Optics and Lasers in Engineering 50 (2012) 170–175

Contents lists available at SciVerse ScienceDirect

Optics and Lasers in Engineering

0143-81

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/optlaseng

A fast method for measuring nanometric displacements by correlatingspeckle interferograms

Lucas P. Tendela a,n, Gustavo E. Galizzi a, Alejandro Federico b, Guillermo H. Kaufmann a,c

a Instituto de Fısica Rosario, Blvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentinab Electronica e Informatica, Instituto Nacional de Tecnologıa Industrial, P.O. Box B1650WAB, B1650KNA San Martın, Argentinac Centro Internacional Franco Argentino de Ciencias de la Informacion y de Sistemas, Blvd. 27 de Febrero 210 bis, S2000EZP Rosario, Argentina

a r t i c l e i n f o

Article history:

Received 26 July 2011

Received in revised form

21 September 2011

Accepted 25 September 2011Available online 8 October 2011

Keywords:

Speckle interferometry

Phase evaluation

Order statistics correlation coefficient

Nanometric displacements

66/$ - see front matter & 2011 Elsevier Ltd. A

016/j.optlaseng.2011.09.011

esponding author. Tel.: þ54 3414853222x12

ail address: tendela@ifir-conicet.gov.ar (L.P. T

a b s t r a c t

A phase evaluation method was recently proposed to measure nanometric displacements by means of

digital speckle pattern interferometry when the phase change introduced by the deformation is in the

range ½0,pÞ rad. To evaluate the phase change, however, it is necessary to record separately the

intensities of the object and the reference beams corresponding to both the initial and the deformed

interferograms. This paper presents a fast approach that overcomes this limitation. The rms phase

errors introduced by the proposed method are determined using computer-simulated speckle inter-

ferograms and its performance is also compared with the results obtained with a phase-shifting

technique. An application of the proposed phase retrieval method to process experimental data is

finally illustrated.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Micro-electro-mechanical systems (MEMS), micro-sensors andmicro-optical devices are found in a wide range of technical compo-nents which are applied from the aerospace industry to medicine.Over the last few years, the tendency toward miniaturization that hasbeen observed in the electro-mechanical industry has generated aconsiderable demand of advanced non-invasive measurement tech-niques to test micron-sized systems, not only during the design timebut also along the entire manufacture process. As mechanical proper-ties determined on much larger specimens cannot be scaled downfrom bulk materials without any experimental verification, testingmethods for deformation measurement in micro- and nanoscaleobjects are of considerable importance for the future developmentof new micro-system devices.

Whole-field optical techniques can be used as useful tools totest micro-system devices due to their advantages, which includerobustness, high processing speed, and also non-contact and non-destructive nature [1–6]. One of these optical techniques is digitalspeckle pattern interferometry (DSPI), which has a high sensitiv-ity and also has been widely used for the measurement ofdisplacement and strain fields generated by rough object sur-faces [7]. This technique is based on the evaluation of the opticalphase changes that are coded in speckle interferograms recorded

ll rights reserved.

0; fax: þ54 3414808584.

endela).

for different states of a specimen, which are usually displayed inthe form of fringe patterns. A review of some applications of DSPIin the inspection of microcomponents can be found in Ref. [8].

In practical applications of DSPI, the phase-shifting and theFourier-transform methods are the most common techniquesused to retrieve the phase distribution introduced by the defor-mation [9]. As it is well known, phase-shifting methods have highaccuracy and the sign ambiguity is resolved automatically due tothe recording of multiple interferograms. However, a mirrordriven by a linear computer-controlled piezoelectric transducermust be introduced in the optical setup, thus generating anadditional technical complexity. Moreover, these algorithmsassume that phase shifts between successive frames are all equal,which can be difficult to obtain experimentally. Phase shiftermiscalibrations and vibrations during the acquisition of multiplespeckle interferograms also produce systematic errors whichmust be appropriately addressed [10]. On the contrary, the Four-ier transform method has the advantage of requiring the acquisi-tion of only two speckle interferograms to be analyzed.Nevertheless, when the phase changes are non-monotonous, thismethod also needs the introduction of spatial carrier fringes toovercome the sign ambiguity [9]. Although there exists simpleways of introducing spatial carrier fringes in the optical setup,such as tilting the reference beam between the acquisition of bothspeckle interferograms to be correlated, this procedure alsocomplicates the automation of the interferometer operation.

Since the development of DSPI, various authors have presenteddifferent approaches to generate speckle correlation fringe patterns.

L.P. Tendela et al. / Optics and Lasers in Engineering 50 (2012) 170–175 171

Among these works, we can mention Ref. [11] in which fringepatterns are generated by calculating the local correlation betweentwo speckle patterns acquired before and after translation of theobject. This approach has the advantage that the illumination over thesurface of the object need not be perfectly uniform. However, in allthese approaches no attempts were done to use a correlationcoefficient to evaluate the phase distribution.

A novel phase evaluation method was recently proposed tomeasure nanometric displacements by means of DSPI when thephase change introduced by the specimen deformation is in therange ½0,pÞ rad, that is when the generated correlation fringesshow less than one fringe [12]. In this case, the wrapped phasemap does not present the usual 2p phase discontinuities, so thatit is not necessary to apply a spatial phase unwrapping algorithmto obtain the continuous phase distribution. It must be noted thatcases of correlation fringe patterns presenting less than one fringecan appear quite frequently when micro-systems are inspected.This phase retrieval method is based on the calculation of thelocal Pearson’s correlation coefficient between the two speckleinterferograms generated by both deformation states of theobject. Although this approach does not need the introductionof a phase-shifting facility or spatial carrier fringes in the opticalsetup, the intensities of the object and the reference beamscorresponding to both the initial and the deformed interfero-grams must be recorded. It should be noted that this limitationcomplicates the automation of the interferometer operation.Moreover, this limitation does not allow the application of thismethod for the analysis of non-repeatable dynamic events byrecording a sequence of interferograms throughout the entiredeformation history of the testing object.

In this paper we present a phase retrieval approach based onthe original method, that is the one reported in Ref. [12], whichovercomes the above mentioned limitation. Therefore, there is noneed to record the intensities of the object and the referencebeams corresponding to both the initial and the deformed inter-ferograms. Moreover, as the order statistics correlation coefficientis used instead of the Pearson’s coefficient, the phase distributioncan be evaluated much faster.

In the following section, a description of the fast phaseretrieval method is presented. Afterwards, the performance ofthe proposed method is analyzed using computer-simulatedspeckle interferograms for the case of out-of-plane displacements.This analysis allows us to evaluate the rms phase errors intro-duced by the fast approach and also to compare its performancewith the one given by a phase-shifting algorithm. Finally, anapplication of the phase retrieval method to process experimentaldata is also illustrated.

2. Fast phase retrieval method

As it is well known, DSPI is based on the recording of thecoherent superposition of two optical fields, being at least one ofthem a speckle field generated by the scattered light coming fromthe rough surface of the specimen. The result of the superpositionis another speckle field called interferogram and its intensity I canbe expressed as [7]

I¼ I1þ I2þ2ffiffiffiffiffiffiffiffiI1I2

pcosðf1�f2Þ ¼ I0þ IM cosðfÞ, ð1Þ

where I1 and I2 are the intensities of the object and the referencefields and f1 and f2 are their associated phases, respectively,I0 ¼ I1þ I2 is the intensity bias, IM ¼ 2ðI1I2Þ

1=2 is the modulationintensity, and f¼f1�f2 accounts for the optical path differencefrom the light source to the observation point considered.

If the scattering surface undergoes a deformation, the resultingintensity changes accordingly. The intensities Ia and Ib corresponding

to the speckle interferograms recorded in the initial (a) and thedeformed states (b), respectively, are determined by

Ia ¼ Ia0þ IaM cos fa ¼ Ia0þ IaM cos fs,

Ib ¼ Ib0þ IbM cos fb ¼ Ib0þ IbM cos ðfsþDfÞ, ð2Þ

where fs ¼fa accounts for the random change in the optical pathdue to the roughness of the scattering surface and Df¼fb�fa

corresponds to the deterministic change in the path introduced bythe underwent deformation.

When the deformation produces a phase change lower than p,in Ref. [12] it is shown that the deterministic phase change Dfcan be related to the Pearson’s correlation coefficient CP betweenthe two interferograms described by Eq. (2), by means of

CPðIa�Ia0,Ib�Ib0Þ ¼/IaMIbMSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/I2

aMS/I2bMS

q cos Df, ð3Þ

being

CP ¼/ðIa�Ia0�/Ia�Ia0SÞðIb�Ib0�/Ib�Ib0SÞS

½ð/ðIa�Ia0Þ2S�/Ia�Ia0S

2Þð/ðIb�Ib0Þ

2S�/Ib�Ib0S2Þ�1=2

, ð4Þ

where the operator / S is evaluated over a sliding window oneach recorded image. It must be pointed out that the spatialcoordinates of the pixel (m, n) at the CCD, with m,n¼ 1, . . . ,Nbeing N the number of pixels along the horizontal and verticaldirections, were omitted intentionally for the sake of clarity.

Taking into account that Ia�Ia0 ¼ IaM cos fa andIb�Ib0 ¼ IbM cos fb, and that the intensity and phase of fullydeveloped and polarized speckle fields are statistically indepen-dent, the phase change Df needed to determine the nanometricdisplacements can be evaluated by inverting Eq. (3) [12]

Df¼ acos ½CPðIa�Ia0,Ib�Ib0Þf �, ð5Þ

where acos½ � is the inverse of the cosine function and f is definedas

f ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/I2

aMS/I2bMS

qIaMIbM

� � : ð6Þ

From Eq. (5) it is clearly seen that the intensities of the object andthe reference beams corresponding to both the initial and thedeformed interferograms must be separately recorded, a proce-dure which complicates the automation of the interferometeroperation.

In order to overcome the above mentioned limitation forretrieving the phase change, here we propose the evaluation ofan estimator gDf which is based on the introduction of twoapproximations in Eq. (5). The first approximation is to ignorethe influence of the intensity bias of both the initial and thedeformed states. The second approximation is to assume thatf¼1. The validity of both approximations will be demonstrated inSections 4 and 5 by using numerical simulations and alsoexperimental data. Finally, to decrease the computing timeneeded to calculate the correlations in Eq. (5), the Pearson’scoefficient is replaced by the the order statistics correlationcoefficient COS [13].

Therefore, taking into account both approximations and usingthe order statistics correlation coefficient COS, the estimated phasechange gDf introduced by the deformation will be given bygDf ¼ acos½COSðIa,IbÞ�: ð7Þ

Below we will describe how the correlation between two seriesðxi,yiÞ of length N can be evaluated by means of the order statisticscorrelation coefficient COS [13]. As this correlation coefficient isbased on order statistics and rearrangement inequality, its calcu-lation implies that the series must be rearranged pairwise first

L.P. Tendela et al. / Optics and Lasers in Engineering 50 (2012) 170–175172

with respect to the magnitudes of x. In this way, two new seriesðxðiÞ,y½i�Þ are obtained. The magnitudes of xðiÞ are called the orderstatistics of x, which are based on ranking those magnitudes inascending order, and y½i� are known as the associated concomi-tants. In a similarly way, the order statistics yðiÞ are obtained fromthe magnitudes of y.

Finally, the order statistics correlation coefficient COSðx,yÞbetween the two series x and y is defined as

COSðx,yÞ ¼

PNi ¼ 1½xðiÞ�xðN�iþ1Þ�y½i�PNi ¼ 1½xðiÞ�xðN�iþ1Þ�yðiÞ

: ð8Þ

To compute the correlation between both recorded interfero-grams Ia and Ib using the order statistics correlation coefficientdefined in Eq. (8), the calculation is performed pixel by pixel. Theprocess consists simply of moving a sliding window of size L� L

from point to point in the horizontal and vertical directions. Foreach pixel, two new subimages IaL and IbL of size L� L are obtainedfrom Ia and Ib, respectively.

Before evaluating the correlation, both subimages must beexpressed in vector form. That is to say, the first L elements of thevector ia are constructed by using the elements in first row of IaL,the next L elements from the second row, and so on. The resultingvector will have dimensions L2

� 1. In the same way, the vector ib

is constructed from the subimage IbL.Once the subimages are expressed in vector form, the ele-

ments of both vectors are rearranged pairwise with respect to theelements of ia and two new vectors are obtained, whose elementsare iaðiÞ and ib½i�. The magnitudes iaðiÞ are the order statistics of ia

and ib½i� are the associated concomitants. Then, the order statisticsibðiÞ are constructed from the vector ib.

Finally, for each pixel (m,n) the correlation coefficient COSðIa,IbÞ

between the two recorded interferograms Ia and Ib can beevaluated as

COSðIa,IbÞ ¼

PL2

i ¼ 1½iaðiÞ�iaðL2�iþ1Þ�ib½i�PL2

i ¼ 1½iaðiÞ�iaðL2�iþ1Þ�ibðiÞ

: ð9Þ

As mentioned before, the number of pixels along the horizontaland vertical directions were omitted for the sake of clarity. It isimportant to note that this method does not generate phasevalues over the first and the last L/2 pixels.

3. Numerical simulations

To evaluate the performance of the fast phase retrievalmethod, computer-simulated speckle interferograms were gener-ated using the same approach applied in Ref. [12]. The numericalmodel was originally developed to generate DSPI fringes pro-duced by an out-of-plane interferometer [14], in which the firstand second statistic of the intensity are well characterised. Thesimulation is based on the imaging properties of a 4f opticalsystem [14]. Any given point in the detector plane receivescontributions from all the scattering centres in the input plane,which are large enough so that the complex amplitude isdescribed by the usual Gaussian first-order speckle statistics.The first lens performs a Fourier transform to the aperture planein which a circular aperture acting as a low pass filter representsthe contribution of the lens pupil in the final intensity distribu-tion. The second lens performs another Fourier transformationwhich gives the complex amplitude at the detector plane. Withthis model, the intensity Ii of the speckle interferograms recordedin the initial (i¼a) and the deformed (i¼b) states can besimulated as

Ii ¼ 9R expðjaÞþF�1HFfexp½jðfsþbDfoÞ�g92, ð10Þ

where R and a are the amplitude and phase of the reference beam,respectively, j is the imaginary unit, fs is a random variable withuniform distribution in ð�p,p�, F and F�1 denote the direct andinverse 2-D Fourier transform, respectively, b¼ 0 when i¼a andb¼ 1 when i¼b, and Dfo is the phase change introduced by thedeformation. H is a circular low pass filter defined as

HðrÞ ¼1, rrD=2,

0, r4D=2,

(ð11Þ

where D is the pupil diameter and r is the modulus of theposition vector in the pupil plane.

In all numerical tests, each speckle interferogram was simu-lated for a resolution of N2

¼ 512� 512 pixels in a scale of 256gray levels with an average speckle size of 1 pixel.

As previously described, the simulation method allows us toknow precisely the original phase distribution. Therefore, theerrors generated by the approximations mentioned in the pre-vious section can be evaluated. The phase change gDf estimatedwith the proposed method was compared with the original phasedistribution Dfo, allowing to evaluate the rms phase error s from

s¼ 1

N2

XN

m ¼ 1

XN

n ¼ 1

½eðm,nÞ� eh i�2( )1=2

, ð12Þ

where the error matrix eðm,nÞ is defined by

eðm,nÞ ¼ 9gDfðm,nÞ�Dfoðm,nÞ9, ð13Þ

being eh i the mean value of the matrix e, and m and n the spatialcoordinates of the pixel along the horizontal and vertical direc-tions, respectively.

Additionally, the performance of the proposed phase retrievalmethod was also compared with the one given by the Carrephase-shifting algorithm [9]. As it is well known, this techniqueinvolves the introduction in the object beam of the interferometerof four additional equal phase shifts g between the successiveframes. Therefore, the intensity of the two sets of four phase-shifted speckle interferograms recorded for the initial anddeformed states were simulated by adding the term gk ¼ ðk�1Þgwith k¼ 1, . . . ,4 to the random variable fs in Eq. (10).

In practice, the actual phase changes gk can differ by a smallamount e from the expected phase shift. In order to representmore realistic experimental conditions, in the simulation we havealso introduced an error due to miscalibrations of the phase-shifting device. Therefore, e was chosen as 0.012 to consider alinear phase-shifter miscalibration of 1% and the phase shiftvalues gk were selected as g1 ¼ 0, g2 ¼ gð1þeÞ, g3 ¼ 2gð1�eÞ,g4 ¼ 3gð1þeÞ.

4. Numerical results

As a typical example, Fig. 1 shows the plot of the retrievedphase distribution estimated with the proposed approach (boldcurve) along a direction crossing the center of the pattern for anout-of-plane parabolic displacement. In this example, the correla-tion was evaluated using a sliding window of size L2

¼ 65� 65pixels and the highest phase change between the initial anddeformed states was chosen as Df¼ p=10 rad. For comparison,the same figure also displays the original input phase map (thincurve) and the phase distribution (solid circles) retrieved bymeans of the Carre phase-shifting technique.

Furthermore, the term f defined in Eq. (6) was also computedfor each pixel (m, n) in the example displayed in Fig. 1. Theobtained results showed that the rms deviation evaluatedbetween f and the approximated value of 1 was 1� 10�5. There-fore, the substitution of f¼1 in Eq. (5) can be considered as very

0 100 200 300 400 500

0.15

0.2

0.25

0.3

Pixels

Pha

se (r

ad)

Fig. 1. Comparison between the original phase change (thin curve), the retrieved

phase distribution estimated with the fast method and a sliding window of size

65�65 (bold curve), and the Carre phase-shifting technique (solid circles), for a

simulated out-of-plane parabolic displacement.

Table 1rms phase error s and processing times t needed to compute the correlation

coefficient for simulated out-of-plane linear tilts.

Df(rad)

Fast method Original method Carre phase-

shifting technique

33�33

pixels

65�65

pixels

33�33

pixels

65�65

pixels

s(rad)

t

(s)

s(rad)

t

(s)

s(rad)

t

(s)

s(rad)

t

(s)

s (rad)

p=10 0.0034 48 0.0022 48 0.0030 67 0.0014 190 0.0033

p=4 0.0095 46 0.0051 49 0.0079 68 0.0043 189 0.0073

p=2 0.0151 47 0.0119 49 0.0127 67 0.0069 189 0.0127

Table 2rms phase error s and processing times t needed to compute the correlation

coefficient for simulated out-of-plane parabolic displacements.

Df(rad)

Fast method Original method Carre phase-

shifting technique

33�33

pixels

65�65

pixels

33�33

pixels

65�65

pixels

s(rad)

t

(s)

s(rad)

t

(s)

s(rad)

t

(s)

s(rad)

t

(s)

s (rad)

p=10 0.0042 48 0.0033 54 0.0037 67 0.0023 191 0.0029

p=4 0.0097 47 0.0080 49 0.0086 67 0.0055 190 0.0057

p=2 0.0176 49 0.0156 50 0.0142 66 0.0090 191 0.0092

L.P. Tendela et al. / Optics and Lasers in Engineering 50 (2012) 170–175 173

accurate. Additionally, the rms phase error s was also used tocompare the performance of the proposed phase retrievalapproach with the one given by the Carre phase-shifting techni-que. In the above mentioned example, it was observed that theperformance given by both phase evaluation methods were quitesimilar, as the rms phase error obtained with the proposedapproach was s¼ 0:0033 rad and with the Carre phase-shiftingtechnique was s¼ 0:0029 rad. For comparison, the rms phaseerror was also computed using the original method, givings¼ 0:0023 rad. Therefore, this example clearly shows that theinfluence of the intensity bias of the initial and the deformedstates in Eq. (5) is quite small.

A last important observation is related to the computing timeneeded to evaluate a phase distribution. This example demon-strated that the fast phase retrieval method that uses the orderstatistics correlation coefficient was approximately four timesfaster than the previous approach presented in Ref. [12].

As mentioned in Section 2, Fig. 1 shows that the proposedmethod does not generate phase values over the first and last L/2pixels. The same limitation is given by the method reported inRef. [12] As it was mentioned in this last paper, it is not a trivialissue to ascertain the window size that must be used as itdepends on the shape of the phase distribution to be evaluated.In general, larger sliding windows are usually preferred to smooththe retrieved phase map to be obtained. However, larger windowsalso tend to reduce small bumps that may appear in the phasemap. It must be taken into account that as the sliding window isdisplaced pixel by pixel in the horizontal and vertical directions,the loss of information that is produced is negligible but theimprovement in the signal-to-noise ratio is quite high.

The performance of the fast phase retrieval approach was alsoanalysed using different phase distributions, phase amplitudesand window sizes. Some of the results obtained from thisnumerical analysis are summarized in Tables 1 and 2. Severalinteresting observations emerge from the numerical tests. First, itshould be noted that the rms phase error strongly depends on thewindow size used to retrieve the phase distribution, that is therms phase error decreases when the size of the sliding windowincreases. As mentioned before, larger sliding windows areusually preferred to smooth the retrieved phase map but theyalso tend to reduce small bumps that may appear. Looking thenumerical results listed in Tables 1 and 2, it is also observed thatthe rms phase error generated by the simplified approach only areslightly higher than the values obtained with the original method.However, in general these errors are quite similar to the onesobtained with the Carre phase-shifting technique. It is also seen

that although the rms phase errors generated by the three phaseretrieval approaches are quite low, they increase with thedeformation amplitude. It is also important to mention thatthrough the numerical analysis performed using computer-simu-lated speckle interferograms, it was demonstrated that the mini-mum and maximum phase values that can be obtained with anacceptable rms phase error by using the fast method are approxi-mately p=100 rad and p=2 rad, respectively.

Finally, Tables 1 and 2 also list the processing times needed forthe proposed approach and the original method to compute thecorrelation coefficient when different sliding windows are used.These results show that when a sliding window of size 65�65pixels is used, the processing time needed by the fast method isapproximately four times shorter than the original method if a2.67 GHz Intel Quad Core personal computer is utilised.

5. Experimental results

To illustrate the performance of the fast phase retrievalmethod when experimental data are processed, a DSPI systemwas used to measure the out-of-plane displacement component w

generated by a circular aluminium plate clamped along its edge,when it was deformed approximately at its center using adifferential micrometer. Fig. 2 shows a schematic view of theDSPI system, which was based on a conventional out-of-planespeckle interferometer illuminated by a Nd:YAG laser with awavelength l¼ 532:8 nm, which was divided into the object andreference beams by a beam-splitter (BS). The reference beam wasexpanded by a microscope objective (L) and directed throughanother beam splitter into the CCD camera (PulnixTM-765),where it was recombined with the light scattered by the objectsurface. In order to obtain a uniform illumination intensity, a pinhole (PH) was used in both object and reference beams. The anglebetween the direction of illumination and the normal to thesurface of the object was g¼ 101. The camera output was fed to aframe grabber located inside a personal computer that digitizes

L

L

PH

PH

Aluminium plate

PT

M

CCD

BS

BS

M

Laser

CL

DM

Fig. 2. Optical arrangement of the out-of-plane digital speckle pattern interfe-

rometer: mirrors (M), piezoelectric transducer (PT), microscope objectives (L),

beam splitters (BS), pin holes (PH), camera lens (CL), differential micrometer (DM).

0 5 10 15 200

5

10

15

20

25

30

35

40

45

x (mm)

w (n

m)

Fig. 3. Out-of-plane displacement component w obtained using the fast method

with a sliding window of size 65�65.

05

1015

20

05

1015

20

0

10

20

30

40

50

x (mm)y (mm)

w (n

m)

Fig. 4. Out-of-plane displacement field w obtained using the fast method with a

sliding window of size 65�65.

L.P. Tendela et al. / Optics and Lasers in Engineering 50 (2012) 170–175174

the images in grey levels with a resolution of 512�512 pixels�8bits. The video camera had a zoom lens which allows to image asmall region of the specimen of approximately 23�23 mm2 insize and for an average speckle size of 1 pixel.

A piezoelectric transducer (PT) attached to one of the mirrors(M) and driven by an electronic unit was used to introduce thephase shifts. This phase-shifting facility enabled to evaluate thephase distribution using the Carre phase-shifting technique,which later was compared with the phase map obtained usingthe fast method. With both phase retrieval methods, the fastapproach and the Carre phase-shifting technique, the out-of-plane displacement component w was determined using [15]

w¼l

2pð1þcosgÞgDf: ð14Þ

Fig. 3 depicts the plot of the out-of-plane displacement compo-nent w measured with the fast method (bold curve) along a linecrossing the center of the aluminium plate. In this case, thecorrelation was evaluated using a sliding window of size 65�65pixels. For comparison, the same figure also displays the out-of-plane displacement component obtained by means of the Carrephase-shifting technique (thin curve). Furthermore, Fig. 4 showsthe 3D plot of the normal displacement component w evaluatedfrom Eq. (14) using the fast method.

Finally, the maximum out-of-plane displacement generated bythe plate can be determined from Figs. 3 and 4, resulting(4373) nm. Here, the displacement uncertainty does not onlydepend on the error introduced by the fast method used toestimate the phase distribution but also on the relative contribu-tion of various error sources, such as the direction of thesensitivity vector and its variation across the specimen, the CCDnoise and the intensity fluctuations of the laser [9,16].

In addition, to evaluate the accuracy of the approximation f¼1when experimental data was processed, the intensities of theobject and the reference beams corresponding to both the initialand the deformed interferograms were separately recorded. Inthis case, the obtained results showed that the rms deviationevaluated between f and the assumed value of 1 was 5� 10�5,demonstrating the validity of the approximation introduced inSection 2.

6. Conclusions

In this paper we propose a fast method to be used in DSPI formeasuring nanometric displacement fields when the phasechange varies in the range ½0,pÞ rad. The proposed phase evalua-tion method is based on the local calculation of the correlationbetween the two speckle interferograms generated by bothdeformation states of the object. This approach does not needthe introduction of a phase-shifting facility or spatial carrierfringes in the optical setup to measure the phase distributionsgenerated by the object deformation without any sign ambiguity.The performance of the proposed method is investigated usingdifferent computer-simulated phase distributions, approachwhich allows to evaluate the rms phase errors. The numericalanalysis shows that the performance of the fast phase retrievalmethod proposed here are quite similar to the one given by theCarre phase-shifting technique. Although the rms errors given bythe previous phase retrieval approach reported in Ref. [12] areslightly lower than those generated by the proposed fast method,when this last simplified approach is applied it is not necessary torecord separately the intensities of the object and the referencebeams corresponding to both the initial and the deformed inter-ferograms. This advantage not only makes easier the automationof the interferometer operation, but also allows the application ofthe fast method to the analysis of non-repeatable dynamic eventsby recording a sequence of interferograms throughout the entiredeformation history of the testing object. Furthermore, when the

L.P. Tendela et al. / Optics and Lasers in Engineering 50 (2012) 170–175 175

proposed fast method is applied using the order statistics correlationcoefficient, each phase distribution is evaluated approximately fourtimes faster than with the original method. Finally, the results givenby the numerical analysis are confirmed by processing experimentaldata obtained from the deformation of a circular aluminium plate.The experimental results demonstrate that the performances of theproposed fast phase retrieval method and the Carre phase-shiftingtechnique are quite similar, and also confirm the simplicity of theproposed phase retrieval method to measure displacement fields inthe nanometer range.

Acknowledgments

L.P. Tendela would like to acknowledge the financial supportprovided by Fundacion Josefina Prats of Argentina.

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