Orthogonality relations for the eigenmodes of lossy anisotropic waveguides (fibres)

Electromagnetic Radiation From Sources Embedded in a Cylindrically Stratified Unbounded Gyrotropic Medium

“…Similarly, two quantities n k , regarded as functions of p and given by (19), can be replaced by the corresponding functions of q if we substitute q and q α (q) for q 1,2 , and p α (q) for p in (19).…”

“…It can be shown that the total field yielded by summing (or integrating) modes over the found values of q will satisfy the radiation condition at infinity (r = (ρ 2 + z 2 ) 1/2 → ∞) [3,19,33]. Note that in the considered case, conditions (43) turn out to be sufficient for finding the eigenvalues q and the corresponding modes in contrast to the case of an isotropic outer medium where, along with the boundedness conditions, certain additional conditions should be imposed on the desired modal fields [10,16,19,33,34].…”

“…Note that in such a case, the superscript (T) may be omitted in formulas representing orthogonality relations and modal expansion coefficients. In addition, it is worth mentioning that the discrete-and continuousspectrum modes in the case discussed in this section can be obtained by an alternative approach based on the so-called S-operator method (see for details [19] and references therein).…”

“…Finally, we note that the above-derived orthogonality relations can be generalized to be valid for transversely unbounded modes, as well. To do this, one should use a method similar to that employed in [19,[37][38][39] for open waveguides located in a nongyrotropic medium. The generalized orthogonality relations turn out to be useful when analyzing leaky modes.…”

“…Open gyrotropic waveguides surrounded by an isotropic outer medium have been discussed in [10,[14][15][16][17][18]. The case where the outer medium is anisotropic is considered in [19]. Recently, open gyrotropic guiding structures located in a gyrotropic background medium have attracted considerable interest [3,20].…”