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Metrology and Sensing
Lecture 9: Speckle methods
2017-12-14
Herbert Gross
Winter term 2017
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Preliminary Schedule
No Date Subject Detailed Content
1 19.10. Introduction Introduction, optical measurements, shape measurements, errors,
definition of the meter, sampling theorem
2 26.10. Wave optics Basics, polarization, wave aberrations, PSF, OTF
3 02.11. Sensors Introduction, basic properties, CCDs, filtering, noise
4 09.11. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry
5 16.11. Interferometry I Introduction, interference, types of interferometers, miscellaneous
6 23.11. Interferometry II Examples, interferogram interpretation, fringe evaluation methods
7 30.11. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods
8 07.12. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight,
Scheimpflug setup
9 14.12. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology
10 21.12. Holography Introduction, holographic interferometry, applications, miscellaneous
11 11.01. Measurement of basic
system properties Bssic properties, knife edge, slit scan, MTF measurement
12 18.01. Phase retrieval Introduction, algorithms, practical aspects, accuracy
13 25.01. Metrology of aspheres
and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms
14 01.02. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous
15 08.02. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
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Content
Spatial coherence
Temporal coherence
Speckle
Speckle properties
Speckly metrology
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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:
Averaged correlation 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
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Coherence in Phase Space
incoherent : every point
radiates in all directions
filled 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
x
u
x
u
x
u
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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:
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
ttrEtrErr ),(),(),,( 2
*
121
z
x
x1
x2
E(x2)
E(x1)Dx
r r r1 2
)()(
),,(),,()(
21
212112
rIrI
rrrr
)(),( rIrr
6
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Spatial Coherence
1
2
starting
plane
receiving
plane
common
area
Area of coherence / transverse coherence length:
Non-vanishing correlation at two points with distance Lc:
Correlation of phase due to common area on source
Radiation out of a coherence cell of
extension Lc guarantees finite contrast
The lateral coherence length
changes during propagation:
spatial coherence grows with
increasing propagation distance
observation
area
O
r2
r1
r r( , )
1 2
Lc
P1
P2
domain of
coherence
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The number of speckles corresponds to the cells of coherence
The number of cells is equivalent to the beam quality
The cells of coherence are the spatial regions in the beam cross section, which
can interfere
cells
speckle spots
beam caustic
propagation
7 spot per cross
section in 1
dimension
2Md
DN
speckle
beamspeckle
Spatial Cells of Coherence
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Coherence Parameter
Heuristic explanation
of the coherence
parameter in a system:
1. coherent:
Psf of illumination
large in relation to the
observation Large s
2. incoherent:
Psf of illumination
small in comparison
to the observation Small s
object objective lenscondensersmall stop of
condenser
extended
source
coherent
illumination
large stop of
condenser
incoherent
illumination
Psf of observation
inside psf of
illumination
Psf of observation
contains several
illumination psfs
extended
source
obs
ill
u
u
sin
sins
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Double Slit Experiment of Young
D
z1
z2
light
source
screen
with slits
detector
x
Dx
First realization:
change of slit distance D
Second realization:
change of coherence parameter s of the source
Visibility / contrast shrinks with growing slit spacind D
D0
1
V
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2
2
0 cos4)(z
xDIxI
l
D
zx 2l
D
screen with
pinholes
detector
source
z2
region of
interference
z2
x
D
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 z2 , intensity
Width of fringes
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Double Slit Experiment of Young
s = 0 s = 0.15 s = 0.25 s = 0.35 s = 0.40s = 0.30
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
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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
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')0,'(1
),,()(')(
2
21
2122
21
rderIez
zrrrrr
z
irr
z
i
l
l
l
V
r
vanishing contrast
1
Van Cittert - Zernike - Theorem
r r r1 2' ' '
( )r a
Jar
z
ar
z
l
l
2
122
2
azr
l 61.0
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 with radius a
Vanishing contrast at radius
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Temporal Coherence
t
U(t)
c
duration of a
single train
Damping of light emission:
wave train of finite length
Starting times of wave trains: statistical
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tDD /1
t
tA
D
D
sin)(
deAtE ti2)()(
Temporal Coherence
I()
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
Finite time of duration: spectral broadening D,
schematic drawing of spectral width
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D
ccl cc
Axial Coherence Length
starting
phase
in phase
l2 time t
l1
phase difference
180°
Two plane waves with equal initial phase and differing wavelengths l1, l2
Idential phase after axial (longitudinal) coherence length
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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
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| ( ) |
c
Time-Related Coherence Function
( ) lim ( ) ( ) ( ) ( )* *
TT
T
TTE t E t dt E t E t
1
2
( ) ( ) ( )*0 E t E t IT
( )( )
( )
( ) ( )
( )
*
02
E t E t
E t
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
Normalized expression
Usually:
decreases with growing symmetrically
Width of the distribution: coherence time c
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0
)( dSI
deS i2)()(
D
1c
dc
2)(
cc cl
Time-Related Coherence Function
Intensity of a multispectral field
Integration of the power spectral density S()
The temporal coherence function and the power
spectral density are Fourier-inverse:
Theorem of Wiener-Chintchin
The corresponding widths in time and spectrum are
related by an uncertainty relation
The Parceval theorem defines the coherence time
as average of the normalized coherence function
The axial coherence length is the space equivalent of
the coherence time
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Michelson-Interferometer
receiverfirst mirrorfrom
source
signal
beam
reference
beam
beam
splitter
second
mirror
moving
overlap
lc
z z
relative
moving
I(z)
wave trains
with finite
length
Michelson interferometer: interference of finite size wave trains
Contrast of interference pattern allows to measure the axial coherence length/time
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z
I
measured
signal
filtered
signal
measured
position
axial length of coherence
m mn
mnmnm IIII cos2
2
42
l
l
zzk
DDD
0,,2)()()( 2121 rrrIrIrI
)()(
),,(
21
21
minmax
minmax
rIrI
rr
II
IIC
Interference Contrast
Superposition of plane wave with initial phase
Intensity:
Radiation field with coherence function :
Reduced contrast for partial coherence
Measurement of coherence in Michelson
interferometer:
phase difference due to path length
difference in the two arms
(Fourier spectroscopy)
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Young Experiment with Broad Band Source
Decreased contrast due to finite spectral bandwidth
Realization with movable triple mirror
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
-400 -300 -200 -100 0 100 200 300 400
x
contrast
contrast
curve
interferogram
x
y
I(x,y)
beam
splitter
laser
reference
mirror
movable
triple mirror
detector
scan
x
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Axial Coherence
Contrast of a 193 nm excimer laser for axial shear
Red line: Fourier transform of spectrum
contrast
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
-0,8 -0,6 -0,4 -0,2 0 0,2 0,4 0,6 0,8
z-shift
in mm
measured
FFT-Data
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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
Speckle Effect
25
incident laser light
surface with roughness
plane of observation
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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
Sum of Random Phasors
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Ref. J. Goodman
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Size of objective speckles:
depends on distance of
observation
Colored speckles
z = 840 mm z = 330 mm
z = 160 mm z = 110 mm
Speckle Pattern
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Creating of speckle pattern:
1. coherent scattering of laser light:
objective speckle
2. imaging of coherent straylight:
subjective speckle
always be visual observation
Subjective / Objective Speckle
Pr
r
1
2
incident laser light
surface with roughness
p > l
point of observation
schreen
lens with focal length f
surface with roughness
z'
d
intensity
D
z
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Incident coherent light
Rough surface with size D
Observation in distance z
Speckle pattern with typical size of cells
screen
incident
coherent laser
light
rough
surface
D
z
d
intensity
2
AiryD
D
zd
l
Objective Speckle Pattern
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Incident coherent light
Rough surface with size D
Observation in distance z
Speckle size in the image:
PSF, Dairy
Speckle pattern with typical size of
cells in the object
m: magnification
Example:
coarse speckle for small NA
Subjective Speckle Pattern
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schreen
lens with focal length f
surface with roughness
z'
d
intensity
D
z
ds
airys DmNA
md )1(2
)1(l
F#= 22 F#= 66
Ref. W. Osten
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Field reflected at rough surface with height h(x,y)
Coherence of reflected waves
Dependence of the coherence from the roughness s of the surface (normal distribution of height assumed)
),(cos1),(),( yxhik
inr eyxEryxE
yxCk
yxhyxhik
hhe
eyx
DD
DD
,1)cos1(
,,cos1
222
1122,
s
Generation of Speckle at a Rough Surface
31
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r/rc
(r)
s = 0.3
s = 1.0
s = 2.0s = 4.0
s = 10
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The area of correlation of the radiation Ac Lc
2
shrinks with the variance s2 of the surface roughness
Signal to noise ratio
Contrast C
0
0.5
1.0
0 5 10s 2
Ac
/ r2
N
s
I
Ir
1 2
1
rC
r
0 5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r
C
Speckle at Rough Surfaces
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Poisson statistics of a single speckle:
Probability of intensity values in the
pattern
Contrast
(full developed speckle)
Necessary: roughness larger than wavelength
Largest probability: darkness I=0
Example
I
I
eI
Iw
1
)(
1ICI
s
Statistics of Single Speckle
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w(I)
1
I
<I>
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Zero intensity points in a speckle pattern
Here often vortex points of the phase
Found in simulation by real and imaginary part
Phase Dislocations
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a) intensity b) Re and Im parts c) phase
circles: zero I(x,y) rotation around zeros
Ref. J. Goodman
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Incoherent superposition of several speckles
Probability has intermediate maximum
Zero probability for darkness
Decreasing contrast
Example
0
2
2
0
4)(
I
I
eI
IIw
Statistics of Superposed Speckles
35
I / Io
w(I)
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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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
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
w(I)
I / Io
n = 2
n = 6
n = 12
n = 20
n = 40
n = 100
n/1
Speckle Statistics for Incoherent Superposition
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Speckle Reduction
Coherent speckles after diffusor plate with different data
starting
phase
spectrum
far field
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Speckle Contrast Changing with Coherence
Contrast of speckle image for changing coherence
a: amplitude lcorr: transverse length of coherence
a/lcorr
= 2.0a/lcorr
= 1.0 a/lcorr
= 4.0
a/lcorr
= 0 a/lcorr
= 0.5a/lcorr
= 0.1
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Reduction of spatial coherence:
- moving scatter plate
- statistical mixing of phases
- temporal integration (time averaging)
- movement should be faster than detector integration time
- diversification of illumination angle (microscopy)
- diversification of wavelength (laser bandwidtz)
incident
coherent
beam
moving
diffusor plate
image plane
x',y'
pupil
ax , ay
s
lens
s'
Reduction of Spatial Coherence
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Reduction of spatial coherence more effective in case of two scatter plates
Only one is moving
incident
coherent laser beam
modulated direction
spatial partial coherent radiation
moved 2nd diffusor
fixed 1st diffusor
propagation distance
Reduction of Spatial Coherence
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Averaging of speckles by time integration
Moving stop in the pupil But: reduced resolution
Moving stop near the image plane
coherent illumination
beam
moving
stop
scattering
object
coherent
speckle
pattern
reduced speckle
by time
integration
coherent
illumination beam
moving
stop
scattering
object
reduced speckle by
time integration
Reduction of Speckle by Temporal Averaging
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Speckle Metrology
Usual: speckle perturbs the imaging for coherent illumination
Speckle is only dependent on spatial roughness: time independent
Different usage of statistical speckle pattern in metrology:
1. Speckle photography:
- recording of two intensity images with small lateral shift of object
- speckle pattern invariant but moved
- comparison/evaluation by
1.1 correlation
1.2 calculation of differences
1.3 Fourier transform evaluation
2. Speckle interferometry:
- superposition of speckle fields (both statistical or one deterministic reference)
- speckles work as statistical strucrured illumination
- imaging of visualization of surface roughness
- referencing by shear: shearography
3. Speckle astronomy:
- recording of many single speckled images (due to atmospheric changed speckles)
- calculation of Fourier transforms
- averaging over all images, autocorrelation, statistics suppressed
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Speckle Photography
Setup with objective speckles
Displacements:
subjective
objective
Example
Ref. W. Osten
F
sub
pdm
Dd
l)(
F
obj
pd
Dd
l)(
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Speckle Photography
Selection of a speckle cell
Shift transform T(u,v):
matched cell
Types:
translation, rotation, shear
Finding the maximum correlation
C(u,v)
speckle cell
matched speckle cell
transform
T(u,v)
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Speckle Photography
Autocorrelation of different shift sizes
16 pix
32 pix
64 pix
128 pix
shift:
Ref. J. Goodman
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Classical setup with deterministic
reference beam
Movement of diffuse object detected
as phase change
Setup for in-plane displacement
Speckle Interferometry
46
displacement
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Coherence cell has finite lateral
and axial dimension
Depth extension of a speckle cell:
- out-of plane displacement
- corresponds to classic structured
illumination for 3D metrology
Speckle Interferometry
47
cells
speckle spots
beam caustic
propagation
7 spot per cross
section in 1
dimension
displacement
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Speckle Shearography
Double exposure after displacement:
shear interferometry
Measurement of phase gradients or slopes
Intensity
Phase difference for x shear
S: sensitivity
Ref. W. Osten
object
lens
wedge
l
D
DxyxS
x
yxyxxyx),(
),(
2
),(),(
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49
Speckle Shearography
Examples
Ref. T. Yoshizawa
raw phase map filtered phase map unwrapped phase unwrapped phase 3D
raw phase map filtered phase map slop map slope map 3D
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Speckle Shearography
Examples for defect detection
Ref. W. Osten