imaging and aberration theory · 2020. 2. 7. · 2 schedule - imaging and aberration theory 2019 1...
TRANSCRIPT
www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 15: Additional topics
2020-02-07
Herbert Gross
Winter term 2019
2
Schedule - Imaging and aberration theory 2019
1 18.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 25.10.Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 01.11. EikonalFermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 08.11. Aberration expansionssingle surface, general Taylor expansion, representations, various orders, stop shift formulas
5 15.11. Representation of aberrationsdifferent types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 22.11. Spherical aberrationphenomenology, sph-free surfaces, skew spherical, correction of sph, asphericalsurfaces, higher orders
7 29.11. Distortion and comaphenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 13.12. Chromatical aberrationsDispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spectrum
10 20.12.Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 17.01. Zernike polynomialsspecial expansion for circular symmetry, problems, calculation, optimal balancing,influence of normalization, measurement
13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio
14 31.01. Transfer function transfer function, resolution and contrast
15 07.02. Additional topicsVectorial aberrations, generalized surface contributions, Aldis theorem, intrinsicand induced aberrations, reversability
1. Generalized systems
2. Vectorial aberration theory
3. Field dependence of aberrations
4. Nodal theory
5. Aldis theorem
6. Induced aberrations
7. Generalized surface contributions
8. Caustics
9. Polarization aberrations
10. 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
4
5
Aberration Theory and Symmetry
Seid
el, w
ith
fie
ld
Ara
ki
Vecto
rial I -
wit
h f
ield
Sh
ack / T
ho
mp
so
n / S
asia
n
Vecto
rial II -
wit
h f
ield
Fu
ers
ch
ba
ch
Ald
is t
he
ory
- o
ne
ra
y o
nly
Welf
ord
- o
ne O
PD
po
int
on
ly
Ole
szko
- Z
ern
ike
Para
xia
l -
axis
Para
basal aro
un
d r
eal O
AR
/ C
R
2 x
2
4 ×
4
5 ×
5
Rotational
symmetric
Photographic lens,
microscope, zoom lens
Double plane-
symmetricAnamorphic
Freeform (plane-
symmetric)Scheimpflug
Freeform (non-
symmetric)Cubic phase plate for EDF
Rotational
symmetric
Schiefspiegler telescope,
spherical TMA, HMD
Double plane-
symmetricAnamorphic prism stretcher
Freeform (plane-
symmetric)
Unobscured telescope, TMA
corrected, HMD
Freeform (non-
symmetric)
Alvarez plate system,
panoramic zoom system
Rotational
symmetricYolo telescope, spherical
Freeform (non-
symmetric)Yolo telescope, corrected
Aberration theory
4th order 6th orderexact -
all orders
OAR
Straight
1D bend
2D bend
Sample systemsSurface
symmetry
ray transfer matrix
Reference
Ref: Y. Zhong
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
6
General Reference and Pupil Re-Scaling
x
y
xP
yp
x'
y'
x'P
y'p
object
plane
entrance
pupil
exit
pupil
image
plane
z
yEnP (rp+Drp)
y'ExP rp
yH
real
ideal
y' (H+DH)
realideal
Wave aberrations: measured as a function of the exit pupil coordinates
Therefore pupil distortion is seen as a change in the entrance
pupil sampling
The changes are in direction (normalized direction vectors)
and in length
7
Wave aberration field
indices
Normalized field vector: H
normalized pupil vector: rp
angle between H and rp: q
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
q
nmj
n
pp
m
p
j
klmp rrrHHHWrHW,,
,
mnlmjk 2,2
y
Hrp
field1
1
pupilj
qcos,, 22 ppppp rHrHrrrHHH
8
Refraction of a ray tube
Non-Axisymmetric Systems: Ray Tube around Axis Ray
xy
q
q'
surface
plane of
incidence
incoming
ray
local
system
axis
Rh1
Rh2
R||
R
R'||
R'
outgoing
ray
'cos'
'cos'cos
'cos'
cos1
'
122
2
|||| q
q
q
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
q
q
Transverse Ray Aberrations
Scaled gradient of wavefront:
transverse ray aberrations (L Lagrange invariant)
Expansion
of orders:
with the relations
Two contributions along field vector
and pupil vector
Alternative:
along field and perpendiccular
corresponds to Seidel convention
WL
Wn
RH
pp rr
D1
'
ppprpr rrrHrH
pp
2,
p
n
pp
m
p
n
pp
m
p
nmj
j
klm
n
ppr
m
p
nmj
j
klm
m
pr
n
pp
nmj
j
klmpr
rrrrHnHrrrHmHHW
rrrHHHW
rHrrHHWrHW
p
pp
11
,,
,,
,,
2
,
khp rkrhr
0kh
nonorthogonal decomposition orthogonal decomposition
DHDH
Drp
Ah
Ch
Brp
10
Vectorial Aberrations
ord j m n Term scalar Name
0 0 0 0 000W uniform Piston
2
1 0 0 HHW
200 2
200 HW quadratic piston
0 1 0 prHW111 qcos111 prHW magnification
0 0 1 pp rrW020 2
020 prW focus
4
0 0 2 2040 pp rrW
4
040 prW spherical aberration
0 1 1 ppp rHrrW131 qcos131 prHW coma
0 2 0 2222 prHW q2
22
222 cosprHW astigmatism
1 0 1 pp rrHHW220
22
220 prHW field curvature
1 1 0 prHHHW311 qcos3
311 prHW distortion
2 0 0 2400 HHW
4
400 HW quartic piston
6
1 0 2 2240 pp rrHHW
42
240 prHW oblique spherical aberration
1 1 1 ppp rHrrHHW331 qcos33
331 prHW coma field 3rd
1 2 0 2422 prHHHW q224
422 cosprHW astigmatism field 4th
2 0 1 pp rrHHW
2
420 24
420 prHW field curvature field 4th
2 1 0 prHHHW
2
511 qcos5
511 prHW distortion field 4th
3 0 0 3600 HHW
6
600 HW piston 6th
0 0 3 3060 pp rrW
6
060 prW spherical aberration 6th
0 1 2 ppp rHrrW
2
151 qcos5
151 prHW coma 6th
0 2 1 2242 ppp rHrrW q242
242 cosprHW astigmatism 6th
0 3 0 3333 prHW q333
333 cosprHW trefoil
11
12
Nodal Aberration Theory
Shift of the nodal points by tilting a surface
Ref: Y. Zhong
13
Vectorial Aberration Contributions
Idea of nodal points:
image points of the tilted component axes
Every component has its individual axis, the aberrations are symmetric around this axis
(circular symmetric sub-system)
The axis are bended towards the image plane
Every circular symmetric component therefore has an individual aberration center sj in
the image plane
The interaction and overlay of the various centers are complicated y
x
lens 2
lens 3
aberration
contribution
lens 3
aberration
contribution
lens 2
s3
s2
lens 1
bended axis
rays
aberration
contribution
lens 1
s1
14
Example system: Yolo Telescope
Extended Yolo telescope with one Freeform at the first surface
System without symmetry
Ref: S. Zhu
M1
M2
M3
Freeform
Sphere
Sphere
𝝈𝒙
𝝈𝒚
Sigma vectors for different
surfaces
M1
M2
M3
Wave aberration
Shift vector sj of symmetry center for every surface j:
aberration field center for surface j
Effective field height relativ to center j
Special case of coma:
- split of wave aberration into
1. field dependent term
2. field independent term
contribution
Relevant:
- distance of surface from pupil
- height y-bar of chief ray at the surface
Systems with Non-Axisymmetric Geometry
q nmj
n
pp
m
pq
j
qqklmp rrrHHHWrHW,,
000,
sss
jjoj HH s
15
131, 0
131, 0 131,
,coma p q q p p p
q
q p p p q q p p p
q q
W H r W H r r r
W H r r r W r r r
s
s
First generalization:
- geometry non-centered
- every surface is a circular symmetric subsystem
- every surface creates aberrations in the final image plane with strength Wklm (coefficient)
- the surface contributions are additive with vectorial character
- every surface has ist individual center point in the image plane
- every field point H in the image plane has an effective height Hho relative
to the surface j
- the contribution of surface j is given by the vector
here the coefficient Wklm is not influences by the centering
- the total aberration in the image is given by the sum
- coefficient aklm: normalized on value of centered system
Second generalization:
- also surfaces not circular symmetric
- use for freeform systems
Vectorial Aberration Theory
16
js
jjoj HH s
, ,klm j klm j jA W s
, ,klm klm j klm j j
j j
A A W s
Superposition of aberrations in the image plane
Vector sum of A-contributions
- center locations influences by surface location/orientation
- weighting/strength Wklm defined by surface shape
Sum of contributions:
influences by both quantities
Vectorial Aberration Contributions
17
js
x
y
s1Wklm,1
A1
s2
Wklm,2
A2
s3
Wklm,3
A3
(0,0)
total
Wklm
H arbitrary field
point (x,y)
HA2
HA3
HA1
Geometry in 2 dimensions
Connection between pupil center P and Center of curvature C:
defines field center
Local system surface j: defines tilt angle q
All properties referenced on optical axis ray (OAR) as parabasal equivalent basis
axis point imaging: Q ---> Q'
field point imaging: A ---> A'
Field Center in Non-Axisymmetric Geometry
18
surface no. j
object no. j
image no. j
optical axis ray
pupilcentre of curvature
of surface no. j
local axis j
vertex
aberration
field
centreR
q
tilt angle
s
A
A‘
Q
Q‘
S
C
P
Basic concept:
- shifted aberration field centers
- valid for circular symmetric surfaces in general geometry
- every surface has an individual centre of symmetry
- vector summation of individual surface aberration fields
- separation in spherical and aspherical contributions
- multi nodal aberrations in case of decentering
Lack of symmetry:
- individual chief ray height at the surfaces
- field dependence of aberrations fields
Surface in pupil plane: field invariant aberrations
surfaces with distance to pupil: field dependent aberration contribution
Recently extended to freeform surfaces
Nodal Aberration Theory
x x
y y
astigmatisma) centered b) decentered
astigmatism
K. Thompson, Proc. SPIE 7652 (2010)
19
Arbitrary variation of performance over the field of view
- one aberration value is not feasible to describe the distribution over the bundle cross section
- uniformity of aberration variation is important property
- single numbers should be defined to summarize the performance by moments, rms,...
- additional uniformity parameters are necessary to be defined in the merit function
Nodal aberration theory
- vectorial approach on aberration theory describes more general geometries without
symmetry
- nodal points are locations with corrected aberrations in the field
20
Freeform Systems: Performance Assessment
y
x
-8
-6
-4
-2
0
2
4
6
8
-8 -6 -4 -2 0 2 4 6 8
binodal
points
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,
sss
jjoj HH s
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
ssss
ss
s
ss
ss
s
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
,
21
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
s
)(
131
,131
,131
,131
131 c
q
j
q
q
W
W
W
W
a
ss
pppo
c
coma rrraHWW 131
)(
131
zero
coma
green zero
coma
blue
zero
coma
total
22
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 s
q
q
q
W
W
a,222
,222
222
s
2
222
,222
,222
2
2
222 aW
W
b
q
q
q
s
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
23
Generalized Nodal Locations
24
Wave aberration terms of can have nodal lines
in case of more complicates surfaces
Example 1:
linear coma node line for coma together
with skew spherical aberration
Example 2:
higher order coma nodal line
Ref: T. Yang / X. Zhang
General OSC
Assumption: pupil spheres are conjugate
General OSC condition
Essential for this simple formulation:
pupil coordinates measured in object/image plane
'( ', ') ' '
'
nW r h h r
RD D D D D
x
yz S
P
Q
rA
x‘y‘
z‘
P‘Q
r
exit
pupil
S‘
A‘entrance
pupil
image
plane
system
surfaces
object
plane
R
R‘
r‘
Dh
Dh‘
n
n‘
25
Low order Zernikes as a function of the field position
Completly different distributions,
Complete characterization gives a huge amount of detailed information.
Also analytical solution for lower orders provided in the literature
26
Zernikes as Function of the Field
R. Gray, C. Dunn, K. Thompson, J. Rolland, Opt. Expres 20(2012) p. 16436, An analytic expression for the field
dependence of Zernike polynomilas in rotational symmetric optical systems
astigmatismcomaspherical aberration
Field Dependent Zernike Coefficients
Approach to introduce Zernike evaluation into aberrations theory of 6th order:
field dependece of Zernike coefficients
Lengthy analytical formulas
Numerical implementation easy
27
R. Gray, C. Dunn, K. Thompson, J. Rolland, Opt. Expres 20(2012) p. 16436, An analytic expression for the field dependence of
Zernike polynomilas in rotational symmetric optical systems
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
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
29
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 coma, 0 ... 0.34 Wrms
, 0.17 ... 0.58
Refractive 3D-system
Free-formed prism
One coma nodal point
Two astigmatism nodal
points
30
Total quality measures, selection depends on application
Circular symmetric systems:
clear definition of optical axis, paraxial range as reference
1. classical primary aberrations, large experience, easy to interpret
2. geometrical representations like transverse or wave aberrations,
fast to calculate, diffraction excluded
3. physical criteria like point spread function (PSF), modulation transfer function (MTF)
General systems with reduced or no symmetry:
1. 3D geometry, freeform surfaces
2. no clear paraxial reference
3. Important: field dependence
Possible general options:
1. geometrical spot diagrams
2. wave aberrations, Zernike representation
3. PSF and MTF
4. Aberration theory of 6th order incorporates field dependence
31
Total Performance Criteria
Sensitivity of surfaces, important for tolerancing,
1. surface contributions, correctability and distribution, induced aberrations
2. structural aberration coefficients (analytic) give relation to system data
Circular symmetric systems:
1. Seidel aberrations in 3rd order (4th order wave aberrations)
2. General aberration theory in 6th order after Shack/Thompson
3. Aldis theorem, but only for one ray
General systems with reduced or no symmetry:
only pure experience on number and location of freeform surfaces as well as complexity and
necessary deviation from circular symmetry
1. 6th order aberration theory for circular symmetric components
2. Wave aberration contribution according to Hopkins / Welford
3. Field dependend Zernike coeffcients
New approaches necessary:
1. Generalization of Aldis theorem
2. Zernike surface contributions
32
Analysis and Sensitivity
Aberration expansion: perturbation theory
Linear independent contributions only in lowest correction order:
Surface contributions of Seidel additive
Higher order aberrations (5th order,...): nonlinear superposition
- 3rd oder generates different ray heights and angles at next surfaces
- induces aberration of 5th order
- together with intrinsic surface contribution: complete error
Separation of intrinsic and induced aberrations: refraction at every surface in the system
Induced Aberrations
PP'0
initial path
paraxial ray
intrinsic
perturbation at
1st surface
y
1 2 3
y'
intrinsic
perturbation at
2st surface
induced perturbation at 2rd
surface due to changed ray height
change of ray height due to the
aberration of the 1st surface
P'
33
34
Induced Aberrations
Example spherical aberration
Induced effects for higher orders
3rd order
spherical
7th order
spherical
5th order
spherical
S(3rd)
intr
intrinsic induced
S(5th)
intr
S(7th)
intr
S(5th)
indu
S(7th)
indu
9th order
spherical S(9th)
intr S(9th)
intr
+2
+4
+6
Surface No. j in the system:
intermediate imaging with object, image, entrance and exit pupil
Case a) : object wave perfect, Welford approach
Case b): real wave impinging onto surface:
incidence angles and coordinates changed, induced aberrations taken into account
a) Intrinsic
b) Intrinsic and
induced
Induced Aberrations
entrance
pupil no. j
wave spherical
intermediate
ideal object no. j
surface
index j
exit pupil no. j
wave with intrinsic
aberrations
intermediate
image no. j
entrance
pupil no. j
grid distorted
wave perturbed
intermediate
real object no. j
surface
index j
exit pupil no. j
intrinsic and induced
aberrations intermediate
image no. j
35
Mathematical formulation:
1. incoming aberrations from
previous surface
2. transfer into exit pupil
surface j
3. complete/total aberration
4. subtraction total/intrinsic:
induced aberrations
Interpretation:
Induced aberration is generated by pupil distortion together with incoming perturbed
3rd order aberration
Similar effects obtained for higher orders
Usually induced aberrations are larger than intrinsic one
Induced Aberrations
1
1
)5()3(
,
j
i
pipipjentr rWrWrW
pjpj
j
i
pipjpipjexit rWrWrWrrWrW )5()3(
1
1
)5()3()3(
,
1
1
)3()3()5()3(
,,,
j
i
pjipjpj
pjentrpjexitpjcompl
rWrWrW
rWrWrW
1
1
)3()3(
,
j
i
pjipjinduc rWrW
36
In 3rd order as first perturbation contributions the total system aberration is independent from
the direction
In higher order due to induced aberrations, this reversability is not fulfilled
Important:
1. the light direction can be inverted
2. the coordinate grids and the reference is changing with direction
37
Reversability of Aberrations
paraxrealparaxial start
real back
real start
real back
real start
real back
Example: chromatical aberration for 2x 4f imaging with high/low dispersing lenses
Consequence:
- for systems with large distance of compensating lens groups: system has to be evaluated in
correct order
- for critical systems not the same performance i both directions
38
Reversability of Aberrations
Ref
= 546 nm
=480 nm
=480 nm
-3.00698 -2.40914 -0.67739 -2.8169
Int -2.40914
Ind +0.07955Int -0.67739
Ind +0.26963
SF39
n1 = 20.4
PK50
n2 = 69.74
39
Connection between Wave and Ray Aberrations
'
'
p
p
R Wx
n x
R Wy
n y
D
D
Relation between wave and transverse aberration:
derivative of the wave front according to the pupil coordinates
scaled with radius of reference sphere and index
Approximation for small aberrations and small aperture angles u
A corresponding relation exists between Aldis transverse ray aberration contributions
and Welfords wave aberration contribution
The changing magnification due to the rear system surfaces must be taken into account
yp
z
real ray
wave front W(yp)
R, ideal ray
C
reference
plane
y'D
reference sphere
q
u
Approach of Hopkins / Welford:
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
- choice of reference surface not trivial
- parabasal ray calculation inaccurate
40
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
41
Options for Surface Contributions
Ord
er
(wav
e a
be
rrat
ion
)
Spat
ial p
up
il re
solu
tio
n
No
n-c
en
tere
d
Fre
efo
rms
Fie
ld d
ep
en
den
ce
Co
mm
en
t
4th order, Seidel 4 N N N Y well known
6th order, Shack/Thompson 6 low Y N Y components circular symmetric
Wavefront, Hopkins/Welford all Y Y Y N one ray only
Aldis all Y N N N one ray only
Aldis generalized all Y Y Y N one ray only
Wavefront all Y Y Y N only numerical, huge information
Zernike high ~ Y Y N problem induced aberrations
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
42
Wave Aberration Additivity
P
arbitrary ray
y
1 2 3
y'
P'
surfaces exit pupil
total Wtot
W1
W2 W3
surface contributions
ray pencil
Surface decomposition of wave aberration
Zernike decomposition of total wave aberration
Zernike decomposition of surface contribution
General relation between coordinates
special case of linear scaling
insertion
case of distortion-free grid projection
For systems with neglectable induced aberrations the bundle diameter scales linear and the
Zernike expansion coefficients are also additive on a normalized bundle radius
If the system suffers from large induced aberrations, the ray grid is distorted and rescaled,
in this case the Zernike coefficients are not exactly additive.
It is also well known, that the Zernikes are changing during propagation
Therefore distance related induced Zernike contributions can be defined in addition
43
Zernike Surface Contributions
G. Dai, Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials,
Appl. Opt. 48 (2009) p.477
s
s
W W
( , ) ( , )p p j p p
j
W x y c Z x y
( , ) ( , )s ps ps js ps ps
j
W x y c Z x y
p p( , y ) , ( , y )ps x p ps y px f x y f x
,p x p p y px m x y m y
( , ) ( , )
( , ), ( , )
p p js ps ps
s j
js x p p y p p
s j
W x y c Z x y
c Z f x y f x y
j js
s
c c
Example system: plane symmetric TMA system
nearly diffraction limited correction for a small field
of view
M1: off axis asphere, M2, M3: freeforms
F-number 1.8, field -1°...+1°
44
TMA System
x = -1° x = +1°
y = +1°
x = 0°
y = -1°
y = 0°
field
angles x/y
M1
circular
symmetric
asphere
M2
pupil
freeform M3
freeform
image
M2
freeform
M3
freeform
M1
asphere
Surface contributions of every mirror with parabasal reference
pupil rescaling neglected
Dominating astigmatism
Sum of wave aberration not exactly additive,
difference due to induced aberrations
45
Wavefront Contribution of every Surface
sum of surface contributions
M1 M2 M3
Contributions of the lower
Zernike coefficients per surface
(Fringe convention)
46
Zernike Coefficients per Surface
0
5
10
15
20
25
-3
-2
-1
0
1
2
3
log cj
zernike
index j
astigmatism
surfaces
sum
M1
M2
M3
comaspherical
tilt defocus
Problem:
in spot diagram the information
of the corresponding pupil
location is lost
Extension of Kingslakes
representation for surface
contributions
Corresponds to wavefront
gradient distribution in the
pupil
Problem:
compaction of high
complexity,
limited clearness
47
Extended Kingslake Diagram
x
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
M3
M1
M2
Caustic phenomena in real world
Caustics
Ref: J. Nye, Natural focusing
Early investigations on caustics:
Leonardo da Vinci 1508
Caustics at mirrors and lenses
Caustics
envelope
caustic curve
envelope
caustic curve
envelope
caustic curve
with cusp
Ref: J. Nye, Natural focusing
More general:
caustic occurs at every wavefront with concave
shape as locus of local curvature
Physically:
- crossing of rays indicates a caustic
- interference with diffraction ripple and ringing
is seen
Caustics
unique wave
front
rays
no unique ray
direction
amplitude variation due to
interference
Ref: J. Nye,
Natural focusingRef: W. Singer
Caustic: envelope of rays
Locus of local curvature
Calculation:
caustic:
ray direction:
rays:
L distance PC
variation of point on wavefront:
solution condition for linear system:
equation of caustic
Caustics
wave
front
rays
caustic
curve
P1
P2
C12
zyx ssss
cccc zyxr
sLrrc
zc
yc
xc
sLz
sLyy
sLxx
0
0
0
Lsyy
sLx
x
sL
Lsyy
sLx
x
sLy
Lsyy
sLx
x
sLx
zzz
y
yy
xxx
0
1
1
1
yx
y
yy
xxx
ss
sy
sL
x
sL
sy
sL
x
sL
Special case of one dimension x-z
Example: spherical aberration for focussing through
plane interface
Ray direction
Variation
Geometry and law
of refraction
Approximation of
small x:
caustic curve
Caustics
x
Wsx
0
01
Lsxx
sL
Lsxx
sL
zz
xx
22 xq
xn
a
xn
x
Wsx
refracting
surface
caustic
x
x
q
a
n
z
sx
2
22
2
)1(1q
xn
n
a
x
s
sL
x
z
3/23/12 )1(2
3cc xqn
nn
qz
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 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
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
58
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
59
Real Image with Different Chromatical Aberrations
original object good image color astigmatism 2
6% lateral color axial color 4
60
Time is Over
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 ?
Paper from 2010:
63
Importance of understanding Aberrations
64
Learning by Books
Mathematical Knowledge
nice analytical solutions are
often of limited practical
benefit
Optics...66
Ref: T. Kaiser
Many mysterious things and new notations
Designer mit ZEMAX
ZEMAX
Ref: H.Zuegge
Feedback
Ref: D. Shafer
nothing clear ?
to complicated ?
to much stuff?
69
Thank you
Thank you for your attention