Design and Correction of optical Systems
Part 12: Correction of aberrations 1
Summer term 2012Herbert Gross
Overview
1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Correction of aberrations 1 2012-07-0413. Correction of aberrations 2 2012-07-1114. Optical system classification 2012-07-18
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12.1 Symmetry principle12.2 Lens bending12.3 Correcting spherical aberration12.4 Coma, stop position12.5 Astigmatism12.6 Field flattening12.7 Chromatical correction12.8 Higher order aberrations
Contents
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Principle of Symmetry
Perfect symmetrical system: magnification m = -1 Stop in centre of symmetry Symmetrical contributions of wave aberrations are doubled (spherical) Asymmetrical contributions of wave aberration vanishes W(-x) = -W(x) Easy correction of:
coma, distortion, chromatical change of magnification
front part rear part
2
1
3
4
Symmetrical Systems
Ideal symmetrical systems: Vanishing coma, distortion, lateral color aberration Remaining residual aberrations:
1. spherical aberration2. astigmatism3. field curvature 4. axial chromatical aberration5. skew spherical aberration
5
Symmetry Principle
Ref : H. Zügge
Application of symmetry principle: photographic lenses Especially field dominant aberrations can be corrected Also approximate fulfillment
of symmetry condition helps significantly:quasi symmetry Realization of quasi-
symmetric setups in nearlyall photographic systems
6
Effect of bending a lens on spherical aberration Optimal bending:
Minimize spherical aberration Dashed: thin lens theory
Solid : think real lenses Vanishing SPH for n=1.5
only for virtual imaging Correction of spherical aberration
possible for:1. Larger values of the
magnification parameter |M|2. Higher refractive indices
Lens Bending
20
40
60
X
M=-6 M=6M=-3 M=3M=0
SphericalAberration
(a)
(b) (c)
(d)-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
Ref : H. Zügge
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Correction of spherical aberration:Splitting of lenses
Distribution of ray bending on severalsurfaces: - smaller incidence angles reduces the
effect of nonlinearity- decreasing of contributions at every
surface, but same sign Last example (e): one surface with
compensating effect
Correcting Spherical Aberration: Lens Splitting
Transverse aberration
5 mm
5 mm
5 mm
(a)
(b)
(c)
(d)
Improvement (a) (b) : 1/4
(c) (d) : 1/4
(b) (c) : 1/2Improvement
Improvement
(e)
0.005 mm
(d) (e) : 1/75Improvement
5 mm
Ref : H. Zügge
8
Splitting of lenses and appropriate bending:1. compensating surface contributions2. Residual zone errors3. More relaxed
setups preferred,although the nominal error islarger
Correcting Spherical Aberration : Power Splitting
Ref : H. Zügge
9
Correcting Spherical Aberration: Cementing
0.25 mm0.25 mm
(d)
(a)
(c)
(b)
1.0 mm 0.25 mm
Crownin front
Filntin front
Ref : H. Zügge
Correcting spherical aberration by cemented doublet: Strong bended inner surface compensates Solid state setups reduces problems of centering sensitivity In total 4 possible configurations:
1. Flint in front / crown in front2. bi-convex outer surfaces / meniscus shape Residual zone error, spherical aberration corrected for outer marginal ray
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Better correctionfor higher index Shape of lens / best
bending changesfrom1. nearly plane convex
for n= 1.52. meniscus shape
for n > 2
Correcting Spherical Aberration: Refractive Index
-8
-6
-4
-2
01.4 1.5 1.6 1.7 1.8 1.9
∆s’
4.0n
n = 1.5 n = 1.7 n = 1.9 n = 4.0
best shape
best shapeplano-convex
plano-convex
1.686
Ref : H. Zügge
11
Better correctionfor high index also for meltiplelens systems Example: 3-lens setup with one
surface for compensationResidual aberrations is quite betterfor higher index
Correcting Spherical Aberration: Refractive Index
-20
-10
0
10
20
30
40
1 2 3 4 5 6 sum
n = 1.5 n = 1.8
SI ()
Surface
n = 1.5
n = 1.8
0.0005 mm
0.5 mm
Ref : H. Zügge
12
Combination of positiv and negative lens :Change of apertur with factor without shift of image plane
Strong impact on spherical aberration
Principle corresponds the tele photo system
Calculation of the focal lengths
d
s
s'
u'u
Bild-ebene
af'
f'b
sddf
sd
df
b
a
1
1'
1
11'
1
Bravais System
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Effect of lens bending on coma Sign of coma : inner/outer coma
Inner and Outer Coma
0.2 mm'y
0.2 mm'y
0.2 mm'y
0.2 mm'y
From : H. Zügge
14
Bending of an achromate- optimal choice: small residual spherical
aberration- remaining coma for finite field size Splitting achromate:
additional degree of freedom: - better total correction possible- high sensitivity of thin air space Aplanatic glass choice:
vanishing coma
Coma Correction: Achromat
0.05 mm'y
0.05 mm'y
0.05 mm'y
Pupil section: meridional meridional sagittal
Image height: y’ = 0 mm y’ = 2 mm
TransverseAberration:
(a)
(b)
(c)
Wave length:
Achromat bending
Achromat, splitting
(d)
(e)
Achromat, aplanatic glass choice
Ref : H. Zügge
15
Perfect coma correction in the case of symmetry But magnification m = -1 not useful in most practical cases
Coma Correction: Symmetry Principle
Pupil section: meridional sagittal
Image height: y’ = 19 mm
TransverseAberration:
Symmetry principle
0.5 mm'y
0.5 mm'y
(a)
(b)
From : H. Zügge
16
Combined effect, aspherical case prevent correction
Coma Correction: Stop Position and Aspheres
aspheric
aspheric
aspheric
0.5 mm'y
Sagittalcoma
0.5 mm'y
SagittalcomaPlano-convex element
exhibits spherical aberration Spherical aberration corrected
with aspheric surface
Ref : H. Zügge
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Bending effects astigmatism For a single lens 2 bending with
zero astigmatism, but remainingfield curvature
Astigmatism: Lens Bending
Ref : H. Zügge
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Petzval Theorem for Field Curvature
Petzval theorem for field curvature:1. formulation for surfaces
2. formulation for thin lenses (in air)
Important: no dependence on bending
Natural behavior: image curved towards system
Problem: collecting systems with f > 0:If only positive lenses: Rptz always negative
k kkk
kkm
ptz rnnnnn
R '''1
j jjptz fnR11
optical system
objectplane
idealimageplane
realimageshell
R
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Petzval Theorem for Field Curvature
Goal: vanishing Petzval curvature
and positive total refractive power
for multi-component systems
Solution:General principle for correction of curvature of image field:1. Positive lenses with:
- high refractive index- large marginal ray heights- gives large contribution to power and low weighting in Petzval sum
2. Negative lenses with: - low refractive index- samll marginal ray heights- gives small negative contribution to power and high weighting in Petzval sum
j jjptz fnR11
fhh
f j
j 11
1
20
Flattening Meniscus Lenses
Possible lenses / lens groups for correcting field curvature Interesting candidates: thick mensiscus shaped lenses
1. Hoeghs mensicus: identical radii- Petzval sum zero- remaining positive refractive power
2. Concentric meniscus, - Petzval sum negative- weak negative focal length- refractive power for thickness d:
3. Thick meniscus without refractive powerRelation between radii
F n dnr r d
' ( )( )
1
1 1
r r d nn2 1
1
21
211'
'1rrd
nn
fnrnnnn
R k kkk
kk
ptz
r2
d
r1
2
2)1('rn
dnF
drr 12
drrndn
Rptz
11
)1(1
0)1(
)1(1
11
2
ndnrrndn
Rptz
21
Group of meniscus lenses
Effect of distance and refractive indices
Correcting Petzval Curvature
r 2r 1
collimated
n n
d
From : H. Zügge
22
Triplet group with + - +
Effect of distance and refractive indices
Correcting Petzval Curvature
r 2r 1
collimated
n1
n2
r 3
d/2
From : H. Zügge
23
Field Curvature
Correction of Petzval field curvature in lithographic lensfor flat wafer
Positive lenses: Green hj large Negative lenses : Blue hj small
Correction principle: certain number of bulges
j j
j
nF
R1
jj
j Fhh
F 1
24
Effect of a field lens for flattening the image surface
1. Without field lens 2. With field lens
curved image surface image plane
Flattening Field Lens25
Compensation of axial colour byappropriate glass choice
Chromatical variation of the sphericalaberrations:spherochromatism (Gaussian aberration)
Therefore perfect axial color correction(on axis) are often not feasable
Axial Colour: Achromate
BK7 F2
n = 1.5168 1.6200= 64.17 36.37
F = 2.31 -1.31
BK7
n= 1.5168 = 64.17F= 1
(a) (b)
-2.5 0z
r p1
-0.20
1
z0.200
r p
486 nm588 nm656 nm
Ref : H. Zügge
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Residual spherochromatism of an achromate Representation as function of apeture or wavelength
Spherochromatism: Achromate
longitudinal aberration
0
rp
0.080.04 0.12 z
defocus variation
pupil height :
rp = 1rp = 0.707rp = 0.4rp = 0
0-0.04 0.080.04
587
656
486z
1
486 nm587 nm656 nm
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Broken-contact achromate:different ray heights allow fofcorrecting spherochromatism
Correction of Spherochromatism: Splitted Achromate
red
blue
y
SK11 SF12
0.4
rp
486 nm587 nm656 nm
0.2
zin [mm]
0
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Non-compact system Generalized achromatic condition
with marginal ray hieghts yj
Use of a long distance andnegative F2 for correction Only possible for virtual imaging
Axial Color Correction with Schupman Lens
first lenspositive
second lenspositive
virtualimage
intermediateimage
wavelengthin m
0.656
0.486
0.571
50 m-50 m 0s
Schupmanlens
achromate
022
22
11
21 F
vyF
vy
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Effect of different materials Axial chromatical aberration
changes with wavelength Different levels of correction:
1.No correction: lens,one zero crossing point
2.Achromatic correction:- coincidence of outer colors- remaining error for center
wavelength- two zero crossing points
3. Apochromatic correction:- coincidence of at least three
colors- small residual aberrations- at least 3 zero crossing points- special choice of glass types
with anomalous partialdispertion necessery
Axial Colour: Achromate and Apochromate30
Choice of at least one special glass
Correction of secondary spectrum:anomalous partial dispersion
At least one glass should deviatesignificantly form the normal glass line
Axial Colour : Apochromate
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Special glasses and very strong bending allows for apochromatic correction
Large remaining spherical zonal aberration
Zero-crossing points not well distributed over wavelength spectrum
Two-Lens Apochromate
-2mmz
656nm
588nm
486nm
436nmz05m
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Cemented surface with perfect refrcative index match No impact on monochromatic aberrations Only influence on chromatical aberrations Especially 3-fold cemented components are advantages Can serve as a starting setup for chromatical correction with fulfilled monochromatic
correction Special glass combinations with nearly perfect
parameters
Nr Glas nd nd d d
1 SK16 1.62031 0.00001 60.28 22.32
F9 1.62030 37.96
2 SK5 1.58905 0.00003 61.23 20.26
LF2 1.58908 40.97
3 SSK2 1.62218 0.00004 53.13 17.06
F13 1.62222 36.07
4 SK7 1.60720 0.00002 59.47 10.23
BaF5 1.60718 49.24
Buried Surface33
Field Lenses
Field lens: in or near image planes Influences only the chief ray: pupil shifted Critical: conjugation to image plane, surface errors sharply seen
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Field Lens im Endoscope
without field lenses
with 2 field lenses
with 1 field lens
Ref : H. Zügge
35
Influence of Stop Position on Performance
Ray path of chief ray depends on stop position
spotstop positions
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Example photographic lens
Small axial shift of stopchanges tranverse aberrations
In particular coma is stronglyinfluenced
stop
Effect of Stop Position
Ref: H.Zügge
37
Distortion and Stop Position
Sign of distortion for single lens:depends on stop position andsign of focal power Ray bending of chief ray defines
distortion Stop position changes chief ray
heigth at the lens
Ref: H.Zügge
Lens Stop location Distortion Examples
positive rear V > 0 tele photo lens negative in front V > 0 loupe positive in front V < 0 retrofocus lens negative rear V < 0 reversed binocular
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Splitted achromate
Aspherical surface
Higher Order Aberrations: Achromate, Aspheres
Ref : H. Zügge
39
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
40
Additional degrees of freedom for correction
Exact correction of spherical aberration for a finitenumber of aperture rays
Strong asphere: many coefficients with high orders,large oscillative residual deviations in zones
Location of aspherical surfaces:1. spherical aberration: near pupil2. distortion and astigmatism: near image plane
Use of more than 1 asphere: critical, interaction andcorrelation of higher oders
Aspherical Surfaces41
Example: Achromate Balance :
1. zonal spherical2. Spot3. Secondary spectrum
Coexistence of Aberrations : Balance
standard achromate
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
compromise :zonal and axial error
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
spot optimizedsecondary
spectrum optimized
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-3
-2
-1
0
1
2
3
sur 1 sur 2 sur 3 sum sur 1 sur 2 sur 3 sur 4 sum
sur 1 sur 2 sur 3 sur 4 sum sur 1 sur 2 sur 3 sur 4 sum
a)
b) c)
sphericalaberration
axialcolor
Ref : H. Zügge
42
Example: Apochromate Balance :
1. zonal spherical2. Spot3. Secondary spectrum
Coexistence of Aberrations : Balance
SSK2 CaF2 F13
0.120
rP
400 nm450 nm500 nm550 nm600 nm650 nm700 nm
-0.04 0.04 0.08z
axis
field0.71°
field1.0°
400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm
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Summary of Important Topics
Nearly symmetrical system are goord corrected for coma, distortion and lateral color Important influence on correction: bending of a lens Correction of spherical aberration: bending, cementing, higher index Correction of coma: bending, stop position, symmetry Correction of field curvature: thick mensicus, field lens, low index negative lenses with
low ray height Achromate: coincidence of two colors, spherical correction, higher order zone remains Apochromatic correction: three glasses, one with anomalous partial dispersion Remaining chromatic error: spherochromatism Field lenses: adaption of pupil imaging Higher orders of aberrations: occur for large angles Whole system: balancing of aberrations and best trade-off is desired
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Outlook
Next lecture: Part 13 – Correction of Aberrations 21. Date: Wednesday, 2012-07-11Contents: 13.1 Fundamentals of optimization
13.2 Optimization in optical design13.3 Initial setups and correction methods13.4 Sensitivity of systems13.5 Miscellaneous
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