Imaging and Aberration Theory

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1 Imaging and Aberration Theory Lecture 7: Distortion and coma Herbert Gross Winter term

2 2 Preliminary time schedule Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems Pupils, Fourier optics, pupil definition, basic Fourier relationship, phase space, analogy optics and Hamiltonian coordinates mechanics, Hamiltonian coordinates Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum 10 Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil isoplanatism aberrations, Herschel condition, relation to Fourier optics Wave aberrations definition, various expansion forms, propagation of wave aberrations Zernike polynomials PSF and transfer function Additional topics special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability

3 3 Contents 1. Geometry of coma spot 2. Coma-dependence on lens bending, stop position and spherical aberration 3. Point spread function with coma 4. Distortion 5. Examples

4 4 Ray Caustic of Coma A sagittal ray fan forms a groove-like surface in the image space coma only sagittal rays Tangential ray fan for coma: caustic bended ray fan

5 5 Building of Coma Spot Coma aberration: for oblique bundels and finite aperture due to asymmetry Special problem: coma grows linear with field size y Systems with large field of view: coma hard to correct Relation of spot circles and pupil zones as shown coma blur chief ray zone 1 zone 2 zone 3 lens / pupil axis

6 6 Coma Coma deviation, elimination of the azimuthal dependence: circle equation Diameter of the circle and position variiation with r p 2 Every zone of the circlegenerates a circle in the image plane All cricels together form a comet-like shape The chief ray intersection point is at the tip of the cone tangential coma y' tan 0 / The transverse extension of the cone shape has a ratio of 2:3 the meridional extension is enlarged and gives a poorer resolution r p =1.0 sag 90 r p = 0.75 sagittal coma r p = x'

7 7 Coma Ray trace properties Double speed azimuthal growth between pupil and image Sagittal coma smaller than tangential coma ytan 3 y sag Pupil Image y p x p T tangential ray sagittal ray y T CT tangential coma sagittal ray tangential ray chief ray S C x S CS sagittal coma C Ref: H. Zügge

8 8 Coma stop surface R sagittal rays upper coma ray tangential coma sagittal coma S T C center of curvature n auxiliary axis n' chief ray lower coma ray astigmatic difference between coma rays pupil intersection point upper coma ray Occurence of coma: skew chief ray and finite aperture Asymmetry between upper and lower coma ray chief ray lower coma ray sagittal rays Bended plane of sagittal coma rays auxiliary axis

9 0.02 mm 9 Coma Typical representations of coma Primary coma Wave aberration tangential sagittal 2 2 Cubic curve in wavefront cross section Quadratic function in transverse aberrations 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 tangential sagittal 0.5 sagittal tangential cyc/mm z/mm Geometrical spot through focus Ref: H. Zügge z/mm

10 10 Bending of a Lens Bending a single lens with stop at the lens Variation of the primary aberrations The stop position is important for the for the off-axis aberrations Typical changes: 1. coma linear 2. chromatical magnification linear 3. spherical aberration quadratically aberration coma 0 distortion transverse chromatical aberration p spherical aberration stop at lens

11 11 Lens with Remote Stop Lens with remote stop Not all of the aberrations spherical, astigmatism and coma can be corrected by bending simultaneously Zero correction for coma and astigmatism possible (depends on stop position) Spherical aberration not correctable astigmatism spherical coma /R 1

12 12 Inner and Outer Coma Effect of lens bending on coma Sign of coma : inner/outer coma y' 0.2 mm y' 0.2 mm outer coma large incidence angle for upper coma ray large incidence angle for lower coma ray inner coma y' 0.2 mm y' 0.2 mm Ref: H. Zügge

13 13 Coma Orientation Orientation of the coma shape: disctinction between 1. outer coma, tip towards optical axis 2. inner coma, tip outside outer coma y' inner coma x' Orientation of the coma spot is always rotating with the azimuthal angle of the considered field point y r j x

14 14 Lens Bending and Natural Stop Position The lens contribution of coma is given by if the stop is located at the lens C 1 n 1 X (2n 1 M lens 4ns' f n 1 ) 2 Therefore the coma can be corrected by bending the lens The optimal bending is given by and corrects the 3rd order coma completly X ( 2n 1)( n 1) M n 1 The stop shift equation for coma is given by with the normalized ratio of the chief ray height to the marginal ray height S II E S h II new E S h h old I If the spherical aberration S I is not corrected, there is a natural stop position with vanishing coma If the spherical aberration is corrected (for example by an aspheric surface), the coma doesn't change with the stop position

15 15 Influence of Stop Position on Coma Achromat 4/100, w = 10, y = 17.6 Ref: H. Zügge

16 16 Coma-free Stop Position Example The front stop position of a single lens is shifted The 3rd order Seidel coefficient as well as the Zernike coefficient vanishes at a certain position of the stop t 1 = 50 t 1 = t 1 = 150 stop shifted 5 A c 0.5 c t t 1

17 17 Coma Correction: Achromate Bending of an achromate - optimal choice: small residual spherical aberration - remaining coma for finite field size Splitting achromate: additional degree of freedom: - better total correction possible - high sensitivity of thin air space Aplanatic glass choice: vanishing coma Cases: a) simple achromate, sph corrected, with coma b) simple achromate, coma corrected by bending, with sph c) other glass choice: sph better, coma reversed d) splitted achromate: all corrected e) aplanatic glass choice: all corrected (a) (b) (c) (d) (e) Achromat bending Achromat, splitting Achromat, aplanatic glass choice Image height: y = 0 mm y = 2 mm Pupil section: meridional meridional sagittal Transverse y' y' y' Aberration: 0.05 mm 0.05 mm 0.05 mm Wave length: Ref : H. Zügge

18 18 Stop Position Influence for Corrected Spherical Achromat 4/100, w = 10, y = 17.6, (Asphäre für Sph. Aberr. = 0) Ref: H. Zügge

19 19 Coma Correction: Stop Position and Aspheres Combined effect, aspherical case prevents correction Plano-convex element exhibits spherical aberration Sagittal coma y' 0.5 mm Spherical aberration corrected with aspheric surface aspheric Sagittal coma y' 0.5 mm aspheric aspheric Ref : H. Zügge

20 20 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 Aberration: y' 0.5 mm y' 0.5 mm (a) (b) Ref: H. Zügge

21 Geometrical Coma Spot Geometrical calculated spot intensity The is a step at the lower circle boundary The peak lies in the apex point The centroid lies at the lower circle boundary The minimal

21 21 Geometrical Coma Spot Geometrical calculated spot intensity The is a step at the lower circle boundary The peak lies in the apex point The centroid lies at the lower circle boundary The minimal rms radius is 3 I ExP a 2Ac R I( x, y) 3 I ExP a A R c x x y 1 3y 2 2 inside largest circle else inside coma shape r rms 3 2 R A a c 3 y I(0,y) [a.u.] x y

22 22 Zonal Curves of Coma with Defocus Transverse aberrations in the case of coma and defocus H L 1 2 W rp H 2 2W W Hr p cos r p Two deviations: 1st term along field vector 2nd term along pupil vector Zonal curve for different defocus values: Limacon of Pascal H 2( b acos ) special cases cardio / double circle in focal point b/a = 2.0 b/a = 1.5 cardiodid b/a = 1.0 b/a = 0.8 b/a = 0.4 defocus b/a = 0.2 b/a = 0.05 double circle focal point b/a = 0.0 chief ray

$diffraction$ ring is

23 23 Psf for Coma Aberration PSF with coma The 1st diffraction ring is influenced very sensitive W 31 = 0.03 W 31 = 0.06 W 31 = 0.09 W 31 = 0.15

24 Psf with Coma I(x) 1 0-0.05 0 0.05 x y x W 13 = 0.3 W 13 = 1.

24 24 Psf with Coma I(x) x y x W 13 = 0.3 W 13 = 1.0 W 13 = 2.4 W 13 = 5.0 W 13 = 10.0 Ref: Francon, Atlas of optical phenomena

25 25 Transversal Psf with Coma Change of Zernike coma coefficient - peak height reduced - peak position constant due to tilt component - distribution becomes asymmetrical I(x) W 13 = 0.0 W 13 = 0.1 W 13 = 0.2 W 13 = 0.3 W 13 = y Change of Seidel coma coefficient - peak height reduced - peak position moving - distribution becomes asymmetrical I(x) Peak W 13 = 0.0 W 13 = 0.2 W 13 = W 13 = 0.4 W 13 = y

26 26 Psf with Coma Separation of the peak and the centroid position in a point spread function with coma From the energetic point of view coma induces distortion in the image c 7 = 0.3 c 7 = 0.5 c 7 = 1 centroid

27 27 Psf with Coma Defocus: centroid moves on a straight line (line of sight) c 8 = 0.3 x I(z) Peak of intensity moves on a curve (bananicity) z z 0.15 y c 8 = c 15 = c 8 = 0.3 c 8 = 0.5 c 15 = 0.5 a coma = 1.7 dashed lines : centroid solid lines : peak z a coma = 1.7

28 28 Line of Sight Centroid of the psf intensity Elementary physical argument: The centroid has to move on a straight line: line of sight x I( x, y, z) dxdy 1 x s ( z) x I( x, y, z) dxdy I( x, y, z) dxdy P Wave aberrations with odd order: - centroid shifted - peak and centroid are no longer coincident y s 2 z ( z) D ExP n1,3,5,... 2( n 1) c n1 exit pupil chief ray centroid ray y z real wavefront reference sphere image plane

$4 in x and y Structure size near the diffraction limit Asymmetry due to coma seen in comparison of edge slopes object Psf with 0.$

29 29 Image Degradation by Coma Imaging of a bar pattern with a coma of 0.4 in x and y Structure size near the diffraction limit Asymmetry due to coma seen in comparison of edge slopes object Psf with 0.4 x-coma image Psf with 0.4 y-coma image psf enlarged image x/y section psf enlarged image x/y section

30 30 Coma Truncation by Vignetting without vignettierung with vignettierung tangential / sagittal Ref: H. Zügge

31 31 Distortion Example: 10% What is the type of degradation of this image? Sharpness good everywhere! Ref : H. Zügge

32 32 Distortion Example: 10% Image with sharp but bended edges/lines No distortion along central directions Ref: H. Zügge

33 33 Distortion Distortion. change of magnification over the field Corresponds to spherical aberration of the chief ray Measurement: relative change of image height V y y real y ideal ideal y' No image point blurr only geometrical shape deviation Sign of distortion: 1. V < 0 : barrel, lens with stop in front 2. V > 0: pincushion, lens with rear stop ideal image h 3 h 1 image height h2 h 3 h 2 h 1 aberration x' real image

34 34 TV Distortion y Conventional definition of distortion V y y Special definition of TV distortion H V TV H Measure of bending of lines y real y ideal H H x Acceptance level strongly depends on kind of objects: 1. geometrical bars/lines: 1% is still critical 2. biological samples: 10% is not a problem Digital detection with image post processing: un-distorted image can be reconstructed

35 35 Distortion Purely geometrical deviations without any blurr Distortion corresponds to spherical aberration of the chief ray Important is the location of the stop: defines the chief ray path Two primary types with different sign: 1. barrel, D < 0 front stop 2. pincushion, D > 0 rear stop Definition of local magnification changes y x lens rear stop y' y' image x' pincussion distortion D > 0 D y' y' y' real ideal ideal object y front stop x' barrel distortion D < 0 x

36 36 Distortion and Stop Position Sign of distortion of a single lens depends on stop position Ray bending of chief ray determines the distorion Lens Stop Distortion Example positive lens rear stop D > 0 Tele lens negative lens front stop D > 0 Loupe positive lens front stop D < 0 Retro focus lens negativ lens rear stop D < 0 reversed Binocular Positive, pincushion Negative, barrel Ref: H. Zügge

37 37 Distortion of Higher Order Combination of distortion of 3rd and 5th order: Bended lines with turning points Typical result for corrected/compensated distortion object 5th order positive 3rd order negative image

38 38 General Distortion Types original Non-symmetrical systems: Generalized distortion types Correction complicated anamorphism, a 10 x shear, a 01 y 1. order linear a 20 x 2 keystone, a 11 xy line bowing, a 02 y 2 2. order quadratic a 30 x 3 a 21 x 2 y a 12 xy 2 a 03 y 3 3. order cubic

39 39 Reasons of Distortion Distortion occurs, if the magnification depends on the field height y In the special case of an invariant location p of the exit pupil: the tangent of the angle of the chief ray should be scaled linear Airy tangent condition: necessary but not sufficient condition for distortion corection: This corresponds to a corrected angle of the pupil imaging form entrance to exit pupil tan w' tan w const y' O 2 EnP ExP real O' 2 y 2 O 1 w 2 w' 2 O' 1 y 2 w o ' 2 p ideal p'

40 40 Reasons of Distortion Second possibility of distortion: the pupil imaging suffers from longitudial spherical aberration The location of the exit pupil than depends on the field height With the simple relations y ptan w, y' p' tan we have the general expression y' p'( y) tan w' po' p'( y) tan w' m( y) for the magnification y p tan w p p tan w For vanishing distortion: 1. the tan-condition is fulfilled (chief ray angle) 2. the spherical aberration of the pupil imaging is corrected (chief ray intersection point) w' O 2 EnP ExP real O' 2 y' y 2 O 1 w w' 2 2 O' 1 y 2 p ideal p' p o '

41 41 Distortion of a Retrofocus System Retro focussystems : 20 % barrel distortion barrel distortion negatives front goup stop positive rear group

42 42 Scheimpflug Imaging real Keystone distortion ' principal plane tilted object h y s system j optical axis h' ideal s' tilted image y'

43 43 Distortion original 20% keystone barrel and pincushion Visual impression of distortion on real images Visibility only at straight edges Edge through the center are not affected 5% barrel 10% barrel 15% barrel 2% pincushion 5% pincushion 10% pincushion

44 44 Fish-Eye-Lens Example lens with 210 field of view y -100% 0 100%

45 45 Fish-Eye-Lens Distortion types y' [a.u.] 2 gnomonic stereographic 1.5 f--projection orthographic 1 aperture related w [ ] a b

46 46 Head Mounted Display Commercial system: Zeiss Cinemizer Critical performance of distortion due to asymmetry

47 47 Head-Up Display Refractive 3D-system Free-formed prism Field dependence of coma, distortion and astigmatism One coma nodal point Two astigmatism nodal points y binodal points x image -6 free formed surface y total internal reflection eye pupil 0-2 x free formed surface -4-6 field angle

Advanced Lens Design

Advanced Lens Design Lecture 3: Aberrations I 214-11-4 Herbert Gross Winter term 214 www.iap.uni-jena.de 2 Preliminary Schedule 1 21.1. Basics Paraxial optics, imaging, Zemax handling 2 28.1. Optical systems