =

plot3(s(subsamp(:),1).*(sig0(subsamp(:))+sigshift),s(subsamp(:),2).*(sig0(subsamp(:))+sig

dhift),s(subsamp(:),3).*(sig0(subsamp(:))+sigshift),'+b');

% % merit functions plot

% norm21 = @(x) (x-min(x(:)))./(max(x(:))-min(x(:)));

% merts = norm21(merts);

% merts0 = norm21(merts0);

% merts1 = norm21(merts1);

% figure(5);

% mertlins = plot(sigshifts,merts,sigshifts,merts0,sigshifts,merts1);

% legend(mertlins,{'Part Size','Raytrace Distance','Surface Normal Angle'});

% curvatures plot

opengl software;

figure(6);

set(6,'position',[194 61 1469 376]);

clohi = round(numpix.*[0.1,0.9]);

C12sort = sort([C1(:);C2(:)]);

C1t2sort = sort(C1(:).*C2(:));

C12lim = C12sort(clohi.*2).';

C1t2lim = C1t2sort(clohi).';

axc(1) = subplot_tight(1,7,1:2,0.05);

im(1) = imagesc(C1,C12lim);

axis image off;

colorbar;

title('Principle Curvature 1 = k_1');


im(2) = imagesc(C2,C12lim);

axis image off;

colorbar;

title('Principle Curvature 2 = k_2');


232

im(3) = imagesc(C1.*C2,C1t2lim);

axis image off;

colorbar;

title('Gaussian Curvature = k_1 x k_2');

linkaxes(axc);

set(im, 'AlphaData', ~isnan(C1));

% save_to_base(1);pause(1);

function [dsp,fidold,fidnew] = fidsig(ds,sig0)

% function calculates new sigshift(="delta sigma prime"=dsp) given current

sigshift(="delta sigma"=ds) and distance between fiducials

% get xyz edge points in camera reference frame

edge3d1 = bsxfun(@times,s(edge1,:),sig0(edge1)+ds);

edge3d2 = bsxfun(@times,s(edge2,:),sig0(edge2)+ds);

fid1 = sum(bsxfun(@times,wedge1,edge3d1),1); % xyz coords of fiducial point 1

fid2 = sum(bsxfun(@times,wedge2,edge3d2),1); % xyz coords of fiducial point 2

fidold = [fid1;fid2];

adist = sqrt(sum((fid2-fid1).^2)); % current distance between fiducials

r = lindist/adist; % ratio of actual distance between fiducials to calculated one

s1pds = norm(fid1);

s2pds = norm(fid2);

cosgam = sum(fid1.*fid2)/(s1pds*s2pds); % cosine of angle between fiducial

vectors (from camera origin)

s1 = s1pds-ds; % distance to fiducial 1 without sigshift(ds)

s2 = s2pds-ds;

a = 2*(1-cosgam); % quadratic formula "a"

b = 2*(1-cosgam)*(s1+s2); % quadratic formula "b"

c = s1^2+s2^2-2*s1*s2*cosgam-r^2*(s1pds^2+s2pds^2-2*s1pds*s2pds*cosgam); %

quadratic formula "c"

dsp = (-b+sqrt(b^2-4*a*c))/(2*a); % "delta sigma prime", take "+" solution since

the "-" solution will put the part behind the camera

edge3d1n = bsxfun(@times,s(edge1,:),sig0(edge1)+dsp);

edge3d2n = bsxfun(@times,s(edge2,:),sig0(edge2)+dsp);

fid1n = sum(bsxfun(@times,wedge1,edge3d1n),1); % xyz coords of new fiducial point

1

fid2n = sum(bsxfun(@times,wedge2,edge3d2n),1); % xyz coords of new fiducial point

2

fidnew = [fid1n;fid2n];

end

function [mert,surfptsx,surfptsy,surfptsz] = mertfid(sigshift,sig0)

% get xyz edge points in mirror reference frame

surfpts0 = (Rmir.'*bsxfun(@minus,bsxfun(@times,s,sig0+sigshift).',Tmir)).'; %

into mirror reference frame

edge3d1 = surfpts0(edge1,:);


fid1 = sum(bsxfun(@times,wedge1,edge3d1),1);

fid2 = sum(bsxfun(@times,wedge2,edge3d2),1);

adist = sqrt(sum((fid2-fid1).^2));

mert = abs(adist-lindist);

233

if nargout > 1

surfptsx = nan(size(nmask));

surfptsy = nan(size(nmask));

surfptsz = nan(size(nmask));

surfptsx(nmask) = surfpts0(:,1);

surfptsy(nmask) = surfpts0(:,2);

surfptsz(nmask) = surfpts0(:,3);

end

end

function [mert,surfptsx,surfptsy,surfptsz,lindat] = mertedge(sigshift,sig0)

% get xyz edge points in mirror reference frame





L1 = mean(edge3d1,1);

L2 = mean(edge3d2,1);

[coeff1] = princomp(edge3d1);

[coeff2] = princomp(edge3d2);

dir1 = coeff1(:,1).';

dir2 = coeff2(:,1).';

dist1 = norm(L1-L2-dot(L1-L2,dir2).*dir2);

dist2 = norm(L2-L1-dot(L2-L1,dir1).*dir1);

adist = (dist1+dist2)/2;

mert = abs(adist-lindist);

if nargout > 1







lindat =

struct('pt1',L1,'dir1',dir1./norm(dir1),'allpts1',edge3d1,'pt2',L2,'dir2',dir2./norm(dir2

),'allpts2',edge3d2);

end

end

function [mert,surfptsx,surfptsy,surfptsz,C1,C2] = mertcurv(sigshift,sig0)

% get xyz surface points in mirror reference frame







234



% find surface curvatures

distx = diff(surfptsx,1,2);

disty = diff(surfptsy,1,1);

fx = diff(surfptsz,1,2)./distx;

fxx = diff(fx,1,2)./distx(:,1:(end-1));

fxx = fxx(1:(size(nmask,1)-2),1:(size(nmask,2)-2));

fy = diff(surfptsz,1,1)./disty;

fyy = diff(fy,1,1)./disty(1:(end-1),:);

fyy = fyy(1:(size(nmask,1)-2),1:(size(nmask,2)-2));

fxy = diff(fx,1,1)./disty(:,1:(end-1));

fxy = fxy(1:(size(nmask,1)-2),1:(size(nmask,2)-2));

fx = fx(1:(size(nmask,1)-2),1:(size(nmask,2)-2));

fy = fy(1:(size(nmask,1)-2),1:(size(nmask,2)-2));

K = (fxx.*fyy-fxy.^2)./(1+fx.^2+fy.^2).^2; % gaussian curvature

Hc = ((1+fx.^2).*fyy+(1+fy.^2).*fxx-2.*fx.*fy.*fxy)./(2.*(1+fx.^2+fy.^2).^(3/2));

% mean curvature

kern = Hc.^2-K;

C1 = Hc+sqrt(kern); % principal curvature 1

C2 = Hc-sqrt(kern); % principal curvature 2

% calculate merit value (sum of minimum absolute curvature)

mert0 = min(abs(C1),abs(C2))./max(abs(C1),abs(C2));

mert = nansum(K(:));

end

function [mert,mert0,surfptsx,surfptsy,surfptsz,monpts,nangs] =

mertslpmtch(sigshift,sig0)

nrefl = slpuv(l,sig0+sigshift,s,0,0);

% get xyz surface points in mirror reference frame



if nargout > 1







end

uvec = Bu*surfpts0;

vvec = Bv*surfpts0;

ndisc0 = cross(uvec,vvec,2);

ndisc = bsxfun(@rdivide,ndisc0,sqrt(nansum(ndisc0.^2,2)));

nangs = acos(nansum(nrefl.*ndisc,2));

mert0 = sqrt(sum(nangs(:).^2));

235

% find reflection vectors

reflvec = s - bsxfun(@times, 2.*sum(ndisc.*s,2), ndisc);

% move S and S+r into the monitor reference frame

S = bsxfun(@times,sig0+sigshift,s);

Spr = S+reflvec;

amon = (R.'*bsxfun(@minus,S.',T)).';

bmon = (R.'*bsxfun(@minus,Spr.',T)).'-amon;

% find positions of reflected rays on monitor (z=0 plane)

xmon = -amon(:,3)./bmon(:,3).*bmon(:,1) + amon(:,1);

ymon = -amon(:,3)./bmon(:,3).*bmon(:,2) + amon(:,2);

if nargout > 4

monpts = bsxfun(@plus,R*[xmon,ymon,zeros(size(xmon))].',T).';

end

% find positional error

err = sqrt( (xmon-unwpx(nmask)).^2 + (ymon-unwpy(nmask)).^2 );

mert = sqrt(nansum(err(:).^2)); % sum of squares

if count==1

assignin('base','ndisc',ndisc);

count = 2;

end

end

end

function [n,r,Su,Sv] = slpuv(l,sig,s,sdu,sdv) % for l, s, sdu, and sdv, rows are x,y,z

components of vectors pertaining to each data pixel, sig is a column vector

rlong = l-bsxfun(@times,s,sig);

r = bsxfun(@rdivide,rlong,sqrt(sum(rlong.^2,2)));

nlong = s-r;

n = bsxfun(@rdivide,nlong,sqrt(sum(nlong.^2,2)));

if nargout > 2

Su = -sum(sdu.*n,2)./sum(s.*n,2);

Sv = -sum(sdv.*n,2)./sum(s.*n,2);

end

end

Dependency: DP_MaskEditor.m

Manual noise mask editor implemented in case of obvious errors in

DP_DualNoiseDeleteFilter.m. User may “paint” new valid regions as well as delete mistakenly

valid regions.

function nmask =

DP_MaskEditor(nmasko,v1234lincrop,h1234lincrop,wpx,wpy,unwpx,unwpy,fidpicrop)

236

% process images to be uint8 rgb and store in a cell array

imnames = {'Original Mask';'Vertical Fringe 1';'Vertical Fringe 2';'Vertical Fringe

3';'Vertical Fringe 4';'Vertical Line';...

'Horizontal Fringe 1';'Horizontal Fringe 2';'Horizontal Fringe 3';'Horizontal Fringe

4';'Horizontal Line';...

'Wrapped X Phase';'Wrapped Y Phase';'Unwrapped X Phase';'Unwrapped Y Phase';'Fiducial

Picture';'None'};

nmask = nmasko;

border = bordcalc(nmask);

numimg = length(imnames);

bgs = cell(numimg,1);%uint8(zeros([size(nmask),3,numimg])); % background images

for im = 1:(numimg-1)

switch im

case 1

imin = nmasko;

case {2,3,4,5,6}

imin = v1234lincrop(:,:,im-1);

case {7,8,9,10,11}

imin = h1234lincrop(:,:,im-6);

case 12

imin = wpx;

case 13

imin = wpy;

case 14

imin = unwpx;

case 15

imin = unwpy;

case 16

imin = fidpicrop;

case 17

imin = zeros(size(nmask));

end

if (max(imin(:))-min(imin(:)))==0

iminsca = uint8(zeros(size(nmask)));

else

imin = double(imin);

iminsca = uint8((imin-min(imin(:)))./(max(imin(:))-min(imin(:))).*255); % scale

to [0,255] and convert to uint8 to save memory

end

bgs{im} = repmat(iminsca,[1,1,3]);

clear imin

end

clear v1234lincrop h1234lincrop unwpx unwpy wpx wpy fidpicrop

% initialize figure

mfig = pptfig('Mask Editor',1);

opengl hardware;

ofigpos = get(mfig,'position');

imsiz = size(nmask);

wid = 250;

237

figsiz = [round(ofigpos(4)/imsiz(1)*imsiz(2)+wid),ofigpos(4)]; % figure size [width,

height]: 'wfactor' times width of image

set(mfig,'position',[ofigpos(1:2),figsiz],'PointerShapeCData',nan(16),'WindowButtonMotion

Fcn',@pointerswitch);

imax =

axes('units','pixels','position',[1,1,round(ofigpos(4)/imsiz(1)*imsiz(2)),ofigpos(4)]);

img = imshow(bgs{1});

axis image

ovax =

axes('units','pixels','position',[1,1,round(ofigpos(4)/imsiz(1)*imsiz(2)),ofigpos(4)],'vi

sible','off');

noverlay =

imshow(uint8(cat(3,zeros(size(nmask)),zeros(size(nmask)),255.*ones(size(nmask)))));

axis image

linkaxes([imax,ovax]);

% hold on

% noverlay =

imshow(uint8(cat(3,zeros(size(nmask)),zeros(size(nmask)),255.*ones(size(nmask)))));

% hold off

alph = 0.5;

panh =

uipanel(mfig,'units','pixels','position',[round(ofigpos(4)/imsiz(1)*imsiz(2))+1,1,figsiz(

1)-round(ofigpos(4)/imsiz(1)*imsiz(2)),ofigpos(4)]);

butg =

uibuttongroup('units','pixels','position',[round(ofigpos(4)/imsiz(1)*imsiz(2))+1,1,figsiz

(1)-round(ofigpos(4)/imsiz(1)*imsiz(2)),ofigpos(4)]);

% controls

ctexts = {'Overlay','Border','Background','Brush Size (px)','Draw','Erase','Crop To

Original','Reset'};

cstyles =

{'checkbox','checkbox','popupmenu','edit','radio','radio','pushbutton','pushbutton'};

cbacks = {@drawoverlay, @drawoverlay, @backswitch, @brushsize, 0, 0, @crop2org, @reset};

cposind = [1,2,3,5,6,7,9,10];

numcon = length(ctexts);

ch = 0.05;

chan = zeros(numcon,1);

for ci = 1:numcon

ctext = ctexts{ci};

cstyle = cstyles{ci};

cback = cbacks{ci};

cposv = 1-(cposind(ci)/max(cposind))+(1/max(cposind)-ch)/2;

if any(strcmp(cstyle,{'pushbutton','checkbox'}));

chan(ci) =

uicontrol(panh,'style',cstyle,'units','normal','position',[0.3,cposv,0.4,ch],'string',cte

xt,'callback',cback);

elseif strcmp(cstyle,'radio');

chan(ci) =

uicontrol(butg,'style',cstyle,'units','normal','position',[0.3,cposv,0.4,ch],'string',cte

xt);

else

238

uicontrol(panh,'style','text','units','normal','position',[0.05,cposv,0.4,ch],'string',ct

ext,'horizontalalignment','right');

chan(ci) =

uicontrol(panh,'style',cstyle,'units','normal','position',[0.5,cposv,0.45,ch],'string',ct

ext,'horizontalalignment','left','min',0,'max',1,'callback',cback);

end

end

set(chan(1:2),'value',1);

set(chan(3),'string',imnames);

brushsiz = [10,10];

set(chan(4),'string',num2str(brushsiz(1)));

brush =

rectangle('hittest','off','edgecolor','none','facecolor','r','position',[1,1,brushsiz],'v

isible','off');

set(chan(3),'value',2);

backswitch(chan(3),0);

drawoverlay();

%debug

set(mfig,'WindowButtonDownFcn',@(~,~) disp(get(ovax,'currentpoint')));

uiwait(mfig);

%% Nested Functions

function reset(~,~)

nmask = nmasko;


drawoverlay();

end

function crop2org(~,~)

nmask = nmask & nmasko;


drawoverlay();

end

function brushsize(~,~)

val = 10; % default brush size

newsiz = str2double(get(chan(4),'string'));

if isnumeric(newsiz) && ~isnan(newsiz)

val = round(abs(newsiz));

val = max(val,1);

val = min(val,50);

end

brushsiz = [val,val];

set(chan(4),'string',num2str(val));

end

function pointerswitch(~,~)

239

cp = get(ovax,'currentpoint');

cp = cp(1,1:2);

axlim = [get(ovax,'xlim');get(ovax,'ylim')].';

if all(cp>=axlim(1,:)) && all(cp<=axlim(2,:))

set(mfig,'pointer','custom','WindowButtonDownFcn',@brushclick);

set(brush,'visible','on','position',[round(cp)-0.5,brushsiz]);

else

set(mfig,'pointer','arrow','WindowButtonDownFcn',[]);

set(brush,'visible','off');

end

end

function brushclick(~,~)

if get(butg,'SelectedObject')==chan(5)

val = 1;

elseif get(butg,'SelectedObject')==chan(6)

val = 0;

end

set(mfig,'WindowButtonMotionFcn',{@paint,val},'WindowButtonUpFcn',@brushrelease);

paint(0,0,val);

end

function brushrelease(~,~)

set(mfig,'WindowButtonMotionFcn',@pointerswitch,'WindowButtonUpFcn',[]);

end

function paint(~,~,val)

cp = get(ovax,'currentpoint');

cp = round(cp(1,1:2));

rows = cp(2):(cp(2)+brushsiz(2)-1);

rows(rows>imsiz(1) | rows<1) = [];

cols = cp(1):(cp(1)+brushsiz(1)-1);

cols(cols>imsiz(2) | cols<1) = [];

nmask(rows,cols) = val;


set(brush,'position',[round(cp)-0.5,brushsiz]);

drawoverlay();

end

function backswitch(handle,~)

picind = get(handle,'value');

set(img,'cdata',bgs{picind});

drawnow expose;

end

function drawoverlay(varargin)

overval = [get(chan(1),'value'),get(chan(2),'value')];

if all(overval==[0,0])

adat = zeros(size(nmask));

elseif all(overval==[1,0])

adat = nmask.*alph;

240


adat = border;


adat = nmask.*alph+border.*(1-alph);

end

set(noverlay,'alphadata',adat);

end

end

function border = bordcalc(nmask)

nmasku = circshift(nmask,[1,0]);

nmaskd = circshift(nmask,[-1,0]);

nmaskl = circshift(nmask,[0,1]);

nmaskr = circshift(nmask,[0,-1]);

nmasku(1,:) = 0;

nmaskd(end,:) = 0;

nmaskl(:,1) = 0;

nmaskr(:,end) = 0;

border = nmask & (~nmasku|~nmaskd|~nmaskl|~nmaskr);

end

Dependency: DP_PhaseUnwrapper.m

Phase unwrapping routine that calls on the Puma algorithm [50].

function [unwpx,unwpy,croprect,xyoffset,nmask,vphir,hphir] =

DP_PhaseUnwrapper(v1234lin,h1234lin,linmm,permm)

v = double(v1234lin);

h = double(h1234lin);

vphi = atan2(v(:,:,4)-v(:,:,2),v(:,:,1)-v(:,:,3));

hphi = atan2(h(:,:,4)-h(:,:,2),h(:,:,1)-h(:,:,3));

% plot phase

figwp = pptfig('Wrapped Phase',1);

set(figwp,'position',[100 100 500 800]);

axv = subplot(2,1,1);

colormap(gray);

imagesc(vphi);

axis equal;

title('Interferogram (vert)');

axh = subplot(2,1,2);

colormap(gray);

imagesc(hphi);

axis equal;

title('Interferogram (horz)');

% pick ROI

241

roiconstraintfcn = makeConstrainToRectFcn('impoly',[1,size(vphi,2)],[1,size(vphi,1)]);

roi = impoly(axv,'PositionConstraintFcn',roiconstraintfcn);

wait(roi);

set(roi,'visible','off');

roimask = createMask(roi); % polygon mask

% vphi(~roimask) = nan; % nan out the area outside the ROI polygon to speed up subsequent

calcs

% hphi(~roimask) = nan; % ^

polypos = getPosition(roi);

croprect = [min(polypos(:,1)), min(polypos(:,2)), max(polypos(:,1))-min(polypos(:,1)),

max(polypos(:,2))-min(polypos(:,2))]; % bounding rectangle of ROI polygon

roimaskcrop = imcrop(roimask,croprect);

[xg,yg] = meshgrid(1:1:size(v1234lin,2),1:1:size(v1234lin,1));

xgc = imcrop(xg,croprect);

ygc = imcrop(yg,croprect);

xyoffset = [min(xgc(:)),min(ygc(:))]-1;

rectangle('Position',croprect,'parent',axv,'edgecolor','b','linestyle','--');

rectangle('Position',croprect,'parent',axh,'edgecolor','b','linestyle','--');

line('xdata',[polypos(:,1);polypos(1,1)],'ydata',[polypos(:,2);polypos(1,2)],'linestyle',

'-','color','b','parent',axv);

line('xdata',[polypos(:,1);polypos(1,1)],'ydata',[polypos(:,2);polypos(1,2)],'linestyle',

'-','color','b','parent',axh);

vphir = imcrop(vphi,croprect);

hphir = imcrop(hphi,croprect);

drawnow expose;

pause(0.5);

% discmask = DP_DualNoiseDeleteFilter(cat(3,vphir,hphir),roimaskcrop);

% % vphir(~discmask) = 0;

% % hphir(~discmask) = 0;

% discmaske = imerode(discmask,strel('disk',5));

% discmaskd = imdilate(discmask,strel('disk',5));

% PUMA processing

p=2; % Clique potential exponent

cliques = [1 0; 0 1];%; 1 1; 1 -1];

figpu = pptfig('PUMA Process',2);

set(figpu,'position',[600 100 500 800]);

axvpu = subplot(2,1,1);

[unwpv,iterv,erglistv] =

DP_puma_ho(vphir,p,axvpu,'schedule',[1],'cliques',cliques,'roi',roimaskcrop);%,'qualityma

ps',repmat(~discmaske,[1,1,size(cliques,1)])

axis equal;

title('Vertical Fringe Unwrap');

axhpu = subplot(2,1,2);

[unwph,iterh,erglisth] =

DP_puma_ho(hphir,p,axhpu,'schedule',[1],'cliques',cliques,'roi',roimaskcrop);%,'qualityma

ps',repmat(~discmaske,[1,1,size(cliques,1)])

axis equal;

title('Horizontal Fringe Unwrap');

242

% nmaskv = DP_NoiseDeleteFilter(unwpv,roimaskcrop);

% nmaskh = DP_NoiseDeleteFilter(unwph,roimaskcrop);

% nmask = nmaskv & nmaskh;

nmask = DP_DualNoiseDeleteFilter(cat(3,unwpv,unwph),roimaskcrop);

% convert phase to mm

unwpx = unwpv./2./pi.*permm(1);

vlin = imcrop(v(:,:,5),croprect);

vlin = vlin.*nmask;

vlinbw = im2bw(vlin, graythresh(vlin));

unwpx = unwpx-mean(unwpx(vlinbw))+linmm(1);

unwpxp = unwpx;

unwpxp(~nmask) = nan;

unwpy = unwph./2./pi.*permm(2);

hlin = imcrop(h(:,:,5),croprect);

hlin = hlin.*nmask;

hlinbw = im2bw(hlin, graythresh(hlin));

unwpy = unwpy-mean(unwpy(hlinbw))+linmm(2);

unwpyp = unwpy;

unwpyp(~nmask) = nan;

% plot unwrapped data

figuw = pptfig('Position of Reflected Ray on Monitor',3);

set(figuw,'position',[1100 100 500 800]);

subplot(2,1,1);

surfl(unwpxp);

shading interp;

colormap(gray);

axis equal;

set(gca,'ydir','reverse');

title('X Monitor Position');

subplot(2,1,2);

surfl(unwpyp);

shading interp;

colormap(gray);

axis equal;

set(gca,'ydir','reverse');

title('Y Monitor Position');

end

Dependency: DP_placepart3d_edge.m

Provides initial estimate of observed surface position and orientation for input into

DP_lsqiter.

243

function [xs,ys,zs,xc,yc,Ropt,Topt,lindist,lmask1,lmask2,shapename] =

DP_placepart3d_edge(f,k1,cxcy,dxy,Rmir,Tmir,nmask,xyoffset,shapename,fidpic,fidist)

close all;

samples = [30,30];

[x0,y0,z0,bordfcn] = DP_SLSurfacePoints(shapename,[10,5]);

pt1ind = 1;

pt2ind = sub2ind(size(x0),size(x0,1),1);

initoffset = [-x0(pt1ind),-y0(pt1ind)];

x0 = x0+initoffset(1);

y0 = y0+initoffset(2);

xb = bordfcn(x0);

yb = bordfcn(y0);

zb = bordfcn(z0);

pt1 = [x0(pt1ind);y0(pt1ind);z0(pt1ind)];


dz = pt2(3)-pt1(3);

dxysq = sum((pt2(1:2)-pt1(1:2)).^2);

o = Rmir.'*(zeros(3,1)-Tmir);

% display 3d part surface model for reference

f1 = figure();

surf(x0,y0,z0,'facecolor','b','facealpha',0.3,'edgecolor','none');

axis equal;

rotate3d on;


line('xdata',xb,'ydata',yb,'zdata',zb,'color','b','linewidth',2);

selecdot =

line('xdata',pt1(1),'ydata',pt1(2),'zdata',pt1(3),'marker','o','markeredgecolor','k','mar

kerfacecolor','r','markersize',10);

% display picture of masked unwrapping results

f2 = figure('position',[56 110 687 797]);

axpic = axes();

impic = imagesc(nmask);

colormap(gray);

axis image off;

contbut = uicontrol(f2,'units','normal','position',[.4 .02 .2

.07],'string','OK','visible','off','fontsize',16,'fontweight','bold');

set(f2,'name','Indicate Position of Point 1');

nerode = imerode(nmask,strel('diamond',1));

nbord = nmask & ~nerode;

[ybord,xbord] = find(nbord);

line('xdata',xbord,'ydata',ybord,'linestyle','none','color','g','marker','.','markersize'

,0.5);

244

% get input for first point

xcam10 = 0;

ycam10 = 0;

set(f2,'WindowButtonDownFcn',{@getcp,axpic},'pointer','crosshair');

uiwait(f2);

set(f2,'WindowButtonDownFcn',[],'pointer','arrow');

xcam1 = xcam10+xyoffset(1);

ycam1 = ycam10+xyoffset(2);

campt1 = [xcam1,ycam1];

campt2 = campt1+1;

% create polygon for displaying surface

polyg =

patch('vertices',[xb+xcam10,yb+ycam10],'faces',1:numel(xb),'facecolor','b','facealpha',0.

3,'edgecolor','b','linewidth',1.5,'parent',axpic);

ptmark =

line('xdata',xb([pt1ind;pt2ind])+xcam10,'ydata',yb([pt1ind;pt2ind])+ycam10,'linestyle','n

one','marker','o','markeredgecolor','k','markerfacecolor','r','markersize',10);

set(selecdot,'xdata',pt2(1),'ydata',pt2(2),'zdata',pt2(3));

% create figure of s1, s2, and surface in mirror reference frame

f3 = figure();

borderline = plot3(xb,yb,zb,'-b','linewidth',2);

axis equal;

grid on;

s1temp = [0;0;1].*abs(o(3));

l1 =

line('xdata',o(1)+[0;s1temp(1)],'ydata',o(2)+[0;s1temp(2)],'zdata',o(3)+[0;s1temp(3)],'co

lor','k','linewidth',2);

l2 =

line('xdata',o(1)+[0;s1temp(1)],'ydata',o(2)+[0;s1temp(2)],'zdata',o(3)+[0;s1temp(3)],'co

lor','r','linewidth',2);

set(gca,'xlimmode','auto','ylimmode','auto','zlimmode','auto','ydir','reverse','zdir','re

verse');

xlabel('X');ylabel('Y');zlabel('Z');

% create callback to draw surface in real time

axes(axpic);

set(f2,'name','Indicate Position of Point 2');

set(f2,'WindowButtonMotionFcn',@setpt2,'WindowButtonDownFcn',@selpt);

set(contbut,'visible','on','callback',@cont);

if isnumeric(shapename) % if surface is flat, let user change dimensions

shapedit(1) = uicontrol(f2,'style','edit','units','normal','position',[.02 .01 .1

.03],'string',num2str(shapename(1)),'userdata',shapename(1),'callback',@editflat);

shapedit(2) = uicontrol(f2,'style','edit','units','normal','position',[.14 .01 .1

.03],'string',num2str(shapename(2)),'userdata',shapename(2),'callback',@editflat);

end

uiwait(f2);

if isnumeric(shapename)

set(shapedit,'visible','off');

245

end

set(contbut,'visible','off');

[Rpart,Tpart] = pts2pos(campt1,campt2);

close([f1 f3]);

xlist0 = 1:size(nmask,2);

ylist0 = 1:size(nmask,1);

[xlist,ylist] = meshgrid(xlist0,ylist0);

%%%%%%%%%% Optimization Start %%%%%%%%%%%%%

% fminsearch (quicker, better if close to right answer)

rpyt0 = [0,0,atan2(Rpart(2),Rpart(1)),Tpart(:).'];

options = optimset('PlotFcns',@optimplotfval,'TolX',1e-2);%,'MaxIter',30);

rpytopt = fminsearch(@edgemert,rpyt0,options);

%%%% OR %%%%%

% % patternsearch (slower, better for bad initial guess?)

% rpyt0 = [0,0,atan2(Rpart(2),Rpart(1)),Tpart(:).'];

% angtol = [10,10,5]./180.*pi; % radians

% tratol = [3,3,20]; % mm

% rpytbnd = [rpyt0(1:3)-angtol,rpyt0(4:6)-tratol;...

% rpyt0(1:3)+angtol,rpyt0(4:6)+tratol];

% psoptions = psoptimset('UseParallel','always','TolMesh',1e-

3,'PlotFcns',{@psplotbestf,@psplotbestx,@psplotmeshsize},'SearchMethod',@MADSPositiveBasi

s2N,'CompletePoll','on');

% psproblem = struct('objective',@(x) posmertsca(x,rpyt0,rpytbnd),...

% 'X0', zeros(1,6),...

% 'Aineq',[],'bineq',[],'Aeq',[],'beq',[],...

% 'lb', -ones(1,6), 'ub', ones(1,6),...

% 'nonlcon',[],...

% 'solver', 'patternsearch',...

% 'options', psoptions,...

% 'rngstate',[]);

% rpytoptsca = patternsearch(psproblem);

% rpytopt = rpytoptsca.*diff(rpytbnd,1,1)./2+rpyt0;

%%%% OR %%%%%

% % two stage fminsearch, high level rotation only optimization with a

% % tranlastion only optimization on every iteration

% % very slow, but gives the good results

% rpyt0 = [0,0,atan2(Rpart(2),Rpart(1)),Tpart(:).'];

% % angtol = [7,7,3]./180.*pi; % radians

% % tratol = [3,3,20]; % mm

% % rpytbnd = [rpyt0(1:3)-angtol,rpyt0(4:6)-tratol;...

% % rpyt0(1:3)+angtol,rpyt0(4:6)+tratol];

% % options =

optimset('PlotFcns',@optimplotfval,'UseParallel','never','algorithm','interior-

point');%,'MaxIter',30);

246

% % ropt = fmincon(@(r)

posmert2stage([r,rpyt0(4:6)]),rpyt0(1:3),[],[],[],[],rpytbnd(1,1:3),rpytbnd(2,1:3),[],opt

ions); % almost always accepts initial point as optimized

% options = optimset('TolX',(1e-2)/180*pi,'display','iter');

% ropt = fminsearch(@(r) posmert2stage([r,rpyt0(4:6)]),rpyt0(1:3),options);

% [~,~,Topt] = posmert2stage([ropt,rpyt0(4:6)]);

% rpytopt = [ropt,Topt(:).'];

%%%%%%%%%%% Optimization End %%%%%%%%%%%%%%%

[mert,Ropt,Topt] = edgemert(rpytopt);

set(f2,'name',num2str(mert));

drawpoly(Ropt,Topt);

[xs,ys,zs,~,lindist,lind1,lind2,lindista,lind1a,lind2a] =

DP_SLSurfacePoints(shapename,samples);

bufs = [max(xs(:))-min(xs(:)),max(ys(:))-min(ys(:))]./2;

xs = [xs(:,1)+bufs(1).*sign(xs(:,1)-xs(:,2)), xs, xs(:,end)+bufs(1).*sign(xs(:,end)-

xs(:,end-1))];

ys = padarray(ys,[0,1],'replicate','both');

zs = [zs(:,1)+bufs(1)./abs(xs(:,2)-xs(:,3)).*(zs(:,1)-zs(:,2)), zs,

zs(:,end)+bufs(1)./abs(xs(:,end-1)-xs(:,end-2)).*(zs(:,end)-zs(:,end-1))];

xs = padarray(xs,[1,0],'replicate','both');

ys = [ys(1,:)+bufs(2).*sign(ys(1,:)-ys(2,:)); ys; ys(end,:)+bufs(2).*sign(ys(end,:)-

ys(end-1,:))];

zs = [zs(1,:)+bufs(2)./abs(ys(2,:)-ys(3,:)).*(zs(1,:)-zs(2,:)); zs;

zs(end,:)+bufs(2)./abs(ys(end-1,:)-ys(end-2,:)).*(zs(end,:)-zs(end-1,:))];

xs = xs+initoffset(1);

ys = ys+initoffset(2);

samptsraw = [xs(:),ys(:),zs(:)].';

samptsposd = bsxfun(@plus,Ropt*samptsraw,Topt);

xr = reshape(samptsposd(1,:),size(xs));

yr = reshape(samptsposd(2,:),size(ys));

zr = reshape(samptsposd(3,:),size(zs));

[samptsx0,samptsy0] = world2cam(xr(:),yr(:),zr(:),Rmir,Tmir,f,k1,cxcy,dxy);

samptsx = samptsx0-xyoffset(1);

samptsy = samptsy0-xyoffset(2);

samplin =

line('xdata',samptsx,'ydata',samptsy,'color','r','marker','+','linestyle','none','parent'

,axpic);

xc = reshape(samptsx,size(xs));

yc = reshape(samptsy,size(ys));

% get sample points on the two opposing straight edges

xcc = xc(2:(end-1),2:(end-1));

ycc = yc(2:(end-1),2:(end-1));

choices = {'long edges','short edges','fiducials'};

if isnumeric(shapename) % if surface is nominally flat

choice = listdlg('PromptString','Choose registering

option','SelectionMode','single','ListString',choices);

if isempty(choice)

247

choice = 1;

end

else % curved surface

choice = listdlg('PromptString','Choose registering

option','SelectionMode','single','ListString',choices([1,3]));

if isempty(choice)

choice = 1;

elseif choice == 2

choice = 3;

end

end

switch choice

case 1

linc = cat(3,[xcc(lind1),ycc(lind1)],[xcc(lind2),ycc(lind2)]);

case 2

lind1 = lind1a;

lind2 = lind2a;

lindist = lindista;

linc = cat(3,[xcc(lind1),ycc(lind1)],[xcc(lind2),ycc(lind2)]);

case 3

lindist = fidist;

set(impic,'cdata',fidpic)

set(samplin,'visible','off');

fidroi = imline(axpic,[50,50;100,50]);

fidpos = wait(fidroi);

lmask12 = zeros([size(nmask),2]);

for ij = 1:2

xfid = fidpos(ij,:).';

pfid = bsxfun(@plus,floor(xfid),[0,0,1,1;0,1,0,1]);

wfid = 1./sqrt(sum(bsxfun(@minus,pfid,xfid).^2,1)).'; % 1/(distance to 4

surrounding vertices)

wfid = wfid./sum(wfid);

for jk = 1:size(pfid,2)

lmask12(pfid(2,jk),pfid(1,jk),ij) = wfid(jk);

end

end

lmask1 = lmask12(:,:,1);

lmask2 = lmask12(:,:,2);

overlayflag = 2;

end

if choice==1 || choice==2

% get distances from aforementioned edges

dlin = zeros(numel(xbord),2);

for li = 1:2

X000 = linc(:,1,li) * ones(1,numel(xbord));

Y000 = linc(:,2,li) * ones(1,numel(ybord));

x000 = ones(size(linc,1),1) * xbord(:).';

y000 = ones(size(linc,1),1) * ybord(:).';

distsl = (x000-X000).^2+(y000-Y000).^2;

[Dl,rowind0l] = min(distsl,[],1);

248

rowind0l(rowind0l==1) = 2;

rowind0l(rowind0l==size(linc,1)) = size(linc,1)-1;

rowindl = [rowind0l-1;rowind0l;rowind0l+1];

colindl = repmat(1:numel(xbord),3,1);

indl = sub2ind(size(x000),rowindl,colindl);

Xl = X000(indl);

Yl = Y000(indl);

xl = repmat(xbord(:).',[2,1]);

yl = repmat(ybord(:).',[2,1]);

% projection

uXl = diff(Xl,1,1);

uYl = diff(Yl,1,1);

LAMBDAl = ((xl - Xl(1:2,:)) .* uXl + (yl - Yl(1:2,:)) .* uYl) ./ (uXl.^2 +

uYl.^2);

PXl = Xl(1:2,:) + LAMBDAl .* uXl;

PYl = Yl(1:2,:) + LAMBDAl .* uYl;

% distance to projections points which is inside a segment

PDl = NaN(size(xl));

bl = ((LAMBDAl > 0) & (LAMBDAl < 1));

PDl(bl) = sqrt((PXl(bl) - xl(bl)).^2 + (PYl(bl) - yl(bl)).^2);

% distance (min of distance to vertex or line segment)

dlin(:,li) = min([Dl; PDl],[],1).';

end

disthresh = 2; % threshold for distance from edge

lmask1 = false(size(nmask));

lmask2 = lmask1;

edgepx1 =

line('xdata',1,'ydata',1,'parent',axpic,'linestyle','none','marker','.','color','g');

edgepx2 =

line('xdata',1,'ydata',1,'parent',axpic,'linestyle','none','marker','.','color','c');

slid = uicontrol(f2,'style','slider','units','normal','position',[.05 .1 .04

.75],'min',0,'max',10,'value',disthresh,'callback',@threshadj);

slidtxt = uicontrol(f2,'style','text','units','normal','position',[.02 .9 .1

.02],'string',sprintf('Threshold\n%0.1f Pixels',disthresh));

threshadj(slid);

set([polyg,ptmark],'visible','off');

overlayflag = 2;

set(contbut,'visible','on','string','Finish');

uiwait(f2);

set([contbut,slid,slidtxt],'visible','off');

end

set(f2,'WindowButtonDownFcn',@toggleoverlay)

% save_to_base(1);

function editflat(~,~)

sntemp = zeros(1,2);

sntemp(1) = str2double(get(shapedit(1),'string'));

sntemp(2) = str2double(get(shapedit(2),'string'));

249

if isnan(sntemp(1))

sntemp(1) = get(shapedit(1),'userdata');

end

if isnan(sntemp(2))

sntemp(2) = get(shapedit(2),'userdata');

end

shapename = sntemp;

[x00,y00,z00,bordfcn] = DP_SLSurfacePoints(shapename,[10,5]);

x00 = x00+initoffset(1);

y00 = y00+initoffset(2);

xb = bordfcn(x00);

yb = bordfcn(y00);

zb = bordfcn(z00);



dz = pt2(3)-pt1(3);

dxysq = sum((pt2(1:2)-pt1(1:2)).^2);

pts2pos(campt1,campt2);

end

function threshadj(slid,~)

disthresh = get(slid,'value');

subs1 = [ybord(dlin(:,1)<=disthresh),xbord(dlin(:,1)<=disthresh)];

subs2 = [ybord(dlin(:,2)<=disthresh),xbord(dlin(:,2)<=disthresh)];

ind1 = sub2ind(size(nmask),subs1(:,1),subs1(:,2));

ind2 = sub2ind(size(nmask),subs2(:,1),subs2(:,2));

lmask1(:) = false;

lmask2(:) = false;

lmask1(ind1) = true;

lmask2(ind2) = true;

set(edgepx1,'xdata',subs1(:,2),'ydata',subs1(:,1));

set(edgepx2,'xdata',subs2(:,2),'ydata',subs2(:,1));

set(slidtxt,'string',sprintf('%0.1f Pixels',disthresh));

end

function getcp(fig,~,ax)

cp = get(ax,'currentpoint');

xcam10 = cp(1,1);

ycam10 = cp(1,2);

uiresume(fig);

end

function setpt2(~,~)

cp = get(axpic,'currentpoint');

xcam2 = cp(1,1)+xyoffset(1);

ycam2 = cp(1,2)+xyoffset(2);

campt2 = [xcam2,ycam2];

pts2pos(campt1,campt2);

end

250

function [Rpts,Tpts,s1,s2] = pts2pos(campt1,campt2)

xcam = [campt1(1);campt2(1)];

ycam = [campt1(2);campt2(2)];

[sxc,syc,szc] = cam2ray(xcam,ycam,f,k1,cxcy,dxy);

s120 = [sxc,syc,szc].';

s12 = Rmir.'*(s120);

s12 = bsxfun(@rdivide,s12,s12(3,:)); % z component normalized to +1

s1 = s12(:,1);

s2 = s12(:,2);

mxy = (s2(1:2)-s1(1:2))./s1(3); % s1(3) = s2(3) ( = 1 )

cxy = (s2(1:2).*(dz-o(3))+s1(1:2).*o(3))./s1(3); % s1(3) = s2(3) ( = 1 )

aq = sum(mxy.^2);

bq = 2.*sum(mxy.*cxy);

cq = sum(cxy.^2)-dxysq;

zsolve1 = (-bq+sqrt(bq^2-4*aq*cq))/(2*aq);

xsolve1 = (zsolve1-o(3))/s1(3)*s1(1)+o(1);

ysolve1 = (zsolve1-o(3))/s1(3)*s1(2)+o(2);

ptsol1 = [xsolve1;ysolve1;zsolve1];

zsolve2 = zsolve1+dz;

xsolve2 = (zsolve2-o(3))/s2(3)*s2(1)+o(1);

ysolve2 = (zsolve2-o(3))/s2(3)*s2(2)+o(2);

ptsol2 = [xsolve2;ysolve2;zsolve2];

cosang = sum((pt2(1:2)-pt1(1:2)).*(ptsol2(1:2)-ptsol1(1:2)))/dxysq;

crossprod = cross([pt2(1:2)-pt1(1:2);0],[ptsol2(1:2)-ptsol1(1:2);0]);

sinang = sign(crossprod(3)).*norm(crossprod)/dxysq;

Rpts = [cosang -sinang 0;...

sinang cosang 0;...

0 0 1];

Tpts = ptsol1-pt1;

drawpoly(Rpts,Tpts,s1,s2);

end

function drawpoly(Rpts,Tpts,varargin)

pts1 = bsxfun(@plus,Rpts*([xb,yb,zb].'),Tpts);

[xbcam0,ybcam0] =

world2cam(pts1(1,:).',pts1(2,:).',pts1(3,:).',Rmir,Tmir,f,k1,cxcy,dxy);

xbcam = xbcam0-xyoffset(1);

ybcam = ybcam0-xyoffset(2);

set(polyg,'xdata',xbcam,'ydata',ybcam);

set(ptmark,'xdata',xbcam([pt1ind,pt2ind]),'ydata',ybcam([pt1ind,pt2ind]));

if nargin>2

s1 = varargin{1};

s2 = varargin{2};

s1t = s1.*abs(o(3));

s2t = s2.*abs(o(3));

set(borderline,'xdata',pts1(1,:).','ydata',pts1(2,:).','zdata',pts1(3,:).');

set(l1,'xdata',o(1)+[0;s1t(1)],'ydata',o(2)+[0;s1t(2)],'zdata',o(3)+[0;s1t(3)]);

set(l2,'xdata',o(1)+[0;s2t(1)],'ydata',o(2)+[0;s2t(2)],'zdata',o(3)+[0;s2t(3)]);

251

end

drawnow expose;

end

function selpt(~,~)

set(f2,'WindowButtonMotionFcn',@pointerswitch,'WindowButtonDownFcn',[]);

axes(axpic);

set(f2,'name','Adjust Control Points');

end

function pointerswitch(~,~)


cp = cp(1,1:2);

if sqrt(sum((cp-(campt1-xyoffset)).^2)) < 10

set(f2,'pointer','fleur','WindowButtonDownFcn',@clickpt1);

elseif sqrt(sum((cp-(campt2-xyoffset)).^2)) < 10

set(f2,'pointer','fleur','WindowButtonDownFcn',@clickpt2);

else

set(f2,'pointer','arrow','WindowButtonDownFcn',[]);

end

end

function clickpt1(~,~)


strtpt1 = cp(1,1:2);

set(f2,'WindowButtonMotionFcn',{@dragpt1,strtpt1,campt1},'WindowButtonUpFcn',@releasept);

end

function dragpt1(~,~,strtpt,campt0)


movec = cp(1,1:2)-strtpt;

campt1 = campt0+movec;

[Rpts,Tpts,s1,s2] = pts2pos(campt1,campt2);

mertit = edgemert([0,0,atan2(Rpts(2),Rpts(1)),Tpts(:).']);

set(f2,'name',num2str(mertit));

end

function clickpt2(~,~)


strtpt = cp(1,1:2);

set(f2,'WindowButtonMotionFcn',{@dragpt2,strtpt,campt2},'WindowButtonUpFcn',@releasept);

end

function dragpt2(~,~,strtpt,campt0)


movec = cp(1,1:2)-strtpt;

campt2 = campt0+movec;

[Rpts,Tpts] = pts2pos(campt1,campt2);

252

mertit = edgemert([0,0,atan2(Rpts(2),Rpts(1)),Tpts(:).']);

set(f2,'name',num2str(mertit));

end

function releasept(~,~)

set(f2,'WindowButtonMotionFcn',@pointerswitch,'WindowButtonUpFcn',[]);

end

function cont(~,~)

set(f2,'WindowButtonMotionFcn',[],'WindowButtonDownFcn',[],'WindowButtonUpFcn',[]);

uiresume(f2);

end

function [mert,R,T,xbcam,ybcam] = edgemert(rpyt)

% extract R and T transformation matricies

roll = rpyt(1);

pitch = rpyt(2);

yaw = rpyt(3);

R = [cos(yaw)*cos(pitch), cos(yaw)*sin(pitch)*sin(roll)-sin(yaw)*cos(roll),

cos(yaw)*sin(pitch)*cos(roll)+sin(yaw)*sin(roll);...

sin(yaw)*cos(pitch), sin(yaw)*sin(pitch)*sin(roll)+cos(yaw)*cos(roll),

sin(yaw)*sin(pitch)*cos(roll)-cos(yaw)*sin(roll);...

-sin(pitch) , cos(pitch)*sin(roll) ,

cos(pitch)*cos(roll) ];

T = rpyt(4:6);

T = T(:);

% find part border in camera frame

pts1 = bsxfun(@plus,R*([xb(1:(end-1)),yb(1:(end-1)),zb(1:(end-1))].'),T);

[xbcam0,ybcam0] =




% calculate merit function value

X00 = xbcam(:) * ones(1,numel(xbord));

Y00 = ybcam(:) * ones(1,numel(ybord));

x00 = ones(numel(xbcam),1) * xbord(:).';

y00 = ones(numel(ybcam),1) * ybord(:).';

dists = (x00-X00).^2+(y00-Y00).^2;

[D,rowind] = min(dists,[],1);

rowind = [rowind-1;rowind;rowind+1];

rowind(rowind<1) = numel(xbcam);

rowind(rowind>numel(xbcam)) = 1;

colind = repmat(1:numel(xbord),3,1);

ind = sub2ind(size(x00),rowind,colind);

X = X00(ind);

Y = Y00(ind);

x = repmat(xbord(:).',[2,1]);

y = repmat(ybord(:).',[2,1]);

253

% projection

uX = diff(X,1,1);

uY = diff(Y,1,1);

LAMBDA = ((x - X(1:2,:)) .* uX + (y - Y(1:2,:)) .* uY) ./ (uX.^2 + uY.^2);

PX = X(1:2,:) + LAMBDA .* uX;

PY = Y(1:2,:) + LAMBDA .* uY;

% distance to projections points which is inside a segment

PD = NaN(size(x));

b = ((LAMBDA > 0) & (LAMBDA < 1));

PD(b) = sqrt((PX(b) - x(b)).^2 + (PY(b) - y(b)).^2);

% distance

d = min([D; PD]);

% weight distance higher if the point is outside the polygon (to push all points

into polygon area)

b = inpolygon(xbord, ybord, xbcam, ybcam);

d(~b) = 20.*d(~b);

mert = sqrt(sum(d.^2));

end

function [mert,R,T,xbcam,ybcam] = posmert(rpyt)

roll = rpyt(1);

pitch = rpyt(2);

yaw = rpyt(3);

R = [cos(yaw)*cos(pitch), cos(yaw)*sin(pitch)*sin(roll)-sin(yaw)*cos(roll),

cos(yaw)*sin(pitch)*cos(roll)+sin(yaw)*sin(roll);...

sin(yaw)*cos(pitch), sin(yaw)*sin(pitch)*sin(roll)+cos(yaw)*cos(roll),

sin(yaw)*sin(pitch)*cos(roll)-cos(yaw)*sin(roll);...

-sin(pitch) , cos(pitch)*sin(roll) ,

cos(pitch)*cos(roll) ];

T = rpyt(4:6);

T = T(:);

pts1 = bsxfun(@plus,R*([xb,yb,zb].'),T);

[xbcam0,ybcam0] =




polymask = inpolygon(xlist,ylist,xbcam,ybcam);

ondif = nnz(~polymask & nmask);

offs = nnz(polymask & ~nmask);

% figure(4); imagesc(~polymask & nmask); axis image; title(int2str(ondif));

% figure(5); imagesc(polymask & ~nmask); axis image; title(int2str(offs));

% figure(6); imagesc(nmask); axis image; title('nmask');

% figure(7); imagesc(polymask); axis image; title('polymask');

mert = 5*ondif + 1*offs;

254

% drawpoly(R,T);

end

function [mert,R,T] = posmertsca(rpytsca,rpytoffset,rpytbnd) % posmert wrapper

rpyt = rpytsca.*diff(rpytbnd,1,1)./2+rpytoffset;

[mert,R,T] = posmert(rpyt);

end

function [mert,R,T] = posmert2stage(rpyt) % posmert wrapper

% [~,~,~,polyx,polyy] = posmert(rpyt);

% if abs(polyarea(polyx,polyy)/nnz(nmask)-1) > 0.05

% mert = 5*nnz(nmask);

% R = 0;

% T = 0;

% return;

% end

t0 = rpyt(4:6);

optionst = optimset('TolX',1e-

2,'PlotFcns',@optimplotfval,'UseParallel','always');%,'MaxIter',30);

topt = fminsearch(@(t) posmert([rpyt(1:3),t]),t0,optionst);

[mert,R,T] = posmert([rpyt(1:3),topt]);

drawpoly(R,T);

end

function toggleoverlay(~,~)

if overlayflag == 1

overlayflag = 2;

set(polyg,'visible','off');

set(samplin,'visible','on');

else

overlayflag = 1;

set(polyg,'visible','on');

set(samplin,'visible','off');

end

end

end

function [sx,sy,sz] = cam2ray(Xf,Yf,f,k1,cxcy,dxy)

Xd = (Xf-cxcy(1)).*dxy(1);

Yd = (Yf-cxcy(2)).*dxy(2);

rsq = Xd.^2+Yd.^2;

sx0 = Xd.*(1+k1.*rsq);

sy0 = Yd.*(1+k1.*rsq);

sz0 = f.*ones(size(sx0));

len = sqrt(sx0.^2+sy0.^2+sz0.^2);

sx = sx0./len;

sy = sy0./len;

sz = sz0./len;

end

255

function [Xw2c,Yw2c,xyzw2c] = world2cam(xw,yw,zw,R,T,f,k1,Cxy,dxy)

% INPUT:

% xw,yw,zw are world coordinates (zw(:) = 0 for planar coordinate points)

% R is 3x3 rotation matrix (world to camera)

% T is three element translation vector [Tx;Ty;Tz] (world to camera)

% f is lens focal length

% k1 is r^2 coefficient of distortion

% Cxy is two element vector [x,y] of pixel position of optical center in computer frame

memory

% dxy is two element vector [x,y] of pixel pitch (units of length)

% OUTPUT:

% Xw2c is an array of X pixel coordinates on camera sensor of real world points, the same

size as xw input

% Yw2c is an array of Y pixel coordinates on camera sensor of real world points, the same

size as xw input

% xyzw2c is an array of [x1,y1,z1;x2,y2,z2;...] 3D camera coordinates of world points

camcoordsw2c = R*[xw(:).'; yw(:).'; zw(:).'] + repmat(T(:),1,numel(xw));

xyzw2c = camcoordsw2c.';

xw2c = reshape(camcoordsw2c(1,:),size(xw));

yw2c = reshape(camcoordsw2c(2,:),size(xw));

zw2c = reshape(camcoordsw2c(3,:),size(xw));

Xdw2c0 = f.*xw2c./zw2c;

Ydw2c0 = f.*yw2c./zw2c;

rsqw2c = Xdw2c0.^2 + Ydw2c0.^2;

Xdw2c = Xdw2c0./(1+k1.*rsqw2c);

Ydw2c = Ydw2c0./(1+k1.*rsqw2c);

Xw2c = dxy(1)^(-1).*Xdw2c + Cxy(1);

Yw2c = dxy(2)^(-1).*Ydw2c + Cxy(2);

end

Dependency: DP_SLSurfacePoints.m

Generates sample points based on the ideal surface shape for comparison to

reconstructed data.

function [x,y,z,justborder,lindist,lind1,lind2,lindista,lind1a,lind2a] =

DP_SLSurfacePoints(shapename,samples)

% function [x,y,z,justborder] = SL_Roch5_Mini_SurfacePoints(samples)

%

% INPUT:

% samples = size of sample grid along x (direction of curvature) and y (direction of

extrusion)

% e.g. samples = [10,2] gives 10 samples along the curve and just the 2 end caps in the

extrusion direction

%

% OUTPUT:

% x,y,z = gridded x, y, and z data

% justborder = anonymous function to extract border data from x, y, or z ( x_border =

justborder(x) )

256

% lindist = distance between opposing straight edges

% lind1 and lind2 = vectors of indicies of sample points along the two opposing straight

edges

shapedata = reflectorshapedatabase(shapename);

shape = shapedata.shape;

xbnd = shapedata.xbnd;

length = shapedata.length; % extruded length of part

tilt = shapedata.tilt; % degrees

thik = shapedata.thik; % perpendicular distance from xbnd(1) point on curve to plane

describing the rear face of the part

if tilt~=0

m = tand(90+tilt); % slope of x sample lines

ybnd = polyval(shape,xbnd);

bbnd = ybnd-m.*xbnd;

b = linspace(bbnd(1),bbnd(2),samples(1)).';

lindist = sqrt(diff(xbnd).^2+diff(ybnd).^2);

x0 = zeros(size(b));

for ij = 1:samples(1)

xraw = roots([shape(1:6),shape(7)-m,shape(8)-b(ij)]);

xraw = xraw(imag(xraw)==0);

x0(ij) = xraw(find(real(xraw)>=min(xbnd)-0.000001 &

real(xraw)<=max(xbnd)+0.000001,1));

end

x0 = real(x0);

else

lindist = abs(diff(xbnd));

x0 = linspace(xbnd(1),xbnd(2),samples(1)).';

end

y0 = polyval(shape,x0);

% % find principle curvatures at each sample point

% fx = polyval(polyder(shape),x0);

% fxx = polyval(polyder(polyder(shape)),x0);

% fy = 0;

% fyy = 0;

% fxy = 0;

% K = (fxx.*fyy-fxy.^2)./(1+fx.^2+fy.^2).^2; % gaussian curvature

% H = ((1+fx.^2).*fyy+(1+fy.^2).*fxx-2.*fx.*fy.*fxy)./(2.*(1+fx.^2+fy.^2).^(3/2)); % mean

curvature

% kern = H.^2-K;

% C1 = H+sqrt(kern); % principal curvature 1

% C2 = H-sqrt(kern); % principal curvature 2

R = [cosd(-tilt),sind(-tilt);-sind(-tilt),cosd(-tilt)];

xzrot = [x0,y0]*R;

x = xzrot(:,1);

x = x-max(x);

x = repmat(x,1,samples(2));

257

z = -xzrot(:,2);

z = z-z(1)-thik;

z = repmat(z,1,samples(2));

y = repmat(linspace(0,length,samples(2)),samples(1),1);

x = x.';

y = y.';

z = z.';

lind1 = find(x==min(x(:)));

lind2 = find(x==max(x(:)));

lindista = length;

lind1a = find(y==min(y(:)));

lind2a = find(y==max(y(:)));

justborder = @(x) [x(1:1:end-1,1);x(end,1:1:end-1).';x(end:-1:2,end);x(1,end:-1:1).'];

% surf(x,y,z,'facecolor','b','facealpha',0.3,'edgecolor','none');

% axis equal;

% set(gca,'ydir','reverse','zdir','reverse');

%

line('xdata',justborder(x),'ydata',justborder(y),'zdata',justborder(z),'color','b','linew

idth',2);

end

Dependency: pptfig.m

Initiates graphical user interfaces figure windows.

function fhan = pptfig(fname,fignum,varargin)

% fname is the figure (base) name

% fignum is the figure number, determines figure screen position

% varargin is optional subtitle to figure name

% size is chosen to fit on a standard 4:3 powerpoint slide without resizing

ss = get(0,'ScreenSize');

fpos = [10+40*(fignum-1),60+10*(fignum-1),960,720];

fogm = findobj('-regexp','Name',fname,'type','figure');

if isempty(fogm);

fhan = figure(); % create figure

else

figure(fogm(1));

fhan = clf('reset'); % clear previous figure

end

set(fhan,'Position',fpos,'color','w','units','pixels','numbertitle','off');

if nargin<3

set(fhan,'Name',fname);

258

else

set(fhan,'Name',[fname,': ',varargin{1}]);

end

fs = get(fhan,'outerposition');

set(fhan,'outerposition',[1+40*(fignum-1), ss(4)-fs(4)-30*(fignum-1), fs(3),

fs(4)],'resize','on');

end

Dependency: targetcoorddatabase.m

Contains a callable database to reference object fiducial coordinates, most notably for

the deflectometer’s reference flat mirror.

function target = targetcoorddatabase(targetname)

switch targetname

case 'paper'

% find coords from distances

ptxi = zeros([4,4]);

ptyi = ptxi;

ptxi(5) = 1.429;

ptxi(9) = 2.953;

ptxi(13) = 4.444;

d1 = [1.324 1.949 3.242 4.610;... % linear distance from point "1"

2.554 2.928 3.911 5.094;...

3.562 3.859 4.665 5.676];

d13 = [4.650 3.295 1.987 1.312;... % linear distance from point "13"

5.128 3.945 2.948 2.559;...

5.693 4.671 3.880 3.612];

thet = acos((ptxi(13)^2 + d1.^2 - d13.^2)./(2.*ptxi(13).*d1));

ptxi(2:end,:) = d1.*cos(thet);

ptyi(2:end,:) = -d1.*sin(thet);

% figure(2);

% plot(ptxi(:),ptyi(:),'+');

% axis equal;

target.ptx = ptxi.*25.4; % convert to mm

target.pty = ptyi.*25.4; % convert to mm

target.names = {'1' '5' '9' '13';...

'2' '6' '10' '14';...

'3' '7' '11' '15';...

'4' '8' '12' '16'};

259

case 'test'

ptx0 = linspace(0,100,3);

pty0 = linspace(0,-100,3);

[ptx1,pty1] = meshgrid(ptx0,pty0);

target.ptx = ptx1;

target.pty = pty1;

target.names = cell(size(ptx1));

for ni = 1:numel(target.names);

target.names{ni} = int2str(ni);

end

case 'phonedots'

target.ptx = [0,50.052,100.104;0,50.052,100.104;0,50.052,100.104;];

target.pty = [0,0,0;-50.052,-50.052,-50.052;-100.104,-100.104,-100.104;];

target.names = {'Top Left', 'Top Center', 'Top Right'; 'Middle Left', 'Center',

'Middle Right'; 'Bottom Left', 'Bottom Center', 'Bottom Right'};

case 'mirror'

% from MirrorDotDataInterpreter.m

target.ptx = [3.5469 43.4622 83.7309;...

3.9823 43.9391 84.1066;...

4.2849 44.3064 84.5811];

target.pty = [4.2152 3.6317 3.0414;...

38.7358 38.2271 37.8273;...

73.4263 73.0113 72.7092];

target.names = {'Top Left', 'Top Center', 'Top Right';...

'Middle Left', 'Center', 'Middle Right';...

'Bottom Left', 'Bottom Center', 'Bottom Right'};

otherwise

disp('Target name not found in database');

target.ptx = 0;

target.pty = 0;

target.names = {'error'};

end

Dependency: camerasensordatabase.m

Contains callable database for sensor specifications from various cameras, most notably

the ‘Marlin’ camera used in the final deflectometer apparatus.

function sensor = camerasensordatabase(camname)

sensor = struct('dx',0,'Ncx',0,'dy',0);

switch camname

case 'RazrMaxx'

sensor.dx = 1.35E-3; % x direction pixel pitch in mm

sensor.Ncx = 3264; % number of pixels in x direction on camera sensor

260

sensor.dy = 1.35E-3; % y direction pixel pitch in mm

case 'NikonD90'




case 'Marlin'




case 'Webcam'

% pixel spacings are just guesses




end

261

Appendix C. Optimization Routine MATLAB Code

This appendix contains the final versions of the code used to optimize the presented

designs. All novel dependencies are included and all other functions are either included in

MATLAB 2012a or are publically available on the MATLAB Central File Exchange website.

Suspended Louver Optimizer

Optimization routine for the suspended louver design.

% function [optx,mert,solutions,optin] = SL_SingleSegment_Optimizer_PatternSearch()

%Optimization of single segment suspended louver using PatternSearch

% clear all;

close all;

global bestmerit bestvangseffs bestx0

d = 2; % optics periodicity

n = 1.49; % refractive index of material

Rmirror = 0.95; % reflectivity of mirrored surfaces

glen = 300; % guide length (mm)

gthik = 13; % guide thickness (mm)

vangmin = 0;

vangmax = 89;

vangnum = 300;

[lat,hitemps,lotemps] = LocationWeatherDatabase('Rochester'); % retrieve location and

weather information

vangs = linspace(vangmin,vangmax,vangnum); % vertical angles

vangwts = VerticalAngleWeightCalculator(vangs,lat,hitemps,lotemps); % vertical angle

weighting function

h = 0.7658; % injection feature size

endptx = -2.22; % LHS endpoint of reflector [x,y]

shape = [9.4971e-05,0.00022229,0.00147395,-0.00083297,0.0053727,0.193836,0.51152,-

0.57171]; % polynomial coefficients describing reflector shape

x0 = [h,endptx,shape]; % initial optimization point

bestmerit = 0;

bestvangseffs = [vangs(:),zeros(size(vangs(:)))];

bestx0 = x0;

ubnds = [0.5, -1, 0.001, 0.001, 0.01, 0.01, 0.02, 0.5, 1, 0];

lbnds = [0.1, -6,-0.001,-0.001,-0.01,-0.01,-0.01,-0.2, 0, -d];

262

rbnds = ubnds-lbnds; % range of raw bounds

x0sca = (x0-lbnds)./rbnds;

lbndsca = zeros(size(lbnds));

ubndsca = ones(size(ubnds));

psoptdef = psoptimset(@patternsearch);

psoptions =

psoptimset(psoptdef,'CompletePoll','on','UseParallel','always','InitialMeshSize',0.2,'Plo

tFcns',{@psplotbestf,@psplotbestx},'Display','iter',...

'TolMesh',1e-3,'MaxIter',100*numel(x0),'MaxFunEvals',2000*numel(x0));

problem =

createOptimProblem('fmincon','x0',x0sca,'lb',lbndsca,'ub',ubndsca,'options',psoptions,...

'objective',

@(x0sca)SL_SingleSegment_MeritFunction(d,x0sca(1).*rbnds(1)+lbnds(1),x0sca(2).*rbnds(2)+l

bnds(2),x0sca(3:end).*rbnds(3:end)+lbnds(3:end),n,Rmirror,glen,gthik,vangs,vangwts,0));

problem.solver = 'patternsearch';

tic;

[optxsca,fval,exitflag,output] = patternsearch(problem);

optx = optxsca.*rbnds+lbnds;

clear bestmerit bestvangseffs bestx0

% try

% txtmsg = sprintf('Runtime of %3.2f hours', toc/3600);

% send_text_message('402-416-8963','Verizon','Optimization Complete',txtmsg);

% catch txtmsgerr

% disp('Text message error...');

% end

%%

optin = {d,optx(1),optx(2),optx(3:end),n,Rmirror,glen,gthik,vangs,vangwts,1};

[merit,effs,injpos] = SL_SingleSegment_MeritFunction(optin{1}, optin{2}, optin{3},

optin{4}, optin{5}, optin{6}, optin{7}, optin{8}, optin{9}, optin{10}, optin{11});

figure();

coeff = (1-numel(shape)):1:0;

hold on;

plot(coeff,optx(3:end),'marker','.','color','r','linewidth',2);

plot(coeff,lbnds(3:end),'-k');

plot(coeff,ubnds(3:end),'-k');

hold off;

title('eff vs. coeffs');

coeffstr = cell(1,numel(coeff));

for jk = 1:numel(coeff)

coeffstr{jk} = ['x^' num2str(abs(coeff(jk)))];

end

set(gca,'XTick',coeff,'XTickLabel',coeffstr);

grid on;

263

% output cell to paste to Excel

outcel = cell(10,max(numel(x0)+1,9));

outcel(1,1) = {mfilename('fullpath')};

outcel(2,2:4) = [{'h'},{'x0'},{'shape...'}];

outcel(3:6,1) = [{'initialx'};{'optx'};{'ubnds'};{'lbnds'}];

outcel(8,1:9) =

[{'d'},{'n'},{'Rmirror'},{'glen'},{'gthik'},{'vangmin'},{'vangmax'},{'vangnum'},{'lat'}];

outcel(3,2:numel(x0)+1) = num2cell(x0);

outcel(4,2:numel(x0)+1) = num2cell(optx);

outcel(5,2:numel(x0)+1) = num2cell(ubnds);

outcel(6,2:numel(x0)+1) = num2cell(lbnds);

outcel(9,1:9) = [{d},{n},{Rmirror},{glen},{gthik},{vangmin},{vangmax},{vangnum},{lat}];

outcel(10,1) = {'Notes:'};

outcel(10,2) = {'Patternsearch options: CompletePoll on, InitialMeshSize 0.2, scaled all

bounds to be between 0 and 1'};

outcel(12,1:3) = [{'vangs'},{'effs'},{'merit'}];

outcel(13:13+vangnum-1,1:2) = num2cell([vangs(:),effs(:)]);

outcel(13,3) = num2cell(merit);

%%

BuildLT = questdlg('Would you like to build this reflector in LightTools?',...

'Build in LT?','Yes','No','No');

if strcmp(BuildLT,'Yes')

SL_SingleSegment_BuildLT(injpos,optx(3:end),optx(2),d,Rmirror,1);

end

%%

% end

Suspended Louver Merit Function

Merit function evaluated for all parameters passed from the optimization routine.

function [merit,effs,injpos] =

SL_SingleSegment_MeritFunction(d,h,x0,shape,n,Rmirror,glen,gthik,vangs,vangwts,plotyn)

% %input arguments

% d = 2.6; % optics periodicity

% h = 0.78; % injection feature size

% endpt = [-4.8 -2]; % LHS endpoint of reflector [x,y]

% shape = [0.0855 0.5623 -1.2713]; % polynomial coefficients describing reflector shape

% n = 1.49;

% Rmirror = 0.95; % reflectivity of mirrored surfaces

% glen = 300;

% gthik = 13;

% plotyn = 1;

% vangs = linspace(0,89,300);

% vangwts = VerticalAngleWeightCalculator(vangs,43.12);

264

global bestmerit bestvangseffs bestx0

global bestinjpos bestrefl

global topax topplot1 topplot2 topplot3 topplot4 topplot5 topplot6

global botax botplot

firstcall = 0;

if sum(bestvangseffs(:,2))==0

firstcall = 1;

end

%%%%%%%%%start of function code

evalpts = 100;

R = 1-h/d;

critang = asind(1/n);


% calculate RHS endpoint components

x1 = 0;

y1 = shape(end);

% calculate injection feature shape

injpos = zeros(4,2);

minj = (sqrt((n^2-1).*((0-x0).^2+(0-y0).^2))-0+x0)./(0-y0-sqrt((0-x0).^2+(0-y0).^2)); %

slope of injection surface to inject ray from endpt to (0,0) at guide critical angle

mcrit = tand(critang);

injpos(2,1) = 2.*h./(minj-mcrit);

injpos(2,2) = minj.*injpos(2,1);

injpos(3,1) = -h./mcrit;

injpos(3:4,2) = h;

if injpos(2,2)>h

injpos(2:3,1) = h./minj;

injpos(2:3,2) = h;

end

if polyval(shape, injpos(3,1)) < (injpos(3,2)-d)

shapeshft = shape;

shapeshft(end) = shapeshft(end)+d-h;

solsinj = roots(shapeshft);

solsinj = round(solsinj.*1e10)./1e10;

solsinj = solsinj( (solsinj<=x1) & (solsinj>=injpos(3,1)) & (imag(solsinj)==0) );

if any(solsinj)

injpos(3,1) = max(solsinj);

injpos(2,1) = (mcrit.*injpos(3,1)-injpos(3,2))./(mcrit-minj);

injpos(2,2) = minj.*injpos(2,1);

end

end

xs = linspace(x0,x1,20);

if firstcall

bestinjpos = injpos;

bestrefl = [xs(:),polyval(shape,xs(:))];

figure(10);

set(gcf,'position',[1049,107,565,806]);

265

topax = subplot(2,1,1);

topplot4 = plot([bestinjpos(:,1);bestinjpos(:,1)],[bestinjpos(:,2)-

d;bestinjpos(:,2)],'-','LineWidth',2,'color',[.8 .8 .8]);

axis equal;

hold on;

topplot5 = plot(bestrefl(:,1),bestrefl(:,2),'-','LineWidth',2,'color',[.8 .8 .8]);

topplot6 = plot(bestrefl(:,1),bestrefl(:,2)+d,'-','LineWidth',2,'color',[.8 .8 .8]);

topplot1 = plot([injpos(:,1);injpos(:,1)],[injpos(:,2)-d;injpos(:,2)],'-

k','LineWidth',2);

topplot2 = plot(xs,polyval(shape,xs),'-r','LineWidth',2);

topplot3 = plot(xs,polyval(shape,xs)+d,'-r','LineWidth',2);

hold off;

else

set(topplot1,'XData',[injpos(:,1);injpos(:,1)],'YData',[injpos(:,2)-d;injpos(:,2)]);

set(topplot2,'XData',xs,'YData',polyval(shape,xs));

set(topplot3,'XData',xs,'YData',polyval(shape,xs)+d);

end

inputs = {d,h,x0,shape,n,Rmirror,glen,gthik,vangs,vangwts,plotyn};

savb(inputs);

% find weights for each incidence angle

aplo = y0;

aphi = min(-tand(vangs).*(x0-x1)+y1, y0+d);

aphi = min(-tand(vangs).*(x0-injpos(2,1))+injpos(2,2), aphi); % so rays don't pass

through the injector on the way to the reflector

wts = max((aphi-aplo)./d, 0);

wtsl = logical(wts); % for logical indexing of non-zero weight incidence angles

wtsmat0 = repmat(wts(wtsl), evalpts, 1)./evalpts;

angsmat = repmat(vangs(wtsl), evalpts, 1);

slpsmat = -tand(angsmat);

% find start point of rays by equally spreading rays across applicable part of the input

aperture for each incidence angle

rayiny = zeros(size(slpsmat));

aphiap = aphi(wtsl);

for ij = 1:size(slpsmat,2)

rayiny(:,ij) = linspace(aplo,aphiap(ij),evalpts);

end

% find where incoming rays intersect segment

segxint = zeros(size(slpsmat));

nosol = segxint;

for jk = 1:numel(slpsmat)

polyadj = zeros(size(shape));

polyadj(end-1:end) = [slpsmat(jk),rayiny(jk)-slpsmat(jk).*x0];

sols = roots(shape-polyadj);

sols = round(sols.*1e10)./1e10;

solsa = sols( (sols<=x1) & (sols>=x0) & (imag(sols)==0) );

if any(solsa)

segxint(jk) = min(solsa);

266

else

nosol(jk) = 1;

end

polyadj2 = polyadj;

polyadj2(end) = polyadj2(end)-d;

sols2 = roots(shape-polyadj2);

sols2 = round(sols2.*1e10)./1e10;

solsa2 = sols2( (sols2<=x1) & (sols2>=x0) & (imag(sols2)==0) );

if any(solsa2)

nosol(jk) = 1;

end

end

segyint = polyval(shape,segxint);

% find slopes/normals of surface where rays strike it

surfslps = polyval(polyder(shape),segxint);

% find directions of reflected rays

rdirvec = refl([ones(size(slpsmat(:))),slpsmat(:)],[-

surfslps(:),ones(size(surfslps(:)))]);

rdirvecx = reshape(rdirvec(:,1),size(slpsmat)); %reflected ray direction unit vectors, x

component

rdirvecy = reshape(rdirvec(:,2),size(slpsmat)); %reflected ray direction unit vectors, y

component

rslpsmat = rdirvecy./rdirvecx;

wtsmat1 = wtsmat0.*Rmirror;

wtsmat1(logical(nosol)) = 0; % zero weights for rays with no reflector intersection

% find rays that intersect with aim region line segment

aimxint = (rslpsmat.*segxint - segyint) ./ (rslpsmat-minj);

aimyint = minj.*(aimxint);

hitinj = find( (aimxint>=injpos(2,1)) & (aimxint<=0) & ~nosol );

if isempty(hitinj)

effs = zeros(size(vangs));

merit = 0;

return

end

wtsmat2 = wtsmat1;

wtsmat2(setdiff(1:numel(wtsmat2),hitinj)) = 0;

% exclude rays that do not successfully inject into the guide

[rrdirvec,aveT] = refractfres([rdirvecx(hitinj),rdirvecy(hitinj)],[-minj,1],1,n);

wtsmat2(hitinj) = wtsmat2(hitinj).*aveT;

rrslps = rrdirvec(:,2)./rrdirvec(:,1);

cutyint = -rrslps.*aimxint(hitinj) + aimyint(hitinj); % y intercept of rays at guide

surface

ibakxint = (h+rrslps.*aimxint(hitinj)-aimyint(hitinj))./rrslps; % x intercept of rays at

injector back reflective surface

hitinjgid = find( (cutyint>=0) & (cutyint<=2*h) & (ibakxint>=injpos(3,1)) );

if isempty(hitinjgid)


267

merit = 0;

return

end

hitinjgiddn = find( (cutyint>h) & (cutyint<=2*h) );

hitind = hitinj(hitinjgid);

hitinddn = hitinj(hitinjgiddn);

wtsmat3 = wtsmat2;

wtsmat3(setdiff(1:numel(wtsmat3),hitind)) = 0;

wtsmat3(hitinddn) = wtsmat3(hitinddn).*Rmirror;

% guide ray tracing

gstrt0 = cutyint; % starting ray positions in guide

gstrt0(hitinjgiddn) = gstrt0(hitinjgiddn)-h; % shift "down rays" to center on origin,

unflipped for calculations

gstrt = gstrt0(hitinjgid)-h/2; % shift center of injector to origin and ignore rays

missing injection feature

gdist = h/2:d:(glen-h/2); % guide sample points (distance from the top edge of the guide)

gdist = gdist + (glen-gdist(end)+h/2)./2; % shift injector positions so they are

symmetric about the center of the guide

loledist = 2.*rrdirvec(hitinjgid+size(rrdirvec,1)).*gthik./rrdirvec(hitinjgid); %

distance rays travel without interacting with lossy guide surface

gideffs = zeros(numel(gstrt),numel(gdist));

for gi = 1:numel(gdist)

dup = gdist(gi); % distance to guide top

gideffs(:,gi) = R.^(floor((dup-gstrt)./loledist));

end

wtsmat4 = wtsmat3;

wtsmat4(hitind) = wtsmat4(hitind).*mean(gideffs,2);


effs(wtsl) = sum(wtsmat4,1);

merit = -sum(effs.*vangwts);

bestflag = 0;

if merit < bestmerit

bestmerit = merit;

bestvangseffs = [vangs(:),effs(:)];

bestx0 = [h,x0,shape];

bestflag = 1;

end

if firstcall

figure(10);

subplot(2,1,1);

botax = subplot(2,1,2);

268

botplot =

plot(bestvangseffs(:,1),bestvangseffs(:,2),vangs,vangwts./max(vangwts),vangs,effs);

set(botplot(1),'color',[.8 .8 .8],'linewidth',3);

set(botplot(2),'color',[0 0 1]);

set(botplot(3),'color',[0 .5 0],'linewidth',3);

if bestflag


end

title(botax,['Merit Function Value = ' num2str(merit) ' ( best = ' num2str(bestmerit)

' )']);

grid on;

else

set(botplot(3),'YData',effs,'color',[0 .5 0]);

if bestflag

set(topplot4,'xdata',[bestinjpos(:,1);bestinjpos(:,1)],'ydata',[bestinjpos(:,2)-

d;bestinjpos(:,2)]);

set(topplot5,'xdata',bestrefl(:,1),'ydata',bestrefl(:,2));

set(topplot6,'xdata',bestrefl(:,1),'ydata',bestrefl(:,2)+d);

set(botplot(1),'YData',bestvangseffs(:,2));


end

title(botax,['Merit Function Value = ' num2str(merit) ' ( best = ' num2str(bestmerit)

' )']);

drawnow expose;

end

%%

if plotyn

pangsmat = angsmat(logical(wtsmat4));

%rays

rayx1 = x0.*ones(1,length(hitind));

rayy1 = transpose(rayiny(hitind));

rayx2 = transpose(segxint(hitind));

rayy2 = transpose(segyint(hitind));

rayx3 = transpose(aimxint(hitind));

rayy3 = minj.*(rayx3-injpos(2,1)) + injpos(2,2);

rayx4 = zeros(size(rayx1));

rayy4 = zeros(size(rayx1));

rayy4(:) = cutyint(hitinjgid);

pangs = unique(pangsmat);

numpangs = numel(pangs);

rgb = zeros(numpangs,3);

rgbind = linspace(0,2,numpangs);

for nw = 1:numpangs

if rgbind(nw)<=1

rgb(nw,2) = rgbind(nw);

rgb(nw,1) = 1-rgbind(nw);

else

269

rgb(nw,3) = rgbind(nw)-1;


end

rgb(nw,:) = rgb(nw,:)./max(rgb(nw,:));

end

rgb = flipud(rgb);

%segment data

segptsx = linspace(x0,x1,50);

segptsy = polyval(shape,segptsx);

rtfig = findobj('type','figure','name','Raytrace');

if isempty(rtfig)

figure('name','Raytrace');

else

figure(rtfig);

end

clf

plot([x0;x0],[y0;y0+d],'Color',[.7 .7 .7],'LineWidth',3); %plot front aperture

axis equal

hold on

plot(segptsx,segptsy,'-k','LineWidth',3);

plot(segptsx,segptsy+d,'-k','LineWidth',3);

pause on

for pa = 1:numpangs

pangind = find(pangsmat==pangs(pa));

rayx1p = rayx1(pangind);

rayy1p = rayy1(pangind);







plot([rayx1p;rayx2p;rayx3p;rayx4p],[rayy1p;rayy2p;rayy3p;rayy4p],'Color',rgb(pa,:));

% pause(0.00000001);

end

pause off

plot([injpos(:,1);injpos(:,1)],[injpos(:,2)-d;injpos(:,2)],'-k','LineWidth',2); %plot

injector

plot(0,2*h,'dk');

hold off

injeffs = zeros(size(vangs));

injeffs(wtsl) = sum(wtsmat3,1);

sefig = findobj('type','figure','name','Segment Efficiency');

if isempty(sefig)

figure('name','Segment Efficiency');

else

270

figure(sefig);

end

clf

effplot = plot(vangs,vangwts./max(vangwts),vangs,effs,vangs,injeffs);

set(effplot(2),'linewidth',3);

title('Segment Efficiency vs. Incidence Angle');

xlabel('Incidence Angle ( ^o )');

ylabel('Efficiency');

legend('Weighting Function','System FoV','Injection

Efficiency','Location','SouthOutside');

grid on;

end

end

function [] = savb(varargin)

save_to_base(1);

end

Immersed Louver Optimizer

Optimization routine for the immersed louver.

function [outcel] = IL_SingleSegment_Optimizer_PatternSearch_Par(locationname)

%Optimization of single segment immersed louver using PatternSearch

close all;

if ~matlabpool('size') > 0

matlabpool

end


n = 1.49; % refractive index of material




vangmin = 0;

vangmax = 89;

vangnum = 300;


[lat,hitemps,lotemps] = LocationWeatherDatabase(locationname); % retrieve location and

weather information


weighting function

h = .4844; % injection feature size

endptx = -2.8066; % LHS endpoint of reflector [x,y]

shape = [0.00008716,0.0007005,0.001474,0.0002608,.02459,.1938,.6287,-.8217]; % polynomial

coefficients describing reflector shape

271

x0 = [h,endptx,shape]; % initial optimization point

%%

% variables for memorization function

mnum = 8; % number of function iterations to memorize

mx = zeros(mnum,numel(x0));

mmerit = zeros(mnum,1);

meffs = zeros(mnum,numel(vangs));

minjpos = zeros(mnum,6);

% variables for output function and plot initialization

dummy = [0;1];

numplot = 4;

numplot = min(numplot,mnum);

figure(10);

set(gcf,'position',[1049,107,565,806]);

subplot(2,1,1);

injplot = plot(repmat(dummy,1,numplot),repmat(dummy,1,numplot),'-m','LineWidth',2); %

injection features

axis equal;

hold on;

refl1plot = plot(repmat(dummy,1,numplot),repmat(dummy,1,numplot),'-r','LineWidth',2); %

reflectors

refl2plot = plot(repmat(dummy,1,numplot),repmat(dummy,1,numplot),'-r','LineWidth',2); %

shifted reflectors

hold off;

subplot(2,1,1);


plot(vangs,vangwts./max(vangwts),'-b'); % vertical angle weights

hold on;

effplot = plot(repmat(dummy,1,numplot),repmat(dummy,1,numplot),'-

','linewidth',3,'color',[0 .5 0]); % efficiency plots

hold off;

title(botax,'Merit Function Value = INITIAL ( best = INITIAL )');

grid on;

greys = linspace(0.9,0.6,numplot-1);

for pj = 1:numplot-1;

set(injplot(pj),'color',greys(pj).*ones(1,3));

set(refl1plot(pj),'color',greys(pj).*ones(1,3));

set(refl2plot(pj),'color',greys(pj).*ones(1,3));

set(effplot(pj),'color',greys(pj).*ones(1,3));

end

%%

% optimization options

ubnds = [1, -1, 0.001, 0.003, 0.01, 0.01, 0.04, 0.5, 1, 0];

lbnds = [0.1, -6,-0.001,-0.001,-0.01,-0.01,-0.01,-0.2, 0, -d];





272

psoptions =

psoptimset('CompletePoll','on','UseParallel','never','InitialMeshSize',0.5,'PlotFcns',{@p

splotbestf,@psplotbestx},'Display','iter',...

'TolMesh',1e-

3,'MaxIter',100*numel(x0),'MaxFunEvals',2000*numel(x0));%,'OutputFcns',@IL_SingleSegment_

PSOutputFunction_Par);

% problem =


% 'objective',

@(x0sca)IL_SingleSegment_MeritFunction(d,x0sca(1).*rbnds(1)+lbnds(1),x0sca(2).*rbnds(2)+l


% problem.solver = 'patternsearch';

problem =

struct('objective',[],'X0',[],'Aineq',[],'bineq',[],'Aeq',[],'beq',[],'lb',[],'ub',[],'no

nlcon',[],'solver','patternsearch','options',[],'rngstate',[]);

problem.objective =

@(x0sca)IL_SingleSegment_MeritFunction_Par_Memorize(d,x0sca(1).*rbnds(1)+lbnds(1),x0sca(2

).*rbnds(2)+lbnds(2),x0sca(3:end).*rbnds(3:end)+lbnds(3:end),n,Rmirror,glen,gthik,vangs,v

angwts,0);

problem.X0 = x0sca;

problem.lb = lbndsca;

problem.ub = ubndsca;

problem.options = psoptions;

tic;



%%

% memorization wrapper function (nested)

function [momerit,moeffs,moinjpos] =

IL_SingleSegment_MeritFunction_Par_Memorize(md,mh,mendptx,mshape,mn,mRmirror,mglen,mgthik

,mvangs,mvangwts,mplotyn)

%record top mnum iterations

% test if input was already calculated

[~,rowloc] = ismember([mh,mendptx,mshape],mx,'rows');

if rowloc==0

% run function

[momerit,moeffs,moinjpos] =

IL_SingleSegment_MeritFunction_Par(md,mh,mendptx,mshape,mn,mRmirror,mglen,mgthik,mvangs,m

vangwts,mplotyn);

% memorize if merit value is good enough

if any(momerit < mmerit)

mmerit(mnum) = momerit;

[mmerit,reind] = sort(mmerit);

mx(mnum,:) = [mh,mendptx,mshape];

mx = mx(reind,:);

273

meffs(mnum,:) = moeffs;

meffs = meffs(reind,:);

minjpos(mnum,:) = moinjpos(:);

minjpos = minjpos(reind,:);

end

else

momerit = mmerit(rowloc);

moeffs = reshape(meffs(rowloc,:),size(vangs));

moinjpos = reshape(minjpos(rowloc,:),size(minjpos,2)./2,[]);

end

% grab data from memorized results

ox = flipud(mx(1:numplot-1,:));

oinjpos = flipud(minjpos(1:numplot-1,:));

oeffs = flipud(meffs(1:numplot-1,:));

for op = 1:numplot-1

otshape = ox(op,3:end);

otxs = linspace(ox(op,2),0,20);

otrefl = polyval(otshape,otxs(:));

otinjpos = reshape(oinjpos(op,:),size(minjpos,2)./2,[]);

set(injplot(op),'XData',[otinjpos(:,1);otinjpos(:,1)],'YData',[otinjpos(:,2)-

d;otinjpos(:,2)]);

set(refl1plot(op),'XData',otxs,'YData',otrefl);

set(refl2plot(op),'XData',otxs,'YData',otrefl+d);

set(effplot(op),'XData',vangs,'YData',oeffs(op,:));

end

moxs = linspace(mendptx,0,20);

morefl = polyval(mshape,moxs(:));

set(injplot(numplot),'XData',[moinjpos(:,1);moinjpos(:,1)],'YData',[moinjpos(:,2)-

d;moinjpos(:,2)]);

set(refl1plot(numplot),'XData',moxs,'YData',morefl);

set(refl2plot(numplot),'XData',moxs,'YData',morefl+d);

set(effplot(numplot),'XData',vangs(:),'YData',moeffs(:));

title(botax,['Merit Function Value = ' num2str(momerit) ' ( Best = '

num2str(mmerit(1)) ' )']);

drawnow expose;

end

%%

% % output function (nested)

% function [stop, optnew, changed] =

IL_SingleSegment_PSOutputFunction_Par(optimValues, prevoptions, flag) % doc psoptimset >

Output Function Options

% stop = false;

% optnew = [];

% changed = false;

%

%

274

% end

%%

optin = {d,optx(1),optx(2),optx(3:end),n,Rmirror,glen,gthik,vangs,vangwts,1};

[merit,effs,injpos] = IL_SingleSegment_MeritFunction_Par(optin{1}, optin{2}, optin{3},

optin{4}, optin{5}, optin{6}, optin{7}, optin{8}, optin{9}, optin{10}, optin{11});

figure();


hold on;




hold off;





end


grid on;




outcel(2,2:4) = [{'h'},{'x0'},{'shape...'}];


outcel(8,1:9) =

[{'d'},{'n'},{'Rmirror'},{'glen'},{'gthik'},{'vangmin'},{'vangmax'},{'vangnum'},{'lat'}];





outcel(9,1:9) = [{d},{n},{Rmirror},{glen},{gthik},{vangmin},{vangmax},{vangnum},{lat}];


outcel(10,2) = {'Patternsearch options: CompletePoll on, InitialMeshSize 0.2, scaled all

bounds to be between 0 and 1'};




%%

BuildLT = questdlg('Would you like to build this reflector in LightTools?',...

'Build in LT?','Yes','No','No');

if strcmp(BuildLT,'Yes')

SL_SingleSegment_BuildLT(injpos,optx(3:end),optx(2),d,Rmirror,1);

end

end

275

Immersed Louver Merit Function


function [merit,effs,injpos] =

IL_SingleSegment_MeritFunction_Par(d,h,x0,shape,n,Rmirror,glen,gthik,vangs,vangwts,plotyn

)

% %input arguments

% d = 2.6; % optics periodicity

% h = 0.78; % injection feature size

% endpt = [-4.8 -2]; % LHS endpoint of reflector [x,y]

% shape = [0.0855 0.5623 -1.2713]; % polynomial coefficients describing reflector shape

% n = 1.49;


% glen = 300;

% gthik = 13;

% plotyn = 1;

% vangs = linspace(0,89,300);

% vangwts = VerticalAngleWeightCalculator(vangs,43.12);

%%%%%%%%%start of function code

evalpts = 100;

R = 1-h/d;

critang = asind(1/n);

vangsn = asind(sind(vangs)./n); % vertical angles after entering material


% calculate RHS endpoint components

x1 = 0;

y1 = shape(end);

% calculate injection feature shape

%%%%%%%%%% changed injpos from size (4,2) to (3,2)

injpos = zeros(3,2);

mcrit = tand(critang);

minj = -mcrit;

injpos(2,1) = h./minj;

injpos(2:3,2) = h;

if shape(end) < (h-d)


merit = 0;

return

elseif polyval(shape, injpos(2,1)) < (h-d)

shapeshft = shape;

shapeshft(end) = shapeshft(end)+d-h;

solsinj = roots(shapeshft);

solsinj = round(solsinj.*1e10)./1e10;

solsinj = solsinj( (solsinj<=x1) & (solsinj>=injpos(2,1)) & (imag(solsinj)==0) );

276

if any(solsinj)

injpos(2,1) = max(solsinj);

minj = (injpos(2,2)-injpos(1,2))./(injpos(2,1)-injpos(1,1));

end

end

% find weights for each incidence angle

aplo = y0;

aphi = min(-tand(vangsn).*(x0-x1)+y1, y0+d); % find from where in the aperture the rays

strike the RHS point of the reflector

aphi = min(-tand(vangsn).*(x0-injpos(2,1))+injpos(2,2), aphi); % so rays don't pass

through the injector on the way to the reflector

wts = max((aphi-aplo)./d, 0); % initial weights based on the fraction of the aperature

that produces rays that hit the reflector

wtsl = logical(wts); % for logical indexing of non-zero weight incidence angles

wtsmat0 = repmat(wts(wtsl), evalpts, 1)./evalpts; % expanding wts into ray dimension,

each ray gets fair share of vang wts

angsmat = repmat(vangsn(wtsl), evalpts, 1); % expand vangs into ray dimension

slpsmat = -tand(angsmat);

% find start point of rays by equally spreading rays across applicable part of the input

aperture for each incidence angle

rayiny = zeros(size(slpsmat));

aphiap = aphi(wtsl);

parfor ij = 1:size(slpsmat,2)

rayiny(:,ij) = linspace(aplo,aphiap(ij),evalpts);

end

% find where incoming rays intersect segment

segxint = zeros(size(slpsmat));

nosol = segxint;

parfor jk = 1:numel(slpsmat)

polyadj = zeros(size(shape));

polyadj(end-1:end) = [slpsmat(jk),rayiny(jk)-slpsmat(jk).*x0];

sols = roots(shape-polyadj);

sols = round(sols.*1e10)./1e10;

solsa = sols( (sols<=x1) & (sols>=x0) & (imag(sols)==0) );

if any(solsa)

segxint(jk) = min(solsa);

else

nosol(jk) = 1;

end

polyadj2 = polyadj;

polyadj2(end) = polyadj2(end)-d;

sols2 = roots(shape-polyadj2);

sols2 = round(sols2.*1e10)./1e10;

solsa2 = sols2( (sols2<=x1) & (sols2>=x0) & (imag(sols2)==0) );

if any(solsa2)

nosol(jk) = 1;

end

end

277

segyint = polyval(shape,segxint);

% find slopes/normals of surface where rays strike it

surfslps = polyval(polyder(shape),segxint);

% find directions of reflected rays

rdirvec = refl([ones(size(slpsmat(:))),slpsmat(:)],[-

surfslps(:),ones(size(surfslps(:)))]);

rdirvecx = reshape(rdirvec(:,1),size(slpsmat)); %reflected ray direction unit vectors, x

component

rdirvecy = reshape(rdirvec(:,2),size(slpsmat)); %reflected ray direction unit vectors, y

component

rslpsmat = rdirvecy./rdirvecx;

wtsmat1 = wtsmat0.*Rmirror;

wtsmat1(logical(nosol)) = 0; % zero weights for rays with no reflector intersection

% find rays that intersect with aim region line segment

aimxint = (rslpsmat.*segxint - segyint) ./ (rslpsmat-minj);

aimyint = minj.*(aimxint);

hitinj = find( (aimxint>=injpos(2,1)) & (aimxint<=0) & ~nosol );

if isempty(hitinj)


merit = 0;

return

end

wtsmat2 = wtsmat1;

wtsmat2(setdiff(1:numel(wtsmat2),hitinj)) = 0; % zero weights for rays that did not make

it to the aim region

% exclude rays that do not successfully inject into the guide

rrdirvec = [rdirvecx(hitinj),rdirvecy(hitinj)];

rrslps = rslpsmat(hitinj);

cutyint = -rrslps.*aimxint(hitinj) + aimyint(hitinj); % y intercept of rays at guide

surface

ibakxint = (h+rrslps.*aimxint(hitinj)-aimyint(hitinj))./rrslps; % x intercept of rays at

injector back reflective surface

hitinjgid = find( (cutyint>=0) & (cutyint<=2*h) & (ibakxint>=injpos(2,1)) );

if isempty(hitinjgid)


merit = 0;

return

end

hitinjgiddn = find( (cutyint>h) & (cutyint<=2*h) );

hitind = hitinj(hitinjgid);

hitinddn = hitinj(hitinjgiddn);

wtsmat3 = wtsmat2;

wtsmat3(setdiff(1:numel(wtsmat3),hitind)) = 0;

wtsmat3(hitinddn) = wtsmat3(hitinddn).*Rmirror;

% guide ray tracing

278

gstrt0 = cutyint; % starting ray positions in guide

gstrt0(hitinjgiddn) = gstrt0(hitinjgiddn)-h; % shift "down rays" to center on origin,

unflipped for calculations

gstrt = gstrt0(hitinjgid)-h/2; % shift center of injector to origin and ignore rays

missing injection feature




loledist = 2.*rrdirvec(hitinjgid+size(rrdirvec,1)).*gthik./rrdirvec(hitinjgid); %

distance rays travel without interacting with lossy guide surface


parfor gi = 1:numel(gdist)



end

wtsmat4 = wtsmat3;

wtsmat4(hitind) = wtsmat4(hitind).*mean(gideffs,2);


effs(wtsl) = sum(wtsmat4,1);

merit = -100*sum(effs.*vangwts)/sum(vangwts(vangwts>0)); % normalized to an absolute

minimum of -100

%%

if plotyn

pangsmat = angsmat(logical(wtsmat4));

%rays

rayx1 = x0.*ones(1,length(hitind));

rayy1 = transpose(rayiny(hitind));

rayx2 = transpose(segxint(hitind));

rayy2 = transpose(segyint(hitind));

rayx3 = transpose(aimxint(hitind));

rayy3 = minj.*rayx3;

rayx4 = zeros(size(rayx1));

rayy4 = zeros(size(rayx1));

rayy4(:) = cutyint(hitinjgid);

pangs = unique(pangsmat);

numpangs = numel(pangs);

rgb = zeros(numpangs,3);

rgbind = linspace(0,2,numpangs);

for nw = 1:numpangs

if rgbind(nw)<=1

rgb(nw,2) = rgbind(nw);


279

else

rgb(nw,3) = rgbind(nw)-1;


end

rgb(nw,:) = rgb(nw,:)./max(rgb(nw,:));

end

rgb = flipud(rgb);

%segment data

segptsx = linspace(x0,x1,50);

segptsy = polyval(shape,segptsx);

rtfig = findobj('type','figure','name','Raytrace');

if isempty(rtfig)

figure('name','Raytrace');

else

figure(rtfig);

end

clf

plot([x0;x0],[y0;y0+d],'Color',[.7 .7 .7],'LineWidth',3); %plot front aperture

axis equal

hold on

plot(segptsx,segptsy,'-k','LineWidth',3);

plot(segptsx,segptsy+d,'-k','LineWidth',3);

pause on

for pa = 1:numpangs

pangind = find(pangsmat==pangs(pa));









plot([rayx1p;rayx2p;rayx3p;rayx4p],[rayy1p;rayy2p;rayy3p;rayy4p],'Color',rgb(pa,:));

% pause(0.00000001);

end

pause off

plot([injpos(:,1);injpos(:,1)],[injpos(:,2)-d;injpos(:,2)],'-k','LineWidth',2); %plot

injector

plot(0,2*h,'dk');

hold off

injeffs = zeros(size(vangs));

injeffs(wtsl) = sum(wtsmat3,1);

sefig = findobj('type','figure','name','Segment Efficiency');

if isempty(sefig)

figure('name','Segment Efficiency');

280

else

figure(sefig);

end

clf



title('Segment Efficiency vs. Incidence Angle');





grid on;

end

end

Extruded Segment Lens Optimizer

Optimization routine for the extruded segment lens.

function [outcel,optin,mx,mmerit] =

EL_SingleSegment_Optimizer_PatternSearch_Par(locationname)

%Optimization of single segment immersed louver using PatternSearch

close all;


matlabpool

end


PMMA = struct('name','PMMA','data',...

[320 407 493 580 667 753 840 927 1013 1100 ; %

wavelengths in nm

1.5136 1.5063 1.4973 1.4922 1.4889 1.4869 1.4852 1.4841 1.4831 1.4831; % index of

refraction

0.134 0.778 1.328 1.333 1.234 1.018 0.898 0.535 0.651 0.278 ]); %

relative weights

Material = PMMA;




vangmin = 0;

vangmax = 89;

vangnum = 200;


ns = Material.data(2,:); % refractive index of material

wavs = Material.data(1,:); % wavelengths for refractive inicies

spectralresponse = Material.data(1,:)./1100;

wavwts = Material.data(3,:).*spectralresponse;

281


weather information


weighting function

h = .5; % injection feature size

y0 = -1; % low endpoint of powered surface

shape = [0 0 0 0 0 0 0 -2]; % polynomial coefficients describing lens shape

x0 = [h,y0,shape]; % initial optimization point

initmeshsize = 0.1;

% optimization options

ubnds = [1.3,-.01, .2, 0.3, 1, 1.2, 1.2, 1.5, 2, 0];

lbnds = [0.1, -2,-.4,-0.3,-1,-1.2,-1.2,-1.5, -2, -5];





%%




mmerit = zeros(mnum,1);


mlensvert = zeros(mnum,10);

% variables for output function and plot initialization

dummy = [0;1];

numplot = 4;


figure(10);

set(gcf,'position',[1049,107,565,806]);

subplot(2,1,1);


injection features

axis equal;

subplot(2,1,1);



hold on;



hold off;


grid on;



282



end

%%

psoptions =

psoptimset('CompletePoll','on','UseParallel','never','InitialMeshSize',initmeshsize,'Plot

Fcns',{@psplotbestf,@psplotbestx},'Display','iter',...

'TolMesh',1e-

3,'MaxIter',100*numel(x0),'MaxFunEvals',2000*numel(x0));%,'OutputFcns',@IL_SingleSegment_

PSOutputFunction_Par);

% problem =


% 'objective',

@(x0sca)IL_SingleSegment_MeritFunction(d,x0sca(1).*rbnds(1)+lbnds(1),x0sca(2).*rbnds(2)+l


% problem.solver = 'patternsearch';

problem =



problem.objective =

@(x0sca)IL_SingleSegment_MeritFunction_Par_Memorize(d,x0sca(1).*rbnds(1)+lbnds(1),x0sca(2

).*rbnds(2)+lbnds(2),x0sca(3:end).*rbnds(3:end)+lbnds(3:end),ns,wavs,wavwts,Rmirror,glen,

gthik,vangs,vangwts,0);

problem.X0 = x0sca;




tic;



%%


function [momerit,moeffs,molensvert] =

IL_SingleSegment_MeritFunction_Par_Memorize(md,mh,my0,mshape,mns,mwavs,mwavwts,mRmirror,m

glen,mgthik,mvangs,mvangwts,mplotyn)



[~,rowloc] = ismember([mh,my0,mshape],mx,'rows');

if rowloc==0

% run function

[momerit,moeffs,molensvert] =

EL_SingleSegment_MeritFunction(md,mh,my0,mshape,mns,mwavs,mwavwts,mRmirror,mglen,mgthik,m

vangs,mvangwts,mplotyn);

% memorize if merit value is good enough and sort all memorized function calls by

merit value

if any(momerit < mmerit)

283

mmerit(mnum) = momerit;

[mmerit,reind] = sort(mmerit);

mx(mnum,:) = [mh,my0,mshape];

mx = mx(reind,:);



mlensvert(mnum,:) = molensvert(:);

mlensvert = mlensvert(reind,:);

end

else

momerit = mmerit(rowloc);


molensvert = reshape(mlensvert(rowloc,:),size(mlensvert,2)./2,[]);

end



olensvert = flipud(mlensvert(1:numplot-1,:));


for op = 1:numplot-1

otshape = ox(op,3:end);

otlensvert = reshape(olensvert(op,:),[],2);

otys = linspace(otlensvert(2,2),otlensvert(3,2),20);

otxs = polyval(otshape,-otys(:));

otxs = [otlensvert(1,1);otxs(:);otlensvert(4:5,1)];

otys = [otlensvert(1,2);otys(:);otlensvert(4:5,2)];

set(injplot(op),'XData',[otxs;otxs],'YData',[otys;otys+d]);

set(effplot(op),'XData',vangs,'YData',oeffs(op,:));

end

moys = linspace(my0,molensvert(3,2),20);

moxs = polyval(mshape,-moys(:));

moxs = [molensvert(1,1);moxs(:);molensvert(4:5,1)];

moys = [molensvert(1,2);moys(:);molensvert(4:5,2)];

set(injplot(numplot),'XData',[moxs;moxs],'YData',[moys;moys+d]);

set(effplot(numplot),'XData',vangs(:),'YData',moeffs(:));

title(botax,['Merit Function Value = ' num2str(momerit) ' ( Best = '

num2str(mmerit(1)) ' )']);

drawnow expose;

end

%%

% % output function (nested)

% function [stop, optnew, changed] =

IL_SingleSegment_PSOutputFunction_Par(optimValues, prevoptions, flag) % doc psoptimset >

Output Function Options

% stop = false;

% optnew = [];

% changed = false;

284

%

%

% end

%%

optin =

{d,optx(1),optx(2),optx(3:end),ns,wavs,wavwts,Rmirror,glen,gthik,vangs,vangwts,1};

[merit,effs,lensvert] = EL_SingleSegment_MeritFunction(optin{1}, optin{2}, optin{3},

optin{4}, optin{5}, optin{6}, optin{7}, optin{8}, optin{9}, optin{10},

optin{11},optin{12},optin{13});

figure();


hold on;




hold off;





end


grid on;




outcel(2,2:4) = [{'h'},{'y0'},{'shape...'}];


outcel(8,1:9) =

[{'d'},{'Material'},{'Rmirror'},{'glen'},{'gthik'},{'vangmin'},{'vangmax'},{'vangnum'},{'

lat'}];





outcel(9,1:9) =

[{d},{Material.name},{Rmirror},{glen},{gthik},{vangmin},{vangmax},{vangnum},{lat}];


outcel(10,2) = {['Patternsearch options: CompletePoll on, InitialMeshSize '

num2str(initmeshsize) ', scaled all bounds to be between 0 and 1']};




end

285

Extruded Segment Lens Merit Function


Includes material dispersion and chromatic effects unlike the reflective louver functions.

function [merit,effs,lensvert] =

EL_SingleSegment_MeritFunction(d,h,y0,shaperot,ns,wavs,wavwts,Rmirror,glen,gthik,vangs,va

ngwts,plotyn)

%%%%%%%%%%%%%%%%%%%%%%

%test inputs

% clear all

% d = 2; % optics periodicity

% PMMA = struct('name','PMMA','data',...

% [320 407 493 580 667 753 840 927 1013 1100 ; %

wavelengths in nm

% 1.5136 1.5063 1.4973 1.4922 1.4889 1.4869 1.4852 1.4841 1.4831 1.4831; % index

of refraction

% 0.134 0.778 1.328 1.333 1.234 1.018 0.898 0.535 0.651 0.278 ]); %

relative weights

% Material = PMMA;


% glen = 300; % guide length (mm)

% gthik = 13; % guide thickness (mm)

% vangmin = 0;

% vangmax = 89;

% vangnum = 200;

% vangs = linspace(vangmin,vangmax,vangnum); % vertical angles

%

% ns = Material.data(2,:); % refractive index of material

% spectralresponse = Material.data(1,:)./1100;

% wavwts = Material.data(3,:).*spectralresponse;

% wavs = Material.data(1,:);

%

% [lat,hitemps,lotemps] = LocationWeatherDatabase('Rochester'); % retrieve location and

weather information

% vangwts = VerticalAngleWeightCalculator(vangs,lat,hitemps,lotemps); % vertical angle

weighting function

%

% h = .46; % injection feature size

% y0 = -0.52; % low endpoint of powered surface

% shaperot = [0 0 -0.7971 -0.9144 0.1431 0.5509 0.4529 -1.7675]; % polynomial

coefficients describing lens shape

% plotyn = 1;

%%%%%%%%%%%%%%%%%%%%%

% y0 < 0

286

evalpts = 100;

critang0(1,1,:) = asind(1./ns);

numwav = numel(ns);

wavwts = wavwts./sum(wavwts(:));

R = 1-h/d;

rnd10 = @(x) round(x.*1e10)./1e10; % round to nearest 1e-10

% set outputs to undesirable values

lensvert = zeros(5,2);

function [] = badmerit()

merit = 0;


end

% find missing endpoint component

x0rot = -y0;

y0rot = polyval(shaperot,x0rot);

if y0rot>0 || (y0rot/x0rot)>1 % return if endoint is inside guide layer or primary

reflector is too steep

badmerit;return;

end

x0 = y0rot;

% find opposite endpoint

bndslp = 2*(-x0rot/y0rot)/(1-(-x0rot/y0rot)^2); % slope of ray (rotated) incident on

primary reflector such that it reflects to be parallel to the guide

if bndslp==Inf || bndslp==-Inf || isnan(bndslp)

x1rot = 0;

else

shapeshft1 = shaperot;

shapeshft1(end-1) = shapeshft1(end-1)-bndslp;

sols1 = roots(shapeshft1);

sols1 = rnd10(sols1);

sols1 = sols1( (sols1<=x0rot) & (imag(sols1)==0) );

x1rot = max([sols1(:);x0rot-d]);

end

if x1rot>-h % return if opposite endpoint is not beyond secondary reflector

badmerit;return;

end

y1rot = polyval(shaperot,x1rot);

% check that lens shape is well defined

testxrot0 = linspace(x1rot,x0rot,100);

testxrot1 = linspace(x1rot,-h,100);

testxrot2 = linspace(0,x0rot,100);

if any(polyval(shaperot,testxrot0)>0) ||

any(polyval(shaperot,testxrot1)>testxrot1.*y1rot./x1rot) ||

any(polyval(shaperot,testxrot2)>testxrot1.*y0rot./x0rot)

badmerit;return;

end

287

% map out lens verticies in unrotated space

lensvert(2:3,1:2) = [x0,y0;y1rot,-x1rot];

lensvert(4,1:2) = [(y1rot./x1rot).*(-h),h];

lensvert(5,2) = h;

% create starting ray positions

vslpsrot = 1./tand(vangs);

aphi = x0rot-y0rot./vslpsrot;

aplo = x1rot-y1rot./vslpsrot;

taphi = x0rot.*ones(size(vangs));

parfor vi = setdiff(1:numel(vangs),find(vangs==0));

% find where each ray angle is tangent to the lens surface

tshape = polyder(shaperot);

tshape(end) = tshape(end)-vslpsrot(vi);

tsols = roots(tshape);

tsols = tsols( (tsols<=x0rot) & (tsols>=x1rot) & (imag(tsols)==0) );

tsolap = tsols-polyval(shaperot,tsols)./vslpsrot(vi);

if any(tsolap(:))

taphi(vi) = max(tsolap(:));

end

end

aphi = max(aphi,taphi);

aplo = max(aplo,max(taphi-d,aphi-d));

apwt = max((aphi-aplo)./d, 0);

% find ray starting positions in rotated space

vslpsrotmat0 = repmat(vslpsrot(:).',evalpts,1);

vslpsrotmat = repmat(vslpsrotmat0,[1,1,numwav]);

rayystrtrot = 1.1*min(polyval(shaperot,linspace(x1rot,x0rot,100))); % starting ray y

positions in rotated space

rayxstrtrot0 = repmat(linspace(0+1/evalpts/100,1-

1/evalpts/100,evalpts).',1,numel(vangs));

rayxstrtrot0 = rayxstrtrot0.*repmat(aphi(:).',evalpts,1) + (1-

rayxstrtrot0).*repmat(aplo(:).',evalpts,1); % starting ray x-intercepts in rotated space

rayxstrtrot = rayystrtrot./vslpsrotmat0 + rayxstrtrot0; % starting ray x positions in

rotated space

rayxstrtrot = repmat(rayxstrtrot,[1,1,numwav]);

%separate array for ray x position, y position, x component of direction

%vector, y component of direction vector

%carry these all the way through the trace

init0 = zeros(evalpts,numel(vangs),numwav);

rayx = struct('start',init0,'lens',init0,'refl1',init0,'refl2',init0,'inj',init0);

rayy = rayx;

rdirx = struct('start',init0,'lens',init0,'inj',init0);

rdiry = rdirx;

wts = struct('start',init0,'lens',init0,'inj',init0,'gid',init0);

rayx.start = rayystrtrot.*ones(evalpts,numel(vangs),numwav);

rayy.start = -rayxstrtrot;

288

rdirx.start = repmat(cosd(vangs(:)).',[evalpts,1,numwav]);

rdiry.start = -repmat(sind(vangs(:)).',[evalpts,1,numwav]);

wavwt(1,1,:) = wavwts(:);

wavwt = repmat(wavwt,[evalpts,numel(vangs),1]);

wts.start = repmat(1./evalpts.*apwt(:).',[evalpts,1,numwav]).*wavwt;

% find where rays intersect lens

rayxlensrot = zeros(size(rayxstrtrot));

wtslensmask = ones(size(wts.lens));

parfor ri = 1:numel(rayxstrtrot)

if vslpsrotmat(ri) == Inf || vslpsrotmat(ri) == -Inf

sols = rayxstrtrot(ri);

else

shapeshft = shaperot;

shapeshft(end-1) = shapeshft(end-1) - vslpsrotmat(ri);

shapeshft(end) = shapeshft(end) + vslpsrotmat(ri).*rayxstrtrot(ri) - rayystrtrot;

sols = roots(shapeshft);

end

sols = rnd10(sols);

sols = sols( (sols<=x0rot) & (sols>=x1rot) & (imag(sols)==0) );

if isempty(sols)

wtslensmask(ri) = 0;

else

rayxlensrot(ri) = min(sols); % non-rotated space

end

end

wts.lens = wts.start.*wtslensmask;

rayy.lens = -rayxlensrot;

rayx.lens = polyval(shaperot,-rayy.lens); % non-rotated space

% find surface normal vectors at ray intersection points

lensslpsrot = polyval(polyder(shaperot),rayxlensrot);

normdiry = lensslpsrot./sqrt(1+lensslpsrot.^2); % surface normal unit vector y component

in unrotated space

normdirx = sqrt(1-normdiry.^2); % surface normal unit vector x component in unrotated

space

% refract rays

n0(1,1,:) = ns(:);

n = repmat(n0,[evalpts,numel(vangs),1]);

wtslensmask = logical(wtslensmask);

[refvec,Tlens] =

refractfres([rdirx.start(wtslensmask),rdiry.start(wtslensmask)],[normdirx(wtslensmask),no

rmdiry(wtslensmask)],1,n(wtslensmask));

rdirx.lens(wtslensmask) = refvec(:,1);

rdiry.lens(wtslensmask) = refvec(:,2);

wts.lens(wtslensmask) = wts.lens(wtslensmask).*Tlens;

wts.lens(rdirx.lens<=0) = 0;

% find goal points

289

% function finds point (pt) reflected over a line defined by point on reflector

(ptonrefl) and vector along reflector (vecalongrefl)

ptrefl = @(pt,ptonrefl,vecalongrefl) -pt + 2.*ptonrefl +

2.*vecalongrefl./norm(vecalongrefl).*dot((pt-ptonrefl),vecalongrefl./norm(vecalongrefl));

% finds point (pt) reflected over a line defined by point on reflector (ptonrefl) and

vector along reflector (vecalongrefl)

g = zeros(4,2); % g1 = left end of secondary, g2 = g1 reflected over primary mirror, g3 =

right end of secondary reflected over primary mirror, g4 = origin reflected over line

defined by g2 and g3

g(1,:) = lensvert(4,:);

g(2,:) = ptrefl(lensvert(4,:),lensvert(1,:),lensvert(2,:));

g(3,:) = ptrefl(lensvert(5,:),lensvert(1,:),lensvert(2,:));

g(4,:) = ptrefl(lensvert(1,:),g(2,:),g(3,:)-g(2,:));

% find rays that are successfully injected into the guide

lineintx = @(pt1x,pt1y,m1,pt2x,pt2y,m2) (pt1y-pt2y+m2.*pt2x-m1.*pt1x)./(m2-m1); % returns

x component of intersection between two lines, each defined by a point and a slope

evalliny = @(x,ptx,pty,m) m.*(x-ptx)+pty; % finds y given x, a point, and a slope

A = (rdiry.lens./rdirx.lens.*(g(1,1)-rayx.lens) + rayy.lens)<=g(1,2); % rays that don't

hit inactive face of lens

yat0 = rdiry.lens./rdirx.lens.*(0-rayx.lens) + rayy.lens;

B = yat0>=0 & yat0<=g(1,2); % rays that go directly to injector (assuming A)

C = yat0>=g(1,2) & yat0<=2*g(1,2); % rays that hit secondary first, then go through

injector (assuming A)

xatrgid = lineintx(rayx.lens,rayy.lens,rdiry.lens./rdirx.lens,0,0,g(3,2)/g(3,1)); % x ray

position at guide surface reflected over primary

D = xatrgid>0 & xatrgid<=g(3,1); % rays that hit primary then go through injector

F = xatrgid>g(3,1) & xatrgid<=g(4,1); % rays that hit primary-secondary-injector,

assuming they hit the secondary at all ('E')

xatrs = lineintx(rayx.lens,rayy.lens,rdiry.lens./rdirx.lens,g(2,1),g(2,2),(g(3,2)-

g(2,2))/(g(3,1)-g(2,1))); % x ray position at secondary mirror surface reflected over

primary

E = xatrs>=g(2,1) & xatrs<g(3,1); % rays that hit primary then secondary

hiti = A & B; % straight to injector

hitsi = A & C; % secondary-injector

hitpi = A & D & ~B; % primary-injector

hitpsi = A & E & F & ~B; % primary-secondary-injector

if any( hiti & hitsi & hitpi & hitpsi )

badmerit;return;

end

%%%%%%%%% debug

% y1 = -x1rot;

% lenptsy = linspace(y0,y1,50);

% lenptsx = polyval(shaperot,-lenptsy);

% lenptsx = [lensvert(1,1),lenptsx,lensvert(4,1),lensvert(5,1)];

% lenptsy = [lensvert(1,2),lenptsy,lensvert(4,2),lensvert(5,2)];

% figure(1);

290

% aps =

plot([0,lenptsx,lenptsx,0],[y0,lenptsy,lenptsy+d,y1+d],[0;1],[0;1],[0;1],[0;1],[0;1],[0;1

]); %plot lens outline and adjacent lens above

% hold on;

% raz = plot([zeros(1,evalpts);ones(1,evalpts)],[zeros(1,evalpts);ones(1,evalpts)],'-r');

% hold off;

% axh = gca;

% axis equal;

% xlim([-3,0.5]);

% ylim([-1,4]);

% set(aps(1),'linestyle','-','color','k','LineWidth',2);

% set(aps(2:3),'linestyle','-','color','b');

% set(aps(4),'linestyle','.','color','r');

% pause on;

% for ij = 1:numel(aphi)

% set(aps(2),'xdata',[0,rayystrtrot],'ydata',[-aplo(ij),-

(rayystrtrot/vslpsrot(ij)+aplo(ij))]);

% set(aps(3),'xdata',[0,rayystrtrot],'ydata',[-aphi(ij),-

(rayystrtrot/vslpsrot(ij)+aphi(ij))]);

% set(aps(4),'xdata',rayx.start(:,ij,1),'ydata',rayy.start(:,ij,1));

% razinds = find(wts.lens(:,ij,1));

%

set(raz(:),'xdata',[rayx.start(razinds(1),ij,1);rayx.lens(razinds(1),ij,1);0],'ydata',[ra

yy.start(razinds(1),ij,1);rayy.lens(razinds(1),ij,1);yat0(razinds(1),ij,1)]);

% for jk = 2:numel(razinds)

%

set(raz(jk),'xdata',[rayx.start(razinds(jk),ij,1);rayx.lens(razinds(jk),ij,1);0],'ydata',

[rayy.start(razinds(jk),ij,1);rayy.lens(razinds(jk),ij,1);yat0(razinds(jk),ij,1)]);

% end

% title([num2str(vangs(ij)) '^o']);

% drawnow expose;

% pause(.1);

% end

% pause off;

%%%%%%%%%

rayy.inj(hiti) = yat0(hiti);

rayy.inj(hitsi) = 2.*g(1,2)-yat0(hitsi);

rayy.inj(hitpi) = xatrgid(hitpi)./g(3,1).*g(1,2);

rayy.inj(hitpsi) = (g(4,1)-xatrgid(hitpsi))./(g(4,1)-g(3,1)).*g(1,2);

rhat = -lensvert(2,:)./norm(lensvert(2,:));

rdirxp = -rdirx.lens + 2.*rhat(1).*(rdirx.lens.*rhat(1) + rdiry.lens.*rhat(2)); % same

basic formula as 'ptrefl' anonymous function

rdiryp = -rdiry.lens + 2.*rhat(2).*(rdirx.lens.*rhat(1) + rdiry.lens.*rhat(2)); % ditto

rdirx.inj = rdirx.lens;

rdiry.inj = rdiry.lens;

rdiry.inj(hitsi) = -rdiry.lens(hitsi);

rdirx.inj(hitpi | hitpsi) = rdirxp(hitpi | hitpsi);

rdiry.inj(hitpi) = rdiryp(hitpi);

rdiry.inj(hitpsi) = -rdiryp(hitpsi);

291

critslp = repmat(tand(critang0),[size(init0,1),size(init0,2),1]);

hitcrit = abs(rdiry.inj)./rdirx.inj >= critslp;

injected = (hiti | hitsi | hitpi | hitpsi) & hitcrit;

wts.inj = wts.lens;

% wts.inj(~(hiti | hitsi | hitpi | hitpsi)) = 0;

wts.inj(~injected) = 0;

wts.inj(hitsi & hitpi) = Rmirror.*wts.inj(hitsi & hitpi);

wts.inj(hitpsi) = Rmirror.^2.*wts.inj(hitpsi);

% guide ray tracing

gstrt = rayy.inj(injected)-lensvert(5,2)./2; % starting ray y positions in guide,

centered on origin

gslps = rdiry.inj(injected)./rdirx.inj(injected); % starting ray slopes in guide

gstrt(gslps<0) = -gstrt(gslps<0); % flip position of "down rays" so calculations can be

made in one direction




loledist = abs(gslps).*2.*gthik; % distance rays travel without interacting with lossy

guide surface


parfor gi = 1:numel(gdist)



end

wts.gid(injected) = wts.inj(injected).*mean(gideffs,2);

effs = reshape(sum( sum(wts.gid,3), 1), size(vangs));

merit = -100*sum(effs.*vangwts)/sum(vangwts(vangwts>0)); % normalized to an absolute

minimum of -100

%%

if plotyn

injeffs = sum(sum(wts.inj,3),1);

sefig = findobj('type','figure','name','Efficiency');

if isempty(sefig)

figure('name','Efficiency');

else

figure(sefig);

end

clf

293

% pickrays = logical(wts.gid); % matrix with ones at all positions that have a

successful ray

% numrayspaw = sum(pickrays,1); % number of successful rays for each wavelength/angle

combination

% rayinds = find(numrayspaw);

% rays2trace = [];

% for jk = 1:numel(rayinds)

% nrays = numrayspaw(rayinds(jk));

% [~,ai,wi] = ind2sub(size(numrayspaw),rayinds(jk));

% posind = find(pickrays(:,ai,wi));

% substruct = ones(size(posind));

% subs = [posind,substruct*ai,substruct*wi];

% inds = sub2ind(size(wts.gid),subs(:,1),subs(:,2),subs(:,3));

% if nrays <= q

% rays2tracetemp = inds(:);

% else

% raysy = rayy.start(inds);

% despts = transpose(linspace(min(raysy),max(raysy),q)); % desired starting

positions of rays

% [DP,RX] = ndgrid(despts,raysy);

% ydiff = abs(DP-RX);

% [~,raysyind] = min(ydiff,[],2);

% rays2tracetemp = inds(raysyind);

% end

% rays2trace = [rays2trace;rays2tracetemp];

% end

%

% [~,pangsind,~] = ind2sub(size(wts.gid),rays2trace);

% pangs = unique(vangs(pangsind));

% numpangs = numel(pangs);

% rgb = zeros(numpangs,3);

% rgbind = (pangs-min(pangs))./(max(pangs)-min(pangs)).*2;

% for nw = 1:numpangs

% if rgbind(nw)<=1

% rgb(nw,2) = rgbind(nw);

% rgb(nw,1) = 1-rgbind(nw);

% else

% rgb(nw,3) = rgbind(nw)-1;

% rgb(nw,2) = 2-rgbind(nw);

% end

% rgb(nw,:) = rgb(nw,:)./max(rgb(nw,:));

% end

% segment data

y1 = -x1rot;

lenptsy = linspace(y0,y1,60);

lenptsx = polyval(shaperot,-lenptsy);

lenptsx = [lensvert(1,1),lenptsx,lensvert(4,1),lensvert(5,1)];

lenptsy = [lensvert(1,2),lenptsy,lensvert(4,2),lensvert(5,2)];

% vangsmat = repmat(vangs(:).',[evalpts,1,numwav]);

294

%

% rtfig = findobj('type','figure','name','Raytrace');

% if isempty(rtfig)

% figure('name','Raytrace');

% else

% figure(rtfig);

% end

% clf

% hold on

% %plot rays

% for pa = 1:numpangs

% traceind = find(vangsmat==pangs(pa));

%

plot([rayx.start(traceind).';rayx.lens(traceind).';rayx.refl1(traceind).';rayx.refl2(trac

eind).';rayx.inj(traceind).';(rayx.inj(traceind)+rdirx.inj(traceind)).'],...

%

[rayy.start(traceind).';rayy.lens(traceind).';rayy.refl1(traceind).';rayy.refl2(traceind)

.';rayy.inj(traceind).';(rayy.inj(traceind)+rdiry.inj(traceind)).'],...

% 'Color',rgb(pa,:));

% end

% plot([0,lenptsx,lenptsx,0],[y0,lenptsy,lenptsy+d,y1+d],'-k','LineWidth',2); %plot

lens outline and adjacent lens above

% hold off

% axis equal

colr = zeros(size(wts.inj));

colg = colr;

colb = colr;

colwts = wts.gid./wts.start;

for vi = 1:numel(vangs)

for wi = 1:numwav

if any(colwts(:,vi,wi)>0)

colwts(find(colwts(:,vi,wi)==0),vi,wi) =

min(colwts(find(colwts(:,vi,wi)>0),vi,wi));

[colr(:,vi,wi),colg(:,vi,wi),colb(:,vi,wi)] = colorfun(colwts(:,vi,wi));

end

end

end

greyval = 0.7;

colr(wts.gid==0) = greyval;

colg(wts.gid==0) = greyval;

colb(wts.gid==0) = greyval;

[mapr,mapg,mapb] = colorfun((1:1:64).');

RaytraceGUI_SingleSegment(d,{lenptsx,lenptsy},vangs,wavs,...

{colr,colg,colb,min(colwts,[],1),max(colwts,[],1),[mapr,mapg,mapb]},...

cat(4,rayx.start,rayx.lens,rayx.refl1,rayx.refl2,rayx.inj,rayx.gid),...

cat(4,rayy.start,rayy.lens,rayy.refl1,rayy.refl2,rayy.inj,rayy.gid));

end

295

end

function [r,g,b] = colorfun(valsorig)

r = zeros(size(valsorig));

g = r;

b = r;

vals = valsorig-min(valsorig);

vals = vals./max(vals).*2; % values range from 0 to 2

b(vals<=1) = 1-vals(vals<=1);

g(vals<=1) = vals(vals<=1);

g(vals>1) = 2-vals(vals>1);

r(vals>1) = vals(vals>1)-1;

end

Tracking Louver Optimizer

Optimization routine for the suspended louver. The position and orientation of the

mobile louver is optimized for each vertical angle for maximum optical efficiency given a

reflector shape. The reflector shape is varied for each call to the louver positioning subroutine.

function [optin,fval,memorizedcalls] = TWB_Optimizer3p1(locationname,varargin)

%PatternSearch Optimization of tracking window blind concentrator

close all;


matlabpool

end

tic;

Material = MaterialDatabase('PMMA');


vangmin = 0;

vangmax = 89;

vangnum = 100;



weather information


weighting function

ns = Material.data(2,:); % refractive index of material

wavs = Material.data(1,:); % wavelengths for refractive inicies

spectralresponse = Material.data(1,:)./1100;

wavwts = Material.data(3,:).*spectralresponse;

296

ndes = ns(4);

extang = linspace(0,.26,5);

extwt = extangwts_posWzero(extang);




ubvang = 90; % vertical angle at which focus lands on the injector in the lowest position

where the incident angle is limited by the critical angle. Should be closer to 90 degrees

than 0 degrees. Sets a base point for where focus is for each reflector position.

% initial conditions

if nargin>1

r = varargin{1};

h = varargin{2};

len = varargin{3};

reflht = varargin{4};

else

r = 0.101; % injection feature radius

h = 0.015; % injection feature size is r+h, h can be negative

len = 3.387; % distance from far endpoint of reflector to focus

reflht = 1.5; % height of reflector measured along the parabola axis

end

x0 = [r,h,len,reflht]; % initial optimization point

lbnds = [0.01, -.1, .5, 0.5];

ubnds = [0.5, .2, 6, 4 ];





initmeshsize = 0.5;

tolmesh = 3e-3;

%%




mmert = zeros(mnum,1);


mp = zeros(mnum,1);

mxrng = zeros(mnum,2);

mfocpos = zeros(mnum,numel(vangs)*2);

mzang = zeros(mnum,numel(vangs));

merrcode = cell(mnum,1);

% plot initialization

plotposnum = 5;

297

plotpos = round(linspace(1,numel(vangs),plotposnum));

dummy = [0;1];

numplot = 3;


figure(10);

set(gcf,'position',[1049,107,565,806]);

topax = subplot(2,1,1);

hold on;

reflplot =

plot(repmat(dummy,1,(numplot+1)*plotposnum),repmat(dummy,1,(numplot+1)*plotposnum),'-

m','LineWidth',2); % reflectors


injection features

hold off;

axis equal;


hold on;




hold off;


grid on;




set(reflplot(plotposnum*(pj-1)+1:plotposnum*pj),'color',greys(pj).*ones(1,3));


end

rotdnx = @(x,y,zenang,xinj,p) x.*cosd(zenang) - (y-p).*sind(zenang) + xinj;

rotdny = @(x,y,zenang,yinj,p) x.*sind(zenang) + (y-p).*cosd(zenang) + yinj;

%%

psoptions =

psoptimset('CompletePoll','on','UseParallel','never','InitialMeshSize',initmeshsize,'Plot

Fcns',{@psplotbestf,@psplotbestx},'Display','iter',...

'TolMesh',tolmesh);

problem =



problem.objective = @(x0sca)TWB_MeritFunction3p1_Memorize(d, x0sca(2).*rbnds(2)+lbnds(2),

x0sca(1).*rbnds(1)+lbnds(1),...

x0sca(3).*rbnds(3)+lbnds(3), ubvang,

x0sca(4).*rbnds(4)+lbnds(4),...

ndes, vangs, vangwts, wavs, wavwts, ns, extang, extwt, glen,

gthik, Rmirror);

problem.X0 = x0sca;




298



%%


function [momert,moeffs,mop,moxrng,mofocpos,mozang,moerrcode] =

TWB_MeritFunction3p1_Memorize(md,mh,mr,mlen,mubvang,mreflht,mndes,mvangs,mvangwts,mwavs,m

wavwts,mns,mextang,mextwt,mglen,mgthik,mRmirror)



[~,rowloc] = ismember([mr,mh,mlen,mreflht],mx,'rows');

if rowloc==0

% run function

[momert,moeffs,mop,moxrng,mofocpos,mozang,moerrcode] =

TWB_MeritFunction3p1(md,mh,mr,mlen,mubvang,mreflht,mndes,mvangs,mvangwts,mwavs,mwavwts,mn

s,mextang,mextwt,mglen,mgthik,mRmirror);

% memorize if merit value is good enough and sort all memorized function calls by

merit value

if any(momert < mmert)

mmert(mnum) = momert;

[mmert,reind] = sort(mmert);

mx(mnum,:) = [mr,mh,mlen,mreflht];

mx = mx(reind,:);



mp(mnum) = mop;

mp = mp(reind);

mxrng(mnum,:) = moxrng;

mxrng = mxrng(reind,:);

mfocpos(mnum,:) = mofocpos(:).';

mfocpos = mfocpos(reind,:);

mzang(mnum,:) = mozang;

mzang = mzang(reind,:);

merrcode{mnum} = moerrcode;

merrcode = merrcode(reind);

end

else

% memorized output

momert = mmert(rowloc);


mop = mp(rowloc);

moxrng = mxrng(rowloc,:);

mofocpos = reshape(mfocpos(rowloc,:),[],2);

mozang = reshape(mzang(rowloc,:),size(vangs));

moerrcode = merrcode{rowloc};

end



299


op = flipud(mp(1:numplot-1,:));

oxrng = flipud(mxrng(1:numplot-1,:));

ofocpos = flipud(mfocpos(1:numplot-1,:));

ozang = flipud(mzang(1:numplot-1,:));

rptnum = 20;

for opi = 1:numplot-1

otr = ox(opi,1);

oth = ox(opi,2);

otlen = ox(opi,3);

otp = op(opi);

otxrng = oxrng(opi,:);

otfocpos = reshape(ofocpos(opi,:),[],2);

otzang = ozang(opi,:);

otxr0 = repmat(linspace(otxrng(1),otxrng(2),rptnum),plotposnum,1);

otyr0 = otxr0.^2./(4.*otp);

% savbase(otxr0,otyr0,otzang(plotpos),otfocpos(plotpos,1));

otxr =

rotdnx(otxr0,otyr0,repmat(otzang(plotpos).',1,rptnum),repmat(otfocpos(plotpos,1),1,rptnum

),otp);

otyr =

rotdny(otxr0,otyr0,repmat(otzang(plotpos).',1,rptnum),repmat(otfocpos(plotpos,2),1,rptnum

),otp);

if oth<-otr

otlbinj = 90;

else

otlbinj = min(90,acosd(-oth/otr));

end

otxi0 = otr.*(-sind(linspace(otlbinj,0,20))); %injector x

otyi0 = -sqrt(otr.^2-otxi0.^2); %injector y

if oth>0

otxi0 = [-otr,otxi0];

otyi0 = [oth,otyi0];

end

otxi = [0;0;otxi0(:);0;otxi0(:);0];

otyi = [d-otr;oth;otyi0(:);oth-d;otyi0(:)-d;min(-(otr+otlen),otyi0(end)-d)];

for ri0 = 1:plotposnum

set(reflplot((opi-1)*plotposnum +

ri0),'XData',otxr(ri0,:),'YData',otyr(ri0,:));

end

set(injplot(opi),'XData',otxi,'YData',otyi);

set(effplot(opi),'XData',vangs,'YData',oeffs(opi,:));

end

moxr0 = repmat(linspace(moxrng(1),moxrng(2),rptnum),plotposnum,1);

moyr0 = moxr0.^2./(4.*mop);

300

moxr =

rotdnx(moxr0,moyr0,reshape(repmat(mozang(plotpos),1,rptnum),size(moxr0)),reshape(repmat(m

ofocpos(plotpos,1),1,rptnum),size(moxr0)),mop);

moyr =

rotdny(moxr0,moyr0,reshape(repmat(mozang(plotpos),1,rptnum),size(moxr0)),reshape(repmat(m

ofocpos(plotpos,2),1,rptnum),size(moxr0)),mop);

if mh<-mr

molbinj = 90;

else

molbinj = min(90,acosd(-mh/mr));

end

moxi0 = mr.*(-sind(linspace(molbinj,0,20))); %injector x

moyi0 = -sqrt(mr.^2-moxi0.^2); %injector y

if mh>0

moxi0 = [-mr,moxi0];

moyi0 = [mh,moyi0];

end

moxi = [0;0;moxi0(:);0;moxi0(:);0];

moyi = [d-mr;mh;moyi0(:);mh-d;moyi0(:)-d;min(-(mr+mlen),moyi0(end)-d)];

for ri0 = 1:plotposnum

set(reflplot((numplot-1)*plotposnum +

ri0),'XData',moxr(ri0,:),'YData',moyr(ri0,:));

set(reflplot((numplot )*plotposnum +

ri0),'XData',moxr(ri0,:),'YData',moyr(ri0,:)+d);

end

set(injplot(numplot),'XData',moxi,'YData',moyi);

set(effplot(numplot),'XData',vangs,'YData',moeffs);

title(botax,['Merit Function Value = ' num2str(momert) ' ( Best = ' num2str(mmert(1))

' )']);

drawnow expose;

end

%%

optin = {d, optx(1), optx(2), optx(3), ubvang, optx(4), ndes, vangs, vangwts, wavs,

wavwts, ns, extang, extwt, glen, gthik, Rmirror};

memorizedcalls =

struct('mx','','mmert','','meffs','','mp','','mxrng','','mfocpos','','mzang','','merrcode

','');

memorizedcalls.mx = mx;

memorizedcalls.mmert = mmert;

memorizedcalls.meffs = meffs;

memorizedcalls.mp = mp;

memorizedcalls.mxrng = mxrng;

memorizedcalls.mfocpos = mfocpos;

memorizedcalls.mzang = mzang;

memorizedcalls.merrcode = merrcode;

%%

301

% write summary to excel

try

xlpath = 'C:\Users\Dan\Documents\Thesis\Tracking Window Blind

Concentrator\OptimizationResults_3p1.xlsx';

line1 = {datestr(now),[mfilename('fullpath'),'.m']};

line2 =

{'d','r','h','len','reflht','ubvang','ndes','glen','gthik','Rmirror','function

value','location','material'};

line3p0 = [d,optx(1),optx(2),optx(3),optx(4),ubvang,ndes,glen,gthik,Rmirror,fval];

line3p1 = {locationname,Material.name};

line4t6 = {'lower bounds';'upper bounds';'initial'};

line8 = {'vangs','zang','xfocpos','yfocpos','effs','wavs','wavwts','extang','extwt'};

[~,sheets] = xlsfinfo(xlpath);

shtnam0 = [locationname '_' sprintf('%3.2f',-fval)];

shtnam = shtnam0;

api = 1;

while any(strcmp(shtnam,sheets))

shtnam = [shtnam0 '(' num2str(api) ')'];

api = api + 1;

end

try

xlswrite(xlpath,line1,shtnam,'A1:B1');

catch openspreadsheet

xlpath = 'C:\Users\Dan\Documents\Thesis\Tracking Window Blind

Concentrator\tempoptresults.xlsx';

xlswrite(xlpath,line1,shtnam,'A1:B1');

end

xlswrite(xlpath,line2,shtnam,'A2:M2');

xlswrite(xlpath,line3p0,shtnam,'A3:K3');

xlswrite(xlpath,line3p1,shtnam,'L3:M3');

xlswrite(xlpath,line4t6,shtnam,'A4:A6');

xlswrite(xlpath,[lbnds;ubnds;x0],shtnam,'B4:E6');

xlswrite(xlpath,line8,shtnam,'A8:I8');

optfocpos = reshape(mfocpos(1,:),[],2);

xlswrite(xlpath,[vangs(:),mzang(1,:).',optfocpos(:,1),optfocpos(:,2),meffs(1,:).'],shtnam

,['A9:E' num2str(9+numel(vangs)-1)]);

xlswrite(xlpath,[wavs(:),wavwts(:)],shtnam,['F9:G' num2str(9+numel(wavs)-1)]);

xlswrite(xlpath,[extang(:),extwt(:)],shtnam,['H9:I' num2str(9+numel(extang)-1)]);

catch xlwritefail

warning('Failed to Write Data to Excel');

end

end

302

Tracking Louver Merit Function


function [mert,errcode] =

TWB_MeritFunctionSingleAng3p1(vang,d,r,h,p,x1,x2,zang,x,y,wavs,wavwts,ns,extang,extwt,gle

n,gthik,Rmirror)

% [mert,effs,reflpos,errcode] =

TWB_MeritFunctionSingleAng3p0(vang,d,r,h,p,x1,x2,zang,x,y,ndes,wavs,wavwts,ns,extang,extw

t,glen,gthik,Rmirror)

% Tracking Window Blind 3.0 solar concentrator merit function for single vertical angle

%

% vang = vertical angle

%

% d = spacing between adjacent optics along guide direction

%

% r = injector radius

%

% h = injector extension distance

%

% p = focus distance of parabolic reflector

%

% x1 = x limit of reflector closest to the vertex

%

% x2 = x limit of reflector furthest from vertex

%

% zang = zenith angle of reflector (degrees from vertical optical axis)

%

% x = x focus shift of rotated reflector

%

% y = y focus shift of rotated reflector

%

% wavs = vector of wavelengths to trace

%

% wavwts = weights for wavs input (must sum to 1)

%

% ns = refractive indicies at each given wavlength (wavs)

%

% extang = angles representing extended source (give explicitly, no

% interpretation is used)

%

% extwt = weights for input extang (must sum to 1)

%

% glen = guide length

%

% gthik = guide thickness

%

% Rmirror = reflectivity of primary reflector and injector's reflective surface

%

303

% Output

% mert = merit function value

%

% effs = efficiencies at each vertical angle (reflector position)

%

% p = 'p' in reflector shape equation, y = x^2/(4*p)

%

% focpos = two column matrix with x,y positions of reflector focus

%

% zang = rotation of reflector at each position relative to a vertical optical axis

%

% errcode = error code

%

% Version 3.1 ignores rays that run into back of adjacent reflector while

% heading towards the injection feature.

% close all;

%%%%%%%%%%%%%%%%%%%%%%%%%%% test inputs (from TWB 2.1, Rochester_66.55)

% vang = 34.88442;

% d = 2;

% r = 0.1;

% h = 0.009961;

% p = 1.5187;

% x1 = -1.4952;

% x2 = -3.3687;

% zang = 53.7676;

% x = -0.0444;

% y = -0.0310;

% Material = MaterialDatabase('PMMA');

% ns = Material.data(2,:); % refractive index of material

% wavs = Material.data(1,:); % wavelengths for refractive inicies

% spectralresponse = Material.data(1,:)./1100;

% wavwts = Material.data(3,:).*spectralresponse;

% extang = [0 0.0675 0.135 0.2025 0.27];

% extwt = [0.276625069 0.267537217 0.238138669 0.177976601 0.039722446];

% glen = 300;

% gthik = 13;

% Rmirror = 0.95;

%

% %interpret extang so it is symmetric about 0 and weights sum to unity

% if all(extang>=0) || all(extang<=0)

% extangs = [extang(:);-extang(:)];

% extwts = [extwt(:);extwt(:)];

% [extangs,exti] = unique(extangs);

% extwts = extwts(exti);

% else

% [extangs,exti] = sort(extang(:));

% extwts = reshape(extwt(exti),size(extangs));

% end

% extangs(extangs==0) = 0;

304

% extwts = extwts./sum(extwts(:));

% extang = extangs;

% extwt = extwts;

%

% wavwts = wavwts./sum(wavwts(:));

%%%%%%%%%%%%%%%%%%%%%%%%%%

numray = 100; % number of rays to trace

numwav = numel(wavs);

numext = numel(extang);

y1 = x1^2/(4*p); % y's for reflector endpoints given x's (rotated space)

y2 = x2^2/(4*p);

dang = (90-vang)-zang; % difference in incident vertical zenith angle and actual

reflector zenith rotation due to collision with guide

% transfer functions between vertical optical axis and actual position

rotdnx = @(x,y,zenang,xinj) x.*cosd(zenang) - (y-p).*sind(zenang) + xinj;

rotdny = @(x,y,zenang,yinj) x.*sind(zenang) + (y-p).*cosd(zenang) + yinj;

rotupx = @(x,y,zenang,xinj,yinj) (x-xinj).*cosd(-zenang) - (y-yinj).*sind(-zenang);

rotupy = @(x,y,zenang,xinj,yinj) (x-xinj).*sind(-zenang) + (y-yinj).*cosd(-zenang) + p;

% parabola intersection functions

parpos = @(x0,y0,m) (m + sqrt(m.^2 - (m.*x0-y0)./p)).*(2*p); % x '+' solution

parneg = @(x0,y0,m) (m - sqrt(m.^2 - (m.*x0-y0)./p)).*(2*p); % x '-' solution

pary = @(x) x.^2./(4*p); % y given x

% error out if bottom point of reflector impinges on guide layer

if rotdnx(x1,y1,zang,x) > 0

%error and exit function

mert = 0;

errcode = 'guide-reflector collision';

return;

end

% error out if reflector infringes on adjacent injector or guide surface

if h<0

lbinj = acosd(-h/r);

else

lbinj = 90;

end

injlimx = rotupx(-r.*sind(lbinj),h-d,zang,x,y);

injlimy = rotupy(-r.*sind(lbinj),h-d,zang,x,y);

parinjx = parneg(injlimx,injlimy,cotd(zang));

botx = rotdnx(x1,y1,zang,x);

if any(parinjx<x1 & parinjx>x2 & parinjx<injlimx) || botx>0

%error and exit function

mert = 0;

errcode = 'injector-reflector or guide-reflector collision';

return;

305

end

% initiate ray position, direction, and weight variables

init0 = zeros([numext,numray,numwav]);

rx = struct('start',init0(:,:,1),'refl',init0(:,:,1),'inj',init0(:,:,1),'gid',init0);

ry = struct('start',init0(:,:,1),'refl',init0(:,:,1),'inj',init0(:,:,1),'gid',init0);

rdx = struct('start',init0(:,:,1)+1,'refl',init0(:,:,1)+1,'inj',init0+1,'gid',init0+1);

rdy = struct('start',init0(:,:,1),'refl',init0(:,:,1),'inj',init0,'gid',init0);

wt = struct('start',init0(:,:,1),'refl',init0(:,:,1),'inj',init0,'gid',init0);

Extangs = repmat(extang(:),[1,numray,numwav]);

Incangs = vang + Extangs;

N = repmat(reshape(ns,1,1,[]),[numext,numray,1]);

% find initial weights based on how filled the aperture is with rays that strike the

reflector

% top point of reflector

topx = rotdnx(x2,y2,zang,x);

topy = rotdny(x2,y2,zang,y);

% bottom point of reflector

boty = rotdny(x1,y1,zang,y);

% find points where incident rays (vang,extang) are tangent to the reflector

tanrx0 = x2 .* ones(size(init0(:,1,1)));

tanrinc = 90 + dang - Extangs(:,1,1);

tanrx0(tanrinc~=Inf) = 2.*p.*tand(tanrinc(tanrinc~=Inf));

tanrx0(tanrx0<x2 | tanrx0>0) = x2;

tanrx0(tanrx0<=0 & tanrx0>x1) = x1;

tanry0 = pary(tanrx0);

tanrx = rotdnx(tanrx0,tanry0,zang,x);

tanry = rotdny(tanrx0,tanry0,zang,y);

% find points where incident rays are tangent to injector

tanix = -r.*sind(Incangs(:,1,1));

taniy = -r.*cosd(Incangs(:,1,1));

% project points back to aperture

apboty = -tand(Incangs(:,1,1)).*(topx-botx)+boty;

aptanry = d-tand(Incangs(:,1,1)).*(topx-tanrx)+tanry;

aptaniy = -tand(Incangs(:,1,1)).*(topx-tanix)+taniy;

% fraction of aperture filled (wt.start) and starting ray positions/directions

aphi = min( topy+d, min(cat(2,apboty,aptanry,aptaniy),[],2) );

aprng = aphi-topy;

wt.start = repmat(max(0,aprng)./d./numray,[1,numray]);

rx.start = topx.*ones(size(wt.start));

rx.start(wt.start==0) = 0;

ystrtseed = repmat(linspace(0+(1/numray/100),1-(1/numray/100),numray),[numext,1]);

ry.start = ystrtseed.*repmat(aprng,[1,numray]) + topy;

ry.start(wt.start==0) = 0;

rdx.start = cosd(Incangs(:,:,1));

rdx.start(wt.start==0) = 1;

rdy.start = -sind(Incangs(:,:,1));

rdy.start(wt.start==0) = 0;

306

% find ray starting positions and slopes in rotated space

rprx = rotupx(rx.start,ry.start,zang,x,y); % Ray Position in Rotated space, X component

rpry = rotupy(rx.start,ry.start,zang,x,y); % Ray Position in Rotated space, Y component

mr = tand(90+dang-Extangs(:,:,1)); % ray slopes in rotated space

% find where rays intersect reflector

pixi = zeros(size(init0(:,:,1))); % ray-Parabola Intersection, X component

pixi(sign(mr)>0) = parneg(rprx(sign(mr)>0),rpry(sign(mr)>0),mr(sign(mr)>0));

pixi(sign(mr)<0) = parpos(rprx(sign(mr)<0),rpry(sign(mr)<0),mr(sign(mr)<0));

pixi(abs(mr)==Inf) = rprx(abs(mr)==Inf);

pix = real(pixi);

piy = pary(pix); % ray-Parabola Intersection, Y component

wt.refl = Rmirror.*wt.start;

wt.refl(pix<x2 | pix>x1 | imag(pixi)~=0) = 0;

rx.refl = rotdnx(pix,piy,zang,x);

ry.refl = rotdny(pix,piy,zang,y);

% find normal vectors at reflector intersection points

nvrx = -sqrt(1./(1+1./(pix./(2.*p)).^2)); % Normal Vector in Rotated space, X component

nvry = -sqrt(1-nvrx.^2);

nvx = nvrx.*cosd(zang) - nvry.*sind(zang); % Normal Vector, X component

nvy = nvrx.*sind(zang) + nvry.*cosd(zang);

% find reflected vectors

rvtemp =

refl([rdx.start(logical(wt.refl)),rdy.start(logical(wt.refl))],[nvx(logical(wt.refl)),nvy

(logical(wt.refl))]);

rdx.refl = rdx.start;

rdy.refl = rdy.start;

rdx.refl(logical(wt.refl)) = rvtemp(:,1);

rdy.refl(logical(wt.refl)) = rvtemp(:,2);

% find if and where rays intersect injection feature

mext = rdy.refl./rdx.refl; % ray slopes

bext = ry.refl - mext.*rx.refl; % ray y-intercepts

aquad = 1+mext.^2; % "a" of quadratic formula

bquad = 2.*mext.*bext; % "b" of quadratic formula

cquad = bext.^2-r.^2; % "c" of quadratic formula

disext = bquad.^2 - 4.*aquad.*cquad; % quadratic formula discriminant

xinj = real((-bquad - sign(rdx.refl).*sqrt(disext))./(2.*aquad));

xinj(abs(mext)==Inf) = rx.refl(abs(mext)==Inf);

yinj = mext.*xinj+bext;

% find rays that strike rear of adjacent reflector

x12ap = rotdnx([x1,x2],[y1,y2],zang,x);

y12ap = rotdny([x1,x2],[y1,y2],zang,y)+d;

% x12a = rotupx(x12ap,y12ap+d,zang,0,0); % x component of adjacent reflector endpoints in

rotated space

% y12a = rotupy(x12ap,y12ap+d,zang,0,0); % y component of adjacent reflector endpoints in

rotated space

ma = diff(y12ap)./diff(x12ap); % slope between reflector endpoints

mdif0 = ma-mext;

307

mdif = mdif0;

mdif(mdif0==0) = 1;

xinta = (ma.*x12ap(1)-y12ap(1)+bext)./mdif;

yinta = ma.*(xinta-x12ap(1))+y12ap(1);

bakrefl = xinta>x12ap(2) & xinta<x12ap(1);

bakrefl(mdif0==0) = false;

bakrefl(yinta>h) = false;

% flag rays that strike the injection feature

hitcyl = xinj<0 & xinj>=-sind(lbinj) & yinj<=0 & disext>=0 & ~bakrefl;

rx.inj(hitcyl) = xinj(hitcyl);

ry.inj(hitcyl) = yinj(hitcyl);

wrep = @(in) repmat(in,[1,1,numwav]);

wt.inj = wrep(wt.refl);

wt.inj(~wrep(hitcyl)) = 0;

% figure();

% plot(x12ap(:),y12ap(:),'linewidth',3,'color','r');

% if any(any(logical(wt.inj(:,:,1))))

%

line([rx.refl(logical(wt.inj(:,:,1))).';rx.inj(logical(wt.inj(:,:,1))).'],[ry.refl(logica

l(wt.inj(:,:,1))).';ry.inj(logical(wt.inj(:,:,1))).']);

% end

% if any(bakrefl(:));

% line([rx.refl(bakrefl).';xinta(bakrefl).'],[ry.refl(bakrefl).';yinta(bakrefl).']);

% end

% title(num2str(vang));

% refract rays

goon = logical(wt.inj);

wrdxrefl = wrep(rdx.refl);

wrdyrefl = wrep(rdy.refl);

wrxinj = wrep(rx.inj);

wryinj = wrep(ry.inj);

[refrtemp,refrT] = refractfres([wrdxrefl(goon),wrdyrefl(goon)],[-wrxinj(goon),-

wryinj(goon)],1,N(goon));

rdx.inj(goon) = refrtemp(:,1);

rdy.inj(goon) = refrtemp(:,2);

wt.inj(goon) = wt.inj(goon).*refrT;

% trace rays to guide surface

ry.gid = wrep(ry.inj)-rdy.inj./rdx.inj.*wrep(rx.inj);

wt.gid = wt.inj;

wt.gid( ry.gid<-r | ry.gid>2*h+r ) = 0;

wt.gid(ry.gid>h) = wt.gid(ry.gid>h).*Rmirror;

rdx.gid = rdx.inj;

rdy.gid = rdy.inj;

rdy.gid(ry.gid>h) = -rdy.gid(ry.gid>h);

ry.gid(ry.gid>h) = 2*h - ry.gid(ry.gid>h);

wt.gidc = wt.gid;

wt.gidc(abs(rdy.gid./rdx.gid)<1./sqrt(N.^2-1)) = 0; % zero weight for rays above guide

critical angle

308

% trace through guide

gR = 1-(r+h)/d; % meta-Reflectance of lossy guide surface

loledist0 = abs(2.*rdy.gid.*gthik./rdx.gid); % distance rays travel without interacting

with lossy guide surface

grays = logical(wt.gidc) & loledist0<=glen;

loledist = loledist0(grays);

gN = floor(glen./loledist); % number of interactions with lossy guide surface while

traversing full guide length

gf = glen./loledist - gN; % remainder of guide length / lossless distance after last

interaction (gN)

wt.end = wt.gidc;

wt.end(grays) = wt.end(grays) .* ((1-gR.^gN)./(1-gR) + gf.*gR.^gN) ./ (gN+gf); % ray

weights averaged over guide length (geometric series

http://en.wikipedia.org/wiki/Summation#Some_summations_involving_exponential_terms)

% find efficiency and function output

eff2 = sum(repmat(reshape(wavwts,1,1,[]),[numext,numray,1]).*wt.end,3); % weighted

average over wavelengths

eff1 = sum(eff2,2); % sum over rays

eff = sum(extwt(:).*eff1); % weighted average over extended angles

mert = -eff; % merit value = negative efficiency

errcode = '';

%

% end

Dependency: VerticalAngleWeightCalculator.m

Calculates vertical angle weighting function for a given location.

function wtdt = VerticalAngleWeightCalculator(vangs,lat,hitemps,lotemps)

% vang = 20; %vertical angle

% lat = 43.12; %latitude

snum = 1000; %number of samples

%create sample points

sang = linspace(0,90,snum);

vang = vangs(:);

[Sang,Vang] = meshgrid(sang,vang);

Ls = sind(Vang-(90-lat)).*cosd(Sang);

%find the two corresponding days of the year (np,nn) for each sample point

np = 365.25./360.*acosd(Ls./(-sind(23.45))) - 10;

np(abs(Ls)>sind(23.45)) = 400;

np = mod(floor(np),365);

np(abs(Ls)>sind(23.45)) = 400;

nn = mod(-(np+10),365)-10;

nn = mod(nn,365);

309

nn(abs(Ls)>sind(23.45)) = 400;

%find temperature for each day

hitemp = [hitemps(end),hitemps,hitemps(1)]; % average monthly highs (farenheit) from

Dec,00 to Jan,02

lotemp = [lotemps(end),lotemps,lotemps(1)]; % average monthly lows (farenheit)

avtemp = (hitemp+lotemp)./2;

ntemp = [-17 15 46 74 105 135 166 196 227 258 288 319 349 380]; % day number for

temperature data points

tp = interp1(ntemp,avtemp,np,'pchip'); %interpolate temperature for each sample point

(np)

tn = interp1(ntemp,avtemp,nn,'pchip');

toutF = (tp+tn)./2; % average temp for both days corresponding to each sample point

toutC = (toutF-32).*5./9; % convert to celcius

% dtC = max(0,20-toutC);

dtC = 20-toutC; % differential temp in celcius/kelvin

dtC(abs(Ls)>sind(23.45)) = 0;

%find projected area for each sample point

pA = cosd(Vang).*cosd(Sang);

%weighted differential temperature output

wtdt = mean(dtC.*pA,2);

wtdt = reshape(wtdt,size(vangs));

save_to_base(1);

end

Dependency: SL_SingleSegment_BuildLT.m

Automates process of creating optimized louver geometry in LightTools (Synopsys)

optical design software.

function [] = SL_SingleSegment_BuildLT(injpos,shape,endptx,d,Rmirror,backrefl)

thickness = 0.02; % thickness of reflector

wid = 100; % extruded length of reflector

gleni = d; % default half length of guide

h = injpos(4,2); % injector height

gthiki = h; % default guide thickness

zpos = endptx./2; % z position of segment

ypos = shape(end); % y position of segment

LTconame = cell(size(shape(1:end-1)));

LTconum = length(shape)-1:-1:1;

for cn = 1:length(shape)-1

LTconame{cn} = ['Y' num2str(length(shape)-cn)];

end

310

% set up activexservers

lt = actxserver('LightTools.LTAPI3');

ltml = actxserver('ltcom64.LTAPI2');

try

%start LightTools interop

retcod = ltml.LTBegin(lt);

%delete old copy of injection feature if it exists

einjLK = ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

ename = 'InjectionFeature_1';

[einjDK, s_einjdk] = ltml.LTListByName(lt, einjLK, ename);

if ~strcmp(s_einjdk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

en = ltml.LTDbKeyStr(lt,einjDK);

ltml.LTCmd(lt, ['\O' en ' ']);

ltml.LTCmd(lt, 'Delete= ');

ltml.LTCmd(lt, '\Q ');

end

ltml.LTListDelete(lt, einjLK);

clear einjLK einjDK

%build extruded injection feature

ipts = [injpos(1,:)+[0.01,0];injpos(1:3,:);injpos(4,:)+[0.01,0]];

ltcmdstr = 'ExtrudedPrism';

for nm = 1:size(ipts,1);

ltcmdstr = [ltcmdstr ' XYZ ' num2str(-wid/2) ',' num2str(ipts(nm,2)) ','

num2str(ipts(nm,1))];

end

for doubleclick = 1:2

ltcmdstr = [ltcmdstr ' XYZ ' num2str(wid/2) ',' num2str(ipts(end,2)) ','

num2str(ipts(end,1))];

end

ltml.LTCmd(lt, ltcmdstr);

%name injection feature

extruLK = ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

extruDK = ltml.LTListLast(lt, extruLK);

ltml.LTDbSet(lt, extruDK, 'Name', 'InjectionFeature_1');

ltml.LTDbSet(lt, extruDK, 'Material', 'PMMA_USER');

%add optical properties if need be

opropLK = ltml.LTDbList(lt,'LENS_MANAGER[1].PROPERTY_MANAGER[Optical Properties

Manager]','PROPERTY');

[fresDK, s_splitdk] = ltml.LTListByName(lt,opropLK,'FresLoss');

[reflDK, s_refldk] = ltml.LTListByName(lt,opropLK,'ReflPart');

if strcmp(s_splitdk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

ltml.LTDbSet(lt, 'LENS_MANAGER[1].PROPERTY_MANAGER[Optical Properties Manager]',

'AddNew', 'FresLoss');

end

if strcmp(s_refldk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

311


'AddNew', 'ReflPart');

end

if strcmp(s_splitdk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString') ||

strcmp(s_refldk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

ltml.LTListDelete(lt, opropLK);

opropLK = ltml.LTDbList(lt,'LENS_MANAGER[1].PROPERTY_MANAGER[Optical Properties

Manager]','PROPERTY');

[fresDK, s_splitdk] = ltml.LTListByName(lt,opropLK,'FresLoss');

[reflDK, s_refldk] = ltml.LTListByName(lt,opropLK,'ReflPart');

end

FresLossFN = 'C:\Users\Dan\Documents\Thesis\Suspended Louver\SL_FresLoss.1.opr';

ReflPartFN = 'C:\Users\Dan\Documents\Thesis\Suspended Louver\SL_ReflPart.1.opr';

ltml.LTDbSet(lt, fresDK, 'Load_Properties', FresLossFN);

ltml.LTDbSet(lt, reflDK, 'Load_Properties', ReflPartFN);

ltml.LTDbSet(lt, reflDK, 'Reflectance_Percent', 100*Rmirror);

%get injection feature zone data keys

zoneLK = ltml.LTDbList(lt,extruDK,'ZONE');

zonenum = ltml.LTListSize(lt,zoneLK);

if ischar(zonenum)

zonenum = LTisStupid(zonenum);

end

zoneDK = cell(zonenum,1);

for ki = 1:zonenum

zoneDK{ki} = ltml.LTListNext(lt, zoneLK);

end

% apply optical properties to surfaces

ltml.LTSetOption(lt, 'DBUPDATE', 0);

for sn = 1:numel(zoneDK)

ltml.LTDbSet(lt, zoneDK{sn}, 'PropertyName', 'FresLoss');

end

ltml.LTDbSet(lt, zoneDK{4}, 'PropertyName', 'ReflPart');


%%

%delete old copy of guide if it exists

egidLK = ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

gname = 'Guide_1';

[egidDK, s_egiddk] = ltml.LTListByName(lt, egidLK, gname);

if ~strcmp(s_egiddk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

gn = ltml.LTDbKeyStr(lt,egidDK);

ltml.LTCmd(lt, ['\O' gn ' ']);



end

ltml.LTListDelete(lt, egidLK);

clear egidLK egidDK

%build guide

312

gpts = [0,h/2; 0,h/2+gleni; gthiki,h/2];

ltcmdstrgid = 'Block3pt';

for mn = 1:size(gpts,1);

ltcmdstrgid = [ltcmdstrgid ' XYZ 0,' num2str(gpts(mn,2)) ','

num2str(gpts(mn,1))];

end

ltml.LTCmd(lt, ltcmdstrgid);

%name guide and assign material

gidLK = ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

gidDK = ltml.LTListLast(lt, gidLK);

ltml.LTDbSet(lt, gidDK, 'Name', 'Guide_1');

ltml.LTDbSet(lt, gidDK, 'Material', 'PMMA_USER');

%set guide width

[gprimLK, s_gprimlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','CUBE_PRIMITIVE');

gprimDK = ltml.LTListLast(lt, gprimLK);

ltml.LTDbSet(lt, gprimDK, 'Width', wid);

ltml.LTListDelete(lt,gprimLK);

clear gprimLK s_gprimlk gprimDK;


gzoneLK = ltml.LTDbList(lt,gidDK,'ZONE');

gzonenum = ltml.LTListSize(lt,gzoneLK);

if ischar(gzonenum)

gzonenum = LTisStupid(gzonenum);

end

gzoneDK = cell(gzonenum,1); % {LeftSurface; BackSurface; TopSurface; FrontSurface;

BottomSurface; RightSurface}

for gi = 1:gzonenum

gzoneDK{gi} = ltml.LTListNext(lt, gzoneLK);

end



for gz = 1:numel(gzoneDK)

ltml.LTDbSet(lt, gzoneDK{gz}, 'PropertyName', 'FresLoss');

end

ltml.LTDbSet(lt, gzoneDK{3}, 'PropertyName', 'Optical Absorber');



%%

%delete currently existing blades if they exist

esolidLK = ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

bname = {'Blade_1';'Blade_2'};

for bd = 1:2

[ebladedk, s_ebladedk] = ltml.LTListByName(lt, esolidLK, bname{bd});

if ~strcmp(s_ebladedk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

313

bn = ltml.LTDbKeyStr(lt,ebladedk);

ltml.LTCmd(lt, ['\O' bn ' ']);



end

end

ltml.LTListDelete(lt, esolidLK);

%create base lens with the correct width and xyz position

pt1 = ltml.LTCoord3(lt, 0, ypos, zpos);

pt2 = ltml.LTCoord3(lt, 0, ypos+abs(endptx)./2, zpos);

pt3 = ltml.LTCoord3(lt, 0, ypos+abs(endptx)./2, zpos+thickness);

pt4 = ltml.LTCoord3(lt, 0, ypos-thickness, zpos);

ltml.LTCmd(lt, ['Sketch3PtLens ' pt1 ' ' pt2 ' ' pt3 ' ' pt4]);

[solidLK, s_solidlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SOLID');

solidDK = ltml.LTListLast(lt, solidLK);

%get datakey of newly created lens

%LK suffix stands for 'listkey'

%DK suffix stands for 'datakey'



solidDK = ltml.LTListLast(lt, solidLK);

[primLK, s_primlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','CIRC_LENS_PRIMITIVE');

primDK = ltml.LTListLast(lt, primLK);

[surfLK, s_surflk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','SPHERICAL_LENS_SURFACE');

[numsurf, s_numsurf] = ltml.LTListSize(lt, surfLK);

if ischar(numsurf)

numsurf = LTisStupid(numsurf);

end

if ischar(numsurf)

ltml.LTCmd(lt, 'Undo');

retcod = ltml.LTEnd(lt);

error('craps')

end

rc_setpos = ltml.LTListSetPos(lt, surfLK, numsurf-1);

surf1DK = ltml.LTListNext(lt, surfLK); % LensFrontSurface

surf2DK = ltml.LTListNext(lt, surfLK); % LensRearSurface

%create reflector optical property if need be

[OpropLK, s_oproplk] = ltml.LTDbList(lt,'LENS_MANAGER[1].PROPERTY_MANAGER[Optical

Properties Manager]','PROPERTY');

[wbreflDK, s_wbrefldk] = ltml.LTListByName(lt,OpropLK,'WBreflector');

if strcmp(s_wbrefldk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

314


'AddNew', 'WBreflector');

ltml.LTListDelete(lt, OpropLK);

[OpropLK, s_oproplk] = ltml.LTDbList(lt,'LENS_MANAGER[1].PROPERTY_MANAGER[Optical

Properties Manager]','PROPERTY');

[wbreflDK, s_wbrefldk] = ltml.LTListByName(lt,OpropLK,'WBreflector');

end

WBreflectorFN = 'C:\Documents and Settings\Dan\My Documents\Rambus Ideas\High FoV

Lightguide\Windowblind Reflector\WBreflector.1.opr';

ltml.LTDbSet(lt, wbreflDK, 'Load_Properties', WBreflectorFN);

ltml.LTListDelete(lt, OpropLK);

clear OpropLK OpropDK

%delete zones and apply correct angle and surface shape to each segment


%delete zones

znf = ltml.LTDbKeyStr(lt, surf1DK);

znb = ltml.LTDbKeyStr(lt, surf2DK);

ltml.LTCmd(lt, ['\O' znf '.ZONE[zone_1] ']);

ltml.LTCmd(lt, '"Delete Zone"= ');


ltml.LTCmd(lt, ['\O' znb '.ZONE[zone_1] ']);

ltml.LTCmd(lt, '"Delete Zone"= ');


%set element shape

ltml.LTDbSet(lt, primDK, 'Element_Shape', 'Rectangular');

ltml.LTDbSet(lt, primDK, 'Width', wid);

%front and rear surface modifications

ltml.LTDbSet(lt, surf1DK, 'Surface_Shape', 'XYPolynomial');

ltml.LTDbSet(lt, surf1DK, 'Concave', 'CC');

ltml.LTDbSet(lt, surf2DK, 'Surface_Shape', 'XYPolynomial');

ltml.LTDbSet(lt, surf2DK, 'Concave', 'CX');

for co = 1:length(LTconame)

ltml.LTDbSet(lt, surf1DK, LTconame{co}, (-1).^LTconum(co).*shape(co));

ltml.LTDbSet(lt, surf2DK, LTconame{co}, shape(co));

end



ltml.LTDbSet(lt, surf1DK, 'Y_Decenter', -zpos);

ltml.LTDbSet(lt, surf2DK, 'Y_Decenter', -zpos);


%get zone data keys and apply optical properties

zoneDK = cell(1,6); %LensFrontSurface:bare, LensRearSurface:bare, TopSurface:bare,

BackSurface:bare, FrontSurface:bare, BottomSurface:bare

segsolid = ltml.LTDbKeyStr(lt, solidDK);

[zoneLK, s_zonelk] = ltml.LTDbList(lt,segsolid,'ZONE'); %List Key for segment using

"ZONE" filter

zoneDK{1} = ltml.LTListNext(lt, zoneLK); %LensFrontSurface

zoneDK{2} = ltml.LTListNext(lt, zoneLK); %LensRearSurface

315

zoneDK{3} = ltml.LTListNext(lt, zoneLK); %TopSurface

zoneDK{4} = ltml.LTListNext(lt, zoneLK); %BackSurface

zoneDK{5} = ltml.LTListNext(lt, zoneLK); %FrontSurface

zoneDK{6} = ltml.LTListNext(lt, zoneLK); %BottomSurface



ltml.LTDbSet(lt, zoneDK{2}, 'PropertyName', 'Optical Absorber');

if backrefl


end


% select new lens

sntemp = ltml.LTDbKeyStr(lt, solidDK);

snpos = strfind(sntemp, '[');

solidname = sntemp(max(snpos)+1:end-1);

ltml.LTListDelete(lt, solidLK);

clear solidLK solidDK

ltml.LTCmd(lt, ['DefaultSelect "' solidname '.tag_0" ']);

ltml.LTCmd(lt, ['CopyVector XYZ 0,0,0 XYZ 0,' num2str(d) ',0 ']);



[numsolid, s_numsolid] = ltml.LTListSize(lt, solidLK);

if ischar(numsolid)

numsolid = LTisStupid(numsolid);

end

rc_setpos1 = ltml.LTListSetPos(lt, solidLK, numsolid-1);

solidDK1 = ltml.LTListNext(lt, solidLK);

solidDK2 = ltml.LTListNext(lt, solidLK);

ltml.LTDbSet(lt, solidDK1, 'Name', 'Blade_1');

ltml.LTDbSet(lt, solidDK2, 'Name', 'Blade_2');

% put parameters into LT variable: 'SLscript'

paramname = 'SLscript';

paramvals = 0;

paramdesc = [sprintf('%s.m, %s, endptx=%.6g, d=%.6g, Rmirror=%.6g, shape=%.6g',

mfilename, datestr(now), endptx, d, Rmirror, shape(1)), sprintf('%.6g,',shape(2:end))];

%find if parameter exists

[paramLK, s_paramlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].PARAMETRIC_CONTROLS[Parametric_Controls]','NUMERIC_PARA

METER');

[paramDK, s_paramdk] = ltml.LTListByName(lt, paramLK, paramname);

if strcmp(s_paramdk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

ltml.LTListDelete(lt, paramLK);

ltml.LTCmd(lt, ['AddParameter "' paramname '" 1 ""']);

[paramLK, s_paramlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].PARAMETRIC_CONTROLS[Parametric_Controls]','NUMERIC_PARA

METER');

316

[paramDK, s_paramdk] = ltml.LTListByName(lt, paramLK, paramname);

end

ltml.LTDbSet(lt, paramDK, 'Value', paramvals);

ltml.LTDbSet(lt, paramDK, 'Description', paramdesc);

%%

%delete old copy of source if it exists

esrcLK =

ltml.LTDbList(lt,'LENS_MANAGER[1].ILLUM_MANAGER[Illumination_Manager]','SOURCES');

sname = 'Source_1';

[esrcDK, s_esrcdk] = ltml.LTListByName(lt, esrcLK, sname);

if ~strcmp(s_esrcdk, 'LTReturnCodeEnum_ltStatusInvalidDataAccessString')

sn = ltml.LTDbKeyStr(lt,esrcDK);

ltml.LTCmd(lt, ['\O' sn ' ']);



end

ltml.LTListDelete(lt, esrcLK);

clear esrcLK esrcDK

%build source

y0 = polyval(shape,endptx);

spts = [endptx-0.1,y0+d/2; endptx-0.1,y0+d; endptx,y0+d/2];

ltcmdstrsrc = 'CubeSource';

for sm = 1:size(spts,1);

ltcmdstrsrc = [ltcmdstrsrc ' XYZ 0,' num2str(spts(sm,2)) ','

num2str(spts(sm,1))];

end

ltml.LTCmd(lt, ltcmdstrsrc);

%name guide and assign material

srcLK =

ltml.LTDbList(lt,'LENS_MANAGER[1].ILLUM_MANAGER[Illumination_Manager]','CUBE_SURFACE_SOUR

CE');

srcDK = ltml.LTListLast(lt, srcLK);

ltml.LTDbSet(lt, srcDK, 'Name', 'Source_1');

ltml.LTDbSet(lt, srcDK, 'Power Extent', 'Aim Region');

ltml.LTDbSet(lt, srcDK, 'Width', wid);

for jk = 1:5

ltml.LTDbSet(lt, srcDK, 'EmittingAt', 'No', jk);

end

srcKS = ltml.LTDbKeyStr(lt,srcDK);

srcasLK = ltml.LTDbList(lt,srcKS,'AIM_SPHERE');

srcasDK = ltml.LTListLast(lt, srcasLK);

ltml.LTDbSet(lt, srcasDK, 'Lower Angle', 0);

ltml.LTDbSet(lt, srcasDK, 'Alpha', -20);

%set guide width

[gprimLK, s_gprimlk] =

ltml.LTDbList(lt,'LENS_MANAGER[1].COMPONENTS[Components]','CUBE_PRIMITIVE');

gprimDK = ltml.LTListLast(lt, gprimLK);

317

ltml.LTDbSet(lt, gprimDK, 'Width', wid);

ltml.LTListDelete(lt,gprimLK);

clear gprimLK s_gprimlk gprimDK;


gzoneLK = ltml.LTDbList(lt,gidDK,'ZONE');

gzonenum = ltml.LTListSize(lt,gzoneLK);

if ischar(gzonenum)

gzonenum = LTisStupid(gzonenum);

end

gzoneDK = cell(gzonenum,1); % {LeftSurface; BackSurface; TopSurface; FrontSurface;

BottomSurface; RightSurface}

for gi = 1:gzonenum

gzoneDK{gi} = ltml.LTListNext(lt, gzoneLK);

end



for gz = 1:numel(gzoneDK)

ltml.LTDbSet(lt, gzoneDK{gz}, 'PropertyName', 'FresLoss');

end




%%


save_to_base(1);

disp('done');

catch LTerror


error('error in LT portion of matlab code');

end

end

Dependency: LocationWeatherDatabase.m

Database of weather data for given location: Rochester, El Paso, and Fairbanks.

function [lat,hitemps,lotemps,epwfn,synDHIconst,cleardayDNI,lon,tmz] =

LocationWeatherDatabase(locstr)

% [lat,hitemps,lotemps,epwfn] = locationweatherdatabase(locstr)

% Accepts location name and returns latitude, average monthly high temperatures,

% average monthly low temperatures (www.weather.com), and a string containing

% the file path to the .epw weather data file

% (http://apps1.eere.energy.gov/buildings/energyplus/weatherdata_about.cfm)

%

% Acceptable locations include 'Rochester', 'El Paso', 'Fairbanks'

318

switch locstr

case 'Rochester'

lat = 43.12; % latitude

hitemps = [32 34 43 56 68 77 81 79 72 60 48 37]; % average monthly highs

(farenheit) www.weather.com

lotemps = [18 19 26 37 46 56 61 60 52 42 33 24]; % average monthly lows


epwfn = 'C:\Users\Dan\Documents\Thesis\Weather Data\USA_NY_Rochester-

Greater.Rochester.Intl.AP.725290_TMY.epw';

synDHIconst = 200; % synthetic sunny case DHI (W/m2) judged by average sunny day

from TMY data

cleardayDNI = 893;

lon = -77.67;

tmz = -5;

case 'El Paso'


hitemps = [58 63 70 79 88 96 95 93 88 78 66 57]; % average monthly highs


lotemps = [33 37 43 51 60 68 71 70 63 52 40 33]; % average monthly lows


epwfn = 'C:\Users\Dan\Documents\Thesis\Weather

Data\USA_TX_El.Paso.Intl.AP.722700_TMY3.epw';

synDHIconst = 200;

cleardayDNI = 1000;

lon = -106.50;

tmz = -7;

case 'Fairbanks'


hitemps = [ 3 11 25 44 61 71 72 66 54 32 12 07]; % average monthly highs


lotemps = [-11 -7 2 21 36 48 51 45 34 16 -2 -8]; % average monthly lows


epwfn = 'C:\Users\Dan\Documents\Thesis\Weather

Data\USA_AK_Fairbanks.Intl.AP.702610_TMY3.epw';

synDHIconst = 200; % re-evaluate for Fairbanks !!!!!!!!!!!!!!!!!!!!!!!!!!

cleardayDNI = 893; % re-evaluate for Fairbanks !!!!!!!!!!!!!!!!!!!!!!!!!!

lon = -147.85;

tmz = -9;

otherwise

error('Location (locstr) not recognized. Note: string is case sensitive.');

end

Dependency: MaterialDatabase.m

Callable database of dispersive dielectric material information.

function [Material] = MaterialDatabase(materialname)

if ~any(strcmp(materialname,{'PMMA','S-BSL7','SBSL7'}))

319

error('Requested material not found in database.')

end

switch materialname

case 'PMMA'

Material = struct('name','PMMA','data',...

[320 407 493 580 667 753 840 927 1013 1100 ; %

wavelengths in nm

1.5136 1.5063 1.4973 1.4922 1.4889 1.4869 1.4852 1.4841 1.4831 1.4831; %

index of refraction

0.134 0.778 1.328 1.333 1.234 1.018 0.898 0.535 0.651 0.278 ]); %

relative weights

case {'S-BSL7','SBSL7'} % Ohara equivalent of N-BK7

Material = struct('name','S-BSL7','data',...

[365 440 516 591 666 742 817 892 968 1043 1118; % wavelengths in nm

1.535787392 1.525798128 1.519927707 1.516187475 1.513557949

1.511557671 1.509987589 1.508672819 1.507509303 1.506469274

1.505497831; % index of refraction

0.418 1.099 1.351 1.317 1.286 1.108 0.790 0.847 0.613 0.628 0.208 ]); %

relative weights

End

Dependency: extangwts_posWzero.m

Creates sample angles and appropriate weights for an extended disc-shaped light

source, simulating sunlight in angular extent.

function extwt = extangwts_posWzero(extang)

sunrad = 0.267;

if ~ismember(0,extang)

error('0 is not a member of the input "extang"');

elseif any(extang<0)

error('some elements of the input "extang" are negative');

end

ahalf = @(h,r) 0.5.*h.*sqrt(r.^2-h.^2) + asin(h./r).*r.^2./2;

[extangsort,sortind] = sort(extang);

[~,revsortind] = sort(sortind);

padext = [-extangsort(2),extangsort,extangsort(end)+(extangsort(end)-extangsort(end-

1))/2];

hhi = mean([extangsort;padext(3:end)],1);

hlo = mean([extangsort;padext(1:end-2)],1);

hhi = max(min(hhi,sunrad),-sunrad);

320

hlo = max(min(hlo,sunrad),-sunrad);

extwt = ahalf(hhi,sunrad)-ahalf(hlo,sunrad);

extwt = extwt(revsortind);

extwt = extwt./sum(extwt);