Fringe Vectors and Observed-Fringe Vectors in Hologram Interferometry Karl A. Stetson

Instrumentation Laboratory, United Aircraft Research Laboratories, East Hartford, Connecticut 06108. Received 18 Octobe 1974. The concept of a fringe vector in hologram interferome­

try has been introduced in a recent paper.1 The fringe-focus function, constant values of which define fringe loci on an object surface, may be defined as the scalar product of the fringe vector and that space vector that defines the coordinates of the object surface. This method of defining fringe loci is very helpful in visualizing how fringes will ap­pear on an object of known shape due to a given object mo-

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tion. For rigid-body motions in particular, it led to the idea of a fringe-projection theorem and to certain useful concepts regarding fringe localization. The object of this paper will be to relate the fringe vector, as previously de­fined,1 to something that may be called an observed-fringe vector.

Let us review the role of the fringe vector in hologram in­terferometry preparatory to explaining the observed-fringe •vector. If the deformation of an object, over a given seg­ment, can be approximated by a homogeneous deformation plus a rigid-body motion, it is possible to define a fringe vector for that segment of the object. If the illumination and observation directions are nearly constant over that segment, the fringe vector will define fringe lamina in space that are equidistant planes normal to the fringe vector. The fringes are seen on the object where its surface inter­sects these lamina. The same fringe pattern, however, could be projected onto the object surface from the direc­tion of the observer. If the object surface is flat and the previous restrictions hold, the fringes will be seen as equi­distant straight lines. If these lines are translated parallel to the direction of observation, they will define another set of equidistant lamina in space that could correspond to an alternative fringe vector, which we may call the observed-fringe vector. If a photograph is made of the fringes as seen on the object, the observed-fringe vector will lie nor­mal to the fringe lines and have a magnitude inversely pro­portional to the fringe spacing for periodic fringe functions. Thus, the observed-fringe vector is easily extracted from photographs, whereas the true fringe vector is not. Once the two vectors are related, however, it may be possible to use the observed-fringe vector together with a knowledge of the object surface, the directions of illumination and obser­vation, etc. to solve for the object motion.

The fringe-locus function may be defined by the equa­tion

where Ω is the fringe-locus function, R0 is the space vector from an arbitrary origin to the coordinates of the object surface, and Kf is the fringe vector. It is possible to define a fringe-line vector H tangent to the fringe loci by the equation

where n is the unit normal vector to the object surface. If we assume that an observed-fringe vector Kf0b, exists, then it must yield the same fringe-line vector when substituted into Eq. (2). Thus we may write

This equation may be solved for Kfob by taking the vector product of both sides with the observation vector K2 and making use of the fact that Kfob is normal to K2 by defini­tion.

which leads to the two alternative solutions,

or

Equation (5) shows that it is possible to express the ob­served-fringe vector as a function of the true fringe vector, but not vice versa. It also indicates the general relation­ship that, if the true fringe vector is perpendicular to the direction of observation, the two fringe vectors are equiva­lent. Equations (4) and (5) would not appear to be of im­mediate value in aiding the extraction of a general defor­mation vector from a holographic interferogram, where no presumption can be made as to the object's motion. As an illustration of what can be done, however, when something is known about the object motion, let us consider rigid-body motion. It has been shown1 that the fringe vector for such motions can be expressed as

where θ is parallel to the axis of the helix that describes the path of an object point and has a magnitude equal to the angular rotation of the object. It is assumed that the sen­sitivity vector K is essentially constant over the region of the object being considered. Let us substitute Eq. (6) into Eq. (4), rather than Eq. .(5), because it leads to a more trac­table expression. We have, for the numerator of the right-hand side of Eq. (4),

which leads to

A previous derivation1 established that when an object is illuminated and viewed retroreflectively from a sufficient distance, fringes would be observed parallel to the helical axis of whatever rigid-body motion the object might under­go. This arose from examination of Eq. (6), which indi­cates that if the sensitivity vector points in the same-direc­tion as the observation vector, the fringe vector lies at right angles to the observation vector. This is shown again in Eq. (8), where the second term of the right-hand side be­comes zero when the vector product of K2 and K is zero. We may recognize, however, that tkis second term may also be zero when the axis of the helix lies parallel to the object surface, i.e., when θ • n = 0. This requires, simply, that the surface not rotate in its own plane. Thus, there exists an even wider class of situations where the direction of the observed fringes can be taken to point along the axis of rotation of an object. For a cosinusoidal fringe function, or a J0 fringe function beyond the first zero, the restrictions above lead to the following relationship between the mag­nitude of the peak-to-peak rotation of the object, i.e., 2|θ|, and the observed fringe spacing dap.

where λ is the wavelength of light at which the hologram was recorded, Φ1 is the angle between K2 and θ, Φ2 is the angle between K and n, Φ3 is the angle between K2 and n, and Φ4 is the half-angle between K2 and the negative of the illumination vector —K1. The vector θ is oriented, of course, parallel to the object surface and parallel to the ob­served fringes. Thus, when the parameter of interest is how much one region of an object may have twisted relative to another, and that part of an object may be assumed not to have twisted in its own plane, Eq. (9) can provide a quick means of evaluation that is insensitive to bulk trans­lation of that region of the object. This method must be regarded as superior to evaluating the vectorial displace-

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ments of two nearby points in that region of the object and calculating their difference, since a small error in the calcu­lation of such a difference would create a large error in the calculation of the rotation of the region of the object.

Reference 1. K. A. Stetson, J. Opt. Soc. Am. 64, 1 (1974).