Exact and Closed-Form Cutoff Wavenumbers of Elliptical Dielectric Waveguides

The cutoff wavenumbers of the elliptical dielectric waveguide are calculated exactly and analytically. Two separate methods are used to solve this problem. The first method is based on the separation of variables technique using Mathieu functions and gives the exact cutoff wavenumbers. The system matrices of which the roots of their determinant should be determined are complicated because of the nonexistence of orthogonality relations for Mathieu functions, due to the different constitutive parameters between the core and the cladding of the fiber. In the second method, the cutoff wavenumbers are obtained through analytical expressions, when the eccentricity
$h$
of the elliptical core is specialized to small values. In the latter case, analytical closed-form algebraic expressions, free of Mathieu functions, are obtained for the expansion coefficients
$g_{sn}^{(2)}$
in the resulting relation
$x_{1,sn}(h)=x_{1,sn}(0)[1+g_{sn}^{(2)}h^{2}+{\cal O}(h^{4})]$
for the cutoff wavenumbers, where
$x_{1,sn}(0)$
are the normalized cutoff wavenumbers of the circular dielectric waveguide. These expressions are valid for every different value of
$s$
and
$n$
, corresponding to every higher order hybrid mode. Numerical results are given for various higher order modes, as well as a comparison with the exact solution.