www.iap.uni-jena.de

Metrology and Sensing

Lecture 9: Speckle methods

2017-12-14

Herbert Gross

Winter term 2017

2

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

3

Content

Spatial coherence

Temporal coherence

Speckle

Speckle properties

Speckly metrology

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

4

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

5

Coherence Function

Coherence function: Correlation

of statistical fields (complex)

for identical locations :

intensity

G 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

G

z

x

x1

x2

E(x2)

E(x1)Dx

r r r1 2

)()(

),,(),,()(

21

212112

rIrI

rrrr

G

)(),( rIrr

G

6

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

G r r( , )

1 2

Lc

P1

P2

domain of

coherence

7

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

8

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

9

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 spacing D

D0

1

V

10

2

2

0 cos4)(z

xDIxI

l

D

zx 2l

D

screen with

pinholes

detector

source

z1

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

11

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

12

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

13

')0,'(1

),,()(')(

2

21

2122

21

rderIez

zrrrrr

z

irr

z

i

G l

l

l

V

r

vanishing contrast

1

Van Cittert - Zernike - Theorem

r r r1 2' ' '

G( )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

14

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

15

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

16

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

17

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

18

| ( ) |

c

Time-Related Coherence Function

G( ) lim ( ) ( ) ( ) ( )* *

TT

T

TTE t E t dt E t E t

1

2

G( ) ( ) ( )*0 E t E t IT

( )( )

( )

( ) ( )

( )

*

G

G 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: G = 0

Vanishing time interval: intensity

Normalized expression

Usually:

G decreases with growing symmetrically

Width of the distribution: coherence time c

19

0

)( dSI

G 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

20

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

21

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

G

)()(

),,(

21

21

minmax

minmax

rIrI

rr

II

IIC

G

Interference Contrast

Superposition of plane wave with initial phase

Intensity:

Radiation field with coherence function G:

Reduced contrast for partial coherence

Measurement of coherence in Michelson

interferometer:

phase difference due to path length

difference in the two arms

(Fourier spectroscopy)

22

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

23

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

24

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:

partial coherent illumination, polarization

Speckle Effect

25

incident laser light

surface with roughness

plane of observation

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

26

Ref. J. Goodman

Creating of speckle pattern:

1. coherent scattering of laser light:

objective speckle

2. imaging of coherent straylight:

subjective speckle

always by 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'

intensity

D

z

27

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

28

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

29

Incident coherent light

Rough surface imaged

lens size D

Observation in distance z

Speckle size in the image:

dominated by PSF, Dairy

Speckle pattern with typical size of

cells in the object

m: magnification

Example:

coarse speckle for small NA

Subjective Speckle Pattern

30

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

Incoherent image:

homogeneous areas, good similarity between object

and image, high fidelity

Coherent image:

Granulation of area ranges, diffraction ripple at

edges

incoherent coherent

Coherent – Incoherent Image Formation

incoherent

coherent

32

Speckle in Imaging

Example with different

illumination setups:

- monochromatic/ broad band

- spatial coherence for

s = 0.05 / 0.1

Ref: R. Guenther

monochromatic

spatial coherence s = 0.05

spatial coherence s = 0.10 broadband

monochromatic through dust

spatial coherence s = 0.10

33

Speckle in Imaging

Sunlight:

- angle 0.25°

- speckle grain size approx. 110 m

- can be resolved under comfortable conditions

Ref: R. Guenther

34

Speckle in Atmospheric Imaging

Imaging though the atmosphere

Turbulence due to statistical changes of

- temperature

- density

- velocity of gas

Imaging point spread function suffers

form speckle

Speckle noise is reduced for longer

exposure times

Ref: J. Goodman

35

Coherent Tissue Imaging

Coherent imaging in scattering tissue

Imaging of a polystyrol spheree embedded

in tissue without (a) and with (b) polarization

Image of blood vessels:

dependence on exposure

time due to averaging

Ref: M. Gu / A. Wax

Field reflected at rough surface with height h(x,y)

Coherence of reflected waves correlation function C

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

36

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

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

37

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

38

w(I)

1

I

<I>

Zero intensity points in a speckle pattern

Here often vortex points of the phase

Found in simulation by real and imaginary part

Phase Dislocations

39

a) intensity b) Re and Im parts c) phase

circles: zero I(x,y) rotation around zeros

Ref. J. Goodman

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

40

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

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

41

Speckle Reduction

Coherent speckles after diffusor plate with different data

starting

phase

spectrum

far field

42

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

43

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 bandwidth)

incident

coherent

beam

moving

diffusor plate

image plane

x',y'

pupil

ax , ay

s

lens

s'

Reduction of Spatial Coherence

44

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

45

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

46

Axial length of coherence:

function of spectral bandwidth

Decorrelation of coherence along direction of propagation:

special delay device

focussing

lens

array of

cylindrical

lenses

distributed

delay

device

l

l

D

2

cl

Axial Length of Coherence

48

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 structured 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

49

Speckle Photography

Setup with objective speckles

Displacements:

subjective

objective

Example

Ref. W. Osten

F

sub

pdm

Dd

l)(

F

obj

pd

Dd

l)(

50

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)

51

Speckle Photography

Autocorrelation of different shift sizes

16 pix

32 pix

64 pix

128 pix

shift:

Ref. J. Goodman

Classical setup with deterministic

reference beam

Movement of diffuse object detected

as phase change

Setup for in-plane displacement

Speckle Interferometry

52

displacement

Coherence cell has finite lateral

and axial dimension:

defines the resolution

Depth extension of a speckle cell:

- out-of plane displacement

- corresponds to classic structured

illumination for 3D metrology

Example:

Vibration detection

Speckle Interferometry

53

cells

speckle spots

beam caustic

propagation

7 spot per cross

section in 1

dimension

displacement

54

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

),(),(

55

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

56

Speckle Shearography

Examples for defect detection

Ref. W. Osten