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Design and Correction of optical Systems
Part 12: Correction of aberrations 1
Summer term 2012Herbert Gross
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Group of meniscus lenses
Effect of distance and refractive indices
Correcting Petzval Curvature
r 2r 1
collimated
n n
d
From : H. Zügge
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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