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Physical Optics
Lecture 13: Scattering
2020-01-30
Herbert Gross / Michael Kempe
Physical Optics: Content
2
K = Kempe G = Gross
No Date Subject Ref Detailed Content
1 17.10. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,
interferometry
2 24.10. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread
function, aberrations
3 07.11. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,
phase imaging
4 14.11.Quality criteria and
resolutionK
Geometric and wave optical critera, Rayleigh and Marechal criteria, Strehl
ratio, lateral and axial point resolution, two-point resolution, contrast
5 21.11. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,
fluorescence, Jablonski diagram, lifetime, quantum yield, FRET
6 28.11. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,
speckle, OCT-principle
7 05.12. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,
components
8 12.12. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,
pulses, power
9 19.12. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,
CARS microscopy, 2 photon imaging
10 09.01. PSF engineering KApodization, superresolution, extended depth of focus, particle trapping,
confocal PSF
11 16.01. Gaussian beams G Basic description, propagation through optical systems, aberrations
12 23.01. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy
beams, applications in superresolution microscopy
13 30.01. Scattering KIntroduction, surface scattering in systems, volume scattering models,
calculation schemes, tissue models, Mie Scattering
14 05.02. Miscellaneous G Coatings, diffractive optics, fibers
Scattering
Introduction
Surface scattering in systems
Volume scattering models
Calculation schemes
Tissue models
3
Contents
4
Scattering as Statistical Propagation of Light
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
Different surface geometries:
Every micro-structure generates
a specific straylight distribution
Light Distribution due to Surface Geometry
Ref.: J. Stover, p.10
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
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
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
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
Empirical model function of BSDF
Notations:
sine of scattering angle
slope parameter m
glance angle
reference and pivot angle : bref
BSDF value at reference: a
Simple isotropic scalar model
m
ref
spec
BSDF aF
b
bbb )(
sqb sin
specspec qb sin
BSDF Model of Harvey-Shack
bspecs b
bref
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
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
Sources of Stray Light
Ref: B. Goerz
Different reasons
Various distributions
Straylight and Ghost Images
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
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
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, a
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
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
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
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
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
Loss of Photons due to Scattering and Absorption
Scattering and Absorption in Tissue
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
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
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
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
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
Modelling Light Scattering in Tissue
Processes
Simulation
Ref: A. Kienle
Bio-medical real sample examples
Real Scatter Objects