optical design with zemax
TRANSCRIPT
www.iap.uni-jena.de
Optical Design with Zemax
Lecture 5: Aberrations II
2012-11-20
Herbert Gross
Winter term 2012
2 5 Aberrations II
Time schedule
1 16.10. Introduction
Introduction, Zemax interface, menues, file handling, preferences, Editors, updates, windows, Coordinate systems and notations, System description, Component reversal, system insertion, scaling, 3D geometry, aperture, field, wavelength
2 23.10. Properties of optical systems I Diameters, stop and pupil, vignetting, Layouts
3 30.10. Properties of optical systems II Materials, Glass catalogs, Raytrace, Ray fans and sampling, Footprints, Types of surfaces, Aspheres
4 06.11. Properties of optical systems III Gratings and diffractive surfaces, Gradient media, Cardinal elements, Lens properties, Imaging, magnification, paraxial approximation and modelling
5 13.11. Aberrations I Representation of geometrical aberrations, Spot diagram, Transverse aberration diagrams, Aberration expansions, Primary aberrations,
6 20.11. Aberrations II Wave aberrations, Zernike polynomials, Point spread function, Optical transfer function
7 27.11. Advanced handling
Telecentricity, infinity object distance and afocal image, Local/global coordinates, Add fold mirror, Vignetting, Diameter types, Ray aiming, slider, multiconfiguration, universal plot, IO of data, Lens catalogs
8 04.12. Optimization I Principles of nonlinear optimization, Optimization in optical design, Global optimization methods, Solves and pickups, variables, Sensitivity of variables in optical systems
9 11.12. Optimization II Systematic methods and optimization process, Starting points, Optimization in Zemax
10 18.12 Imaging Fundamentals of Fourier optics, Physical optical image formation, Imaging in Zemax
11 08.01. Illumination Introduction in illumination, Simple photometry of optical systems, Non-sequential raytrace, Illumination in Zemax
12 15.01. Correction I Symmetry principle, Lens bending, Correcting spherical aberration, Coma, stop position, Astigmatism, Field flattening, Chromatical correction, Retrofocus and telephoto setup, Design method
13 22.01. Correction II Field lenses, Stop position influence, Aspheres and higher orders, Principles of glass selection, Sensitivity of a system correction, Microscopic objective lens, Zoom system
14 29.01. Physical optical modelling I Gaussian beams, POP propagation, polarization raytrace, polarization transmission, polarization aberrations
15 05.02. Physical optical modelling II coatings, representations, transmission and phase effects, ghost imaging, general straylight with BRDF
5 Aberrations II
Rays and Wavefronts
Rays and Wavefront forms an orthotomic system
Any closed path integral has zero value
Corresponds to law of Malus and Fermats principle
Ref: W. Singer
3
5 Aberrations II
Wave Aberration in Optical Systems
Definition of optical path length in an optical system:
Reference sphere around the ideal object point through the center of the pupil
Chief ray serves as reference
Difference of OPL : optical path difference OPD
Practical calculation: discrete sampling of the pupil area,
real wave surface represented as matrix
Exit plane
ExP
Image plane
Ip
Entrance pupil
EnP
Object plane
Op
chief
ray
w'
reference
sphere
wave
front
W
y yp y'p y'
z
chief
ray
wave
aberration
optical
systemupper
coma ray
lower coma
ray
image
point
object
point
4
AP
OE
OPL rdnl
)0,0(),(),( OPLOPLOPD lyxlyx
R
y
WR
y
y
W
p
''
p
pp
pp y
yxW
y
R
u
yy
y
Rs
),(
'sin
'''
2
5 Aberrations II
Relationships
Concrete calculation of wave aberration:
addition of discrete optical path lengths
(OPL)
Reference on chief ray and reference
sphere (optical path difference)
Relation to transverse aberrations
Conversion between longitudinal
transverse and wave aberrations
Scaling of the phase / wave aberration:
1. Phase angle in radiant
2. Light path (OPL) in mm
3. Light path scaled in l
)(2
)(
)(
)()(
)()(
)()(
xWi
xki
xi
exAxE
exAxE
exAxE
OPD
5
5 Aberrations II
Wave Aberration
Definition of the peak valley value
exit
aperture
phase front
reference
sphere
wave
aberration
pv-value
of wave
aberration
image
plane
6
5 Aberrations II
Wave Aberrations
Mean quadratic wave deviation ( WRms , root mean square )
with pupil area
Peak valley value Wpv : largest difference
General case with apodization:
weighting of local phase errors with intensity, relevance for psf formation
dydxAExP
ppppmeanpp
ExP
rms dydxyxWyxWA
WWW222 ,,
1
pppppv yxWyxWW ,,max minmax
pppp
w
meanppppExPw
ExP
rms dydxyxWyxWyxIA
W2)(
)(,,,
1
7
0),(1
),( dydxyxWF
yxWExP
5 Aberrations II
Wave Aberrations
x
z
s' < 0
W > 0
reference sphere
ideal ray
real ray
Wave front
R
C
y'
reference
plane
(paraxial)
U'
Wave aberration: relative to reference sphere
Choice of offset value: vanishing mean
Sign of W :
- W > 0 : stronger
convergence
intersection : s < 0
- W < 0 : stronger
divergence
intersection : s < 0
8
y
z
W < 0
Wave aberrationy'p
y'
Reference
sphere
Wave front
Transverse
aberration
Pupil
plane
Image
plane
Tilt angle
y
W
p
'Re
yR
yW
f
p
tilt
Change of reference sphere:
tilt by angle
linear in yp
Wave aberration
due to transverse
aberration y‘
ptilt ynW
5 Aberrations II
Tilt of Wavefront
9
uznzR
rnW
ref
p
Def
2
2
2
sin'2
1'
2
Paraxial defocussing by z:
Change of wavefront
y
z
W > 0
Wave aberrationy'p
z'Reference sphere
Wave front
Pupil
planeImage
plane
Defocus
5 Aberrations II
Defocussing of Wavefront
10
01
0cos
0sin
)(),(
mfür
mfürm
mfürm
rRrZ m
n
m
n
''
0'*
'
1
0
2
0)1(2
1),(),( mmnn
mm
n
m
nn
drrdrZrZ
n
n
nm
m
nnm rZcrW ),(),(
1
0
*
2
00
),(),(1
)1(2drrdrZrW
nc m
n
m
nm
5 Aberrations II
Zernike Polynomials
Expansion of the wave aberration on a circular area
Zernike polynomials in cylindrical coordinates:
Radial function R(r), index n
Azimuthal function , index m
Orthonormality
Advantages:
1. Minimal properties due to Wrms
2. Decoupling, fast computation
3. Direct relation to primary aberrations for low orders
Problems:
1. Computation oin discrete grids
2. Non circular pupils
3. Different conventions concerning indeces, scaling, coordinate system ,
11
1. Fringe - representation
- CodeV, Zemax, interferometric test of surfaces
- Standardization of the boundary to ±1
- no additional prefactors in the polynomial
- Indexing accordint to m (Azimuth), quadratic number terms have circular symmetry
- coordinate system invariant in azimuth
2. Standard - representation
- CodeV, Zemax, Born / Wolf
- Standardization of rms-value on ±1 (with prefactors), easy to calculate Strehl ratio
- coordinate system invariant in azimuth
3. Original - Nijboer - representation
- Expansion:
- Standardization of rms-value on ±1
- coordinate system rotates in azimuth according to field point
k
n
n
gerademn
m
m
nnm
k
n
n
gerademn
m
m
nnm
k
n
nn mRbmRaRaarW0 10 10
0
000 )sin()cos(2
1),(
5 Aberrations II
Zernike Polynomials: Different Nomenclatures
12
5 Aberrations II
Zernike Polynomials
+ 6
+ 7
- 8
m = + 8
0 5 8764321n =
cos
sin
+ 5
+ 4
+ 3
+ 2
+ 1
0
- 1
- 2
- 3
- 4
- 5
- 6
- 7
Zernike polynomials orders by indices:
n : radial
m : azimuthal, sin/cos
Orthonormal function on unit circle
Expansion of wave aberration surface
Direct relation to primary aberration types
Direct measurement by interferometry
Orthogonality perturbed:
1. apodization
2. discretization
3. real non-circular boundary
n
n
nm
m
nnm rZcrW ),(),(
01
0cos
0sin
)(),(
mfür
mfürm
mfürm
rRrZ m
n
m
n
13
drrdrZrWc jj
),(*),(1
1
0
2
0
min)(
2
1 1
i
ijj
N
j
i rZcW
WZZZcTT 1
5 Aberrations II
Calculation of Zernike Polynomials
Assumptions:
1. Pupil circular
2. Illumination homogeneous
3. Neglectible discretization effects /sampling, boundary)
Direct computation by double integral:
1. Time consuming
2. Errors due to discrete boundary shape
3. Wrong for non circular areas
4. Independent coefficients
LSQ-fit computation:
1. Fast, all coefficients cj simultaneously
2. Better total approximation
3. Non stable for different numbers of coefficients,
if number too low
Stable for non circular shape of pupil
14
5 Aberrations II
Zernike Polynomials: Explicite Formulas
n m Polar coordinates
Interpretation
0 0 1 1 piston
1 1 r sin x
Four sheet 22.5°
1 - 1 r cos y
2 2 r 2
2 sin 2 xy
2 0 2 1 2
r 2 2 1 2 2
x y
2 - 2 r 2
2 cos y x 2 2
3 3 r 3
3 sin 3 2 3
xy x
3 1 3 2 3
r r sin 3 2 3 3 2
x x xy
3 - 1 3 2 3
r r cos 3 2 3 3 2
y y x y
3 - 3 r 3
3 cos y x y 3 2
3
4 4 r 4
4 sin 4 4 3 3
xy x y
4 2 4 3 2 4 2
r r sin 8 8 6 3 3
xy x y xy
4 0 6 6 1 4 2
r r 6 6 12 6 6 1 4 4 2 2 2 2 x y x y x y
4 - 2 4 3 2 4 2
r r cos 4 4 3 3 4 4 4 2 2 2 2
y x x y x y
4 - 4 r 4
4 cos y x x y 4 4 2 2
6
Cartesian coordinates
tilt in y
tilt in x
Astigmatism 45°
defocussing
Astigmatism 0°
trefoil 30°
trefoil 0°
coma x
coma y
Secondary astigmatism
Secondary astigmatism
Spherical aberration
Four sheet 0°
15
Deviation in the radius of normalization of the pupil size:
1. wrong coefficients
2. mixing of lower orders during fit-calculation, symmetry-dependent
Example primary spherical aberration:
polynomial:
Stretching factor of the radius
New Zernike expansion on basis of r
166)( 24
9 Z
r
14
24
44
2
949
23
)(13
)(1
Z
rZrZZ
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
c4
c1
c9 / c
9
5 Aberrations II
Zernike Coefficients for Wrong Normalization
16
5 Aberrations II
Zernike Expansion of Local Deviations
Small Gaussian bump in
the topology of a surface
Spectrum of coefficients
for the last case
model
error
N = 36 N = 64 N = 100 N = 144 N = 225 N = 324 N = 625
original
Rms = 0.0237 0.0193 0.0149 0.0109 0.00624 0.00322 0.00047
PV = 0.378 0.307 0.235 0.170 0.0954 0.0475 0.0063
0 100 200 300 400 500 6000
0.01
0.02
0.03
0.04
17
Orthogonalization of Zernike
Polynomilas for ring shaped
pupil area
Basis function depends on
obsuration parameter e:
no easy comparisons
possible
5 Aberrations II
Tatian Polynomials for Ring Pupils
41 2 3 5 6
3431 32 33 35 36
107 8 9 11 12
1613 14 15 17 18
2219 20 21 23 24
2825 26 27 29 30
18
5 Aberrations II
Polynomial for Rectangular Pupil Areas
Systems with rectangular pupil:
Use of Legendre polynomials Pn(x)
1. Factorized representation
Problem: zero-crossing lines
2. Definition of 2D area-orthogonal
Legendre functions
General shape of the pupil area:
Gram-Schmidt-orthogonalization
drawback:
1. Individual function for every pupil shape
2. no intuitive interpretation
3. no comparability between different systems possible
W x y A P x P ynm n m
mn
( , ) ( ) ( )
mnwenn
n
mnwenn
dxxPxP mn
12
2
0
)()(
1
1
19
2D-Legendre polynomials
for rectangular areas
Application:
Spectrometer slit aperture
5 Aberrations II
Legendre Polynomials
y
x
20
5 Aberrations II
Testing with Twyman-Green Interferometer
detector
objective
lens
beam
splitter 1. mode:
lens tested in transmission
auxiliary mirror for auto-
collimation
2. mode:
surface tested in reflection
auxiliary lens to generate
convergent beam
reference mirror
collimated
laser beam
stop
Short common path,
sensible setup
Two different operation
modes for reflection or
transmission
Always factor of 2 between
detected wave and
component under test
21
5 Aberrations II
Interferograms of Primary Aberrations
Spherical aberration 1 l
-1 -0.5 0 +0.5 +1
Defocussing in l
Astigmatism 1 l
Coma 1 l
22
5 Aberrations II
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
23
5 Aberrations II
Diffraction at the System Aperture
Self luminous points: emission of spherical waves
Optical system: only a limited solid angle is propagated, the truncaton of the spherical wave
results in a finite angle light cone
In the image space: uncomplete constructive interference of partial waves, the image point
is spreaded
The optical systems works as a low pass filter
object
point
spherical
wave
truncated
spherical
wave
image
plane
x = 1.22 l / NA
point
spread
function
object plane
5 Aberrations II
Fraunhofer Point Spread Function
Rayleigh-Sommerfeld diffraction integral,
Mathematical formulation of the Huygens-principle
Fraunhofer approximation in the far field
for large Fresnel number
Optical systems: numerical aperture NA in image space
Pupil amplitude/transmission/illumination T(xp,yp)
Wave aberration W(xp,yp)
complex pupil function A(xp,yp)
Transition from exit pupil to
image plane
Point spread function (PSF): Fourier transform of the complex pupil
function
1
2
z
rN
p
Fl
),(2),(),( pp yxWi
pppp eyxTyxA
pp
yyxxR
i
yxiW
pp
AP
dydxeeyxTyxEpp
APpp
''2
,2,)','(
l
''cos'
)'()('
dydxrr
erE
irE d
rrki
I
l
0
2
12,0 I
v
vJvI
0
2
4/
4/sin0, I
u
uuI
-25 -20 -15 -10 -5 0 5 10 15 20 250,0
0,2
0,4
0,6
0,8
1,0
vertical
lateral
inte
nsity
u / v
Circular homogeneous illuminated
Aperture: intensity distribution
transversal: Airy
scale:
axial: sinc
scale
Resolution transversal better
than axial: x < z
Ref: M. Kempe
Scaled coordinates according to Wolf :
axial : u = 2 z n / l NA2
transversal : v = 2 x / l NA
5 Aberrations II
Perfect Point Spread Function
NADAiry
l
22.1
2NA
nRE
l
5 Aberrations II
Ideal Psf
r
z
I(r,z)
lateral
Airy
axial
sinc2
aperture
cone image
plane
optical
axis
focal point
spread spot
5 Aberrations II
Abbe Resolution and Assumptions
Assumption Resolution enhancement
1 Circular pupil ring pupil, dipol, quadrupole
2 Perfect correction complex pupil masks
3 homogeneous illumination dipol, quadrupole
4 Illumination incoherent partial coherent illumination
5 no polarization special radiale polarization
6 Scalar approximation
7 stationary in time scanning, moving gratings
8 quasi monochromatic
9 circular symmetry oblique illumination
10 far field conditions near field conditions
11 linear emission/excitation non linear methods
Abbe resolution with scaling to l/NA:
Assumptions for this estimation and possible changes
A resolution beyond the Abbe limit is only possible with violating of certain
assumptions
I(r)
DAiry / 2
r0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 4 6 8 10 12 14 16 18 20
Airy function :
Perfect point spread function for
several assumptions
Distribution of intensity:
Normalized transverse coordinate
Airy diameter: distance between the
two zero points,
diameter of first dark ring 'sin'
21976.1
unDAiry
l
2
1
2
22
)(
NAr
NAr
J
rI
l
l
'sin'sin2
l
ak
R
akrukr
R
arx
5 Aberrations II
Perfect Lateral Point Spread Function: Airy
log I(r)
r0 5 10 15 20 25 30
10
10
10
10
10
10
10
-6
-5
-4
-3
-2
-1
0
Airy distribution:
Gray scale picture
Zeros non-equidistant
Logarithmic scale
Encircled energy
5 Aberrations II
Perfect Lateral Point Spread Function: Airy
DAiry
r / rAiry
Ecirc
(r)
0
1
2 3 4 5
1.831 2.655 3.477
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2. ring 2.79%
3. ring 1.48%
1. ring 7.26%
peak 83.8%
Axial distribution of intensity
Corresponds to defocus
Normalized axial coordinate
Scale for depth of focus :
Rayleigh length
Zero crossing points:
equidistant and symmetric,
Distance zeros around image plane 4RE
22
04/
4/sinsin)(
u
uI
z
zIzI o
42
2 uz
NAz
l
'sin' 22 unRE
l
5 Aberrations II
Perfect Axial Point Spread Function
-4 -3 -2 -1 0 1 2 3 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
I(z)
z/
RE
4RE
z = 2RE
5 Aberrations II
Defocussed Perfect Psf
Perfect point spread function with defocus
Representation with constant energy: extreme large dynamic changes
z = -2RE z = +2REz = -1RE z = +1RE
normalized
intensity
constant
energy
focus
Imax = 5.1% Imax = 42%Imax = 9.8%
Spherical aberration Astigmatism Coma
c = 0.2
c = 0.3
c = 0.7
c = 0.5
c = 1.0
5 Aberrations II
Psf with Aberrations
Zernike coefficients c in l
Spherical aberration,
Circular symmetry
Astigmatism,
Split of two azimuths
Coma,
Asymmetric
5 Aberrations II
Comparison Geometrical Spot – Wave-Optical Psf
aberrations
spot
diameter
DAiry
exact
wave-optic
geometric-optic
approximated
diffraction limited,
failure of the
geometrical model
Fourier transform
ill conditioned
Large aberrations:
Waveoptical calculation shows bad conditioning
Wave aberrations small: diffraction limited,
geometrical spot too small and
wrong
Approximation for the
intermediate range:
22
GeoAirySpot DDD
0,0
0,0)(
)(
ideal
PSF
real
PSFS
I
ID
2
2),(2
),(
),(
dydxyxA
dydxeyxAD
yxWi
S
Important citerion for diffraction limited systems:
Strehl ratio (Strehl definition)
Ratio of real peak intensity (with aberrations) referenced on ideal peak intensity
DS takes values between 0...1
DS = 1 is perfect
Critical in use: the complete
information is reduced to only one
number
The criterion is useful for 'good'
systems with values Ds > 0.5
5 Aberrations II
Strehl Ratio
r
1
peak reduced
Strehl ratio
distribution
broadened
ideal , without
aberrations
real with
aberrations
I ( x )
35
Approximation of
Marechal:
( useful for Ds > 0.5 )
but negative values possible
Bi-quadratic approximation
Exponential approach
Computation of the Marechal
approximation with the
coefficients of Zernike
2
241
l rms
s
WD
N
n
n
m
nmN
n
ns
n
c
n
cD
1 0
2
1
2
0
2
12
1
1
21
l
5 Aberrations II
Approximations for the Strehl Ratio
22
221
l rms
s
WD
2
24
l
rmsW
s eD
defocusDS
c20
exac t
Marechal
exponential
biquadratic
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
36
pppp
pp
vyvxi
pp
yxOTF
dydxyxg
dydxeyxg
vvH
ypxp
2
22
),(
),(
),(
),(ˆ),( yxIFvvH PSFyxOTF
pppp
pp
y
px
p
y
px
p
yxOTF
dydxyxP
dydxvf
yvf
xPvf
yvf
xP
vvH
2
*
),(
)2
,2
()2
,2
(
),(
llll
5 Aberrations II Optical Transfer Function: Definition
Normalized optical transfer function
(OTF) in frequency space
Fourier transform of the Psf-
intensity
OTF: Autocorrelation of shifted pupil function, Duffieux-integral
Absolute value of OTF: modulation transfer function (MTF)
MTF is numerically identical to contrast of the image of a sine grating at the
corresponding spatial frequency
I Imax V
0.010 0.990 0.980
0.020 0.980 0.961
0.050 0.950 0.905
0.100 0.900 0.818
0.111 0.889 0.800
0.150 0.850 0.739
0.200 0.800 0.667
0.300 0.700 0.538
5 Aberrations II
Contrast / Visibility
The MTF-value corresponds to the intensity contrast of an imaged sin grating
Visibility
The maximum value of the intensity
is not identical to the contrast value
since the minimal value is finite too
Concrete values:
minmax
minmax
II
IIV
I(x)
-2 -1.5 -1 -0.5 0 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Imax
Imin
object
image
peak
decreased
slope
decreased
minima
increased
5 Aberrations II
Optical Transfer Function of a Perfect System
Aberration free circular pupil:
Reference frequency
Cut-off frequency:
Analytical representation
Separation of the complex OTF function into:
- absolute value: modulation transfer MTF
- phase value: phase transfer function PTF
ll
'sinu
f
avo
ll
'sin222 0
un
f
navvG
2
000 21
22arccos
2)(
v
v
v
v
v
vvHMTF
),(),(),( yxPTF vvHi
yxMTFyxOTF evvHvvH
/ max
00
1
0.5 1
0.5
gMTF
Due to the asymmetric geometry of the psf for finite field sizes, the MTF depends on the
azimuthal orientation of the object structure
Generally, two MTF curves are considered for sagittal/tangential oriented object structures
5 Aberrations II
Sagittal and Tangential MTF
y
tangential
plane
tangential sagittal
arbitrary
rotated
x sagittal
plane
tangential
sagittal
gMTF
tangential
ideal
sagittal
1
0
0.5
00.5 1
/ max