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Imaging and Aberration Theory

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Imaging and Aberration Theory Imaging and Aberration Theory Lecture 8: Astigmastism and field curvature 2013-12-19 Herbert Gross Winter term 2013 www.iap.uni-jena.de 2 Preliminary time schedule 1 24.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems 2 07.11. Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and Hamiltonian coordinates mechanics, Hamiltonian coordinates 3 14.11. Eikonal Fermat Principle, stationary phase, Eikonals, relation rays‐waves, geometrical approximation, inhomogeneous media 4 21.11. Aberration expansion single surface, general Taylor expansion, representations, various orders, stop shift formulas 5 28.11. Representations of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations 6 05.12. Spherical aberration phenomenology, sph‐free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders 7 12.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options 8 19.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options 9 09.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum 10 16.01. Further reading on aberrations sensitivity in 3rd order, structure of a system, analysis of optical systems, lens contributions, Sine condition, isoplanatism, sine condition, Herschel condition, relation to coma and shift invariance, pupil aberrations, relation to Fourier optics 11 23.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations, relation to PSF and OTF 12 30.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, recalculation for offset, ellipticity, measurement 13 06.02. Miscellaneous Intrinsic and induced aberrations, Aldi theorem, vectorial aberrations, partial symmetric systems 3 Contents 1. Geometrical astigmatism 2. Point spread function for astigmatism 3. Field curvature 4. Petzval theorem 5. Correcting field curvature 6. Examples 4 Astigmatism . Reason for astigmatism: chief ray passes a surface under an oblique angle, the refractive power in tangential and sagittal section are different . The astigmatism is influenced by the stop position . A tangential and a sagittal focal line is found in different distances . Tangential rays meets closer to the surface . In the midpoint between both focal lines: circle of least confusion 5 Astigmatism . Beam cross section in the case of astigmatism: - Elliptical shape transforms its aspect ratio - degenerate into focal lines in the focal plane distances - special case of a circle in the midpoint: smallest spot y tangential aperture sagittal aperture z x sagittal circle of least focus tangential confusion focus 6 Primary Aberration Spot Shape for Astigmatism . Seidel formulas for field point only in y‘ considered 3 2 2 3 ' ' ' cosy S 'p r ' ' pC ( 2 yp r cos 2P A ) P ( 2 y 'p ' r ) P D ' ' y cos ' ' 3 2 2 ' ' ' x sin Sp r ' pC ' ' y p r sin P P 2 y p ' r ' P ' sin 2 2 . Field curvature:' 'y ' P ' y cosp r P ,x P' ' yp ' r ' P sin circle 2 2 2 4 2 x''''' y P yp r . Astigmatism: 'y 2 ' A ' ' 2 y cosr x , ' 0 focal line p P . General spot with defocus: 2 2 2 wave aberration W x(,) y c20 x y 22 c y W W ransverse aberrationsT x R Rc2 x , y R Ry2 c c x 20 y 20 22 2 2 Circle/ring in the pupil rxy22 2 x y 2 2 1 delivers an elliptical spoit in the image ( 2Rrc20 ) Rr2 20 c 22 c Special case for c20 = - c22/2: x2 y2 1 circle of least confusion 2 2 ()()Rrc22 Rrc22 7 Astigmatism Primary astigmatism at sagittal focus Wave aberration tangential sagittal . At sagittal focus 2 2 . Defocus only in tangential cross section Transverse ray aberration y x y 0.01 mm 0.01 mm 0.01 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm 1 1 tangential sagittal sagittal 0.5 tangential 0.5 0 0 0100200 -0.04 0.0 0.04 cyc/mm z/mm Geometrical spot through focus 0.02 mm 0.02 -0.04 -0.02 0.0 0.02 0.04 Ref: H. Zügge z/mm 8 Astigmatism Primary astigmatism at medial focus Wave aberration tangential sagittal . At median focus 2 2 . Anti-symmetrical defocus in T-S-cross section Transverse ray aberration y x y 0.01 mm 0.01 mm 0.01 mm Pupil: y-section x-section x-section Modulation Transfer Function MTF MTF at paraxial focus MTF through focus for 100 cycles per mm 1 1 tangential sagittal tangential 0.5 sagittal 0.5 0 0 0100200 -0.04 0.0 0.04 cyc/mm z/mm Geometrical spot through focus 0.02 mm -0.04 -0.02 0.0 0.02 0.04 Ref: H. Zügge z/mm 9 Astigmatism . Imaging of a polar grid in different planes . Tangential focus: - blur in azumuthal direction - rings remain sharp . Sagittal focus: - blur in radial direction - spokes remain sharp 10 Astigmatism . Behavior of a local position of the field: 1. good resolution of horizonthal structures 2. bad resolution of vertical structures . Imaging of polar diagram shows not the classical behaviour of the complete field 11 Coddington Equations . fective curvatures, the efrayFor an oblique surface depend on azimuth of the spherical . There are two focal points for sagittal / tangential aperture rays . This splitting occurs already for infinitesimal aperture angles around the chief ray . equationsIntersection lengths along the chief ray: Coddington 'n coscos2 i n ' 2 n 'i cos i n ' i cos n' 'n n cos i ' n cos i l'sag lsag R l'tan ltan R . Right side: oblique power of the spherical interface . The sagittal image must be located axon the auxiliary is by symmetry 12 Wavefront for Astigmatiusm with Defocus . Astigmatic wavefront with defocus . Purely cylindrical for focal lines in x/y . Purely toroidal without defocus: circle of least confusion 13 Psf for Astigmatism . Zernike coefficients c in . Pure astigmatism . Shape is not circular symmetric due to finite width of focal lines 14 Psf bei Astigmatismus Ref: Francon, Atlas of optical phenomena 15 Astigmatism: Lens Bending . Bending effects astigmatism . For a single lens 2 bending with zero astigmatism, but remaining field curvature Ref : H. Zügge 16 Field Curvature and Image Shells . Imaging with astigmatism: Tangential and sagittal image shell sharp depending on the azimuth . Difference between the image shells: astigmatism . Astigmatism corrected: It remains one curved image shell, Bended field: also called Petzval curvature . System with astigmatism: Petzval sphere is not an optimal surface with good imaging resolution . No effect of lens bending on curvature, important: distribution of lens powers and indices 17 Astigmatisms and Curvature of Field . Image surfaces: 1. Gaussian image plane 2. tangential and sagittal image shells (curved) 3. mean image shell of best sharpness 4. Petzval shell, arteficial, not a good image . Seidel theory: ss''tan pet 3 ss ''sag pet 3s ' s ' s' sag tan pet 2 . Astigmatism is difference sss'''ast tan sag . Best image shell ss'' s' sag tan best 2 18 Correction of Astigmatism and Field Curvature . Different possibilities for the correction of astigmatism and field curvature . Two independend aberrations allow 4 scenarious 19 Petzval Shell . The Petzval shell is not a desirable image surface . It lies outside the S- and T-shell: 3s ' s ' s' sag tan pet 2 . The Petzval curvature is a result of the Seidel aberration theory 20 Field Curvature . The image splits into two curved shells in the field . The two shells belong to tangential / sagittal aperture rays . There are two different possibilities for description: 1. sag and tan image shell 2. difference (astigmatism) and mean (medial image shell) of sag and tan Ref: H. Zügge 21 Field curvature . Basic observation: A plane object gives a curved image 22 Field Curvature . Focussing into different planes of a system with field curvature . Sharp imaged zone changes from centre to margin of the image field focused at field boundary focused in field zone focused in center (mean image plane) (paraxial image plane) y' receiving planes z image sphere 23 Field for Single Surface . The image is generated on a curved shell . In 3rd order, this is a sphere . For a single refracting surface, the Petzval radius is given by nr r p nn' . For a system of several lenses, the Petzval curvature is given by 11 n' rpkk nf k 24 Petzval Theorem for Field Curvature . curvature:theorem for field Petzval 1 nk' n k 1. formulation for surfaces nm ' Rptz knk n k' k r 1 1 (in air)2. formulation for thin lenses Rptz j nj j f . on dependence Important: no bending . Natural behavior: image curved towards system . systems Problem: collecting with f > 0: lenses: positive If only Rptz always negative 25 Petzval Theorem . Elementary derivation by a momocentric system of three surfaces: , object and image surfaceinterface surface with r . Consideration of a skew auxiliary axis a s r,'' a s r n' n n' n . Imaging condition s' s r . with gives of a flat object case For the special a', Rp a 1n ' n Rp nr 26 Petzval Theorem for Field Curvature . Goal: vanishing Petzval curvature 1 1 Rptz j nj j f 1h 1 and positive total refractive power j f j h1 j 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 27 LensesFlattening Meniscus .
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