JOSA LETTERS

Use of fringe vectors in hologram interferometry to determine fringe localization Karl A. Stetson

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

The problem of fringe localization in hologram inter­ferometry has been treated to a remarkable number of analyses by various authors. Recently, however, the phenomenon has been codified by a rather simple theo­rem, first published by Stetson1 and confirmed shortly thereafter by Schumann and Dubas.2 At regions in space where the fringes of hologram interferometry exhibit complete localization (that is , they may be ob­served with highest contrast using a large two-dimen-sionally apertured optical system), the apparent motion of the object is at right angles to the apparent orienta­tion of the fringes. In both treatments, the conclusion is the result of rather substantial analysis, the magni­tude of which is perhaps not in proportion with the s im­plicity of the result.

Recent work in the interpretation of the fringes of hologram interferometry has yielded the concept of fringe vectors, 1 , 3 - 5 which has made it quite practical to extract the homogeneous components of strain and rota­tion from fringes seen on three-dimensional objects. It would appear, however, that the concept of fringe vec­tors can greatly shorten the derivation of the fundamen­tal theorem of fringe localization. It is the purpose of this Letter to present this shortened derivation.

Fringes on the surface of an object are generated by singular values of a fringe-locus function, Ω, which may be defined by the scalar product of the vectorial displacement of an object surface, L, and the sensitivity vector, K, which is equal to the vector difference be­tween the observation propagation vector, K2, and the illumination propagation vector, K1. That is , K = K 2 - K 1 , and

where Ro is a vector from the origin to points on the object surface. If we consider a sufficiently small r e ­gion of the object, the variations of surface displace­ment and variations of the sensitivity vector may be r e ­garded as approximately linear. Under this condition, the fringes may be characterized by a fringe vector, Kf, a s follows:

where Lo is the vectorial displacement of whatever o r i ­gin is chosen. The fringes will be seen on the object as the intersection of the object surface and laminae per ­pendicular to Kf. For fringe localization, however, it i s necessary to describe the fringe-locus function in three-dimensional space, and this i s done by assuming it to have constant values in the observation direction, i. e . , that of K2.6 If the object surface is flat, the fringe loci on that surface will be equidistant, straight l ines.

Projection of these lines along the observation direction will generate a second set of laminae perpendicular to an observed-fringe vector, Kfob.3 Thus, we have

It is possible to express Kfob in t e rms of Kf, however3:

where n is the unit normal to the object surface. If we substitute Eq. (4) into Eq. (3), and also substitute R = Ro + (R - Ro) for R, we get

(5) which becomes

The vector in square brackets has the value of zero.

Complete fringe localization implies that the value of the fringe-locus function does not change for small but arbi trary changes in K2. Substituting K2 + ΔK2 for K2 and retaining only f i rs t -order t e rms , we get

Because the vector in square brackets is zero, the vari­ation in Kf caused by ΔK2 need not be evaluated. If we recognize that (n • R)/(n • K2) = λD/2π, where D is the d is ­tance from the object surface to the region of complete localization, we may write

The vector in square brackets, however, is K fob. We have, therefore,

Because K2 i s perpendicular to the observation direction, any component of L in the observation direction does not contribute to the condition prescribed by Eq. (9). The apparent displacement of the object, Lob., is the projec­tion of L onto a plane perpendicular to the observation direction and this may be substituted for L in Eq. (9). The vector Kfob is not, however, the projection of Kf

onto such a plane and, therefore, Kf may not be sub­stituted for Kfob in Eq. (10). Complete localization r e ­quires, therefore,

Localization with a one-dimensional aperture, i . e . , a slit, requires

626 J. Opt. Soc. Am., Vol. 66, No. 6, June 1976 Copyright © 1976 by the Optical Society of America 626

which leads to the usual expression for the relationship between fringe spacing, localization distance, and ob­ject displacement in the direction of the aperture s l i t .6

1K. A. Stetson, J . Opt. Soc. Am. 64, 1 (1974). 2M. Dubas and W. Schumann, Opt. Acta 21, 547 (1974). 3K. A. Stetson, Appl. Opt. 14, 272 (1975). 4K. A. Stetson, Appl. Opt. 14, 2256 (1975). 5R. Pryputniewicz and K. A. Stetson, Appl. Opt. (in press). 6K. A, Stetson, Optik (Stuttgart) 31, 576 (1970).

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