TM-modes in a planar optical waveguide with a graded index of the symmetric Epstein type

The eigenvalue problem for the TM-modes in a planar optical waveguide is rigorously solved by analytical and numerical methods in the case of a symmetric Epstein-type variation of the dielectric permittivity. It is shown that there exists a finite number of guided TM-modes. The TM-modal fields are expressed in terms of Heun functions and the corresponding eigenvalues (or propagation coefficients) are numerically determined from the minimal solution of a three-term recurrence relation. Upper and lower bounds are established for the difference between corresponding eigenvalues of the TE- and TM-modes