15 February 1999

Ž .Optics Communications 160 1999 195–200

Measuring velocity of speckle fieldswith digital speckle pattern interferometry

Claudia Carletti a,) ,1, Roberto Torroba a,2, Rodrigo Henao b,3

a ( )Centro de InÕestigaciones Opticas CIOp , Casilla de Correo 124, 1900 La Plata, Argentinab Departamento de Fısica, UniÕersidad de Antioquia, A.A. 1226 Medellın, Colombia´ ´

Received 17 June 1998; revised 27 October 1998; accepted 2 December 1998

Abstract

We investigate a digital method for detecting the velocity of a diffusing object. The technique is based on Digital SpeckleŽ .Pattern Interferometry DSPI . A set of reference fringes is generated externally through the reference beam in a digital

interferometer. As the object moves, subsequent frames are acquired and subtracted according to the normal DSPI procedureand stored. By means of the theory of first order speckle statistics applied to speckle intensity correlation, we relate thevisibility variations in the reference fringes with the object velocity. Thus, by measuring the fringe visibility variation in theresulting DSPI stored frames the mean object velocity can be obtained. The theoretical results are experimentally verified.q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 42.30.MSKeywords: Velocity measurement; Speckle correlation; Fringe pattern visibility

1. Introduction

w xSince the early work of Jakeman and coworkers 1,2on velocity measurements of particles undergoing Brown-ian motion on the basis of moving speckle phenomenon,much research has been conducted. Examples are found influid-velocity measurements, where a fluid is seeded witha scattering medium, resembling the behavior of a specklepattern. These experiments were performed using a com-puter based system, indicating that this as a viable tool forthe measurement of relatively small fluid velocities.Rigid-body motions were also studied by illuminating the

) Corresponding author. E-mail: claudiac@odin.ciop.unlp.edu.ar1 Also at the Departamento de Fısica, Universidad Nacional del´

Sur, Bahıa Blanca, Argentina.´2 Also at OPTIMO, Departamento de Fisicomatematica, Facul-´

tad de Ingenierıa, Universidad Nacional de La Plata, Argentina.´E-mail: robertot@odin.ciop.unlp.edu.ar.

3 E-mail: rhenao@pegasus.udea.edu.co.

rough object with laser light. As the laser speckle patternmoves, due to the motion of a diffusing object undercoherent light illumination, the displacement and velocitycan be measured using several techniques. The dynamicalbehavior of speckles may be analyzed by first order andsecond order statistical methods by using interferometricor non-interferometric techniques. The last techniques aremore sensible to velocities relatively high, in the range ofcmrs or higher. Among these methods, we can name thetime-integrated speckle and spatially integrated speckle

w x w xtechniques 3,4 , spatial filtering technique 5 and specklew xpattern correlation technique 6,7 . The application of a

coherent reference beam leads to techniques more suitableto measurement of very small velocities. In the presentwork, we propose a novel interferometric method, which isvery sensitive to small velocities, based on the Digital

Ž .Speckle Pattern Interferometric technique DSPI . This is amodern technique based on the correlation of speckles.The dynamical behavior of the speckle patterns depends onthe relation of the object surface and the temporal fluctua-

0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 98 00651-8

( )C. Carletti et al.rOptics Communications 160 1999 195–200196

w xtions of the speckle 8 . These fluctuations, detected andprocessed in the time domain, take the form of a Gaussianrandom process. This assumption enables the use of thetheory on intensity correlation. This approach is based onthe statistics of light intensity variations of time-varyinglaser speckles produced by an illuminated area of a mov-ing diffuse object and integrated over a fixed time interval.Full advantage is taken of the statistical independencebetween the object light and reference light in our pro-posed experimental scheme, as well as the identities forzero-mean complex Gaussian random processes. However,in the present discussion, we do not make use of aprobability density function as usual in previous papers,where this function is automatically produced in the dataprocessing of integrated speckle pattern intensity varia-

Žtions. From the beginning of these techniques basically.photographic , the sampling rate severely limited the appli-

cability of the procedures, so that electronic-based methodswere considered as a solution to this problem as they arewell-suited for studying dynamic processes. Thinking in away to make intensity variations detectable, we make useof the DSPI as a means to display that information. Thistechnique, also known as TV holography, presents as amain advantage easy recording and analyzing of interfero-grams and the possibility of observing objects in near realtime. A CCD camera is used as the recording device,allowing changes to be followed at the video rate. Thispotential enables rapid changes to be recorded beforespeckle decorrelation occurs. The main operation behind

DSPI is intensity subtraction. As the object moves, andcorrespondingly the speckle field, correlation fringes areformed due to the subtraction operation, simultaneouslydisplayed at the TV monitor. For a complete description of

w xDSPI, refer to the literature 9–13 . In this paper, we limitourselves by giving a brief description of the set-up used.We demonstrate in this contribution that the visibility ofthe fringes depends, among other parameters, on the veloc-ity of the moving diffuser. We briefly outline the theoreti-cal explanation and we present experimental results con-firming our proposal.

2. Theoretical approach

The basic set-up for measurement of the velocity, foran in-plane movement of a diffuse object, is depicted inFig. 1. A beam splitter BS1 divides the laser beam in two

Ž . Ž .in order to 1 illuminate the object and 2 conform thereference beam. The image beam is brought together withthe plane reference beam via the beamsplitter BS2 into the

Ž .imaging system CCD camera . We employ coherent lightillumination of wavelength l, and we assume the object tomove with a constant speed Õ in the plane of the object.The intensity at a given point P at the image plane isdetermined by coherent superposition of the uniform refer-ence beam and the object beam. The complex field ampli-

Ž .tude, A r,t , observed at a point P with coordinate vector1Ž .r for a given instant t and the corresponding A r,tqt2

Fig. 1. Experimental set-up used for the display technique: BS1 and BS2, beam splitters, M, M1 and M2, mirrors BE1 and BS2, beamexpander devices, L collimator lens and CCD camera.

( )C. Carletti et al.rOptics Communications 160 1999 195–200 197

after the object was displaced a distance D rszt , with t

the time interval between frames, are respectively ex-pressed by:

A r ,t sa r ,t exp if r ,t qa r ,t 1Ž . Ž . Ž . Ž . Ž .Ž .1 0 0 R

A r ,tqt sa r ,tqt exp if r ,tqtŽ . Ž . Ž .Ž .2 0 0

qa r ,tqt exp id 2Ž . Ž . Ž .R

where a and f are the amplitude and phase of the0 0

speckle field, and a corresponds to the field amplitude ofR

the uniform reference beam, d is the phase differenceintroduced additionally to generate a reference set offringes. The latter is accomplished by slightly tilting the

Ž .reference mirror M Fig. 1 after acquisition of the first2

frame. This frame is used as reference for all the subtrac-tions performed in the process.

The corresponding recorded intensities, for each frame,are:

1r22< < w xI r s A r s I q I q2 I I cos f rŽ . Ž . Ž .Ž .1 1 0 R 0 R 0

3Ž .

< < 2I ryzt s A ryztŽ . Ž .2 2

1r2w xs I q I q2 I I cos f ryzt ydŽ .Ž .0 R 0 R 0

4Ž .< < 2 < < 2with I s a and I s a .0 0 R R

The intensity distribution in the resulting speckle fieldafter the DSPI process is represented by the cross-correla-tion of intensities I and I . That is, the quantity represent-1 2

ing the observed speckle pattern is:

² :J zt s I r I ryzt 5Ž . Ž . Ž . Ž .s 1 2

Assuming that the rough object obeys the stochasticGaussian random processes, which means that the specklefield behaves according to the properties of the complexcircular Gaussian statistics, then:

² :² : <² ) : < 2J zt s I I q A r A ryztŽ . Ž . Ž .s 1 2 1 2

<² : < 2q A r A ryzt 6Ž . Ž . Ž .1 2

² Ž . )Ž .:where A r A ryzt is the cross correlation func-1 2² Ž . Ž .:tion of the complex amplitude and A r A ryzt is1 2

different from zero since the amplitude distribution of thereference beam is uniform, and therefore A and A do1 2

not obey the Gaussian random process of zero mean.Explicitly,

² :² : <²J zt s I I q a r exp if r qa rŽ . Ž . Ž . Ž .Ž .Ž .s 1 2 0 0 R

= a) ryzt exp yif ryztŽ . Ž .Ž .Ž 0 0

) : < 2qa ryzt exp yidŽ . Ž ..R

<²q a r exp if r qa rŽ . Ž . Ž .Ž .Ž .0 0 R

= a ryzt exp if ryztŽ . Ž .Ž .Ž 0 0

: < 2qa ryzt exp id 7Ž . Ž . Ž ..R

And making use of the statistical independence betweenthe object beam and the reference beam and assuming theamplitude a of the speckle uniform,0

² :² :J zt s I IŽ .s 1 2

2<² < <q a r exp i f r yf ryztŽ . Ž . Ž .Ž .0 0 0

< < 2q a r exp yidŽ . Ž .R

2) : <q2Re a r a r exp i f r ydŽ . Ž . Ž .� 4Ž .0 R 0

2<²q a r expi f r qf ryztŽ . Ž . Ž .Ž . Ž .0 0 0

2q a r exp idŽ . Ž .Ž .R

qa r a r exp i f r qdŽ . Ž . Ž .Ž .Ž0 R 0

: < 2w xqexp if ryzt 8Ž . Ž ..0

Calculating, rearranging terms and taking into account that² w Ž Ž . .x: Ž .exp i f r "d s0, Eq. 8 then reduces to:0

² :2 2 ² :J zt s I q3 I q2 I IŽ .s 0 R 0 R

2 2² : ² :q I exp i f r yf ryztŽ . Ž .Ž .0 0 0

² : ²q2 I I exp i f rŽ .Ž0 R 0

:yf ryzt cosd 9Ž . Ž ..0

² :where I and I represent the average intensity distribu-0 R

tion of the object and reference beam, respectively.The correlation fringes originated from the last term ofŽ .Eq. 9 . They are associated to the phase difference d

introduced externally between the two correlated images.As a result of the shift of the object between frames, thespeckles are partially correlated, and, as a consequence,the visibility of the fringes is reduced.

Now we will briefly analyze the ensemble averageŽ .contained in Eq. 9 . It represents a measure of the correla-

tion of the speckle field scattered in the neighborhood ofthe object surface. The surface roughness properties are

Ž .described by the surface profile function h r that repre-sents the fluctuations of the surface about the mean posi-

w xtion 7,14 . We assume that the surface high fluctuationsare wide-sense stationary Gaussian random processes with

Ž .normalized autocorrelation function r zt . In general,h

the characteristics of the surface are given in terms ofŽ .standard deviation of phase s r and the correlationf

Ž . Ž .coefficient, r zt , of the function h r . The ensembleh

that was taken with respect to the Gaussian random vari-Ž .able f r has the following form:

² :G zt s exp i f r yf ryztŽ . Ž . Ž .Ž .0 0

22sexp ys 1yr zt 10w xŽ . Ž .ž /f h

where we supposed that the variance of the phase, s , isf

larger than 2p . Furthermore, for isotropic random condi-Ž .tions r zt is also of Gaussian form:h

2ztŽ .

r zt sexp y 11Ž . Ž .h 2ž /a0

( )C. Carletti et al.rOptics Communications 160 1999 195–200198

where a is the spatial correlation length. For values of zt0Ž .such that ztFa r2, then Eq. 10 is reduced to:0

2< <z t

2G zt sexp ys 12Ž . Ž .f ž /ž /a0

Ž . Ž .Eq. 12 can be employed instead of Eq. 10 since weneglect the difference introduced for s G2p , as is thef

case with the surfaces that we study here. Therefore, Eq.Ž .9 can now be expressed as:

22 2² : < <J zt s I 1q3r q2 rqG z tŽ . Ž .s 0

< <q2 rG z t cosd 13Ž . Ž .where r is the reference to the object seam intensity ratio,

² : ² :rs I r I .R 0

As we have already mentioned, the last term containsinformation about the correlation fringes. Therefore, wecan affirm that the visibility is given by:

2< <z t

2< <V z srexp ys 14Ž . Ž .f ž /ž /a0

As we can observe, for a given t , the visibility expres-sion is given as a function of the velocity. Due to thisreason, by reverse operation, it is possible to obtain themodulus of the velocity of a diffuse object undergoing amovement at a constant speed by measuring the visibilityof the fringes obtained following the DSPI process. In Fig.3, the comparison between theoretical and experimentalvisibility curve is shown.

3. Experiment and discussion

The basic configuration used to test the theoreticalresults is shown in Fig. 1. The laser beam is divided intotwo by the beam splitter BS1, then both beams are ex-

Žpanded and collimated by expander beam devices EB1.and EB2 . One beam is used to illuminate the object under

test and the corresponding scattered speckle field is im-aged onto a detector array of a CCD camera by the lens L,whereas the plane reference beam is overlapped at thedetector by means of the beam splitter BS2. The focallength of the lens is 20 cm and the magnification factor ofthe optical system is Ms1. A pupil in contact with thelens L is used to control the amount of light arriving to thedetector in order to maintain the same level of intensity inboth beams, i.e., rs1. The object in our experiments wasa metal plate mounted on a computer-controlled translationstage. Similarly, the orientation of the mirror introducingthe reference fringes was adjusted with a precision drivenmechanism.

Although this array is frequently used to detect out-of-plane displacements, the method we proposed here onlyallows the measurement of the transversal component ofvelocity vector since the changes of the visibility in the

correlation fringe are introduced for an in-plane shift. Itmust be noted that no restrictions are found on the specklesize. If the object moves along its own plane, no fringefrequency changes are seen. This is confirmed by the

Ž .Fig. 2. DSPI correlation fringes for different velocity values: acorresponds to a zero object velocity and shows high visibility

Ž .fringes while in b , which corresponds to a velocity of 6 mmrsof the object, clearly shows a decrease of the visibility of thecorrelation fringes.

( )C. Carletti et al.rOptics Communications 160 1999 195–200 199

Ž .results of Eq. 10 where the parameter d that sets theseparation between fringes does not depend on the timevariable. On the other side, if that is not the case, bymeasuring the pitch change, it is possible to get the angleof the displacement with respect to the object plane. Noisereduction techniques need not be implemented in our case.

Ž . Ž .According to Eqs. 9 and 12 , the averaged intensityŽ .2fluctuations are affected by the ratio s ra , that is thef 0

roughness properties of the object surface. In turn, the timecorrelation length of speckle intensity variations due to themoving diffuse object with constant velocity Õ is tsŽ < <.Ž .1r z a rs . The measured speed is independent of the0 f

magnification of the optical system.The velocity is directly calculated by analyzing the

visibility of the correlation fringes produced by the stan-dard DSPI technique. As it is known, the visibility dependson several parameters. Among them, it depends on thecorrelation degree between the speckle patterns stored inthe different frames. Consequently, if the object is movedafter the first exposure along a fixed time interval t , thevisibility of the speckle pattern resulting from the subtrac-tion of the first frame from the last one will depend on thevelocity due to the in-plane object displacement, which is

< < < <given by D r s z t . In Fig. 2 we can observe the differ-ence of visibility between the speckle pattern of pictureŽ . Ž .a , in this case the object velocity is zero and picture bwhich corresponds to a velocity of 6 mmrs of the object.Therefore, for each value of velocity we achieve a value ofvisibility and conversely. Following this procedure forvarious values of velocity we have calculated the curveshown in Fig. 3. In order to increase the accuracy of theexperiment, all measurements were repeated 40 times andthe resulting average value was recorded in the graph. Theintensity values were normalized with respect to theirmaximum value. The standard deviation of the visibilityvalues ranged between 0.04 and 0.09 and, as can beobserved in Fig. 3, there exists good agreement between

Ž . Žthe experimental symbols and theoretical results solid.line . We observe that the visibility of the fringes decays

rapidly when the object displacement is higher than ap-proximately 40 mm. On the other hand, the method be-comes unpractical in the range of velocities correspondingto displacements of a rough surface greater than 40 mm.For these displacements, the accuracy in the determinationof the velocities decreases.

The visibility of the fringes was calculated using thew xFast Fourier transform method 15 . This digital method,

which lets us determine the visibility value of a noisyfringe pattern, is based on the intensity peak ratio obtainedby digitally Fourier transforming the fringe pattern. As it iswell known, the Fourier transform has the property toseparate the different spatial frequencies constituting thesignal. Therefore, since the spatial frequency of the specklenoise is higher than those corresponding to the fringes, it ispossible with this method to calculate the visibility in spiteof the speckle noise. Previous to the execution of the FFT

Fig. 3. Comparison between experimental data of visibility versusŽ . Ž .velocity v and the theoretical curve solid line . The parameters

Ž .s and a in Eq. 14 take the values of 6.2 and 57 mm,f 0

respectively.

algorithm, we obtained the intensity fringe profile byaveraging the intensity in a direction parallel to the fringes.The relative error in this process depends on the ratio

Ž .between the spatial frequency in cycle per frame and thenumber of pixels in each cycle.

We set the time, t , between exposure as 2.6 s and thevelocities measurements ranged from 0 to 20 mmrs. Thevelocity measurements are limited to the decorrelationlength of the recorded speckle field and to the value of t ,which itself is restricted by the time response of the imageprocessing system.

Acknowledgements

We wish to acknowledge financial support of the Multi-Ž .purpose Optical Network MON., Trieste, Italy , CON-

ICET grant PIA 6825 and PICT 249, and Departamento deŽFisicomatematica Facultad de Ingenierıa, UNLP, Ar-´ ´

.gentina .

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