Lens Design I. Lecture 8: Aberrations III Herbert Gross. Summer term

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Lens Design I Lecture 8: Aberrations III 17-6-1

1 Lens Design I Lecture 8: Aberrations III Herbert Gross Summer term 17

2 Preliminar Schedule - Lens Design I Basics Proerties of otical sstems I Proerties of otical sstems II Proerties of otical sstems III Introduction, Zema interface, menues, file handling, references, Editors, udates, windows, coordinates, Sstem descrition, 3D geometr, aerture, field, waelength Diameters, sto and uil, ignetting, Laouts, Materials, Glass catalogs, Ratrace, Ra fans and samling, Footrints Tes of surfaces, cardinal elements, lens roerties, Imaging, magnification, araial aroimation and modelling, telecentricit, infinit object distance and afocal image, local/global coordinates Comonent reersal, sstem insertion, scaling of sstems, asheres, gratings and diffractie surfaces, gradient media, soles Adanced handling I Miscellaneous, fold mirror, uniersal lot, slider, multiconfiguration, lens catalogs Aberrations I Reresentation of geometrical aberrations, Sot diagram, Transerse aberration diagrams, Aberration eansions, Primar aberrations Aberrations II Wae aberrations, Zernike olnomials, measurement of qualit Aberrations III Point sread function, Otical transfer function Otimization I Otimization II Adanced handling II Princiles of nonlinear otimization, Otimization in otical design, general rocess, otimization in Zema Initial sstems, secial issues, sensitiit of ariables in otical sstems, global otimization methods Sstem merging, ra aiming, moing sto, double ass, IO of data, stock lens matching Correction I Correction II Smmetr rincile, lens bending, correcting sherical aberration, coma, astigmatism, field curature, chromatical correction Field lenses, sto osition influence, retrofocus and telehoto setu, asheres and higher orders, freeform sstems, miscellaneous

3 3 Contents 1. Diffraction at an aerture. Ideal PSF 3. Real PSF and Strehl 4. OTF

$Diffraction at the Sstem Aerture Self luminous oints: emission of sherical waes Otical sstem: onl a limited solid angle is roagated, the truncaton of the sherical wae results in a finite angle light$

4 Diffraction at the Sstem Aerture Self luminous oints: emission of sherical waes Otical sstem: onl a limited solid angle is roagated, the truncaton of the sherical wae results in a finite angle light cone In the image sace: uncomlete constructie interference of artial waes, the image oint is sreaded The otical sstems works as a low ass filter sherical wae image lane object oint truncated sherical wae oint sread function object lane = 1. / NA

5 Fraunhofer Point Sread Function Raleigh-Sommerfeld diffraction integral, Mathematical formulation of the Hugens-rincile ik r r ' i e EI ( r E( r' r r' cos d' d' d Fraunhofer aroimation in the far field for large Fresnel number N F r z 1 Otical sstems: numerical aerture NA in image sace Puil amlitude/transmission/illumination T(, Wae aberration W(, comle uil function A(, Transition from eit uil to image lane E( ', ' Point sread function (PSF: Fourier transform of the comle uil function iw (, A(, T(, e AP T i ' ' iw, RAP, e e d d

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

7 Abbe Resolution and Assumtions Abbe resolution with scaling to /NA: Assumtions for this estimation and ossible changes A resolution beond the Abbe limit is onl ossible with iolating of certain assumtions Assumtion Resolution enhancement 1 Circular uil ring uil, diol, quadruole Perfect correction comle uil masks 3 homogeneous illumination diol, quadruole 4 Illumination incoherent artial coherent illumination 5 no olarization secial radiale olarization 6 Scalar aroimation 7 stationar in time scanning, moing gratings 8 quasi monochromatic 9 circular smmetr oblique illumination 1 far field conditions near field conditions 11 linear emission/ecitation non linear methods

Perfect Lateral Point Sread Function: Air Air distribution: Gra scale icture Zeros non-equidistant Logarithmic scale Encircled energ 1 1 1 1 1-1 -

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

9 Perfect Aial Point Sread Function Aial distribution of intensit Corresonds to defocus Normalized aial coordinate NA z u z 4 Scale for deth of focus : Raleigh length I(z z sin sin u / 4 I( z I Io z u / 4 R E n'sin u' Zero crossing oints:. equidistant and smmetric,.1 Distance zeros around image lane 4R E z/ R E z = R E 4R E

10 Defocussed Perfect Psf Perfect oint sread function with defocus Reresentation with constant energ: etreme large dnamic changes z = -R E z = -1R E focus z = +1R E z = +R E normalized intensit I ma = 5.1% I ma = 9.8% I ma = 4% constant energ

11 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.

12 Comarison Geometrical Sot Wae-Otical Psf Large aberrations: Waeotical calculation shows bad conditioning Wae aberrations small: diffraction limited, geometrical sot too small and wrong Aroimation for the intermediate range: sot diameter D Sot D Air D Geo eact wae-otic D Air geometric-otic aroimated aberrations diffraction limited, failure of the geometrical model Fourier transform ill conditioned

13 13 Strehl Ratio Imortant citerion for diffraction limited sstems: Strehl ratio (Strehl definition Ratio of real eak intensit (with aberrations referenced on ideal eak intensit D S I I ( real PSF ( ideal PSF,, D D S takes alues between...1 D S = 1 is erfect S A(, e iw (, A(, dd dd I( 1 Critical in use: the comlete information is reduced to onl one number The criterion is useful for 'good' sstems with alues D s >.5 distribution broadened eak reduced Strehl ratio ideal, without aberrations real with aberrations r

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

15 Contrast / Visibilit The MTF-alue corresonds to the intensit contrast of an imaged sin grating Visibilit The maimum alue of the intensit is not identical to the contrast alue since the minimal alue is finite too Concrete alues: I I ma V V I I ma ma I( 1 I I min min eak decreased sloe decreased object image I ma I min minima increased

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

17 Sagittal and Tangential MTF Due to the asmmetric geometr of the sf for finite field sizes, the MTF deends on the azimuthal orientation of the object structure Generall, two MTF cures are considered for sagittal/tangential oriented object structures g MTF tangential lane 1 ideal.5 sagittal tangential sagittal tangential.5 1 / ma arbitrar rotated tangential sagittal sagittal lane

PSF in Zema Onl far field model (Fraunhofer Two different algorithms aailable: 1. FFT-based - fast - equidistant eit uil samling assumed - high resolution PSF needs man oints.

18 PSF in Zema Onl far field model (Fraunhofer Two different algorithms aailable: 1. FFT-based - fast - equidistant eit uil samling assumed - high resolution PSF needs man oints. elementar integration (Hugens - slow (N 4 - indeendence of uil and image samling - alid also for calculation of uil distortion - gies correct Strehl number Different otions for reresentation ossible 18

19 PSF in Zema Logarithmic reresentation cross section 19

$geometrical aroimation ia sot calculation for not diffraction limited sstems Different reresentation settings: - maimun satial frequenc - olume relief - MTF / PTF -$

20 OTF in Zema Various otions: 1. FFT based calculation. reresentation as a function of - field size - defocus 3. Hugens PSF integral based 4. geometrical aroimation ia sot calculation for not diffraction limited sstems Different reresentation settings: - maimun satial frequenc - olume relief - MTF / PTF - changes oer the field size

21 OTF in Zema Various MTF reresentations 1

22 OTF in Zema Various MTF reresentations / geometrical aroimation

Medical Photonics Lecture Optical Engineering

Medical Photonics Lecture Optical Engineering Medical Photonics Lecture Otical Engineering Lecture 6: Diffraction 6-- Herbert Gross Winter term 6 www.ia.uni-jena.de Contents No Subject Ref Date Detailed Content Introduction Gross.. Materials, disersion,