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Physical Optics

Lecture 11: Scattering

2019-01-17

Herbert Gross

Physical Optics: Content

2

No Date Subject Ref Detailed Content

1 18.10. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,

interferometry

2 25.10. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread

function, aberrations

3 01.11. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,

phase imaging

4 08.11.Quality criteria and

resolutionG

Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point

resolution, criteria, contrast, axial resolution, CTF

5 22.11. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,

fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

6 29.11. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,

speckle, OCT-principle

7 06.12. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,

components

8 13.12. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,

pulses, power

9 20.12. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,

CARS microscopy, 2 photon imaging

10 10.01. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,

confocal PSF

11 17.01. Scattering GIntroduction, surface scattering in systems, volume scattering models,

calculation schemes, tissue models, Mie Scattering

12 24.01. Gaussian beams G Basic description, propagation through optical systems, aberrations

13 31.01. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy

beams, applications in superresolution microscopy

14 07.02. Miscellaneous G Coatings, diffractive optics, fibers

K = Kempe G = Gross

Scattering

Introduction

Surface scattering in systems

Volume scattering models

Calculation schemes

Tissue models

3

Contents

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

Different surface geometries:

Every micro-structure generates

a specific straylight distribution

Light Distribution due to Surface Geometry

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

Phenomenology of Surface Scattering

dP

dLog

q

qscattering angle

special polishing

Normal polishing

specular angle

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

Area under PSD-curve

Meaningful range of frequencies

Polished surfaces are similar and have fractal structure,

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

13

PSD Ranges

Typical impact of spatial frequency

ranges on PSF

Low frequencies:

loss of resolution

classical Zernike range

High frequencies:

Loss of contrast

statistical

Large angle scattering

Mid spatial frequencies:

complicated, often structured

light distributions

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

cos

s sBSDF

i s i

dL dPF

dP dA dP dq

W

ndW

solid angle

areaelement

scattered power

dPs

normal

incidentpower

dPi

qi

sq

( , ) , , , ( , ) coss BSDF i i i i i sP F P dA dq q q q q W

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

dW s

dW 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

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

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

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

Sources of Stray Light

Ref: B. Goerz

Different reasons

Various distributions

Straylight and Ghost Images

Scattering of Light

Scattering of light in diffuse media like frog

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

Model Options

3 major approaches

Analytical vs. numerical

solutionsRigorous

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) g,ms,ma

volume

density of

scattering centers

single

particlesinteraction

multiple

scattering

continuous

inhomogen media

microscopic

sample

blood

eye

cataractOCT

imaging

n(x,y,z) g,ms,ma

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

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), x = l/20

Pseudospectral method (PSTD) , x = 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

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

Simulation of Diffuse Light Propagation

1. Illumination

2. Diffuse

propagation of

fluorescence

light

Excitation of tissue by focussed laser illumination

Propagation of fluorescencec light in tissue by diffusion

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 W

WW '',' dssPLdVdNdP ssscat

sss N m

dVdLdxdP text W m

dVdtsrSdPsrc W ,,

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

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

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

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

qq

qqq

2/3

1

2

2

2

2

cos21

cos1

2

1

2

3),(

q

qq

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

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

Modelling Light Scattering in Tissue

Processes

Simulation

Ref: A. Kienle