optical design with zemax for phd

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

qq

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),(

qq

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)(

qq

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