Imaging and Aberration Theory

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1 Imaging and Aberration Theor Lecture 5: Aberration Reresentations Herbert Gross Winter term 1

2 Preliminar time schedule Paraxial imaging araxial otics, fundamental laws of geometrical imaging, comound sstems 6.1. Puils, Fourier otics, Hamiltonian coordinates uil definition, basic Fourier relationshi, hase sace, Hamiltonian coordinates, analog otics and mechanics Eikonal Fermat Princile, relation ras-waves, inhomogeneous media, geometrical aroximation Aberration exansion single surface, general Talor exansion, various orders Reresentations of aberrations different tes of reresentations, fields of alication, limitations and itfalls Sherical aberration henomenolog, sh-free surfaces, skew sherical, correction of sh, asherical surfaces, higher orders Distortion and coma henomenolog, relation to sine condition, effect of sto osition, various toics, correction otions Astigmatism and curvature henomenolog, Coddington equations, Petzval law, correction otions Sine condition, isolanas sine condition, Herschel condition, isolanas, relation to coma and shift invariance, general conditions, relation to Fourier otics and hase sace Surface contributions sensitivit in 3rd order, structure of a sstem, suerosition and induced aberrations, analsis of otical sstems, lens contributions Wave aberrations definition, relation to transverse aberrations, relation to PSF and OTF, various exansion forms, roagation of wave aberrations Zernike olnomials secial exansion for circular smmetr, roblems, calculation, otimal balancing, influence of normalization, recalculation for offset, elliticit, measurement Miscellaneous Aldi theorem, telecentric case, afocal case, aberration balancing Vectorial aberrations Introduction, secial cases, actual research Statistical aberrations atmosheric turbulence, calculation, satial frequencie contents Phsical model of imaging coherent and incoherent imaging, Fourier descrition, artial coherence

3 3 Contents 1. Sto shift formulas. Lens aberration contributions 3. Incidence angle effects and examles for use of Seidel coefficients 4. Reresentation of geometrical aberrations 5. Reresentation of wave aberrations

4 Sto Shift Formulas If the sto is moved, the chief ra takes a modified wa through the sstem Aroach: exansion of the surface coefficient formulas for small changes in the uil osition, The sto shift formulas shows the change of the Seidel coefficients due to this effect Also ossible: set of formulas for object or image shift alicable for curved objects ra image lane chief ra 1 z r max chief ra 1 ExP 1 ExP

5 Sto Shift Formulas, Sto shift formulas exclicite with the hel of the moving arameter E h new h h old old osition new osition sh S I S I hnew h old coma ast S S II S II E S III SIII E SII E I S I old chief ra new chief ra curv S IV S IV dist S 3S III SIV 3E SII E SI V SV E 3 Mix of aberration tes due to sto shift: induced aberrations Examles: 1. sherical aberration induces coma. coma induces astigmatism

6 Lens Contributions of Seidel In 3rd order (Seidel) : Additive contributions of thin lenses (equal w) to the total aberration value (sto at lens osition) Sherical aberration X: lens bending M: osition arameter S lens 1 3n( n 1) f 3 3 n n 1 n n 1 X ( n 1) M n n ( n 1) M n Coma C 1 n 1 X (n 1 M lens 4ns f n 1 ) Astigmatisms Field curvature A lens P lens 1 f s 4 n nf 1 s Distortion D lens

7 Lens Contributions of Seidel Sherical aberration S lens 1 3n( n 1) f 3 3 n n 1 n n 1 X ( n 1) M n n ( n 1) M n Secial imact on correction: 1. Secial quadratic deendence on bending X Minimum at n 1 Xsh min n M W sh n = 1.5. No correction for small n and M 3. Correction for large n: infrared materials M: virtual imaging Limiting value nn M s n 1 M = - 6 M = - 3 M = M = 3 M = 6 X

8 Photograhic lens Incidence angles for chief and marginal ra Field dominant sstem Quasi smmetr can be seen at the surface contributions of field aberrations Smmetr disturbed for sherical aberration.1 sherical coma -.1 astigmatism curvature Photograhic lens distortion marginal ra axial chromatic. chief ra incidence angle -. lateral.1 chromatic sum

9 Microscoic Objective Lens Incidence angles for chief and marginal ra Aerture dominant sstem marginal ra microscoe objective lens chief ra incidence angle

10 Microscoic Lens Large distance sstem Problems with large diameters Alanatic front grou Not corrected for curvature and distortion Astigmatic contributions of cemented surfaces corrected b rear grou Sign of lateral chromatic aberration in front grou sherical coma astigmatism curvature distortion axial chromatic lateral chromatic sum

11 Lithograhic Lens Large effect of mirror on field curvature intermediate image concave mirror. intermediate image Tical bulge structure shows the correction of field curvature according to the Petzval theorem 1. bulge 1. waist. bulge. waist 3. bulge

12 Primar Aberration Sot Shaes Simlified set of Seidel formulas: field oint onl in considered Sherical aberration S: circle Coma: shifted circle Astigmatism: focal line Field curvature: circle Distortion: shifted oint P P P P r P r C r S x D r P A r C r S sin sin sin cos ) ( ) cos ( cos sin, cos r S x r S x r S 4 sin, ) cos ( P P r C r C x r C x r C, cos x r A P 4 sin, cos P P r P x r P x r P, 3 x D

13 Primar Aberration Sot Shaes Schematicall:

14 Otical Image Formation Perfect otical image: All ras coming from one object oint intersect in one image oint Real sstem with aberrations: 1. tranverse aberrations in the image lane. longitudinal aberrations from the image lane 3. wave aberrations in the exit uil object lane wave aberrations image lane otical sstem transverse aberrations longitudinal aberrations

15 Reresentation of Geometrical Aberrations Longitudinal aberrations s reference ra ra logitudinal aberration along the reference ra l Gaussian image lane ra Gaussian image lane U reference oint otical axis l o otical axis sstem longitudinal aberration rojected on the axis sstem s Transverse aberrations longitudinal aberration reference ra (real or ideal chief ra) transverse aberration ra U otical axis reference lane sstem

16 Reresentation of Geometrical Aberrations Angle aberrations u ideal reference ra U angular aberration real ra otical axis sstem Wave aberrations W x reference shere wavefront W > Gaussian reference lane araxial ra real ra U C z R s <

17 Aerture Deendence of Longitudinal Aberration Tical reresentation: Longitudinal aberration as function of aerture (uil coordinate) If correction at the edge: maximum residuum at the zone 1/ Tical: largest gradients at the edge r l = 644 nm l = 546 nm l = 48 nm s r Correcting asheres or high NA of higher order: oscillator behavior -3.E-3-.4E-3-1.8E-3-1.E-3-6.E-4.E+6.E-41.E-31.8E-3.4E-3 s

18 Longitudinal Aberration Chart sherical aberration 4 colors coma in zone 4 colors coma in full field 4 colors secondar chromatic chromatical difference in magnification distortion main color image shells/astigmatism 4 colors

19 Transverse Aberrations Tical low order olnomial contributions for: defocus, coma, sherical aberration, lateral color This allows a quick classification of real curves K r cos S r 3 cos C r ( cos ) P linear: defocus quadratic: coma cubic: sherical offset: lateral color

20 Transverse Aberrations Classical aberration curves Strong relation to sot diagram Usuall onl linear samling along the x-, -axis no information in the quadrant of the aerture tangential 5 m sagittal x 5 m x D x D l= 486 nm l= 588 nm l= 656 nm

21 Best Image lane Best resolution: - bright central eak in sot - tangent at transverse aberration curve max best matching in the centre Best contrast: - mean straight line over comlete uil of transverse aberration curve - smallest maximal deviation Different criteria give slightl different best image lanes minimal difference min u max z W rms D s z

22 Transverse Aberrations Characteristic chart for the reresentation of transverse aberrations axis field zone full field meridional deviation wavelengths: 365 nm 48 nm 546 nm 644 nm x x x x sagittal deviation

23 Puil Aberration Characteristic chart for the reresentation of uil aberration Distortion of the uil grid from the entrance to the exit uil Puil aberration can be interreted as the sherical aberration of the chief ra for the uil imaging / %.4.3 ideal real 1% r x

24 Variation of Chromatical Aberrations Reresentation of the image location as a function of the wavelength axial chromatical shift 6 nm l 5 nm 4 nm s [m] Reresentation of the chromatical magnification difference with field height lateral chromatical aberration Air field height / max [m]

25 Sot Diagram All ras start in one oint in the object lane The entrance uil is samled equidistant In the exit uil, the transferred grid ma be distorted In the image lane a sreaded sot diagram is generated object lane oint entrance uil equidistant grid otical sstem exit uil transferred grid image lane sot diagram o x o x x x z

26 Sot Diagram Variation of field and color Scaling of size: 1. Air diameter (small circle). nd moment circle (larger circle, scales with wavelength) 3. surrounding rectangle 486 nm 546 nm 656 nm axis field zone full field

diameter Generalized: Ras with weighting factor g j : corresonds to aodization M G j

27 Gaussian Moment of Sot Sot attern with transverse aberrations x j and j 1. centroid xs 1 x j S 1 j N N. nd order moment M 3. diameter Generalized: Ras with weighting factor g j : corresonds to aodization M G j x x r 1 N G j S j S j D M G Worst case estimation: size of surrounding rectangle D x =x max, D = max j 1 r g x x N G j j j S j S

erformance with the distance to the axis x Examle HMD without smmetr image 8 free formed

28 Aberrations of a Single Lens Single lane-convex lens, BK7, f = 1 mm, l = 5 nm Sot as a function of field osition Coma shae rotates according to circular smmetr Decrease of erformance with the distance to the axis x Examle HMD without smmetr image 8 free formed surface 6 4 total internal reflection x ee uil - -4 free formed surface -6 field angle

aberration intrafocal: ring structurewith bright center extrafocal: ring structure with

29 Caustic of Sherical Aberration and Coma negativ sherical aberration intrafocal: comact broadened sot with bright edge extrafocal: ring structure sherical negativ ositiv sherical aberration intrafocal: ring structurewith bright center extrafocal: ring structure with bright outer ring sherical ositiv coma bending of caustic shifted center of gravit coma z Ref: W. Singer

30 . mm Sherical Aberration Tical chart of aberration reresentation Reference: at araxial focus Primar sherical aberration at araxial focus Wave aberration tangential sagittal l l Transverse ra aberration x.1 mm.1 mm.1 mm Puil: -section x-section x-section Modulation Transfer Function MTF MTF at araxial focus MTF through focus for 1 ccles er mm cc/mm -... z/mm Geometrical sot through focus Ref: H. Zügge z/mm

31 Wave Aberration Definition of the eak valle value v-value of wave aberration wave aberration image lane hase front exit aerture reference shere

32 Wave Aberrations Classification of wave aberrations for one image oint: Zernike olnomials Mean root square of wave front error W rms Normalization: size of uil area A ExP W dxd W Worst case / eak-valle wave front error v 1 A x, W x, Generalized for aodized uils (non-uniform illumination) ExP W x, W x W max W, W rms 1 A max min mean ( w) x, W x, W x I w ExP mean, ( ) ExP dx dx d d

33 Primar Aberrations Te Wave aberration Geometrical sot Relation : wave / geometrical aberration Sherical aberration Smmetr to Periodicit W c 1 r x axis constant 3 W c r cos 4 x c 1 r c 1 r 3 3 sin cos oint 1 eriod cr (cos x c r sin ) Coma Smmetr to Periodicit x one lane 1 eriod W c3 r cos one straight line eriods x c3 r cos Astigmatism Smmetr to Periodicit Field curvature (sagittal) x two lanes eriod W c4 r two straight lines 1 eriod x c4 r sin c cos 4 r Smmetr to Periodicit x axis constant 3 W c5 r cos oint 1 eriod x c 5 3 Distortion Ref: H. Zügge Smmetr to Periodicit x one lane 1 eriod one straight line constant

34 Tical Variation of Wave Aberrations Microscoic objective lens: Changes of rms value of wave aberration with wavelength W rms [l].3.4 Achrolan 4x.65 Common ractice: 1. diffraction limited on axis for main art of the sectrum. Requirements relaxed in the outer field region 3. Requirement relaxed at the blue edge of the sectrum Reresentation of the wave aberration with field osition field zone diffraction limit field edge on axis l [m] W rms [l] diffraction limit l= 48 nm l= 546 nm l= 644 nm field w [ ]

Tical Variation of Wave Aberrations Reresentation

field oints - decrease of erformance with field

35 Tical Variation of Wave Aberrations Reresentation of the wave aberration for defocussing at several field oints - decrease of erformance with field height - field curvature Wavefront over the uil as surface W rms [l] axis zone full field axis field zone full field +1 l -1 l

$diffraction limited correction near to axis for medium wavelength range. no flattening 3. blue edge more critical than red edge.4.3..1 diffraction limit.48.5.5.54.56.58.6.6.64.6.714 l [m]$

36 Tical Variation of Wave Aberrations Reresentation of the wave aberration as a function of field and wavelength for a microscoic lens Analsis: 1. diffraction limited correction near to axis for medium wavelength range. no flattening 3. blue edge more critical than red edge diffraction limit l [m]

37 Satial Frequenc of Wavefront Aberrations The satial frequenc determines the effect of the wave front aberration Characteristic ranges, scaled on the diameter of the uil: - figure error : Zernike causes resolution loss - midfrequenc range - high frequenc : roughness causes contrast loss log I(r) g/d = 4 g/d = g/d = 1 g/d =.5 g/d =.1 g/d = g/d = r/r air

38 Satial Frequenc of Surface Perturbations Power sectral densit of the erturbation Three tical frequenc ranges, scaled b diameter D log A Four oscillation of the olishing machine limiting line sloe m = long range low frequenc figure Zernike mid frequenc 1/D 1/D 4/D micro roughness 1/l

Imaging and Aberration Theory

Imaging and Aberration Theory Imaging and Aberration Theor Lecture 5: Reresentations of Aberrations 7--3 Herbert Gross Winter term 7 www.ia.uni-jena.de Preliminar time schedule 6.. Paraxial imaging araxial otics, fundamental laws of