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: n : radial - 3 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
Z (r) 2r 2 1 4 4 2 Z9 (r) 6r 6r +1 6 4 2 Z16 (r) 20r 30r +12r 1 Z (r) 70r8 140r 6 + 90r 4 20r 2 +1 25 Z (r) 252r10 630r8 + 560r 6 210r 4 + 30r 2 1 36 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 Z (r) 12870r16 51480r14 + 84084r12 72072r10 + 34650r8 9240r 6 +1260r 4 72r 2 +1 81 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 . Constant rms-value for all terms k n 2 2 easy estimation possible Wrms cnm n1 mn
. Indexing of standard polynomials: 1. increasing radial index n 2. increasing absolut value of azimuthal index |m|
sin cos . Therefore irregulary n/m -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 growing spatial 0 1 1 2 3 frequency 2 5 4 6 3 9 7 8 10 4 14 12 11 13 15 5 20 18 16 17 19 21
6 27 25 23 22 24 26 28
7 35 33 31 29 30 32 34 36
8 44 42 40 38 37 39 41 43 45 15 Azimuthal Dependence of Zernike Polynomials
r3 . Azimuthal spatial r2
frequency r1 r3 r1
r2
1 2 3 4
16
Orthogonality
. Expansion of the wave aberration on a circular area n m W (r,) cnmZn (r,) n mn 1 2 (1+ ) Z m (r,)Z m'* (r,) drdr m0 . Orthonormality for Fringe n n' 2(n +1) nn' mm' convention 0 0 1 . Orthogonality of radial functions Rm(r) Rm(r)r dr nn' n n' 2(n +1) 0 . Determination of coefficients 2(n +1) 1 2 c W (r,) Z m*(r,)drdr nm (1+ ) n m0 0 0 . Necessary requirements for orthogonality: 1. pupil shape circular 2. uniform illumination of pupil (corresponds to constant weighting) 3. no discretization effects (finite number of points, boundary)
. Orthogonality perturbed in reality by: 1. real non-circular boundary (vignetting) 2. apodization (laser illumination) 3. discretization (calculation by a discrete finite ray set) , 17
Orthogonality
. Usual found new sets of orthogonal functions: 1. discretized finite sampling grid 2. apodization due to gaussian illumination (Mahajan) 3. elliptical deformed pupil shape (Dai) 4. ring-shaped pupil (Tatian) 5. rectangular pupil (2D Legendre)
. General orthogonalization of polynomials by Gram-Schmidt algorithm from Zernikes possible: 1 - Definition of inner product of two functions F ,F F F rdrd 1 2 1 2
- first new function Y, normalized Y'1 Y'1 Z1 Y1 Y1,Y1
- second new function, linear combination of Y'2 Z2 +T21 Y1 Z2 Z2,Y1 Y old function and lower order new functions Y'2 normalized Y2 Y ,Y 2 2 n1 n1 - general step, analogoues Y'n Zn + Tnm Ym Zn Zn ,Ym Ym m1 m1
Y'n Yn Yn ,Yn 18 Mathematical Properties
m m+1 m+1 . Recurrence formulas 2(n +1)r Rn (n + m+ 2) Rn+1 + (n m) Rn1
2 2 2 2 m n + 2 2 (n + m) (n m + 2) m n m m Rn+2 2 2 4(n +1)r Rn Rn2 (n + 2) m n n + 2 n
nm 2 (n q)! . Explicite formula Rm (r) (1)q r n2q n n + m n m q0 q! q! q! 2 2
. Symmetry m m Rn Rn
. Value at the edge m Rn (1) 1
. Value range: m Zn (1) 1...+1 19 Mathematical Properties
nm m . Fourier transform relationship: i (1) 2 cos(m) for m 0 frequency spectrum of Zernike J (2k) nm ˆ n+1 m 2 functions FZnm (r,)U nm (k,) i (1) sin(m) for m 0 k n (1) 2 for m 0
. Distributed frequency 0.7 content 0.6 n = 2 0.5 n = 6 n = 10 . Higher radial order polynomials 0.4 n = 16 n = 24 has higher spatial frequency 0.3 support 0.2 0.1 . In the spatial domain, the high 0 -0.1 frequency are located at the -0.2 edge 0 2 4 6 8 10 12 20 Zernikepolynomials: Different Conventions
. Different standardizations used concerning: 1. indexing 2. scaling / normalization 3. sign of coordinates (orientation for off-axis field points)
. Fringe - representation 1. CodeV, Zemax, interferometric test of surfaces 2. Standardization of the boundary to ±1 3. no additional prefactors in the polynomial 4. Indexing according to m (Azimuth), quadratic number terms have circular symmetry 5. coordinate system invariant in azimuth
. Standard - representation - CodeV, Zemax, Born / Wolf - Standardization of rms-value on ±1 (with prefactors), easy to calculate Strehl ratio - coordinate system invariant in azimuth
. Original - Nijboer - representation - Expansion: k k n k n 1 0 m m W (r,) a00 + a0n Rn + anmRn cos(m) + bnmRn sin(m) 2 n0 n0 m1 n0 m1 nm nm - Standardization of rms-value on ±1 gerade gerade - coordinate system rotates in azimuth according to field point 21 Zernikepolynomials: Performance Criteria
. Mean value vanishes 1 1 2 Z Z (r,) rddr 0 j j 0 0
1 2 . Rms value of the wave aberration 1 2 W 2 W (r,) W rddr 1. Fringe convention: rms 0 0
k c2 1 k n c2 n0 + nm n1 n +1 2 n1 mn n +1
k n 2 2 2. Standard convention Wrms cnm n1 mn
2 N 2 N n 2 . Marechal approximation for large 2 cn0 1 cnm Ds 1 + Strehl numbers n1 n +1 2 n1 m0 n +1
. Marechal criteria for single indices Aberration Coefficient Limit
Defocus c20 0.125
Spherical c40 0.161
Astigmatism c22 0.177
Coma c31 0.210 22 Balance of Lower Orders by Zernike Polynomials
. Mixing of lower orders to get the minimal Wrms
. Example spherical aberration: th 4 2 1. Spherical 4 order according to W(rp ) 6rp 6rp +1 Seidel 2. Additional quadratic expression: Optimal defocussing for edge correction W 3. Additional absolute term 4 Minimale value of Wrms 4th order (Seidel) 3 . Special case of coma:
Balancing by tilt contribution, 2 4th, 2nd and 0th order (Zernike) rms is minimal corresponds to shift between peak 1 and centroid + + 1 rp 0 + _
-1 _ 4th and 2nd order -2 23 Calculation of Zernike Polynomials
. Assumptions: 1. Pupil circular 2. Illumination homogeneous 3. Neglectible discretization effects /sampling, boundary)
. Direct computation by double integral: 1. Time consuming 1 1 2 2. Errors due to discrete boundary shape c W (r,)Z *(r,) drdr j j 3. Wrong for non circular areas 0 0 4. Independent coefficients
2 . LSQ-fit computation: N Wi c j Z j (ri ) min 1. Fast, all coefficients cj simultaneously obtained i1 j1 2. Better total approximation 3. Non stable for different numbers of coefficients, 1 if number too low c(Z T Z) Z TW 4. Stable for non circular shape of pupil
2irk 2 . Calculation by Fourier transform A(k,q) W(r,)e d r r1
2 1 N 1 c A (k,q )U * (k,q ) U (k,q) Z (r,)e2irk d 2r nm l nm nm nm l0 24 Radial Sampling
Z(r) . Spatial frequency grows towards the edge of 1 the pupil radius 0. 8 n = 30 0. 6 . The outer zeros are denser distributed and grows 0. 4 0. with index n 2
0 (zero) 1 - r 2 0.2 n - 0.4 - . The sampling is dominated at the edge 0.6 - 0.8 - 0. 0. 0. 0. 0. 0. 0. 0. 0. r 1 0 1 . Possible maximum radial order for a 1 2 3 4 5 6 7 8 9
given equidistant sampling number N rn 30 zeros N over N over 25 n radius diameter 20 32 64 4 64 128 5 15 128 256 8 10 256 512 11 512 1024 16 5
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r 25 Truncation Error
. If the series expansion is truncated before convergence is obtained:
- errors in wavefront description
- errors of lower order coefficients for NLSQ-calculation: balancing to minimize the rms-error
. Example: c/c
- circular symmetric wavefront
- coefficients c4=c9=c16=... = 0.2 0.3 - error of lower order coefficients c , c for 9 16 growing number of terms c9 0.2 c16
0.1
0 0 2 4 6 8 10 12 14 26 Calculation of Zernike Polynomials
. Correlation of modes 1 1 2 C Z (r,)Z *(r,) drdr jj' j j' . Numerical residual errors due to 0 0
discretization j . Main errors are caused by the log C jj' corrugated boundary
. Largest errors for same symmetry:
no cancellation
j' 27 Conversion to Monomials
. Cartesian representation of Zernike functions m m Zn Rn cos m nm nm m j 1 if n even q 2 2 n m i+ j m j (n j)! 2(i+k ) n2(i+ j+k ) 2 (1) x y q 2 m 1 i0 j0 k0 2i n m n m if n odd k j! !n j! 2 2 2 2
. Equalization of two expansion representations
n p W(x, y) c Z m (x, y) q pq nm n W (x, y) apq x y n0 mn p0 q0 gives mapping matrix T for conversion of coefficients
p cnm apq Tpqnm p0 q0
explicit
( p q)!q! q 1 pq g( p,q,m,t,t') Tpqnm p 2 t0 t!(q t)! t'0 t'!( p q t')! (n m ) / 2 s (1) n +1(n s)! s0 s!(n + p 2s + 2)(n + m) / 2 s!(n m) / 2 s!
( pq) / 2+t' 2(1) m( p2t2t') + m( p2q+2t2t') if m 0 ( pq) / 2+t' g( p,q,m,t,t') (1) 0( p2t2t') + 0( p2q+2t2t') if m 0 2(1)( pq1) / 2+t' if m 0 m ( p2q+2t2t') m (2q p2t+2t') 28 Conversion to Monomials
. Matrix H for linear indexed functions
n . Matrix is 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 m sparse 1 1 -1 1 2 1 -2 3 3 1 -2 3 4 2 1 -6 -3 5 2 -6 6 2 -1 -6 3 7 3 1 -12 -4 8 3 3 -12 -12 9 3 -3 -12 12 10 -1 3 4 -12 11 6 4 1 12 4 8 13 12 -4 -6 14 -4 8 15 6 -4 1 16 10 5 1 17 5 15 10 18 20 -15 -10 19 -10 -5 20 20 10 -15 5 21 1 -5 10 22 23 6 24 25 -20 26 27 6 29 Cartesian Derivatives of Zernikes
. Cartesian derivatives of the Zernike function according to x, y R'sin für m 0 R'cos für m 0 mR mR Z x R'sin sin(m) + cos cos(m) für m0 Z y R'cos sin(m) sin cos(m) für m0 r r mR mR R'sin cos(m) cos sin(m) für m 0 R'cos cos(m) + sin sin(m) für m 0 r r
. Most measuring techniques: 1 2 3 4 5 6 7 8 9 Z Primary the gradients of the
wave are measured
. Direct fit of derivatives is Zx
appropriate
Z . Calculation / integration of y
coefficients via expansion 10 11 12 13 14 15 16 17 18 Z
Zx
Zy 30 Vectorial Zernike Functions
. Composition of the gradients in a vectorial function S' e Z +e Z j x x j y y j . Normalization and expansion into original functions
. Describes elementary decomposition of orientation fields S j ex a j Z j +ey bj Z j j j . Applications: polarization aberrations S e Z S 2 x 1 S2 3 S4 S3 ey Z1 1 S (e Z + e Z ) 4 2 x 2 y 3 1 S (e Z + e Z ) 5 2 x 3 y 2 1 S e Z e Z S 6 ( x 2 y 3 ) 5 S6 S7 2 1 S7 exZ5 + ey ( 2 Z4 Z6 ) 2 1 S8 ex ( 2 Z4 + Z6 )+ ey Z5 2 31
Zernike Coefficients for Changed Conditions
. Changes of conditions changes the Zernike coefficients
. Special small changes of practical relevance can be calculated analytically by expansion:
1. change of radius of normalization
2. ellipticity of the aperture
3. lateral shift of the pupil center
4. azimuthal rotation
5. change of exact z-position of the pupil
. Point 5. corresponds to the propagation changes of Zernikes:
1. Zernikes are no invariant eigenfunctions 2 2 W(x, y) W(x, y) 2. Linear approach: s(x, y) + - direction vector s depends on position x y
- change of wavefront in geometrical z approximation W '(x', y') W (x, y) + s(x, y) 2r2
z a + b P (x, y) j 2 j j 3. Problem: additional change of pupil size j1 2r
due to convergence/divergence needs
re-normalization j1 s z N (1) ( j s)(2 j s)! r' r + 2c2 j,0 2 j +1) 2 r j1 s0 s!( j s)! 32
Zernike Coefficients in Different z-Planes
object pupil exit rear . Changes of z-distance changes Zernikes plane pupil stop
. Relevant applications: chief 1. Human eye, iris pupil not accessible ray
2. Microscopic lens, exit pupil not accessible
. Possible solution to determine the exact temporal posterior chamber vitreous pupil phase front: anterior humor chamber 1. Calculation of Zernike changes by numerical fovea
propagation optical disc crystalline lens blind spot cornea 2. Pupil transfer relay optical system lens capsule
iris retina nasal . For a phase preserving transfer, a well corrected
4f-system is necessary
A simple one-lens imaging generates a quadratic
phase in the image plane starting final plane plane
f f 2 2 f f 1 1 d d' 33
Zernike Coefficients for Change of Radius
. Change of normalization radius,
Problem, if pupil edge is not well known or badly defined
. Deviation in the radius of normalization of the pupil size: 1. wrong coefficients 2. mixing of lower orders during fit-calculation, symmetry-dependent
. Example primary spherical aberration: polynomial: 0.9
Z ( )6 4 6 2 +1 9 0.8 0.7
Stretching factor of the radius 0.6
r c / c 0.5 9 9 0.4 New Zernike expansion on basis of r c4 1 3(1 2 ) 0.3 Z () Z 9 (r) + Z 4 (r) 9 4 4 0.2 c1 4 2 3 + 2 0.1 + 4 Z 1 0 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 34
Zernike Coefficients for Pupil Decenter
. Change pupil center position, lateral shift a x x a . Mixing of Zernike coefficients
. Example: 2 1. initially spherical aberration Z (x, y)6 x 2 + y 2 6 x 2 + y 2 +1 9 ( ) ( )
2. finally: 2 2 - coma, grows linear with a Z9 Z9 + 8a Z7 +12a Z4 12a Z6
- astigmatism, grows quadratic with a + 24a3 + 4a Z +12a2 Z ( ) 2 1 - defocus, grows quadratic with a
- tilt, grows linear with a 0.8
0.7
0.6
0.5 coma c7 0.4 tilt c2
0.3 astigmatism 0.2 and defocus c / c 0.1 4 6
0 a 0 0.02 0.04 0.06 0.08 0.1 35 Zernike Expansion of Local Deviations
original Small Gaussian bump in the topology of a surface
N = 36 N = 64 N = 100 N = 144 N = 225 N = 324 N = 625 Rms = 0.0237 0.0193 0.0149 0.0109 0.00624 0.00322 0.00047 PV = 0.378 0.307 0.235 0.170 0.0954 0.0475 0.0063 model
error
Spectrum of coefficients for the last case 0.04
0.03
0.02
0.01
0 0 100 200 300 400 500 600 36 Zernike Representation of a Spherical Wave
. Defocus coefficient of Zernike c4: parabolic approximation parabolic r . Exact expansion of a sphere: approximation by Z4 2 2 a z(r) R R r NA R correction 2 z by a a r NA r NA higher z(r) 1 1 x NA a orders a z a 1 2 1 4 1 6 5 8 7 10 21 12 z(r) x + x + x + x + x + x NA 2 8 16 128 256 1024 exact 33 14 429 16 715 18 + x + x + x +... spherical wave 2048 32768 65536
. Important aspect for high numerical aperture
37 Zernike Representation of a Spherical Wave
. Comparing coefficients with circular symmetric cj 0 Zernike functions: 10 - linear relationship between Zernikes -2 - corresponds to a Zernike expansion of the 10 spherical wave
-4 10
1 715 17 c100 NA 48620 65536 -6 10 1 429 15 c NA c 218790 81 100 ( ) 12870 32768 -8 10 4 9 16 25 36 49 64 81 100 1 33 13 c64 NA c100 (411840) c81 ( 51480) 3432 2048
1 21 11 c49 NA c100 ( 420420) c81 (84084) c64 (12012) 924 1024 ...... Accuracy depends on radius, NA and number of terms
. Special case: Defocus on the high NA side:
causes a linear change of all orders c4, c9,c16,...
38 Zernike Representation of a Spherical Wave
z . Example:
-2 Accuracy as a function of NA and 10 radius for 7 terms as a function of
position -4 10 NA = 0.9
-6 10 NA = 0.8
-8 10 NA = 0.6
- 10 10 r -1 -0.5 0 0.5 1 39 Axial Intensity for Circular Symmetric Zernikes
. Circular symmetric Zernikes: - corresponds to a chirped radial phase grating . Increasing Talbot-effect along the axis causes a split of the axial intensity into separated peaks . Effect grows with order and size of coefficient . The spreading of the side-lobes increases with the order . Corresponds to a multi-focus image
I(z) I(z) I(z) 1 1 1 n = 10 c = 0.3 c = 0.5 n = 10 c = 0.7 n = 10 n = 14 n = 14 n = 14 n = 18 n = 18 n = 18 n = 24 n = 24 n = 24 n = 30 n = 30 n = 30
0.5 0.5 0.5
zRE
0 0 0 zRE -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 40 Zernike Expansion of a Vignetted Pupil
. Direct numerical (SVD optimization) expansion of a vignetted amplitude in the pupil. . Apodization due to surface vignetting: r 24% relative area. V
. Modelling of this profile by N Zernike 24% coefficients on a 128x128 grid
r . Errors in Apodization and corresponding Psf:
xM Apod Psf Peak N Apod PV Psf Rms Psf PV zern Rms error [%] 36 0.1268 0.593 0.00113 0.0185 1.8 64 0.1083 0.568 6.4 10-4 0.0108 1.1 100 0.0962 0.531 3.8 10-4 0.0064 0.63 225 0.0765 0.512 6.2 10-5 8.5 10-4 0.13
41 Zernike Expansion of a Vignetted Pupil
N = 36 N = 64 N = 100 N = 225
Apodization
Error of apodization
Error of Psf
42 Zernike Expansion of a Vignetted Pupil
. The improvement of the apodization itself grows slowly with the number of terms . The accuracy of the Psf in increased quite better
rms 0 10
-1 apodization 10
-2 10
-3 10
-4 Psf 10
-5 N 10 zern 50 100 150 200 250 43 Zernike Expansion of a Vignetted Pupil
. Cross section of the modelled apodization Apodization amplitude A(y) and corresponding error 1.2
1 A(y) 0.8
0.6
0.4
0.2 . Poor convergence of the Zernike coefficients 0 error A(y) -0.2
c -0.4 y j -1 -0.5 0 0.5 1 0.5
0.4
0.3
0.2
0.1
0 50 100 150 200 250 44 Performance Description by Zernike Expansion
. Vector of cj linear sequence with runnin g index
cj
. Sorting by symmetry 0.5 0.4 0.3
0.2
0.1
0
-0.1
-0.2
m 0 1 2 3 4 -1 -2 -3 -4 circular symmetric cos terms sin terms m = 0 m > 0 m < 0 45 Field Dependence of Zernike Coefficients
. Usually the system quality changes with the field position
. The natural behavior is x = 0 x = 20% x = 40% x = 60 % x = 80 % x = 100 % a decrease in quality from the center to the edge
. For spatial variant PSF
deconvolution or image cj in calculation applications, 4
a robust interpolation for c4 defocus c astigmatism arbitrary field points is desirable 5 3 c8 coma c spherical 9 c11 trefoil c astigmatism 5. order . An interpolation of the PSF-intensity 2 12 c15 coma 5. order distribution is nearly impossible c16 spherical 5. order
1 . The individual Zernike coefficients vary rather smoothly with the field location 0
. An interpolation of the individual -1 Zernike coefficients therefore is quite relative field good -2 position 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 46 Zernike Surface Contributions
. The wave aberrations surface contributions of Hopkins/Welford are additive corresponding to the summation of phase
. For systems with neglectable induced aberrations the bundle diameter scales linear and the Zernike expansion coefficients are also additive on a normalized bundle radius
. If the system suffers from large induced aberrations, the ray grid is distorted and rescaled, in this case the Zernike coefficients are not exactly additive. It is also well known, that the Zernikes are changing during propagation
. Therefore distance related induced Zernike contributions can be defined in addition
. For most real systems, the Zernikes are rather good additive
G. Dai, Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials, Appl. Opt. 48 (2009) p.477 47 Zernike Surface Contributions
WW . Surface decomposition of wave aberration s s
W(,)(,) x y c Z x y Zernike decomposition of total wave aberration p p j p p j
Ws(,)(,) x ps y ps c js Z x ps y ps Zernike decomposition of surface contribution j
xps f x( x p , ypp ) , y ps f y ( x p , y ) General relation between coordinates x m x, y m y p x p p y p special case of linear scaling W(,)(,) xp y p c js Z x ps y ps sj insertion cjs Z f x( x p , y p ), f y ( x p , y p ) sj cc case of distortion-free grid projection j js s . For systems with neglectable induced aberrations the bundle diameter scales linear and the Zernike expansion coefficients are also additive on a normalized bundle radius
. If the system suffers from large induced aberrations, the ray grid is distorted and rescaled, in this case the Zernike coefficients are not exactly additive. It is also well known, that the Zernikes are changing during propagation
. Therefore distance related induced Zernike contributions can be defined in addition
G. Dai, Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials, Appl. Opt. 48 (2009) p.477 48 Zernike Coefficients per Surface
. Contributions of the lower Zernike coefficients per surface (Fringe convention)
tilt defocus log cj astigmatism coma 3 spherical
2
1
0
-1
-2
-3 0 5 10 zernike 15 sum index j M 20 3 M2 surfaces 25 M1 49 Zernike Coefficients per Surface
. Contributions of the lower Zernike coefficients per surface, In logarithmic scale not additive (Fringe convention) Log cjs 3 M1 M1 M3 2 1 0 -1 -2 tilt def ast com sph -3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 index j cj [] 0.14 . Error in additivity due to 0.12 numerical reasons for 0.1 astigmatism 0.08 . Effect of induced aberrations 0.06 and grid distortion in the 0.04 range of / 20 in this case 0.02
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 index j 50 Extended Zernike Approach
. Conventional usage of Zernike coefficients:
- description of wave front in pupil
- determines the PSF intensity in the reference plane
. More general approach due to Braat (2005) according to an old idea of Zernike (1930)
- expansion of the intensity I(x,y,z) in the image domain in all dimensions
- lateral expansion into Zernikes
- axial Taylor expansion
p 2J (r) m iz l1 nm Jm+l+2 j (r) E(r, , z)2 1 + 4 i c i cos(m)e ( 2iz ) b r nm lj l rl n,m0 l1 j0
l 1 with coefficients nm p m + j + l 1 j + l 1 p j blj (1) (m + l + 2 j) l 1 l 1 q + l + j l n m n + m p q 2 2
. cnm: classical Zernike coefficients
. This gives an analytical representation of the volume distribution of intensity
51 Extended Zernike Approach
. Example:
- intensity z-stack for coma
- calculation with diffraction integral / extended Zernike approach
- nearly perfect result without differences
ccoma = 0.05 z = -1.5 -1.0 -0.5 0 0.5 1.0 1.5
diffraction integral
extended Nijboer- Zernike 52 Extended Zernike Accuracy
. Problems with extended Zernikes:
1. circular coordinates
2. no apodization
3. truncation of expansion critical, in particular along z
finite range of convergence . Example calculation:
accuracy as a function of growing coefficients for fixed number of terms
Irms correlation
0.5 1 defocus defocus astigmatism astigmatism 0.4 0.9 coma coma spherical spherical
0.3 0.8
0.2 0.7
0.1 0.6
0 cj 0.5 cj 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1