Holographic strain analysis by fringe-localization planes Karl A. Stetson

Instrumentation Laboratory, United Technologies Research Center, East Hartford, Connecticut 06108 (Received 24 November 1975)

One of the continuing problems in hologram interferom-etry is the determination of surface strain from the fringes that appear on the surface of an object in the reconstruction of a hologram, exposed before and after subjecting the object to a s t r e s s . Of the many methods that have been proposed and demonstrated, none offer a direct method free of sophisticated calculations. The purpose of this Letter is to point out the existence of one such method, and describe its use. The method makes use of the planes of fringe localization that oc­cur when a hologram reconstruction is observed by means of an optical system with a slit aperture. It will be shown that the surface strains, shears , and in-plane rotations can be obtained from the angles between these planes and the object surface.

When a hologram reconstruction is observed by means of an optical system with a slit aperture, fringes will be observed with high contrast in a plane defined by the equation1

where Rap is a unit vector pointing along the slit of the aperture, Lob is the observed vectorial displacement of the point on the object surface intersected by a ray from the center of transmittance of the aperture in the view­ing direction, Kfob is the observed fringe vector at the plane of localization, D is the distance from the object surface to the localization plane, and λ is the wave­length of light. The fringe-locus function, Ω, may be defined in te rms of the observed fringe vector by

where R is a vector denoting position in space relative to an arbitrary origin of coordinates. For cosinusoidal fringes (e .g . , due to double-exposure) the magnitude of the fringe vector is

where d is the observed fringe spacing.

Let us return to Eq. (1) and impose some geometrical restr ic t ions. Assume the surface is locally flat and is being observed at normal incidence. The observed components of object motion will be Lx and Ly, i. e . , the components that lie in the surface of the object.

Let us assume further that the aperture direction is labeled x. Equation (1) becomes

where we have made use of Eq. (3). Let us take the partial derivatives of Eq. (4) with respect to x and y (which we indicate by superscripts) . This gives

Similarly, if the aperture is turned 90° so that it points along the y direction, we can obtain

Dxy and Dy

x a re merely the slopes of the localization plane in the x and y direction. Similarly Dx

y and Dyy are

the slopes of the localization plane when the aperture is pointed in the y direction. dx and dy a re the fringe spacings in the x and y directions, respectively. Thus, Eqs. (5) and (6) give the four parameters needed to characterize in-surface rotation, strain, and shear in te rms of fringe spacings and slopes of localization planes.

It is curious to note that the solutions obtained by Eqs. (5) and (6) are independent of bulk translation of the object, out-of-plane rotations of the object, and curvature of illumination. At normal incidence, out-of-plane rotations of the object will not affect fringe localization, whereas bulk translations or curved i l ­lumination will give a uniform shift of the localization plane and leave its slope unaffected.

The foregoing suggests a considerable simplification in holographic strain analysis for objects with flat sur ­faces that can be viewed normally. It is likely that the technique of reconstructing the hologram with the con­jugate to the original reference beam, so as to obtain a distortion-free real image, would offer the best op­portunity for making the necessary measurements. For this method, a collimated reference beam allows the simplest setup, because the hologram plate need only be rotated by 180° for reconstruction.

1K. A. Stetson, J. Opt. Soc. Am. 66, 626 (1976) (this issue).

627 J. Opt. Soc. Am., Vol. 66, No. 6, June 1976 Copyright © 1976 by the Optical Society of America 627