imaging and aberration theory - uni-jena.de · 2014. 1. 15. · usually, the reference is the...
TRANSCRIPT
www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 13: Miscellaneous
2014-02-06
Herbert Gross
Winter term 2013
2
Preliminary time schedule
1 24.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 07.11. Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 14.11. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 21.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 05.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 12.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 19.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 09.01. Chromatical aberrations
Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 16.01. 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
11 23.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF
12 30.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement
13 06.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, vectorial aberrations, partial symmetric systems
1. Aldis theorem
2. Induced aberrations
3. Aberrations of diffractive elements
4. Vectorial aberrations
5. Polarization aberrations
6. Why aberration theory ?
3
Contents
Aldis Theorem
Aldis theorem: surface contribution of transverse aberration of all orders
Calculation by tracing two rays: 1. paraxial marginal ray 2. finite ray
H: Lagrange invariant
A: Paraxial refraction invariant
Transverse aberrations
)( jjjjjjj uchninA
yunH kk
k
j
yjxj
zjzj
jj
yjjj
zkkk
k
j
yjxj
zjzj
jj
xjjj
zkkk
ssss
HyAszA
suny
ssss
xAszA
sunx
1
22
1
22
''
1
''
1
object image
paraxial
marginal ray
arbitrary
finite ray
surfaces 1 2
3 4
5 6
y
y'
u u'
x',y'
P
P'
Aldis Theorem
Advantage of Aldis theorem: contain all orders
Larger differences for surfaces/cases with higher order contributions
Usually, the reference is the paraxial ray, therefore distortion is taken into account
A known formulation is available for aspherical surfaces in centered systems
A specialized equation must be used for the case of image in infinity
More general 3D geometries are not supported
More general formulations are possible (Brewer)
Disadvantage of Aldis theorem: only for one ray
Example Achromate - Seidel and Aldis contributions at everey surface and in summary
Differences to Seidel terms due to higher
order at cemented surface for larger pupil radii
Aldis Theorem
312
?y’
0.5
-0.5
Transverse
spherical aberrationF/2 Achromat, f’=100
Ref: H. Zügge
- 2
- 1
0
1
2
3
rp
1
Δy'
Surfaces
Sum
1 to 3
- 2
- 1
0
1
2
3
1
Δy'
rp
Surface 1
- 2
- 1
0
1
2
3
Seidel
Aldis
1
Δy'
higher
orders
rp
Surface 2
- 2
- 1
0
1
2
3
1
Δy'
Surface 3
rp
Expansion approach for aberrations: cartesian product of invariants of rotational symmetry
Third order aberrations
exponent sum 4
Fifth order aberrations exponent sum 6
Higher Order Aberrations
22
22
222
past
pppcoma
ppsph
yyAW
yxyyCW
yxSW
6223
1
2222
2
2222
1
222
322
pppcomaellcoma
pppskewsphsph
ppskewsphsph
ppplinearcoma
ppzonesphsph
yxyyCW
yxyySW
yxySW
yxyyCW
yxSW
2,,
2
2222pp
pp
yxwyyxxv
yxu
6
5
224
24
33
1
yPW
yyDW
yxyCW
yyAW
yyCW
sphpupspP
pdistdist
pppetzptz
pastast
pcomaellcoma
4
3
222
yPW
yyDW
yxyCW
spP
pdist
ppptz
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'
Surface No. j in the system: - intermediate imaging with object, image, entrance and exit pupil
- separate calculations with ideal/real perturbed object point
- pupil distortion must be taken into account
Induced Aberrations
entrance
pupil no. j
grid distorted
wave spherical
intermediate
ideal object no. j
surface
index j
exit pupil no. j
grid uniform
wave with intrinsic
aberrationsintermediate
image no. j
entrance
pupil no. j
grid distorted
wave perturbed
intermediate
real object no. j
surface
index j
exit pupil no. j
grid uniform
wave with intrinsic
aberrationsintermediate
image no. j
Mathematical formulation:
1. incoming aberrations form
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
Example Gabor telescope - a lens pre-corrects a spherical mirror to obtain vanishing spherical aberration
- due to the strong ray deviation at the plate, the ray heights at the mirror changes
significantly
- as a result, the mirror has induced
chromatical aberration, also the
intrinsic part is zero by definition
Surface contributions and chromatic difference (Aldi, all orders)
Induced Aberrations
1 2 3-1.5
-1
-0.5
0
0.5
1
1.5
l = 400 nm
1 2 3-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
l = 700 nmmirror
contribution
to color
surfaces surfaces
difference
heigth
difference
with
wavelengthl= 400 nm
l= 700 nm
Deviation of Light
reflection
mirror
scattering
scatter plate
refraction
lens
diffraction
grating
Mechanisms of light deviation and ray bending
Refraction
Reflection
Diffraction according to the grating equation
Scattering ( non-deterministic)
'sin'sin nn
'
g mo sin sin l
Diffractive Elements
z
h2
hred(x) : wrapped
reduced profile
h(x) :
continuous
profile
3 h2
2 h2
1 h2
hq(x) : quantized
profile
Original lens height profile h(x)
Wrapping of the lens profile: hred(x) Reduction on
maximal height h2
Digitalization of the reduced profile: hq(x)
Diffractive Optics
Local micro-structured surface
Location of ray bending :
macroscopic carrier surface
Direction of ray bending :
local grating micro-structure
Additional degree of freedom: independent determination of 1. ray bending location (carrier surface) 2. ray bending direction (local grating)
First effect corresponds to asphere Second effect corresponds to plane grating
macroscopic
surface
curvature
local
grating
g(x,y)
lens
bending
angle
m-th
order
thin
layer
Lens with diffractive structured
surface: hybrid lens
Refractive lens: dispersion with
Abbe number n = 25...90
Diffractive lens: equivalent Abbe
number
Combination of refractive and
diffractive surfaces:
achromatic correction for compensated
dispersion
Usually remains a residual high
secondary spectrum
Broadband color correction is possible
but complicated
refractive
lens
red
blue
green
blue
red
green
red
bluegreen
diffractive
lens
hybrid
lens
R D
453.3
CF
dd
ll
ln
Achromatic Hybrid Lens
Principle of achromatic correction
Ratio of Abbe numbers defines refractive power distribution
Diffractive element: Abbe number n = -3.45
Diffractive element gets only
approx. 5% of the refractive
power
diffglas
glas
refr FFnn
n
diffglas
diffs
refr FFnn
n
Color Correction of a Hybrid System
first refractive
power
short l1
long l2
image plane
corrected
second powerrefractive /diffractive
refractivesolution
diffractivesolution
1
2'1
'2
bending angles
Dispersion by grating diffraction:
Abbe number: small and negative !
Relative partial dispersion
Consequence :
Large secondary spectrum
n-P-diagram
Diffractive Optics: Dispersion
330.3''
CF
ee
ll
ln
2695.0''
'
',
CF
Fg
FgPll
ll
normal
line
ne
0.8
0.6
0.4
0.2-20 0 4020 60 80
DOE
SF59SF10
F4KzFS1
BK7
PgF'
Spherical Hybrid Achromate
Classical achromate:
- two lenses, different glasses
- strong curved cemented surface
Hybride achromate:
- one lens
- one surface spherical with
diffractive structure
- tolerances relaxed
1 2 3
refractive
solution
1 2
diffractive
solution
1
2
3
tilt
sensitivity
1 2
tilt
sensitivity
Expansion of the optical pathlength for one field point:
Primary Seidel aberrations:
No field curvature
No distortion (stop at lens)
Ray bending in a plane corresponds to linear collineation
Equivalent bending of lens
Primary Aberrations of a Diffractive Lens
f
r
w
DOE
chief
rayimage
plane
f
rw
f
wr
f
r
f
rrW
4
3
282
2)(
22
2
3
3
42
l
2
2am
ccfX
diff
diff
l
Straylight suppression by proper doe location and rear stop
Diffractive Optics: Hybrid Lens
0. order
1. order
2. order
transmitted
light
blocked light
DOE
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
pupil
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 prHW111
magnification
0 0 1 pp rrW020 focus
4
0 0 2 2040 pp rrW
spherical aberration
0 1 1 ppp rHrrW131
coma
0 2 0 2222 prHW
astigmatism
1 0 1 pp rrHHW220
field curvature
1 1 0 prHHHW311
distortion
2 0 0 2400 HHW
quartic piston
6
1 0 2 2240 pp rrHHW
oblique spherical aberration
1 1 1 ppp rHrrHHW331
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
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
reference
axis
plane of
symmetry
q
p
pn
p
m
pp
k
qpnmk
qpnqnmpnk
reHerHrrHH
WerHW
,,,,
,,,2,2,,
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
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
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
43
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
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
Thank you for attending
the lecture