Physical Optics. Lecture 7: Coherence Herbert Gross.

Transcription

1 Physical Optics Lecture 7: Coherence Herbert Gross

2 Physical Optics: Content No Date Subject Ref Detailed Content Wave optics G Complex fields, wave equation, k-vectors, interference, light propagation, interferometry.04. Diffraction B Slit, grating, diffraction integral, diffraction in optical systems, point spread function, aberrations Fourier optics B Plane wave expansion, resolution, image formation, transfer function, phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point B resolution resolution, criteria, contrast, axial resolution, CTF Polarization G Introduction, Jones formalism, Fresnel formulas, birefringence, components Photon optics D Energy, momentum, time-energy uncertainty, photon statistics, fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Coherence G Temporal and spatial coherence, Young setup, propagation of coherence, speckle, OCT-principle Laser B Atomic transitions, principle, resonators, modes, laser types, Q-switch, pulses, power Gaussian beams D Basic description, propagation through optical systems, aberrations Generalized beams D Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy beams, applications in superresolution microscopy PSF engineering G Apodization, superresolution, extended depth of focus, particle trapping, confocal PSF.06. Nonlinear optics D Basics of nonlinear optics, optical susceptibility, nd and 3rd order effects, CARS microscopy, photon imaging Scattering D Introduction, surface scattering in systems, volume scattering models, calculation schemes, tissue models, Mie Scattering Miscellaneous G Coatings, diffractive optics, fibers D = Dienerowitz B = Böhme G = Gross

3 3 Contents Coherence Introduction Young's experiment Spatial coherence Temporal coherence Partial coherent imaging Speckle OCT principle

4 4 Coherence in Phase Space coherent : every point radiates in one direction line in phase space partial coherent :every point has an individuell angle characteristic finite area in the phase space incoherent : every point radiates in all directions filled phase space u u u x x x

5 5 Coherence in Optics Statistical effect in wave optic: start phase of radiating light sources are only partially coupled Partial coherence: no rigid coupling of the phase by superposition of waves Constructive interference perturbed, contrast reduced Mathematical description: Averagedcorrelation between the field E at different locations and times: Coherence function G Reduction of coherence:. Separation of wave trains with finite spectral bandwidth Dl. Optical path differences for extended source areas 3. Time averaging by moved components Limiting cases:. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length. Incoherence: no correlation, light source with independent radiating point like molecules

6 Coherence Function Coherence function: Correlation of statistical fields (complex) for identical locations : intensity normalized: degree of coherence In interferometric setup, the amount of describes the visibility V Distinction:. spatial coherence, path length differences and transverse distance of points. time-related coherence due to spectral bandwidth and finite length of wave trains t t r E t r E r r ), ( ), ( ),, ( * z x x x E(x ) E(x ) Dx r r r ) ( ) ( ),, ( ),, ( ) ( r I r I r r r r ) ( ), ( r I r r 6

7 7 Axial Coherence Length of Lightsources Light source l c Incandescent lamp Hg-high pressure lamp, line 546 nm Hg-low pressure lamp, line 546 nm Kr-isotope lamp, line at 606 nm HeNe - laser with L = m - resonator HeNe - laser, longitudinal monomode stabilized.5 m 0 m 6 cm 70 cm 0 cm 5 m

8 8 Double Slit Experiment of Young First realization: change of slit distance D Second realization: change of coherence parameter s of the source light source screen with slits detector z V D x z Dx 0 D

9 9 Double Slit Experiment of Young Young interference experiment: Ideal case: point source with distance z, ideal small pinholes with distance D Interference on a screen in the distance z, intensity Width of fringes D lz D x I Dx ( x ) 4 I cos 0 l z x detector source D region of interference z screen with pinholes z

10 0 Double Slit Experiment of Young Partial coherent illumination of a double pinhole/double slit Variation of the size of the source by coherence parameter s Decreasing contrast with growing s Example: pinhole diameter D ph = D airy / distance of pinholes D = 4D airy s = 0 s = 0.5 s = 0.5 s = 0.30 s = 0.35 s = 0.40

11 Coherence Measurement with Young Experiment Typical result of a double-slit experiment according to Young for an Excimer laser to characterize the coherence Decay of the contrast with slit distance: direct determination of the transverse coherence length L c

12 Spatial Coherence Area of coherence / transverse coherence length: Non-vanishing correlation at two points with distance L c : Correlation of phase due to common area on source observation area P L c ( r, r ) Radiation out of a coherence cell of extension L c guarantees finite contrast domain of coherence r P O r The lateral coherence length changes during propagation: starting plane receiving plane spatial coherence grows with increasing propagation distance common area

13 3 Spatial Coherence Incoherent source with diameter D = a Receiver plane indistance z Cone of observation l / a Source is coherent in the distance z a / l Transverse length of coherence (zeros of -function) L c l z a l

14 4 Near- and Farfield of Excimer Lasers Parameter of real Excimer lasers Near field (spatial) Far field (angle) 57 nm - laser 93 nm - laser 48 nm - laser

15 ' ',0) ( ),, ( ) '( ) ( dr e r I e z z r r r r r z i r r z i l l l V r vanishing contrast Van Cittert - Zernike - Theorem r r r ' ' ' ( ) r a J ar z ar z l l a z r l 6 0. Propagation of coherence function: in special case Van Cittert-Zernike theorem: Coherence function of an incoherent source is the Fourier transform of the intensity profile Example: circular light source wirh radius a Vanishing contrast at radius 5

16 ) 0, ( c L r r e z r r c o L w c o L w M Gauß-Schell Beam: Definition Partial coherent beams:. intensity profile gaussian. Coherence function gaussian Extension of gaussian beams with similar description Additional parameter: lateral coherence length L c Normalized degree of coherence Beam quality depends on coherence Approximate model do characterize multimode beams 6

17 7 Gauss-Schell Beams Due to the additional parameter: Waist radius and divergence angle are independent. Fixed divergence: waist radius decreases with growing coherence w / wo z / zo w / wo Fixed waist radius: divergence angle decreases with growing coherence z / zo

18 8 Temporal Coherence Damping of light emission: wave train of finite length Starting times of wave trains: statistical U(t) t c duration of a single train

19 9 Temporal Coherence Radiation of a single atom: Finite time Dt, wave train of finite length, No periodic function, representation as Fourier integral with spectral amplitude A() Example rectangular spectral distribution E( t) A( ) A( ) e i t sin Dt Dt d Finite time of duration: spectral broadening D, schematic drawing of spectral width I() D / Dt

20 0 Axial Coherence Length Two plane waves with equal initial phase and differing wavelengths l, l Idential phase after axial (longitudinal) coherence length l c c c c D l l time t starting phase phase difference 80 in phase

21 Time-Related Coherence Function Time-related coherence function: Auto correlation of the complex field E at a fixed spatial coordinate For purely statistical phase behaviour: = 0 Vanishing time interval: intensity T ( ) lim * ( ) ( ) * ( ) ( ) T T E t E t dt E t E t T * ( 0) E ( t) E( t) I T T Normalized expression ( ) Usually: ( ) ( 0) decreases with growing symmetrically Width of the distribution: coherence time c * E ( t) E( t ) E( t) ( ) c

22 Time-Related Coherence Function Intensity of a multispectral field Integration of the power spectral density S() I 0 S( ) d The temporal coherence function and the power spectral density are Fourier-inverse: Theorem of Wiener-Chintchin S( ) ( ) e i d The corresponding widths in time and spectrum are related by an uncertainty relation c D The Parceval theorem defines the coherence time as average of the normalized coherence function c ( ) d The axial coherence length is the space equaivalent of the coherence time l c c c

23 3 Michelson-Interferometer Michelson interferometer: interference of finite size wave trains Contrast of interference pattern allows to measure the axial coherence length/time second mirror moving z signal beam wave trains with finite length relative moving z reference beam overlap I(z) first mirror from source beam splitter l c receiver

24 4 Interference Contrast Superposition of plane wave with initial phase Intensity: I m I m nm I n I m cos n m Radiation field with coherence function : Reduced contrast for partial coherence Imax Imin ( r, r, ) K I I I( r ) I( r ) max min Measurement of coherence in Michelson I I( r) I( r) I( r) r, r,0 filtered signal measured signal interferometer: phase difference due to path length difference in the two arms (Fourier spectroscopy) D Dk z 4 Dl z l measured position axial length of coherence z

25 5 Axial Coherence Contrast of a 93 nm excimer laser for axial shear Red line: Fourier transform of spectrum contrast 0,9 0,8 measured FFT-Data 0,7 0,6 0,5 0,4 0,3 0, 0, 0-0,8-0,6-0,4-0, 0 0, 0,4 0,6 0,8 z-shift in mm

26 6 Coherence Parameter Heuristic explanation of the coherence parameter in a system:. coherent: Psf of illumination large in relation to the observation Large s coherent illumination extended source small stop of condenser condenser object objective lens Psf of observation inside psf of illumination. incoherent: Psf of illumination small in comparison to the observation Small s incoherent illumination extended source large stop of condenser Psf of observation contains several illumination psfs s sin u sin u ill obs

27 7 Coherence Parameter Finite size of source : aperture cone with angle u ill Observation system: aperture angle u obs Definition of coherence parameter s: Ratio of numerical apertures Limiting cases: coherent s = 0 u ill << u obs s sin u sin u ill obs incoherent s = u ill >> u obs source object x o, y o lens image x i, y i u ill u obs illumination observation

28 8 Partial Coherence Simulation of partial coherent illumination: Finite size of light source Corresponding finite size of illuminated area in aperture plane Every point in this area is considered to emit independent (incoherent) Off-axis point in aperture plane generates an inclined plane wave in the object Angular spectrum illumination of the object describes partial coherence Estimated sampling of illumination points: Ls s arcsin f c D y s 0.6l n sin aperture plane condenser f c object plane source extension L s s angular spectrum of object illumination Dy s size of coherence cell

29 9 Partial Coherent Imaging Image intensity:. Correlation of two points in the object. Integration over all points in the incoherent light source * * * xi Isxs hobsxi, xo, xs hobsxi, xo, xs h illxo, xs hillxo, xs Oxo O xo dxo dxo dxs Ii light source object plane pupil image y s y o y p y i x s x o x p xi z o (x o,x o ) ' o (x o,x o ) P(x p ) h ill (x L,x ) h obs (x,x') I s (x s ) I i (x i ) O(x ) Simplification: thin object, transmission does not change for moderate inclination angles I i * * xi oxo, xo hobsxi, xo hobsxi, xo O xo O xo dxo dxo

30 Transmission Cross Correlation Function (TCC) Integration overlap of pupils a) partial coherent Typical chnage of transfer capability b) incoherent pupil light source x H MTF () coherent s = 0 x 0.5 incoherent s = contrast increased partial coherent 0 s < contrast decreased threshold 0 o loss of resolution o

31 Example: Partial Coherent Edge Image solid line : exact, dashed : approximation Image formation of a phase edge s = 0. s = 0.5 s = 0.8 under partial coherent illumination : phase step = s : degree of coherence = = = =

32 3 Example: Partial Coherent Imaging of Bar-Pattern object m s = 0.08 pupil intensity s = 0.50 s =.0 object spectrum image

33 33 Example: Partial Coherent Imaging of Siemens Star coherent partial coherent incoherent frequency o = sinu / l frequency o = sinu / l

34 34 Speckle Effect Generation of speckles: Coherent light is refracted / reflected at a rough surface Roughness creates phase differences Interference of all partial waves: granulation, signature for a local surface patch Transmission of random media in a volume is also possible (atmosphere, biological) Higher order effects: patial coherent illumination, polarization incident laser light plane of observation surface with roughness

35 35 Sum of Random Phasors Sum of random phasors due to field superposition:. nearly zero result, dominant destructive. large result, dominant constructive 3. special case of one large contribution Ref. J. Goodman

36 36 Speckle Pattern Size of objective speckles: depends on distance of observation z = 840 mm z = 330 mm Colored speckles z = 60 mm z = 0 mm

37 Subjective / Objective Speckle 37 Creating of speckle pattern:. coherent scattering of laser light: objective speckle incident laser light. imaging of coherent straylight: subjective speckle always be visual observation r P lens with focal length f surface with roughness p > l r point of observation D intensity d surface with roughness z z' schreen

38 Objective Speckle Pattern 38 Incident coherent light Rough surface with size D Observation in distance z Speckle pattern with typical size of cells d z l D D Airy incident coherent laser light intensity D rough surface d screen z

39 39 Subjective Speckle Pattern Incident coherent light Rough surface with size D lens with focal length f Observation in distance z Speckle size in the image: PSF, D airy d s D intensity d Speckle pattern with typical size of cells in the object d l ( m) ( m) NA s D airy m: magnification Example: coarse speckle for small NA surface with roughness z F#= F#= 66 z' schreen Ref. W. Osten

40 Statistics of Superposed Speckles 40 Incoherent superposition of several speckles w(i) 0.9 Probability has intermediate maximum w( I) Zero probability for darkness Decreasing contrast Example 4I I 0 e I I I / I o

41 4 Speckle Statistics for Incoherent Superposition Reduction of speckle contrast by incoherent superposition Overlay of large number of individual fully modulated images Many images necessary to get a uniform illumination Reduction of variance goes with / n w(i) n = n = 6 n = n = 0 n = 40 n = I / I o

42 4 Speckle Reduction Coherent speckles after diffusor plate with different data starting phase spectrum far field

43 Speckle Contrast Changing with Coherence 43 Contrast of speckle image for changing coherence a: amplitude lc: transverse lenght of coherence a/l corr = 0 a/l corr = 0. a/l corr = 0.5 a/l corr =.0 a/l corr =.0 a/l corr = 4.0

44 44 Scattering in Turbid Media Different strengths of interaction a) ballistic photon b) snake photons c) multiple scattered photons Ref: M. Gu

45 45 Scattering in Turbid Media Change of light properties a) spectral shift b) spatial broadening Dw frequency D frequency x x snake c) temporal broadening d) polarization snake time t Dt time t Ref: M. Gu

46 46 Resolution in OCT. Axial resolution limited by spectral bandwidth Low NA High NA D z coh ln l l Dl Dl Dx Dx Dz diff Dz coh Dz diff. Lateral resolution: diffraction limited, improvement by confocal setup 3. Usually low NA

47 47 Principle of OCT Basic method of optical coherence tomography: - short pulse light source creates a coherent broadband wave - white light interferometry allows for interference inside the axial coherence length Measured signal: - low pass filtering - maximum of envelope gives the relative length difference between test and reference arm lo Dl Dl For Gaussian beam D 4ln l o l Dl High frequency oscillation depends on z I(z) filtered signal signal measured D Dk z 4 Dl z l measured position z coherence length

48 48 Fiber Based OCT Interferometer I source spectrum Basic setup LED source fiber coupler measuring arm surface under test fiber fiber I signal fiber fiber detector z reference arm z-scan

49 49 Optical Coherence Tomography Example: sample with two reflecting surfaces. Spatial domain. Complete signal 3. Filtered signal high-frequency content removed Ref: M. Kaschke

50 50 Optical Coherence Tomography Achronyms in literature Ref: R. Leach

51 5 Resolution of OCT Lateral resolution: Airy profile Dx 4l f l sin u lateral resolution Log Dx Penetration depth: axial resolution 00 m ln l Dl D z res 0 m confocal microscopy OCT ultra sound m Log Dz 00 m mm cm 0 cm depth

52 5 Example of OCT Imaging Example: Fundus of the human eye

53 53 White Light Interferometry Examples Ref: R. Kowarschik

54 54 Spectral Domain OCT Spectral Domain-OCT: - broad band source - reference mirror fixed in position, no A-scan necessary - signal splitted by spectrometer The high-frequency content of the signal is analyzed The frequency is proportional to the depth z, measured is the overlay beat-signal of all scatterers reference mirror Sample in z fixed Broadband source z interferogramm frequency domain Fourier transform spatial domain spectrometer data processing z

55 55 Fourier Domain OCT Fourier Domain-OCT: setup Signals: a) intensity spectrum b) spatial intensity distribution Ref: M. Kaschke

56 56 Fourier Domain OCT Signal Signal complexity depends on scatter-distribution scatterer scatterer r S (z S ) reflectivity reference sample scatterer A-scan z R I D (z) z S zs z S DC term cross correlation auto correlation mirror image artifacts (z R -z S ) (z R -z S ) 0 -(z R -z S ) -(z R -z S ) z