No Slide Title 5.pdf · the somb is the function generating Airy’s disk. It drops to half the initial value at x=±0.71, and thus, the speckle size at 50% correlation is s t = 2Δ=

1SPECKLE PATTERNfrom: 'Electro-Optical Instrumentation' by S.Donati, 2004, © Prentice Hall (USA)

SUMMARY

SpeckleSpeckle--patternpattern InstrumentsInstruments

• Speckle properties ◊ statistical analysis, ◊ speckle size◊ joint statistical distributions◊ speckle phase errors

• Speckle in single-point interferometers◊ vibration meas. ◊ displacement meas.◊ phase error correction

• Electronic Speckle-Patter Interferometry (ESPI)◊ development of instrumentation


Coherent light projected on a diffusing surface looks grainy, like a spatial white-noise distribution, and this aspect is

called speckle pattern

Speckle-pattern


Speckle pattern is the fieldradiated from a diffuser, that is (top left), a surfacewith roughness larger thanλ, Δz>>λ. The field at point P is the summation of many individual vectorswith a random phase (top center). Moving away fromP to P+ΔP’ and P+ΔP’’, the field gradually loses corre-lation. A speckle grain iswhere correlation is C≥0.5.Grains are cigar shaped and point to the diffuser center.

How speckle-pattern is formed

laser beam

D

Δz

diffuser

z

s t

ls

P

P+ΔP'P+ΔP''

E(P) E(P+ΔP')E(P+ΔP'')

ζ

x

y

z

__

_


◊ Speckles are charactrerized by a longitudinal size sl (along z) and a transversal size st (along x or y). ◊ A circular diffuser of diameter D, uniformly illuminated, has:

st = λ z/D, sl = λ (2z/D)2

◊ Speckles are cigar shaped, with longitudinal size larger thantransversal, and become more and more elongated as we move awayfrom the diffuser. ◊ Speckles point to the diffuser center, and the projection of off-axisspeckles is equal to the on-axis speckle. ◊ Speckle sizes are easily seen to be the single-mode spatial regions, writing the acceptance as a=AΩ =λ2

Speckle-pattern size


◊ Let N=1 for a single mode in a=AΩ = N λ2

◊ radiation at z is received under the solid angle Ω =π(D/2z)2 andarea A= π(st/2)2 associated with tranvesal size st. Inserting, we getλ2= π(st /2)2π(D/2z)2 whence st =(4/π)λz/D.

◊ Second, the bundle of rays in Ω keeps limited transversally to ≤stfor a size st /θ, where θ=D/2z is the angular aperture of the bundle.

◊ Combining expressions gives longitudinal size: sl =(2/π)λ(2z/D)2 .◊ Expressions of st and st are the same as found with correlationanalysis, except for a minor ≈1 multiplying factor.

Speckle-pattern size from acceptance

D

zsource

Ω

s l

st


Subjective speckle pattern: a lens looking at the sourcecollects N speckles and, because of mode invarian-ce, forms N (smaller) speckles in the image.

Objective and subjective speckle

FDL

z

image

source

speckle

LD

projected speckle

Along its diameter, the lens accommodates a number of speckles N= DL/st, ratio of lens diameter to speckle transversal size. Image diameter is D(FL/z) and contains N=DL/st speckles. Then, the transversal size st(fp) of speckles in the focal plane is:

st(fp) = D(FL/z) (st /DL) = λ FL/DL

This size is the same as produced by an objective speckle from diffuser of diameter DL at distance FL. From the focal plane we see a virtual image of the source magnified by z/FL. Thus, apparent transversal size st(pr) on the diffuser is:

st(pr) = (λ FL/DL) (z/FL) = λz /DL


In the Fresnel approximation, the field at a point P2 (x,y,z ) as due to the source at z=0, a uniform illuminated disk of aperture D in the (ξ,η) plane of P1 is written as (Rayleigh-Sommerfeld approximation):

E(x,y,z) = (E1/λz) ∫∫−∞,+∞ dξdη rect(ξ,η,D/2) (z/r12) exp ikr12 exp iφ(ξ,η)Expanding r12 = [z2+(x-ξ)2+(y-η)2]1/2 ≈ z {1+[(x-ξ)/z]2+[(y-η)/z]2}1/2 ≈ ≈ z +(x2+y2)/2z +(ξ2+η2)/2z -(xξ+yη)/z,

and with z ≈ r12 (Fresnel approximation), we get:

Speckle-pattern: Statistical Analysis

yxξ

η

z

P1

P2θ

r12


E(x,y,z) = (E1/λz) exp ikz exp ik(x2+y2)/2z ×

× ∫∫−∞,+∞ dξdη rect(ξ,η,D/2) exp ik(ξ2+η2)/2z

× exp-ik(xξ+yη)/z exp iφ(ξ,η)

Here, after the amplitude E1/λz, the first term is phase shift of propagationdown to distance z, and second is the field curvature term. The integral withboundary truncation term (rect) is for the source. The fourth term is again a field curvature term, the fifth is the mixed-coordinate term that originates the Fourier transform kernel. A last term is added, the diffuser random-phaseterm, a white noise in ξ,η.

As a first property, the field E is a complex quantity with a real ER and animaginary EI part. Both are affected by the random part of the phase φ(ξ,η), and as φ= kΔz >>2π, we have ⟨cosφ⟩ =⟨sinφ⟩ =0 or

⟨ER⟩ = ⟨EI⟩ =0

Speckle-pattern: Statistical Analysis


Second, in the Δz profile, the number of scatterers is very large and therefore, (central-limit theorem) Δz obeys a normal (or Gauss) distribution.

At the same way, φ=kΔz and its sine and cosine functions are normaldistributions, too. Thus, the real and imaginary parts of the field, E =ER +i EIare both normal distributed and uncorrelated and written as:

p(ER) = (2πσE2)–1/2exp-ER

2/2σE2, p(EI) = (2πσE

2)–1/2exp-EI2/2σE

2

here, σE2 are the variance of the fields, the same for ER and EI for symmetry

reasons. Writing σE

2 as the mean of squares reveals that it is coincident to the meanintensity:

σE2=⟨ER

2⟩ =⟨EI2⟩, 2σE

2=⟨ER2⟩+⟨EI

2⟩=⟨ER2+EI

2⟩ =⟨I⟩

About the field amplitude⎟ E⎟=√(ER2+EI

2), by transforming variables it iseasily found that ⎟ E⎟ is Rayleigh distributed, or :

p(⎟ E⎟ ) = (2πσE2)–1/2⎟ E⎟ exp-(⎟ E⎟ 2/2σE

2)

Speckle-pattern: Statistical Analysis 2


Intensity I=⎟E⎟2=ER2+EI

2 is now found from ER and EI are normal: p(I) = ⟨I⟩–1 exp -I / ⟨I⟩

Because of the negative exponential distribution, weak (or dark) speckles are more probable than intense (or bright) ones.

For example, 10% of speckles have ≤10% the average intensity, 1% have ≤1% the average intensity, and so on.

This is just amplitude fading, a feature hampering measurements thatwe shall appropriately tackle in all systems working on a diffusingsurface rather than a mirror.

Last, the phase of the electric field at a given point P is a randomdistribution with no information on the initial distance-related phase kz.

Computing the phase as ϕ= atan EI/ER gives as a result a uniformdistribution of phase on 0-2π as:

p(ϕ) = 1/(2π)

Speckle-pattern: Statistical Analysis 3


Summary of the specklepattern statistics: field components ER and EI, amplitude IEI,intensity I and phase ϕ, and joint distribution of E1 and E2

basic properties of speckle

P

P+ΔP

IER

E ,0

p(E )R p(E )I,

Gaussian distribution for both real and imaginary part of the field

Rayleigh distribution for the amplitude ⎮E⎮

⎮E⎮

p( )⎮E⎮

0

Negative exponential distribution for intensity I= E2

0

p(I)

I⟨ I ⟩ 0

Uniform distribution for phase ϕ

ϕ-π π

Eσ = ⟨Ι⟩ /2

p(ϕ)

Statistical properties of the speckle field at a generic point P

Statistics of the speckle field at two points P =P and P =P+ ΔP:1 2

:

E2

1E

joint Gaussian with coherence factor μ=μ(Δ)


For the longitudinal size, we consider a displacement Δ along z and write:

⟨Est(x,y,z+Δ)E*st(x,y,z)⟩ = exp ikΔ -ikΔ(x2+y2)/2z2 ∫ ∫ ∫ ∫rect(ξη,D) rect(ξ’η’,D )

exp ik(ξ2+η2)/2(z+Δ) exp-ik(ξ’2+η’2)/2z2 exp-ik(xξ+yη)/(z+Δ) ×

× exp+ik(xξ’+yη’)/z ⟨expiφ(ξ,η)exp-iφ(ξ’,η’)⟩ =

= exp ikΔ-ikΔ(x2+y2)/2z2 ∫ ∫rect(ξη,D) exp -ikΔ(xξ+yη)/z2 -ikΔ(ξ2+η2)/2z2

Again, first term is a pure phase term ϕ that includes the correct delay kΔ=k[(z+Δ)-z] as well as the curvature (deterministic) error kΔ(x2+y2)/2z2, which issmall with respect to kΔ but shows up when we perform an interferometricmeasurement in the speckle regime. The term of integral is real and is given by:⎟ ⟨Est(x,y,z+Δ) Est*(x,y,z)⟩⎟ = somb (Δ/λz2)[D2/4+D√(x2+y2)]

≈ somb (ΔD2/4λz2) for x2+y2<<D2

As already noted, the somb function drops to 0.5 at x=±0.71, and thus the longitudinal size is

sl = λ (2z/D)2

speckle size calculation


writing the correlation for two points with a displacement Δ along x as:

⟨Est (x+Δ,y,z)Est*(x,y,z)⟩ = exp ik(Δ2+2Δx)/2z ×

× ∫∫ rect(ξη,D) exp-ik[(x+Δ)ξ+yη]/z exp+ik(xξ+yη)/z =

= exp ik(Δ2+2Δx)/2z ∫ ∫ rect(ξη,D) exp-ikΔξ/zthe integral is recognized as the Fourier transform of rect(ξ,η,D/2) of diameterD, calculated in the plane ξ ,η, and specialized at r=Δ. The multiplying factoris a pure delay for a phase ϕ=kΔ(Δ+2x)/2z≈kΔ(x/z). This is the deterministicerror because of field curvature. The integral is readily evaluated as

⎟⟨Est(x+Δ,y,z)Est*(x,y,z)⟩⎟ = somb ΔD/λzthe somb is the function generating Airy’s disk. It drops to half the initialvalue at x=±0.71, and thus, the speckle size at 50% correlation is st = 2Δ =2×0.71/(D/λz) =1.42 λz/D. In conclusion, we simply take as reference valuefor the transversal size:

st = λz/D

speckle size calculation 2


Further statistical properties are supplied by with the bivariate or jointprobability function, relating speckles in two points in space, ie P and P+ΔP. Starting point for second-order statistical properties of the speckles is the jointprobability of intensity and phase in two points P1 and P2, that is:

p(I1,I2,ϕ1,ϕ2) = [16π2σE4(1-μ2)]–1 ×

× exp-{[I1+I2-2(I1I2)1/2μcos(ϕ1-ϕ2-ψ)]/2σE2(1-μ2)}

here, μ exp iψ=μc is the complex coherence factor for the fields at P1 and P2. μc is defined as the ratio of the mutual intensity normalized to the product of rms values of fields at points P1 and P2:

μc = ⟨E(P1)E*(P2)⟩/ [⟨⎟E(P1)⎟2⟩ ⟨⎟ E*(P2)⎟2⟩]1/2

where ⟨..⟩ indicates ensemble average on the speckle set, and * is complexconjugate. μc has a modulus going from 0 (complete uncorrelation) to 1 (full correlation), and is complex. The numerator is the correlation of the field, which is sometimes called mutual intensity Cμ = ⟨E(P1)E*(P2)⟩

the joint distribution


Using p(I1,I2,ϕ1,ϕ2), we can calculate the mean value ⟨X⟩ and variance σX2=

⟨[X - ⟨X⟩] 2⟩ of any variable like intensity I1 and I2, or phase ϕ1 and ϕ2. To eliminate an unused variable, we integrate the joint probability on it.Changing ϕ1 and ϕ2 to sum and difference and eliminating the sum we get:p(ϕ1-ϕ2) = [(1-μ2)/4π2](1-β2)–3/2 [√(1-β2)+β arcsinβ+(π/2)β],

where β=μcos(ϕ1-ϕ2-ψ), and ψ is the argument of factor μc=μexp iψ.

the joint statistics

PRO

BABI

LITY

DEN

SITY

p(ϕ

)

0.7

0

0.25

0

0.5

1.0

0

21PHASE DIFFERENCE ϕ= ϕ -ϕ -ψ

−π −π/2 +π/2 +π

1/2π

μ=0.9

The variance of phase is thenfound as:σϕ

2 = π2/3 - π arcsin μ +

arcsin2 μ - (1/2) Σ n=1,∞ μ2n/n2

and its diagram peaks the more μ is close to 1, whereasis flat for μ =0.


We can also calculate mean and variance if we derive the conditionalprobabilities from p(I1,I2,ϕ1,ϕ2), using Bayes' formula, for example:p(I1, ϕ1⎟ I2, ϕ2)= p(I1,I2,ϕ1,ϕ2)/ p(I2,ϕ2), etc.In the diagram below, we plot mean value (left) and variance (right) of intensity I2 conditioned to I1, as a function of μ, and for some values of I1 asa parameter. The ‘free’ values of mean and variance of the unconditioneddistribution are also indicated.

the joint statistics 2

COHERENCE FACTOR μ COHERENCE FACTOR μ 0

00.5 1.0

1.0

1.2

1.4

0.8

1.6

0.6

0.4

0.2

I / =1

ME

AN

INTE

NS

ITY

I /



2

INTE

NS

ITY

STA

ND

AR

D D

EV

IATI

ON

σ

/σ2 I 2

2

52

1

1.5

0.5

0.2

0

FREE

0.5 1.0

I / =1 5

2

1

0.5

0.20

00

1.0

1.2

1.4

0.8

1.6

0.6

0.4

0.2

FREE


Mean value (left) and variance (right) of intensity I2 conditioned to I1 and tothe phase difference ϕ, as a function of the coherence factor and for some values of the projected intensity I1cos2ϕ as a parameter, given by expressions.The abscissa is the modulus of the coherence factor extended to negative values by the multiplication to the sign of cosϕ to account for anti-correlationwhen ϕ≈π. We can see that, similar to intensity, also the phase becomes more regular in correspondence to bright speckles.

the joint statistics 3

00

00 0.5 0.5-0.5-0.5 1.0 1.0-1.0 -1.0

1.01.0

2.0

2.0

0.2

0

0.5

1

1.5

2

4I cos θ /12 =

0.2

0.5

1

0

2

0

1 24

5I cos ϕ /12 =

COHERENCE FACTOR μ sign(cos θ) COHERENCE FACTOR μ sign(cos θ)

ME

AN

INTE

NS

ITY

I /



2

.2

0

1 25

.2

FREE

INTE

NS

ITY

STA

ND

AR

D D

EV

IATI

ON

σ

/σ2 I 2

2

FREE


Variance of the phase difference σϕ2 normalized to π2/3, as a function of the

coherence factor μ (left), and of the dynamic range factor μ=1-ζ2 (right), with the relative intensity √( I1I2)/ as a parameter. The ‘free’ variance isalso plotted for comparison. The right hand diagram is an expansion nearμ≈1, the region of high coherence, and is plotted for μ=1-ζ2.

phase statistics conditioned to intensity

COHERENCE FACTOR μ

PH

AS

E V

AR

IAN

CE

σ

/(π

/3)

ϕ22

RE

LATI

VE

STA

ND

AR

D D

EV

IATI

ON

C

= σ

/ζ

ϕ

FREEFREE

DYNAMIC RANGE ζ

¦I I / =1 2

¦I I 1 2

= 5

1.5

0.5

0.25

0.05

0.1

1.0

10

10-7 10-5 10-3 10-10

0.5

1.0

0.5 1.000

1

2

3

4

5

6

7


When μ≈1, we let μ=1-ζ2 and obtain the asymptotic behavior for ζ≈0

σϕ2⎪free = ζ2 (3 - 2 lnζ2) σϕ

2⎪I2,I1 = ζ2 [√(I1I2)/⟨I⟩]–1

Main dependence of both free and conditioned phase variances is on ζ2. The multiplying factor steadily increases for ζ→0 for the former and isgiven by the inverse of the relative speckle intensity √I1I2/⟨I⟩ for the latter. The ratio of rms phase error σϕ to dynamic range factor is:

C = σϕ⎪I2,I1 / ζ (see previous slide)

Introducing the NED (noise-equivalent-displacement) of the interfero-metric measurement, we get the expressive form

NED = σϕ/k = C ζ/k

speckle phase error conditioned to intensity


Now let us recall that the coherence factor μ=1-ζ2 in the two cases of speckle, oflongitudinal and transversal displacements from P to P+Δ, is given by⎟⟨Est(P) Est*(P+Δ)⟩⎟=somb X , where X=ΔD/λz is for tranversal and X=ΔD2/λz2 is for longitudinal. In termsof transversal and longitudinal speckle sizes st and sl it is alsoX=Δ/λst and X=Δ/sl, or, in generalized form: X=Δ/sspckl.Now, let's develop the somb function at the third order in X and obtain μ as: The factor ζ is found from the coherence factor as.

μ = somb X = 2J1(πX)/πX = 2 [πX/2 –(πX)3/16]/πX = 1-(πX)2/8By comparing with μ=1-ζ2 we get: ζ = πX /2√2

Going back to find the NED due to the speckle we find the expressive form:

NED = C ζ/k = (C/4√2) λ Δ/sspckl

This equation tells us that the speckle error, in wavelength #, is given by the ratio of displacement Δ to speckle size sspckl, either longitudinal or transversal.

speckle phase error conditioned to intensity 2


In addition to the basic phase errors, several other sources are found when dealing with a diffuse surface,

Errors due to target movement

Errors due to beam movement

Focussing lens effects

Errors due to detector size

other speckle errors


A first prblem with speckle statistics is amplitude fading: • in a vibrometer aimed to a target, we may eventually fall on a weakspeckle, with an intensity I much smaller than average ⟨I⟩. Remedy:

we may incorporate an intensity level sorter in the instrument, warning the operator that speckle intensity is too small.

a second approach, Automatic Gain Control can restore signalamplitude. If AGC dynamic range is improved by M , the threshold of signal loss is moved to a lower value, η⟨I⟩/M, but not eliminated.

a third strategy is sensor duplication, a technique known in radio asdiversity. It uses two receivers, slightly apart in space to get differentrealization of statistics. Then, probability of fading is η2, a smallervalue than a single channel, yet not zero.

A last possibility, is a superdiffuser as the target surface to providea gain ≈ 20 to 50 (typ.) in the back-reflected signal reaching the sensor. Application of the tape or varnish on target is a quasi-invasiveoperation that cannot be accepted in all cases

Speckle in Single-Point Interferometers


Another concern is speckle phase error, affecting the accuracy of the measurement performed on a diffusing surface.

The error is usually small, but, as we go down to measure veryminute amplitudes of vibration, it becomes important.

In a vibrometer, the speckle size can be made much larger than the dynamic range Δmax and, with sl(trg)>>Δmax, the phase error is madenegligible.

As a practical example, assuming λ=1μm, DL=30mm, and z=300mm, the speckle longitudinal size is sl(trg)=λ(2z/DL)2=400 μm. The phase error λΔmax/sl(trg) is 2.5 nm for a swing of 1μm, is 0.25 μm for a swing of 100 μm, and so on.Moreover, if we choose a bright speckle, we could be able reduce the phase error further.

Speckle in Single-Point Interferometers 2


Arrangement for the dynamical tracking of speckle. Two smallbars of PZT ceramic are mounted in the lens fixture and move the

lens along the X and Y axes, changing the spot position on the target just of a few micrometers to track maximum intensity

Speckle regime in displacement measurement


Block scheme of the speckle-tracking circuit. Signal from the photo-diode is rectified peak-to-peak and demodulated respect to the ditherfrequency, in phase and quadrature. Results are the X and Y error signals that, after low-pass filter, are sent to the piezo-actuators X and Y to track the maximum amplitude or stay locked on the bright speckle

Speckle tracking technique


Pdf p(IEI) of the field amplitude IEI, normalized to 1 for a value C=1 of the SM parameter. Left: thin line is the Rayleigh distribution for an ideal diffuser, and thick line is the result of a numerical simulation for a real surface, with 2% reflection and 98% diffusion, and fully developed speckle statistics. Bars are experimental data for white paper (z=50cm, wlas=2 mm). Right: with the bright-speckle tracking on, the experimental distribution moves to larger IEI (dark bars) as predicted by simulations, respect to off (gray bars).

Speckle tracking probability


Two samples of simulations of amplitude (top curves) and phase (bottomcurves) of the returning field. The diffusing target moves from 108 to 51 cm. In amplitude diagrams, top curves are with the tracking circuit on, and bottomwith circuit off. In phase diagrams, the curves that vary less are with the tracking circuit on. Abrupt jumps (near z=87cm on the left and 95 cm on the right) are correct switches decided by the tracking system, which skips fromone speckle becoming too weak to the next adjacent speckle, being brighter.

Speckle tracking performance


Top: comparison between the signal amplitude with (thin trace) and without(thick trace) bright-speckle tracking. Bottom: the displacement measured bythe interferometer shows an error near 76 cm because of the speckle fading, whereas the error is removed with the bright-speckle tracking system. The target was moved from 70 to 80 cm at a 1-cm/sec speed.

Speckle tracking performance 2


Left: Transversal movement of the target worsens the fading problem in a normal interferometer, but can be tolerated when the bright-speckle tracker is added (top); loss of counts and error without tracker versus no error with the bright-speckle tracker. Right: Typical experimental samples of signalamplitude with and without the bright-speckle tracker.



Pdf of the field amplitude from a superdiffusing target, that is, Scotchlite™tape with a superdiffuser gain G=20. Amplitude on the abscissa is normalizedto C=1. Line with aces is for the normal speckle statistics, and dotted line isfor speckle with tracking on. Gray bars are for AGC on and speckle trackingoff. Dark bars are for both AGC and speckle tracking on.



Because of the causality of effects in the propagation integral of the field, the real ER and imaginary EI parts of the field at a point z are Hilbert transformsof each other. Also the log amplitude of the field, log IEI= log√(ER

2+EI2) and

phase ϕ = atan (EI /ER) = kz+ϕsp are Hilbert pairs. The HT transform is givenby: H [f(z)] = π-1 ∫C f(ζ)/(z-ζ) dζ.

Speckle Phase Error Correction by HT

We can try using HT for correcting the phase error due to speckle pattern.Shown here are the real (top) and imaginary (bottom) part of the fieldE(z) at distance z, for a speckle field at λ=1μm generated by a square diffuser, 100-μm by side. Lines are results of simulation, and points in the bottomdiagram are the results of a Hilberttransform of the real part.


Intensity IEI (left) and phase ϕsp (right) of the speckle-pattern field from a diffuser. The lines represent the results of a numerical simulation, and the dotsin the right-hand diagram are the result of the Hilbert transform of log-

intensity log IEI data shown in the left diagram.

Speckle Phase Error Correction by HT - 2


If the intensity versus z has a point where it falls to zero (left), then itslog has a singularity and the Hilbert transform of log intensity does notgive the correct phase any more, as shown at left by the thick line HT(ln I) compared to the thin line, the correct phase.



In conclusion:

• the Hilbert transform actually supplies the phase correction to beapplied to measured phase asϕ = atan (EI /ER) = kz+ϕsp =kz+H [log IEI] , but under the condition that IEI never goes to zero along z, (no intensity fading) otherwise errors spoil the correction

• to make the correction, we need theoretically to swing thetarget upon a distance from z=–∞ to z=+∞. In practice, thetruncation error becomes large (>>λ) as soon as Δz is a fewlongitudinal-speckle size sl long.

• it is unclear how to apply the correction if the motion of thetarget is arbitrary (i.e. not uniform from z=–∞ to z=+∞).



ESPI for pickup of vibration in subjective mode. Light from a He-Ne laser illuminates the object. The objective lens formes subjective speckles of sizeslfp=λF2/DL

2 at the image plane. Seen projected onto the object, speckles havea size slpr=λs2/DL

2. When object vibrates by Δs> slpr , speckles are washed out. To improve contrast, a circular stop or a double-slit stop is inserted intoobjective lens. The trade-off between size slpr, resolution λs/DL and light collection efficiency is difficult, a practical limit of the method.

Electronic Speckle Pattern Interferometry (ESPI)

He Ne laser

pinhole filter

object

image plane

viewing objective lens

image pickup device

illuminating objective

ir

s

display

laser line interference

filter

diameter D stop or double slit filter

Lslits


Setup of ESPI for pickup of displacement and vibration with interferometric sensitivity. Light from a He-Ne laser is shed on the object. By means of twobeam splitters BS, part of the beam is superposed to the image formed by the viewing objective lens. Spatial filtering of the speckle is again provided at the viewing objective lens by a circular stop or a double-slit stop.

Electronic Speckle Pattern Interferometry (ESPI)

He Ne laser

pinhole filter

object

BS

image plane

BS


image pickup device


ir

s

display

laser line interference filter

diameter D stop or double slit filter

Lslits


ESPI with reference beam in the time-averaging mode. Here, the target is a loudspeaker covered with a plain cardboard sheet and is driven into vibrationat increasing frequency (in the audio range). At nodes the speckles areunchanged whereas at anti-nodes are washed out. A few patterns of the manycoming out from 200 to 2000 Hz are shown, revealing vibrations modes and device resonances. Upper left picture is the speckle image without excitation.

ESPI - 2


ESPI image of a loudspeaker taken in the reference mode with a circular stop and a video high-pass filter (left) and the same with the double-slitstop, high pass, and peak-to-peak rectifier (right).

ESPI - 3

ESPI with reference and frame subtraction. The image of analuminum plate is shown before(left) and after (center) the application of a deformation in the center of it. The image at rightis the difference of the two and unveils the displacement fringes.


Operation of ESPI in framesubtraction mode. The high-frequency spatial content isenhanced with the appropriate lens stop and by time differentiation of the video signal. The initial speckleimage is stored in the recorder so that we can subtract it fromthe live image presented on the display. As a generalization of the reference beam, an opticalcorrelation block can be used.

ESPI - 4

laser source

pinhole filter


image pickup device


display

time diffe- rentiator

video recorder

signal subtractor

IN

OUTvideo in(-)

(+)

initial image switch

opt

ical

co

rrela

tion


ESPI - 4

The measuring systems provide comprehensive capability for research of new materials and development of new products. The unique full-field

and three-dimensional measurements enable users to obtain real-time,

complete, reliable static and dynamic measurements of their products.

Users get a complete view, including graphic images, movies, and

numerical results, of stress/strain, vibration modal/amplitude/phase and

non-destructive testing.The measuring principle is based on the very sensitive Electronic Speckle

Pattern Interferometry (ESPI) technique.

courtesy of Dantec Dynamics


Optical correlation adds enhancement of ESPI. Left: A partially ground glassprovides the reference for the average position of the target. Center: A tiltedbeamsplitter in front of the objective lens superposes contributions of imagepixels shifted of Δx for strain measurement. Right: A mirror pair allowssuperpose images with different fields of view to get a shearing interferometer.

ESPI - 5

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objective lens

Δx

partially ground glass

image plane

image plane

target

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1 2r - r

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mirrormirror

targettarget


♦ Speckle statistics comes across every time a laser beam shinesa diffuser – destroying its local spatial coherence

♦ nevertheless, spatial coherence remains in the propagated field along speckle regions (or grains) – of size sl and st

♦ projected speckle is objective, observed speckle is subjective♦ the speckle field has zero-mean, is Gaussian distributed in the

two components, Raileigh distributed in amplitude, negativeexponential in intensity, uniform in phase

♦ conditional distributions allows us to compute the accuracy ofsubset of measurements (of intensity and phase)

Summary and ConclusionsSummary and Conclusions


♦ in single-point interferometers, we face amplitudeamplitude--fadingfading and phase errorsphase errors

♦ amplitude fading is cured with: AGC, space diversity, use of superdiffuser target, last-speckle (sl>z, or z>D2/λ) condition

♦ amplitude fading is readily cured in vibrometers, and can beeffectively cured with speckle-tracking in interferometers

♦ phase errors are composed of a deterministic part, the curvatureerror, and a random part, the speckle-field error

♦ curvature error can be corrected easily (e.g. a-posteriori )♦ speckle-field phase error is given by NED=λΔs/ssp - not so severe

usually. It is cured by e.g. last-speckle and tracking techniques

Summary and Conclusions 2


♦ in image interferometers, each speckle is a separate measure-ment channel, though starting from a random phase

♦ image interferometers, or ESPI, may work on subjectivespeckles as low-resolution (20..50μm) vibrometers, or ashigh resolution (λ/2≈0.3μm) vibrometers and strain-sensors, when a reference beam is superposed on the detector

♦ respect to holography, ESPI offers less operation constraints,but has much less spatial resolution- respect to White-Light-Interferometry it covers easily largersurfaces and is faster– but again with less spatial resolution

- respect to 3-D contouring has more sensitivity

Summary and Conclusions 3

Documents

No Slide Title 5.pdf · the somb is the function generating Airy’s disk. It drops to half the initial value at x=±0.71, and thus, the speckle size at 50% correlation is s t = 2Δ=