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Optical configuration for TV holographymeasurement of in-plane and out-of-plane deformations

Nandigana Krishna Mohan, Angelica Andersson, Mikael Sjodahl, and Nils-Erik Molin

A novel TV holography method is proposed for parallel evaluation of in-plane and out-of-plane deforma-tion fields. The method permits a trade-off between in-plane and out-of-plane measuring sensitivity. Afour-exposure, four-frame phase shifting technique is used in the experiments; the experimental resultsfor an aluminum specimen subjected to both rotation in its own plane and a bending couple load at thecenter are presented. © 2000 Optical Society of America

OCIS codes: 120.3940, 120.6160, 120.4290.

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1. Introduction

TV holography and TV shearography are simple, fast,and contact-free techniques for the measurement ofdeformation fields and gradients of deformation fieldsin real time.1 Krishna Mohan et al.2 recently re-

orted an optical configuration that multiplexesolography and shearography for simultaneous mea-urement of out-of-plane deformation and its slopehange. They accomplished this by dividing theCD into two parts, which enabled them to extract

wo entirely different pieces of information from aingle setup. The multiplexing technique has alsoeen extended to a holocomparative configuration.3,4

A similar concept is adopted in this paper for simul-taneous quantitative evaluation of in-plane and out-of-plane deformation fields from a single opticalarrangement.

Few methods for real-time evaluation of deforma-tion fields have been reported in recent years.5,6

Pedrini et al.5 reported a two-dimensional spatial car-rier phase-shifting method to determine in-plane andout-of-plane deformations. Sjodahl and Saldner6

combined TV holography with digital–electronicspeckle photography for parallel evaluation of bothin-plane and out-of-plane displacement components

N. Krishna Mohan is with the Applied Optics Laboratory, De-partment of Physics, Indian Institute of Technology, Madras 600036, India. A. Anderson ~angelica.andersson@mt.luth.se!, M. Sjo-dahl, and N.-E. Molin are with the Division of Experimental Me-chanics, Luleå University of Technology, SE-97187, Luleå,Sweden.

Received 8 June 1999; revised manuscript received 5 October1999.

0003-6935y00y040573-05$15.00y0© 2000 Optical Society of America

about a crack tip. In their method the in-plane com-ponents were measured with the sensitivity ofdigital–electronic speckle photography and the out-of-plane component was obtained with interferomet-ric accuracy.

An optical system for parallel evaluation of in-plane and out-of-plane deformation components of adeformation vector is demonstrated here. In thismethod the object is illuminated symmetrically withtwo collimated beams and the scattered light is ob-served along the direction of one of the illuminatingbeams for sensing the in-plane displacement compo-nent. The scattered light from the same illuminat-ing beam in the specular direction is combined with asmooth reference beam for parallel evaluation of theout-of-plane deformation component. In the experi-ments, a four-exposure, four-frame phase-shiftingtechnique is employed in a commercially availableimage-processing system developed by Karl StetsonAssociates and Recognition Technology, Inc.7

2. Experimental Arrangement and Theory

Figure 1 shows the schematic arrangement of a dual-beam illumination–observation setup. The opti-cally rough surface is illuminated at an angle u fromoth sides of the surface normal by two collimatedeams. The scattered light is collected back alonghe direction of the illuminating beams by beam split-er BS1 and mirror M1 and by beam splitter BS2 and

mirror M2. The average speckle size is adjusted tobe within the resolution of the detector. The twoscattered fields enter half of the field of view of a zoomvideo lens and relay lenses and are imaged as twoseparate images ~A and B! onto the photosensitiveurface plate of a CCD camera. For each illuminat-ng beam there are two scattered fields, namely, a

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backscattered field along same direction of illumina-tion and a scattered field in the specular reflectiondirection, that enter half of the lens.8,9 A smoothreference wave derived from the same laser source isadded with the help of a single mode fiber2 to imageB. The CCD and a phase-stepped mirror ~PZM! areconnected to an image-processing system that is in-terfaced to the host computer.7 In the interferome-ter, three mechanical shutters are introduced to blockthe beams between successive exposures. We recordthe first exposure, which represents the initial stateof the object, by closing shutter S1, which allows onlyfour frames of image A to be stored, with a phase stepof py2 between the frames. The intensity distribu-tion of each frame for image A can be expressed as10

In~initial 1! 5 I1 1 I2 1 2ÎI1I2 cos~w 1 npy2!, (1)

where I1 and I2 are the intensities of the backscat-tered light along the direction of illuminating beam 1and the light scattered in the same direction owing tospecular reflection of illuminating beam 2, respec-tively. w is the random phase between the twowaves, and n 5 0, 1, 2, 3.

Similarly, we record the second exposure, whichepresents the initial state of the object, by openinghutter S1 and closing shutters S2 and S3. Blocking

shutters S2 and S3 permits only the light scattered inthe specular reflection direction from illuminatingbeam 1 and the smooth reference beam to be recordedin image B. The phase steps can be introduced byuse of the same phase-stepped mirror. The inten-sity distribution of each frame for image B can bewritten as10

In~initial 2! 5 I19 1 Ir 1 2ÎI19Ir cos~w9 1 npy2!, (2)

Fig. 1. Schematic arrangement for simultaneous measurementmirrors; PZM, piezoelectric mirror; S’s, mechanical shutters; kW, pr

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where I19 and Ir are the intensities of the light scat-tered in the specular direction from illuminatingbeam 1 along blocked illuminating beam 2 and thereference light, respectively.

The subsequent frames, which represent the de-formed state of the object under observation condi-tions similar to those described above, are stored, andthe final intensity distributions can be written as10

In~final 1! 5 I1 1 I2 1 2ÎI1I2 cos~w 1 Df 1 npy2!, (3)

In~final 2! 5 I19 1 Ir 1 2ÎI19Ir cos~w9 1 Df9 1 npy2!,(4)

where Df and Df9 are the phase changes introducedinto images A and B, respectively, by object deforma-tion.

One can see the phase term Df in image A, whichis responsible for fringe formation, by viewing in thedirections of the illuminating and observation beams.For illumination beam 1, which has a propagationvector k1, there is a backscattered beam along 2k1.In addition, there is also a component that is due tospecular reflection of illuminating beam 2 that has apropagation vector k2 in the direction of illuminationbeam 1. Following the analysis in Refs. 8 and 9, theresultant phase change can be expressed as

Df 5 $@~2k1! 2 ~k1!#L 2 @~2k2! 2 ~k1!#L%

5 ~k2 2 k1!L, (5)

where L is the deformation vector.Similarly, in image B the phase term Df9 is due to

the interference between the smooth reference waveand the scattered light generated by illuminatingbeam 1 with propagation vector k1 in the symmetri-

plane and out-of-plane deformations: BS’s, beam splitters; M’s,ation vectors.

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cally opposite direction ~along the direction of blockedilluminating beam 2, i.e., along k2!. The net phasechange Df9 can be written as8

Df9 5 @~2k2! 2 ~k1!#L 5 @2~k2 1 k1!#L. (6)

In the image processor, four sets of four phase-shifted images corresponding to image A @Eqs. ~1! and~3!# and image B @Eqs. ~2! and ~4!# are stored to re-trieve phase terms Df and Df9. Operating the sys-tem in time-lapse display mode,7 we observeinterference patterns at video rate speed that aremodulated by cosinusoidal functions of the form

I~Image A! 5 u8ÎI1I2 cos~Dfy2!u2,

I~Image B! 5 u8ÎI19Ir cos~Df9y2!u2. (7)

Assuming that the illuminating beams lie in the x–zplane, we can express propagation vectors k1 and k2and deformation vector L in terms of unit vectors as10

k1 52p

l~2ı sin u 2 k cos u!,

k2 52p

l~ı sin u 2 k cos u!,

L 5 ~ıu 1 jv 1 kw!. (8)

Substituting Eq. ~8! into Eqs. ~5! and ~6!, we obtainhe following expressions for phase terms Df andf9:

Df 54p

lu sin u, (9)

Df9 54p

lw cos u. (10)

Equations ~9! and ~10! clearly indicate that theinterferograms obtained from individual optical con-figurations A and B contain the information that per-tains to in-plane deformation component ~u! andut-of-plane deformation ~w!. The incremental dis-

placement between the adjacent fringes is ly2 sin ufor in-plane deformation and ly2 cos u for out-of-plane deformation. It is interesting to note thatboth components of the deformation can be extractedfrom the setup with identical measuring sensitivitieswhen the illumination angle is u 5 45°.

For quantitative evaluation of the deformationfields from the individual configurations, the imagesare further processed by a computer to produce quan-titative results.7,10

3. Experimental Results

The experiments are conducted on an aluminumplate ~110 mm 3 75 mm 3 1 mm! supported alongwo sides. The specimen is coated with matte whitepray paint. The object is simultaneously subjectedo both in-plane rotation and out-of-plane deflection.or the in-plane motion the object is fixed upon arecision rotation stage; for the out-of-plane deflec-

tion a screw is fixed at the center of the plate tointroduce a bending couple load. The 40-mm-diameter collimated illuminating beams, from afrequency-doubled diode-pumped 80-mW laser ~l 5532 nm!, strike the object at an angle of 20° withrespect to the surface normal. To receive the scat-tered fields in the directions of the illuminationbeams, two beam splitters with coatings with atransmission-to-reflection ratio of approximately50:50 at 532 nm and two mirrors are used in theexperimental setup. The scattered fields are col-lected and imaged onto an NEC TI-324A CCD cameravia a combination of a zoom video lens ~ fy11! andrelay lenses. The magnification of the imaging sys-tem is adjusted such that each of the two images ~And B! occupies one half of the detector plane. A

Fig. 2. ~a! In-plane rotation fringes ~u component!, ~b! phase mapfor in-plane rotation. We obtain the experimental results by com-bining the set of phase-shifted images from exposure 1 with expo-sure 3 from image A.

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smooth reference wave extracted from the same lasersource is added to image B with the help of a variablebeam splitter and a single-mode fiber. A computer-controlled piezoelectric-driven mirror is provided inilluminating beam 1, as shown in Fig. 1.

With the image-processing system operating intime-lapse save mode, the computer first stores twoexposures, images A and B, which represent the ini-tial state of the object, with the beams blocked be-tween exposures as explained in Section 2. Thealuminum plate is subjected to both rigid body in-plane rotation about the z axis and out-of-plane de-flection by means of a bending couple load applied atthe center of the plate from the back side. The thirdand fourth exposures are captured by the same pro-cedure. Using the time-lapse mode allows one tosave a series of static holograms of an object and

Fig. 3. ~a! Out-of-plane deformation fringes ~w component!, ~b!hase map for out-of-plane deformation. We obtain the experi-ental results by combining the set of phase shifted images from

xposure 2 with exposure 4 from image B.

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generates either cosine fringes or wrapped phasemaps in a controlled manner.

Figure 2 shows the interferograms and phase mapsthat we obtained from image A by combining the setof phase-shifted images generated from exposure 1with those obtained from exposure 3. Similarly Fig.3 shows fringe patterns and phase maps obtainedfrom image B by use of the set of phase-shifted im-ages in exposures 2 and 4, respectively. The phasemaps are unwrapped and scaled by Eqs. ~9! and ~10!to yield the deformation values. The three-dimensional profiles obtained from the present anal-ysis from in-plane rotation and out-of-planedeflection are shown in Figs. 4~a! and 4~b!, respec-tively. The experimental results clearly show thatthe proposed configuration yields both of the defor-mation components with comparable measuring sen-sitivities.

4. Conclusions

An optical configuration combined with an image-processing system has been described and demon-strated experimentally for quantitative evaluation ofin-plane and out-of-plane components of a deforma-tion vector. Advantages of the proposed method are

Fig. 4. ~a! Measured in-plane deformation component, ~b! mea-sured out-of-plane deformation.

ative TV holography for vibration analysis,” Opt. Eng. 34,

that it affords one the flexibility to control the mea-suring sensitivity of the in-plane and out-of-planecomponents of the deformation vector by changingthe angle between the illumination beams and thatthe in-plane and out-of-plane components are re-corded separately.

The experiments were performed at Luleå Univer-sity of Technology, where N. Krishna Mohan was avisiting scientist with the Division of ExperimentalMechanics in May 1999. The Swedish Foundationfor Strategic Research–Integrated Vehicle Structuresresearch program in Sweden and the Swedish Re-search Council for Engineering Sciences, supportsthis project.

References1. C. Joenathan, “Speckle photography, shearography, and

ESPI,” in Optical Measurement Techniques and Applications,P. K. Rastogi, ed. ~Artech House, Boston, Mass., 1997!, Chap.6, pp. 151–182.

2. N. Krishna Mohan, H. O. Saldner, and N.-E. Molin, “Electronicspeckle pattern interferometry for simultaneous measurementof out-of-plane displacement and slope,” Opt. Lett. 18, 1861–1863 ~1993!.

3. H. O. Saldner, N. Krishna Mohan, and N.-E. Molin, “Compar-

486–492 ~1995!.4. N. Krishna Mohan, H. O. Saldner, and N.-E. Molin, “Recent

applications of TV holography and shearography,” in LaserInterferometry VIII: Applications, R. J. Pryputniewicz, G. M.Brown, and W. O. Juptner, eds., Proc. SPIE 2861, 248–256~1996!.

5. G. Pedrini, Y.-L. Zou, and H. J. Tiziani, “Simultaneous quan-titative evaluation of in-plane and out-of-plane deformationsby use of a multidirectional spatial carrier,” Appl. Opt. 36,786–792 ~1997!.

6. M. Sjodahl and H. O. Saldner, “Three-dimensional deformationfield measurements with simultaneous TV holography and elec-tronic speckle photography,” Appl. Opt. 36, 3645–3648 ~1997!.

7. K. A. Stetson, W. R. Brohinsky, J. Wahid, and T. Bushman,“An electro-optic holography system with real-time arithmeticprocessing,” J. Nondestruct. Eval. 8, 69–76 ~1989!.

8. A. Sohmer and C. Joenathan, “Twofold increase in sensitivitywith a dual-beam illumination arrangement for electronicspeckle pattern interferometry,” Opt. Eng. 35, 1943–1948~1996!.

9. N. Krishna Mohan, “Measurement of in-plane displacementwith twofold sensitivity using phase reversal technique,” Opt.Eng. ~to be published!.

10. R. J. Pryputniewicz, “Electro-optic holography,” in Holo-graphic Interferometry—Principles and Methods, P. K. Ras-togi, ed. ~Springer-Verlag, Berlin, 1994!, Chap. 3, pp. 59–74.

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