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www.iap.uni-jena.de
Design and Correction of Optical
Systems
Lecture 5: Wave aberrations
2016-05-11
Herbert Gross
Summer term 2016
2
Preliminary Schedule
1 06.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 13.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 20.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 27.04. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 04.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 11.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 18.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 25.05. Further Performance Criteria Line of sight, apodization, edges and lines, pupil aberrations, sine condition,
induced aberrations, vectorial aberrations Fourier imaging, caustics
9 01.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 08.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 15.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 22.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
13 29.06. Special System Examples Zoom systems, confocal systems
14 06.07. Further Topics New system developments, modern aberration theory,...
1. Rays and wavefronts
2. Wave aberrations
3. Expansion of the wave aberrations
4. Zernike polynomials
5. Performance criteria
6. Non-circular pupil areas
7. Measurement of wave aberrations
Contents
3
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
Rays and waves carry the same information
Wave surface is perpendicular on the rays
Wave is purely geometrical and has no diffraction properties
5
Ray-Wave Equivalent
Fermat principle:
the light takes the ray path, which corresponds
to the shortest time of arrival
The realized path is a minimum and therefore
the first derivatives vanish
Several realized ray pathes have the same
optical path length
The principle is valid for
smooth and discrete
index distributions
Fermat Principle
P1
x
z
P1
Cds
2
1
0),,(
P
P
dszyxnL
2
1
.
P
P
constrdsnL
P'
Pn d1 1
n d2 2
n d3 3
n d4 4
n d5 5
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
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
8
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 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
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
10
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
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
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
15
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
16
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
17
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
18
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
19
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
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 ,
20
W
rp1
4th order
4th and 2nd order
4th, 2nd and 0th order
4
1
-1
3
2
0
166)( 24 ppp rrrW
Balance of Lower Orders by Zernike Polynomials
Mixing of lower orders to get the minimal Wrms
Example spherical aberration:
1. Spherical 4th order according to
Seidel
2. Additional quadratic expression:
Optimal defocussing for edge
correction
3. Additional absolute term
Minimale value of Wrms
21
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
22
Azimuthal Dependence of Zernike Polynomials
Azimuthal spatial
frequency
r1
r3
r2
r1r3
r2
31
2 4
23
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°
24
Zernike Polynomials
Advantages of the Zernike polynomials:
- de-coupling due to orthogonality
- stable numerical computation
- direct relation of lower orders to classical aberrations
- optimale balancing of lower orders (e.g. best defocus for spherical aberration)
- fast calculation of Wrms and Strehl ratio in approximation of Marechal
Necessary requirements for orthogonality:
- pupil shape circular
- uniform illumination of pupil (corresponds to constant weighting)
- no discretization effects (finite number of points, boundary)
Different standardizations used concerning indexing, scaling, sign of coordinates
(orientation for off-axis field points):
- Fringe representation: peak value 1, normalized
- Standard representation Wrms normalized
- Original representation according to Nijbor-Zernike
Norm ISO 10110 allows Fringe and Standard representation
25
drrdrZrWc jj
),(*),(1
1
0
2
0
min)(
2
1 1
i
ijj
N
j
i rZcW
WZZZcTT 1
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
26
Performance Description by Zernike Expansion
Vector of cj
linear sequence with runnin g index
Sorting by symmetry
0 1 2 3 4 -1 -2 -3 -4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
circular
symmetric
m = 0cos terms
m > 0
sin terms
m < 0
cj
m
27
Different standardizations used concerning:
1. indexing
2. scaling / normalization
3. sign of coordinates (orientation for off-axis field points)
Fringe - representation
1. CodeV, Zemax, interferometric test of surfaces
2. Standardization of the boundary to ±1
3. no additional prefactors in the polynomial
4. Indexing according to m (Azimuth), quadratic number terms have circular symmetry
5. coordinate system invariant in azimuth
Standard - representation
- CodeV, Zemax, Born / Wolf
- Standardization of rms-value on ±1 (with prefactors), easy to calculate Strehl ratio
- coordinate system invariant in azimuth
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),(
Zernikepolynomials: Different Conventions
28
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
29
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
30
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
31
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
32
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
33
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
34
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
35
Interferograms of Primary Aberrations
Spherical aberration 1 l
-1 -0.5 0 +0.5 +1
Defocussing in l
Astigmatism 1 l
Coma 1 l
36
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
37
Interferogram - Definition of Boundary
Critical definition of the interferogram boundary and the Zernike normalization
radius in reality
38