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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
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),(
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
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