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Imaging and Aberration Theory
Lecture 14: Vectorial aberrations
2013-02-08
Herbert Gross
Winter term 2012
2
Preliminary time schedule
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. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 09.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 16.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 23.11. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 07.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 14.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 21.12. Chromatical aberrations
Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 11.01. Further reading on aberrations sensitivity in 3rd order, structure of a system, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics and phase space
11 18.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF
12 25.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement
13 01.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, telecentric case, afocal case, aberration balancing, Delano diagram, Scheimpflug imaging, Fresnel lenses, statistical aberrations
14 08.02. Vectorial aberrations Introduction, special cases, actual research, anamorphotic, partial symmetric
1. Systems with poor symmetry
2. Vectorial aberration theory
3. Aberration fields
4. Anamorphotic systems
5. Examples
6. Polarizations
Why aberration theory ?
3
Contents
Classes according to remaining symmetry
Non-Axisymmetric Systems: Classes and Types
axisymmetric
co-axial
double plane symmetric
anamorphotic
plane symmetric
non-symmetrical
eccentric
off-axis
rot-sym components
3D tilt and decenter
Vectorial description
Axis ray as reference
System description by
4-4-matrix
More general : 5x5-calculus
Non-Axisymmetric Systems: Matrix description
image
object
mirror
lens
optical axis
ray
d1
d2
d3
R
DDCC
DDCC
BBAA
BBAA
RR
yyyxyyyx
xyxxxyxx
yyyxyyyx
xyxxxyxx
M'
v
u
y
x
R
Ray tube around axis ray
Propagation of curvature according to Coddington equations
Differential ray trace
Non-Axisymmetric Systems: Pilot Axis ray
z
x
y
Rx
Ry
Cy
Cx
wavefront
toric shape
R||
R'||
dR
R'
z
Refraction of a ray tube
Non-Axisymmetric Systems: Ray Tube around Axis Ray
xy
'
surface
plane of
incidence
incoming
ray
local
system
axis
Rh1
Rh2
R||
R
R'||
R'
outgoing
ray
'cos'
'cos'cos
'cos'
cos1
'
122
2
||||
nR
nn
n
n
RR s
2
2
1
2
||
sincos1
hh RRR
'
'cos'cos
'
1
'
1
nR
nn
n
n
RR s
1
2
2
2 sincos1
hh RRR
||
21
'/1'/1
/1/1
'cos'
cos´sincos2'2tan
RR
RR
n
n hh
Wave aberration field
indices
Normalized field vector: H normalized pupil vector: rp
angle between H and rp:
Expansion according to the invariants for circular symmetric components
Vectorial Aberrations
x
yrp
s
p
s'
p'
xP
yp
x'
y'
x'P
y'p
object
plane
entrance
pupil
exit
pupil
image
plane
z
system
surfaces
P'
P
H
nmj
n
pp
m
p
j
klmp rrrHHHWrHW,,
,
mnlmjk 2,2
y
Hrp
field1
1
pupilj
cos,, 22 ppppp rHrHrrrHHH
Wave aberration field
until the 6th order
Analogue:
transverse aberrations
with
Vectorial Aberrations
ord j m n Term Name
0 0 0 0 000W uniform Piston
2
1 0 0 HHW
200 quadratic piston
0 1 0 prHW
111 magnification
0 0 1 pp rrW
020 focus
4
0 0 2 2
040 pp rrW
spherical aberration
0 1 1 ppp rHrrW
131 coma
0 2 0 2222 prHW
astigmatism
1 0 1 pp rrHHW
220 field curvature
1 1 0 prHHHW
311 distortion
2 0 0 2400 HHW
quartic piston
6
1 0 2 2
240 pp rrHHW
oblique spherical aberration
1 1 1 ppp rHrrHHW
331 coma
1 2 0 2422 prHHHW
astigmatism
2 0 1 pp rrHHW
2
420 field curvature
2 1 0 prHHHW
2
511 distortion
3 0 0 3600 HHW
piston
0 0 3 3060 pp rrW
spherical aberration
0 1 2 ppp rHrrW
2
151
0 2 1 2242 ppp rHrrW
0 3 0 3333 prHW
Wn
RH
pr'
'
Wave aberration
with shift vector
In 3rd order:
1. spherical
2. coma
3. astigmatism
4. defocus
5. distortion
Systems with Non-Axisymmetric Geometry
q nmj
n
pp
m
pq
j
qqklmp rrrHHHWrHW,,
000,
jjoj HH
p
q
q q
qqqqq
q q
qqqq
q
q
p
q q
q q
qqqq
q q
p
q
q
q
q
ppp
q
q
q
pp
q
qp
rWHW
HWHHWHHW
r
WW
HWWHWW
rWHWHW
rrrWHW
rrWrHW
2
,3110,311
0
2
,31100,3110
2
0,311
2
2
,222,220
0,222,22
2
0,222,220
22
,2220,222
2
0,220
,1310,131
2
,040
2
2
2
1
2
12
2
1
2
1
2
1
,
Aberration center point
Systems with Non-Axisymmetric Geometry
field
point
y
j
optical
axis ray
image
plane
r
pupil
point
y
H
pupil
plane
e
symmetry
vector
y
aberration
field centreH
o
x
H
jjoj HH
Aberration field center point:
connection of center of curvature and center of pupil: H
Optical axis without relevance
Systems with Non-Axisymmetric Geometry
surface no. jobject no. j
image no. j
optical axis ray
pupil
centre of
curvature of
surface no. j
local
axis
vertex
aberration
field centre
R
tilt angle
H
Expanded and rearranged 3rd order expressions:
- aberrations fields
- nodal lines/points for vanishing aberration
Example coma:
abbreviation: nodal point location
one nodal point with
vanishing coma
Nodal Theory
ppp
q
q
q
o
q
qcoma rrrW
W
HWW
,131
,131
,131
)(
131
,131
,131
,131
131 c
q
j
q
q
W
W
W
W
a
pppo
c
coma rrraHWW
131
)(
131
zero
coma
green zero
coma
blue
zero
coma
total
Example astigmatism:
abbreviations
General: two nodal points
possible
Special cases
Nodal Theory
q
poq
q
pqoqast rbaHWrHWW22
222
2
222,222
22
,2222
1
2
1
q
q
q
W
W
a,222
,222
222
2
222
,222
,222
2
2
222 aW
W
b
q
q
q
y
x
a222
ib222
-ib222
nodal point 1,
astigmatism corrected
nodal point 2,
astigmatism corrected
constant
astigmatism
image plane
focal
surfaces :
planes
image plane
focal
surfaces :
cones
linear
astigmatism
image plane
focal
surfaces :
parabolas
centered
quadratic
astigmatsim
image plane
focal
surfaces :
complicated
binodal
astigmatism
Different forms of distortion fields
General Distortion
original
anamorphism, a10
x
keystone, a11
xy
1. order
linear
2. order
quadratic
3. order
cubic
line bowing, a02
y2
shear, a01
y
a20
x2
a30
x3 a21
x2y a12
xy2 a03
y3
More general case with residual symmetry plane:
plane symmetric systems
Components are allowed to be non-circular symmetric
More easy formulation of shift vector
Wave aberration expression
Plane-Symmetric Systems
field
point
Hrp
e
pupil
pointunit
vector
j
reference
axis
plane of
symmetry
q
p
pn
p
m
pp
k
qpnmk
qpnqnmpnk
reHerHrrHH
WerHW
,,,,
,,,2,2,,
Characteristic zero-order property of non-symmetrical systems: Anamorphism, different magnifications in two azimuthal cross sections of the image
Calculation:
Simple example: well know Scheimpflug imaging setup
Results in keystone distortion
Anamorphism
object
plane
lens
image
plane
'
s
s'
j
jN
j
y
x
I
I
m
m
'cos
cos1
sin1'
o
ox
mhs
msm
'sin
sin
sin1'
2
o
oy
mhs
msm
(34-198)
ideal
real
Anamorphotic imaging:
different magnifications in x- and y-cross section,
tangential and sagittal magnification
Identical image location in both sections
Anamorphotic factor
t
sanamoph
m
mF
ktk
t
tun
unm
,
1,1
ksk
s
sun
unm
,
1,1
Anamorphotic Imaging Setup
cylindrical
lens 1
us
ut
cylindrical
lens 2
18
fx
fy
x'
x
y
y'
Crossed Cylindrical Lenses
Example of two aspherical cylindrical
lenses with different focal lengths
Due to difference, the numerical apertures
are different
The wavefront shows the deviations
in the 45° directions
The spot diagram has extreme small
diameters only along the axes
50 l
y-z-section
NA = 0.4
x-z-section
NA = 0.57
Lithographic Lens
X-Design
Lithographic Lens
I-Design
EUV-Mirror Systems
8-mirrors, NA = 0.4
All surfaces centered
Relaxed distribution of incidence
angles
M1
M2
M3
M4
M5
M6
M8
M6
intermediate
image
No of mirror Angle in [°]
1 10.5
2 15.0
3 14.9
4 11.0
5 10.6
6 25.6
7 15.7
8 4.7
3-mirror Schiefspiegler telescope without central obscuration
Correction of finite field aberrations by free form surfaces
Freeform Telescope
General 3D-System:
Yolo-telescope
Aberration fields:
1. spot 2. coma 3. astigmatism
Schiefspiegler
incoming
light
image
mirror M1
tilted around y
deviation in x
mirror M2
tilted around x and y
deviation in y and x
y
z
mirror M3
tilted around x and y
deviation in y and x
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20y
x
y
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
x
y
-20 -15 -10 -5 0 5 10 15 20-20
-15
-10
-5
0
5
10
15
20
Pseudo-3D-layouts:
eccentric part of axisymmetric system
common axis
Remaining symmetry plane
Schiefspiegler-Telescopes
mirror M1
mirror M3
mirror M2
image
used eccentric subaperture
M1
M3M
2
y
x
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
field points of figure 34-143
HMD Projection System
Special anatomic requirements
Aspects:
1. Eye movement
2. Pupil size
3. Eye relief
4. Field size
5. See-through / look-around
6. Brightness
7. Weight and size
8. Stereoscopic vision
9. Free-forme surfaces and DOE
spectacles
eye
balleye
axis
earfree space
for HMD
retina
iris
j
y
L
HMD Projection Lens
eye
pupil
image
total
internal
reflection
free formed
surface
free formed
surface
field angle 14°
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
binodal
points
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
astigmatism, 0 ... 1.25 l coma, 0 ... 0.34 l Wrms
, 0.17 ... 0.58 l
Refractive 3D-system
Free-formed prism
One coma nodal point
Two astigmatism nodal points
Pair of lenses with one plane surface and one asphere / free form surface
Equation of the aspheric surface:
Small shift of one lens along x: Defocus
Small shift of one lens in y: Astigmatism
Application in ophthalmoscopy for pre-correction of eye aberrations
Alvarez Lens
x
y
x
x32
3
1),( xyxyxP
232 23
22),( axaaxyxPx
yxbyxPy 4),(
32 )(3
1)(
),(),(),(
xxyxx
yxPyxxPyxPx
Polarization
If polarization effects have influence on the performance of a system, the pure
geometrical aberration model is no longer sufficient
The main reasons for polarization effects in optical systems are
1. Coatings
2. stress induced birefringence
3. intrinsic birefringend in crystaline materials
4. mixing of field component in high-NA systems without x-y-decoupling
coatings
stress induced
birefringence
intrinsic
birefringence
high NA
geometry
Polarization
The understanding of the intensity distribution of the point spread function and
image formation needs the consideration of the physical field E
In the most general case, in the exit pupil we have a field with 3 orthogonal components,
that can not interfere
In the coherent case, the intensity
in the image plane is the sum of
the 3 intensity contributions
In the case of small numerical
apertures, only 2 transverse field
components must be considered
To determine polarization effects
in the image, first the propagation
of the polarization through the
system must be calculated
system exit
pupilimage
EyEx
I'=| E'x2+E'y
2+E'z
2 |
2
Ez
Embedded local 2x2 Jones matrix
Matrices of refracting surface
and reflection
Field propagation
Cascading of operator matrices
Transfer properties
1. Physical changes
2. Geometrical bending effects
Polarization Raytrace
1,
1,
1,
,
,
,
1
jz
jy
jx
zzyzxz
zxyyxy
zxyxxx
jz
jy
jx
jjj
E
E
E
ppp
ppp
ppp
E
E
E
EPE
121 .... PPPPP MMtotal
100
00
00
,
100
00
00
s
p
rs
p
t r
r
Jt
t
J
100
0
0
2221
1211
,1 jj
jj
J refr
1
,1,1,11
inrefrout TJTP
1
,1,1,11
inbendout TJTQ
Change of field strength:
calculation with polarization matrix,
transmission T
Diattenuation
Eigenvalues of Jones matrix
Retardation: phase difference
of complex eigenvalues
To be taken into account:
1. physical retardance due to refractive index: P
2. geometrical retardance due to geometrical ray bending: Q
Retardation matrix
Diattenuation and Retardation
EE
EPPE
E
EPT
T
*
*
2
2
minmax
minmax
TT
TTD
2/12/12/12/12/1 wewwJ
i
ret
21 argarg
totaltotalPQR
1
System Model
The field must be decomposed in components
1. in the object
2. in the entrance pupil
3. at every surface in the system
4. in the exit pupil
The transfer is established by coordinate transforms and Jones matrices
yp y'p
x'p
Eyi
y'
x'
u
entrancepupil image
plane
exitpupil
y
x
objectplane
si
Exi
Eyp
sp
Exp
xp
E'yp
s'p
E'xp
(flat) (curved)
),(
),(
),(),(
),(),(
),(
),(
pp
in
y
pp
in
x
ppyyppyx
ppxyppxx
pp
out
y
pp
out
x
yxE
yxE
yxJyxJ
yxJyxJ
yxE
yxE
Any change of the polarization state from the object to the image space can be considered
as an aberrtion of polarization
The changes of the field can be decomposed in components
The vectoirial Zernikes can be used to describe these changes
From a practical point of view, phase and amplitude changes should be distinguished
Therefore usually the detailed assessment is divided into
1. retardance
2. diattenuation
Physically this corresponds to the phase and the size of the complex eigenvalues of
the system Jones matrix
System Quality Assessment for Polarizing Systems
11
),(Z),(
0
0
),(,(
j
ppjj
j ppyj
jy
ppxj
jxpp yxEyxE
ZyxE
Z)yxE
Vectorial Zernike Functions
Composition of the gradients in a vectorial function
Normalization and expansion into original functions
Describes elementary decomposition of orientation fields
Applications: polarization aberrations
35
jyyjxxj ZeZeS
'
j
jjy
j
jjxj ZbeZaeS
5648
6457
326
235
324
13
12
22
1
22
1
2
1
2
1
2
1
ZeZZeS
ZZeZeS
ZeZeS
ZeZeS
ZeZeS
ZeS
ZeS
yx
yx
yx
yx
yx
y
x
S2 S4
S5 S7
S3
S6
Change of incoming linear polarization
in the pupil area
Total or specific decomposition
Polarization Performance Evaluation
negative
positive
piston defocustilt
Polarization
Polarization of a donat mode in the focal region:
1. In focal plane 2. In defocussed plane
Ref: F. Wyrowski
Polarization Point Spread Function
The understanding of the intensity distribution of the point spread function and
image formation needs the consideration of the physical field E
In the most general case, in the exit pupil we have a field with 3 orthogonal components,
that can not interfere
In the coherent case, the intensity
in the image plane is the sum of
the 3 intensity contributions
In the case of small numerical
apertures, only 2 transverse field
components must be considered
system exit
pupilimage
EyEx
I'=| E'x2+E'y
2+E'z
2 |
2
Ez
Polarization for High NA
Especially for high numerical aperture angles, ther z-component must be taken into
account as well
The polarization breaks the symmetry
The point spread function can have non-circular symmetry for rotational symmetric
systems
Ref: M. Totzeck
Ix (0.95)exit pupil Iy (0.01)
a) linearIz (0.15) Isum (1.0)
b) circular Ix (0.5) Iy (0.5) Iz (0.1) Isum (1.0)
Vectorial diffraction integral in Fraunhofer representation
All components x,y,z must be considered
In particular due to the large bending angle an axial component EZ occurs
High-NA-Systeme
E r F P x y e
n u x y
n u y
n u y
y n u x y
n u y
p p
iz n u x y
p p
p
p
p p p
p
p p
' ,
' sin '
' sin '
' sin '
' sin '
' sin '
' ' sin '
1
1
2
2 2
2 2 2
2 2 2
2 2 2 2
2 2 2
2 2 2 2
1
1
1
1
l
Polarization in High-NA Lenses
High NA :
Polarization effects important
Effects:
1. Large angles
2. Material birefringence
3. Coatings
Complete interference only for
s-component
Exposure process better for
polarized light
a) low NA b) high NA
exposure
latitude [%]
20
15
10
5
00 0.2 0.4 0.6 1.00.8 1.2 1.4
z
[a.u.]
polarized
dryunpolarized
dry
unpolarized
immersion
polarized
immersion
More general geometries and free shaped surfaces breaks symmetry of the systems
A pilot ray and an infinitesimal ray tube substitutes the optical axis and the paraxial rays
A vectorial aberration theory describes the performance
Easier to formulate: special cases:
1. circular symmetric shifted/tilted components
2. plane symmetric systems
The aberrations are field with certain corrected nodal points/lines
Applications: mirror systems, Lithography, HMD
Polarization aberrations: change of polarization in the system by coatings, materials, high-NA
geometry
Jones matrix algorithm feasible
For high NA: geometric effect, also z-component of field due to large angles
Decomposition into phase (retardance) and amplitude (diattenuation)
Decomposition with orientation Zernikes
Conclusion
Understanding optical systems is only possible with aberration theory
Correction of systems is efficient with detailed analysis of aberrations and
the methods to prevent or compensate them after a proper classification
Especially the decomposition of the total aberrations into the surface contributions helps
for analyzing and improving systems
Allows qualified performance assessment
But:
1. the classical aberration theory is restricted to the geometrical picture
2. the classical aberrations theory mostly assumes circular symmetry
3. complete general geometries are complicate to implement,
the single numbers becomes matrices and are hard to interprete
4. the digital image processing approaches of today reduce the necessity of perfectly
corrected analogue systems
4. the application to real human image perception is still complicated
Why Aberration Theory ?
Fourier Filtering
Digital optics with pupil phase mask
Primary image blurred
Digital reconstruction with the help of
the system transfer function
Objective tube lens
digital image
Iimage(x') Pupil with
phase mask
transfer function ImageComputer
image digital
restored
Object
image
a) object
Image quality with Real Objects
b) good image c) defocussed d) axial chromatic
aberration
e) lateral chromatic
aberration
g) chromatical
astigmatism
f) sphero-
chromatism
Real Image with Different Chromatical Aberrations
original object good image color astigmatism 2 l
6% lateral color axial color 4 l
Aberration theory is not easy to understand
Aberrations theory is necessary for the fundamental understanding of correction and setup
of optical systems
Many assumptions and approximations are necessary
Real world mostly is more complicated
A minimum amount of analytical calculation helps to understand relationships
Lengthy analytical formulas are not of relevance today, mostly calculations are numerically
The aberration theory is still a research topic
Summary
Thank you for attending
the lecture
