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Imaging and Aberration Theory
Lecture 4: Aberrations Expansions
2012-11-09
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. Fermat principle and eikonals Fermat principle, method of 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 30.11. Distortion and coma phenomenology, relation to sine condition, aplanatic systems, effect of stop position,
various topics, correction options
8 07.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 14.12. Chromatical aberrations,
Sine condition, isoplanatism
Dispersion, axial chromatical aberration, transverse chromatical aberration,
sine condition, Herschel condition, isoplanatism, relation to coma and shift,
Pupil aberrations, invariants, relation to Fourier optics and phase space
10 21.12. Surface contributions sensitivity in 3rd order, structure of a system, superposition and induced aberrations,
analysis of optical systems, lens contributions
11 04.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF
and OTF
12 11.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing,
influence of normalization, recalculation for offset, ellipticity, measurement
13 18.01. Miscellaneous Aldi theorem, telecentric case, afocal case, aberration balancing, Delano
diagram, Scheimpflug imaging, Fresnel lenses
14 25.01. Vectorial aberrations Introduction, special cases, actual research, anamorphotic, partial symmetric
15 01.02. Statistical aberrations atmospheric turbulence, calculation, spatial frequency contents
16 08.02. Physical model of imaging coherent and incoherent imaging, Fourier description, partial coherence
Repetition...
Single refracting surface
Adaptation on optical system terms
Primary monochromatic aberration surface contributions
Extension on several surfaces
Pupil shift formula
Lens contributions
Examples
3
Contents
Wave Aberration
Exact relation between wave aberration and ray deviation
General expression from geometry
a describes the lateral aberration
Substitution of angle by scalar
product
Exact relation is quadratic in R
Approximation for large R
with
4
real ray
C
R
s
sr
reference
sphere
a
A
W
wave
front
z
yp
cos1
'1
'
22
0
R
asan
ss
assssnWW
r
rr
2
2222
41
2'
R
asa
R
asanWW
'2
''
22
zR
yx
yR
yx
R
xW
pp
pp
'
'
'
z
y
x
a
Fermat principle: the light takes the optical
path of the smallest time of flight
The derivative on variation parameters vanishes
Method of stationary phase:
The light propagates along rays, which
are locally perpendicular to the wave
5
Repetition
2
1
0),,(
P
P
dszyxnL
wave front
contributions
of stationary
phase in P
point of
calculation P
P1
P2
P3
P4
P5
A
B
initial path
modified path
dh
variation
0
h
L
A description of the light field by
rays or waves is equivalent
The eikonal function L describes the optical path length
A ray is fully described in one
plane by 4 parameters x, y, sx, sy
Due to the Fermat principle, there are only 4
independent parameters for one ray in a system,
they are coupled by the eikonal function
6
Repetition
object
plane
image
plane
z0 z1
y1
y0
phase
L = const
L = const
srays
),,,(',',',' yxyx ssyxLssyx
optical
system
s
angle
(direction)
point
r
object
spaceimage
space
s'
angle
(direction)
point r'
Considering spatial coordinates or directions, there are four different eikonals possible
A change in the object space
fully determines athe change
in the image space by the
eikonal characteristic function
The integration of the 4 derivative conditions fully describes the remaining parameters
Example angle eikonal:
Perfect imaging is only possible for infinitesimal field sizes:
ellipse, special gradient lenses, aplanatic surface,...
7
Repetition
optical
system
dr
ss'
dr'
P
B
Q
P'
B'
Q'
''')',( srdnsrdnrrdLP
xns
L
x
A
''
'xn
s
L
x
A
yn
s
L
y
A
''
'yn
s
L
y
A
Example eikonal
of refracting surface:
optical path length
Second order expansion: relation of paraxial optics
Fourth order expansion: aberrations
Ray-wave conversion: always possible, if no ray crossing occurs (outside caustics)
Approximation of geometrical optics:
separation of amplitude Eo and phase L in wave equation:
For small wavelength l, small gradients of phase and amplitude:
geometrical eikonal equation
8
Repetition
P
surface
Rrn n' P'
z
s'(-)s
uu'
aa'
Q
xx'
2222 )'()'(')()()( xrsznxrsznrL
02 0000
22
0
2
0 ELELEikLnEk
)(22rnL
Refracting Surface I: Basic Eikonal Approach
P, P' points on ray
A, A' arbitrary points on axis
s,s' direction unit vectors of the rays
relationship
Real surface equation
contains system parameters R, b
Angle eikonal, gives the optical path length change in P', if the point P is varied
In coordinate
representation
...)1(
82 3
22222
bR
yx
R
yxz
P
surface
R
Q
A
P'
A'
M
y
z
n n'
ss'
r
a
O origin
a'z
'''')',( sdarnsdarnssdLA
zyxzyxA dsazdsydsxndsazdsydsxndL '''''
2222 ''1',1 yxzyxz ssssss
Refracting Surface II: Eikonal
Integration of the differential to get the total optical path length
Result of 4th order Taylor approximation
Elimination of intermediate variables x, y of the intersection point with the help of the
law of refraction
222
2222222
3
222
2222222
3
2222
2222
''8
'
4
''
8
1'
848
1
''2
'
2'''
22
''
yx
yx
yx
yx
yxyx
yxyx
A
ssa
R
ssyxyx
R
bn
ssa
R
ssyxyx
R
bn
ssa
R
yxsysxn
ssa
R
yxsysxn
ananL
zyxzyxA sazsysxnsazsysxnL '''''
Refracting Surface III: Law of Refraction
Implicite surface equation
Normal unit vector in the intersection point
Law of refraction
Formulation in components
Solving for x, y in approximation of 4th order
3
22222
8
)1(
2),,(
R
yxb
R
yxzzyxF
1,
2
)1(1,
2
)1(1
),,(
2
22
2
22
zyx eR
yxb
R
ye
R
yxb
R
xe
zyxFe
0)''( esnsn
0
2
)1(1)''(
2
)1(1)''(
02
)1(1)''()''(
02
)1(1)''()''(
2
22
2
22
2
22
2
22
R
yxb
R
xsnns
R
yxb
R
ysnns
R
yxb
R
xsnnssnns
R
yxb
R
ysnnssnns
yyxx
zzxx
zzyy
'
'',
'
''
nn
snnsRy
nn
snnsRx
yyxx
Refracting Surface IV: Eikonal
Resulting Eikonal function of sx, sy, s‘x, s‘y only
Further re-arrangements for better practical usage:
1. introduction of the pupil coordinates xp, yp instead of the image sides directions s‘x, s‘y
2. switch to the image space ray parameters
3. optional switch to circular coordinates
4. according to Seidel substitution of the system parameter by the paraxial properties of the
system and the marginal ray
5. Further approximation in 4th order to get a perturbation representation of the paraxial
imaging. This allows a decoupling of the surface contributions
22
3
222222
222222
2
222222
)4()2()0(
'''')'(8
)1(''''
8
1
''''''')'(4
''2
''
2''''
)'(2''
yyxxyxyx
yxyxyyxx
yxyxyyxx
AAAA
snnssnnsnn
Rbssanssna
ssnssnsnnssnnsnn
R
ssan
ssna
snnssnnsnn
Ranna
LLLL
Refracting Surface V: Paraxial Optics
2nd order: paraxial optics
Eliminating sx, sy
Imaging condition: x‘, y‘ independent from ray directions sx, sy
1. imaging equation
2. Magnification
''
'''
''
1'
''
'''
''
1'
'
''
'
1
'
''
'
1
)2(
)2(
)2(
)2(
nn
nRs
nn
Rnas
s
L
ny
nn
nRs
nn
Rnas
s
L
nx
nn
Rns
nn
nRas
s
L
ny
nn
Rns
nn
nRas
s
L
nx
yy
y
A
xx
x
A
yy
y
A
xx
x
A
R
nn
a
n
a
n '
'
'
an
na
x
xm
an
naxx
'
'',
'
''
R
nn
a
n
a
ns
n
aa
R
nn
a
n
n
ayy
R
nn
a
n
a
ns
n
aa
R
nn
a
n
n
axx
y
x
'
'
'
'
''
'
'
'
''
'
'
'
'
''
'
'
'
''
Refracting Surface VI: Pupil Coordinates
Change to pupil coordinates
for optical systems
Relations
Approximated to 2nd order
object
plane
entrance
pupil
system
surface
s
p
y
y yp
yparcsin(sz)
arcsin(sy)
z
p
y
z
yp
y
z
p
x
z
xp
x
z
p
y
z
yp
y
z
p
x
z
xp
x
sp
yys
s
s
p
yyu
sp
xxs
s
s
p
xxu
sp
yys
s
s
p
yyu
sp
xxs
s
s
p
xxu
''
''',
'
'
'
'''tan
''
''',
'
'
'
'''tan
,tan
,tan
'
''',
'
'''
,
p
yys
p
xxs
p
yys
p
xxs
p
y
p
x
p
y
p
x
Refracting Surface VII: Eikonal
Eikonal of 4th order
Coefficients
1. Spherical aberration
2. Astigmatism
3. Field curvature
4. Distortion
5. Coma
pppppp
ppppA
yyxxyxp
pssCyyxxyx
p
pssDyyxxyx
p
pssP
yyxxp
pssAyx
p
sSyx
p
psKL
22
4
322
4
322
4
22
2
4
22222
4
4222
4
4)4(
2
)(
2
)(
4
)(
2
)(
88
)(
''
11)'(
''
11)(
''
11)'(
''
11)(
''
11)'(
''
11)'(
''
11)'(
'
1
'
1''
11)'(
3
2
3
3
2
3
2
3
2
2
2
23
snnsQQ
R
bnnC
snnsQQQ
snnsQ
R
bnnD
snnsQQQ
snnsQQ
R
bnnP
snnsQ
R
bnnA
snnsQ
R
bnnS
sRssn
sRsns
R
bnnK
p
pp
ppp
pp
pp
p
pp
Refracting Surface VIII: Aberrations
Corresponding differential equations of the angle eikonal determine the changes in the
spatial coordinates
Calculation of the transverse aberrations:
separation of the paraxial and the perturbation part
Used paraxial abbreviations:
1. pupil imaging magnification
2. Abbe invariants
3. usual special Picht-operator: difference before / after the refraction surface
''
1'',
''
1''
y
A
x
A
s
L
nyymy
s
L
nxxmx
''
1',
''
1'
)4()4(
y
A
x
A
s
L
nymy
s
L
nxmx
ps
ps
y
ym
p
p
p
'
''
p
psR
nQsR
nQ11
,11
...,,'
','
s
n
s
n
s
nnnn
Refracting Surface IX: Transverse Aberrations
Transverse aberrations
Due to the derivative, only 5 terms remain for the primary aberrations,
The transverse aberrations are of 3rd order in the coordinates
For the special cases of
1. image in infinity
2. exit pupil in infinity
there are particular sets of formulas
Possible variables:
1. x‘,y‘, x‘p,y‘p, s‘, s‘p, p‘
2. A,A‘, i, i‘
'sps'
pp
pp
pp
p
pp
pppp
pp
p
pp
pp
pp
p
pp
pppp
pp
p
yyxxpn
ssyCyx
pn
ssyDyx
pn
ssyA
yyxxpn
ssyCyx
pn
ssyPyx
pn
sySy
yyxxpn
ssxCyx
pn
ssxDyx
pn
ssxA
yyxxpn
ssxCyx
pn
ssxPyx
pn
sxSx
''''''
'''''
''2
'''''
''2
'''
''''''
'''''
''2
'''''
''2
'''
''''''
'''''
''2
'''''
''2
'''
''''''
'''''
''2
'''''
''2
'''
3
22
22
3
3
22
3
3
3
3
22
3
22
22
3
4
3
22
22
3
3
22
3
3
3
3
22
3
22
22
3
4
Generalization on Optical Systems
Paraxial optics: small field and aperture angles,
Aberrations occur for larger angle values
Two-dimensional Taylor expansion shows field
and aperture dependence
Expansion for one meridional field point y
Pupil: cartesian or polar grid in xp / yp
field
point
optical
axis
axis
point
entrance
pupil
coma rays
outer rays of
aperture cone
chief
ray
object
height y
xp
O
xp
yp
r
yp
ray
object
plane
meridional
plane
sagittal
plane
x, y object coordinates, especially object height
x', y' image coordinates, especially image height
xp,yp coordinates of entrance pupil
x'p, y'p coordinates of exit pupil
s object distance form 1st surface
s' image distance form last surface
p entrance pupil distance from 1st surface
p' exit pupil distance from last surface
x' sagittal transverse aberration
y' meridional transverse aberration
19
Notations for an Optical System
x
y
y
s
p
s'
p'
xP
yp
y'
x'
y'
x'P
y'p
object
plane
entrance
pupil
exit
pupil
image
plane
zx'
y'
system
surfaces
xP
yp
x'P
y'p
Sequence of refracting surfaces:
the optical path length contributions are additive
The contributions of every surface are summed up
Every surface contribution is imaged and magnified by the following surfaces
The successive surfaces fulfill sj+1 = s'j,...
The height ratios help to express the magnifications
Individual magnifications dependencies for the various aberration types
20
Sequence of Surfaces
1
1
21
1
1
21 '...'','...''N
k
kNkkN
N
k
kNkkN xmmmxxymmmyy
j
jh
h
1
pj
pj
p
h
h
1
P1 P2 P3P4 P5
Q1 Q2
Q3 Q4
Q5
Transverse Aberrations of Seidel
Decomposition of transverse aberrations
k
k
p
k
k
pppp
k
k
ppp
k
k
pppppp
k
k
ppp
Dpn
ssyxx
Ppn
ssyxxA
pn
ssyyxxx
Cpn
ssyxxyyxxxS
pn
syxxx
3
322
3
2222
3
22
3
322
3
422
''2
'''''
''2
'''''
''
'''''''
''2
''''''''''2
''2
'''''
k
k
p
k
k
pppp
k
k
ppp
k
k
pppppp
k
k
ppp
Dpn
ssyxy
Ppn
ssyxyA
pn
ssyyxxy
Cpn
ssyxyyyxxyS
pn
syxyy
3
322
3
2222
3
22
3
322
3
422
''2
'''''
''2
'''''
''
'''''''
''2
''''''''''2
''2
'''''
Surface Contributions
Spherical aberration
Coma
Astigmatisms
Field curvature
Distortion
jjjj
jjjsnsn
QS1
''
124
11
124 111
''
1
pjj
pjpj
jjjj
jjjssQ
Qn
snsnQC
2
11
2
124 111
''
1
pjj
pjpj
jjjj
jjjssQ
Qn
snsnQA
jjrpjj
pjpj
jjjj
jjjnnrssQ
Qn
snsnQP
1
'
11111
''
12
11
2
124
11
1
2
11
2
124 111
'
11111
''
1
pjj
pjpj
jjrpjj
pjpj
jjjj
jjjssQ
Qn
nnrssQ
Qn
snsnQD
Rotational Invariants
General case : two coordinates in object plane and pupil
Rotational symmetry: 3 invariants
1. Scalar product of field vector and pupil vector
2. Square of field height
3. Square of pupil height
Therefore:
Only special power
combinations are
physically meaningful
Object
Pupilx
y
P
F
y
x
xp
yp
z
xp
yp
F
P
222 yxFFFu
222
pp yxPPPv
yyxxFPFPw pp )cos( PF
Power Series Expansion of Aberrations
General case of Taylor expansion
Expansion with selection rules:
only powers of the rotational invariants can occur
Simple expansion according
to this scheme
Explicite equation in real coordinates
nmlk
nml
p
k
pklmn yxyxaW,,,
),,( wvuWW
...)(
)()()()()(
)()()(
)(
6
10
5
9
224
8
24
7
33
6
223
5
2222
4
2222
3
222
2
322
1
4
6
3
5
222
4
22
3
22
2
222
1
2
32
22
10
ydyydyxydyydyyd
yxyydyxydyxyydyxyydyxd
ycyycyxycyycyxyycyxc
ybyybyxbaW
ppppp
ppppppppppppp
ppppppppp
ppp
...3
10
2
9
2
8
2
7
3
65
2
4
2
3
2
2
3
1
2
654
2
32
2
1
3210
udwudvuduwdwduwvduvdvwdwvdvd
ucuwcuvcwcwvcvc
ubwbvbaW
Higher Order Aberrations
Relevance of higher order expansion terms
Perfect geometrical imaging possible in the special cases:
1. small aperture
2. small field
(29-6)
field y/w
aperture
DAP/NA2nd order
in NA
4th order
in NA
6th order
in NA....
....
2nd
order
in y
4th
order
in y
6th
order
in y
paraxial
optics
conic mirrors
telescopes
microscopy
high NA
monocentric
systems
photography
wide angle
micro
lithography
Circular Coordinates
Introduction of circular coordinates
according to the geometry
Transverse aberrations
Aberration curves:
Considering a rinf with constant r'p in the pupil,
- eliminating 'p
- delivers curve in image plane for x', y'
22''' ppp yxr
ppp
ppp
ry
rx
'cos'
'sin''
ppp
pp
p
p
pppp
p
pp
p
pp
ppp
pp
p
p
pppp
p
pp
p
pp
p
yxrpn
ssyCyx
pn
ssyDr
pn
ssyA
yxrpn
ssCyxr
pn
ssPr
pn
sSy
yxrpn
ssxCyx
pn
ssxDr
pn
ssxA
yxrpn
ssCyxr
pn
ssPr
pn
sxSx
'cos''sin''''
'''''
''2
''''
''2
'''
'cos''cos'sin''''
'''''cos'
''2
'''cos'
''2
''
'cos''sin''''
'''''
''2
''''
''2
'''
'cos'sin''sin''''
'''''sin'
''2
'''sin'
''2
'''
3
22
22
3
3
2
3
3
22
3
3
22
3
22
3
3
4
3
22
22
3
3
2
3
3
22
3
3
22
3
22
3
3
4
Polynomial Expansion of Aberrations
Representation of 2-dimensional Taylor series vs field y and aperture r
Selection rules: checkerboard filling of the matrix
Constant sum of exponents according to the order
Field y
Spherical
y0 y 1 y 2 y3 y 4 y 5
Distortion
r
0
y
y3
primary
y5
secondary
r
1
r 1
Defocus
Aper-
ture
r
r
2
r2y Coma primary
r 3
r 3 Spherical
primary
r
4
r
5
r 5 Spherical
secondary
DistortionDistortionTilt
Coma Astigmatism
Image
location
Primary
aberrations /
Seidel
Astig./Curvat.
cos
cos
cos2
cos
Secondary
aberrations
cos
r 3y 2 cos
2
Coma
secondary
r4y cos
r2y
3 cos
3
r2y
3 cos
r1 y
4
r1 y
4 cos
2
r 3y 2
r12
yr
12 y
Power Expansion of Aberrations
Wave aberration
Transverse aberration
Longitudinal aberration Aberration Coefficient
Seidel sum
Aperture Field Aperture Field Aperture Field
Spherical aberr. c1 IS 4 0 3 0 2 0
Coma c2 IIS 3 1 2 1 1 1
Astigmatism c3 IIIS 2 2 1 2 0 2
Field curvatures (sagittal)
c4
(Petzval)
IVS 2 2 1 2 0 2
Distortion c5 VS 1 3 0 3 - -
Axial color 1
~b
IC 2 0 1 0 0 0
Lateral color 2
~b
IIC 1 1 0 1 - -
From : H. Zügge
Orders of field and aperture dependencies for different representations of primary
aberrations
29
5th Order Aberrations
No
k Field
power
l, Pupil power
m, Azimuthal
power Term Name
1 0 6 0 W rp060
6 Secondary spherical aberration
2 1 5 1 W y rp151
5' cos Secondary coma
3 2 4 2 W y rp242
2 4 2' cos Secondary astigmatis, wing error
4 3 3 3 W y rp333
3 3 3' cos trefoil error, arrow error
5 2 4 0 W y rp240
2 4' Skew spherical aberration
6 3 3 1 W y rp331
3 3' cos Skew coma
7 4 2 2 W y rp422
4 2 2' cos Skew astigmatism
8 4 2 0 W y rp420
4 2' Secondary field curvature
9 5 1 1 W y rp511
5' cos Secondary distortion
Power Series Expansion of Aberrations
If the stop is moved, the chief ray takes a modified way through the system
The stop shift formulas shows the change of the Seidel coefficients due to this
effect
II sS
IIIII sA
AsS
IIIIIIII sA
AsS
IVIV sS
,
imageplane
ray
y'chief ray 1
chief ray 2
ExP 1ExP 2
p'1
rpmax
z
p'2
IVIIIVV ss
A
AsS
Power Series Expansion of Aberrations
Stop shift formulas excplicite with the help of the
moving parameter
,
h
hhE oldnew
old position
new position
old chief ray
new chief ray
oldh
new
h
II SS
IIIII SESS
IIIIIIIII SESESS 2
IVIV SS
IIIIVIIIVV SESESSESS 3233
There is a large number of different notations for the thrid order representation:
Haferkorn, Welford, Seidel, Berek, Köhler,...
The differences are the choice of the parameter and some of the approximations
The so called reduced representations are of a special form without considering the
field dependence explicite
Example: Zernike expansion
The third order theory is limited on systems with not too high angles of marginal and chief
ray
Mostly the third order describes the leading term
Higher orders than 3 can not be decomposed as simple into the surface contributions:
1. the magnification of the following surfaces acts non-linear and can not be neglected
2. the second order perturbation theory has no decoupling of the contributions
(induced aberrations)
32
Notations
Welford‘s Notation
Abbreviations
bar: chief ray
Seidel aberrations
1. spherical aberration
2. coma
3. astigmatism
4. Petzval curvature
5. distortion
Representation in normalized circular coordinates
'')( ininuhcnA
'')( ininuchnA
n
uhAS I
2
n
uhAAS II
n
uhAS III
2
ncHS IV
12
ncH
A
A
n
uh
A
ASV
123
maxmax
,y
y
r
r
coscoscos),,(32222234
2
1
4
1
2
1
2
1
8
1VIVIIIIIIIII SSSSSSW
Surface Contributions: Example
Seidel aberrations:
representation as sum of
surface contributions possible
Gives information on correction
of a system
Example: photographic lens
1
23 4
5
6 8 9
10
7
Retrofocus F/2.8
Field: w=37°
SI
Spherical Aberration
SII
Coma
-200
0
200
-1000
0
1000
-2000
0
2000
-1000
0
1000
-100
0
150
-400
0
600
-6000
0
6000
SIII
Astigmatism
SIV
Petzval field curvature
SV
Distortion
CI
Axial color
CII
Lateral color
Surface 1 2 3 4 5 6 7 8 9 10 Sum
Graphical supported representation of the Seidel surface contributions of a photographic
lens
35
Seidel Surface Contributions
Microscopic Objective Lens
Incidence angles for chief and
marginal ray
Aperture dominant system
marginal ray
chief ray
incidence angle
0 5 10 15 20 2560
40
20
0
20
40
60
microscope objective lens
Photographic lens
Incidence angles for chief and
marginal ray
Field dominant system
incidence angle
chief
ray
Photographic lens
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1560
40
20
0
20
40
60
marginal
ray
Microscope Objective Lens
Seidel surface contributions
for 100x/0.90
No field flattening group
Lateral color in tube lens corrected
5 10
-0.5
0
0.5
-0.02
0
0.02
-4
-2
0
2
4
-5
0
5
-2
0
2
-0.02
0
0.02
-1
0
1
spherical
coma
astigmatism
curvature
distortion
axial
chromatic
lateral
chromatic
1
518
11
13
sum
Zoom lens
Three moving groups
Zoom Lens
e)
f' = 203 mm
w = 5.64°
F# = 16.6
d)
f' = 160 mm
w = 7.13°
F# = 13.7
c)
f' = 120 mm
w = 9.46°
F# = 10.9
b)
f' = 85 mm
w = 13.24°
F# = 8.5
a)
f' = 72 mm
w = 15.52°
F# = 7.7
group 1 group 2 group 3
Performance Variation over z
Seidel
surface
contrib.
coma distortion axial chromatical lateral chromatical
lens 1
lens 2
lens 3
sum
spherical aberration
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
-0.1
0
0.1
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
-0.2
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2
-0.2
-0.1
0
0.1
0.2
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-5
0
5
1 2 3 4 5
-0.5
0
0.5
1 2 3 4 5
-0.5
0
0.5
1 2 3 4 5
-0.5
0
0.5
1 2 3 4 5
-
0.5
0
0.5
Lens Contributions of Seidel
In 3rd order (Seidel) :
Additive contributions of thin lenses (equal ) to the total aberration value
(stop at lens position)
Spherical aberration
X: lens bending
M: position parameter
Coma
Astigmatisms
Field curvature
Distortion
2
22
23
3 2
)1(
2
)1(2
1
2
1)1(32
1M
n
nnM
n
nX
n
n
n
n
fnnSlens
MnX
n
n
fnsClens )12(
1
1
'4
12
2'2
1
sfAlens
2'4
1
snf
nPlens
0lensD
Lens Contributions of Seidel
Spherical aberration
Special impact on correction:
1. Special quadratic dependence on
bending X
Minimum at
2. No correction for small n and M
3. Correction for large
n: infrared materials
M: virtual imaging
Limiting value
2
22
23
3 2
)1(
2
)1(2
1
2
1)1(32
1M
n
nnM
n
nX
n
n
n
n
fnnSlens
sphW
X
M = 6M = - 6
M = - 3M = 0
M = 3
n = 1.5
X
n
nMsph min
2 1
2
2
M
n n
ns
0
2
2
2
1
