Phase shifting applied to four-wave holographic interferometers Pramod K. Rastogi

The author is with the Laboratory of Stress Analysis, Swiss Federal Institute of Technology, Lausanne 1015, Switzerland. Received 10 December 1991. Sponsored by P. Hariharan, National Measurement Lab­oratory, Sydney, Australia. 0003-6935/92/111680-02$05.00/0. © 1992 Optical Society of America.

A phase-shifting procedure that is applicable to four-wave holographic inteferometry is described. The technique can be used to obtain whole-field phase maps with all methods that use holographic moire fringes.

The technique of phase-shifting holographic interferome­try1-6 is now well established and is used regularly in the nondestructive testing of objects with diffusely reflecting surfaces. The attractive features of the phase-shifting procedure are the ease of data analysis and high accuracy of the results obtained. Over the last few years the technique has been applied to the measurement of object deforma­tions,1'2 the measurement of vibrations by stroboscopic illumination,3 and the measurement of surface shapes by two refractive-index4 and multiple-source contouring6,7 tech­niques. All these applications involve fringe patterns that are obtained by two-wave interferometry.

The use of the phase-shifting technique was extended recently to cover four-wave holographic interferometers.8"10

Four-wave interference results in an increase in flexibility over two-wave interference since it permits a manipulation of measured optical phase differences to isolate certain deformation components11-16 that would be difficult to mea­sure otherwise. The phase-shifting holographic technique is applied to the measurement of difference displacement components,8 in-plane displacements,9 and the derivatives of out-of-plane displacements.10 The methods that are employed require basically the sequential acquisition of two sets of interference fringes that are subsequently treated by image-processing techniques to provide phase contours that correspond to the required component. These methods therefore need three steps to extract the desired informa­tion.

The need to acquire two sets of fringes sequentially by phase stepping is a major drawback. First it requires more

data storage space and increases data acquisition and processing times. Second, and more important, any changes in the displacement fields that are due to mechanical or environmental changes in the time interval between the acquisition of the two sets of interference contours necessar­ily leads to erroneous results.

The purpose of this Communication is to describe a phase-shifting procedure that can be used with four-wave interferometers to yield a quantitative display of phase information that corresponds to a specific displacement component. The method is described here as applied to holographic moire interferometry for the measurement of in-plane displacements13-15 of a flat surface. Figure 1 is a schematic of the interferometric setup. The object is illumi­nated by means of two coherent collimated beams that are situated symmetrically with respect to the optical axis in the x-z plane. The two beams are directed on the object by mirrors that are mounted on piezoelectric transducers (PZT's). The intensity in the interference pattern is given by the relation

where φ1 and φ2 are phase differences that are introduced in the two arms of the interferometer by the object deforma­tion, I0 ( x , y ) is the average intensity, V is the fringe contrast, and φ1(x, y) -φ 2 (x , y) is the phase distribution we are interested in measuring.

The phase-stepping procedure consists of introducing a series of specific pairs of phase shifts in the two arms of the interferometer. The corresponding values of the intensities at each point on the interference pattern are recorded. In the method proposed, three successive phase shifts in pairs of ( - 2 Π / 3 , - 4 Π / 3 ) , (0, 0) and ( 2 Π / 3 , 4 Π / 3 ) are introduced in the two arms of the interferometer by providing con­trolled movements to the two PZT-driven mirrors. The intensities I1 I2, and I3 that correspond to these three pairs of phase shifts are given by the relations

Fig. 1. Schematic of the experimental arrangement for obtaining phase maps that correspond to the in-plane displacement component sx.

1680 APPLIED OPTICS / Vol. 31, No. 11 / 10 April 1992

These intensities are stored in the computer memory. The solution of Eqs. 2(a)-2(c) yields the following algorithm for calculating the phase difference φ1(x, y) - φ2(x,y):

It is of interest to note that Eq. (3) can also be used for frame intensities I1 I2, and I3 measured with pairs of phase shifts of (2π/3, - 2 Π / 3 ) , (0, 0) and (4π/3, - 4 π / 3 ) , respec­tively. The result is the modulo 2Π phase values at each point on the interference pattern. The measured wave front can then be reconstructed by removing the 2Π phase ambiguities between adjacent pixels.1-6 The four-wave phase-shifting procedure is, of course, subject to the same sam­pling conditions as two-wave phase-shifting techniques, namely, that the phase change between two adjacent pixels must remain less than Π.

The in-plane displacement component sx(x, y) is related to the measured phase φ (x , y ) = φ1(x,y) - φ2(x,y) by the expression

where 0 is half the angle between the two beams in the x-z plane.

To summarize, this Communication describes a phase-shifting procedure intrinsically application for four-wave holographic interferometry. The procedure is applicable to a range of interferometers that use holographic moire11-16

techniques. The rapid acquisition and analysis of the information obtained with a four-wave holographic interfer­ometer promises to remove many of the drawbacks of earlier methods,8-10 not the least of which is their vulnerabil­ity to time-dependent loading.

The financial support of the Swiss National Science Foundation is gratefully acknowledged.

References 1. P. Hariharan, B. F. Oreb and N. Brown, "A digital phase-

measurement system for real-time holographic interferome­try," Opt. Commun. 41, 393-396 (1982).

2. P. Hariharan, "Quasi-heterodyne holographic interferometry,'' Opt. Eng. 24, 632-638 (1985).

3. P. Hariharan and B. F. Oreb, "Stroboscopic holographic interferometry: application of digital techniques,'' Opt. Com­mun. 59, 83-86 (1986).

4. P. Hariharan and B. F. Oreb, "Two index holographic contour­ing: application of digital techniques," Opt. Commun. 51, 142-144 (1984).

5. K. Creath, "Phase-measurement interferometry," in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 5, pp. 349-393.

6. K. Creath, "Holographic contour and deformation measure­ment using a 1.4 million element detector array," Appl. Opt. 28,2170-2175(1989).

7. P. K. Rastogi and L. Pflug, "A holographic technique featuring broad range sensitivity to contour diffuse objects," J. Mod. Opt. 38, 1673-1683 (1991).

8. P. K. Rastogi, M. Barillot and G. Kaufmann, "Comparative phase shifting holographic interferometry," Appl. Opt. 30, 722-728 (1991).

9. P. K. Rastogi and E. Denarie, "Visualization of in-plane displacement fields using phase shifting holographic moire: application to crack detection and propagation," Appl. Opt. (to be published).

10. P. K. Rastogi, "Visualization and measurement of slope and curvature fields using holographic interferometry: an applica­tion to flaw detection," J. Mod. Opt. 38, 1251-1263 (1991).

11. P. K. Rastogi, "Comparative holographic moire interferome­try," Appl. Opt. 23, 924-927 (1984).

12. P. K. Rastogi, "Comparative holographic interferometry: a nondestructive inspection system for detection of flaws," Exp. Mech. 25, 325-337 (1985).

13. C. A. Sciammarella and J. A. Gilbert, "A holographic moire technique to obtain separate patterns for components of displacements," Exp. Mech. 16, 215-220 (1976).

14. C. A. Sciammarella, P. K. Rastogi, P. Jacquot, and R. Nara­yanan, "Holographic moire in real time," Exp. Mech. 22, 52-63 (1982).

15. P. K. Rastogi, M. Spajer, and J. Monneret, "In-plane deforma­tion measurement using holographic moire," Opt. Lasers Eng. 2, 79-103 (1981).

16. P. K. Rastogi, "A real-time holographic moire technique for the measurement of slope change," Opt. Acta 31, 159-167 (1984).

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