www.iap.uni-jena.de
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