Interferometric Measurements Using the Wavefront Reconstruction Technique B. P. Hildebrand and K. A. Haines

Willow Run Laboratories, Insti tute of Science and Technology, The University of Michigan, Ann Arbor, Michigan. Received 26 July 1965.

This letter outlines a technique for making real time deforma­tion measurements of arbitrary three-dimensional objects under stress. I t is well known that interferometric techniques yield extremely accurate measurements of minute surface deformities in mirrors, lenses, or highly polished objects. I t has not been possible, however, to use these methods on arbitrary diffuse reflecting objects since a reference wavefront exactly duplicating the object has not been available. With the advent of the wavefront reconstruction technique as described by Leith and Upatnieks,1 it is now possible to record and reconstruct a wavefront exactly representative of the wavefront reflected by the object. This reconstructed wavefront may now be used as a reference against which the real object may be compared.

The work described here is part of an over-all study being carried out a t this laboratory under the direction of E. N. Leith to deter­mine areas of applicability for the wavefront reconstruction process. One phase of this study has already been reported on by Stetson and Powell, of this laboratory, in two papers. The first of these was the application of the wavefront reconstruction technique to vibration analysis.2 The second paper describes the generation of fringe patterns.3 The present letter describes a specific application of the fringing phenomenon to strain analysis.

Consider the diagram shown in Fig. 1. The object to be stressed is placed in its testing apparatus and illuminated with a coherent monochromatic source of light. A second beam of light, known as the reference beam, is used to illuminate a photographic plate. The light scattered from the object interferes with the reference beam to form a density pattern on the film. This plate, a hologram, constitutes a recording of the amplitude and phase of the light reflected by the object. When the plate is developed and then illuminated with a coherent light beam, the density pattern on the plate diffracts the light into several orders just as does a diffraction grating. If the plate is inserted in a light

Fig. 1. Wavefront reconstruction interferometer.

172 APPLIED OPTICS / Vol. 5, No. 1 / January 1966

Fig. 2. Experimental results. The photographs are arranged in order of increasing deformation.

beam exactly duplicating the geometry in which the recording was made, one of the first-order diffraction patterns will form a complete three-dimensional virtual image in exactly the position the object occupied in the recording situation. This, then, is the means for obtaining a reference wavefront for interferometric measurements.

In practice, the developed recording is placed in the same posi­tion in which it was made. The reference beam now acts as the illumination for the recording. The image produced by the recording falls upon the object. The light reflected by the object will thus interfere with the light wave representing the image. If the object is deformed in any way, interference fringes will form across the object. These fringes form a complete record of the amount of deformation the object has suffered and stress concentrations are immediately obvious.

As in any interferometric work, the fringes may be localized in space. That is, for certain types of object motion, the fringes will appear not on the object but somewhat in front of or behind the object. The complete analysis of fringe localization and the in­formation which may be obtained from a measurement of the fringe position, distribution, and motion when the observation angle is changed will appear in a more complete paper.

Figure 2 is a series of photographs showing preliminary ex­perimental verification of the method outlined above. The object in this case was an aluminum disk which could be pulled in at the center. The disk was painted a flat white to eliminate all specular reflection. Figure 2(a) shows the fringe pattern under no stress. Notice that some deformation of the object or the photographic record must have occurred since some fringing is visible. Figures 2(b) and 2(c) show the fringe pattern with successively higher deformation amplitude. By counting the number of fringes between any two points on the disk, the amount of deformation may be calculated.

The authors acknowledge many fruitful discussions with K. Stetson, R. Powell, and E. N. Leith.

References 1. E. N . Leith and J. Upatnieks, J. Opt. Soc. Am. 52, 1123

(1962); 53,1377(1963); 54, 579A, 1295 (1964). 2. R. Powell and K. Stetson, J. Opt. Soc. Am. (to be published). 3. K. Stetson and R. Powell, J. Opt. Soc. Am. (to be published).

January 1966 / Vol. 5, No. 1 / APPLIED OPTICS 173