IAT13_Imaging and aberration Theory Lecture 10 Further reading

Imaging and Aberration Theory
Lecture 10: Further reading aberrations
2014-01-16
Herbert Gross
Winter term 2013
www.iap.uni-jena.de
2
Preliminary time schedule
1
2
24.10. Paraxial imaging
07.11. Pupils, Fourier optics,
Hamiltonian coordinates
3
14.11. Eikonal
4
5
6
paraxial optics, fundamental laws of geometrical imaging, compound systems
pupil definition, basic Fourier relationship, phase space, analogy optics and
mechanics, Hamiltonian coordinates
Fermat Principle, stationary phase, Eikonals, relation rays-waves, geometrical
approximation, inhomogeneous media
21.11. Aberration expansion
single surface, general Taylor expansion, representations, various orders, stop
shift formulas
28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls,
measurement of aberrations
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
9
19.12. Astigmatism and curvature
09.01. Chromatical aberrations
phenomenology, Coddington equations, Petzval law, correction options
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
23.01. Wave aberrations
definition, various expansion forms, propagation of wave aberrations, relation to
PSF and OTF
11
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
3
Contents
1. Sensitivity in 3rd order
2. Structure of a system
3. Pupil aberrations
4. Sine condition
5. Isoplanatism
6. Herschel condition
7. Relation to Fourier optics and phase space
Field-Aperture-Diagram
 Classification of systems with
field and aperture size
 Scheme is related to size,
correction goals and etendue
of the systems
w
photographic
Biogon
40°
lithography
Braat 1987
36°
32°
Triplet
Distagon
28°
 Aperture dominated:
Disk lenses, microscopy,
Collimator
 Field dominated:
Projection lenses,
camera lenses,
Photographic lenses
 Spectral widthz as a correction
requirement is missed in this chart
24°
Sonnar
20°
projection
16°
12°
double
Gauss
split
triplet
projection
projection
Gauss
8°
lithography
2003
diode
collimator
achromat
0
0.2
0.4
micro
100x0.9
micro
40x0.6
micro
10x0.4
4°
0°
constant
etendue
Petzval
disc
0.6
0.8
microscopy
collimator
focussing
NA
5
Sensitivity of a System
4
0
3
0
2
0
1
0
Sph
0
Sp
h
 Sensitivity/relaxation:
Average of weighted surface contributions
of all aberrations
-10
-20
-30
-40
-50
 Correctability:
Average of all total aberration values
1
2
3
4
5
6
7
8
1
0
9
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
3
0
2
0
1
0
Coma
0
Kom
a
 Total refractive power
-10
-20
-30
-40
-50
1
2
3
4
5
6
7
8
1
0
9
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
k
F  F1    j F j
4
j 2
j 
2
Ast
 Important weighting factor:
ratio of marginal ray heights
3
Ast
1
0
-1
-2
1
hj
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
4
0
3
0
2
0
CH
L
h1
CHL
1
0
0
-10
-20
1
2
3
4
5
6
7
8
1
0
9
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
5
0
4
0
3
0
InzWi
incidence
angle
2
0
1
0
0
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
6
Sensitivity of a System
 Quantitative measure for relaxation
Aj   j 
with normalization
k
A
j 1
j
Fj
F

h j  Fj
h1  F
1
 Non-relaxed surfaces:
1. Large incidence angles
2. Large ray bending
3. Large surface contributions of aberrations
4. Significant occurence of higher aberration orders
5. Large sensitivity for centering
 Internal relaxation can not be easily recognized in the total performance
 Large sensitivities can be avoided by incorporating surface contribution of aberrations
into merit function during optimization
7
Sensitivity of a System
Representation of wave
Seidel coefficients [l]
Double Gauss 1.4/50
surfaces
Ref: H.Zügge
8
Microscopic Objective Lens
microscope objective lens
 Incidence angles for chief and
marginal ray
marginal ray
 Aperture dominant system
 Primary problem is to correct
spherical aberration
chief ray
incidence angle
60
40
20
0
20
40
60
0
5
10
15
20
25
9
Photographic lens
Photographic lens
 Incidence angles for chief and
marginal ray
 Field dominant system
 Primary goal is to control and correct
field related aberrations:
coma, astigmatism, field curvature,
lateral color
marginal
ray
chief
ray
incidence angle
60
40
20
0
20
40
60
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
System Structure
|wj|
0.7
 Distribution of refractive power
good: small W
W
1
N
N
 w2j
j 1
0.6
0.5
wj  
n' j  n j
1 m

yj
nN ' u ' N
power : W = 0.273
j
0.4
 Symmetry content
good: large S
1
S
N
N
s
j 1
2 0.3
j
0.2
nj ij
1
sj 

1  m n  istop  n N ' u ' N
 u' j u j 

 
 n' n 
j 
 j
0.1
 General trend :
Cost of small W and large S : - long systems
- many lenses
0
5
10
15
20
25
30
35
j
|sj|
0.9
symmetry : S = 0.191
0.8
 Advantage of wj, sj-diagram :
Identification of strange surfaces
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30
35
j
System Structure
|wj|
|sj|
2
1
S = 0.147
W = 0.912
0.8
1.5
 Example:
optimizing W and S with one additional lens
 Starting system:
0.6
1
0.4
0.5
0.2
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
|wj|
1 2
|sj|
2
3 4
5
7
8 9
1
10 11
S = 0.147
W = 0.912
0.8
6
1.5
|wj|
|sj|
2
0.6
1
S = 0.182
W = 0.586
1
0.8
0.4
1.5
0.6
0.5
1
0.2
0.4
0.5
 Final design
0
1
2
3
4
5
6
7
0
0.2
8
9
10 11 12 13
0
1
2
1 2
3
4
5
6
7
8
9
10 11
12 13
1
2
3
|wj|
4
6
7
6
79
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
|sj|
3 4
5
12 13
5
8
1 2
2
3 4
5
7
W = 0.586
10 11
8 9
6
1.5
10
111
S = 0.182
0.8
0.6
1
0.4
0.5
0.2
0
1
2
3
4
5
6
7
8
9
10 11 12 13
0
1
2
1 2
3 4
12 13
3
4
5
6
7
8
9
10 11
12 13
Relaxed System
 Example: achromate with cemented/splitted setup
 Equivalent performance
 Inner surfaces of splitted version more sensitive
a) Cemented achromate f = 100 mm , NA = 0.1
10
5
0
1
2
3
-5
Seidel coefficient
spherical aberration
spot enlargement for
0.2° surface tilt
-10
-15
b) Splitted achromate f = 100 mm , NA = 0.1
surface
index
10
5
0
1
-5
-10
-15
Ref: H. Zügge
2
3
4
surface
index
Sine Condition
ny sin U  n' y ' sin U '
 Lagrange invariante for paraxial angles U, U‘
 sin-condition:
extension for finite aperture angle u
ny sin u  n' y ' sin u '
nU n sin u
m

n'U ' n' sin u'




Corresponds to energy conservation in the system
Constant magnification for alle aperture zones
Pupil shape for finite aperture is a sphere
Definition of violation of the sine condition:
OSC (offense against sine condition)
 OSC = 0 means correction of sagittal coma (aplanatic system)
y
y
y'
U
U'
z
y'
n
n'
Optical Sine Condition
nu
n sin U

nu  n sin U 

Condition for finite angles
m

Condition for object at infinity
f

Condition for afocal system
H 1 h1

H k hk

h
h

u
sinU 
y'
In the formulation
ms 
y'p
y s
n sin U

y n  sin U 
U’
x'p
xp
y
the sagittal magnification
is used
x'
yp
x
U
y
U
U’
y's
z'
15
Abbe Sine Condition
 If for example a small field area and a widespread ray bundle is considered, a perfect
imaging is possible
 


dL  n' s 'dr '  ns  dr
 
 
n  s  dr  n's 'dr '
n  dr  cos q  n'dr ' cos q '
n  cos q  n'  cos q '
 The eikonal with the expression
can be written for dL=0 as
 In the special case of an angle 90°we get with cos(q)=sin(u) the Abbe sine condition
m
n sin u
n' sin u '
with the lateral magnification

dr '
m 
dr
Q
q
dr
P
s
s'
q
u
u'
Q'
dr'
P'
16
Derivation of the Sine Condition
surface
P
Q
i
marginal
ray
i'
sagittal
image
plane
ideal
image
plane
M'o
P'o
y
u
w
Po
C
u'
S
w
y's
s
R
chief ray
M's
s'M
P'
s'
 From geometry
sin u sin(  i)

R
Rs
 Refraction
n sin i  n' sin i'
 Division and
substitution
s 'M  R
sin i' sin u
n sin u


 
Rs
sin i sin u '
n' sin u '
y'
s 'M  R
 s
Rs
y

sin u sin i '

R
s 'M  R
yn sin u  y ' n' sin u '

y 's
y

s 'M  R R  s
M 'o M 's PPo

CM 'o
PoC
Transfer of Energy in Optical Systems
 Conservation of energy
d 2 P  d 2 P'
 Invariant local differential flux
d 2 P L sin u  cos u dA du d
 Assumption: no absorption
T1
 Delivers the sine condition
n y sin u  n' y'  sin u'
y
y'
F
F'
n'
n
dA
u
u'
s
s'
EnP
ExP
dA'
Vectorial Sine Condition
 General vectorial sine condition:
spatial frequencies / direction cosines are linear related from entrance to exit pupil
s xi  mx  s xo  c x
s yi  m y  s yo  c y
image
plane
object
plane
so
q0
yo
sy0
syi
si
qi
zi
zo
system
 Generalization can be applied for anamorphic systems
yi
Pupil Sphere
 Sine condition fulfilled: linear scaling from entrance to exit pupil
 Pupil surface must be sperical
 The pupil height scales with
the sine of the angle
entrance pupil
sphere
object
image
exit pupil
sphere
yo
y'
u
hEnP=REnP
sin(u)
hExP=RExP
sin(u')
u'
REnP
RExP
object
yo
spherical pupil
surface
equidistant
h =R sin(u)
angle not
equidistant
Pupil Distortion
 Sine condition fulfilled: linear scaling from entrance to exit pupil
 Offence against the sine condition (OSC):
Exit pupil grid is distorted
Dp 
 Consequences:
xap
f  n  sin u
1
1. Photometric effect causes apodization
2. Wave aberration colud be calculated wrong
3. Spatial filtering on warped grid
exit pupil
xo
optical
system
object
sphere
distorted entrance
pupil surface
sx
u
xp
distorted
exit
pupil
grid
xp
Pupil Distortion
 Afocal system:
n  x p  w  n'x' p w'
Lagrange invariant
(classical Lagrange invariant for pupil imaging)
w
m
 Magnification
w'
m
 Sine condition
n' x ' p
n  xp
 Fulfillment of the sine condition: linear scaling of entrance ti exit pupil
xp
entrance
pupil
optical
system
exit
pupil
x'p
w
field
angle
w'
x'p
xp
z
Sine Condition in Microscopic Objective Lens
 Typical high-NA system
 Virtual pupil located inside
pupil
object
plane
 Typical grid distortion
exit
pupil
rear
stop
chief
ray
1
x 10
-3
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
OSC and Apodization
 Photometric effect of pupil distortion:
illumination changes at pupil boundary
intensity
1.8
1.6
-0.05 - barrel
 Effect induces apodization
1.4
 Sign of distortion determines the effect:
1.2
no
distortion
1
outer zone of pupil brighter / darker
 Additional effect: absolute diameter of
pupil changes
0.8
+0.05 - pincushion
0.6
0.4
0.2
0
-50 mm
-20 mm
rp
0
10
focused
20
+20 mm
30
40
+50 mm
50
Pupil Aberrations
 Spherical aberration of the chief ray / pupil imaging
 Exit pupil location depends on the field height
yobject
chief rays
sP
pupil position
pupil
location
Pupil Aberration



Interlinked imaging of field and pupil
Distortion of object imaging corresponds to spherical aberration of the pupil
imaging
tan w'
Corrected spherical pupil aberration: tangent condition
tan w
optical system
O’
O
object
image
exit pupil
stop and
entrance pupil
Object imaging
Pupil imaging
Blue rays
Marginal rays
Chief rays
Red rays
Chief rays
Marginal rays
 const.
Pupil Aberration
 Eyepiece with pupil aberration
eyepiece
instrument
pupil
 Illumination for decentered pupil :
dark zones due to vignetting
lens and
pupil of
the eye
retina
caustic of the pupil
image enlarged
Sine Condition and Coma
 Linear coma

Wc ( x p , y p , y' )   cm y'y p x 2p  y 2p

m
m
 Transverse aberrations
y'c  


R  Wc
R

   y'  cm x 2p  (2m  1) y 2p  x 2p  y 2p
n' y p
n'
m

m 1
 Sagittal coma
R
y 'c , sag    y '  cm a 2 m
n'
m
R

 Wc ,max (0, a, y ' )
n' a
pupil
surface R
n
n'
tangential
coma
sagittal rays
sagittal
coma
upper coma ray
S
S
chief ray
T
C
 Sine condition fulfilled:
linear sagittal coma
vanishes
 If in addition spherical
aberration is corrected
(aplanatic):
also tangential coma vanishes
center of
curvature
auxiliary axis
upper coma ray
astigmatic
difference between
coma rays
lower coma ray
chief ray
sagittal
rays
chief
ray
lower
coma ray
auxiliary axis
S
spot
T
Skew Spherical aberration
 Decomposition of coma:
1. part symmetrical around
chief ray: skew spherical
aberration
 yskewsph 
y'
p
chief
ray
upper
coma ray
sagittal image
point
S
T
tangential
image point
 yupcom   ylowcom
2
2. asymmetrical part:
tangential coma
 ytangcoma 
 yupcom   ylowcom
2
lower
coma ray
ideal image
location
exit
pupil
y'
p
 Skew spherical aberration:
- higher order aberration
- caustic symmetric around
chief ray
upper
coma ray
common
intersection
point
chief
ray
lower
coma ray
exit
pupil
ideal image
plane
Aplanatic and Perfect Imaging
Dspot mm]
 Perfect imaging on axis due to conic section
- not aplanatic:
linear grows of coma with field size
100
50
0
 Aplanatic:
- Perfect stigmatic imaging
on axis, spherical corrected
- linear coma vanishes:
good correction off-axis
but near to axis
- quadratic grows of spot size
due to astigmatism
- aplanatic and perfect
marginal ray quite different
ideal
lens
0
1
2
1
2
w in °
Dspot mm]
ideal
rays
real rays
real
lens
100
sin uideal' = 0.707
sin ureal' = 0.894
50
0
0
w in °
30
Spherical Corrected Surface
 Seidel contribution of spherical aberration
with
 1 1
hj
Qj  nj    
j 
R s 
h1
j 
 j
 1
1 

Sj  Q

 n' s ' n s 
j j 
 j j
4
j
2
j
 1 1
h 
S j   j   n 2j    
R s 
j 
 h1 
 j
4
 Result
 Vanishing contribution:
1. first bracket: vertex ray
hj  0
2. second bracket: concentric
Rj  s j
3. bracket: aplanatic surface
n' j s ' j  n j s j
 Discussion with the Delano formula
i'u
2
i

sin
n
n U sin u1
i 'i
2
s'SPH  sSPH  1 1
 j  h  sin

n'k U 'k sin u 'k j n' j
2 U ' j sin u ' j
2. concentric corresponds to i' = i
3. aplanatic condition corresponds to i' = u
2
 1
1 


 n' s ' n s 
j j 
 j j
Aplanatic Surfaces with Vanishing Spherical Aberration
 Aplanatic surfaces: zero spherical aberration:
3. Aplanatic
s'  s
u  u'
und
ns  n' s'
 Condition for aplanatic
surface:
ns
n' s'
ss'
r


n  n' n  n' s  s'
 Virtual image location
oblate ellipsoid
+ power series
hyperboloid
+ power series
oblate ellipsoid
+ power series
sphere
2. concentric
s'  s  0
sphere
1. Ray through vertex
prolate ellipsoid
+ power series
s'
0.1
aplanatic
0
vertex
concentric
-0.1
-0.2
 Applications:
1. Microscopic objective lens
2. Interferometer objective lens
-0.3
-0.4
-0.5
0
50
100
150
200
250
300
S
Aplanatic Lenses
 Aplanatic lenses
 Combination of one concentric and
one aplanatic surface:
zero contribution of the whole lens to
spherical aberration
 Not useful:
1. aplanatic-aplanatic
2. concentric-concentric
bended plane parallel plate,
nearly vanishing effect on rays
A-A :
parallel offset
A-C :
convergence enhanced
C-A :
convergence reduced
C-C :
no effect
33
General Aplanatic Surface
 General approach
of Fermat principle:
aplanatic surface
r
oval
surface
n  r 2  ( z  s)2  n' r 2  (s' z )2  s' n'ns
 Cartesian oval,
4th order
P'
P
S
z
s
 Special case OPD = 0:
s' n'  ns
Solution is spherical aplanatic surface
n  r 2  ( z  s) 2  n' r 2  (n / n' s  z ) 2  0
n / n'2  r 2  z 2  s 2  2 zs  r 2  z 2  n / n'2 s 2  2 zsn / n'
2
z 2 n 2 / n'2 1  2 zsn / n'  n / n' r 2 n 2 / n'2 1  0
2
sn 

2
z


r
 n  n' 
 sn 

 n  n' 
2
s'
34
Isoplanatism
 General definition of isoplanatism:
- Invariance of performance for small lateral shifts of the field position
- spherical aberration not necessarily corrected
 Usual simple case: near to axis
 Consequences:
- vanishing linear growing coma
- caustic symmetrical arounf chief ray
Isoplanatism Condition of Staeble-Lihotzky
 Sagittal coma aberration:
from the geometry of the figure and Lagrange invariant

S ' s p '
y '  n sin u
y 's   

 m
m  n' sin u ' S ' s p ' ssph '

 Condition of Staeble-Lihotzky
s' s p ' 
 Problems:
- no quantitative measure
- only tangential rays are considered
- integral criterion
S ' s p '  n sin u


 m
m  n' sin u '

P'
t
Q't
marginal ray
ys'
chief ray
Q's
projection of
sagittal coma ray
optical axis
ys'
y'
u'
Q'
s'
P'
s'
S'
sp'
exit pupil
last
surface
real
tangential
image plane
ideal
gaussian
image plane
36
Isoplanatism from Wave Aberrations
W
 Lateral shift of object point
dW  dy  n  sin u
dy
 Change in image
dy' '  dy'
R' ds'
R' ds'
 m  dy 
R'
R'
u
chief ray
 Isoplanatism
R' ds'
n' sin u '
R'
1 n  sin u R' ds'


1  0
m n' sin u '
R'
n  sin u   m 
dy'
P
s'
R
 Change of wave aberration must be equal
dW '  dy' 'n' sin u '   m  dy 
dy''
R'ds'
n' sin u '
R'
R'
37
Isoplanatism
 Berek's condition of proportionality
S ' s p '
m

 Berek's coincidence condition
s'

S ' s'
n sin u
m
n' sin u '
S ' s p ' n sin u
m n' sin u '
 s p '  const.
 Isoplanatism in case of defocussing:
can only be fulfilled in one plane or for telecentricity
 s' p
 sp


s ' p 
s p 




d  1  cos u '  
 1
 1
  1  cos u   



 s' p  z '  s' p  z ' 
 s p  z  s p  z 
Piecewise Isoplanatism
 Invariance of PSF: to be defined
MO not plane 40x0.85
isoplanatic patch size in
mm
Strehl 1%
Psf
correlation
0.5%
Strehl 1%
Psf
correlation
0.5%
on axis
70
72
81
100
half field
3.8
3.8
27
3.1
field zone
2.5
2.5
29
39
full field
45
3.8
117
62
System
 Possible options:
1. relative change of Strehl
2. correlation of PSF's
 Examples for microscopic lenses
with and without flattening
correction
 In medium field size:
small isoplanatic patches
 On axis:
large isoplanatic area
MO plane 100x1.25
isoplanatic patch size in
mm
1
0.9
 Criteria not useful at the
edge for low performance
Plane MO 100x1.25
0.8
0.7
Strehl
correlation
0.6
0.5
0.4
correlation
Strehl
0.3
no plane MO
40x0.85
0.2
0.1
0
normalized
field position
0
0.2
0.4
0.6
0.8
1
Offence Against the Sine Condition
 Conradys OSC (offense against sine condition):
- measurement of deviation of sagittal coma
- quantitative validation of the sine condition
yp
CR
P'1
Q'1
y's
y ' y '
n sin u S ' s p '
 OSC  t s  1

yt '
m  n' sin u ' s ' s p '
y'
Q's
y't
y's
ExP
z
P'
Q'
ideal
 Only sagittal coma considered
in case of OSC=0 the Staeble-Lihotzkycondition is automatically fulfilled
Wcoma ( y, rp ,0)  rp  yt   OSC
n sin u 

 yt '  3 y   m 

n' sin u ' 

P't
Q't
marginal ray
y'
t
chief ray
Q's
sagittal
coma ray
 OSC allows for the definition of surface
contribution
y
o
y'
s
optical axis
Q'
s'
P'
s'
S'
(Qk  Q'k )  nk ik(CR)
sin w1
 OSC 

sin u1 k
h' k n ' k u ' k
real
ideal
tangential
gaussian
image plane image plane
exit pupil
OSC
 Coma and isoplanatism are strongly connected
exit
pupil
yp
tangential
coma rays
y'
coma
spot
xp
x'
sagittal
coma rays
chief ray
image
plane
 Vectorial OSC:
linear scaling of spatial frequencies:
perturbation of the linearity

1 
v 'apl 
 vapl
mp
  
1
v  v '  v 'apl     x , yW
l
41
General Invariant of Welford
 Rotation around axis for small angle
calculation of change in wave aberration
 Welfords condition
  
  
dW  n'd ' p', D', e '  n  d   p, D, e
 All other conditions can be obtained as special cases:
1. sine condition
2. off axis isoplanatism
3. Herrschel condition
4. Smith cos-invariant
Surface
p'
axis of
rotation
d'
p
D'
ray
s'
q'
d
D
s
q
n
n'
Cos-Condition of Smith
 From Eikonal theory:
General condition of Smith:
Invariance of the scalar product
 
 
n  d s  en'd s 'e '
n  dr  cos q n'dr ' cos q '
Q'
Q
dr
P
dr'
q
 Special case:
P on axis, q = 90°:
Abbe sine condition, invariant transverse magnification
 Special case:
P on axis, q = 0°:
Herschel condition, invariant axial magnification
P'
q'
Herschel Condition
 Herschel condition:
Invariance of the depth magnification
 z  n  sin 2
u
u'
  z 'n' sin 2
2
2
 In principle not compatible with the sine condition
 Therefore a perfect imaging of a volume is impossible
u
P
dz
Q
u'
P'
dz' Q'
44
Overview Aplanatism-Isoplanatism
 Overview on conditions for aberrations and aplanatism-isoplanatism
Nr
Sine
cond.
Isoplanat
cond.
1
#
#
2a
#

2b
#

3a

3b

Isoplanatism
condition
Spherical
aberration
Sagittal
coma
Tangential
coma
Imaging system
#
#
#
general
OSC=0, Conrady
#
0
#
isoplanatic-I
Staeble-Lihotzky /
Berek
#
0
0
isoplanatic-II

0
0

0 (skew)
0
Isoplanatism
Conrady
OSC
Tangential coma
Sagittal coma
0
Isoplanatism
StaebleLihotzky
sine condition sine condition
off-axis
axial
Aplanatism
Aplanatism
0
0
0
0
Spherical aberration
Skew Spherical aberration
0
0
0
0
axial aplanatic
0
off-axis aplanatic
45
Overview
 Overview on invariants and conditions
general invariance
(Welford)
only
translation
cos-law
(Smith)
change
object
position dy
off axis isoplanatism
special on
axis y'=0
axis isoplanatism
(Staeble-Lihotzki /
Berek)
only
transverse
translation
special for
sph=0
sine condition
(Abbe)
translation
along z
off axis z-invariance
(Herrschel
special on
axis x'=y'=0
on axis z-invariance
(classical Herrschel)
Phase Space: 90°-Rotation
 Transition pupil-image plane: 90° rotation in phase space
 Planes Fourier inverse
 Marginal ray: space coordinate x ---> angle q'
 Chief ray: angle q ---> space coordinate x'
image
location
Fourier plane
pupil
marginal ray
x'
x
q'
q
chief
ray
f
Nearfield - Farfield
 2f-setup:
Fourier-conjugated
planes
 Angle and spatial
frequency are equivalent
q v l
x'
x
 Angle- and spatial coordinate are interchanged:
x ---> q'
q ---> x'
 Corresponds to nearfield <---> farfield
 Relationship:
q'
x
f
, x'  f  q
pupil
Fourier domain
angle/frequency
object space
spatial domain
coordinate x
u
u
f
f
Helmholtz-Lagrange Invariant
 Product of field size y and numercial aperture is invariant in a paraxial system
L  n  y  u  n' y'u'
 The invariant L describes to the phase space volume (area)
 The invariance corresponds to
1. Energy conservation
2. Liouville theorem
3. Constant transfer of information
marginal ray
object
y'
u
y
u'
chief ray
system
and stop
image
Helmholtz-Lagrange Invariant
arbitrary z
pupil
image
y
yp
y'
 Basic formulation of the Lagrange
invariant:
Uses image heigth,
only valid in field planes
O'
Q
chief
ray
y'CR
yMR
marginal
ray
 General expression:
1. Triangle SPB
S
y
w' CR
s'ExP
2. Triangle ABO'
y'CR  w's's'ExP 
3. Triangle SQA
u' 
4. Gives
L  n'u ' y'CR  n'
B
yCR
A
P
s'Exp
s'
yMR
s'
yMR
w's' s'ExP   n' yMR w'u ' w' s'ExP 
s'
5. Final result for arbitrary z:
L  n'w' yMR ( z )  u' yCR ( z )
z
50
Photometry in Phase Space
p
 Radiation transport in optical systems
 Phase space area changes its shape
2sinu'
 Finite chief ray angle:
parallelogram geometry
y
sinw'
2sinu
y'
y
lens
2y
stop
2y'
y'
U
U'
w'
y
s
s'
Aberrations in Phase Space
pupil
u
u
• Angle diviations due to aberrations in the pupil
x
focus
u
• Increased spatial extention in the focus region
x
x
Ray Caustic
• Special case of vanishing determinante of Jacobian matrix: ray caustic
• Singulare solution of the wave equation
• Two ray directions in one point
• Special characterization with Morse- and
Maslov index
u
xc
x
caustic
Uncertainty Relation in Optics
1. Slit diffraction
q
q
Diffraction angle inverse to slit
D
D
width D
q
l
D
2. Gaussian beam
x
Constant product of waist size wo
and divergence angle qo
w0q 0 
l

qo
wo
z
Light Sources in Phase Space
 Angle u is limited
u
 Typical shapes:
Ray : point (delta function)
Coherent plane wave: horizonthal line
Extended source : area
Isotropic point source: vertical line
Gaussian beam: elliptical area with
minimal size
point
source
LED
gaussian
beam
spherical
wave
plane wave
(laser)
ray
x
u
quasi continuum
 Range of small etendues: modes, discrete
structure
 Range of large etendues: quasi continuum
discrete
mode
points
x