Imaging and Aberration Theory

Lecture 4: Aberration expansions 2018-11-08 Herbert Gross

Winter term 2019 www.iap.uni-jena.de 2 Schedule - Imaging and aberration theory 2019

1 18.10. 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 25.10. Hamiltonian coordinates mechanics, Hamiltonian coordinates Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical 3 01.11. Eikonal approximation, inhomogeneous media single surface, general Taylor expansion, representations, various orders, stop 4 08.11. Aberration expansions shift formulas different types of representations, fields of application, limitations and pitfalls, 5 15.11. Representation of aberrations measurement of aberrations phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical 6 22.11. Spherical aberration surfaces, higher orders phenomenology, relation to sine condition, aplanatic sytems, effect of stop 29.11. 7 Distortion and coma position, various topics, correction options 8 06.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options Dispersion, axial chromatical aberration, transverse chromatical aberration, 9 13.12. Chromatical aberrations spherochromatism, secondary spectrum Sine condition, aplanatism and Sine condition, isoplanatism, relation to coma and shift invariance, pupil 10 20.12. isoplanatism aberrations, Herschel condition, relation to Fourier optics 11 10.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations special expansion for circular symmetry, problems, calculation, optimal balancing, 12 17.01. Zernike polynomials influence of normalization, measurement 13 24.01. Point spread function ideal psf, psf with aberrations, Strehl ratio 14 31.01. Transfer function transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic 15 07.02. Additional topics and induced aberrations, reversability 3 Contents

. Repetition... . Single refracting surface . Adaptation on optical system terms . Primary monochromatic aberration surface contributions . Extension on several surfaces . Lens contributions . Examples 4 Eikonal Overview

. Fermat principle: light takes the fastest way P2 between two points  L   n(x, y, z) ds 0 The variance of the OPL is zero, stationary point P1

. The OPL is the Lagrange function in optics L L  n s ,  n s the equation of motion is x' x y' y  . This is the Eikonal equation of geometrical optics d  dr  n  n ds  ds 

. One ray is fixed by 4 parameters object image optical space angle space system (direction) angle . The general 4x4 functional relation (direction) between object and image space is point s s' restricted by the Fermat principle: r point r' only 4 independent variables

. A Legendre transform allows to switch between the variables

L(x, y, sx , sy )  nsx  x'nsy  y' L(x, y, x', y')

. Every selection of eikonal variables has at least one case, which is singular 5 Eikonal Overview

. Point eikonal: spatial coordinates in dL (x, y, x', y')  n's 'dx's 'dy' ns dx  s dy object and image space are fixed P x y x y the corresponding angles can be calculated  LP  LP  ns  nsy by the differential equations  x x  y  L P  LP  n'sx '  n's '  x'  y' y . Small change of initial ray data in the object space: Fermat principle allows to calculate the corresponding change in the image space A' A R'     Q L  n's'dr' ns dr s' q' Q' dr s q P' dr' R P

. Special case q = 90°delivers the Abbe sine condition nsin u y' m   n'sin u' y 6 Refracting Surface I: Basic Eikonal Approach

. P, P' points on ray . A, A' arbitrary points on axis 2 2 2 2 . s,s' direction unit vectors of the rays sz  1 sx  sy , s'z  1 s'x s'y relationship 2 2 2 2 2 . Real surface equation x  y x  y  z   3 (1 b)... contains system parameters R, b 2R 8R . Angle eikonal, gives the optical path length change in P', if the point P is varied         dLA (s,s')  n r  a ds  n'r'a' ds' In coordinate representation dLA  n x  dsx  y  dsy  z  a dsz  n'x  ds'x  y  ds'y z  a' ds'z 

y Q surface

s' s P P' R n r n'

M z A a z a' A'

O origin 7 Refracting Surface II: Eikonal

.Integration of the differential to get the total optical path length

    dLA n x ds x y ds y z a ds z   n'''''  x ds x  y ds y  z a ds z 

.Result of 4th order Taylor approximation

LA   na  n'a' 2 2  x  y a 2 2   n x  sx  y  sy   sx  sy   2R 2  2 2  x  y a' 2 2   n'x  s'x  y  s'y   s'x s'y   2R 2  2 2 2 2 1 b 2 2 2 x  y sx  sy  a 2 2 2   n  3 x  y    sx  sy    8R 4R 8  2 2 2 2 1 b 2 2 2 x  y s'x s'y  a' 2 2 2   n' 3 x  y    s'x s'y    8R 4R 8 

. Elimination of intermediate variables x, y of the intersection point with the help of the law of refraction 8 Refracting Surface III: Law of Refraction

2 x2  y 2 (1 b)x2  y 2  . Implicite surface equation F(x, y, z)   z   0 2R 8R3 . Normal unit vector in the intersection point  e F(x, y, z) x  (1 b) x2  y2  y  (1 b) x2  y2  ex   1 2  , ey   1 2  , ez 1 R  2R  R  2R     . Law of refraction (n s n's')e  0 . Formulation in components y  (1 b) x2  y2   (nsy  n's'y )(nsz  n's'z )   1 2  0 R  2R  x  (1 b) x2  y2   (nsx  n's'x )(nsz  n's'z )   1 2  0 R  2R  y  (1 b) x2  y2  x  (1 b) x2  y2  (nsx  n's'x )   1 2   (nsy  n's'y )   1 2  0 R  2R  R  2R  . Solving for x, y in approximation of 4th order ns  n's' ns  n's' x  R  x x , y  R  y y n  n' n  n' 9 Refracting Surface IV: Eikonal

. Resulting Eikonal function of sx, sy, s‘x, s‘y only

(0) (2) (4) LA  LA  LA  LA

R 2 2 na 2 2 n'a' 2 2   na  n'a' nsx  n's'x   nsy  n's'y    sx  sy  s'x s'y  2(n  n')  2 2 

R 2 2 2 2 2 2  nsx  n's'x   nsy  n's'y   nsx  sy  n's'x s'y  4(n  n')2

1 2 2 2 2 2 2 (1 b)R 2 2  na sx  sy   n'a's'x s'y   nsx  n's'x   nsy  n's'y   8 8(n  n')3

. Further re-arrangements for better practical usage:

1. introduction of the pupil coordinates xp, yp instead of the image sides directions s‘x, s‘y 2. switch to the image space ray parameters 3. optional switch to circular coordinates 4. according to Seidel substitution of the system parameter by the paraxial properties of the system and the marginal ray 5. Further approximation in 4th order to get a perturbation representation of the paraxial imaging. This allows a decoupling of the surface contributions 10 Refracting Surface V: Paraxial Optics

(2) . 2nd order: paraxial optics 1  LA  nR  n' R x   sx a    s'x  n  sx  n  n'  n  n' (2) 1  LA  nR  n' R y   sy a    s'y  n  sy  n  n'  n  n' (2) 1  LA  n' R  nR x'   s'x a'   sx  n'  s'x  n  n'  n  n' (2) 1  LA  n' R  nR y'   s'y a'   sy  n'  s'y  n  n'  n  n'

a'  n' n  n'  aa'  n' n n  n'  x' x      s     n' a' R n' x a' a R . Eliminating s'x, s'y     a'  n' n  n'  aa'  n' n n  n'  y' y      sy     n'  a' R  n'  a' a R 

. Imaging condition: x‘, y‘ independent from ray directions sx, sy n n' n  n' 1. imaging equation   a a' R 2. magnification na' x' na' x' x , m   n'a x n'a 11 Refracting Surface VI: Pupil Coordinates

y yp . Change to pupil coordinates object entrance system for optical systems plane pupil surface

. Relations arcsin(sy) x  x s x  x tanu  p  x , s  p  s x x z arcsin(sz) y p sz p p y y  y p sy y  y p tanu y   , sy   sz p sz p p s x'x' x'x' p p s'x p s tanu'x   , s'x   s'z p' s'z p'

y'y' p s'y y'y' p tanu'y   , s'y   s'z p' s'z p'

. Approximated to 2nd order x  x y  y s  p , s  p x p y p x'x' y'y' s'  p , s'  p x p' y p' 12 Refracting Surface VII: Eikonal

. Eikonal of 4th order 4 4 2 2 (s  p) 2 s 2 s (s  p) 2 L(4)  K  x2  y 2   S  x2  y 2   A xx  yy  A 8p4 8p4 p p 2 p4 p p s2 (s  p)2 s (s  p)3 s3 (s  p)  P  x2  y 2 xx  yy  D  x2  y 2 xx  yy  C  x2  y 2 xx  yy  4 p4 p p 2 p4 p p 2 p4 p p

2 2 (n'n)b  1 1   1 1  . Coefficients K    ns     n's'   3  2   2  R  Rs s p   Rs' s' p  1. Spherical aberration (n'n)b  1 1  S    Q2    R3  ns n's' 

2. Astigmatism (n'n)b 2  1 1  A    Qp    R3  ns n's'  (n'n)b  1 1   1 1  3. Field curvature P    QQ     Q(Q  Q )   3 p p   R  ns n's'   ns p n's p '  (n'n)b  1 1   1 1  4. Distortion D    Q2     Q (Q  Q )   3 p p p   R  ns n's'   ns p n's p '  (n'n)b  1 1  5. Coma C    QQp    R3  ns n's'  13 Refracting Surface VIII: Aberrations

. Corresponding differential equations of the angle eikonal determine the changes in the spatial coordinates

. Calculation of the transverse aberrations: separation of the paraxial and the perturbation part 1  L 1  L x'm  x  x'  A , y'm  y  y'  A n'  sx ' n'  sy ' 11LL(4) (4) xy','   AA     n'''' sxy n s

. Used paraxial abbreviations: y' p s  p' 1. pupil imaging magnification mp   y p s'p

 1 1   1 1  2. Abbe invariants Q  n  , Q  n     p    R s   R s p  3. usual special Picht-operator: difference before / after the refraction surface

 n  n n' n n  n' ,    , , ...  s  s s' 14 Refracting Surface IX: Transverse Aberrations

. Transverse aberrations x' s'4 x' s'2 s'2 x' s'3 s' x' S  p x'2  y'2  P  p p x'2  y'2  C  p p x' x'  y' y'  2n' p'3 p p 2n' p'3 n' p'3 p p x's'3 s' x's's'3 x's'2 s'2  A p x'2  y'2  D  p x'2  y'2  C  p x' x'  y' y'  2n' p'3 p p 2n' p'3 n' p'3 p p y' s'4 y' s'2 s'2 y' s'3 s' y' S  p x'2  y'2  P  p p x'2  y'2  C  p p x' x'  y' y'  2n' p'3 p p 2n' p'3 n' p'3 p p y's'3 s' y's's'3 y's'2 s'2  A p x'2  y'2  D  p x'2  y'2  C  p x' x'  y' y'  2n' p'3 p p 2n' p'3 n' p'3 p p

. Due to the derivative, only 5 terms remain for the primary aberrations,

the transverse aberrations are of 3rd order in the coordinates (sum of powers in x‘,y‘,s‘x,s‘y)

. For the special cases of s' 1. image in infinity s' p  2. exit pupil in infinity there are particular sets of formulas

. Possible variables: different nomenclatures

x‘,y‘, x‘p,y‘p, s‘, s‘p, p‘, A,A‘, i, i‘ 15 Generalization on Optical Systems

. Paraxial optics: small field and aperture angles, Aberrations occur for larger angle values optical . Two-dimensional Taylor expansion shows field yp axis entrance and aperture dependence pupil xp . Expansion for one meridional field point y xp coma rays chief . Pupil: cartesian or polar grid in xp / yp outer rays of ray aperture cone r

q ray yp

O field sagittal point plane

object height y meridional plane

axis point

object plane 16 Notations for an Optical System

x, y object coordinates, especially object height x', y' image coordinates, especially image height

xp,yp coordinates of entrance pupil y' x'p, y'p coordinates of exit pupil s object distance form 1st surface x' x' P' s' image distance form last surface P'0 z y' p entrance pupil distance from 1st surface y' p' exit pupil distance from last surface y'p s' x' sagittal transverse aberration image y' meridional transverse aberration x'P p' yp plane

y' xP p

system x'P surfaces y exit p pupil y xP p entrance x s pupil

object plane y

P 17 Sequence of Surfaces

. Sequence of refracting surfaces: the optical path length contributions are additive . The contributions of every surface are summed up . Every surface contribution is imaged and magnified by the following surfaces N 1 N 1 y' y'N mk1mk2...mN y'k , x' x'N mk1mk2...mN x'k k1 k1

. The successive surfaces fulfill sj+1 = s'j,... h h . The height ratios help to express the magnifications   j   pj j h pj 1 hp1

. Individual magnification dependencies for the various aberration types

Q3 Q4 Q1 Q2 Q5

P P4 1 P2 P3 P5 18 Seidel Approach

. Special idea of Seidel to consider the 3rd order as a perturbation of the paraxial ray . Independent changes/contributions of every surface aberration to the final transverse aberration . Therefore special reference on paraxial fundamental properties

y y' perturbation at 2st surface perturbation at 3rd surface

perturbation at y'(2) y' initial path 1st surface (3) y' paraxial ray perturbation at (1) 4th surface y'(4) P'0 P y'0 1 2 3 4 19 Transverse Aberrations of Seidel

. Decomposition of transverse aberrations x' x'2  y'2 s'4 2x' x' x'  y' y'  x'x'2  y'2 s'3 s' x' p p p  S  p p p p p p  C 3  k 3  k 2n' p' k 2n' p' k x'x' x'  y' y' s'2 s'2 x' x'2  y'2 s'2 s'2  p p p  A  p p p p  P 3  k 3  k n' p' k 2n' p' k x'x'2  y'2 s's'3  p  D 3  k 2n' p' k

y' x'2  y'2 s'4 2y' x' x'  y' y'  y'x'2  y'2 s'3 s' y' p p p  S  p p p p p p  C 3  k 3  k 2n' p' k 2n' p' k y'x' x'  y' y' s'2 s'2 y' x'2  y'2 s'2 s'2  p p p  A  p p p p  P 3  k 3  k n' p' k 2n' p' k y'x'2  y'2 s's'3  p  D 3  k 2n' p' k 20 Surface Contributions

. Spherical aberration  1 1  S   4Q2    j j j    n' j s' j n j s j 

 1 1  n  Q  1 1  C   4Q2    1 pj pj    . Coma j j j      n' j s' j n j s j   jQ j  s1 s p1 

2 2 . Astigmatisms  1 1   n  Q   1 1  A   4Q2    1 pj pj     j j j        n' j s' j n j s j    jQ j   s1 s p1 

2 2 . Field curvature  1 1   n  Q   1 1  1  1 1  P   4Q2    1 pj pj         j j j          n' j s' j n j s j    jQ j   s1 s p1  rr  n' j n j  . Distortion

2 2   1 1   n  Q   1 1  1  1 1  n  Q  1 1  D   4Q2    1 pj pj          1 pj pj    j  j j  n' s' n s    Q   s s  r  n' n   Q  s s    j j j j   j j   1 p1  r  j j  j j  1 p1  21 Circular Coordinates

. Introduction of circular coordinates 2 2 x'  r' sin' according to the geometry r'p  x'p  y'p p p p

y p  r'p cos'p . Transverse aberrations

x' s'4 s'2 s'2 s'3 s' x' S  p r'3 sin' P  p r' sin' x'2  y'2  C  p r'2 x'sin 2 '  y'sin' cos'  2n' p'3 p p 2n' p'3 p p n' p'3 p p p p x's'3 s' x's's'3 x's'2 s'2  A p r'2 D  p x'2  y'2  C  p r' x'sin'  y'cos'  2n' p'3 p 2n' p'3 n' p'3 p p p s'4 s'2 s'2 s'3 s' y' S  r'3 cos' P  p r' cos' x'2  y'2  C  p r'2 x'sin' cos'  y'cos2 '  2n' p'3 p p 2n' p'3 p p n' p'3 p p p p y's'3 s' y's's'3 y's'2 s'2  A p r'2 D  p x'2  y'2  C  p r' x'sin'  y'cos'  2n' p'3 p 2n' p'3 n' p'3 p p p

. Aberration curves:

. Considering a ring with constant r'p in the pupil,

- eliminating q'p - delivers curve in image plane for x', y' 22 Simplified Formulas

. Simplified set of formulas: - field point only in y‘ considered - all system constants hidden in modified coefficients

3 2 2 3 y' S'r'p cos p  C'y'r'p (2  cos2P )(2A'P')  y' r'p cosP  D'y' 3 2 2 x' S'r'p sin p  C'y'r'p sin 2P  P'y' r'p sinP

. Simple discussion of aberrations types: 1. spherical aberration S‘: no field dependence 2. coma C‘: grows linear with field 3. astigmatism A‘: double azimuthal angle dependence 4. field curvature P‘: interrelated with astigmatism 5. distortion D‘: no dependence on aperture, no blur

. Special case astigmatism and field curvature: can alternatively be considered as tangential and sagittal image shell 23 Polynomial Expansion of Aberrations

. Representation of 2-dimensional Taylor series vs field y and aperture r . Selection rules: checkerboard filling of the matrix . Constant sum of exponents according to the order Image Primary location aberrations / Seidel Field y Secondary Spherical Com a Astigm atism aberrations y0 y 1 y 2 y 3 y 4 y 5 y cos q y 3 co sq y 5 c osq 0 Distortion r Tilt Distortion Distortion primary secondary 2 1 r 1 y r co s2q y 4 r 1 2 1 2 1 cos q r Defocus y r y 4 r 1 Aper- Astig./Curvat. 2 y r 3 2 3 ture cosq y r cos q r 2 Coma r 3 2 primary y r co sq r 3 y 2 r 3 c o s2q 3 r Spherical 2 3 primary y r y r 4 c o sq r 4 Coma secon dary r 5 r 5 Spherical secondary

24 NA dependency of Primary Aberrations

. Every primary aberration has an individual NA-dependence

. The representation of the aberrations (longitudinal/transversal/wave) have different NA- dependence

. Example field curvature: longitudinal: constant

transverse: linear growing constant wave: quadratically growing s'

exit

pupil 2y'full real full NA wave 2y'half ideal half NA wave

chief ray

Whalf

Wfull real image image plane shell 25 Power Expansion of Aberrations

. Orders of field and aperture dependencies for different representations of primary aberrations

Wave Transverse Longitudinal Seidel Aberration Coefficient aberration aberration aberration sum Aperture Field Aperture Field Aperture Field

Spherical aberr. c1 SI 4 0 3 0 2 0

Coma c2 S II 3 1 2 1 1 1

Astigmatism c3 SIII 2 2 1 2 0 2 (sagittal) (Petzval) Field curvatures S 2 2 1 2 0 2 c4 IV Distortion c SV 1 3 0 3 - - ~ 5 Axial color b1 CI 2 0 1 0 0 0 ~ b2 C Lateral color II 1 1 0 1 - -

From : H. Zügge 26 Rotational Invariants

. General case : two coordinates in object plane and pupil . Rotational symmetry: 3 invariants 1. Scalar product of field vector and pupil vector   w  P  F  P  F cos(F P )  x p  x  y p  y yp 2. Square of field height    z u  F  F  F 2  x2  y 2 xp P  yp xp P 3. Square of pupil height   2 2 2 y v  P  P  P  x p  y p  . Therefore: F x P upil  Only special power y F combinations are x physically meaningful

Ob ject 27 Power Series Expansion of Aberrations

k l m n . General case of Taylor expansion W  aklmnxp yp x y k,l,m,n . Expansion with selection rules: only powers of the rotational invariants can occur W  W (u,v, w)

. Simple expansion according to this scheme

W  a0  b1v  b2w  b3u 2 2 2  c1v  c2wv  c3w  c4uv  c5uw  c6u 3 2 2 2 3 2 2 2 3  d1v  d2wv  d3w v  d4uv  d5uwv  d6w  d7uw  d8u v  d9u w  d10u ...

. Explicite equation in real coordinates

2 2 2 W  a0  b1(x p  y p )  b2 yy p  b3 y 2 2 2 2 2 2 2 2 2 2 3 4  c1(x p  y p )  c2 yy p (x p  y p )  c3 y y p  c4 y (x p  y p )  c5 y y p  c6 y 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2  d1(x p  y p )  d2 yy p (x p  y p )  d3 y y p (x p  y p )  d4 y (x p  y p )  d5 y y p (x p  y p ) 3 3 4 2 4 2 2 5 6  d6 y y p  d7 y y p  d8 y (x p  y p )  d9 y y p  d10 y ... 28

Higher Order Aberrations (29-6)

. Relevance of higher order expansion terms . Nearly perfect geometrical imaging possible in the special cases: 1. small aperture 2. small field field y/w

....

6th photography micro order wide angle lithography in y

4th monocentric order systems in y

2nd paraxial conic mirrors microscopy order optics telescopes high NA in y aperture 2nd order 4th order 6th order DAP/NA .... in NA in NA in NA 29 5th Order Aberrations

k m, l, Pupil No Field Azimuthal Term Name power power power W r6 Secondary spherical 1 0 6 0 060 p aberration

5 2 1 5 1 W151 y'rp cosq Secondary coma

W y'2 r4 cos2 q Secondary astigmatis, wing 3 2 4 2 242 p error

3 3 3 4 3 3 3 W333 y' rp cos q trefoil error, arrow error

2 4 5 2 4 0 W240 y' rp Skew spherical aberration

3 3 6 3 3 1 W331 y' rp cosq Skew coma

4 2 2 7 4 2 2 W422 y' rp cos q Skew astigmatism

4 2 8 4 2 0 W420 y' rp Secondary field curvature

5 9 5 1 1 W511 y' rp cosq Secondary distortion

30 Higher Order Aberrations

. Oblique sagittal spherical aberration y

24 W c2  y  rp 1 x'   4 c  y23  r  sin nu'' 1 pp x 1 y'   c  y23  r  cos nu''2 pp

. Oblique meridional spherical aberration y

W c  y2  r 2  y 2  x 3 pp 1 2 3 2 x'   c  y  r  sin cos nu''3 p p p

1 2 3 2 y'   c3  y  rp  cos p 1  cos p  nu'' 31 Higher Order Aberrations

y y

. Linear coma 5th order

4 W c1  y  ypp  r

x x 1 4 x'   2 c  y  r  sin 2 nu'' 1 pp

1 4 y'   c1  y  rpp  3  2cos 2 

nu'' Coma 3th order

y y . Elliptical coma

x

W c  y3  r 2  y  c  y 3  y 3 45p p p

1 32 x'   c  y  r  sin 2 4 pp x nu'' 1 3 2 3 2 2 y' c45  y rp 2 cos 2 p  3 c y r p cos p nu''

32 Notations and General View

. There is a large number of different notations for the third order representation: Haferkorn, Welford, Seidel, Berek, Köhler,... . The differences are the choice of the parameter and some of the approximations . The so called reduced representations are of a special form without considering the field dependence explicite Example: Zernike expansion . The third order theory is limited on systems with not too high angles of marginal and chief ray . Mostly the third order describes the leading term . Higher orders than 3 can not be decomposed as simple into the surface contributions: 1. the magnification of the following surfaces acts non-linear and can not be neglected 2. the second order perturbation theory has no decoupling of the contributions (induced aberrations) . Also available: analytical aberration coefficients for gradient media (F. Bociort) 33 Welford‘s Notation

. Abbreviations A  n(hc  u)  n i  n'i' bar: chief ray A  n(hc  u)  n i  n'i '

. Seidel aberrations 2  u  S I   A  h    1. spherical aberration  n   u  2. coma S II   AA h     n  3. astigmatism 2  u  S III   A  h     n 

4. Petzval curvature 2  1  S IV   H c     n 

3  A  u  A 2  1  5. distortion SV    h      H  c     A  n  A  n  r y . Representation in normalized circular coordinates   ,   rmax ymax

1 1 1 1 1 W(  , , )  S  4  S  3 cos  S  2  2 cos2   S  S  2  2  S  3 cos 8 I 2 II 2 III 4 III IV 2 V 34 Aberration Theory and Symmetry

Aberration theory Reference

exact - 4th order 6th order ray transfer matrix all orders

Surface OAR Sample systems symmetry

Seidel, with field Araki Vectorial I - with field Shack / Thompson / Sasian Vectorial II - with field Fuerschbach only - one ray Aldis theory Welford - one OPD point only Oleszko - Zernike Paraxial - axis Parabasal around real OAR / CR 2 x 2 4 × 4 5 × 5 Rotational Photographic lens, symmetric microscope, zoom lens Double plane- Anamorphic symmetric Straight Freeform (plane- Scheimpflug symmetric) Freeform (non- Cubic phase plate for EDF symmetric)

Rotational Schiefspiegler telescope, symmetric spherical TMA, HMD

Double plane- Anamorphic prism stretcher symmetric 1D bend Freeform (plane- Unobscured telescope, TMA symmetric) corrected, HMD

Freeform (non- Alvarez plate system, symmetric) panoramic zoom system

Rotational Yolo telescope, spherical symmetric 2D bend Freeform (non- Yolo telescope, corrected symmetric) 35 Vectorial Aberrations

    . Expansion of the wave  j  m   n W F,rp  Wklm F  F  F rp  rp rp  aberration field j,m,n . Normalized field vector: F

normalized pupil vector: rp

yp angle between H and rp: q  F  p . Due to circular symmetry: restriction of a field point only in y  z . Invariants for circular symmetric systems x   p P 2  yp u  F  F  F xp P   2 v  rp rp  rp   y w F rp  F rp cosq  y F x P upil  1 y F x F rp q pupil field 1

Ob ject 36 Generalized Aberration Types

ord j m n Term scalar Name uniform Piston . Wave aberration terms 0 0 0 0 W000   2 quadratic piston 1 0 0 W  H  H  W200  H . Vectorial or polar-scalar 200   magnification 2 0 1 0 W H  r  W111  H  rp cosq representation 111 p   2 focus 0 0 1 W020 rp  rp  W020 rp   2 W r 4 spherical . New aberrations types 0 0 2 W040 rp  rp  040 p  aberration    3 coma 0 1 1 W131 rp  rp H rp  W131  H rp cosq   2 2 2 astigmatism 0 2 0 W  H r cos q W222  H  rp  222 p 2 4     2 2 field curvature 1 0 1 W220 H  H rp rp  W220  H rp     3 distortion 1 1 0 W311 H  H H  rp  W311  H  rp cosq   2 4 quartic piston 2 0 0 W H  H  W400  H 400     2 2 4 oblique W  H  H  r r W  H r 1 0 2 240    p p  240 p spherical aberration       3 3 coma field 3rd 1 1 1 W331 H  H rp  rp H rp  W331  H rp cosq     2 42 astigmatism 1 2 0 W422  H rp cos 2q W422  H  H H  rp  field 4th   2   4 2 field curvature 2 0 1 W420  H rp W420  H  H   rp  rp  field 4th   2   5 distortion field 2 1 0 W511  H rp cosq 6 W511  H  H   H  rp  4th   3 6 piston 6th 3 0 0 W  H W600 H  H  600   3 6 spherical 0 0 3 W  r  r W r 060  p p  060 p aberration 6th     5 2 coma 6th 0 1 2 W151 rp rp  H rp  W151  H  rp cosq     2 2 4 2 astigmatism 6th 0 2 1 W  H r cos q W242 rp  rp H  rp  242 p   3 W  H 3r 3 cos3 q trefoil 0 3 0 W333  H  rp  333 p

General Distortion

. Different forms of distortion fields

original

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 General Reference and Pupil Re-Scaling

. Wave aberrations: measured as a function of the exit pupil coordinates . Therefore pupil distortion is seen as a change in the entrance pupil sampling y'

. The changes are in direction (normalized direction vectors) x' and in length z

y'p y' (H+H)

image x'P ideal real plane

yp y'ExP rp

yEnP (rp+rp) xP exit y ideal pupil real x entrance yH pupil

object plane 39 Surface Contributions: Example

200

. Seidel aberrations: SI 0 representation as sum of Spherical Aberration -200 surface contributions possible 1000

. Gives information on correction SII 0 of a system Coma -1000 . Example: photographic lens 2000

SIII 0 Astigmatism

-2000 1000

SIV Retrofocus F/2.8 0 Petzval field curvature Field: w=37° -1000 6000

SV 0 Distortion

-6000 150 5 10 3 4 CI 2 0 Axial color 1 6 7 8 9 -100 600

CII 0 Lateral color -400 Surface 1 2 3 4 5 6 7 8 9 10 Sum 40 Seidel Surface Contributions

. Graphical supported representation of the Seidel surface contributions of a photographic lens 41 Microscope Objective Lens

0.5 . Seidel surface contributions spherical 0 -0.5

for 100x/0.90 0.02

coma 0

. No field flattening group -0.02

4 . Lateral color in tube lens corrected 2 astigmatism 0 -2 -4

5 curvature 0 -5

2 distortion 0 -2

0.02

axial 0 chromatic -0.02

1 lateral 0 chromatic -1

1 5 10 sum

13 11

8 1 5 Microscope Objective Lens: High NA 100x/0.93

. Objective lens 100x/0.9 . No effort for field flattening . Small range of diffraction limit depending on wavelength and field size

y'

1 0.64 0.4 0.9 0.56

0.8 0.48

Wrms [] 0.7 0.32 0.5 480 nm 0.6 546 nm 0.24 0.4 644 nm 0.5 poly

0.4 0.3 0.16 0.0714

0.3 diffraction 0.2 0.08 limit 0.2 0.1 0.1 diffraction limit  0 y [mm] 0 0 2 4 6 8 0.48 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 [m] 43 Zoom Lens

group 1 group 2 group 3

. Zoom lens e) f' = 203 mm w = 5.64° . Three moving groups F# = 16.6

d) f' = 160 mm w = 7.13° F# = 13.7

c) f' = 120 mm w = 9.46° F# = 10.9

b) f' = 85 mm w = 13.24° F# = 8.5

a) f' = 72 mm w = 15.52° F# = 7.7 44 Performance Variation over z

Seidel spherical aberration coma distortion axial chromatical lateral chromatical surface 0.2 5 5 0.1 0.1 0.5 contrib. lens 1 0 0 0 0 0

-0.1 -0.1 -0.5 -5 -5 -0.2 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0.2 5 5 0.1 0.1 0.5 lens 2 0 0 0 0 0

-0.1 -0.1 -0.5 -5 -5 -0.2 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0.2 5 5 0.1 0.1 0.5 lens 3 0 0 0 0 0

-0.1 -0.1 -0.5 -5 -5 -0.2 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 0.2 5 5 0.1 0.1 0.5 sum 0 0 0 0 0 - -0.1 -5 -0.1 0.5 -5 1 2 3 4 5 -0.2 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 45 Even Aberrations in Symmetrical Systems

. Aberrations with even symmetry are doubled . Spherical aberration, Astigmatism, field curvature, axial chromatical aberration

spherical aberration in an symmetrical system

W c  Z  c  Z 4 4 9 9 W c4  Z 4  c 9  Z 9

doubled values W22 c4  Z 4  c 9  Z 9

Ref: M. Seesselberg 46 Odd Aberrations in Symmetrical Systems

. Aberrations with odd symmetry are vanishing . Coma, distortion, transverse chromatical aberration

coma in an symmetrical system

W c  Z  c  Z 8 8 15 15 W  c8  Z 8  c 15  Z 15

vanishing values W  0

Ref: M. Seesselberg