Design and Correction of optical Systems
Part 12: Correction of aberrations 1
Summer term 2012 Herbert Gross 2 Overview
1. Basics 2012-04-18 2. Materials 2012-04-25 3. Components 2012-05-02 4. Paraxial optics 2012-05-09 5. Properties of optical systems 2012-05-16 6. Photometry 2012-05-23 7. Geometrical aberrations 2012-05-30 8. Wave optical aberrations 2012-06-06 9. Fourier optical image formation 2012-06-13 10. Performance criteria 1 2012-06-20 11. Performance criteria 2 2012-06-27 12. Correction of aberrations 1 2012-07-04 13. Correction of aberrations 2 2012-07-11 14. Optical system classification 2012-07-18 3 Contents
12.1 Symmetry principle 12.2 Lens bending 12.3 Correcting spherical aberration 12.4 Coma, stop position 12.5 Astigmatism 12.6 Field flattening 12.7 Chromatical correction 12.8 Higher order aberrations 4 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 3
1 5 Symmetrical Systems
Ideal symmetrical systems: . Vanishing coma, distortion, lateral color aberration . Remaining residual aberrations: 1. spherical aberration 2. astigmatism 3. field curvature 4. axial chromatical aberration 5. skew spherical aberration 6 Symmetry Principle
. 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 nearly all photographic systems
Ref : H. Zügge 7 Lens Bending
. Effect of bending a lens on spherical aberration . Optimal bending: Spherical Minimize spherical aberration Aberration . Dashed: thin lens theory Solid : think real lenses 60 . Vanishing SPH for n=1.5 only for virtual imaging 40 . Correction of spherical aberration
possible for: 20
1. Larger values of the (b) (c)
magnification parameter |M| X (a) 2. Higher refractive indices -7-6-5-4-3-2-101234567 (d) M=0 M=-3 M=3 M=-6 M=6
Ref : H. Zügge 8 Correcting Spherical Aberration: Lens Splitting
Transverse aberration . Correction of spherical aberration: 5 mm
Splitting of lenses (a)
. Distribution of ray bending on several 5 mm surfaces: Improvement (b) - smaller incidence angles reduces the (a) (b) : 1/4 effect of nonlinearity
- decreasing of contributions at every 5 mm surface, but same sign Improvement (c) . Last example (e): one surface with (b) (c) : 1/2 compensating effect
5 mm
Improvement (d) (c) (d) : 1/4
0.005 mm
Improvement (e) (d) (e) : 1/75
Ref : H. Zügge 9 Correcting Spherical Aberration : Power Splitting
Splitting of lenses and appropriate bending: 1. compensating surface contributions 2. Residual zone errors 3. More relaxed setups preferred, although the nominal error is larger
Ref : H. Zügge 10 Correcting Spherical Aberration: Cementing
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 front 2. bi-convex outer surfaces / meniscus shape . Residual zone error, spherical aberration corrected for outer marginal ray
1.0 mm 0.25 mm Crown (a) (b) in front
0.25 mm 0.25 mm Filnt (c) (d) in front
Ref : H. Zügge 11 Correcting Spherical Aberration: Refractive Index
∆s’ . Better correction for higher index 1.4 1.5 1.6 1.7 1.8 1.9 4.0 0 n . Shape of lens / best bending changes -2 from 1.686 -4 best shape 1. nearly plane convex plano-convex for n= 1.5 -6 2. meniscus shape -8 for n > 2
best shape
n = 1.5 n = 1.7 n = 1.9 n = 4.0
plano-convex
Ref : H. Zügge 12 Correcting Spherical Aberration: Refractive Index
n = 1.5 n = 1.8 . Better correction SI () for high index also for meltiple 40 lens systems 30 20 . Example: 3-lens setup with one 10 surface for compensation 0 Residual aberrations is quite better -10 -20 for higher index Surface 123456sum
0.5 mm
n = 1.5
0.0005 mm
n = 1.8
Ref : H. Zügge 13 Bravais System
. 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 1 1 d f'a 1 f' f 'a d s b 1 1 f ' d Bild- b d 1 ebene u s u'
s'
d
s 14 Inner and Outer Coma
y' 0.2 mm . Effect of lens bending on coma . Sign of coma : inner/outer coma
y' 0.2 mm
y' 0.2 mm
y' 0.2 mm
From : H. Zügge 15 Coma Correction: Achromat
Image height: y’ = 0 mm y’ = 2 mm Achromat Pupil section: meridional meridional sagittal bending . Bending of an achromate Transverse y' y' y' - optimal choice: small residual spherical Aberration: 0.05 mm 0.05 mm 0.05 mm aberration (a) - remaining coma for finite field size . Splitting achromate: additional degree of freedom: (b) - better total correction possible - high sensitivity of thin air space . Aplanatic glass choice:
vanishing coma (c)
Achromat, splitting
(d)
Achromat, aplanatic glass choice
(e)
Wave length:
Ref : H. Zügge 16 Coma Correction: Symmetry Principle
. Perfect coma correction in the case of symmetry . But magnification m = -1 not useful in most practical cases
Image height: y’ = 19 mm Symmetry principle Pupil section: meridional sagittal Transverse y' y' Aberration: 0.5 mm 0.5 mm
(a)
(b)
From : H. Zügge 17 Coma Correction: Stop Position and Aspheres
. Combined effect, aspherical case prevent correction
Sagittal Sagittal Plano-convex element coma Spherical aberration corrected coma exhibits spherical aberration y' with aspheric surface y' 0.5 mm 0.5 mm aspheric
aspheric
aspheric
Ref : H. Zügge 18 Astigmatism: Lens Bending
. Bending effects astigmatism . For a single lens 2 bending with zero astigmatism, but remaining field curvature
Ref : H. Zügge 19 Petzval Theorem for Field Curvature
. Petzval theorem for field curvature: 1 nk 'nk 1. formulation for surfaces nm ' Rptz k nk nk 'rk 1 1 2. formulation for thin lenses (in air) Rptz j n j f j . Important: no dependence on bending
. Natural behavior: image curved object towards system plane
. Problem: collecting systems with f > 0: If only positive lenses:
Rptz always negative R
ideal optical system real image image plane shell 20 Petzval Theorem for Field Curvature
. Goal: vanishing Petzval curvature 1 1 Rptz j n j f j 1 h 1 and positive total refractive power j f j h1 f 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 21 Flattening Meniscus Lenses
. Possible lenses / lens groups for correcting field curvature . Interesting candidates: thick mensiscus shaped lenses
r2 2 1 n 'n 1 n 1 d k k d Rptz k nk nk 'rk n f n r1r2 r1
1. Hoeghs mensicus: identical radii (n 1)2 d - Petzval sum zero F' 2 nr - remaining positive refractive power
2. Concentric meniscus, r r d - Petzval sum negative 2 1 - weak negative focal length 1 (n 1)d ()nd1 F' - refractive power for thickness d: R n r r d ptz 1 1 nr11() r d
3. Thick meniscus without refractive power 1 (n 1)2 d n 1 0 Relation between radii rrd21 n Rptz n r1 nr1 d (n 1) 22 Correcting Petzval Curvature
. Group of meniscus lenses n n
d collimated
r 1 r 2
. Effect of distance and refractive indices
From : H. Zügge 23 Correcting Petzval Curvature
r r . Triplet group with + - + 1 2
r 3
d/2 collimated
n2
n1
. Effect of distance and refractive indices
From : H. Zügge 24 Field Curvature
1 F . Correction of Petzval field curvature in lithographic lens j for flat wafer R j n j
h j . Positive lenses: Green hj large F Fj j h1 . Negative lenses : Blue hj small
. Correction principle: certain number of bulges 25 Flattening Field Lens
Effect of a field lens for flattening the image surface
1. Without field lens 2. With field lens
curved image surface image plane 26 Axial Colour: Achromate
(a ) (b ) . Compensation of axial colour by appropriate glass choice
. Chromatical variation of the spherical aberrations: spherochromatism (Gaussian aberration)
. Therefore perfect axial color correction BK7 BK7 F2 (on axis) are often not feasable n = 1.5168 n = 1.5168 1.6200 =64.17 = 64.17 36.37 F= 1 F= 2.31 -1.31
r p r p 1 1
486 nm 588 nm 656 nm
z z -2.5 0 -0.20 0 0.20
Ref : H. Zügge 27 Spherochromatism: Achromate
. Residual spherochromatism of an achromate . Representation as function of apeture or wavelength
longitudinal aberration defocus variation rp 1 656
pupil height :
rp = 1 r = 0.707 587 p rp = 0.4 rp = 0
486 nm 587 nm 656 nm 486 z z 0 0.04 0.08 0.12 -0.04 0 0.04 0.08 28 Correction of Spherochromatism: Splitted Achromate
. Broken-contact achromate: different ray heights allow fof y red
correcting spherochromatism blue
rp
SK11 SF12
486 nm 587 nm 656 nm
z in [mm] 0 0.2 0.4 29 Axial Color Correction with Schupman Lens
wavelength . Non-compact system in m 0.656 . Generalized achromatic condition Schupman lens with marginal ray hieghts yj achromate 2 2 y1 y2 F1 F2 0 v1 v2 0.571 . Use of a long distance and
negative F2 for correction . Only possible for virtual imaging
0.486 s -50 m050 m
first lens second lens positive positive
intermediate virtual image image 30 Axial Colour: Achromate and Apochromate
. 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 partial dispertion necessery 31 Axial Colour : Apochromate
. Choice of at least one special glass
. Correction of secondary spectrum: anomalous partial dispersion
. At least one glass should deviate significantly form the normal glass line
32 Two-Lens Apochromate
. Special glasses and very strong bending allows for apochromatic correction
. Large remaining spherical zonal aberration
. Zero-crossing points not well distributed over wavelength spectrum
656nm
588nm
486nm
z 436nm -2mm 5m 0 z 33 Buried Surface
. 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
34 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 35 Field Lens im Endoscope
without field lenses
with 1 field lens
with 2 field lenses
Ref : H. Zügge 36 Influence of Stop Position on Performance
. Ray path of chief ray depends on stop position
stop positions spot 37 Effect of Stop Position
stop . Example photographic lens
. Small axial shift of stop changes tranverse aberrations
. In particular coma is strongly influenced
Ref: H.Zügge 38 Distortion and Stop Position
. Sign of distortion for single lens: depends on stop position and Lens Stop location Distortion Examples sign of focal power positive rear V > 0 tele photo lens . Ray bending of chief ray defines negative in front V > 0 loupe distortion positive in front V < 0 retrofocus lens . Stop position changes chief ray heigth at the lens negative rear V < 0 reversed binocular
Ref: H.Zügge 39 Higher Order Aberrations: Achromate, Aspheres
. Splitted achromate
. Aspherical surface
Ref : H. Zügge 40 Higher Order Aberrations: Merte Surface
. Merte surface: - low index step Transverse spherical aberration - strong bending - mainly higher aberrations O.25 generated
(a) 1
O.25
(b) 1
Merte surface
From : H. Zügge 41 Aspherical Surfaces
. 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 42 Coexistence of Aberrations : Balance
compromise : standard achromate a) zonal and axial error . Example: Achromate 0.03 0.8
. Balance : axial 0.6 0.02 color 1. zonal spherical 0.4 0.01 2. Spot 0.2 3. Secondary spectrum 0 0 -0.2 -0.01 -0.4 -0.02 spherical aberration -0.6
-0.03 -0.8 sur 1 sur 2 sur 3 sum sur 1 sur 2 sur 3 sur 4 sum
secondary b) spot optimized c) spectrum optimized 0.6 3
0.4 2
0.2 1 0 0 -0.2 -1 -0.4
-0.6 -2
-0.8 -3 sur 1 sur 2 sur 3 sur 4 sum sur 1 sur 2 sur 3 sur 4 sum
Ref : H. Zügge 43 Coexistence of Aberrations : Balance
r SSK2 CaF2 F13 P . Example: Apochromate . Balance :
1. zonal spherical 400 nm 450 nm 2. Spot 500 nm 3. Secondary spectrum 550 nm 600 nm 650 nm 700 nm
400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm
z axis -0.040 0.04 0.08 0.12
field 0.71°
field 1.0° 44 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 45 Outlook
Next lecture: Part 13 – Correction of Aberrations 2 1. Date: Wednesday, 2012-07-11 Contents: 13.1 Fundamentals of optimization 13.2 Optimization in optical design 13.3 Initial setups and correction methods 13.4 Sensitivity of systems 13.5 Miscellaneous