optical design with zemax for phd
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www.iap.uni-jena.de
Optical Design with Zemax
for PhD
Lecture 15: Scattering
2016-04-22
Herbert Gross
Winter term 2015 / Summer term 2016
2
Preliminary Schedule
No Date Subject Detailed content
1 11.11. Introduction Zemax interface, menus, system description, editors,, coordinate systems, aperture,
field, wavelength, layouts, raytrace, stop and pupil, solves, ray fans, paraxial optics
2 02.12. Basic Zemax handling surface types, quick focus, catalogs, vignetting, footprints, system insertion, scaling,
component reversal
3 09.12. Properties of optical systems aspheres, gradient media, gratings and diffractive surfaces, special types of
surfaces, telecentricity, ray aiming, afocal systems
4 16.12. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave
aberrations
5 06.01. Aberrations II PSF, MTF, ESF
6 13.01. Optimization I algorithms, merit function, variables, pick up’s
7 20.01. Optimization II methodology, correction process, special requirements, examples
8 27.01. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro
language
9 03.02. Correction I simple and medium examples
10 10.02. Correction II advanced examples
11 02.03. Illumination simple illumination calculations, non-sequential option
12 23.03. Physical optical modelling Gaussian beams, POP propagation
13 08.04 Tolerancing I Sensitivities, Tolerancing
14 15.04. Tolerancing II Adjustment, thermal loading
15 22.04. Scattering Introduction, surface scattering, diffraction and empirical, optical, systems, volume
scattering, models, tissue, scattering in Zemax
16 13.05. Coatings Introduction, matrix calculus, properties, single and double layer, miscellaneous,
coatings in Zemax
1. Introduction
2. Fourier description and PSD of surface scattering
3. Diffraction scattering models
4. Empirical models and BSDF
5. False light in optical systems
6. Calculation of straylight and examples
7. Volume scattering
8. Modelling
9. Tissue scattering
10. Scattering in Zemax
3
Contents
Definition of Scattering
basic description of scattering
Different surface geometries:
Every micro-structure generates
a specific straylight distribution
Light Distribution due to Surface Geometry
x
y
Smooth
surface
Specular
reflex
No scattering
x
y
Regular
grating
Discrete
pattern of
diffraction
orders
x
y
Irregular
grating
Continuous
linear scatter
pattern
x
yStatistical
isotropic
surface
Broad scatter
spot
Ref.: J. Stover, p.10
Scattering at rough surfaces:
statistical distribution of light scattering
in the angle domain
Angle indicatrix of scattering:
- peak around the specular angle
- decay of larger angle distributions
depends on surface treatment
is
dP
dLog
q
qscattering angle
special polishing
Normal polishing
specular angle
Phenomenology of Surface Scattering
Definition of Scattering
• Physical reasons for scattering:
- Interaction of light with matter, excitation of atomic vibration level dipols
- Resonant scattering possible, in case of re-emission l-shift possible
- Direction of light is changed in complicated way, polarization-dependent
• Phenomenological description (macroscopic averaged statistics)
1. Surface scattering:
1.1 Diffraction at regular structures and boundaries:
gratings, edges (deterministic: scattering ?)
1.2 Extended area with statistical distributed micro structures
1.3 Single micros structure: contamination, imperfections
2. Volume scattering:
2.1 Inhomogeneity of refractive index, striae, atmospheric turbulence
2.2 Ensemble of single scattering centers (inclusions, bubble)
Therefore more general definition:
- Interaction of light with small scale structures
- Small scale structures usually statistically distributed (exception: edge, grating)
- No absorption, wavelength preserved
- Propagation of light can not be described by simple means (refraction/reflection)
1. Surface scattering
1.1 Edge diffraction
1.2 Scattering at topological small structures of a surface
Continuous transition in macroscopic dimension: ripple due to manufacturing,
micro roughness, diffraction due to phase differences
1.3 Scattering at defects (contamination, micro defects), phase and amplitude
2. Scattering at single particles:
2.1 Rayleigh scattering , d << l
2.2 Rayleigh-Debye scattering, d < l
2.3 Mie scattering, spherical particles d > l
3. Volume scattering
3.1 Scattering at inhomogeneities of the refractive index,
e.g. atmospheric turbulence, striae
3.2 Scattering at crystal boundaries (e.g. ceramics)
3.3 Scattering at statistical distributed dense particles
e.g. biological tissue
Scattering Mechanisms and Models
Geometry simplified
Boundaries simplified, mostly at infinity
Isotropic scattering characteristic
Perfect statistics of distributed particles
Multiple scattering neglected
Discretization of volume
Angle dependence of phase function simplified
Scattering centers independent
Scatterers point like objects
Spatially varying material parameters ignored
Field assumed to be scalar
Decoherence effects neglected
Absorption neglected
Interaction of scatterers neglected
l-dispersion of material data neglected
Approximations in Scattering Models
Definition of Scattering
Fourier description and PSD
Autocorrelation function of a rough surface
Correlation length tc :
Decay of the correlation function,
statistical length scale
Value at difference zero
Special case of a Gaussian distribution
2)0( rmsC
dxxxhxhL
xxhxhxC )()(1
)()()(
2
2
1
2)(
c
x
rms exCt
C( x)
x
tc
rms
2
Autocorrelation Function
Surface Characterization
h(x)
x
C(x)
x
PSD(k)
k
A(k)
k
FFT FFT
| |2
< h1h
2 >
correlation
square
dxeyxhkA ikx
L
0
),()(
dxxxhxhL
xC )()(1
)(
topology
spectrum
autocorrelation
power spectral
density
2
)(1
)( dxexhL
kF ikx
PSD
Fourier transform of a surface
spectral amplitude density
PSD power spectral density
relative power of frequency
components
Arae under PSD-curve
Meaningful range of frequencies
Polished surfaces are similar and have fractal sgtructure,
PSD has slope 1.5 ... 2.5
Relation to auto-correlation function of the surface
dxeyxhkA ikx
L
0
),()(
2
21 1(v , v ) ( , )
2
x yi x v y v
PSD x yF h x y e dx dyA
2 1, vrms PSD x y x yF v dv dv
A
1 1...v
D l
0
1ˆ(v) ( ) ( ) cosPSDF F C x C x xv dx
PSD of a Surface
Spatial Frequency of Surface Perturbations
Power spectral density of the perturbation
Three typical frequency ranges,
scaled by diameter D
1. Long range, figure error
deterministic description
resolution degradation
2. Mid frequency, critical
model description complicated
3. Micro roughness
statistical description
decrease of contrast
limiting lineslope m = -1.5...-2.5
log A2
Four
long range
low frequency
figure
Zernike
mid
frequencymicro
roughness
1/l
oscillation of
the polishing
machine
12/D1/D 40/D
Definition of Scattering
diffraction scattering models
Scalar model for straylight calculation with Kirchhoff diffraction integral
Surface as phase mask
Approximations:
-no obscuration
- smooth surface limit
dydxeeEyxEyyxx
R
ik
yxhki
s
''),(2
0)','(
q
q
i
s
h1
h2
r
ki
ks
Kirchhoff Theory of Scattering
l
q 81
cos i rms
Description of scattering by linear system theory
L: ray density
Transfer function
Angle distribution:
1. specular part
2. scattering part
Scattering contribution corresponds to BSDF
Harvey-Shack Theory
L H Lout in
i
s
rmsrmsir
y
r
xC
S
OTF eeyxHqq cos
,1
1cos4)(
22
),(
),(),(
),(ˆ),(
SBA
yxHFL SPSF
ssiiBSDF
ii
FRPR
IS qq
q
,,,
1
cos
),(),(
H(x,y)
y
A
B
specular contribution
scattering contribution
L ( )
specular contribution
scattering contribution
PSF
i
Definition of Scattering
BSDF and empirical scattering models
Description of scattering characteristic of a surface: BSDF
(bidirectional scattering distribution function)
Straylight power into the solid angle dW
from the area element dA relative to
the incident power Pi
The BSDF works as the angle response
function
Special cases: formulation as convolution
integral
W
ddP
dP
dP
dLF
i
s
i
sBSDF
qcos
ndW
solid angle
areaelement
scattered power
dPs
normal
incidentpower
dPi
qi
sq
W iiiiiiBSDF dPFP qqqqq cos),(,,,),(
BSDF of a Surface
Exponential correlation
decay
PSD is Lorentzian function
Gaussian coerrelation
Fractal surface with
Hausdorf parameter D
K correlation model
parameter B, s
Model Functions of Surfaces
C x erms
x
c( )
t2
F s
sPSD
rms c
c
( )
1
1
2
2
t
t
C x erms
x
c( )
t2
1
2
2
F s ePSD
c rms
s c
( )
t
t2
2
4
2
F s
n
n
K
sPSD
n
n( )
1
2
21
2 2
1
F s
A
s BPSD C
( )/
12
2
Empirical model function of BSDF
Notations:
sine of scattering angle
slope parameter m
glance angle
reference and pivot angle : ref
BSDF value at reference: a
Simple isotropic scalar model
m
ref
spec
BSDF aF
)(
sq sin
specspec q sin
BSDF Model of Harvey-Shack
specs
ref
Definition of Scattering
Data of technical surfaces
Roughness
of optical
surfaces,
Dependence of
treatment
technology
10 4
10 2
10 0
10 -2
10 -610 -4 10 -2 10 0
10 +2
Grinding
Polishing
Computercontrolledpolishing
Diamond turning
Plasmaetching
Ductilemanufacturing
Ion beam finishing
Magneto-rheological treatment
roughnessrms [nm]
material removalqmm / s
Roughness of Optical Surfaces
Roughness of Optical Surfaces
l
0.2
0.5
1.0
2.0
5.0
10.0
0.2 0.5 1.0 2.0 5.0 10.0 20.0
super
polish
normal
polishmetal
TIS = 10-1
TIS = 10-2
TIS = 10-3
TIS = 10-4
TIS = 10-5
TIS = 10-6
Maximum BRDF at angle of reflection
Larger BRDF
for skew incidence
BRDF of Black Lacquer
q
BRDF
10-1
10-2
10-3
10-4
10-5
10-6
0° 30° 60° 90° 120° 150° 180° 210° 240°
0°30°
10°20°
40°
50°
60°
70°
80°
qi
x
z
qi
q
Ref.: A. Bodemann
Definition of Scattering
false light in optical systems
Sources of Stray Light
Ref: B. Goerz
Different reasons
Various distributions
Straylight and Ghost Images
a b
Scattering of Light
Scattering of light in diffuse media like frog
Ref: W. Osten
Definition of Scattering
Calculation of straylight and examples
Photometrical calculation of the transfer of energy density
Integration of the solid angle by raytrace
in the system model
g : geometry factor
surface response : BSDF
T : transmission
Practical Calculation of Straylight
W ddALdP qcos
W BRDFs FTgEP
incident ray
mirror
next
surfaceFBRDF
real used solid
angle
Decomposition of the system
into different ray paths
Properties:
- extrem large computational effort
- important sampling guaratees quantitative results for large dynamic ranges
- mechanical data necessary and important
often complicated geometry and not compatible with optical modelling
- surface behavior (BRDF) necessary with large accuracy
Practical Calculation of Straylight
source
diffraction
scattering
scattering
detector
Straylight Calculation
1. Mechanical 2. Simplified mechanics
system for calculation
3. Critical
straylight
paths
Ref: R. Sand
Straylight calculation in a telescope
Contributions form:
1. surfaces
2. mechanical parts
3. diffraction at edges
Example for Straylight in a Telescope
Log I(q)
q
10-5
10-7
10-9
collimator
spider
diffraction
primary
mirror
secondary
mirror
aperture
diffraction
Scattering Theory in volumes
Model Options
3 major approaches
Analytical vs. numerical
solutions Rigorous
Maxwell solutions
numerical
analytical
FDTDPSTD
T-matrix
Mie
spheresGLMT
Cylinders
Radiation transport
equation
analytical
numericalFD - grid-
based
SH expansion
spheres
DDA
Features: - polarization(PMC)
- electric field (EMC)
- particles fixed
- time resolved
Monte Carlo
Diffusion
equation
numerical
analytical
FD-grid based
layered
bricks
Finite
elements
cylinder
FE
Model Validity Ranges
Typical tissue features
Model validity ranges
0.01 m 0.1 m 1.0 m 10 m
cell membranes
macromolecule
aggregates,
stiations in
collagen fibrils
mitochondria cells
cell nucleivesicles,
lysosomes
typical scale
size l = d
single: Rayleigh
volume: RTE
single: Mie
volume: Maxwell
Problem:
- Exact solutions of scattering: Maxwell equations
- volume sampling requires large memory
- realistic simulations: small volumes ( 2 m3 )
- real sample volumes can not be calculated directly
Approach:
- Calculation of response function of microscopic scattering particles with Maxwell
equations
- empiricial approximation of scattering phase function p(q)
- solution of transport theory with approximated scattering function
The Volume Dilemma
Ref: A. Kienle
Model Validity Ranges
Simple view: diagram volume vs. density
volume
density of
scattering centers
single
particlesinteraction
multiple
scattering
continuous
inhomogen media
rigorous
Maxwell
radiation
transport
equationdiffusion
equation
n(x,y,z) gs,a
volume
density of
scattering centers
single
particlesinteraction
multiple
scattering
continuous
inhomogen media
microscopic
sample
blood
eye
cataractOCT
imaging
n(x,y,z) gs,a
Maxwell solution in the nearfield
Rigorous Scattering at Sphere
nout=1.33 / nin=1.59
Ref: J. Schäfer
nout=1.59 / nin=1.33 nout=1.33 / nin=1.59 + 5 i
r = 1 m
r = 2 m
l = 600 nm
Ref: J. Schäfer
Rayleigh-Scattering
22
22
4
44
23
128
nn
nnaQ
s
ss
l
Scattering at particles much smaller
than the wavelength
Scattering efficiency decreases with
growing wavelength
Angle characteristic depends on
wavelength
Phase function
Example: blue color of the sky
ld
q
q 2cos116
3)( p
Mie Scattering
Result of Maxwell equations for spherical dielectric
particles, valid for all scales
Interesting for larger sizes
Macroscop interaction:
Interference of partial waves,
complicated angle distribution
Usuallay dominating: forward scattering
Parameter: n, n', d, l
Example: small water droplets ( d=10 m)
Limitattion: interaction of neighboring particles
Approximation of parameter
cross section
ld
ll 5025 an nnn s 1.1
09.237.0
1'2
28.3
n
nnd
l
Ref.: M. Möller
Transport Theory
Radiative transport equation: photon density model (gold standard for large volumes),
Purely energetic approach, no diffraction
Integration of PDE by raytracing or expansion in spherical harmonics
Options:
1. time, space and frequency domain
2. fluorescence
3. polarization
4. flexible incorporation of boundaries and surfaces, voxel based
Analytical solutions for special geometries:
1. several source geometries
2. space extended to infinity
3. Already some minor differences to Monte-Carlo approach due to assumptions
Not included features:
1. diffraction, no description of speckles, interference
2. no coherent back scattering
3. no dependencies of neighboring scatterers
Transport Theory
Radiance Transport Equation
Description of the light prorpagtion with radiance transport equation
for photon density balance:
1. incoming photons
2. outgoing photons
3. absorption, extinction
4. emission, source
Numerical solution approach:
Expansion into spherical harmonics
srcscatextdiv dPdPdPdPdP
strQdsspstrLstrLstrLs
t
strL
cssa
,,',,,,,,,,,1
Scattering in Tissue
Large scale / cells: macroscopic range
Diffusion equation, isotropic
Some analytical solutions, numerical
with spherical harmonics expansion
Parameters: effective 's, a, n
Medium scale / cell fine structure:
mesoscopic range
transport theory, radiation propagation
only numerical solutions, scalar anisotropic
Prefered: Monte-Carlo raytrace
Some analytical solutions
Parameters: s,a,n, p(,q)
Fine scale: microscopic range
Only small volumes, with polarization
Maxwell equation solver, FDTD, PSTD, Some analytical solutions
Parameter: complex index n(r)
Correct scaling: feature size vs. wavelength, depends on application
Approaches of Biological Straylight Simulation
10 m
1 m
0.1 m
0.01 m
cell
Mie
scattering
Rayleigh
scattering
l visible
cell core
mitochondria
lysosome, vesikel
collagen fiber
membrane
aggregated macro molecules
scale
of size
typical
structures
scattering
mechanism
Henyey-Greenstein Scattering Model
Henyey-Greenstein model for human tissue
Phase function
Asymmetry parameter g:
Relates forward / backward scattering
g = 0 : isotropic
g = 1 : only forward
g = -1: only backward
Rms value of angle spreading
Typical for human tissue:
g = 0.7 ... 0.9
)1(2 grms q
2/32
2
cos21
1
4
1),(
gg
ggpHG
forward
30
210
60
240
90
270
120
300
150
330
180 0
g = 0.5
g = 0.3
g = 0.7
g = 0.95
g = -0.5
g = -0.8
g = 0 , isotrop
z
qqqqqqqq0
sincos)(2)(coscos)(cos dpdpg
Bio-medical real sample examples
Real Scatter Objects
cancer cell
muscle fibers
dentin
blood vessel wood
cell complex
Light Scattering in Beer
The absorption in beer is dispersive
A longer path length changes the spectral composition of white light illumination
Daylight illumination from the side gives the characteristic color
Illumination from the bottom changes the color form yellow to red depending on
the height
Ref: A. Kienle
Scattering in Zemax
Definition of scattering at every surface
in the surface properties of sequential mode
Possible options:
1. Lambertian scattering indicatrix
2. Gaussian scattering function
3. ABg scattering function
4. BSDF scattering function (table)
5. User defined
More complex problems only make sense in
the non-sequential mode of Zemax,
here also non-optical surfaces (mechanics) can be included
Surface and volume scattering possible
Optional ray-splitting possible
Relative fraction of scattering light can be specified
52
Scattering in Zemax
Definition of scattering at every surface
in the surface properties of non-sequential mode
Options:
1. Scatter model
2. Surface list for important sampling
3. Bulk scattering parameters
53
Scattering in Zemax
Definition of scattering at a surface
in the non-sequential mode
1. selection of scatter model
2. for some models:
to be fixed:
- fraction of scattering
- parameter
- number of scattered rays for ray splitting
54
Scattering in Zemax
Surface scattering:
Projection of the scattered ray on the surface, difference to the specular ray: x
Lambertian scattering:
isotropic
Gaussian scattering
ABg model scatter
BSDF by table
Volume scattering: Angle scattering description by probability P
Henyey-Greenstein volume scattering
(biological tissue model)
Rayleigh scattering
Scattering Functions in Zemax
2
2
)(
x
BSDF eAxF
gBSDFxB
AxF
)(
2/32
2
cos214
1)(
gg
gP
ql
q 2
4cos1
8
3)( P
( )BSDFF x A
Data file with scattering functions: ABg-data.dat
File can be edited
56
Scattering Tables in Zemax
Tools / Scatter / ABg Scatter Data Catalogs
Specification and definition of scattering
parameters for a new ABg-modell function:
wavelength, angle, A, B, g
Analysis / Scatter viewers / Scatter Function Viewer
Graphical representation of the scattering function
57
Scattering Input and Viewing in Zemax
Acceleration of computational speed:
1. scatter to - option, simple
2. Importance sampling with energy normalization
Importance sampling:
- fixation of a sequence of objects of interest
- only desired directins of rays are considered
- re-scaling of the considered solid angle
- per scattering object a maximum
of 6 target spheres can be
defined
58
Scattering with Importance Sampling
Definition of bulk scattering at the surface
menue
Wavelength shift for fluorescence is possible
Typically angle scattering is assumed
Some DLL-model functions are supported:
1. Mie
2. Rayleigh
3. Henyey-Greenstein
59
Bulk Scattering
Simple example: single focussing lens
Gaussian scattering characteristic at
one surface
Geometrical imaging of a bar pattern
Image with / without Scattering
Scattering must be activated in settings
Blurring increases with growing -value
60
Scattering Example I
Example from samples with non-sequential mode
Important sampling accelerates the calculation
61
Scattering Example II
Volume scattering example
Stokes shift is possible for fluorescence
62
Scattering Bulk Example