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IAT14 Imaging and Aberration Theory Lecture 12 Zernike Polynomials.Pdf Imaging and Aberration Theory Lecture 12: Zernike polynomials 2015-01-29 Herbert Gross Winter term 2014 www.iap.uni-jena.de 2 Preliminary time schedule 1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and 2 06.11. Hamiltonian coordinates mechanics, Hamiltonian coordinates Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical 3 13.11. Eikonal approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 4 20.11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 5 27.11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 6 04.12. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 7 11.12. Distortion and coma position, various topics, correction options 8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 9 08.01. Chromatical aberrations spherochromatism, secondary spoectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 10 15.01. isoplanatism aberrations, Herschel condition, relation to Fourier optics 11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations special expansion for circular symmetry, problems, calculation, optimal balancing, 12 29.01. Zernike polynomials influence of normalization, measurement ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and 13 05.02. PSF and transfer function contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic 14 12.02. Additional topics and induced aberrations, revertability 3 Contents 1. Definition 2. Properties 3. Calculation 4. Application in optical performance description 5. Aberration balancing 6. Sampling 7. Relation to power expansion 8. Change influences 9. High NA case 10. Miscellaneous 4 Zernike Polynomials . Expansion of wave aberration surface into elementary functions / shapes m = + 8 cos n + 7 W (r,) c Z m (r,) nm n + 6 n mn + 5 . Zernike functions are defined in circular + 4 coordinates r, + 3 sin (m) for m 0 + 2 m m + 1 Zn (r,) Rn (r) cos(m) for m 0 0 1 for m 0 - 1 . Ordering of the Zernike polynomials by - 2 indices: - 3 n : radial m : azimuthal, sin/cos - 4 - 5 . Mathematically orthonormal function - 6 on unit circle for a constant weighting function - 7 sin - 8 . Direct relation to primary aberration types n = 0 1 2 3 4 5 6 7 8 5 Zernike Polynomials . Alternative representation m = + 8 cos + 7 + 6 + 5 + 4 + 3 + 2 + 1 0 - 1 - 2 - 3 - 4 - 5 yp - 6 xp - 7 sin - 8 n = 0 1 2 3 4 5 6 7 8 6 Zernike Polynomials . Advantages of the Zernike polynomials: 1. usually good match of circular symmetry to most optical systems 2. de-coupling of coefficients due to orthogonality 3. stable numerical computation 4. direct measurement by interferometry possible 5. direct relation of lower orders to classical aberrations 6. optimale balancing of lower orders (e.g. best defocus for spherical aberration) 7. fast calculation of Wrms and Strehl ratio in approximation of Marechal . Problems and disadvantages of the Zernike polynomials: 1. computation on discrete grids 2. non circular pupils often occur in practice 3. different conventions can be found, conversion is quite confusing 4. calculation not stable for very high orders 5. Zernike functions are no eigenfunctions of wave propagation, if the measurement is not made exactly in the pupil, the coefficients are erroneous 7 Indexing and Azimuthal Periodicity rotational in cosq in cos2q in cos3q of symmetry: linear : quadratic : 3rd power : . Index m: defocus tilt, coma astigmatisms trefoil azimuthal periodicity spherical . Constant spatial frequency: sum of n+|m| azimuthal index m empty same spatial frequency radial index n 8 Zernike Polynomials: Fringe Convention Nr Cartesian representation Circular representation 1 1 1 2 x r sin 3 y r cos 4 2 x2 + 2 y2 - 1 2 r² - 1 5 2 x y r² sin 2 6 y2 - x2 r² cos 2 7 ( 3x2 + 3 y2 - 2 ) x ( 3r3 - 2r ) sin 8 ( 3x2 + 3 y2 - 2 ) y ( 3r3 - 2r ) cos 9 6 (x2+y2)2-6 (x2+y2) +1 6r4 - 6r² + 1 10 ( 3y2-x2 ) x r³ sin 3 11 ( y2-3x2) y r³ cos 3 12 (4x2+4y2-3) 2xy ( 4r4 - 3r² ) sin 2 13 (4x2+4y2-3) (y2 - x2) ( 4r4 - 3r² ) cos 2 14 [10(x2+y2)2-12(x2+y2)+3] x ( 10r5 - 12r³ + 3r ) sin 15 [10(x2+y2)2-12(x2+y2)+3] y ( 10r5 - 12r³ + 3r ) cos 16 20 (x2+y2)3 - 30 (x2+y2)2 + 12 (x2+y2) - 1 20r6 - 30r4 + 12r² - 1 17 (y2-x2) 4xy R4 sin 4 18 y4+x4-6x2y2 R4 cos 4 9 Zernike Polynomials: Fringe Convention 19 (5x2+5y2-4) (3y2-x2)x ( 5r5 - 4r³ ) sin 3 20 (5x2+5y2-4) (y2-3x2)y ( 5r5 - 4r³ ) cos 3 21 [15(x2+y2)2-20(x2+y2)+6] 2xy ( 15r6 - 20r4 + 6r² ) sin 2 22 [15(x2+y2)2-20(x2+y2)+6] (y2-x2) ( 15r6 - 20r4 + 6r² ) cos 2 23 [35(x2+y2)3-60(x2+y2)2+30(x2+y2)-4] x ( 35r7 - 60r5 + 30r³ - 4r ) sin 24 [35(x2+y2)3-60(x2+y2)2+30(x2+y2)-4] y ( 35r7 - 60r5 + 30r³ - 4r ) cos 25 70(x2+y2)4-140(x2+y2)3+90(x2+y2)2-20(x2+y2)+1 70r8 - 140r6 + 90r4 - 20r² + 1 26 (5y4-10x2y2+x4)x R5 sin 5 27 (y4-10x2y2+5x4)y R5 cos 5 28 (6x2+6y2-5) (y2-x2)2xy ( 6r6 - 5r4 ) sin 4 29 (6x2+6y2-5) (y4-6x2y2+x4) ( 6r6 - 5r4 ) cos 4 30 [21(x2+y2)2-30(x2+y2)+10] (3y2-x2)x ( 21r7 - 30r5 + 10r3 ) sin 3 31 [21(x2+y2)2-30(x2+y2)+10] (y2-3x2)y ( 21r7 - 30r5 + 10r3 ) cos 3 32 [ 56(x2+y2)3-105(x2+y2)2+60(x2+y2)-10] 2xy ( 56r8 – 105r6 + 60r4 - 10r2 ) sin 2 33 [ 56(x2+y2)3-105(x2+y2)2+60(x2+y2)-10] (y2-x2) ( 56r8 – 105r6 + 60r4 - 10r2 ) cos 2 34 [ 126(x2+y2)4-280(x2+y2)3+210(x2+y2)2-60(x2+y2)+5] x ( 126r9 – 280r7 + 210r5 – 60r3 + 5r ) sin 35 [ 126(x2+y2)4-280(x2+y2)3+210(x2+y2)2-60(x2+y2)+5] y ( 126r9 – 280r7 + 210r5 – 60r3 + 5r ) cos 36 252(x2+y2)5-630(x2+y2)4+560(x2+y2)3- ( 252r10 – 630r8 + 560r6 – 210r4 + 30r2 - 1 ) 210(x2+y2)2+30(x2+y2)-1 10 Zernike Polynomials: Meaning of Lower Orders n m Polar coordinates Cartesian coordinates Interpretation 0 0 1 1 piston 1 1 r sin x tilt in x 1 - 1 r cos y tilt in y 2 2 r 2 sin 2 2 xy Astigmatism 45° 2 2 2 2 0 2 r 1 2 x + 2 y 1 defocussing 2 - 2 r 2 cos 2 y 2 x 2 Astigmatism 0° 3 3 r 3 sin 3 3 xy 2 x 3 trefoil 30° 3 1 3 3 2 (3 r 2 r )sin 3 x 2 x + 3 xy coma x 3 2 3 - 1 (3 r 3 2r )cos 3 y 2 y + 3 x y coma y 3 - 3 r 3 cos 3 y 3 3 x 2 y trefoil 0° 4 4 r 4 sin 4 4 xy 3 4 x 3 y Four sheet 22.5° 4 2 4 2 3 + 3 (4 r 3r )sin 2 8 xy 8 x y 6 xy Secondary astigmatism 4 4 2 2 2 2 4 0 6r 4 6r 2 + 1 6x + 6 y + 12x y 6x 6 y + 1 Spherical aberration 4 - 2 4 2 4 4 + 2 2 2 2 (4 r 3r )cos 2 4 y 4 x 3 x 3 y 4 x y Secondary astigmatism 4 - 4 r 4 cos 4 y 4 + x 4 6 x 2 y 2 Four sheet 0° 11 Radial Zernike Polynomials . Radial polynomial functions 2 Z4 (r) 2r 1 4 2 Z9 (r) 6r 6r +1 6 4 2 Z16 (r) 20r 30r +12r 1 8 6 4 2 Z25 (r) 70r 140r + 90r 20r +1 10 8 6 4 2 Z36 (r) 252r 630r + 560r 210r + 30r 1 12 10 8 6 4 2 Z49 (r) 924r 2772r + 3150r 1680r + 420r 42r +1 14 12 10 8 6 4 2 Z64 (r) 3432r 12012r +16632r 11550r + 4200r 756r + 56r 1 16 14 12 10 8 6 4 2 Z81(r) 12870r 51480r + 84084r 72072r + 34650r 9240r +1260r 72r +1 Z (r) 48620r18 218790r16 + 411840r14 420420r12 + 252252r10 90090r8 +18480r 6 1980r 4 + 90r 2 1 100 . Oscillating signs corresponds to compensating effects . Large coefficients for higher orders cause numerical inaccuracies for explicite calculations in particular for points near to the edge . Recurence formulas preferred, but residual errors are propagated 12 Indices of Zernike Fringe Polynomials . Indexing of Fringe polynomials . Principle: growing spatial frequency of variations 1. radial 2. azimuthal . Meaningful: truncation at quadratic numbers 4: image location 9: 4th order, primary aberrations 16: 6th order, secondary aberrations ..... Indexing: 1. starting with m=0 2. growing absolute value of m 2 n + m 1 sgn(m) . Running index j +1 2 m + 2 2 13 Indices of Zernike Fringe Polynomials 14 Zernike Standard Polynomials sin(mq ) for m 0 . Normalization of standard Zernike m 2(n +1) m Zn (r,q) Rn (r)cos(mq ) for m 0 polynomials 1+ m0 1 for m 0 1 2 . Orthogonality Z m (r,) Z m'* (r,) drdr n n' nn' mm' 0 0 .
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