Inhomogeneous matching system for single-mode waveguides

The problem of single-frequency matching of two solid rod waveguides of different cross section for minimum reflection coefficient is considered. Matching may be achieved using a rod of constant cross section, of length $l_0=\lambda _{0}(2n +1)/4$, and of area $S_0=(\rho _{1},c_1,\rho _{2}c_2S_1S_2/\rho ^2c^2)^{1/2}$. If the form of the matching rod is close to cylindrical, the controlling function may be written in the for·m of a linear expression in terms of the small parameter $\Delta = k_0\Delta _x + l_0\Delta _{k}:\tilde{u}_{\Delta} (x)=\Delta \cdot \tilde{u}_1 (x)$, where $k_0 = 2\pi / \lambda _0$. The continuous function $\tilde{u}_1 (x)$ is represented by the polynomial $\sum_{i=0}^{N} a_i x^i$, and N = 2 is chosen. As an example, the matching of rods with an area ratio $S_1 : S_2 = 1:10$ by means of a quarter-wave cylindrical wave and rods of varying cross section is considered.