IAT13 Imaging and Aberration Theory Lecture 7 Coma and Distortion.Pdf

Imaging and Aberration Theory

Lecture 7: Coma and Distortion 2013-12-12 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. 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

Coma

stop surface R tangential sagittal rays sagittal upper coma ray coma coma

C center of curvature

chief ray astigmatic difference between coma rays auxiliary axis lower coma ray pupil intersection upper n n' point coma ray

sagittal rays . Occurence of coma: chief skew chief ray and finite aperture ray

. Asymmetry between upper and lower coma ray lower coma ray . Bended plane of sagittal coma rays auxiliary axis

5 Ray Caustic of Coma

. A sagittal ray fan forms a groove-like only sagittal surface in the image space coma rays

. Tangential ray fan for coma: bended ray caustic fan

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 chief ray zone 1 zone 2 coma zone 3 blur

lens / pupil

axis

Coma

. Coma deviation, elimination of the azimuthal dependence: circle equation

y' 2 . Diameter of the circle and position variiation with rp Every zone of the circlegenerates a circle in the tan 0°/180° image plane tangential coma . All cricels together form a comet-like shape

. The chief ray intersection point is at the tip of

the cone 45° 135°

. The transverse extension of the cone shape has a ratio of 2:3 the meridional extension is enlarged and gives rp =1.0 a poorer resolution

sag 90°

rp = 0.75

sagittal coma rp = 0.5

30° x'

Coma

. Ray trace properties . Double speed azimuthal growth between pupil and image . Sagittal coma smaller than tangential coma

ytan 3 ysag Pupil Image

xp T

tan genti al ray y’ CT sagit tal ray tangential coma T chief ray S S sagittal ray C l ray x’ CS entia tang sagittal coma C

Ref: H. Zügge

Coma

Primary coma Wave aberration tangential sagittal . Typical representations of coma 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 1 1

sagittal sagittal 0.5 0.5 tangential tangential

0 0 0 100 200 -0.02 0.0 0.02 cyc/mm z/mm

Geometrical spot through

focus 2

0 .

0 -0.02 -0.01 0.0 0.01 0.02 z/mm Ref: H. Zügge

Coma Orientation

y' . Orientation of the coma shape: disctinction between 1. outer coma, tip towards optical axis 2. inner coma, tip outside outer inner coma coma

. Orientation of the coma spot is always y rotating with the azimuthal angle of the considered field point

j x

Bending of a Lens

. Bending a single lens

. Variation of the primary aberrations

. The stop position is important aberration for the for the off-axis distortion aberrations . Typical changes: 1. coma linear transverse coma 2. chromatical magnification chromatical linear aberration p 3. spherical aberration 0 quadratically

stop at lens spherical aberration

12 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 200 and astigmatism possible astigmatism (depends on stop position) 170

140 . Spherical aberration not correctable 110 80 spherical 50

-10

-40 coma -70

-100 1/R1 -0.03 -0.014 0.002 0.018 0.034 0.05

Inner and Outer Coma

y' 0.2 mm . Effect of lens bending on coma . Sign of coma : inner/outer coma

y' 0.2 mm outer coma

large y' incidence angle 0.2 mm for lower coma ray

large incidence angle for upper coma ray inner coma y' 0.2 mm

Ref: H. Zügge 14 Lens Bending and Natural Stop Position

1 n 1  . The lens contribution of coma is given by C   X  (2n 1)M lens 2   if the stop is located at the lens 4ns' f n 1  . Therefore the coma can be corrected by bending the lens . The optimal bending is given by (2n 1)(n 1) X   M and corrects the 3rd order coma completly n 1

. The stop shift equation for coma is given by  SII  SII E  SI with the normalized ratio of the chief ray height h  h to the marginal ray height E  new old h

. If the spherical aberration SI 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 Coma-free Stop Position

stop shifted . Example t1 = 50 . The front stop position of a single lens is shifted t = 114.3 . The 3rd order Seidel coefficient 1 as well as the Zernike coefficient

vanishes at a certain position of the stop t1 = 150

Ac c8 5 0.5 4 0.4 3 0.3 2 0.2 1 0.1 114.3 114.5 0 0 -1 -0.1 -2 -0.2 -3 -0.3 -4 -0.4 t t1 -5 1 -0.5 50 60 70 80 90 100 110 120 130 140 150 50 60 70 80 90 100 110 120 130 140 150

Influence of Stop Position on Coma

Achromat 4/100, w = 10°, y‘ = 17.6

Tranverse aberr. +- 2.0 Spot w = 0° w= 10° w = 10° y‘ y‘ y‘ y‘

Ref: H. Zügge

Coma Correction: Achromate

Image height: y’ = 0 mm y’ = 2 mm . Bending of an achromate Achromat Pupil section: meridional meridional sagittal bending - optimal choice: small residual spherical Transverse y' y' y' Aberration: 0.05 mm 0.05 mm 0.05 mm aberration (a) - remaining coma for finite field size . Splitting achromate: additional degree of freedom:

- better total correction possible (b) - high sensitivity of thin air space . Aplanatic glass choice: vanishing coma . Cases: (c) a) simple achromate, sph corrected, with coma Achromat, splitting b) simple achromate, (d) coma corrected by bending, with sph c) other glass choice: sph better, coma reversed Achromat, aplanatic glass choice

d) splitted achromate: all corrected (e) e) aplanatic glass choice: all corrected

Wave length:

Ref : H. Zügge

Stop Position Influence for Corrected Spherical

Achromat 4/100, w = 10°, y‘ = 17.6, (Asphäre für Sph. Aberr. = 0)

Transverse aberr. +- 2.0 Spot Asphäre w = 0° w= 10° w = 10° y‘ y‘ y‘ y‘

Ref: H. Zügge

Coma Correction: Stop Position and Aspheres

. Combined effect, aspherical case prevents 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

Ref : H. Zügge

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)

Ref: H. Zügge

Geometrical Coma Spot

I a3 1 ExP  inside largestcircle . Geometrical calculated spot  2 2  2Ac  R x  3y intensity I(x, y)   I a3 1 The is a step at the lower circle boundary  ExP  else inside coma shape  A  R 2 2 . The peak lies in the apex point  c x  3y . The centroid lies at the lower circle boundary . The minimal rms radius is

2 R Ac rrms   3 a y I(0,y) [a.u.] 10 3 9

-1 2 1 7

4 1 3

1 x 0 0 y 0 1 2 3

Psf for Coma Aberration

. PSF with coma . The 1st diffraction ring is influenced very sensitive

W31 = 0.03  W31 = 0.06  W31 = 0.09  W31 = 0.15 

Psf with Coma

I(x) 1

x 0 y x -0.05 0 0.05

W13 = 0.3  W13 = 1.0  W13 = 2.4  W13 = 5.0  W13 = 10.0 

Ref: Francon, Atlas of optical phenomena

Transversal Psf with Coma

I(x)

. Change of Zernike coma coefficient 1

W13 = 0.0 - peak height reduced 0.9 W13 = 0.1 - peak position constant due to tilt 0.8 W13 = 0.2 component 0.7 - distribution becomes asymmetrical 0.6 W13 = 0.3 0.5 0.4 W13 = 0.4 0.3 0.2 0.1

0 y -15 -10 -5 0 5 10 15 I(x) Peak 1 W = 0.0 . Change of Seidel coma coefficient 0.9 13

0.8 - peak height reduced W13 = 0.2 - peak position moving 0.7 - distribution becomes asymmetrical 0.6 W13 = 0.3 0.5 W13 = 0.4 0.4 W13 = 0.5 0.3

0.2

0.1

0 y -15 -10 -5 0 5 10 15

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

c7 = 0.3 c7 = 0.5 c7 = 1

centroid

Psf with Coma

x I(z) . Defocus: centroid moves on a straight

line (line of sight) c8 = 0.3 

. Peak of intensity moves on a curve z (bananicity) z

y c8 = 0.5  0.15

0.1

0.05 c15 = 0.5 

c8 = 0.3 

c8 = 0.5 

-0.05 c15 = 0.5 

acoma = 1.7  dashed lines : centroid acoma = 1.7  solid lines : peak -0.1 z -8 -6 -4 -2 0 2 4 6 8

Line of Sight

. Centroid of the psf intensity x I(x, y, z)dxdy 1 x (z)   x I(x, y, z)dxdy s  . Elementary physical argument:  I(x, y, z)dxdy P The centroid has to move on a straight line: line of sight

. Wave aberrations with odd order: - centroid shifted exit y - peak and centroid are no longer pupil coincident chief ray 2 z ys (z)    2(n 1) cn1 centroid ray DExP n1,3,5,... z

reference sphere real image wavefront plane

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

Coma Truncation by Vignetting

without vignettierung

with vignettierung tangential / sagittal

Ref: H. Zügge

Distortion Example: 10%

. What is the type of degradation of this image ? . Sharpness good everywhere !

Ref : H. Zügge

Distortion Example: 10%

. Image with sharp but bended edges/lines . No distortion along central directions

Ref: H. Zügge

Distortion

. Distortion. change of magnification over the field . Corresponds to spherical aberration of the chief ray . Measurement: relative change of image height

y  y y' V  real ideal yideal

h3 ideal h 3 . No image point blurr image aberration only geometrical shape deviation h 2 h . Sign of distortion: 2 h 1. V < 0 : barrel, 1 x' image h1 lens with stop in front height 2. V > 0: pincushion, lens with rear stop

real image

TV Distortion

y . Conventional definition of distortion H y V  y y y . Special definition of TV distortion real ideal H VTV  H H . Measure of bending of lines 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

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 y' . Two primary types with different sign: 1. barrel, D < 0 rear stop pincussion front stop y lens x' distortion 2. pincushion, D > 0 rear stop D > 0 . Definition of local x magnification image changes y'

y' y' front D real ideal object y stop barrel x' distortion y'ideal D < 0 x

Distortion and Stop Position

. Sign of distortion of a single lens Distor- depends on stop position Lens Stop Example tion

. Ray bending of chief ray positive lens rear stop D > 0 Tele lens determines the distorion 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

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

General Distortion Types

original . Non-symmetrical systems: Generalized distortion types . Correction complicated

anamorphism, a10x shear, a01y

1. order linear

2 2 a20x keystone, a11xy line bowing, a02y

2. order quadratic

3 2 2 3 a30x a21x y a12xy a03y

3. order cubic

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: tan w'  const necessary but not sufficient condition for distortion corection: tan w . This corresponds to a corrected angle of the pupil imaging form entrance to exit pupil

y' O'2 O2 EnP ExP real

y2 w' y2 O1 w2 2 O'1

wo'2

ideal p p'

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  ptanw , y' p'tanw'

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)

y' O'2 O2 EnP ExP real

y2 w'2 y2 O1 w2 O'1

ideal p p ' p' o

Distortion of a Retrofocus System

Retro focus- 20 % systems :

barrel distortion

negatives positive front goup stop rear group

41 Scheimpflug Imaging

real

Keystone distortion

 '

principal plane

tilted ideal object y h system optical axis j h' s s' tilted image

Distortion

original 20% keystone barrel and pincushion . Visual impression of distortion on real images

. Visibility only at straight edges

. Edge through the center are 5% barrel 10% barrel 15% barrel not affected

2% pincushion 5% pincushion 10% pincushion

Fish-Eye-Lens

. Example lens with 210°field of view y

-100% 0 100%

Fish-Eye-Lens

y' [a.u.] gnomonic stereographic 2 . Distortion types

f--projection 1.5 orthographic

aperture 1 related

0.5

w 0 0 10 20 30 40 50 60 70 80 90 [°]

a b

Head Mounted Display

. Commercial system: Zeiss Cinemizer

. Critical performance of distortion due to asymmetry

Head-Up Display

binodal y points 8 . Refractive 3D-system 6 . Free-formed prism 4 . Field dependence of coma, distortion and 2 astigmatism 0 x . One coma nodal point -2 . Two astigmatism nodal points -4

-6 image

-8 free formed -8 -6 -4 -2 0 2 4 6 8 surface y 8

6 total internal 4 reflection 2

eye 0 x pupil -2

free formed -4 surface -6 field angle 14° -8 -8 -6 -4 -2 0 2 4 6 8