Appendix 5.B: Solution of the Eigenvalue Equations

We discuss in this appendix the solution of the transcendental equation (5.A14) and (5.A26) for the eigenvalues and , respectively. Equations of this type, which are commonly encountered in the analysis of inhomogeneously filled waveguides, must be solved numerically. In view of the fact that the transcendental functions involved have poles interspersed between the zeros, it is rather difficult to write a reliable computer program for the computation of the eigenvalues. As Maystre et al. [17] point out, missing just one eigenvalue can lead to completely erroneous final results in the mode-matching method. These authors, who only consider the TE case, describe a systematic procedure for the solution of (5.A26), which is, however, more complicated and less efficient than the procedure given below.

Referring to Fig. 5.2, we assume that the middle layer has the highest dielectric constant, i.e., ? ₂ > ? ₁ and ? ₂ > ? ₃. Since in the mode matching procedure the middle region will correspond to the high- ? dielectric sample, this condition will be satisfied in most practical cases.

Turning attention to the solution of (5A.14), we consider the integral:

(5.B1)

where, in the integration by parts, the integrated terms vanish in view of (5.22b). (Here, as elsewhere in this chapter, the prime denotes a derivative with respect to the argument.) Evidently, since the eigenfunctions are real, < 0. On the other hand, by using (5.22a) we can...