Quantum-mechanical analogy and supersymmetry of electromagnetic wave modes in planar waveguides

Abstract

We study an analog of the equations describing transverse electric (TE) and transverse magnetic (TM) modes in a planar waveguide with an arbitrary continuous dependence of the electric permittivity and magnetic permeability on coordinates with the stationary Schr\"odinger equation. The effective potential energies involved in the Schr\"odinger equation for TE and TM modes are found. In general, the effective potential energies for TE and TM modes are different but in the limit of a weak dependence of the permittivity and permeability on coordinates they coincide. In the case when the product of a position-dependent permittivity and permeability is constant, it means that the refractive index is constant, and we find that the TE and TM modes are described by the supersymmetric quantum mechanics.