Design and Correction of Otical Systems Lecture 5: Wave aberrations 017-05-19 Herbert Gross Summer term 017 www.ia.uni-jena.de
Preliminary Schedule - DCS 017 1 07.04. Basics 1.04. Materials and Comonents 3 8.04. Paraxial Otics 4 05.05. Otical Systems 5 1.05. Geometrical Aberrations 6 19.05. Wave Aberrations 7 6.05. PSF and Transfer function 8 0.06. Further Performance Criteria 9 09.06. Otimization and Correction 10 16.06. Correction Princiles I 11 3.06. Correction Princiles II 1 30.06. Otical System Classification Law of refraction, Fresnel formulas, otical system model, raytrace, calculation aroaches Disersion, anormal disersion, glass ma, liquids and lastics, lenses, mirrors, asheres, diffractive elements Paraxial aroximation, basic notations, imaging equation, multi-comonent systems, matrix calculation, Lagrange invariant, hase sace visualization Puil, ray sets and samling, aerture and vignetting, telecentricity, symmetry, hotometry Longitudinal and transverse aberrations, sot diagram, olynomial exansion, rimary aberrations, chromatical aberrations, Seidels surface contributions Fermat rincile and Eikonal, wave aberrations, exansion and higher orders, Zernike olynomials, measurement of system quality Diffraction, oint sread function, PSF with aberrations, otical transfer function, Fourier imaging model Rayleigh and Marechal criteria, Strehl definition, -oint resolution, MTF-based criteria, further otions Princiles of otimization, initial setus, constraints, sensitivity, otimization of otical systems, global aroaches Symmetry, lens bending, lens slitting, secial otions for sherical aberration, astigmatism, coma and distortion, asheres Field flattening and Petzval theorem, chromatical correction, achromate, aochromate, sensitivity analysis, diffractive elements Overview, hotograhic lenses, microscoic objectives, lithograhic systems, eyeieces, scan systems, telescoes, endoscoes 13 07.07. Secial System Examles Zoom systems, confocal systems
3 Contents 1. Rays and wavefronts. Wave aberrations 3. Exansion of the wave aberrations 4. Zernike olynomials 5. Performance criteria 6. Non-circular uil areas 7. Measurement of wave aberrations
4 Ray-Wave Equivalent Rays and waves carry the same information Wave surface is erendicular on the rays Wave is urely geometrical and has no diffraction roerties
Law of Malus-Duin 5 Law of Malus-Duin: - equivalence of rays and wavefronts - both are orthonormal - identical information wave fronts Condition: No caustic of rays rays Mathematical: Rotation of Eikonal vanish rotn s Otical system: Rays and sherical waves orthonormal 0 object lane y 0 hase L = const L = const image lane rays s y 1 z 0 z 1
6 Fermat Princile Fermat rincile: the light takes the ray ath, which corresonds to the shortest time of arrival The realized ath is a minimum and therefore the first derivatives vanish x ds C P 1 L P P 1 n( x, y, z) ds 0 Several realized ray athes have the same otical ath length P 1 z L P P1 n s dr const. The rincile is valid for smooth and discrete index distributions P n d 1 1 n d n d 3 3 n d 4 4 n d 5 5 P'
AP OE OPL r d n l (0,0) ), ( ), ( OPL OPL OPD l y x l y x R y W R y y W ' ' y y x W y R u y y y R s ), ( ' sin ' ' ' Relationshis Concrete calculation of wave aberration: addition of discrete otical ath lengths (OPL) Reference on chief ray and reference shere (otical ath difference) Relation to transverse aberrations Conversion between longitudinal transverse and wave aberrations Scaling of the hase / wave aberration: 1. Phase angle in radiant. Light ath (OPL) in mm 3. Light ath scaled in l ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( x W i x k i x i e x A x E e x A x E e x A x E OPD 7
8 Relationshi to Transverse Aberration Relation between wave and transverse aberration Aroximation for small aberrations and small aerture angles u Ideal wavefront, reference shere: W ideal Real wavefront: W real Finite difference Angle difference Transverse aberration Limiting reresentation W W W real W ideal W tan y y' R W y y' R W y' R y reference shere wave front W(y ) reference lane q R, ideal ray u real ray C y' z
9 Wave Aberration in Otical Systems Definition of otical ath length in an otical system: Reference shere around the ideal object oint through the center of the uil Chief ray serves as reference Difference of OPL : otical ath difference OPD Practical calculation: discrete samling of the uil area, real wave surface reresented as matrix y y y' y' uer coma ray otical system W wave aberration chief ray w' image oint z chief ray wave front object oint lower coma ray reference shere Object lane O Entrance uil EnP Exit lane ExP Image lane I
10 Puil Samling All rays start in one oint in the object lane The entrance uil is samled equidistant In the exit uil, the transferred grid may be distorted In the image lane a sreaded sot diagram is generated object lane oint entrance uil equidistant grid otical system exit uil transferred grid image lane sot diagram y o y y' y' x o x x' x' z
11 Wave Aberrations Quality assessment: - eak valley value (PV) - rms value, area average - Zernike decomosition for detailed analysis v-value of wave aberration wave aberration image lane hase front exit aerture reference shere
1 Wave Aberrations Sign and Reference Wave aberration: relative to reference shere Choice of offset value: vanishing mean Sign of W : 1 W ( x, y) W ( x, y) dx dy 0 F ExP - W > 0 : stronger convergence intersection : s < 0 - W < 0 : stronger divergence intersection : s < 0 x reference shere wave front W > 0 ideal ray reference lane (araxial) real ray U' C z R y' < 0 s' < 0
13 Tilt of Wavefront W Change of reference shere: tilt by angle q linear in y W tilt n y q y Wave aberration due to transverse aberration y W tilt R y Re f y' Is the usual descrition of distortion y wave aberration W < 0 reference shere q tilt angle y' transverse aberration y' z wave front uil lane image lane
14 Defocussing of Wavefront Paraxial defocussing by z: Change of wavefront W Def nr R ref z' 1 nz' sin u y wave aberration W > 0 y' wave front z reference shere z' defocus uil lane image lane
15 Secial Cases of Wave Aberrations Wave aberrations are usually given as reduced aberrations: - wave front for only 1 field oint - field deendence reresented by discrete cases y y' Secial case of aberrations: 1. axial color and field curvature: reresented as defocussing term, Zernike c 4. distortion and lateral color: reresented as tilt term, Zernike c, c 3 uil lane wave front Reference shere axial chromatic z ideal image lane y y' y curved image shell y' wave front chief ray image height difference wave front chief ray defocus due to field curvature z z reference shere reference shere uil lane ideal image lane uil lane ideal image lane
16 Secial Cases of Wave Aberrations y 3. afocal system - exit uil in infinity - lane wave as reference wave front reference lane z image in infinity 4. telecentric system chief ray arallel to axis y y' wave front reference shere z uil lane axial chromatic ideal image lane
17 Exansion of the Wave Aberration Table as function of field and aerture Selection rules: checkerboard filling of the matrix Image location Primary aberrations / Seidel y r 6 Field y Sherical Coma Astigmatism y 0 y 1 y y 3 y 4 y 5 y r cosq 3 r cosq y 5 r cosq Distortion r 1 Tilt Distortion Distortion rimary secondary r y r cos q y 4 r cos q r Defocus y r Aerture Astig./Curvat. y 4 r y r 3 cosq y 3 r r 3 r 3 cos 3 q Coma rimary y 3 r 3 cosq r 4 y r 4 cos q r 4 Sherical rimary y r 4 y r 5 cosq r 5 Coma secondary Sherical r 6 secondary Secondary aberrations
18 Polynomial Exansion of Wave Aberrations Taylor exansion of the wavefront: y' Image height index k r Puil height index l q Puil azimuth angle index m Symmetry invariance: 1. Image height. Puil height 3. Scalar roduct between image and uil vector k l m W( y', r, q) Wklm y' rcos q k, l, m y ' r y' r y' r cosq Number of terms sum of indices in the exonent i sum i sum N i number of terms Tye of aberration image location 4 5 rimary aberrations, 4th order 6 9 secondary aberrations, 6th order 8 14 8th order
19 Taylor Exansion of the Primary Aberrations Exansion of the monochromatic aberrations First real aberration: rimary aberrations, 4 th order as wave deviation W( y', r, y ) A r A y' r y A y' y A y' r A s 4 c a d y' 3 y Coefficients of the rimary aberrations: A S : Sherical Aberration A C : Coma A A : Astigmatism A P : Petzval curvature A D : Distortion Alternatively: exansion in olar coordinates: Zernike basis exansion, usually only for one field oint, orthogonalized
0 Zernike Polynomials Exansion of the wave aberration on a circular area m ) cnmzn ( r, c nm W ( r, ) n n mn Zernike olynomials in cylindrical coordinates: Radial function R(r), index n Azimuthal function, index m Orthonormality Advantages: 1. Minimal roerties due to W rms. Decouling, fast comutation 3. Direct relation to rimary aberrations for low orders ( n 1) 1 Z m n m0 1 0 0 ( r, ) R W ( r, ) Z m n m* n sinm für m0 ( r) cosm für m0 1 für m 0 1 m m'* 1 m0 Zn ( r, ) Zn' ( r, ) drdr nn' mm' ( n 1) 0 0 Problems: 1. Comutation oin discrete grids. Non circular uils 3. Different conventions concerning indeces, scaling, coordinate system, ( r, ) drdr
1 Zernike Polynomials Exansion of wave aberration surface into elementary functions / shaes m W ( r, ) c Z ( r, ) n n mn nm n m = + 8 + 7 + 6 + 5 cos Zernike functions are defined in circular coordinates r, Z m n ( r, ) R m n sin ( m) for ( r) cos( m) for 1 for m 0 m 0 m 0 Ordering of the Zernike olynomials by indices: n : radial m : azimuthal, sin/cos + 4 + 3 + + 1 0-1 - - 3-4 Mathematically orthonormal function on unit circle for a constant weighting function Direct relation to rimary aberration tyes - 5-6 - 7-8 n = sin 0 1 3 4 5 6 7 8
Azimuthal Deendence of Zernike Polynomials Azimuthal satial frequency r 1 r r 3 r 1 r 3 r 1 3 4
3 Zernike Polynomials: Meaning of Lower Orders n m Polar coordinates 0 0 1 1 iston 1-1 r sin x 1-1 r cos y - r sin xy 0 r 1 x y 1 r cos y x 3-3 r 3 sin 3 3 3 xy x 3 3 r r sin 3 x x 3 xy 3-1 3-1 3 3 r r 3 3 cos 3 y y 3 x y 3 3 r 3 cos 3 3 y 3 x y 4-4 r 4 sin 4 3 3 4 xy 4 x y 4 3 3 3 r r sin 8 xy 8 x y 6 xy 4-4 0 6 6 4 4 r r 1 6 x 6 y 1 x y 6 x 6 y 1 4 3 4 4 r r cos 4 y 4 x 3 x 3 y 4 x y 4 Cartesian coordinates 4 4 r 4 cos 4 y x 6 x y 4 4 Interretation tilt in x tilt in y Astigmatism 45 defocussing Astigmatism 0 trefoil 30 coma x coma y trefoil 0 Four sheet.5 Secondary astigmatism Sherical aberration Secondary astigmatism Four sheet 0
4 Zernike Fringe vs Zernike Standard Polynomials In radial symmetric system for y-field (meridional) sinus-terms vanish Fringe coefficients: Z 4,9,16 = n² sherical azimutal order f grows if number increases Standard coefficients - different term numbers Z N = 0.05 RMS = 0.05l SR ~ 1 40 Z N ² = 0.9 Elimination of tilt No Elimination of defocus @ Zemax 4
5 Balance of Lower Orders by Zernike Polynomials Mixing of lower orders to get the minimal W rms Examle sherical aberration: 1. Sherical 4 th order according to Seidel. Additional quadratic exression: Otimal defocussing for edge correction 3. Additional absolute term Minimale value of W rms Secial case of coma: Balancing by tilt contribution, corresonds to shift between eak and centroid W 4 3 1 0 W( r 4th, nd and 0th order (Zernike) rms is minimal + ) 6r 4 6r 1 4th order (Seidel) _ + + 1 r -1 4th and nd order _ -
6 Zernike Polynomials Advantages of the Zernike olynomials: - de-couling due to orthogonality - stable numerical comutation - direct relation of lower orders to classical aberrations - otimale balancing of lower orders (e.g. best defocus for sherical aberration) - fast calculation of W rms and Strehl ratio in aroximation of Marechal Necessary requirements for orthogonality: - uil shae circular - uniform illumination of uil (corresonds to constant weighting) - no discretization effects (finite number of oints, boundary) Different standardizations used concerning indexing, scaling, sign of coordinates (orientation for off-axis field oints): - Fringe reresentation: eak value 1, normalized - Standard reresentation W rms normalized - Original reresentation according to Nijbor-Zernike Norm ISO 10110 allows Fringe and Standard reresentation
7 Calculation of Zernike Polynomials Assumtions: 1. Puil circular. Illumination homogeneous 3. Neglectible discretization effects /samling, boundary) Direct comutation by double integral: 1. Time consuming. Errors due to discrete boundary shae 3. Wrong for non circular areas 4. Indeendent coefficients c j 1 1 0 0 W ( r, ) Z j *( r, ) drdr LSQ-fit comutation: 1. Fast, all coefficients c j simultaneously. Better total aroximation 3. Non stable for different numbers of coefficients, if number too low Stable for non circular shae of uil W N i i1 j1 1 c c T T Z Z Z W j Z j ( ri ) min
8 Performance Descrition by Zernike Exansion Vector of c j linear sequence with runnin g index c j Sorting by symmetry 0.5 0.4 0.3 0. 0.1 0-0.1-0. 0 circular 1 3 4-1 - -3-4 symmetric cos terms sin terms m = 0 m > 0 m < 0 m
9 Zernikeolynomials: Different Conventions Different standardizations used concerning: 1. indexing. scaling / normalization 3. sign of coordinates (orientation for off-axis field oints) Fringe - reresentation 1. CodeV, Zemax, interferometric test of surfaces. Standardization of the boundary to ±1 3. no additional refactors in the olynomial 4. Indexing according to m (Azimuth), quadratic number terms have circular symmetry 5. coordinate system invariant in azimuth Standard - reresentation - CodeV, Zemax, Born / Wolf - Standardization of rms-value on ±1 (with refactors), easy to calculate Strehl ratio - coordinate system invariant in azimuth Original - Nijboer - reresentation - Exansion: k k n k n 1 0 m m W ( r, ) a00 a0nrn anmrn cos( m) bnmrn sin( m) n 0 m n 1 m gerade - Standardization of rms-value on ±1 - coordinate system rotates in azimuth according to field oint n 0 n0 m n 1 m gerade
30 Wave Aberration Criteria Mean quadratic wave deviation ( W Rms, root mean square ) W rms W with uil area 1 W W x, y Wmean x, y AExP A ExP dxdy Peak valley value W v : largest difference v x, y W x y W max W, max min General case with aodization: weighting of local hase errors with intensity, relevance for sf formation W rms 1 A ( w) x, y W x, y W x y I w ExP mean, ( ) ExP dx dx dy dy
31 Rayleigh Criterion The Rayleigh criterion W PV l 4 gives individual maximum aberrations coefficients, deends on the form of the wave a) otimal constructive interference b) reduced constructive interference due to hase aberrations c) reduced effect of hase error by aodization and lower energetic weighting d) start of destructive interference for 90 or l/4 hase aberration begin of negative z-comonent Examles: aberration tye coefficient defocus Seidel a 0 0. 5 defocus Zernike c 0. 0 15 sherical aberration Seidel a 0. 40 5 sherical aberration Zernike c 40 0. 167 astigmatism Seidel a 0. 5 astigmatism Zernike c 0. 15 coma Seidel a 31 0. 15 coma Zernike c 0. 31 15
3 Criteria of Rayleigh and Marechal Rayleigh criterion: 1. maximum of wave aberration: W v < l/4. beginning of destructive interference of artial waves 3. limit for being diffraction limited (definition) 4. as a PV-criterion rather conservative: maximum value only in 1 oint of the uil 5. different limiting values for aberration shaes and definitions (Seidel, Zernike,...) Marechal criterion: 1. Rayleigh crierion corresonds to W rms < l/14 in case of defocus Rayleigh W rms l 19 l 13.856 l 14. generalization of W rms < l/14 for all shaes of wave fronts 3. corresonds to Strehl ratio D s > 0.80 (in case of defocus) 4. more useful as PV-criterion of Rayleigh
33 PV and W rms -Values PV and W rms values for different definitions and shaes of the aberrated wavefront Due to mixing of lower orders in the definition of the Zernikes, the W rms usually is smaller in comarison to the corresonding Seidel definition
34 Tyical Variation of Wave Aberrations Microscoic objective lens Changes of rms value of wave aberration with 1. wavelength. field osition 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 W rms [l] 0.30 0.4 0.18 0.1 field zone Achrolan 40x0.65 field edge 0.06 diffraction limit on axis 0 0.480 0.56 0.644 l [m]
35 Rms of Wavefront as Function of the Field Rms of the wavefront changes with field osition scaled in l Only one scalar number er field oint: fine structure suressed
36 Zernike Calculation on distorted grids Conventional calculation of the Zernikes: equidistant grid in the entrance uil Real systems: Puil aberrations and distorted grid in the exit uil Deviating ositions of hase gives errors in the Zernike calculation Additional effect: re-normalization of maximum radius
37 Zernike Calculation on distorted grids 5% radial distorted grid with both signs: +5%: incushion -5% barrel Wavefront of only one selected Zernike is re-analyzed on distorted re-normalized grid: Effects of distortion: - errors in the range of some ercent - cross-mixing to other zernikes of same symmetry - neighboring Zernikes artly influenced until 0% - larger effects of higher order Zernikes - slightly larger effects for incushion distortion
38 Zernike Coefficients for Change of Radius Change of normalization radius, Problem, if uil edge is not well known or badly defined Deviation in the radius of normalization of the uil size: 1. wrong coefficients. mixing of lower orders during fit-calculation, symmetry-deendent Examle rimary sherical aberration: olynomial: 4 Z9( ) 6 6 1 Stretching factor of the radius r New Zernike exansion on basis of r Z 9 1 3 1 Z 9( r) 4 4 4 3 Z 4 1 Z 4 ( r) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0.1 c 9 / c 9 c 1 0 0.9 0.91 0.9 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 c 4
39 Zernike Exansion of Local Deviations original Small Gaussian bum in the toology of a surface model N = 36 N = 64 N = 100 N = 144 N = 5 N = 34 N = 65 Rms = 0.037 0.0193 0.0149 0.0109 0.0064 0.003 0.00047 PV = 0.378 0.307 0.35 0.170 0.0954 0.0475 0.0063 error 0.04 Sectrum of coefficients for the last case 0.03 0.0 0.01 0 0 100 00 300 400 500 600
40 Measurement of Wave Aberrations Wave aberrations are measurable directly Good connection between simulation/otical design and realization/metrology Direct hase measuring techniques: 1. Interferometry. Hartmann-Shack 3. Hartmann sensor 4. Secial: Moire, Holograhy, hase-sace analyzer Indirect measurement by inversion of the wave equation: 1. Phase retrieval of PSF z-stack. Retrieval of edge or line images Indirect measurement by analyzing the imaging conditions: from general image degradation Accuracy: 1. l/1000 ossible, l/100 standard for rms-value. Rms vs. individual Zernike coefficients
41 Testing with Twyman-Green Interferometer reference mirror Short common ath, sensible setu Two different oeration modes for reflection or transmission Always factor of between detected wave and comonent under test collimated laser beam beam slitter objective lens sto 1. mode: lens tested in transmission auxiliary mirror for autocollimation. mode: surface tested in reflection auxiliary lens to generate convergent beam detector
4 Interferograms of Primary Aberrations Sherical aberration 1 l Astigmatism 1 l Coma 1 l -1-0.5 0 +0.5 +1 Defocussing in l
43 Interferogram - Definition of Boundary Critical definition of the interferogram boundary and the Zernike normalization radius in reality
44 Hartmann Shack Wavefront Sensor Lenslet array divides the wavefront into subaertures Every lenslet generates a simgle sot in the focal lane The averaged local tilt roduces a transverse offset of the sot center Integration of the derivative matrix delivers the wave front W(x,y) array detector wavefront D array h f D meas u x sot offset D sub refractive index n
45 Hartmann Shack Wavefront Sensor Tyical setu for comonent testing detector test surface lenslet array fiber illumination Lenslet array collimator beamslitter telescoe for adjustment of the diameter a) b) subaerture oint sread function -dimensional lenslet array