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 r2 O
. The lateral coherence length starting receiving plane plane changes during propagation: spatial coherence grows with increasing propagation distance 1
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 w 1 o L c 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 waist radius decreases with 0.50 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: divergence angle decreases with 2 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 related by an uncertainty relation D
. 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 l c the coherence time c c
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 Imax Imin (r1,r2 , ) I K 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
. Contrast of a 193 nm excimer laser for axial shear . Red line: Fourier transform of spectrum
contrast
1
0,9 measured 0,8 FFT-Data
0,7
0,6
0,5
0,4
0,3
0,2
0,1 z-shift in mm 0 -0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8
26 Coherence Parameter
Heuristic explanation coherent small stop of condenser object objective lens of the coherence illumination condenser parameter in a system: Psf of observation extended inside psf of 1. coherent: source illumination Psf of illumination large in relation to the observation
Large s
incoherent illumination large stop of condenser 2. incoherent: Psf of observation Psf of illumination extended contains several small in comparison source illumination psfs to the observation
Small s
sinu s ill sinuobs
27 Coherence Parameter
. Finite size of source : aperture cone with angle uill . Observation system: aperture angle uobs
. Definition of coherence parameter s: sinu Ratio of numerical apertures s ill sinuobs . Limiting cases: coherent s = 0
uill << uobs
incoherent source object lens image x , y s = 1 o o xi , yi
uill >> uobs
uill uobs
illumination observation
28 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 Ls 2s 2arcsin . Estimated sampling of illumination points: 2 fc
0.61l aperture condenser object Dys plane plane
n sin fc angular source spectrum of 2 extension s object illumination Ls
Dys size of coherence cell
29 Partial Coherent Imaging
. Image intensity: 1. Correlation of two points in the object 2. Integration over all points in the incoherent light source
I x I x h x , x , x h* x , x , x h x , x h* x , x O x O* x dx dx dx i i s s obs i o1 s obs i o2 s ill o1 s ill o2 s o1 o2 o1 o2 s
light object pupil image source plane
ys yo y yi p
xo xs xp xi z
P(xp) o(xo1,xo2) 'o(xo1,xo2)
hill(xL,x1) hobs(x1,x') Ii(xi) Is(xs) O(x1)
. Simplification: thin object, transmission does not change for moderate inclination angles
I x x , x h x , x h* x , x O x O* x dx dx i i o o1 o2 obs i o1 obs i o2 o1 o2 o1 o2
Transmission Cross Correlation Function (TCC)
. Integration overlap of pupils a) partial coherent
pupil light . Typical chnage of x source transfer capability b) incoherent
HMTF()
1 coherent x s = 0
incoherent partial s = 1 0.5 coherent contrast 0 s < 1 increased contrast decreased
threshold 0 o loss of 2o resolution
Example: Partial Coherent Edge Image
solid line : exact , dashed : approximation s = 0.2 = 0.5 = 0.8 . Image formation of a phase edge s s 1 1 1 under partial coherent illumination = 0° . : phase step 0.5 0.5 0.5 0 0 0 s : degree of coherence -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 1 1 1 = 30° 0.5 0.5 0.5
0 0 0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
1 1 1 = 60° 0.5 0.5 0.5
0 0 0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
1 1 1 = 90° 0.5 0.5 0.5
0 0 0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
1 1 1 = 180° 0.5 0.5 0.5
0 0 0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2
32 Example: Partial Coherent Imaging of Bar-Pattern
12 m s = 0.08 s = 0.50 s = 1.0 object pupil intensity
object image spectrum 33 Example: Partial Coherent Imaging of Siemens Star
coherent partial coherent incoherent
frequency frequency o = sinu / l o = 2sinu / l 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 Sum of Random Phasors
. Sum of random phasors due to field superposition: 1. nearly zero result, dominant destructive 2. large result, dominant constructive 3. special case of one large contribution
Ref. J. Goodman
36 Speckle Pattern
. Size of objective speckles: depends on distance of z = 840 mm z = 330 mm observation
. Colored speckles
z = 160 mm z = 110 mm 37 Subjective / Objective Speckle
. Creating of speckle pattern: 1. coherent scattering of laser light: incident laser light objective speckle
2. imaging of coherent straylight: subjective speckle always be visual observation r1 P point of r2 observation surface with lens with focal roughness length f p > l
intensity D d
surface with roughness schreen z z' 38 Objective Speckle Pattern
. Incident coherent light . Rough surface with size D . Observation in distance z l z D . Speckle pattern with typical size of cells d Airy D 2
incident coherent laser light intensity
D rough surface
d
screen
z 39 Subjective Speckle Pattern
. Incident coherent light lens with focal . Rough surface with size D length f . Observation in distance z . Speckle size in the image: intensity ds D d PSF, Dairy . Speckle pattern with typical size of cells in the object l surface with roughness ds (1 m) (1 m) Dairy schreen 2NA z z' m: magnification F#= 22 F#= 66 . Example: coarse speckle for small NA
Ref. W. Osten
40 Statistics of Superposed Speckles
w(I) . Incoherent superposition of several speckles 1
0.9
0.8 . Probability has intermediate maximum 0.7 I 0.6 2 4I I 0 0.5 w(I) 2 e I0 0.4 . Zero probability for darkness 0.3 0.2
. Decreasing contrast 0.1
0 I / Io 0 0.5 1 1.5 2 2.5 3 . Example
41 Speckle Statistics for Incoherent Superposition
. Reduction of speckle contrast by incoherent superposition
w(I) . Overlay of large number of individual fully modulated 1 images 0.9
0.8 . Many images necessary to n = 2 get a uniform illumination 0.7 n = 6
0.6 n = 12
. Reduction of variance goes 0.5 n = 20 with 1/ n 0.4 n = 40
0.3 n = 100
0.2
0.1
0 I / I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 o 42 Speckle Reduction
. Coherent speckles after diffusor plate with different data
starting phase
spectrum
far field 43 Speckle Contrast Changing with Coherence
. Contrast of speckle image for changing coherence . a: amplitude lc: transverse lenght of coherence
a/lcorr = 0 a/lcorr = 0.1 a/lcorr = 0.5
a/lcorr = 1.0 a/lcorr = 2.0 a/lcorr = 4.0 44 Scattering in Turbid Media
. Different strengths of interaction a) ballistic photon
b) snake photons
c) multiple scattered photons
Ref: M. Gu 45 Scattering in Turbid Media
. Change of light properties
a) spectral shift b) spatial broadening
Dw
snake frequency D x x frequency
c) temporal broadening d) polarization snake
time t Dt time t
Ref: M. Gu 46 Resolution in OCT
2ln 2 l2 l2 1. Axial resolution limited by spectral bandwidth Dzcoh 0.4413 Low NA High NA Dl Dl
2Dx 2Dx
2Dzdiff
2Dz coh 2Dz diff
2. Lateral resolution: diffraction limited, improvement by confocal setup 3. Usually low NA
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 I(z) arm l2 Dl o filtered signal Dl signal measured . For Gaussian beam 4ln 2 l2 Dl o Dl . High frequency oscillation depends on z 4 Dl z D 2Dk z z l2 measured position coherence length
48 Fiber Based OCT Interferometer
I Basic setup source spectrum
surface measuring under test LED arm source fiber coupler
fiber fiber
fiber I signal fiber
detector reference arm z-scan z
49 Optical Coherence Tomography
. Example: sample with two reflecting surfaces . 1. Spatial domain
2. Complete signal
3. Filtered signal high-frequency content removed
Ref: M. Kaschke
50 Optical Coherence Tomography
. Achronyms in literature
Ref: R. Leach
51 Resolution of OCT
. Lateral resolution: Airy profile
4l f 2l lateral Dx resolution sin u Log Dx
. Penetration depth: axial resolution 100 m
ultra sound 2ln 2 l2 Dzres Dl 10 m confocal microscopy OCT
1 m
Log Dz 100 m 1 mm 1 cm 10 cm depth
52 Example of OCT Imaging
Example:
Fundus of the human eye
53 White Light Interferometry
. Examples
Ref: R. Kowarschik
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
z Broadband source
interferogramm Fourier transform frequency domain spatial domain
spectrometer
data z processing
55 Fourier Domain OCT
. Fourier Domain-OCT: setup
. Signals: a) intensity spectrum b) spatial intensity distribution
Ref: M. Kaschke
56 Fourier Domain OCT Signal
1.4
1.2 1 scatterer
1
. Signal complexity depends on 0.8 scatter-distribution 0.6 0.4
0.2
0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.4
1.2 2 scatterer
1
0.8
0.6
0.4
0.2
0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.4 rS(zS) reflectivity reference sample 1.2 3 scatterer
1
0.8
0.6
0.4
0.2 zS zR zS1 zS2 0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
A-scan ID(z) DC term
cross correlation mirror image artifacts auto correlation z 2(z -z ) 2(z -z ) -2(z -z ) -2(z -z ) R S2 R S1 0 R S1 R S2