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www.iap.uni-jena.de
Optical Design with Zemax
for PhD
Lecture 12: Physical Optics
2016-03-23
Herbert Gross
Winter term 2015
2
Preliminary Schedule
No Date Subject Detailed content
1 11.11. Introduction
Zemax interface, menus, file handling, system description, editors, preferences,
updates, system reports, coordinate systems, aperture, field, wavelength, layouts,
raytrace, diameters, stop and pupil, solves, ray fans, paraxial optics
2 02.12. Basic Zemax handling surface types, quick focus, catalogs, vignetting, footprints, system insertion, scaling,
component reversal
3 09.12. Properties of optical systems aspheres, gradient media, gratings and diffractive surfaces, special types of
surfaces, telecentricity, ray aiming, afocal systems
4 16.12. Aberrations I representations, spot, Seidel, transverse aberration curves, Zernike wave
aberrations
5 06.01. Aberrations II PSF, MTF, ESF
6 13.01. Optimization I algorithms, merit function, variables, pick up’s
7 20.01. Optimization II methodology, correction process, special requirements, examples
8 27.01. Advanced handling slider, universal plot, I/O of data, material index fit, multi configuration, macro
language
9 03.02. Correction I simple and medium examples
10 10.02. Correction II advanced examples
11 02.03. Illumination simple illumination calculations, non-sequential option
12 23.03. Physical optical modelling Gaussian beams, POP propagation
13 ?? Tolerancing Sensitivities, Tolerancing, Adjustment
Content
Gaussian beams
Gauss-Schell beams
Non-fundamental modes
Propagation methods
Numerical issues
POP in Zemax
POlarization in Zemax
Scattering in Zemax
2
2
)(
w
r
oeIrI
Gaussian Beams, Transverse Beam Profile
I(r) / I0
r / w
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-2 -1 0 1 2
0.135
0.0111.5
0.589
1.0
Transverse beam profile is gaussian
Beam radius w at 13.5% intensity
Expansion of the intensity distribution around the waist I(r,z)
Gaussian Beams
z
asymptotic
lines
x
hyperbolic
caustic curve
wo
w(z)
R(z)
o
zo
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
z / z
r / w
o
o
asymptotic
far field
waist
w(z)
o
intensity
13.5 %
Geometry of Gaussian Beams
2
2
2
1
2
2
2 1
2),(
o
To
z
zzw
r
o
To
e
z
zzw
PzrI
2
0 1)(
o
T
z
zzwzw
00000
zzw
o
o
Gaussian Beams, Definitions and Parameter
Paraxial TEM00 fundamental mode
Transverse intensity is gaussian
Axial isophotes are hyperbolic
Beam radius at 13.5% intensity
Only 2 independent beam parameter of the set:
1. waist radius wo
2. far field divergence angle o
3. Rayleigh range zo
4. Wavelength o
Relations
f
zfz
zzz
TT
oTT
1
1
1112'
Transform of Gaussian Beams
Diffraction effects are taken into account
Geometrical prediction corrected in the waist region
No singulare focal point: waist with finite width
Focal shift: waist located towards the system, intra focal shift
Transform of paraxial beam
propagation
z'T / f
zT / f
-6 -5 -4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
5
6
zo / f = 0.1
zo / f = 0.2
zo / f = 0.5
zo / f = 1
zo / f = 2
geometrical
limit zo / f = 0
Transform of Gaussian Beam
R
w
w'
R'
starting
plane
receiving
plane
paraxial
segment
A B
C D
incoming
Gaussian beam
z
outgoing
Gaussian beam
Transfer of a Gaussian beam by a paraxial ABCD system
w wB
wA
B
R'
2
2 2
R
AB
R
B
w
AB
RC
D
RD
B
w
'
2
2
2
2
2
-2
0
2-8
-6
-4
-2
0
2
4
6
8
0
0.5
1
z
intensity I
[a.u.]
x
Gaussian Beam Propagation
Paraxial transform of
a beam
Intensity I(x,z)
2
)(2
2 )(
2),(
zw
r
ezw
PzrI
2
21
2
1
21 )0,(
cL
rr
ezrr
2
1
2
1
c
o
L
w
2
2 11
c
o
L
wM
Gauß-Schell Beam: Definition
Partial coherent beams:
1. intensity profile gaussian
2. Coherence function gaussian
Extension of gaussian beams with similar description
Additional parameter: lateral coherence length Lc
Normalized degree of coherence
Beam quality depends on coherence
Approximate model do characterize multimode beams
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
0 z / zo
w / wo
0.25
0.50
1.0
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
0 z / zo
w / wo
1.0
0.50 0.25
Gauß-Schell Beams
Due to the additional parameter:
Waist radius and divergence angle are
independent
1. Fixed divergence:
waist radius decreases with
growing coherence
2. Fixed waist radius:
divergence angle decreases with
growing coherence
f
zfz
zzz
TT
oTT
1
1
1112'
s'/f'
1
1
s / f
= 0 geometrical
optic
0 < < 1 Gauss-
Schell beams
= 1 Gaussian
beams
Gauß-Schell Beam Transform
Similiar to gaussian beam propagation
Smooth transition between:
1. coherent gaussian beam
= 1
2. incoherent geometrical
optic = 0
a/wo = 1
0 1 2 3 4 5 610
-12
10-10
10-8
10-6
10-4
10-2
100
a/wo = 2
a/wo = 3
a/wo = 4
x
Log |A|
Truncated Gaussian Beams
Untruncated gaussian beam: theoretical infinite extension
Real world: diameter D = 2a = 3w with 1% energy loss acceptable
Truncation: diffraction ripple occur, depending on ratio x = a / wo
Focussed Gaussian beam with spherical aberration
Asymmetry intra - extra focal
depending on sign of spherical aberration
Gaussian profile perturbed
Gaussian Beam with Spherical Aberration
c9 = -0.25
c9 = 0.25
c9 = 0
Solution Methods of the Maxwell Equations
Maxwell-
equations
diffraction
integrals
asymptotic
approximation
Fresnel
approximation
Fraunhofer
approximation
finite
elements
finite
differences
exact/
numerical1st
approximation
direct
solutions of
the PDE
spectral
methods
plane wave
spectrum
vector
potentials
2nd
approximation
finite element
method
boundary
element
method
hybrid method
BEM + FEM
Debye
approximation
Kirchhoff-
integral
Rayleigh-
Sommerfeld
1st kind
Rayleigh-
Sommerfeld
2nd kind
mode
expansion
boundary
edge wave
Method Calculation Properties / Applications
Kirchhoff
diffraction integral E r
iE r
e
r rdF
i k r r
FAP
( ) ( ' )'
'
Small Fresnel numbers,
Numerical computation slow
Fourier method of
plane waves )(ˆˆ)'(21 xEFeFxE zvi
II
Large Fresnel numbers
Fast algorithm
Split step beam
propagation
Wave equation: derivatives approximated
En
yxnkE
z
Eik
z
E
o
1
),(2
2
222
2
2
Near field
Complex boundary geometries
Nonlinear effects
Raytracing Ray line law of refraction
r r sj j j j 1 sin '
'sini
n
ni
System components with a aberrations
Materials with index profile
Coherent mode
expansion
Field expansion into modes
n
nn xcxE )()( dxxxEc nn )()( *
Smooth intensity profiles
Fibers and waveguides
Incoherent mode
expansion
Intensity expansion into coherent modes
n
nn xcxI2
)()(
Partial coherent sources
Wave Optical Coherent Beam Propagation
Sampling of the Diffraction Integral
x-6 -4 -2 0 2 4
0
10
20
30
40
50
quadratic
phase
wrapped
phase
2
smallest sampling
intervall
phase
Oscillating exponent :
Fourier transform reduces on 2-
period
Most critical sampling usually
at boundary defines number
of sampling points
Steep phase gradients define the
sampling
High order aberrations are a
problem
Sampling of the Diffraction Integral
c9 = 3
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
= c9(6x4-6x2)
r
r
Re[ U ]
= c9(6x4)
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
r
= c9(6x4-9x2)
a
b
c
Wave with spherical aberration
Real part of the electric field for different values of defocussing
Optimal defocussing means minimal slope of the difference between wavefront and reference sphere
In optimal defocussing the sampling requirements are strongly reduced
The important measure is not the absolut aberration value, but the slope
Propagation by Plane / Spherical Waves
Expansion field in simple-to-propagate waves
1. Spherical waves 2. Plane waves
Huygens principle spectral representation
rdrErr
erE
rrik
2
'
)('
)'(
x
x'
z
E(x)
eikr
r
)(ˆˆ)'( 1 rEFeFrE xy
zik
xyz
x
x'
z
E(x)
eik z z
Kirchhoff diffraction integral in Fresnel approximation
Fourier transform: plan wave expansion
Equivalent form
Curvature removed
Calculation in spherical coordinates
Fresnel Propagation with Equivalence Transform
222 )1(
1'
)1(
)(ˆˆ)'(x
z
Miv
M
zix
Mz
MiM
zik
OO exEFeFeM
exE
x
R<0
z1
z2
starting
plane
observation
plane 1
focus
observation
plane 2
R'<0
R2'>0
x''
x x'
a
a
xxz
i
dxexECxE
2'
)()'(
Optimal Conditioning of the Fresnel-Propagator
Four different cases of propagation in a caustic
Starting plane / final plane inside/outside the focal region
Flattening transform only necessary outside focal range
z
y
inside waist region:
weak curvature
outside waist region:
strong curvature
case 1 : I - I
case 42 : O - O
case 41 : O - O
case 2 : O - I case 3 : I - O
waist
plane
R < 0
R > 0
Sampling and Phase Space
x
u
with
curvature
plane
f
radius of
curvatureR
xp xsDumax << 1
Dumax large
Wave front with spherical curvature:
large angle interval to be sampled
Quasi collimated beam:
very small angle interval
Phase space consideration:
smaller number of sampling points
necessary
Individual control of parameters
at every surface
Model of calculation:
1. propagator from surface to
surface
2. estimation of sampling by pilot
gaussian beam
3. mostly Fresnel propagator with
near-far-selection
4. re-sampling possible
5. polarization, finite transmission,...
possible
Beam Propagation in Zemax
Model:
1. definition of a starting polarization
2. every ray carries a Jones vector of polarization, therefore a spatial variation of polarization
is obtained.
3. at any interface, the field is decomposed into s- and p-component in the local system
4. changes of the polarization component due to Fresnel formulas or coatings:
- amplitude, diattenuation
- phase, retardance
Spatial variations of the polarization phase accross the pupil are aberrations,
the interference is influenced and Psf, MTF, Strehl,... are changed
Polarization in Zemax
Starting polarization
Polarization influences:
1. surfaces, by Fresnel formulas or coatings
2. direct input of Jones matrix surfaces with
Polarization in Zemax
EJE
'
y
x
imreimre
imreimre
y
x
y
x
E
E
DiDCiC
BiBAiA
E
E
DC
BA
E
E
'
'
Analysis of system polarization:
1. pupil map shows the spatial variant
polarization ellipse
2. The transmission fan shows the variation of
the transmission with the pupil height
3. the transmission table showes the mean
values of every surface
Polarization in Zemax
Single ray polarization raytrace:
detailed numbers of
- angles
- field components
- transmission
- reflection
at all surfaces
Polarization in Zemax
Detailed polarization analyses are possible at the individual surfaces by using the coating
menue options
Polarization 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
33
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
34
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 s
- number of scattered rays for ray splitting
35
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
)( s
x
BSDF eAxF
gBSDFxB
AxF
)(
2/32
2
cos214
1)(
gg
gP
2
4cos1
8
3)( P
( )BSDFF x A
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
38
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
39
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
40
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 s-value
41
Scattering Example I
Example from samples with non-sequential mode
Important sampling accelerates the calculation
42
Scattering Example II