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Time-averaged electronic speckle patterninterferometry in the presence of ambient motion.Part I: Theory and experimentsThomas R. MooreDepartment of Physics, Rollins College, TMOORE@rollins.edu

Jacob J. SkubalDepartment of Physics, Rollins College

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Published InMoore, Thomas R., and Jacob J. Skubal. 2008. Time-averaged electronic speckle pattern interferometry in the presence of ambientmotion. Part I. Theory and experiments. Applied Optics 47 (25): 4640-8.

Time-averaged electronic speckle patterninterferometry in the presence of ambientmotion. Part I. Theory and experiments

Thomas R. Moore* and Jacob J. SkubalDepartment of Physics, Rollins College, Winter Park, Florida 32789, USA

*Corresponding author: tmoore@rollins.edu

Received 25 June 2008; revised 6 August 2008; accepted 8 August 2008;posted 8 August 2008 (Doc. ID 97876); published 29 August 2008

An electronic speckle pattern interferometer is introduced that can produce time-averaged interfero-grams of harmonically vibrating objects in instances where it is impractical to isolate the object fromambient vibrations. By subtracting two images of the oscillating object, rather than the more commontechnique of subtracting an image of the oscillating object from one of the static objects, interferogramsare produced with excellent visibility even when the object is moving relative to the interferometer. Thisinterferometer is analyzed theoretically and the theory is validated experimentally. © 2008 OpticalSociety of America

OCIS codes: 120.6160, 120.7280, 330.4150.

1. Introduction

Electronic speckle pattern interferometry is a well-established technique and is commonly used to studythe deflection of diffusely reflecting objects. These de-flections may be due to static displacement, transientvibrations, or continuous harmonic motion [1,2]. Thistechnique is often a desirable method of vibrationanalysis since it is sensitive to submicrometer mo-tion, is both noncontact and nondestructive, andcan be relatively inexpensive to implement.Although high sensitivity is usually a desirable

feature of interferometers, it is also problematic inspeckle pattern interferometry, because increasedsensitivity enhances the susceptibility to decorrela-tion of the interfering beams due to ambient vibra-tions. Low-frequency vibrations are notoriouslydifficult to suppress, and this problem is accentuatedwhen studying large or weakly supported objects.The problems associated with electronic speckle

pattern interferometry in the presence of ambient vi-brations have been the subject of discussion for some

time. However, the discussions usually center onmethods to mitigate the effects of vibration eitherby isolation of the system or by limiting the samplingtime of the detector [3]. While these methods aresometimes successful, implementation is usuallycostly and difficult, especially outside a laboratoryenvironment.

Recently the problem of decorrelation of the twobeams of an electronic speckle pattern interferometerdue to low-frequency ambient vibrations was ad-dressed within the context of studying the deflectionshapes of a harmonically vibrating piano soundboard[4]. In these experiments the observation of fringepatterns corresponding to the deflection shapes ofthe harmonically vibrating object was complicatedby ambient vibrations. These low-frequency vibra-tions resulted in the motion of the entire object de-spite semi-active vibration isolation of the supportmechanisms. This motion was primarily due to thefact that the interferometer and object under studyweremounted on separate supports,whichwasneces-sary due to the large size of the object. The resultingindependent motion of the object and the interferom-eter caused the speckle pattern to change during thesampling time, resulting in the decorrelation of the

0003-6935/08/254640-09$15.00/0© 2008 Optical Society of America

4640 APPLIED OPTICS / Vol. 47, No. 25 / 1 September 2008

two speckle patterns, which must be subtracted toform the interferogram.It was reported in [4] that the adverse effects attri-

butable to the decorrelation of the object and refer-ence beams could be overcome to some extent bymodifying the image subtraction algorithm. In fact,the reported algorithm only produces interferencefringes if a time-dependent phase shift exists be-tween the two interfering beams. It was also sug-gested that for more stable objects, it may beadvantageous to purposefully introduce such a phaseshift. The authors also posited a theory explaininghow this time-dependent phase shift of the beamscould be advantageous but did not fully develop it.We do so here.In what follows we discuss an electronic speckle

pattern interferometer that images harmonically vi-brating objects. After introducing the most simpleversion of the interferometer, we propose a designfor an electronic speckle pattern interferometer thatimages harmonic vibrations in the presence of ambi-ent motion. We derive a theory that describes the vis-ibility of the interferograms produced by such aninterferometer, followed by experimental evidencethat the theory accurately predicts the output. Final-ly we discuss how this interferometer can be appliedto objects undergoing harmonic motion both in thepresence of and absence of ambient motion.

2. Theory

Electronic speckle pattern interferometers have beenused for more than 35 years, and there are severaldifferent arrangements described in the literature.Here we concentrate only on the design useful forimaging harmonically vibrating objects and specifi-cally on the most simple design. Those interestedin the many imaginative and useful variations onthis simple arrangement should consult [1,2] andthe references therein.

A. Electronic Speckle Pattern Interferometer

The most simple arrangement of an electronicspeckle pattern interferometer is shown schemati-cally in Fig. 1 [5,6]. Two coherent beams are simul-taneously imaged onto a detector: the object beam,which is derived from light reflected from a diffuselyreflecting, harmonically vibrating object, and the re-

ference beam, which is directly transmitted to thedetector without being reflected from the object. Ty-pically the detector is a charge-coupled device (CCD)array with an integration time that is much greaterthan the period of the harmonic motion of the object.

In the following analysis we consider the responseof only an infinitesimal portion of the recording de-vice, and therefore we can treat both the objectand the reference beams as plane waves. Thus theintensity at a point on the detector may be describedby the usual equation for two-beam interference, i.e.,

I ¼ Iobj þ Iref þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qcosϕ; ð1Þ

where Iobj and Iref are the intensities of the object andreference beams, respectively, and ϕ is the phase dif-ference between the two beams.

Speckle pattern interferometric studies of harmo-nically vibrating objects involve a time-varyingphase difference between the object and the refer-ence beams. We follow the usual method of analysisby assuming that the surface of the object illumi-nated by the object beam moves with simple harmo-nic motion at some angular frequency ω0. The phasedifference between the object and the referencebeams during deformation of the surface of the objectcan then be expressed as

ϕ ¼ ϕ0 þ ξ sinω0t; ð2Þ

where ϕ0 is the initial phase difference between thetwo beams. Parameter ξ is the amplitude of the time-varying phase due to the motion of the object’s sur-face given by

ξ ¼ 2πΔzλ ðcos θi þ cos θrÞ; ð3Þ

where λ is the wavelength of the illuminating light,Δz is the amplitude of the object’s surface displace-ment, and θi and θr are the incident and reflected an-gles of the object beam, respectively.

Substituting Eq. (2) into Eq. (1) yields an equationfor the time-varying intensity of a point on thedetector:

I1 ¼ Iobj þ Iref þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qcos½ϕ0 þ ξ sinω0t�: ð4Þ

While Eq. (4) describes the instantaneous intensityat the detector, the interference pattern resultingfrom the coincidence of the object and the referencebeams is normally recorded by a device that has anintegration time that is long compared to the periodof the motion of the object. Therefore, the recordedimage represents a time average of the interference.Provided that the intensities of the two beams aretime independent, the intensity recorded by the de-tector can be described as

Fig. 1. Simple schematic of an electronic speckle pattern interfe-rometer. The object beam with intensity Iobj originates with lightreflected from a harmonically vibrating object. The reference beamwith intensity Iref is coherent with the object beam but has onlystatic speckle.

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hI1i ¼ Iobj þ Iref þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qhcosðϕ0 þ ξ sinω0tÞi;

ð5Þ

where the angled brackets are used to denote anaverage over the integration time of the detector. Un-less the period of the motion of the object is compar-able to or greater than the integration time of thedetector, the intensity recorded by the detector canbe well described as a time average of the two-beaminterference over the period of the object’s motion.The time-averaged portion of Eq. (5) can be expandedto yield

hcosðϕ0 þ ξ sinω0tÞi ¼cosðϕ0Þ

T

ZT

0cosðξ sinω0tÞdt

−sinðϕ0Þ

T

ZT

0sinðξ sinω0tÞdt;

ð6Þ

where T is the integration time of the detector. Thesecond integral in Eq. (6) vanishes, and the integra-tion of the remaining term reduces Eq. (6) to

hI1i ¼ Iobj þ Iref þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qcosðϕ0ÞJ0ðξÞ; ð7Þ

where J0 is the zero-order Bessel function of the firstkind [7].Although each image recorded by the detector can

be expressed in terms of simple two-beam interfer-ence, to extract the information both the object andthe reference beams must have a constant phase re-lationship. Unless this is true, phase difference ϕ0will vary as a function of position, and the resultingimage will not contain the characteristic fringe pat-terns typically associated with coherent two-beaminterference. Thus, to correctly interpret the image,one must know ϕ0 at each point on the detector.Since the object under investigation must be diffu-

sely reflecting and illuminated by coherent light, theimage will contain a random speckle pattern due tothe fact that ϕ0 varies randomly across the imageplane, and any information about the motion of ob-ject will be difficult to obtain. To alleviate this pro-blem it is normal to obtain an image before theonset of harmonic motion and subtract it from thesubsequent image described by Eq. (7). The intensityat a point on the image plane before the object is setinto simple harmonic motion is given by

I2 ¼ Iref þ Iobj þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIref Iobj

qcosϕ0; ð8Þ

and upon subtraction of the two images and subse-quent rectification, the intensity of the final interfer-ogram at a point is given by

I1;2 ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qj cosðϕ0Þ½1 − J0ðξÞ�j: ð9Þ

At this point it is useful to consider more than aninfinitesimal point on the detector. All detectors sam-ple a finite area of the interfering waves, and thephase difference between the two waves varies asa function of position, resulting in the presence ofspeckle. Even when the experimental arrangementassures that the average speckle size is equal tothe size of the detector (which is usually a single pixelon a CCD array), the size of the speckle is randomlydistributed about this mean value so that the re-sponse of the detector is normally described by anaverage over some spatially varying function ofphase angle ϕo. Additionally it is common to averageseveral measurements of the intensity made at dif-ferent times to reduce the annoying visual effects at-tributable to speckle. Since it is reasonable to assumethat the value of ϕ0 will be uniformly distributed, therecorded intensity can be approximated by an aver-age over all possible values. Alternatively, one couldrecord several images and select only the peak inten-sity from each part of the image for display. In eithercase the term cosϕo in Eq. (9) can be replaced by apositive constant.

In practice the final values derived from the detec-tor are multiplied by a constant and then displayedon a computer monitor or printed page. Therefore,from a practical standpoint, the multiplicative con-stants are not important unless they are very closeto zero. It is useful to collect all the constants intoa single parameter β, which then emphasizes thefunctional form of the final interferogram, i.e.,

I1;2 ¼ βj1 − J0ðξÞj: ð10Þ

According to Eq. (10), interferograms recorded usingthe technique described here display points with nodisplacement (ξ ¼ 0) as black, while contours of equalnonzero displacement are denoted by white or gray.Thus the maximum contrast is always unity. How-ever, due to the nature of the Bessel function, the vis-ibility of contiguous contour lines describing anantinodal region decreases as the amplitude of themotion of the surface increases. A plot of Eq. (10)is shown in Fig. 2, which clearly demonstrates thatthe fringe visibility decreases significantly with in-creasing displacement of the object.

There are numerous reports of methods to enhancethe contrast and usefulness of this type of interfe-rometer. Of particular importance are techniquesthat shift the phase of one of the beams or the objectbetween images [8–11]. Recording multiple imageswith known phase shifts between them can not onlyincrease fringe visibility, but it also allows one to de-termine the relative phases of different parts of a vi-brating object unambiguously.

One of the problems with this interferometer isthat motion attributable to ambient vibrations cansignificantly shift the phase of the object beam overthe integration time of the detector or between thetime of acquisition of the first image and any subse-quent image. Such phase shifts decorrelate the object

4642 APPLIED OPTICS / Vol. 47, No. 25 / 1 September 2008

and the reference beams, spoiling the visibility of thefinal interferograms; therefore, careful isolation fromambient vibrations is an important requirement forthis interferometer. de Groot has theoretically inves-tigated the errors that result from ambient vibrationand shown that if the noise spectrum is known, it issometimes possible to perform phase-shifting inter-ferometry without vibration isolation [12]; however,as a general rule, ambient vibrations that result insignificantmotion of the objectwill render the electro-nic speckle pattern interferometer useless. Normally,time-averaged electronic speckle pattern interfero-metry is not used for objects with significant nonhar-monic motion; instead, high-speed cameras or pulsedlasers are used to minimize the adverse effects [13].

B. Electronic Speckle Pattern Interferometer in thePresence of Ambient Motion

The necessity for careful isolation is one of the pri-mary drawbacks of the electronic speckle patterninterferometer. The relative displacement of the in-terferometer and the object (apart from the intendedharmonicmotion under investigation)must be signif-icantly less than a wavelength of the illuminatinglight, or the speckle becomes decorrelated. A lack ofsufficient isolation results in the visibility of thefringes approaching zero, and therefore, as time pro-gresses after obtaining the initial image, it is commonfor the fringes to disappear. The time between the in-itial image and the final image can often be up to sev-eral seconds, depending on the level of isolation, butwhen the object and interferometer are not mountedon the same support mechanism, the time can be re-duced to milliseconds. In this latter case the ability toview fringes becomes highly unlikely, simply becauseit takes more time to perform the necessary imageprocessing than is available before the object and re-ference beams become decorrelated.

In what follows we describe a method that allowsthe electronic speckle pattern interferometer to beused in some situations where ambient vibrationscause the relative motion of the interferometer andobject to exceed the subwavelength limit. We assumethat the relative motion of the object is due to low-frequency vibrations so that the motion over the in-tegration time of the detector is linear. That is, whilethe relative motion between the object and the inter-ferometer may be periodic, the period of the motion ismuch longer than the integration time of the detec-tor. Likewise, should the ambient motion be random,we assume that the mean time between changes inthe direction of motion is much longer than the inte-gration time of the detector.

With this assumption, the analysis presented in theprevious section ismodified by the addition of a linearterm in the phase difference between the object andthe interferometer so that it is now described by

ϕ ¼ ϕ0 þ γtþ ξ sinω0t; ð11Þ

where

γ ¼ 2ð~v ·~kÞ; ð12Þ

~v is the velocity of the object due to ambientmotion, and~k is the wave vector of the incident light.Substituting Eq. (11) into Eq. (1), we find that theaverage intensity recorded at a point on the detectoris given by

hIni ¼ Iobj þ Iref þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiIobjIref

qhcosðϕn þ γtþ ξ sinω0tÞi:

ð13Þ

Since the initial phase angle will change with eachmeasurement, we have introduced subscript n to in-dicate thatwe are referring to a specificmeasurementby the detector. In an experiment this would indicatethe specific pixel value in the nth image.

By expanding the cosine term in Eq. (13) and car-rying out the time average over the integration timeof detector T, the functional form of the interferencebecomes

hcosðϕ0 þ γtþ ξ sinω0tÞi ¼cosðϕnÞ

TZT

0cosðγtþ ξ sinω0tÞdt −

sinðϕnÞTZ

T

0sinðγtþ ξ sinω0tÞdt; ð14Þ

which is similar to Eq. (6), except that there is a lin-ear term within the arguments of the trigonometricfunctions. Unfortunately the second integral does notidentically vanish as it does without the linear termin the phase. Furthermore when γT=2π is not an in-teger, Eq. (14) has no known closed-form solution.

Fig. 2. Plot of the intensity of an electronic speckle pattern inter-ferogram versus the displacement of the object for the interferom-eter described in Subsection 2.A.

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Even without solving the integrals in Eq. (14), it isapparent that subtracting an image described byEq. (13) from an image derived before the onset ofharmonic motion will not generally result in interfer-ence fringes corresponding to contour lines of equaldisplacement of the object. The term that is linear intime serves to change the initial phase angle at theonset of integration as well as changing the phaseduring the integration process. Therefore one wouldexpect that the nonharmonic motion of the object willdecorrelate the object and the reference beams, re-sulting in an interferogram that is difficult, if not im-possible, to interpret. However, if one changes theimage processing technique, information about theharmonic motion of the object may be retrieved.To ensure that high-quality interferograms result

under these conditions, the two images that are to besubtracted must be recorded while the object is un-dergoing harmonic motion. That is, rather than sub-tracting an image of the vibrating object from onerecorded before the onset of the motion, both imagesmust be recorded after the onset of harmonic motion.In this case both images have intensities describedby Eq. (13).The ability to observe interference fringes when

both images are recorded while the object is harmoni-cally vibrating rests on theassumption that speckle inthe nodal portions of the object will have different in-tensities at different times due to the ambientmotion,while the portions of the object that are movingwith harmonic motion can still be approximated byEq. (10). Thus portions of an object whereΔz ¼ 0willappear bright, while harmonically vibrating portionswill appear as contours of dark rings.To investigate the results of this process, we con-

sider the intensity recorded by a detector at two dif-ferent times, t and t0, both occurring after the onset ofharmonic vibration of the object. The two intensitiesdenoted as Im and In are subtracted, and after recti-fication the resulting intensity is given by

Imn ¼ jhImi − hInij¼ βjhcosðϕm þ γtþ ξ sinω0tÞi

− hcosðϕn þ γt0 þ ξ sinω0t0ij; ð15Þ

where as before β is a constant that depends on thedetails of the display. Assuming that the two inten-sities are recorded by the detector contiguously intime, the change in the phase angle can be closelyapproximated by the phase difference due to the dis-placement of the object between the two images.Therefore

ϕn ¼ ϕm þ γT: ð16Þ

Under this assumption Eq. (15) can be rewritten as

Imn ¼ βjhcos½ϕm þ γTτ þ ξ sinðω0TτÞ�i− hcos½ϕm þ γTð1þ τÞ þ ξ sinfω0Tð1þ τÞg�ij;

ð17Þ

where τ ¼ t=T and is bounded by zero and unity.For the reasons cited in Subsection 2.A, it is neces-

sary to either integrate over all values of initialphase angles ϕm or assume the phase angle that pro-duces the maximum intensity, depending on how theimage is processed. Assuming that several of the fi-nal images will be averaged, Eq. (17) can be explicitlywritten as

Imn ¼ βZ

2π

0

����1TZ

1

0cos½ϕm þ γTτ þ ξ sinfω0TðτÞg�dτ

−1T

Z1

0cos½ϕm þ γTð1þ τÞ

þ ξ sinfω0Tð1þ τÞg�dτ����dϕm: ð18Þ

Should one wish to record only the highest value ofthe intensity, the average over ϕm can be replaced byassuming ϕm ¼ 0. Once this is decided the timeaverages in Eq. (17) can be numerically integratedto determine the resulting intensity at the detector,but the result depends critically on the value of γT. Aplot of Eq. (18) versus Δz=λ for several different va-lues of γT is shown in Fig. 3. The results are insen-sitive to the frequency of the harmonic vibration aslong as ωo is significantly greater than 2π=T. For thepurposes of Fig. 3, the integration time of the detec-tor is assumed to be five times the period of the har-monic vibration of the object so that ωo ¼ 10π=T.

The results shown in Fig. 3 demonstrate that de-spite the linear motion of the object, interferencefringes due to the harmonic motion will still be visi-ble. Furthermore, when compared to Fig. 2, it be-comes clear that both the precision and the

Fig. 3. Plot of the intensity of an electronic speckle pattern inter-ferogram versus the displacement of the object for the interferom-eter described in Subsection 2.B for three different values of γT.

4644 APPLIED OPTICS / Vol. 47, No. 25 / 1 September 2008

contrast of this interferometer are superior to one inwhich there is no relative motion between the objectand the interferometer.In the analysis of the system reported in [4] it was

assumed that γT ≪ ξ and ωo ≫ T−1; therefore thedifference between two contiguous measurementsof intensity could be approximated simply by repla-cing the initial phase ϕn in Eq. (15) by ϕm þ γT sothat both images can be described by Eq. (4) but withdifferent initial phases. That is, if γ is small enough,the motion of the object can be ignored during the in-tegration time of the detector, but the shift in positionof the object between measurements results in aphase shift of γT. The validity of this approximationclearly depends on the value of both γ and T, but forsmall values of γT, it can be a good approximation.Figure 4 contains plots of Eq. (18) for several valuesof small γT along with plots of the same equationwhere it is assumed that γt ¼ 0 and ϕn ¼ ϕm þ γT.Note that the approximation is extremely good forγT < 1, and depending on the application, it maybe acceptable for values exceeding γT ∼ 2.The results shown in Fig. 3 also demonstrate that a

tolerably wide range of values of γT exists that willproduce acceptable interferograms. In fact, an exam-ination of Eq. (18) reveals that the visibility of thefringes is unity for all γT > 0. However, this doesnot mean that the interferometer is useful for thatentire range, because the intensity of the interfer-ence fringes changes drastically.

The visibility remains high for values of γT > 0 dueto the fact that the minima of Eq. (18) are consis-tently zero regardless of the value of γT, althoughthe maximum intensity is cyclic in nature. Figure 5contains a plot of the maximum value of Imn as afunction of γT normalized to the maximum value,which occurs at γT ¼ 2:32. As one would expect,when γT is a multiple of 2π, the maximum intensityis reduced to zero; however, with the exception of va-lues that are close to integer multiples of 2π, the in-terferometer can be used for almost any value of γT.The ability to actually utilize the interferometer forlarge values of γT depends on the noise floor of thedetector. Therefore one may assume that the highestquality interferogram is the one in which the inten-sity is the greatest whenΔz ¼ 0. This is easily calcul-able and also provides a relatively simple method forverifying the validity of Eq. (18).

By considering only the special case of viewing no-dal regions of the object (Δz ¼ 0), Eq. (17) reduces to

ImnðΔz ¼ 0Þ ¼ βjhcos½ϕm þ γTτ�i− hcos½ϕm þ γTð1þ τÞ�ij: ð19Þ

Equation (19) indicates that if there is no linearmotion of the object, γ ¼ 0, the pixel intensity of in-terferograms at nodal regions is identically zero re-gardless of the value of Δz. However, any nonzerovalue of γT will result in a nonzero value of Eq. (19).

Fig. 4. Comparison of the exact and approximate solution to Eq. (18) for four different values of γT.

1 September 2008 / Vol. 47, No. 25 / APPLIED OPTICS 4645

Since the visibility remains high regardless of thevalue of γT, optimizing the output of the interferom-eter becomes an issue of maximizing Eq. (19). Carry-ing out the time averages in Eq. (19) explicitly leadsto an equation for the intensity at nodal regions thatis only dependent upon γT:

Imn ¼ βγT

Z2π

0j2 sinðϕm þ γTÞ − sinðϕm þ 2γTÞ

− sinðϕmÞjdϕm: ð20Þ

Note that, as before, we have integrated Eq. (20) overall possible values of ϕm. This describes the intensityof the region of interest when the speckle is notresolved by the detector, or equivalently, when theregion of interest is comprised of many detectors(as may be the case when the detector is a two-dimensional array).Maximizing Eq. (20) in terms of γT provides an in-

dication of the optimum value for actual use. Also,plotting Imn versus γT provides a description of thebehavior of the interferometer that can be easilymea-sured. This will be discussed in the following section.

3. Experiments

To verify the validity of the above analysis, we con-structed an electronic speckle pattern interferometershown schematically in Fig. 6 [5,6]. The beam from aHe–Ne laser with a wavelength of 632:8nm was splitinto two beams by directing it toward a half-waveplate and polarizing beam splitter (PBS). One ofthe beams was directed through an expanding lensand toward a metal plate sprayed with white paint.The second beam was used as a reference beam, andafter the plane of polarization was rotated by a half-wave plate, it was directed toward a beam expandinglens and then onto a piece of opal glass. The referencebeam and the object were imaged onto a CCD arraysimultaneously using a commercially available lens

and a beam splitter. The integration time of the arrayprovided by the manufacturer was T ¼ 0:0313 s.

To introduce a known time-varying phase shiftonto one of the beams, one of the mirrors used to re-flect the reference beam was mounted on a piezoelec-tric transducer (pzt), which was driven by atriangular waveform. Thus, although the objectwas not moving relative to the interferometer, themovement could be simulated by a movement ofthe mirror, and the value of γT could be known pre-cisely. Under these conditions the intensity of the in-terferogram of the object is given by Eq. (20).

The actuated mirror was driven over a range of ve-locities, and the average intensity of a region of therecorded interferograms was determined for each ve-locity. To determine the average intensity of the in-terferogram, a rectangular region containing 6622pixels was used. Intensities of pixels within the se-lected region were averaged, and the average pixelintensity over this region was recorded for each inter-ferogram. For each velocity of the mirror, the averagepixel intensity over the selected region was deter-mined for fifteen separate interferograms. Figure 7contains a plot of these measurements versus γT,along with the theoretical predictions of Eq. (20).

Fig. 6. Diagram of the experimental arrangement.

Fig. 5. Normalized plot of the maximum value of Imn versus γT.Note that there will almost always be some steady-state interfer-ence observable as long as γT is not an even-integer multiple of π.

Fig. 7. Comparison of the average intensity of a nodal region ofan object (Δz ¼ 0) to the predictions of Eq. (20).

4646 APPLIED OPTICS / Vol. 47, No. 25 / 1 September 2008

Note the excellent agreement between the measure-ments and the theory.It is also useful to compare the images of this type

of interferometer with the more common type de-scribed in Subsection 2.A. This provides a vivid ex-ample of the ability of the interferometer to workunder the conditions described above as well as de-monstrating the advantages of inserting a time-vary-ing phase shift in one beam if there is little motion ofthe object due to effective vibration isolation.Figure 8 contains two images of a 13 cm diameter

flat circular plate vibrating in one of the fundamentalmodes. The oscillation of the plate was driven acous-tically by a speaker placed approximately 1m away.The amplitude of the vibrations of the antinodesare approximately 0:5 μm. Figure 8(a) contains an in-terferogramobtained in themanner described inSub-section 2.A; Fig. 8(b) contains an interferogramobtained in the manner described in Subsection 2.Bwith the total phase shift of thebeambeing equivalentto a linear displacement of the plate of approximatelyλ=4 over the time of the exposure. The increased pre-cision and visibility are obvious.

4. Discussion

The work discussed above clearly demonstrates thatit is possible to perform time-averaged electronicspeckle pattern interferometry in the presence ofambient motion, provided that the motion can be as-sumed to be linear over the integration time of thedetector. Experience has proved this to be the usualcase for many objects that are not actively isolatedfrom the environment when the integration time of

the detector is of the order of 0:03 s. Furthermoreif low-frequency ambient vibrations are not present,it can be advantageous to introduce a time-varyingphase shift into one beam of the interferometer toincrease the fringe visibility and precision of theinterferometer. The data shown in Fig. 7 indicatethat the maximum intensity of a nodal region is pro-duced for γT ¼ 2:4� 0:1, which is in good agreementwith the value of γT ¼ 2:32 predicted by Eq. (20).These results also demonstrate the importance ofthe noise floor of the detector. The noise floor ofthe detector used for the experiments describedabove is normal for an inexpensive, uncooled CCD ar-ray. However, despite the obvious noise inherent inthe detector, the images shown in Fig. 8 demonstratethat excellent interferograms can be produced.

Numerical integrations of Eq. (18) have also shownthat it is not necessary to completely restrict the am-bientmotion such that it is linear over the integrationtime of the detector. Depending on the magnitude ofthemotion, interference fringesmay be visible for am-bient harmonic motion up to a large fraction of ωo,although the relationship between the fringes andthe displacement of the object may become compli-cated. Likewise, if the ambient motion is complexor random, but so rapid that Eq. (11) is not a good ap-proximation, averaging several interferograms willstill produce an image with fringes correspondingto contours of equal displacement.

Finally, due to the manner in which most detectorsoperate, the theory presented in Subsection 2.B as-sumes that the initial phase angle between thetwo subtracted images is γT [Eq. (16)]; however, thisneed not necessarily be the case. In some cases de-coupling the initial phase angle from the period ofintegration of the detector can offer possibilities tomaximize the contrast of the final image. By not as-suming the phase angle is defined by Eq. (16), butinstead that the initial phase angle ϕn can be con-trolled through a judicious choice of time delay be-tween images, the initial phase of the nth imagecan be described by

ϕn ¼ ϕm þ γT0; ð21Þwhere T0 need not be related to the integration timeof the detector. In some circumstances this may offeran additional opportunity to maximize the intensityof nodal regions of the interferogram, but if the finalresult is the average of many images, the time delaybetween images becomes unimportant, and the re-sults do not depend critically on the value of T0.

5. Conclusion

In the past the use of electronic speckle pattern inter-ferometry has typically been restricted to laboratorysettings due to the stringent requirements for vibra-tion isolation. The experimental arrangement de-scribed here relaxes this requirement and makes itpossible to use speckle pattern interferometry in avariety of situations where vibration isolation is

Fig. 8. Electronic speckle pattern interferograms of a flat circularplate oscillating in one of the normal modes obtained using theinterferometers described in (a) Subsection 2.A and (b) Subsec-tion 2.B. The numbers in the lower right corner of the interfero-grams indicate the frequency of oscillation (1045:69� 0:01Hz).

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complicated or expensive. This work does not indicatethat the need for vibration isolation can alwaysbe eliminated; high-frequency ambient vibrationsshould still be minimized to produce the maximumfringe visibility. However, because much of thedifficulty in using electronic speckle pattern inter-ferometry to study large, weakly supported, orindependently supported objects can be traced tolow-frequency motion, the method described herecan often be used to produce interferograms in situa-tions heretofore deemed too difficult to attempt.We note in closing that there is no reason to limit

the application of this interferometer to cases wheredegradation of the image is due to terrestrial motionas has been the emphasis here. This interferometermay have application in instances where the pre-sence of any source of optical path variation makesinterferometry of a harmonically vibrating object dif-ficult. Provided the path variation meets the criteriaspecified above, the interferometer can be used in thepresence of environmental disturbances such as airflow or thermal drift and can eliminate the deleter-ious effects attributable to such things as the inad-vertent motion of optical components.

References1. R. Jones and C. Wykes, Holographic and Speckle Interferome-

try, 2nd Ed. (Cambridge, 1989).2. P. K. Rastogi, Speckle Pattern Interferometry and Related

Techniques (Wiley, 2001).

3. D. Findeis, D. R. Rowland, and J. Gryzagoridis, “Vibration iso-lation techniques suitable for portable electronic speckle pat-tern interferometry,” Proc. SPIE 4704, 159–167 (2002).

4. T. R. Moore and S. A. Zietlow, “Interferometric studies of apiano soundboard,” J. Acoust. Soc. Am. 119, 1783–1793(2006).

5. T. R. Moore, “A simple design for an electronic speckle patterninterferometer,” Am. J. Phys. 72, 1380–1384 (2004).

6. T. R. Moore, “Erratum: a simple design for an electronicspeckle pattern interferometer,” Am. J. Phys. 73, 189(2005).

7. M. Abramowitz and I. A. Stegun, eds.,Handbook of Mathema-tical Functions (Dover, 1972).

8. S. Nakadate, “Vibration measurement using phase-shiftingspeckle-pattern interferometry,” Appl. Opt. 25, 4162–4167(1986).

9. K. A. Stetson and W. R. Brohinsky, “Fringe-shifting techniquefor numerical analysis of time-averaged holograms of vibrat-ing objects,” J. Opt. Soc. Am. A 5, 1472–1476 (1988).

10. C. Joenathan, “Vibration fringes by phase stepping on an elec-tronic speckle pattern interferometer: an analysis,” Appl. Opt.30, 4658–4665 (1991).

11. C. Joenathan and B. M. Khorana, “Contrast of the vibrationfringes in time-averaged electronic speckle-pattern interfero-metry: effect of speckle averaging,” Appl. Opt. 31, 1863–1870(1992).

12. P. J. de Groot, “Vibration in phase-shifting interferometry,”J. Opt. Soc. Am. A 12, 354–365 (1995).

13. A. J. Moore, J. D. C. Jones, and J. D. R. Valera, “Dynamicmeasurements,” in Speckle Pattern Interferometry andRelated Techniques, P. K. Rastogi, ed. (Wiley, 2001),pp. 225–288.

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