There are two books on functions which are special cases of the Heun Equation: Mathieusche Funktionen und Sphaeroidfunktionen mit anwendungen auf physikalische und technische Probleme by Joseph Meixner and Friedrich Wilhelm Schaefke, published by Springer Verlag in 1954 [16] and a Dover reprint of a book first published in 1946, Theory and Applications of Mathieu Functions by N.

The wave equation for the scalar particle in the background of the Eguchi-Hanson metric [3] in four dimensions has hypergeometric functions as solutions [26], whereas the Nutku helicoid [27, 28] metric, the next higher one, gives us Mathieu functions [29], a member of the Heun function set, if the method of separation of variables is used to get a solution.

In the next sections we will outline the one coupled mode approximation (OCMA), the Mathieu functions approximation, and the theory of finite periodic systems (TFPS).

In the next sections we will present the Mathieu functions approach and the theory of finite periodic systems and, with the purpose of comparing with the results shown in Figure 1, we will calculate the same physical quantities for a system similar to that considered here.

The results obtained will be compared with those obtained by using Kogelnik's coupled wave theory and a matrix method which gives exact solutions in terms of Mathieu functions. In Sections 3, 4, and 5, the general solution is particularized for the cases of a homogeneous dielectric slab (no index modulation), an index-matched dielectric grating, and a partially index-matched dielectric grating, respectively.

The matrix method gives exact results for the efficiency of the first diffracted order in terms of Mathieu functions. The parameters used in the simulations throughout the text, such as the refractive index, the refractive index modulation, or the thickness of the layer, were chosen to represent reflection diffraction gratings recorded in photopolymers [20-23].

The elements of the layer matrix which characterizes the reflection grating are calculated in terms of Mathieu functions. In addition the band structure for a general non-slanted reflection grating is obtained by using the layer matrix corresponding to one single period demonstrating that no stop bands exist and that the band edges coincide with the Bragg angles of Kogelnik's Theory.

Portugal, "Algebraic methods to compute Mathieu functions," J.

The solution of such an equation invariably leads to Mathieu functions [Alhargan 1996; Alhargan and Judah 1992; McLachlan 1947; Morse and Feshbach 1953; NBS 1967].

Many algorithms have been devised to compute Mathieu functions with various degrees of success [Alhargan 1996; Arscott and Shymansky 1978; Blanch 1966; Leeb 1979; McLachlan 1947; Morse and Feshbach 1953; NBS 1967; Rengarajan and Lewis 1980; Shirts 1993a; 1993b; Toyama and Shogan 1984; Wimp 1984].

This article presents the outline of a set of routines that enable the computation of all Mathieu functions of integer orders for large range of the order n and the parameter h.

The accuracy of the results can be checked using the Wronskians for Mathieu functions, as the algorithms for computing Mathieu functions are not based on the Wronskian of the functions.