Lens Design II. Lecture 1: Aberrations and optimization Herbert Gross. Winter term

Transcription

1 Lens Design II Lecture 1: Aberrations and otimization Herbert Gross Winter term 16

2 Preliminar Schedule Aberrations and otimization Reetition 6.1. Structural modifications Zero oerands, lens slitting, lens addition, lens removal, material selection Asheres Correction with asheres, Forbes aroach, otimal location of asheres, several asheres Freeforms Freeform surfaces Field flattening Astigmatism and field curvature, thick meniscus, lus-minus airs, field lenses Chromatical correction I Achromatization, aial versus transversal, glass selection rules, burried surfaces Chromatical correction II secondar sectrum, aochromatic correction, sherochromatism Secial correction toics I Smmetr, wide field sstems,sto osition Secial correction toics II Anamorhotic lenses, telecentricit Higher order aberrations high NA sstems, broken achromates, induced aberrations Further toics Sensitivit, scan sstems, eeieces Mirror sstems secial asects, double asses, catadiotric sstems Zoom sstems mechanical comensation, otical comensation Diffractive elements color correction, ra equivalent model, stralight, third order aberrations, manufacturing Realization asects Tolerancing, adjustment

3 3 Contents 1. Geometrical aberrations. Wave aberrations and Zernikes 3. Point sread function 4. Modulation transfer function 5. Princiles of otimization

4 4 Otical Image Formation Perfect otical image: All ras coming from one object oint intersect in one image oint Real sstem with aberrations: 1. tranverse aberrations in the image lane. longitudinal aberrations from the image lane 3. wave aberrations in the eit uil object lane wave aberrations image lane otical sstem transverse aberrations longitudinal aberrations

5 5 Notations for an Otical Sstem, object coordinates, eseciall object height ', ' image coordinates, eseciall image height, coordinates of entrance uil ', ' coordinates of eit uil s object distance form 1st surface s' image distance form last surface entrance uil distance from 1st surface ' eit uil distance from last surface D' sagittal transverse aberration D' meridional transverse aberration ' P ' ' ' s' ' P' ' D' P' D' image lane z P ' sstem surfaces s P entrance uil ' P eit uil object lane P

6 6 Polnomial Eansion of Aberrations Reresentation of -dimensional Talor series vs field and aerture r Selection rules: checkerboard filling of the matri Constant sum of eonents according to the order Image location Primar aberrations / Seidel r 5 Field Sherical Coma Astigmatism cos 3 cos 5 cos Distortion r Tilt Distortion Distortion rimar secondar r 1 r 1 cos 4 r 1 cos r 1 Defocus r 1 Aerture Astig./Curvat. 4 r 1 r cos 3 r r r cos 3 Coma rimar 3 r cos r 3 r 3 cos r 3 Sherical rimar r 3 r 4 cos r 4 Coma secondar Sherical r 5 secondar Secondar aberrations

7 7 Puil Samling Ra lots Sot sagittal ra fan tangential aberration D sagittal aberration D diagrams tangential ra fan D whole uil area

8 . mm Sherical Aberration 8 Tical chart of aberration reresentation Reference: at araial focus Primar sherical aberration at araial focus Wave aberration tangential sagittal Transverse ra aberration D D D.1 mm.1 mm.1 mm Puil: -section -section -section Modulation Transfer Function MTF MTF at araial focus MTF through focus for 1 ccles er mm cc/mm -... z/mm Geometrical sot through focus Ref: H. Zügge z/mm

9 Surface Contributions: Eamle 9 Seidel aberrations: reresentation as sum of surface contributions ossible Gives information on correction of a sstem Eamle: hotograhic lens Retrofocus F/.8 Field: w= S I Sherical Aberration S II Coma S III Astigmatism S IV Petzval field curvature S V Distortion C I Aial color C II Lateral color -4 Surface Sum

10 1 Wave Aberration in Otical Sstems Definition of otical ath length in an otical sstem: Reference shere around the ideal object oint through the center of the uil Chief ra serves as reference Difference of OPL : otical ath difference OPD Practical calculation: discrete samling of the uil area, real wave surface reresented as matri ' ' uer coma ra otical sstem W wave aberration chief ra w' image oint z chief ra wave front object oint lower coma ra reference shere Object lane O Entrance uil EnP Eit lane EP Image lane I

11 11 Puil Samling All ras start in one oint in the object lane The entrance uil is samled equidistant In the eit uil, the transferred grid ma be distorted In the image lane a sreaded sot diagram is generated object lane oint entrance uil equidistant grid otical sstem eit uil transferred grid image lane sot diagram o ' ' o ' ' z

12 1 Zernike Polnomials Eansion of wave aberration surface into elementar functions / shaes m W ( r, ) c Z ( r, ) n n mn nm n m = 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 m m Ordering of the Zernike olnomials b indices: n : radial m : azimuthal, sin/cos Mathematicall orthonormal function on unit circle for a constant weighting function Direct relation to rimar aberration tes n = sin

13 13 Perfect Lateral Point Sread Function: Air Air distribution: I(r,z) Gra scale icture Zeros non-equidistant Logarithmic scale Encircled energ z E circ (r) 1 r D Air ring 1.48%. ring.79% 1. ring 7.6% eak 83.8% r / r Air

14 intensit 14 Perfect Point Sread Function Circular homogeneous illuminated Aerture: intensit distribution transversal: Air scale: 1. D Air NA aial: sinc scale n R E NA Resolution transversal better than aial: D < Dz 1,,8,6,4, vertical lateral, J1 v Iu I, v I v u / v sin u / 4, I u / 4 Scaled coordinates according to Wolf : aial : u = z n / NA transversal : v = / NA Ref: M. Keme

15 15 Psf with Aberrations Zernike coefficients c in Sherical aberration Astigmatism Coma Sherical aberration, Circular smmetr Astigmatism, Slit of two azimuths Coma, Asmmetric c =. c =.3 c =.5 c =.7 c = 1.

16 16 Otical Transfer Function: Definition Normalized otical transfer function (OTF) in frequenc sace Fourier transform of the Psfintensit H H OTF OTF ( v, v ) g(, ( v, v ) Fˆ I (, ) PSF ) g( e, i v v ) d d d d OTF: Autocorrelation of shifted uil function, Duffieu-integral H OTF ( v, v ) P( f v, f v P( * ) P (, ) f v d d, f v ) d d Absolute value of OTF: modulation transfer function (MTF) MTF is numericall identical to contrast of the image of a sine grating at the corresonding satial frequenc

17 17 Otical Transfer Function of a Perfect Sstem Aberration free circular uil: Reference frequenc v o a f sinu' 1.5 g MTF Maimum cut-off frequenc: v ma v na f nsin u' Analtical reresentation.5 1 / ma H MTF v ( v) arccos v v v 1 Searation of the comle OTF function into: - absolute value: modulation transfer MTF - hase value: hase transfer function PTF v v H OTF ( v, v PTF ( v, v ) H ( v, v ) e MTF ih )

18 18 Basic Idea of Otimization Toolog of the merit function in dimensions Iterative down climbing in the toolog toolog of meritfunction F start iteration ath 1

19 19 Nonlinear Otimization Mathematical descrition of the roblem: n variable arameters m target values Jacobi sstem matri of derivatives, Influence of a arameter change on the various target values, sensitivit function Scalar merit function Gradient vector of toolog f () J i j f m F( ) w g j i1 F j i j i f ( ) i Hesse matri of nd derivatives H jk F j k

20 Otimization in Otical Design Merit function: Weighted sum of deviations from target values Formulation of target values: 1. fied numbers. one-sided interval (e.g. maimum value) 3. interval g f ist j j f j1, m soll j Problems: 1. linear deendence of variables. internal contradiction of requirements 3. initail value far off from final solution Tes of constraints: 1. eact condition (hard requirements). soft constraints: weighted target Finding initial sstem setu: 1. modification of similar known solution. Literature and atents 3. Intuition and eerience

21 1 Otimization Merit Function in Otical Design Goal of otimization: Find the sstem laout which meets the required erformance targets according of the secification Formulation of erformance criteria must be done for: - Aertur ras - Field oints - Wavelengths - Otional several zoom or scan ositions Selection of erformance criteria deends on the alication: - Ra aberrations - Sot diameter - Wavefornt descrition b Zernike coefficients, rms value - Strehl ratio, Point sread function - Contrast values for selected satial frequencies - Uniformit of illumination Usual scenario: Number of requirements and targets quite larger than degrees od freedom, Therefore onl solution with comromize ossible

22 Lens bending und shift of rincial lane Ra ath at a lens of constant focal length and different bending Quantitative arameter of descrition X: The ra angle inside the lens changes X R R 1 R R 1 The ra incidence angles at the surfaces changes strongl The rincial lanes move For invariant location of P, P the osition of the lens moves P P' F' X = -4 X = - X = X = + X = +4

23 3 Correcting Sherical Aberration: Lens Slitting Transverse aberration Correction of sherical aberration: Slitting of lenses (a) 5 mm Distribution of ra bending on several surfaces: - smaller incidence angles reduces the effect of nonlinearit - decreasing of contributions at ever surface, but same sign Last eamle (e): one surface with comensating effect (b) (c) 5 mm 5 mm Imrovement (a)à(b) : 1/4 Imrovement (b)à(c) : 1/ 5 mm (d) Imrovement (c)à(d) : 1/4.5 mm (e) Imrovement (d)à(e) : 1/75 Ref : H. Zügge