Imaging and Aberration Theory

Transcription

1 maging and Aberration Theory Lecture 3: PSF and transfer function 5--5 Herbert Gross Winter term 4

2 Preliminary time schedule 3.. Paraial imaging araial otics, fundamental laws of geometrical imaging, comound systems Puils, Fourier otics, uil definition, basic Fourier relationshi, hase sace, analogy otics and 6.. Hamiltonian coordinates mechanics, Hamiltonian coordinates Eikonal Fermat rincile, stationary hase, Eikonals, relation rays-waes, geometrical aroimation, inhomogeneous media 4.. Aberration eansions single surface, general Taylor eansion, reresentations, arious orders, sto shift formulas Reresentation of aberrations different tyes of reresentations, fields of alication, limitations and itfalls, measurement of aberrations Sherical aberration henomenology, sh-free surfaces, skew sherical, correction of sh, asherical surfaces, higher orders 7.. Distortion and coma henomenology, relation to sine condition, alanatic sytems, effect of sto osition, arious toics, correction otions Astigmatism and curature henomenology, Coddington equations, Petzal law, correction otions Chromatical aberrations Disersion, aial chromatical aberration, transerse chromatical aberration, sherochromatism, secondary soectrum Sine condition, alanatism and Sine condition, isolanatism, relation to coma and shift inariance, uil 5.. isolanatism aberrations, Herschel condition, relation to Fourier otics.. Wae aberrations definition, arious eansion forms, roagation of wae aberrations 9.. Zernike olynomials PSF and transfer function 4.. Additional toics secial eansion for circular symmetry, roblems, calculation, otimal balancing, influence of normalization, measurement ideal sf, sf with aberrations, Strehl ratio, transfer function, resolution and contrast Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, reertability

3 3 Contents. deal oint sread function. PSF with aberrations 3. Strehl ratio 4. High NA ectorial PSF 5. Two-oint-resolution 6. Otical transfer function 7. Resolution and contrast

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

5 5 PSF by Huygens Princile Huygens waelets corresond to ectorial field comonents The hase is reresented by the direction The amlitude is reresented by the length Zeros in the diffraction attern: destructie interference Aberrations from sherical wae: reduced conctructie suerosition ideal reference shere oint sread function side lobe eak zero intensity central eak maimum constructie interference wae front uil sto reduced constructie interference due to hase aberration

6 Fraunhofer Point Sread Function 6 Rayleigh-Sommerfeld diffraction integral, Mathematical formulation of the Huygens-rincile ik r r ' i e E r E r' r r' cos d' dy' d Fraunhofer aroimation in the far field for large Fresnel number N F r z Otical systems: numerical aerture NA in image sace Puil amlitude/transmission/illumination T,y Wae aberration W,y comle uil function A,y Transition from eit uil to image lane E ', y' Point sread function PSF: Fourier transform of the comle uil function iw, y A, y T, y e AP T i ' y y' iw, y RAP, y e e d dy

7 intensity 7 Perfect Point Sread Function Circular homogeneous illuminated Aerture: intensity distribution transersal: Airy scale:. D Airy NA aial: sinc scale n R E NA Resolution transersal better than aial: < z,,8,6,4, ertical lateral, J u, u / sin u / 4, u / 4 Scaled coordinates according to Wolf : aial : u = z n / NA transersal : = / NA Ref: M. Keme

8 8 deal Psf r,z focal oint sread sot z aial sinc otical ais aerture cone r lateral Airy image lane

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

10 Perfect Lateral Point Sread Function: Airy Airy function : Perfect oint sread function for seeral assumtions Distribution of intensity: r J NA r r NA Normalized transerse coordinate ar R akr krsin u' ak sin' R Airy diameter: distance between the two zero oints, diameter of first dark ring r D Airy / D Airy.976 n' sin u' r

11 Perfect Lateral Point Sread Function: Airy Airy distribution: Gray scale icture Zeros non-equidistant Logarithmic scale Encircled energy log r E circ r r D Airy ring.48%. ring.79%. ring 7.6% eak 83.8% r / r Airy

12 Perfect Aial Point Sread Function Aial distribution of intensity Corresonds to defocus Normalized aial coordinate NA z u z 4 Scale for deth of focus : Rayleigh length z z sin sin u / 4 z o z u / 4 R E n' sin u' n' NA Zero crossing oints:. equidistant and symmetric,. Distance zeros around image lane 4R E z/ R E z = R E 4R E

13 3 Defocussed Perfect Psf Perfect oint sread function with defocus Reresentation with constant energy: etreme large dynamic changes z = -R E z = -R E focus z = +R E z = +R E normalized intensity ma = 5.% ma = 9.8% ma = 4% constant energy

14 4 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j =...7 The eak intensities are renormalized trefoil coma 5. order astigmatism 5. order sherical 5. order c =. c =. c =. c =.3 c =.4 c =.5 c =.7 coma astigmatism sherical defocus

15 5 Strehl Ratio mortant citerion for diffraction limited systems: Strehl ratio Strehl definition Ratio of real eak intensity with aberrations referenced on ideal eak intensity D S real PSF ideal PSF,, D D S takes alues between... D S = is erfect S A, y e iw, y A, y ddy ddy Critical in use: the comlete information is reduced to only one number The criterion is useful for 'good' systems with alues D s >.5 distribution broadened eak reduced Strehl ratio ideal, without aberrations real with aberrations r

16 6 Aroimations for the Strehl Ratio Aroimation of Marechal: useful for D s >.5 but negatie alues ossible Bi-quadratic aroimation Eonential aroach D s Wrms 4 Wrms D s D s e 4 W rms D S Marechal defocus eonential biquadratic eac t c Comutation of the Marechal aroimation with the coefficients of Zernike D s N n cn n N n n m c nm n

17 7 Strehl Ratio Criterion n the case of defocus, the Rayleigh and the Marechal criterion deliers a Strehl ratio of D S The criterion D S > 8 % therefore also corresonds to a diffraction limit This alue is generalized for all aberration tyes aberration tye coefficient Marechal aroimated Strehl eact Strehl 8 defocus Seidel a defocus Zernike c sherical aberration Seidel a sherical aberration Zernike c astigmatism Seidel a astigmatism Zernike c coma Seidel a coma Zernike c

18 8 Quality Criteria for Point Sread Function Criteria for measuring the degradation of the oint sread function:. Strehl ratio. width/threshold diameter 3. second moment of intensity distribution 4. area equialent width 5. correlation with erfect PSF 6.ower in the bucket a Strehl ratio b Standard deiation c Light in the bucket d Equialent width SR / D s STDEV LB EW e Second moment f Threshold width g Correlation width h Width enclosed area SM FWHM Ref P=5% WEA CW

19 High-NA Focusing Transfer from entrance to eit uil in high-na:. Geometrical effect due to rojection hotometry: aodization with A A s sin u s 4 NA n Tilt of field ector comonents A s r A r s r cos y E r E E E dy y dy/cosu y y' e Ey e y s E r R s u u y' ' entrance uil eit uil R image lane

20 High-NA Focusing Total aodization corresonds to astigmatism Eamle calculations A lin r, A r, s r s s r 4 r cos NA =.5 NA =.8 NA =.9 NA =.97 Ar.5 y r

21 Vectorial Diffraction for high-na Vectorial reresentation of the diffraction integral according to Richards/Wolf Auiliary integrals General: aial and cross comonents of olarization cos sin cos ' ', / 4sin i i e E E E E z r E iu z y o d e kr J z r ikz 'cos sin ' cos sin cos ', o d e kr J z r ikz 'cos sin ' sin cos ' ', o d e kr J z r ikz 'cos sin ' cos sin cos ' ',

22 Vectorial Diffraction at high NA Linear Polarization Puil

23 High NA and Vectorial Diffraction Relatie size of ectorial effects as a function of the numerical aerture Characteristic size of errors: / o error aial lateral aial lateral NA

24 4 Point Resolution According to Abbe Transerse resolution of an image: - Detection of object details / fine structures - basic formula of Abbe k Fundamental deendence of the resolution from:. waelength. numerical aerture angle 3. refractie inde 4. refactor, deends on geometry, coherence, olarization, illumination,... nsin Basic ossibilities to increase resolution:. shorter waelength DUV lithograhy. higher aerture angle eensie, 75 in microscoy 3. higher inde immersion 4. secial olarization, otimal artial coherence,... Assumtions for the alidity of the formula:. no eanescent waes no near field effects. no non-linear effects -hoton

25 5 ncoherent -Point Resolution : Rayleigh Criterion Rayleigh criterion for -oint resolution Maimum of sf coincides with zeros of neighbouring sf D Airy.6 nsin u Contrast: V =.5 Decrease of intensity between eaks = sum of PSF.6.4. PSF PSF / r airy

26 6 ncoherent -Point-Resolution: Sarrow Criterion Criterion of Sarrow: anishing deriatie in the center between two oint intensity distribution, corresonds to anishing contrast d d Modified formula Sarrow D nsin u.77 Usually needs a riory information Alicable also for non-airy distributions Used in astronomy Rayleigh Airy / r airy

27 7 ncoherent -oint Resolution Criterions Visual resolution limit: Good contrast isibility V = 6 % : nsin u D Airy Total resolution: Coincidence of neighbouring zero oints of the Airy distributions: V = Etremly conseratie criterion D Airy. nsin u Contrast limit: V = : ntensity = between eaks nsin u D Airy

28 8 -Point Resolution Distance of two neighboring object oints Distance scales with / sinu Different resolution criteria for isibility / contrast V =. / sinu total V = =.68 / sinu isual V =.6 =.6 / sinu Rayleigh V =.5 =.474 / sinu Sarrow V =

29 9 -Point Resolution ntensity distributions below % for oints with different scaled on Airy =. =. =. =.83 =.6 =.474 =.388 =.5

30 3 ncoherent Resolution: Deendence on NA Microscoical resolution as a function of the numerical aerture NA =. NA =.3 NA =.45 NA =.9

31 3 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 Airy D Geo eact wae-otic D Airy geometric-otic aroimated aberrations diffraction limited, failure of the geometrical model Fourier transform ill conditioned

32 3 Resolution of Fourier Comonents object detail high satial frequencies numerical aerture resoled frequencies object oint low satial frequencies object sum image for low NA decomosition of Fourier comonents sin waes image for high NA object high satial frequencies Ref: D.Aronstein / J. Bentley

33 33 Otical Transfer Function: Definition Normalized otical transfer function OTF in frequency sace Fourier transform of the Psfintensity H H OTF OTF, y g, y, Fˆ, y y PSF g e, y i y d y dy d dy OTF: Autocorrelation of shifted uil function, Duffieu-integral H OTF, y P f, y f y P P *, y f d dy, y f y d dy Absolute alue of OTF: modulation transfer function MTF MTF is numerically identical to contrast of the image of a sine grating at the corresonding satial frequency

34 MTF and Contrast Object Contrast Object sectrum mage sectrum mage cos a c obj c a a c a c a c a c V min ma min ma ˆ a a c F obj obj H MTF obj ima cos ˆ ˆ ˆ ' H a c e H a e H a H c H a H a H c F H F F MTF i MTF i MTF MTF MTF MTF MTF MTF obj ima ima 34

35 35 nterretation of the Duffieu integral nterretation of the Duffieu integral: oerla area of th and st diffraction order, interference between the two orders objectie uil y' direct light The area of the oerla corresonds to the information transfer of the structural details ' Frequency limit of resolution: areas comletely searated y o at object diffracted light in st order y L object o q y L shifted uil areas y condenser conjugate to object uil f f y light source area of integration

36 8 36 Duffieu ntegral and Contrast Searation of uils for. and +-. Order attern eriod /NA MTF function uil image contrast % mage contrast for sin-object NA/ V=% Ref: W. Singer V=5% V=3% min =.3 min =.5

37 37 Otical Transfer Function of a Perfect System Aberration free circular uil: Reference frequency o a f sinu' Cut-off frequency:.5 g MTF G na nsin u' f Analytical reresentation.5 / ma H MTF arccos Searation of the comle OTF function into: - absolute alue: modulation transfer MTF - hase alue: hase transfer function PTF H OTF, PTF y, H, e y MTF y ih

38 38 Contrast / Visibility The MTF-alue corresonds to the intensity contrast of an imaged sin grating Visibility The maimum alue of the intensity is not identical to the contrast alue since the minimal alue is finite too Concrete alues: ma V V ma ma min min eak decreased sloe decreased object image ma min minima.5 increased

39 39 Sagittal and Tangential MTF Due to the asymmetric geometry of the sf for finite field sizes, the MTF deends on the azimuthal orientation of the object structure Generally, two MTF cures are considered for sagittal/tangential oriented object structures g MTF tangential lane y ideal.5 sagittal tangential sagittal tangential.5 / ma arbitrary rotated tangential sagittal sagittal lane

40 4 Real MTF Real MTF of system with residual aberrations:. contrast decreases with defocus. higher satial frequencies hae stronger decrease g MTF z,f.75.5 g MTF z = Zernike coefficients: c 5 =. c 7 =.5 c 8 =.3 c 9 =.5 ma =.5 ma =. ma =. ma =.3 ma =.4 ma =.5 ma =.6 ma =.7 ma =.8.5 z =. R u z =. R u z =.3 R u z =. R u z =.5 R -.5 u z in R U

41 4 Resolution Test Chart: Siemens Star a. original b. good system c. defocus d. sherical e. astigmatism f. coma

42 4 Contrast and Resolution Contrast s contrast as a function of satial frequency Tyical: contrast reduced for increasing frequency V Comromise between resolution and isibilty is not triial and deends on alication H MTF Contrast sensitiity H CSF c

43 43 Otical Transfer Function of a Perfect System Loss of contrast for higher satial frequencies.9.8 ideal MTF / ma contrast / ma

44 44 Contrast / Resolution of Real mages Degradation due to. loss of contrast. loss of resolution resolution, sharness contrast, saturation