Analytic treatment of the polariton problem for a smooth interface

We study the polariton problem for smooth boundaries, i.e. whether or not there exist some localized solutions of Maxwell's equations for different types of smooth spatial variation of complex dielectric- and/or magnetic-permittivity tensors. For a particular dielectric permittivity profile varying according to the hyperbolic tangent law, the singular term is mathematically strongly taken into account in Maxwell's equations, not in the boundary conditions. The problem is then reduced to the canonical form of Heun's equation possessing four regular singular points. The solution to Heun's equation as a power series is constructed, and an approximate solution involving a combination of two incomplete beta-functions is derived. Further, the exact eigenvalue solution to the polariton problem as a series in terms of incomplete beta-functions or, equivalently, Gauss hypergeometric functions is constructed. It is shown that the dispersion relation for the polariton wavenumber does not depend on the interface transition layer width, i.e. it is always exactly the same as the one derived in the limit of abrupt interface. We conjecture that the polariton wavenumber eigenvalue depends on either zeros of the dielectric permittivity variance profile or the poles of the logarithmic derivative of the latter.