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Lens Design II
Lecture 10: Higher order aberrations
2016-12-21
Herbert Gross
Winter term 2016
2
Preliminary Schedule
1 19.10. Aberrations and optimization Repetition
2 26.10. Structural modifications Zero operands, lens splitting, lens addition, lens removal, material selection
3 02.11. Aspheres Correction with aspheres, Forbes approach, optimal location of aspheres, several aspheres
4 09.11. Freeforms Freeform surfaces
5 16.11. Field flattening Astigmatism and field curvature, thick meniscus, plus-minus pairs, field lenses
6 23.11. Chromatical correction I Achromatization, axial versus transversal, glass selection rules, burried surfaces
7 30.11. Chromatical correction II secondary spectrum, apochromatic correction, spherochromatism
8 07.12. Special correction topics I Symmetry, wide field systems,stop position
9 14.12. Special correction topics II Anamorphotic lenses, telecentricity
10 21.12. Higher order aberrations high NA systems, broken achromates, induced aberrations
11 04.01. Further topics Sensitivity, scan systems, eyepieces
12 11.01. Mirror systems special aspects, double passes, catadioptric systems
13 18.01. Zoom systems mechanical compensation, optical compensation
14 25.01. Diffractive elements color correction, ray equivalent model, straylight, third order aberrations, manufacturing
15 01.02. Realization aspects Tolerancing, adjustment
1. Higher order in an achromate
2. Aplanatic-concentric surfaces
3. Merte surface
4. Aspheres
5. High-NA systems
6. Induced aberrations
3
Contents
Taylor expansion of the
sin-function
Definition of allowed error
10-4
Deviation of the various
approximations:
- linear: 5°
- cubic: 24°
- 5th order: 54.2°
4
Paraxial approximation
0 10 20 30 40 50 60 70 80 900
0.2
0.4
0.6
0.8
1
sin(x)
x [°]
exact sin(x)
linear
cubic
5th order
x = 5° x = 24° x = 52° deviation 10-4
Contribution of various orders of the sin-expansion
5
Paraxail Approximation
1st order i
0 5 10 15 20 25 30 35 40 4510
-12
10-10
10-8
10-6
10-4
10-2
100
incidence angle i
3rd order i3/6
5th order i5/120
7th order i7/5040
9th order i9/362880
...36288050401206
sin9753
iiii
ii
Higher Order Aberrations
Relevance of higher order expansion terms
Nearly 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
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
8
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
Spherical Correction / Higher Orders
Partial correction of residual spherical aberration by 5th order or 5th and 7th order
Different (alternating) sign of coefficients
Residual total error significantly smaller
Large gradients at the edge
3rd and 5th order compensation: residual zonal error at of the pupil radius
9
total
rp
Dz [a.u.]
primary
3rd order
secondary
5th order
1
0.5
0.71
0
total
rp
Dz [a.u.]
primary
3rd order
secondary
5th order
1
0.89
0
a) 2 orders b) 3 orders
0.46
tertiary
7th order
707.02
1
Sine condition fulfilled:
linear coma removed
Remaning aberrations of higher
order:
1. Oblique tangential spherical
2. Oblique sagittal spherical
3. Elliptical coma
10
Higher Order Aberrations
coma 3rd
Dy‘
Dx‘
linear coma 5th
Dy‘
Dx‘
oblique sagittal
spherical 5th
Dy‘
Dx‘
oblique tangential
spherical 5th
Dy‘
Dx‘
elliptical coma 5th
Dy‘
Dx‘
Dy‘
Dx‘
2 2 2
3
2 3 2
3
2 3 2
3
1' sin cos
' '
1' cos 1 cos
' '
p p
p p p
p p p
W c y r y
x c y rn u
y c y rn u
D
D
2 4
2
2 3 2 3
1 2
1 1' 4 sin , ' cos
' ' ' '
p
p p p p
W c y r
x c y r y c y rn u n u
D D
3 2 3 3
4 5
3 2
4
3 2 3 2 2
4 5
1' sin 2
' '
1' 2 cos 2 3 cos
' '
p p p
p p
p p p p
W c y r y c y y
x c y rn u
y c y r c y rn u
D
D
Splitted achromate
Aspherical surface
Higher Order Aberrations: Achromate, Aspheres
achromat broken contact
0.0250-0.025
1
aspheric
a
0.010-0.01
1
0.0020-0.002
1
1
0.0002
1
0.00001
1
0.002
zone
zone zone
air space
ray
Ref : H. Zügge
11
Relaxed System
Example: achromate with cemented/splitted setup
Equivalent performance
Inner surfaces of splitted version more sensitive
Ref: H. Zügge
a ) Cemented achromate f = 100 mm , NA = 0.1
b ) Splitted achromate f = 100 mm , NA = 0.1
-15
-10
-5
0
5
10
1 2 3
-15
-10
-5
0
5
10
1 2 3 4
Seidel coefficient
spherical aberration
spot enlargement for
0.2 ° surface tilt
surface
index
surface
index
Aplanatic Surface
Spherical aberration vanishes for all orders
Aplanatic lens at high NA:
effective real NA is higher than paraxial
Further possible aplanatic lenses
1. less practical importance
2. used in microscopic objective front lens
13
concentric at vertex aplanatic at vertex
real rays
ideal
lensideal
rays
real
lens
sin ureal' = 0.894
sin uideal' = 0.707
Merte Surface
Small difference in refractive
index
Growing higher order
contributions
14
n2n1
Merte
surface
radius
concentric
NA
R
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510
-5
10-4
10-3
10-2
10-1
100
Dn
NA = 0.20NA = 0.05 NA = 0.10
c16/c9
c25/c9
c16/c9
c25/c9
c16/c9
c25/c9
Merte surface:
- low index step
- strong bending
- mainly higher aberrations
generated
Higher Order Aberrations: Merte Surface
1
1
O.25
O.25
Merte surface
(a)
(b)
Transverse
spherical aberration
From : H. Zügge
15
Additional degrees of freedom for correction
Exact correction of spherical aberration for a finite
number of aperture rays
Strong asphere: many coefficients with high orders,
large oscillative residual deviations in zones
Location of aspherical surfaces:
1. spherical aberration: near pupil
2. distortion and astigmatism: near image plane
Use of more than 1 asphere: critical, interaction and
correlation of higher oders
SPH
tan u'
corrected
points
residual
spherical
aberration
Aspherical Surfaces
16
Microscope Objective Lens Types
Medium magnification system
40x0.65
High NA system 100x0.9
without field flattening
High NA system 100x0.9
with flat field
Large-working distance
objective lens 40x0.65
Microscope Objective Lens
Examples of large-working-
distance objective lenses
Aplanatic-concentric shell-lenses
in the front group
Large diameter of the lens L1 M
2 M3 M
4
Microscope Lens
Zernike spectrum shows
large higher contributions
Spherical Aberration of Higher Order
(a)
(b) (c)
wavefront surface wavefront section
λ0.1
1 3 5 7 9 11 13 15 17-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
order
sherical aberration
l
Ref: H. Zügge
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
2nd surface
induced perturbation at 2nd
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 vanishinh 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
Dialyt achromate:
Induced axial color at 2nd lens
Correction by distance t
24
Induced Axial Color Aberration
Ref: A. Berner
Several relay systems:
accumulation of residual
aberrations
Growing induced aberrations
Induced Aberrations in Endoscopes
25
W0n0(r)
1 2 3 4 5 6 7 8 9 10 11 125th intrinsic
5th extrinsic
5th total
3rd order
0
0.02
0.04
0.06
0.08
0.1
0.12
surface
index
lens index1 2 3 4 5 60
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
5th order intrinsic
5th order
extrinsic
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
26
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
27
Reversability of Aberrations
Ref
l = 546 nm
l =480 nm
l =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
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