Method of Stationary Phase for Analysis of Fringe Functions in Hologram Interferometry

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Abstract
In this paper, a mathematical technique called the method of stationary phase is presented for the purpose of obtaining approximate solutions for characteristic fringe functions in hologram interferometry. The cases of greatest concern are where holograms are formed of objects that vibrate simultaneously in different geometrical patterns according to sinusoidal time functions whose frequencies are related by rational numbers. The method assumes that the object motion is sufficiently large that contributions to the fringe function come only from the holographic recordings of the object at positions where its velocity is zero. At such position, the phase of the light scattered from an object point is stationary with respect to time, and the method permits the approximate calculation of the relative intensity of the partial recording at these positions. The fringe functions become expressions similar to those found in multiple-beam interferometry, and fringe patterns can be understood more easily. The case of two sinusoidal vibrations at a frequency ratio of 2:1 is considered experimentally and theoretically according to the method presented.