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Imaging and Aberration Theory
Lecture 11: Wave aberrations
2019-01-11
Herbert Gross
Winter term 2018
2
Schedule - Imaging and aberration theory 2018
1 19.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 26.10.Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 02.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 09.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas
5 16.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 23.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders
7 30.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 14.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 21.12.Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 11.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 18.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement
13 25.01. Point spread function ideal psf, psf with aberrations, Strehl ratio
14 01.02. Transfer function transfer function, resolution and contrast
15 08.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
1. Definition of wave aberrations
2. Performance criteria
3. Primary aberrations
4. Order expansions
5. Non-circular pupil shapes
6. Statistical aberrations
7. Measurement of wave aberrations
3
Contents
Law of Malus-Dupin:
- equivalence of rays and wavefronts
- both are orthonormal
- identical information
Condition:
No caustic of rays
Mathematical:
Rotation of Eikonal
vanish
Optical system:
Rays and spherical
waves orthonormal
wave fronts
rays
Law of Malus-Dupin
object
plane
image
plane
z0 z1
y1
y0
phase
L = const
L = const
srays
rot n s
0
4
R
y
WR
y
y
W
p
''
Relationship to Transverse Aberration
Relation between wave and transverse aberration
Approximation for small aberrations and small aperture angles u
Ideal wavefront, reference sphere: Wideal
Real wavefront: Wreal
Finite difference
Angle difference
Transverse aberration
Limiting representation
5
yp
z
real ray
wave front W(yp)
R, ideal ray
C
reference
plane
y'
reference sphere
q
u
idealreal WWWW
py
W
tan
Ry'
Wave Aberration
Exact relation between wave aberration and ray deviation
General expression from geometry
a describes the lateral aberration
Substitution of angle by scalar
product
Exact relation is quadratic in R
Approximation for large R
with
6
qcos1
'1
'
22
0
R
asan
ss
assssnWW
r
rr
2
2222
41
2'
R
asa
R
asanWW
'2
''
22
zR
yx
yR
yx
R
xW
pp
pp
'
'
'
z
y
x
a
real ray
C
R
s
sr
reference
sphere
q
a
A
W
wave
front
z
yp
center of
reference
sphere
real
intersection
point
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
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
7
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
8
Pupil Sampling
y'p
x'p
yp
xp x'
y'
z
yo
xo
object plane
point
entrance pupil
equidistant grid
exit pupil
transferred grid
image plane
spot diagramoptical
system
All rays start in one point in the object plane
The entrance pupil is sampled equidistant
In the exit pupil, the transferred grid
may be distorted
In the image plane a spreaded spot
diagram is generated
9
Wave Aberration
Definition of the peak valley value WPV
Reference sphere corresponds to perfect imaging
Rms-value is more relevant for performance evaluation
exit
aperture
phase front
reference
sphere
wave
aberration
pv-value
of wave
aberration
image
plane
10
Wave Aberration Criteria
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
11
0),(1
),( dydxyxWF
yxWExP
Wave Aberrations – Sign and Reference
x
z
s' < 0
W > 0
reference sphere
ideal ray
real ray
wave front
R
C
reference
plane
(paraxial)
U'
y' < 0
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
12
y
z
W < 0
wave aberrationy'p
y'
q
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 q
linear in yp
Wave aberration
due to transverse
aberration y‘
Is the usual description
of distortion
q ptilt ynW
Tilt of Wavefront
13
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
Defocussing of Wavefront
14
Representation of primary aberrations
Seidel terms
Surface in pupil plane
Special case of chromatical aberrations
Primary Aberrations
Spherical aberration Coma
Primary monochromatic wave aberrations
Astigmatism Field curvature (sagittal)
Distortion
ypxp
ypxp
ypxp
ypxp
ypxp
4
1
222
1 rcyxcW pp cos3
2
22
2 yrcyxyycW ppp
222
3
22
3 cosrycyycW p 22
4
222
4 rycyxycW pp
cos3
5
3
5 rycyycW p
Axial color Lateral color
Primary chromatic wave aberrations
2
1
22
1
~~rbyxbW pp
ypxp
ypxp
cos~~
22 yrbyybW p
Ref: H. Zügge
15
Relation :
wave / geometrical aberration
Primary Aberrations
Type
ypxp
ypxp
ypxp
ypxp
ypxp
4
1 rcW
cos3
2 yrcW
222
3 cosrycW
22
4 rycW
cos3
5 rycW
sin' 3
1 rcx
cos' 3
1 rcy
2sin' 2
2 yrcx
0'x
cos' 2
3 rycy
sin' 2
4 rycx
cos' 2
4 rycy
0'x
3
5' ycy
Spherical
aberration
Symmetry to
Periodicity
axis
constant
point
1 period
Wave aberration Geometrical spot
Coma
Astigmatism
Field
curvature
(sagittal)
Distortion
Symmetry to
Periodicity
Symmetry to
Periodicity
Symmetry to
Periodicity
Symmetry to
Periodicity
one plane
1 period
two planes
2 period
one plane
1 period
axis
constant
point
1 period
one straight line
constant
one straight line
2 periods
two straight lines
1 period
)2cos2('2
2 yrcy
Ref: H. Zügge
16
Relation :
wave / geometrical aberration
Primary Aberrations
ypxp
ypxp
Axial color
Lateral color
Symmetry to
Periodicity
Symmetry to
Periodicity
one plane
1 period
axis
constant
point
1 period
one straight line
constant
Type Wave aberration Geometrical spot
2
1
~rbW
cos~
2 yrbW
sin~
' 1 rbx
cos~
' 1 rby
0'x
yby 2
~'
Ref: H. Zügge
17
Special Cases of Wave Aberrations
Wave aberrations are usually given as reduced
aberrations:
- wave front for only 1 field point
- field dependence represented by discrete
cases
Special case of aberrations:
1. axial color and field curvature:
represented as defocussing term,
Zernike c4
2. distortion and lateral color:
represented as tilt term, Zernike c2, c3
18
yp
z
y'
Reference sphere
wave front
pupil
plane
ideal
image
plane
axial
chromatic
yp
z
y'
reference
sphere
wave front
pupil
plane
ideal
image
plane
defocus
due to
field
curvaturechief ray
curved
image
shell
yp
z
y'
reference
sphere
wave front
pupil
plane
ideal
image
plane
image
height
differencechief ray
Special Cases of Wave Aberrations
3. afocal system
- exit pupil in infinity
- plane wave as reference
4. telecentric system
chief ray parallel to axis
19
yp
z
reference
plane
wave front
image in
infinity
yp
z
y'
reference
sphere
wave front
pupil
planeideal
image
plane
axial
chromatic
Expansion of the Wave Aberration
Table as function of field and aperture
Selection rules:
checkerboard filling of the matrix
Field y
Spherical
y0 y 1 y 2 y3 y 4 y 5
Distortion
r
1
y
y3
primary
y5
secondary
r
2
r 2
Defocus
Aper-
ture
r
r
3
r3y Coma primary
r 4
r 4 Spherical
primary
r
5
r
6
r 6 Spherical
secondary
DistortionDistortionTilt
Coma Astigmatism
Image
location
Primary
aberrations /
Seidel
Astig./Curvat.
r cosq
cosq
cos2q
r cosq
Secondary
aberrations
r cosq
r 4y 2 cos
2q
Coma
secondary
r5y cosq
r3y
3 cos
3q
r3y
3 cosq
r2 y
4
r2 y
4 cos
2q
r 4y 2
r22
yr
22 y
20
mlk
ml
p
k
klmp ryWryW,,
cos'),,'( qq
isum Ni number of terms
Type of aberration
2 2 image location
4 5 primary aberrations, 4th order
6 9 secondary aberrations, 6th order
8 14 8th order
Polynomial Expansion of Wave Aberrations
Taylor expansion of the wavefront:
y' Image height index k
rp Pupil height index l
q Pupil azimuth angle index m
Symmetry invariance:
1. Image height
2. Pupil height
3. Scalar product between image and pupil vector
Number of terms
sum of indices in the
exponent isum
y r y rp p' ' cos q
pr
'y
21
type of aberration wave aberration transverse aberration longitudinal aberration
u w u w u w
spherical 4 0 3 0 2 0
coma 3 1 2 1
astigmatism 2 2 1 2 0 2
Petzval curvature 2 2 1 2 0 2
distortion 1 3 0 3
axial chromatical 2 0 1 0 0 0
chromatical magnif. 1 1 0 1
The exponents of the Taylor expansion on the aperture depends on the kind of
representation of the aberrations.
The exponent grows by 1 in the sequence longitudinal-transversal-wave aberrations
The Seidel term ‚3rd order‘ is valid only for transverse aberrations
Dependence on aperture and fiels size for the primary aberrations:
Taylor - Expansion
22
pdpppappcpspp yyAryAyyAyryArAyryW 3222224 ''''),,'(
Taylor Expansion of the Primary Aberrations
Expansion of the monochromatic aberrations
First real aberration: primary aberrations, 4th order as wave deviation
Coefficients of the primary aberrations:
AS : Spherical Aberration
AC : Coma
AA : Astigmatism
AP : Petzval curvature
AD : Distortion
Alternatively: expansion in polar coordinates:
Zernike basis expansion, usually only for one field point, orthogonalized
23
4
lPVW
Rayleigh Criterion
The Rayleigh criterion
gives individual maximum aberrations
coefficients,
depends on the form of the wave
Examples:aberration type coefficient
defocus Seidel 25.020 a
defocus Zernike 125.020 c
spherical aberration
Seidel 25.040 a
spherical aberration
Zernike 167.040 c
astigmatism Seidel 25.022 a
astigmatism Zernike 125.022 c
coma Seidel 125.031 a
coma Zernike 125.031 c
24
a) optimal constructive interference
b) reduced constructive interference
due to phase aberrations
c) reduced effect of phase error
by apodization and lower
energetic weighting
d) start of destructive interference
for 90° or l/4 phase aberration
begin of negative z-component
Rayleigh criterion:
1. maximum of wave aberration: Wpv < l/4
2. beginning of destructive interference of partial waves
3. limit for being diffraction limited (definition)
4. as a PV-criterion rather conservative: maximum value only in 1 point of the pupil
5. different limiting values for aberration shapes and definitions (Seidel, Zernike,...)
Marechal criterion:
1. Rayleigh crierion corresponds to Wrms < l/14 in case of defocus
2. generalization of Wrms < l/14 for all shapes of wave fronts
3. corresponds to Strehl ratio Ds > 0.80 (in case of defocus)
4. more useful as PV-criterion of Rayleigh
Criteria of Rayleigh and Marechal
14856.13192
lllRayleigh
rmsW
25
PV and Wrms-Values
PV and Wrms values for
different definitions and
shapes of the aberrated
wavefront
Due to mixing of lower
orders in the definition
of the Zernikes, the Wrms
usually is smaller in
comparison to the
corresponding Seidel
definition
26
Typical Variation of Wave Aberrations
Microscopic objective lens
Changes of rms value of wave aberration with
1. wavelength
2. field position
Common practice:
1. diffraction limited on axis for main
part of the spectrum
2. Requirements relaxed in the outer
field region
3. Requirement relaxed at the blue
edge of the spectrum
Achroplan 40x0.65
on axis
field zone
field edge
diffraction limit
l [m]0.480 0.6440.562
0
Wrms [l]
0.12
0.06
0.18
0.24
0.30
27
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
Expansion of wave aberration surface
Zernike polynomials orders by indices:
n : radial
m : azimuthal, sin/cos
Orthonormal function on unit circle
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
0)(cos
0)(sin
)(),(
mfor
mform
mform
rRrZ m
n
m
n
28
Orthogonalization of Zernike
Polynomilas for ring shaped
pupil area
Basis function depends on
obsuration parameter e:
no easy comparisons
possible
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
29
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
( , ) ( ) ( )
mnif
n
mnif
dxxPxP mn
12
2
0
)()(
1
1
30
Orthogonalization of Zernike polynomials on a unit square
Gram-Schmidt-Orthogonalization procedure
2-Dimensional Legendre Polynomials
j
m
j
mjm ZcQ 1
kjjk dydxQQ
1
1
1
1
mmk
m
k
km ZaQaQ
1
1
lj
j
j
l
l
jlmm yxdQ
max max
0 0
,
1
1,...3,2,1m
kj
jkjmk mkcac
1
1
1
1 1
2 2
1
m
k
m
k
k
j
mjkjkmkmmm
mmm
PcTTP
ca
31
2D-Legendre polynomials
for rectangular areas
Application:
Spectrometer slit aperture
First few polynomials:
quite similar to Zernikes
Legendre Polynomials
y
x
32
At any surface the wavefront can be compared with the ideal one before and after the
refraction/reflection
Due to the additivity of the phase, at any surface the contribution can be calculated
Practical problems:
- collimated intermediate ray paths
- change of normalization radii and grid distortion
- change of wavefront/Zernikes during propagation
- choice of reference surface not trivial
- parabasal ray calculation inaccurate
33
Wave Aberration Generated at a Surface
real
ray
z
paraxial
ray
intermediate ideal
object plane
intermediate ideal
image plane
spherical image wave
surface
spherical object wave
real waves
n‘n
W. Welford, Aberrations of optical systems, Hilger 1986
W. Welford, Opt Acta 19 (1972) p.719, A new total aberration formula
Complete system:
Additivity of phase delay at every surface is obvious
Practical problems:
- change of normalization radii
- grid distortion
- huge amount of information, systematic analysis complicated
- analytical representation not possible
34
Wave Aberration Additivity
P
arbitrary ray
y
1 2 3
y'
P'
surfaces exit pupil
total Wtot
W1
W2 W3
surface contributions
ray pencil
Real distortion of the grid in at the various mirror positions
35
Grid Distortion2
20
mm
M1
11
0 m
m
26
0 m
m
M2 M3
Complex field with statistical phase
Correlation of the phase:
structural function
Coherence function
For gaussian statistics
Auto covariance function
PSD, power spectral density
)()()( ri
o erErE
)()()(
2
1
2112 ririrD
ee
)(2
1
2112
12
)()()(rD
oo erErEz
2
2
12)( 2 ca
r
erD
2
22
2),(),(),( ca
yx
eyyxxyxyxC
222222),(ˆ),( yxc vva
cyx eayxCFvvS
Statistical Wave Aberrations
36
Description:
1. in the spatial domain: topology of the rough surface
2. in the spatial frequency domain
Statistical Wave Aberrations
h(x) C(x)
x
PSD()A()
Fourier
transform
| |2
< h1h
2 >
correlation
square
height,
topologycorrelation
function
amplitude
spectrumpower spectral
density
Fourier
transform
x
37
Spatial Frequency of Surface Perturbations
Power spectral density of the perturbation
Three typical frequency ranges, scaled by diameter D
limiting line
slope m = -1.5...-2.5
log A2Four
long range
low frequency
figure
Zernike
mid
frequencymicro
roughness
1/l
oscillation of
the polishing
machine
12/D1/D 40/D
38
Atmospheric turbulence:
statistical phase screen
Scale below 1 cm:
Tatarski regime, viscosity
PSD
Scale from 1 cm to 5 m:
Kolmogorov regime
PSD
Greater length scales:
Karman regime
PSD
2
ik
k
Ta eb
3
11
kbTa
6
11
2
22 4
o
KaL
kb
Atmospheric Turbulence
const
Karman
large scale
Kolmogorov
inertial subrange
viscose
dissipation
Tatarski
k
= b
Tae-k
2
ko outer scale ki
inner scale
LogCn
2
(k)
=bKo
k 311
39
Measurement of Wave Aberrations
Wave aberrations are measurable directly
Good connection between simulation/optical design and realization/metrology
Direct phase measuring techniques:
1. Interferometry
2. Hartmann-Shack
3. Hartmann sensor
4. Special: Moire, Holography, phase-space analyzer
Indirect measurement by inversion of the wave equation:
1. Phase retrieval of PSF z-stack
2. Retrieval of edge or line images
Indirect measurement by analyzing the imaging conditions:
from general image degradation
Accuracy:
1. l/1000 possible, l/100 standard for rms-value
2. Rms vs. individual Zernike coefficients
40
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
41
Interferograms of Primary Aberrations
Spherical aberration 1 l
-1 -0.5 0 +0.5 +1
Defocussing in l
Astigmatism 1 l
Coma 1 l
42
Real Measured Interferogram
Problems in real world measurement:
Edge effects
Definition of boundary
Perturbation by coherent
stray light
Local surface error are not
well described by Zernike
expansion
Convolution with motion blur
Ref: B. Dörband
43
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
44
Wave front determines local direction of propagation
Propagation over distance z : change of transverse intensity distribution
Intensity propagation contains phase information
Principle of Psf Phase Retrieval
z
intensity caustic
I(x,y,z)
wave front : W(x,y)
local Poynting vector
direction of propagation
propagation
45
Transport of intensity equation couples phase and intensity
Solution with z-variation of the intensity delivers start phase at z = 0
Determine phase from intensity distribution.
- Inverse propagation problem : ill posed
- Boundary condition : measured z-stack I(x,y,z)
Algorithm for numerical solution
- IFTA / Gerchberg Saxton ( error reduction )
- Acceleration ( conjugate gradients, Fienup,...)
- Modal non least square-methods
- Extended Zernike method
Applications:
- Calculation of diffractive components for given illumination distribution
- Wave front reconstruction
- Phase microscopy
),(),,(),,(
yxWzyxIz
zyxIk
Propagation of Intensity
46
Principle of phase retrieval for metrology of optical systems
Measurement of intensity caustic z-stack
Reconstruction of the phase in the exit pupil
Phase Retrieval
z
x
y
opticalsystem
exit
pupil
image plane
z
x
3D- intensity image
stack I (x,y,z)
W( ,y)
wave front
x
47
Gerchberg-Saxton-Algorithm
Iterative reconstruction of the
pupil phase with back-and-forth
calculation between image and
pupil:
IFTA / Gerchberg-Saxton
Substitution of known intensity
Problems with convergence:
Twin-image degeneration
Modified algorithms:
1. Fienup-acceleration
2. Non-least-square
3. Use of pupil intensity
48
Phase Space Interpretation
Known measurement of intensity in defocussed planes:
- Several rotated planes in phase space
- Information in and near the spatial domain
Calculation of distribution in the Fourier plane
Wave equation is valid
Principle : Tomography
xp
minimal
defocus plane
pupil plane
maximal
defocus plane
desired
distribution
measuring
planes
phase
space
angle
image
plane
x'
49
Example Phase Retrieval
Evaluation of real data psf-stack
model phasemeasurement
50
Retrieval
without / with
Apodization
Correlation
over z
Phase Retrieval with Apodization
M
R no A
Dif
R wi A
Dif
51
Object Space Defocussing
a) defocussing in
image space
z
object
plane
several
detector planes
intensity caustic I(x,y,z)
pupil
one
wavefront
lens
b) defocussing in
object space
z
several
object
planes
one fixed
detector plane
intensity caustic I(x,y,zobj)
pupil
several
wavefronts
lens
52
Comparion phase
retriev vs Hartmann
test
Case of coma
Measurements
Zernike-
coefficients
0
0.0
1
0.0
2
0.0
3
0.0
4
0.05
0.0
6
0.0
7Coma
a13
a15
| A13 |
| A15 |HartmannRetrieval
+field- field -zone + zoneaxis
Ref: B. Möller
53