Imaging and Aberration Theory

Transcription

1 maging and Aberration Theory Lecture 4: Transfer function 6-- Herbert Gross Winter term 5

2 Preliminary time schedule.. Paraial imaging araial otics, fundamental laws of geometrical imaging, comound systems Puils, Fourier otics, uil definition, basic Fourier relationshi, hase sace, analogy otics and 7.. 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 secial eansion for circular symmetry, roblems, calculation, otimal balancing, influence of normalization, measurement PSF ideal sf, sf with aberrations, Strehl ratio 4.. Transfer function Transfer function, resolution and contrast Additional toics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, reertability

3 3 Contents Fourier method Otical Transfer function Contrast and resolution Quantitatie erformance assessment ncoherent image formation Cascaded systems Coherent image formation 3D transfer theory and deth resolution

4 Definitions of Fourier Otics Phase sace with satial coordinate and. angle. satial frequency in mm - 3. transerse waenumber k k Fourier sectrum k k A(, Fˆ E(, y y object structure k / g diffracted ray direction k T corresonds to a lane wae eansion i k yk y, y, (,, A k k z E y z e d dy g = / Diffraction at a grating with eriod g: deiation angle of first diffraction order aries linear with = /g sin g

5 5 Grating Diffraction and Resolution a resoled incident light object b not resoled diffracted orders otical system Arbitrary object eaneded into a satial frequency sectrum by Fourier transform Eery frequency comonent is considered searately To resole a satial detail, at least two orders must be suorted by the system -. off-ais illumination g g sin m sin NA g Ref: M. Keme NA

6 6 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

7 7 Otical Transfer Function: Definition Normalized otical transfer function (OTF in frequency sace H OTF (, y N sf (, y e i y y d dy Fourier transform of the Psfintensity H OTF (, Fˆ (, y y PSF OTF: Autocorrelation of shifted uil function, Duffieu-integral (general: D H OTF ( P f P * f d Transfer roerties: PSF: resonse answer of a oint object OTF: resonse answer of an etended cosine grating Absolute alue of OTF: modulation transfer function (MTF MTF is numerically identical to contrast of the image of a cosine grating at the corresonding satial frequency

8 Number of Suorted Orders A structure of the object is resoled, if the first diffraction order is roagated through the otical imaging system The fidelity of the image increases with the number of roagated diffracted orders. / +. / -. order. / +. / / -. order. / / +. / -. / +3. / -3. order

9 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 9

10 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

11 8 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

12 Resolution and Satial Frequencies Grating object uil maging with NA =.8 maging with NA =.3 Ref: L. Wenke

13 3 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

14 4 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

15 5 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

16 6 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

17 7 Polychromatic MTF Polychromatical MTF: Cut off frequency deends on Sectral incoherent weighted suerosition of monochromatic MTF s g ( oly OTF ( S( g (, d OTF g MTF ideal 35 nm = 35 nm = 4 nm = 45 nm = 5 nm = 55 nm = 6 nm = 65 nm = 7 nm olychromatic. ma / 7.5 L/mm ma 55 L/mm

18 8 Hokins Factor Resolution/contrast criterion: Ratio of contrasts with/without aberrations for one selected satial frequency g MTF ( real gmtf ( ( ( ideal g ( MTF g MTF Real systems: Choice of seeral alication releant frequencies e.g. hotograhic lens: L/mm, L/mm, 4 L/mm.5 real ideal g MTF ideal real g MTF

19 9 Contrast s Resolution Balance between contrast and resolution: not triial Otimum deends on alication Receier: minimum contrast cure seres as real reference Most detector needs higher contrast to resole high frequencies CSF: contrast sensitiity function g MTF : high contrast threshold contrast b : is better threshold contrast a : is better :high resolution

20 MTF [%] bei,, 4 L/mm... tan sag Modulation Transfer Function Photograhic lenses with different erformance Objekti f/ 3.5 Objekti Lens f/3.5 Lens ma. MTF Bildhöhe [mm] ma. MTF mage height c/mm c/mm 4 c/mm

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

22 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

23 3 Otical Transfer Function of a Perfect System Loss of contrast for higher satial frequencies.9.8 ideal MTF / ma contrast / ma

24 4 Contrast / Resolution of Real mages Degradation due to. loss of contrast. loss of resolution resolution, sharness contrast, saturation

25 5 Resolution Estimation with Test Charts Measurement of resolution with test charts: bar attern of different sizes two different orientations calibrated size/satial frequency

26 6 Test: Siemens Star Determination of resolution and contrast with Siemens star test chart: Central segments b/w Growing satial frequency towards the center Gray ring zones: contrast zero Calibrating satial feature size by radial diameter Nested gray rings with finite contrast in between: contrast reersal, seudo resolution

27 7 Contrast and Resolution High frequent structures : contrast reduced Low frequent structures: resolution reduced contrast brillant blurred shar milky resolution

28 Fourier Otics Point Sread Function Point sread function amlitude in an otical system with magnification m Puil function P, ik ' Puil coordinates,y z g (, y, ', y' N P, y e sf m y y' my d dy PSF is Fourier transform of the uil function (scaled coordinates source oint object lane g (, y N Fˆ P, y sf image lane oint image distribution

29 Fourier Theory of ncoherent mage Formation Transfer of an etended object distribution obj (,y ima ( ', y' (, ', y, y' (, y d dy sf obj n the case of shift inariance (isolanatism: incoherent conolution ntensities are additie n frequency sace: - roduct of sectra - linear transfer theory - sectrum of the sf works as low ass filter onto the object sectrum - Otical transfer function H otf image (, FT (, y y PSF object intensity (, y Hotf (, y obj(, y ima ima object lane ( ', y' ( ', y y' (, y d dy ( ', y' (, y* (, y sf sf obj obj image intensity single sf image lane

30 ncoherent mage Formation Eamle: incoherent imaging of bar attern near the resolution limit object resoled not resoled

31 ncoherent mage Formation Eamle: incoherent imaging of attern near the resolution limit with aberrations object ideal astigmatism coma sherical aberration PSF

32 Cascaded Otical Systems Cascaded systems with indiidual transfer functions:. Diffusing screen in intermediate image lane: ncoherent conolution of indiidual Psfs incoher aro ( sf ( sf( object( sf ( Asf ( ( A. Usual case: Second system works coherent Wae aberrations can be balanced Conolution of amlitude Psfs eact sf ( Asf ( ( A

33 Cascaded Otical Systems Cascaded systems with indiidual transfer functions: Behaiour of the transfer functions in the limiting cases a Single systems MTF MTF c def = -c def c def = -c def real erfect real b Comlete cascaded system MTF coh correction c def = -c def MTF incoh suerosed broadening c def, c def real coherent real incoherent

34 Cascaded Otical Systems ncoherent case:. Psf conolution. MTF roduct incoher incoher incoher incoher ( sf ( sf( object( ( sf ( sf( object( ( object( sf( sf ( ( object( sf( sf ( object lane system intermediate system image and sf, diffusing sf, screen image lane

35 Cascaded Otical Systems Coherent case:. Amlitude Psf conolution eact ( A sf ( A ( * * A ( A ( A ( A ( sf sf sf sf sf. MTF roduct eact iw iw iw iw T ( e T ( e T ( e T ( e iw iw T ( e T ( e ( object lane system A sf, intermediate image system A sf, image lane c def, c def,

36 Fourier Theory of Coherent mage Formation Transfer of an etended object distribution E obj (,y Eima ( ', y' Asf, y, ', y' E(, y d dy n the case of shift inariance (isolanasie: coherent conolution of fields Comle fields additie n frequency sace: - roduct of sectra - linear transfer theory with fields - sectrum of the sf works as low ass filter onto the object sectrum - Coherent otical transfer function H E ctf ima ima (, FT E (, y y PSF object amlitude distribution (, y Hctf (, y Eobj (, y (, y Hctf (, y Eobj (, y object lane Eima ( ', y' Asf ', y y' Eobj (, y d dy E ima, y E (, ( ', y' A y sf obj image lane image amlitude distribution single oint image

37 Comarison of OTF and CTF deal coherent transfer function: - corresonds to scaled uil function - cut of at uil edge - full contrast until edge deal incoherent transfer function: - conolution of shifted scaled uils - smooth decrease to cut-off frequence H H otf ctf ( P f ( arccos ncoherent case: - higher satial frequencies resoled - lower contrast at low frequencies H coherent incoherent.5 c

38 38 Ewald Shere Assuming an object as grating with eriod L k obj L Scattering of a wae at the object with - conseration of energy k in k out - conseration of momentum The outgoing k-ector must be on a shere: Ewald's shere for ossible scattered wae ectors k k k in obj out grating Ewald shere k obj k out k out k in k obj k in

39 39 McCutchen Formula and Aial Resolution maging of a lane wae at a olume object : minimum alue resolution : maimum interal Uncertainty relation: = P( z transerse uil z Radius of the Ewald shere generalized 3D uil: red area Transerse resolution due to Abbe R sin n / NA / n NA n R ca light cone z aial uil P( z Aial resolution: - height of the ca of the cone - McCutchen formula z z n R Rcos n / n NA n NA sin n/ Ewald shere

40 4 3D Transfer Function maging as 3D scattering henomen Only secial satial frequencies are allowed due to energy conseration and momentum reseration Green circle: suorted satial frequencies of the transmitted wae ector z obj = s - i o-ma n/ i s obj i backward forward Ewald shere

41 4 3D Transfer Function - Missing Cone Realistic case: finite numerical aerture illumination i Blue cone: ossible incoming wae direction due to illumination cone scattered wae 3D coherent transfer function: limited green area, that fulfills all conditions s obs z transfer function object Missing cone: certain range of satial aial satial frequencies can not be seen in the image, Eamle: interfaces of thin coatings are not seen missing cone transfer function object z

42 3D Point Sread Function 3D Fourier transform of uil with defocussing M : magnification A 3D sf (, y P(, y e ik d ik z M z' y yy d e d dy y y y' object t(,y,z uil P(,y,z field E(',y',z' z z z' ' d d'

43 General 3D Transfer Theory General coherent 3D transfer function Fourier transfer of sf field distribuition H cft ( A sf ( r ' e ir ' dr ' maging of an transarent object with transmission function T E ima ( ', y', z' e ikz' Tobj, y, z Hctf, y, z e i y y ( z M z d d y d z Offset / : otical ath length in thick object r Secial case of a single lens with uil function P..5 H ( rotsym r ctf r, z P r z with z z sin H CTF P( r z