Derivatives of displacement obtained by directmanipulation of phase-shifted interferograms

Mauro Facchini and Paolo Zanetta

Themaps of phase derivatives are extracted here by direct manipulation of phase-shifted interferograms.There are three main advantages: There is no need for prior phase evaluation or unwrappingprocedures, and only a short processing time is needed. By digital integration of the derivatives theabsolute phase map can also be retrieved without unwrapping procedures. A general description of themethod is presented and discussed. For example, the proposed technique has been applied to the studyof the deformation of a test object by the manipulation of four phase-shifted interferograms.Key words: Phase shifting, interferometry, derivatives, unwrapping. r 1995 Optical Society of

America

1. Introduction

Digital-processing techniques are increasingly beingapplied to perform quantitative analysis of interfero-grams.1 In particular, phase-shifting algorithms arewidely used for the retrieval of the interferometricphase.2 The main reasons for their success are thehigh precision of the measurements and the simpleexperimental arrangement. Because phase data areextracted in the 0–2p range, final unwrapping proce-dures are necessary for the absolute phase map to beobtained.3,4 With phase-shifting techniques a preci-sion of 2p@100 rad, which is equivalent to one hun-dredth of the fringe period, is typical.2In some cases, however, the phase derivatives are

also required for a full understanding of the physicalproperties being investigated.5 In strain analysis,for example, the phase of the interferograms obtainedwith holographic interferometry or speckle tech-niques is proportional to displacement, whereas thespatial derivatives of displacement are necessary forthe description of the strain tensor.6 Phase partialderivatives can be calculated by application of digitalalgorithms to the phasemap.7,8 Alternatively, speckleshearing interferometry9 is applied to produce inter-ferograms whose phase is directly proportional todisplacement derivatives. In this case, however,speckle decorrelation effects reduce fringe visibility,

The authors are with the European Commission, Joint ResearchCentre, Institute for Systems Engineering and Informatics, T.P.680, Ispra 1VA2 21020, Italy.Received 26 June 1995.0003-6935@95@317202-05$06.00@0.

r 1995 Optical Society of America.

7202 APPLIED OPTICS @ Vol. 34, No. 31 @ 1 November 1995

and the application of digital phase extraction tech-niques is limited.10In this paper we show how the displacement deriva-

tives can be calculated by direct manipulation ofinterferograms, and the advantages of this procedureare highlighted.

2. Theory

Phase-shifting techniques are based on the analysisof at least three fringe patterns, differing by theknown amount of phase introduced between the twoarms of the interferometer.2,11–14 Each intensity pat-tern Iai

1x, y2 can be described by

Iai1x, y2 5 Ib1x, y251 1 g1x, y2cos3f1x, y2 1 ai46, 112

wheref1x, y2 is the interferometric phase, Ib1x, y2 repre-sents the background, g1x, y2 is the contrast modula-tion, and ai is the introduced phase shift.By the computational point of view, a variety of

versions of the method exist, all sharing the ability toeliminate the background and contrast terms bysimple arithmetic or trigonometric operations on theacquired images. When a number N of interfero-grams is considered, the interferometric phase can beexpressed as

f1x, y2 5 arctanoi51

N

aiIai1x, y2

oi51

N

biIai1x, y2

, 122

where ai and bi are constants that depend on the

phase-shift values. If the derivatives of Eq. 122 aretaken along the x sensitive direction, the followingequation is obtained:

≠f1x, y2

≠x5

o ai≠Iai

1x, y2

≠x o biIai1x, y2 2 o aiIai

1x, y2 o bi≠Iai

1x, y2

≠x

3o aiIai1x, y242 1 3o biIai

1x, y242. 132

The derivatives ≠Iai1x, y2@≠x of each interferogram can

be replaced with their finite difference approxima-tion:

≠Iai1x, y2

≠x<Iai1x 1 h, y2 2 Iai

1x, y2

h, 142

where h indicates the distance between two adjacentpoints along x, yielding

≠f1x, y2

≠x51

ho aiIai

1x 1 h, y2 o biIai1x, y2 2 o aiIai

1x, y2 o biIai1x 1 h, y2

3o aiIai1x, y242 1 3o biIai

1x, y242. 152

We can readily calculate each term of Eq. 152 bydigitizing the interferograms shifted in phase. There-fore phase derivatives can be obtained without anevaluation of the absolute phase distribution. It isclear that, to obtain fine derivative maps, it is betterto apply the technique to interferograms that have asufficiently good quality. For this reason a prefilter-ing of interferometric patterns can often be useful.When the maps of the partial derivatives along two

orthogonal directions x and y are known, the interfero-metric phase term f1x, y2 can readily be retrieved by afinite integration procedure. In fact, by letting f bethe phase at P 5 P1x, y2, we can approximate thephase of a point Q 5 Q1x 1 h, y 1 k2 in the neighbor-hood of P by

f1Q2 5≠f1P2

≠xh 1

≠f1P2

≠yk 1 f1P2, 162

where h and k represent the distance between twopoints along x and y, respectively. One obtains thephase map by performing the calculation in Eq. 162from point to point along a path that covers the wholeinterferogram. However, note that any error occur-ring between two points affects the phase values ofthe other points to be processed.

3. Results

The method proposed has been applied to processingholographic and speckle pattern phase-shifted inter-

ferograms, as described in Ref. 15. Here we studythe static deformation of a circular steel plate with adiameter of 10 cm by using holographic interferom-

etry. Between the first and second exposures theplate was loaded at its center with a pointlike pushingforce. A sketch of the optical setup is presented inFig. 1. The source of coherent light was a He–Necontinuous-wave laser of 10-mW power, emitting at awavelength l 5 633 nm, whose output beam was splitinto reference and illumination beams. Along thereference path a mirror mounted on a piezoelectrictransducer was inserted to introduce the desired

phase shifts. A holocamera 1Rottenkolber HSB-852with a thermoplastic film was used to record thehologram. After the first exposure the film wasdeveloped and then reilluminated with the referencebeam. The object was then deformed, and real-timedeformation fringes were observed. After deforma-tion the piezoelectric mirror was translated by fixedamounts corresponding to phase shifts of p@2 of theholographic fringes. Four interferograms were digi-tized and stored in a frame grabber memory installedin a personal computer. An interferogram of theplate obtained after application of the load is pre-sented in Fig. 2.

Fig. 1. Optical setup. B.S., beam splitter; T.p., thermoplasticfilm.

1 November 1995 @ Vol. 34, No. 31 @ APPLIED OPTICS 7203

In the case of four images 1n 5 42 shifted by p@2, Eq.122 becomes

f1x, y2 5 arctanI3p@21x, y2 2 Ip@21x, y2

I01x, y2 2 Ip1x, y2. 172

The expression for the phase derivatives along x 1oralternatively y2 has been calculated according to Eq.132:

≠f

≠x5

1≠I3p@2

≠x2

≠Ip@2

≠x 21I0 2 Ip22 1I3p@2 2 Ip@221≠I0≠x2

≠Ip

≠x 21I3p@2 2 Ip@22

2 1 1I0 2 Ip22

;

182

and substituting the finite difference approximationof Eq. 142 yields

Fig. 2. Interferogram of the plate centrally loaded.

7204

≠f

≠x51

h

3I3p@21x 1 h, y2 2 Ip@21x 1 h, y243I01x, y2 2 Ip1x, y24 2 3I3p@21x, y2 2 Ip@21x, y243I01x 1 h, y2 2 Ip1x 1 h, y24

3I3p@21x, y2 2 Ip@21x, y242 1 3I01x, y2 2 Ip1x, y242.

192

A similar expression can be obtained for calculatingthe phase derivative along the y direction. Themapsof the x and y phase derivatives are presented in Fig.3, and the corresponding three-dimensional 13D2 plotsare shown in Fig. 4. According to Eq. 162, the phasemap, proportional to the static displacement of thecircular plate, has been retrieved. The phase andthe corresponding 3D plot are displayed relatively inFigs. 5 and 6.

4. Conclusions

The method proposed here for calculation of thepartial derivatives of the interferometric phase offersthree main advantages over classical procedures:

c Prior evaluation of displacement is not necessary.

APPLIED OPTICS @ Vol. 34, No. 31 @ 1 November 1995

Fig. 3. Gray-level maps of partial derivatives of phase along1a2 the horizontal and 1b2 the vertical directions.

For this reason real-time evaluation of strain can beforeseen. This feature can be useful when the reli-ability of components must be tested in real time.

c The derivatives are calculated with simple arith-metic operations without the trigonometric functionbeing evaluated. For this reason the results areobtained directly, and there is no need for unwrappingprocedures. All uncertainties related to the unwrap-ping process 1discontinuities, independent areas, etc.2are therefore avoided.

c With the technique presented here just one stepis needed to calculate the partial derivatives from theinterferograms. Therefore the time of evaluation ofpartial derivatives is considerably faster than theordinary procedure in which the following three stepsare involved: phase retrieval, phase unwrapping,derivative calculation.

Finally, it has been demonstrated that it is possibleto retrieve the displacement information from thederivative maps. The precision and reliability ofresults depend on the quality of the interferogramsacquired experimentally.A direct application of the technique described in

this paper should be the real-time inspection of strain

Fig. 4. 3D meshes of partial derivatives of phase along 1a2 thehorizontal and 1b2 the vertical directions.

Fig. 5. Gray-level map of phase 1displacement2 obtained after thefinite integration process.

distribution 1rather than displacement evaluation2 ina structure under a dynamic load.

We acknowledge the helpful discussions and sup-port of D. Kerr and C. Forno, the assistance of D.Albrecht with the image processing, the technicalassistance of M. Franchi and M. Puccia, and A. C.Lucia for encouragement and valuable advice.

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Fig. 6. 3D mesh of phase 1displacement2.

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