Generalization

Abstract

In this chapter we describe the method of obtaining the various generalizations of the Bremmer series by a successive diagonalization procedure of the transport equations for the u and v beams obtained at each stage [1]. This was carried out by various authors [2], [3], and [4] for the wave equation, without the source term. We study these methods and see how a unified picture of these procedures emerges. We also illustrate how these are related to the method of applying the Liouville transformations successively to the wave equation [5]. We also treat the wave equation with a source term and demonstrate the technique of arriving at different types of generalized Bremmer series. The diagonalizing matrix at each successive stage of the process represents the manner in which the wave function can be split up at that stage. As we have seen already in Chapter V, better and better approximations can be obtained for the wave function even with the first term of the generalized Bremmer series, if the splitting of the wave function involves higher and higher terms of the Eikonal approximation [6]. It is seen that for each term to be finite, it is imperative that the k(x) should be nonzero, i.e., the refractive index should remain positive in the region of interest.