Metrology and Sensing

Transcription

1 Metrolog and Sensing Lecture : Wave otics Herbert Gross Winter term 016

2 Preliminar Schedule No Date Subject Detailed Content Introduction Introduction, otical measurements, shae measurements, errors, definition of the meter, samling theorem Wave otics ACP Basics, olarization, wave aberrations, PSF, OTF Sensors Introduction, basic roerties, CCDs, filtering, noise Fringe rojection Moire rincile, illumination coding, fringe rojection, deflectometr Interferometr I ACP Introduction, interference, tes of interferometers, miscellaneous Interferometr II Eamles, interferogram interretation, fringe evaluation methods Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods Geometrical methods Tactile measurement, hotogrammetr, triangulation, time of flight, Scheimflug setu Seckle methods Satial and temoral coherence, seckle, roerties, seckle metrolog Holograh Introduction, holograhic interferometr, alications, miscellaneous Measurement of basic sstem roerties Bssic roerties, knife edge, slit scan, MTF measurement Phase retrieval Introduction, algorithms, ractical asects, accurac Metrolog of asheres and freeforms Asheres, null lens tests, CGH method, freeforms, metrolog of freeforms OCT Princile of OCT, tissue otics, Fourier domain OCT, miscellaneous Confocal sensors Princile, resolution and PSF, microsco, chromatical confocal method

3 3 Content Basic wave otics Polarization Wave aberrations Point sread function Transfer function

4 4 Basic Wave Otics Scalar wave A r a r e i r hase function, r, A 0 Phase surface: - fied hase for one time k r r e const - hase surface erendicular to unit vektor e Ref: W. Osten

5 5 Plane and Sherical Waves Plane wave wave vector k Sherical wave, t r k i Ae t r E, t r k i e r A t r E Ref.: B. Dörband

6 6 Basic Wave Otics Scalar wave Different tes of waves: hase function amlitude function Plane wave Phase Amlitude in D Sherical wave A r a r e i r r, ar A r a r e r k r o, i kr cos sin,,z e i a z A z r k r o o o A r a r e i kr Parabolic wave A a e i sin R

7 7 Basic Wave Otics Sherical wave interference o A r a r e i kr sherical wave 1 sherical wave sensor maima herbola

8 8 Coherence Observabilit of interference and coherence Ref: R. Kowarschik

9 9 Coherence Coherence: caabilit to interfere Satial coherence: - defined b size of light source - measurement rocedure: Young interferometer Temoral coherence: - finite wave train, aial length of coherence Dl c - finite bandwidth D - no interference for long ath differences - Measurement rocedure: Michelson interferometer - tical values: table Ref: R. Kowarschik

10 10 Interference of Waves The main roert is the hase difference between two waves Interference of two waves jk j k I I I I I cos secial case of equal intensites I I 1cos 0 1 Maima of intensit at even hase differences Minima of intensit at odd hase differences Interference of lane waves Interference of sherical waves: 1. outgoing waves rotational herboloids jk jk k r N N 1 k r 1 k r r 1. one outgoing, one incoming wave rotational ellisoids Ref: W. Osten k r r 1

11 11 Intensit CCD is not able to detect hase due to time averaging Measuring of intensit with simle detector Measured intensit is time average Interferometr and holograh: coding of hase information into measurable intensit variation 1 I P t 0 r E A o Conrast / visibilit: normalized difference of two different intensities ticall maimum / minimum values Value between I I ma min C I ma I min General case of two-wave interference 1 cos I I I C 1 1 Ref: W. Osten

12 1 Interference of Two Plane Waves Two lane waves with normals e k angles against -ais Equations of interference k1r kr 1 z 1 z cos1 cos zsin1 sin cos sin cos sin lane wave normals z Location of maima: straight lines cos cos sin sin z N 1 1 sin sin N cos cos cos cos 1 z 1 1 fringe maima Distance of maima: along / z / angle D, Dz cos cos sin sin 1 1 Dz tan D 1 tan fringe distance Ref: W. Osten h Dzcos 11 sin

13 Scalar: Helmholtz equation Vectorial: Mawell equations Scalar / vectorial Otics 0 D r E n k o k E H k 0 B k i D k B E k j i D H k E J J k M H B P E D r r 0 0

14 Basic Notations of Polarization Descrition of electromagnetic fields: - Mawell equations - vectorial nature of field strength Decomosition of the field into comonents Proagation lane wave: - field vector rotates - rojection comonents are oscillating sinusoidal E A cos te A cos t e z

15 Basic Forms of Polarisation 1. Linear comonents in hase E z E. circular hase difference of 90 between comonents E E z 3. ellitical arbitrar but constant hase difference E E z

16 Polarization Ellise Elimination of the time deendence: Ellise of the vector E Different states of olarization: - sense of rotation - shae of ellise E A E A E E A cos A sin

17 Descritions of Polarization Parameter Proerties 1 Polarization ellise Elliticit, orientation onl comlete olarization Comle arameter Parameter onl comlete olarization 3 Jones vectors Comonents of E onl comlete olarization 4 Stokes vectors Stokes arameter S o... S 4 comlete or artial olarization 5 Poincare shere Points on or inside the Poincare shere onl grahical reresentation 6 Coherence matri - matri C comlete or artial olarization

18 Polarizer Polarizer with attenuation c s/ J LIN 1 c 0 s 0 1 c Rotated olarizer J P cos sincos sincos sin Polarizer in -direction 0 J P TA z

19 Pair Polarizer-Analzer Polarizer and analzer with rotation angle Law of Malus: Energ transmission TA E TA E cos z I I cos o I linear olarizer linear olarizer olarizer analzer arallel erendicular

20 Retarder Phase difference between field comonents Retarder late with rotation angle Secial value: / 4 - late generates circular olarized light 1. fast ais 1 J V 0, / 0 0 i J RET i e 0 e 0 i cos sin, sincos i i e sincos 1 e i i 1 e sin cos e J V SA z. fast ais 45 J V i 11 / 4, / i i 1 LA

21 Rotator Rotate the of lane of olarization Realization with magnetic field: Farad effect B LV J ROT cos sin sin cos Verdet constant V z

22 Law of Malus-Duin Law of Malus-Duin: - equivalence of ras and wavefronts - both are orthonormal - identical information wave fronts Condition: No caustic of ras ras Mathematical: Rotation of Eikonal vanish rotn s Otical sstem: Ras and sherical waves orthonormal 0 object lane 0 hase L = const L = const image lane ras s 1 z 0 z 1

23 AP OE OPL r d n l 0,0,, OPL OPL OPD l l D R W R W ' ' D D W R u R s, ' sin ' ' ' D D D Relationshis Concrete calculation of wave aberration: addition of discrete otical ath lengths OPL Reference on chief ra 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 W i k i i e A E e A E e A E OPD D 3

24 4 Relationshi to Transverse Aberration Relation between wave and transverse aberration Aroimation 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 DW W W real W ideal W tan D' R W D' R W D' R reference shere wave front W reference lane R, ideal ra u real ra C D ' z

25 5 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

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

27 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, Wave 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

28 PSF b Hugens Princile Hugens wavelets corresond to vectorial field comonents The hase is reresented b the direction The amlitude is reresented b the length Zeros in the diffraction attern: destructive interference Aberrations from sherical wave: reduced conctructive suerosition ideal reference shere oint sread function side lobe eak zero intensit central eak maimum constructive interference wave front uil sto reduced constructive interference due to hase aberration

29 intensit 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 1,0 0,8 0,6 0,4 0, vertical lateral 0, J1 v Iu 0 I 0, v I v 0 u / v sin u / 4, I u / 4 Scaled coordinates according to Wolf : aial : u = z n / NA transversal : v = / NA 0 Ref: M. Keme

30 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 violating 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, moving gratings 8 quasi monochromatic 9 circular smmetr oblique illumination 10 far field conditions near field conditions 11 linear emission/ecitation non linear methods

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

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

33 33 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 0,0 0,0 D D S takes values between 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 values D s > 0.5 distribution broadened eak reduced Strehl ratio ideal, without aberrations real with aberrations r

34 34 Aroimations for the Strehl Ratio Aroimation of Marechal: useful for D s > 0.5 but negative values ossible Bi-quadratic aroimation Eonential aroach D s Wrms 1 4 Wrms D 1 s D s e 4 W rms D S Marechal defocus eonential biquadratic eac t c 0 Comutation of the Marechal aroimation with the coefficients of Zernike D s 1 N n1 cn0 1 n 1 N n n1 m0 c nm n 1

35 35 Psf with Aberrations Psf for some low oder Zernike coefficients The coefficients are changed between c j = The eak intensities are renormalized trefoil coma 5. order astigmatism 5. order sherical 5. order c = 0.0 c = 0.1 c = 0. c = 0.3 c = 0.4 c = 0.5 c = 0.7 coma astigmatism sherical defocus

36 Resolution of Fourier Comonents object detail high satial frequencies numerical aerture resolved frequencies object oint low satial frequencies object sum image for low NA decomosition of Fourier comonents sin waves image for high NA object high satial frequencies Ref: D.Aronstein / J. Bentle

37 v v i OTF d d g d d e g v v H,,,, ˆ, I F v v H PSF OTF OTF d d P d d v f v f P v f v f P v v 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 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

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

39 Number of Suorted Orders A structure of the object is resolved, if the first diffraction order is roagated through the otical imaging sstem The fidelit of the image increases with the number of roagated diffracted orders 0. / +1. / -1. order 0. / +1. / / -. order 0. / / +. / -. / +3. / -3. order

40 40 Polchromatic MTF Polchromatical MTF: Cut off frequenc deends on Sectral incoherent weighted suerosition of monochromatic MTF s g ol OTF 0 v S g v, d OTF g MTF ideal 350 nm = 350 nm = 400 nm = 450 nm = 500 nm = 550 nm = 600 nm = 650 nm = 700 nm olchromatic ma / 7.5 L/mm ma 55 L/mm

41 Otical Transfer Function of a Perfect Sstem Aberration free circular uil: Reference frequenc v o a f sinu' g MTF Maimum cut-off frequenc: v ma v 0 na f nsin u' Analtical reresentation / 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 0 0 v v 0 H OTF v, v PTF v, v H v, v e MTF ih

42 Interretation of the Duffieu Iintegral Interretation of the Duffieu integral: overla area of 0th and 1st diffraction order, interference between the two orders objective uil ' direct light The area of the overla corresonds to the information transfer of the structural details ' Frequenc limit of resolution: areas comletel searated o at object diffracted light in 1st order L object o q L shifted uil areas condenser conjugate to object uil f f light source area of integration

43 43 Contrast and Resolution High frequent structures : contrast reduced Low frequent structures: resolution reduced contrast brillant blurred shar milk resolution

44 44 Contrast and Resolution Contrast vs contrast as a function of satial frequenc Tical: contrast reduced for increasing frequenc V Comromise between resolution and visibilt is not trivial and deends on alication 1 H MTF Contrast sensitivit H CSF / c