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Optical Design with Zemax

for PhD - Advanced

Seminar 6 : Physical Modelling IV - Scattering

2015-01-14

Herbert Gross

Winter term 2014

2

Preliminary Schedule

No Date Subject Detailed content

1 12.11. Repetition Correction, handling, multi-configuration

2 19.11. Illumination I Simple illumination problems

3 26.11. Illumination II Non-sequential raytrace

4 03.12. Physical modeling I Gaussian beams, physical propagation

5 10.12. Physical modeling II Polarization

6 07.01. Physical modeling III Coatings

7 14.01. Physical modeling IV Scattering

8 21.01. Tolerancing I Sensitivity, practical procedure

9 28.01. Tolerancing II Adjustment, thermal loading, ghosts

10 04.02. Additional topics Adaptive optics, stock lens matching, index fit, Macro language,

coupling Zemax-Matlab

1. Basic description of surface scattering

2. Surface measurement

3. Fourier description and PSD

4. Diffraction scattering models

5. Empirical models and BSDF

6. Data of technical surfaces

7. False light in optical systems

8. Calculation of straylight and examples

3

Contents

1. Scattering theory in volumes

2. Exact Maxwell gheory

3. Transport theory

4. Diffusion transport

5. Comparison of models

6. Scattering in tissue

7. Examples

1. Scattering in Zemax

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 regular - statistical distributed

Single - multi scattering

Density of scatterers low - high, independence, saturation, change of illumination

Near - far field

Scaling, size of scatterers vs. wavelength, micro - macro

Coherence, scattering vs. re-emission

Polarization dependence

Discret scatterers vs. continuous n-variations

Absorption

Diffraction vs. geometrical approach

Steady state vs time dependence

Wavelength dispersion of material parameters

Finite volume size - boundary conditions

Aspects of Scattering

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

Analytical solutions:

Spherical particles

1. generalized Lorentz-Mie theory, near and far field

2. multi sphere configurations

3. layered structures

Spheroids

Cylinders

1. single cylinders, with oblique incidence, near and far field

2. stacked cylinders

3. multi cylinder configurations, perpendicular incidence

Numerical solutions in time domain:

Arbitrary geometies

Finite difference time domain method (FDTD), only small volumes ( 2mm3), Dx = l/20

Pseudospectral method (PSTD) , Dx = l/4

Stationary solutions:

Discrete dipole approximation for arbitrary geomtries

T-matrix method

Available Solutions of Maxwells Theory

Ref: A. Kienle

Definition of Scattering

surface measurement

TIS value ( total integrated scattering ) :

total scattered straylight relativ to incoming power

Measurement of TIS by Ulbricht sphere

Approximation for small roughness and statistical

height distribution for nomal incidence

qqqq2

0

2/

0

sincos),(1

ddFdPP

I BSDFs

i

TIS

2

)( 4

l

rmsnormal

TISI

TIS-Measurement with Ulbricht Sphere

detector

baffle

sample

beam stop

beam stop

Ulbricht

sphere

entrance

pupil

output

aperture

collimated

input beam

straylight

direct

reflected

light

transmitte

d

light

baffle

Distribution into spatial frequency domains

10 10 10 10 10+2+10-1-2

10-3

spatial frequencyLog s in 1 / my

scattering at 632 nm

Mirau interferometer

optical heterodyn profilometer

mechanical profilometer

atomic force microscope

figure waviness micro roughness

classical interferometer

1 mm 1 m 10 nmm

DIC interference microscope

10-4

Measurement of Surface Defects

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

DDD dxxxhxhL

xxhxhxC )()(1

)()()(

2

2

1

2)(

c

x

rms exCt

C( x)D

xD

tc

rms

2

Autocorrelation Function

Surface Characterization

h(x)

x

C(Dx)

Dx

PSD(k)

k

A(k)

k

FFT FFT

| |2

< h1h

2 >

correlation

square

dxeyxhkA ikx

L

0

),()(

DD 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

PSD of an Optical Surface

l

2...

2

k

TISPSDrms PdkkFA

12

PSD function, range of spatial frequency:

Area under curve is proportional to PTIS:

Typical: limiting straight

line, fractal surface

Wide scale of sizes

Different measuring tools

necessary

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

D r

ki

ks

Kirchhoff Theory of Scattering

l

q 81

cos i rms

Angle distribution in the far field as diffraction integral

Special case of sine grating:

- corresponds to one Fourier component

- phase difference D

Fraunhofer Farfield Diffraction

''),( '

'sincos'sin)','('cos'

2)','('cos

4

dydxeessE x

zysyxzsxs

f

iyxz

i

zx

iiyixixiqqqq

l

q

l

qi

x'

z'

z'(x',y')

z

x)2sin()( xsaxz

Dn

nnscat JPP q22

0 cos)(

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 d

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

ddP

dP

dP

dLF

i

s

i

sBSDF

qcos

nd

solid angle

areaelement

scattered power

dPs

normal

incidentpower

dPi

qi

sq

iiiiiiBSDF dPFP qqqqq cos),(,,,),(

BSDF of a Surface

3D description of a surface

Large angles:

consideration of the cosines

Example

distribution

BSDF of a Surface

z

areaelement dA

normal

direction

qi

sq

y

x

d s

d i

s

i

incidence

power dPiscattered

power dPs

q sin sins s

q q sin cos sins s spec

s

s

0

+1

+1

-1

-1

i

FBSDF

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

Scattering from Optical Surfaces

Scattering BSDF from polished optical surfaces

: rms rouhness

lc : autocorrelation length

o : scatter angle / deviation

Increases with nearly l4

Increases with square of surface roughness

Strong dependence on angle of incidence

Ref: B. Görtz / Linos

2

224

/21

/22

oc

c

l

lBSDF

l

l

Mirror surface after diamond turning:

sinusoidal regular ripple

corresponds to radial phase error

Strehl definition for small amplitude a by direct evaluation of the diffraction integral in

the far field

Shape:

central peak with sidelobes

stronger effect on short wavelength

side lobe energy is related to a2

General approach:

Fourier decomposition of the surface

Regular Ripple of Mirrors

2

0

2

l

aJDS

Shift invariance around reflection angle

Symmetric behavior in cos-space

Harvey-Shack Theory

qscattering anglei

intensity of straylight

qi qi difference of cosines

straylight intensity

is

Sinusoidal ripple on mirror surface

- regular phase perturbation

- typoical from Diamont turning without polishing

Sinusoidal waveiness with amplitude a:

Strehl ratio

General case:

Fourier decomposition of the

surface topology

Larger impact on shorter wavelengths

Regular Ripple on Diamont Turned Surfaces

2

0

2

l

aJDS

Log r

l

E = 50 %

E = 80 %1st zero

2nd zero

ideal

only figure

figure and

ripple

Scattering from Optical Surfaces

Scattering BSDF from polished optical surfaces

: rms rouhness

lc : autocorrelation length

o : scatter angle / deviation

Increases with nearly l4

Increases with square of surface roughness

Strong dependence on angle of incidence

Ref: B. Görtz / Linos

2

224

/21

/22

oc

c

l

lBSDF

l

l

Scattering from Mechanical Surfaces

Scattering BRDF from strongly scattering (rms>>l) mechanical surfaces

: rms rouhness

lc : autocorrelation length

Ref: B. Görtz / Linos

22

3

2

2

32

sinsinsincossin1

coscoscos

coscossincoscos1

coscos2

)(2

sssss

sii

ssisi

si

c

Lf

iD

f

F

lL

eL

FRBRDF

qqql

qqq

qqqq

l

lq

Scattering by Diffraction

Diffraction scattering BDDF

D : diameter of stop

Approximation of smeared out

diffraction rings:

asymptotic expression

Ref: B. Görtz / Linos

2

1

2

2

/sin

/sin2)(

lq

lq

lq

D

DJDBDDF

log BDDF

o

exact

asymptotic

D = 1 mm / l = 587 nm

q

lq

sin)(

3

DBDDF

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

Scattering from Optical Surfaces

Scattering BSDF from polished optical surfaces

: rms rouhness

lc : autocorrelation length

o : scatter angle / deviation

Increases with nearly l4

Increases with square of surface roughness

Strong dependence on angle of incidence

Ref: B. Görtz / Linos

2

224

/21

/22

oc

c

l

lBSDF

l

l

Straylight Calculation

Measurement of BRDF measurement incidence / observation angle

Ref: A. Bodemann

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

BR

DF

observation angle Phi

Alpha=0°

Alpha=-10°

Alpha=-20°

Alpha=-30°

Alpha=-40°

Alpha=-50°

Alpha=-60°

Alpha=-70°

Alpha=-80°

Lastina

Straylight Measurement

Measurement of BRDF functions

Ref: A. Bodemann

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

BR

DF

observation angle Phi

Alpha=0°

Alpha=-10°

Alpha=-20°

Alpha=-30°

Alpha=-40°

Alpha=-50°

Alpha=-60°

Alpha=-70°

Alpha=-80°

2K-lacquer

Straylight Calculation

Measurement of BRDF functions

From: A. Bodemann

1,0E-06

1,0E-05

1,0E-04

1,0E-03

1,0E-02

1,0E-01

0

10

20

30

40

50

60

70

80

90

10

0

11

0

12

0

13

0

14

0

15

0

16

0

17

0

18

0

19

0

20

0

21

0

22

0

23

0

24

0

25

0

26

0

27

0

Beobachtungswinkel Phi

BR

DF

Alpha=0°

Alpha=-10°

Alpha=-20°

Alpha=-30°

Alpha=-40°

Alpha=-50°

Alpha=-60°

Alpha=-70°

Alpha=-80°

Finapon

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

BSDF

Logarithmic scale in value and angle distance to specular case

Ref: B. Görtz / Linos

0.00001

0.0001

0.001

0.01

0.1

BSDF

0.0001 0.001 0.01 0.1 1 10

|sinqscat-sinqspec|

34'3.4'0.34' distance from specular

Scattering BSDF Decomposition

Bidirectional scattering distribution function

Decomposition into three types

Several reasons for scattering

BSDF is additive

Ref: B. Görtz / Linos

BSDF

(scattering)

BRDF

(reflection)

BRTF

(transmission)

BDDF

(diffraction)

roughness

surface defects

particle contamination

coating irregularitieslog BSDF

BSDF1

10

1

0.1

0.01

0.001

0.001 10.10.01o

BSDF

BSDF2

j

jsurf BSDFBSDF qq ,,

Particles on Optical Surfaces

Model of Mie scattering at particle contamination

Ref: B. Görtz / Linos

measured

BRDF in 1/sr

210°

30°

180°

150°

300°

0°

l=350nm,

D=10mm

l=600nm,

D=2mm

Mie

theory

Particles on Optical Surfaces

Cleaned surface

Ref: B. Görtz / Linos

size of

particles

in mm

0.6 0.9 1.1 1.4 1.7 2.1 2.9 3.1 3.6 7.1 10

number

of particles

0

20

40

60

80

cleaned surface

dark field

microscopic image

Definition of Scattering

false light in optical systems

Sources of Stray Light

Ref: B. Goerz

Ghost Images

Ghost image in photographic lenses:

Reflex film / surface

Ref: K. Uhlendorf, D. Gängler

Different reasons

Various distributions

Straylight and Ghost Images

a b

Scattering of Light

Scattering of light in diffuse media like frog

Ref: W. Osten

Calculation of reflected light

Colour effects due to coatings

Straylight and Ghost Images

Ref.: M. Peschka

sequence 5 - 3

3

5

sequence 6 - 4

4

6

6 - 4

5 - 3

9 - 3

7 - 2

14 - 1115 - 11

20 - 18sequence 13 - 4 sequence 13 - 5

sequence 7 - 2

sequence 20 - 18

sequence 6 - 4

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

ddALdP qcos

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

Optimal design of straylight

suppressing diaphragms

Suppression of Straylight by Baffles

s/a*tanq

n

5

4

3

2

1

0 1 20.5 2/3

q

q

for-

ward

back

ward

backward

forward

q

q

s

a

Design and geometry of

baffle diaphragms

Baffle Design

comfortable better uncomfortable

baffles

further

optical

system

incident direction

of secondary light

source

no direct reflected

light into signal

direction

double reflected

Clever geometry of lens boundary

Appropriate coatings with reflectivity r on surfaces

Provoque n-times multiple scattering events with lower probability

Suppression of Straylight

diffuse bounday

cylinder

false

light

signal

light

rj

0.001 0.01 0.1 1

10-8

10-6

10-4

10-2

1

Rges

n = 1

n = 2

n = 3

n = 4 n = 5

r decreased

n

increased

Straylight Calculation

1. Mechanical 2. Simplified mechanics

system for calculation

3. Critical

straylight

paths

Ref: R. Sand

Selection of the relevant parts of a full CAD mechanical model

Straylight at Mechanical Parts

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 mm 0.1 mm 1.0 mm 10 mm

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 mm3 )

- 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) gms,ma

volume

density of

scattering centers

single

particlesinteraction

multiple

scattering

continuous

inhomogen media

microscopic

sample

blood

eye

cataractOCT

imaging

n(x,y,z) gms,ma

Transition form single scatterers to extended volumes

Sample types are essential

Approximations are necessary

Aggregation to Extended Samples

Single particlesaggregates of

single particles

dense media resolved

into single particles

(with interaction)

dense media with continuous

random index distribution

Rayleigh

Spheres: Mie

Cylinders

Maxwell FDTD

Maxwell PSTD

Maxwell T-matrix

Maxwell Multipol expansion

Spheres: multiple Mie

Multiple Cylinders

Maxwell PSTD

Maxwell T-matrix

Maxwell Multipol expansion

Single scattering particles

exact

Aggregation in RTE

approximation

Beam propagation

RTE approximation

Diffusion approximation

Approximations and assumptions:

1. low density, no interaction of scatter events

2. no absorption

3. statistical distribution of many isolated small scatter centers

Approach: description with BSDF function

Cs cross section area

rs density

p(q) phase function

Scattering Theories

ss

si

BSDF pCF rqqq

)(coscos4

1

Exact Maxwell Theory

Analytical solutions:

Spherical particles

1. generalized Lorentz-Mie theory, near and far field

2. multi sphere configurations

3. layered structures

Spheroids

Cylinders

1. single cylinders, with oblique incidence, near and far field

2. stacked cylinders

3. multi cylinder configurations, perpendicular incidence

Numerical solutions in time domain:

Arbitrary geometies

Finite difference time domain method (FDTD), only small volumes ( 2mm3), Dx = l/20

Pseudospectral method (PSTD) , Dx = l/4

Stationary solutions:

Discrete dipole approximation for arbitrary geomtries

T-matrix method

Available Solutions Maxwell Theory

Ref: A. Kienle

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 mm

r = 2 mm

l = 600 nm

Ref: J. Schäfer

Change of scattering cross section due to

shadowing effects

Phase function depends on

neighboring particle

Multiple Scattering

Ref: J. Schäfer

Larger aggregates of simple single scatter particels

Can be treated rigorous for moderate numbers/volumes

Aggregation to Extended Samples

Ref: S. Tseng

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 mm)

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

Shape of the phase function due to Mie scattering:

- growing complexity with radius of sphere

- interference condition complicated

with several points of stationary phase

Mie Scattering Phase Function

Ref: J. Schäfer

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

mmm

Diffusion Theory

Diffuse Light Propagation: General Model

General radiation transfer equation (RTE, Boltzmann)

Local balance of photon numbers:

1. Incoming photons, divergence

2. Outgoing photons, scattering

3. Absorbed photons, extinction

4. Emitted photons, source

Problem: direction dependence of scattering

Approximative phase function p of the

scattering process: Henyey-Greenstein

g : mean of scattering anisotropy

strQdsspstrLstrLstrLs

t

strL

cssa

,,',,,,,,,,,1

mmm

dVdLsdPdiv

'',' dssPLdVdNdP ssscat

sss N m

dVdLdxdP text m

dVdtsrSdPsrc ,,

srcscatextdiv dPdPdPdPdP

s1

s

s2 s

3

s4

s5

s6

s7...

2/32

2

'21

1

4

1',

ssgg

gsspHG

Severe approximations assumed:

- perfect isotropic scattering

- no wave optical effects

- description of light as photon density evolution

Time dependent or steady state

Analytical solutions for special geometries

1. Infinity

2. semi-infinity

3. bricks

4. Layered structures

5. cylinder

6. Spheres

DiffusionTheory

Diffusion Equation

Approximation:

- scattering dominates gradient effects

- scattering interaction is isotropic

Radiance:

Diffusion equation

Isotropic diffusion constant

Mean free path

Stationary and isotropic

trsDtrstrL ,3,4

1,,

),(),(),(),(1

trStrtrDt

tr

ca

m

),(),(),(),(1 2 trStrtrD

t

tr

ca

m

)1(3

1

gD

sa

mm

D

trStr

Dtr a ),(

),(),(2

m

g

DL

saaaeff

13

11

mmmmm

Scattering Theory Comparisons

Modelling Fluorescence

Ref: A. Kienle

Models:

1. Maxwell theory simulation

2. Monte Carlo calculation

3. Diffusion theory

Different approaches

Under investigation

Analytical solutions with Maxwell solver

Multiple cylinder geometry

RTE with Maxwell analytic

Multiple spheres

Comparison of Methods

Ref: A. Kienle

RTE with Maxwell analytic

with polarization

Multiple spheres

Comparison of Methods

Ref: A. Kienle

RTE with Maxwell numeric

Multiple spheres

Comparison of Methods

Ref: J. Schäfer

Diffusion versus Monte Carlo method

Spatial domain

Diffusion versus Monte Carlo method

Time domain

Comparison of Methods

Ref: A. Kienle

Scattering in Tissue

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

Henyey-Greenstein Model

Simple model for phase function

in the case of scattering in

biological tissue:

Values for human tissue

0 20 40 60 80 100 120 140 160 18010

10

10

10

10

10

-3

-2

-1

0

1

2

q

Log p

g = 0.5

g = 0.3

g = 0.7

g = 0.95

g = -0.5

g = -0.8

g = 0 , isotrop

l ma [mm-1

] ms [mm-1

] g

Human dermis 635 0.18 24.4 0.775

Liver 635 0.23 31.3 0.68

Lung 635 0.081 32.4 0.75

Fundus, healthy 514 0.423 5.312 0.79

Fundus, diseased 514 0.689 13.43

Retinal pigment 800 14

1000 9

1200 5

Ciliar body 694 2.52

1060 2.08

Skin epidermis 800 4.0 42.0 0.852

Henyey-Greenstein Model

Extended representation 1:

superposition of two terms

Extended representation 2:

Two parameters

More realistic

),()1(),(

cos21

1

4

1)1(

cos21

1

4

1),,,(

21

2/3

2

2

2

2

2

2/3

1

2

1

2

121

gpagpa

gg

ga

gg

gaggap

HGHG

DHG

qqq

2/3

1

2

2

2

2

cos21

cos1

2

1

2

3),(

q

ggg

ggpCS

30

210

60

240

90

270

120

300

150

330

180 0

g = 0 , isotrop

g = 0.2

g = 0.3

g = 0.5

g = 0.7

forward

zg = 0.9

LSM-Simulation: Cell Model

cell model after Starosta & Dunn

(3D Computation of Focused Beam

Propagation through Multiple Biological Cells,

OE 17, 12455, 2009)

- cell: ellipsoid with n = 1.36

- nucleus: sphere with n = 1.4

- 100 mitochondria:

ellipsoids with n = 1.4

- 4 fluorescence beads

(zero Stokes-shifts)

Ref: S. Siegler

Bio-medical real sample examples

Real Scatter Objects

cancer cell

muscle fibers

dentin

blood vessel wood

cell complex

Large scale / cells: macroscopic range

Diffusion equation, isotropic

Some analytical solutions, numerical

with spherical harmonics expansion

Parameters: effective m's, ma, 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: ms,ma,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 mm

1 mm

0.1 mm

0.01 mm

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

Scattering in Tissue

Scattering in biological tissue:

Relevant for therapeutic spectral

window l = 650 nm...1.3 mm

Definition / typical numbers:

- coefficient of absorption:

µa 0.01 ... 1 mm-1

- coefficient of scattering:

µs 10 ... 100 mm-1

dominating : scattering with forward

direction

- total attenuation of ballistic photons

µt = µa + µs

- Albedo (scattering contribution on attenuation)

a = µs / µt 0.99 ... 0.999

- mean free path of photons

s = 1 / µs 10 ... 100 µm

Ref.: M. Möller

Scattering in Tissue

Best approximation depends on ratio µs' / µa ab (ms': forward scattering)

µa >> µ's: Lambert-Beer law (l < 300nm, l > 2000nm)

µa << µ's: Diffusion approximation (650nm < l < 1150nm)

µa µ's: RTE equation, Monte-Carlo-simulations

(300nm< l <650nm, 1150nm < l < 2000nm)

In therapeutic windowr:

Diffusion approximation is good,

equivalent scattering coefficient

penetration depth

Diffusion constant

)'(3 saaeff µµµµ

mm51)'(3

1

saa µµµ

)'(3

1

sa µµD

Ref.: M. Möller

Modelling Light Scattering in Tissue: Backscattering

Processes

Simulation

Ref: A. Kienle

Examples

Application: OCT

• Optical coherence tomography:

• Backscattering of light with

broad spectrum.

• Depth discrimination by axial

coherence length

• Generation of 3D images in

tissue

Example :

Image of fundus

receiverfirst mirrorfrom

source

signal

beam

reference

beam

beam

splitter

second

mirror

moving

overlap

lc

z z

relative

moving

I(z)

wave trains

with finite

length

-4 -2 0 2 4

0

0,2

0,4

0,6

0,8

1,0

-4 -2 0 2 4

0

0,2

0,4

0,6

0,8

1,0

primary

signal

filtered

signal

t

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

Modelling of Volume Scattering

Calculation:

1. Geometrical approximation:

Raytrace with Monte-Carlo-method

time consuming, non-smooth results

2. Wave optical with beam propagation

Scalar approach only for large scales

3. Diffusion theory:

In strongly scattering media with isotropic behavior; tissue

off axis

bubble

collimated

gaussian beam

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

107

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

108

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

109

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

111

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

112

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

113

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

114

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

115

Scattering Example I

Example from samples with non-sequential mode

Important sampling accelerates the calculation

116

Scattering Example II

Volume scattering example

Stokes shift is possible for fluorescence

117

Scattering Bulk Example