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 .