Chapter I. Introduction. The Method of Stationary Phase One of the early conclusive experiments verifying the wave nature of light was the double mirror experiment of Fresnel. In this experiment, two plane mirrors are placed so as to form an angle of slightly less than 180 degrees between them. If light is incident from a source S, then interference fringes are observed in the region common to the two beams reflected by the mirrors. The two beams can be assumed as coming from Sx and S2, the images of S in the two mirrors. These sources can be thought of as synchronous and homogeneous since they are derived from the same source S. If a screen is placed in the region common to the two beams, then the points equidistant from Sl and S2 are highly illuminated, and dark and light regions then alternate giving the interference pattern. It was discovered by Gouy (Comp. Rend. 110 (1890), p. 1251) that if one of the plane mirrors is replaced by a concave mirror, and the screen placed beyond the focus, then the center is dark rather than light and, in fact, the entire interference pattern is reversed. Thus it appears that the light goes through a phase shift of m when passing through a focus. A focus can be thought of as the coincidence of two focal lines. Thus, as Gouy points out, we can formulate the above result as saying that light goes through a phase shift of TT/2 when passing through a focal line. This phenomenon was explained by Poincare in his lectures of 1891-1892 on the theory of light, cf. [2]. His method was to apply an asymptotic evaluation of certain integrals arising in the solution of the wave equation. This asymptotic evaluation was invented earlier by Kelvin and is known as the method of stationary phase. We now describe this discussion. Instead of dealing with mirrors, we shall first assume that we have a surface emitting radiation of high frequency. Also, to simplify the discussion, we will treat the scalar wave equation rather than the vector equations of Maxwell. The discussion extends easily to the vector case. Let us consider spherically symmetric solutions of the wave equation / a 2 A \ U A a2 a2 92 I —z - A )u = 0 where A = —= + —= + —r \dt2 I dx2 dy2 dz2 http://dx.doi.org/10.1090/surv/014/01
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