Physical Optics Lecture 7: Coherence 2017-05-17 Herbert Gross www.iap.uni-jena.de 2 Physical Optics: Content No Date Subject Ref Detailed Content Complex fields, wave equation, k-vectors, interference, light propagation, 1 05.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 12.04. Diffraction B function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 19.04. Fourier optics B phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 26.04. B resolution resolution, criteria, contrast, axial resolution, CTF Introduction, Jones formalism, Fresnel formulas, birefringence, 5 03.05. Polarization G components Energy, momentum, time-energy uncertainty, photon statistics, 6 10.05. Photon optics D fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 7 17.05. Coherence G speckle, OCT-principle Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 24.05. Laser B pulses, power 9 31.05. Gaussian beams D Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 10 07.06. Generalized beams D beams, applications in superresolution microscopy Apodization, superresolution, extended depth of focus, particle trapping, 11 14.06. PSF engineering G confocal PSF Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 12 21.06. Nonlinear optics D CARS microscopy, 2 photon imaging Introduction, surface scattering in systems, volume scattering models, 13 28.06. Scattering D calculation schemes, tissue models, Mie Scattering 14 05.07. Miscellaneous G Coatings, diffractive optics, fibers D = Dienerowitz B = Böhme G = Gross 3 Contents Coherence . Introduction . Young's experiment . Spatial coherence . Temporal coherence . Partial coherent imaging . Speckle . OCT principle 4 Coherence in Phase Space coherent : every point partial coherent :every point has incoherent : every point radiates in one direction an individuell angle characteristic radiates in all directions line in phase space finite area in the phase space filled phase space u u u x x x 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: 1. Separation of wave trains with finite spectral bandwidth Dl 2. Optical path differences for extended source areas 3. Time averaging by moved components . Limiting cases: 1. Coherence: rigid phase coupling, quasi monochromatic, wave trains of infinite length 2. Incoherence: no correlation, light source with independent radiating point like molecules 6 Coherence Function . Coherence function: Correlation x of statistical fields (complex) * (r ,r , ) E(r ,t )E (r ,t) x1 1 2 1 2 t E(x ) Dx 1 z for identical locations : x 2 E(x ) r1 r2 r 2 intensity (r,r) I(r) . normalized: degree of coherence (r1,r2 , ) 12( ) (r1,r2 , ) I(r1)I(r2 ) . In interferometric setup, the amount of describes the visibility V . Distinction: 1. spatial coherence, path length differences and transverse distance of points 2. time-related coherence due to spectral bandwidth and finite length of wave trains 7 Axial Coherence Length of Lightsources Light source lc Incandescent lamp 2.5 m Hg-high pressure lamp, line 546 nm 20 m Hg-low pressure lamp, line 546 nm 6 cm Kr-isotope lamp, line at 606 nm 70 cm HeNe - laser with L = 1 m - resonator 20 cm HeNe - laser, longitudinal monomode stabilized 5 m 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 z1 detector V 1 D x z2 Dx 0 D 9 Double Slit Experiment of Young . Young interference experiment: . Ideal case: point source with distance z1, ideal small pinholes with distance D . Interference on a screen in the distance z , intensity 2 Dx 2 I(x) 4I0 cos l z2 lz . Width of fringes Dx 2 D x detector source D region of interference screen with z2 pinholes z2 10 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 Dph = Dairy / distance of pinholes D = 4Dairy s = 0 s = 0.15 s = 0.25 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 Lc 12 Spatial Coherence observation . Area of coherence / transverse coherence length: area Non-vanishing correlation at two points with distance Lc: Lc Correlation of phase due to common area on source ( r , r ) P1 1 2 domain of coherence . Radiation out of a coherence cell of P2 r1 extension Lc guarantees finite contrast r 2 O . The lateral coherence length starting receiving plane plane changes during propagation: spatial coherence grows with 1 increasing propagation distance 2 common area 13 Spatial Coherence . Incoherent source with diameter D = 2a . Receiver plane indistance z . Cone of observation l / a . Source is coherent in the distance z a 2 / l . Transverse length of coherence (zeros of -function) l z l L c a 14 Near- and Farfield of Excimer Lasers . Parameter of real Excimer lasers Near field (spatial) Far field (angle) 157 nm - laser 193 nm - laser 248 nm - laser 15 Van Cittert - Zernike - Theorem 2 . Propagation of coherence function: i 2 2 i 1 (r1 r2 ) r '(r1 r2 ) (r ,r , z) e lz I(r',0)e lz dr' in special case r 'r 'r ' 1 2 1 2 lz . 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 V 2 ar 2J 1 l z (r) a2 1 2 ar l z vanishing contrast Vanishing contrast at radius l r 0.61 z a r 16 Gauß-Schell Beam: Definition . Partial coherent beams: 2 1 r r 1 2 1. intensity profile gaussian 2 Lc 2. Coherence function gaussian (r1 r2 , z 0) e . Extension of gaussian beams with similar description . Additional parameter: lateral coherence length Lc 1 2 2 . Normalized degree of coherence wo 1 Lc 2 2 1 wo . Beam quality depends on coherence M 1 L c . Approximate model do characterize multimode beams 17 Gauss-Schell Beams . Due to the additional parameter: w / wo Waist radius and divergence angle are 4 independent 3 0.25 1. Fixed divergence: 2 0.50 waist radius decreases with 1 growing coherence 1.0 0 z / zo -4 -3 -2 -1 0 1 2 3 4 w / wo 4 0.25 0.50 3 2. Fixed waist radius: 2 divergence angle decreases with 1.0 growing coherence 1 0 z / zo -4 -3 -2 -1 0 1 2 3 4 18 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 Temporal Coherence . Radiation of a single atom: Finite time Dt, wave train of finite length, 2 i t No periodic function, representation as Fourier integral E(t) A()e d with spectral amplitude A() sin Dt . Example rectangular spectral distribution A( ) Dt . Finite time of duration: spectral broadening D, D1/Dt schematic drawing of spectral width I() 20 Axial Coherence Length . Two plane waves with equal initial phase and differing wavelengths l1, l2 . Idential phase after axial (longitudinal) coherence length c l c c c D l1 l 2 time t starting phase difference in phase phase 180° 21 Time-Related Coherence Function . Time-related coherence function: 1 T ( ) lim E*(t) E(t )dt E*(t) E(t ) Auto correlation of the complex field E T T 2T T at a fixed spatial coordinate . For purely statistical phase behaviour: = 0 . Vanishing time interval: intensity (0) E*(t) E(t) I T . Normalized expression * ( ) E (t) E(t ) ( ) 2 (0) E(t) | ( ) | . Usually: decreases with growing symmetrically Width of the distribution: coherence time c c 22 Time-Related Coherence Function . Intensity of a multispectral field Integration of the power spectral density S() I S( )d 0 . The temporal coherence function and the power 2i spectral density are Fourier-inverse: S( ) ( )e d Theorem of Wiener-Chintchin 1 . The corresponding widths in time and spectrum are c D related by an uncertainty relation . The Parceval theorem defines the coherence time ( ) 2d c as average of the normalized coherence function . The axial coherence length is the space equaivalent of lc c c the coherence time 23 Michelson-Interferometer . Michelson interferometer: interference of finite size wave trains . Contrast of interference pattern allows to measure the axial coherence length/time second mirror signal beam relative moving moving z wave trains z with finite length I(z) reference overlap beam lc first mirror beam from splitter receiver source 24 Interference Contrast . Superposition of plane wave with initial phase I Im 2 In Im cosn m Intensity: m nm . Radiation field with coherence function : I(r) I1(r)I2 (r)2r1,r2,0 . Reduced contrast for partial coherence I I (r ,r , ) K max min 1 2 I filtered Imax Imin I(r1)I(r2 ) signal measured signal . Measurement of coherence in Michelson interferometer: phase difference due to path length difference in the two arms (Fourier spectroscopy) 4 Dl z z D 2Dk z 2 measured l position axial length of coherence 25 Axial Coherence .