The KdV hierarchy in optics

Abstract

There is a well explored relationship between quantum mechanical scattering from a potential and the Korteweg–de Vries (KdV) equation of fluid dynamics: if the potential is ‘evolved’ according to the KdV equation then it will have the same reflectivity and transmissivity as a function of energy, for each snapshot in time. In this work we explore this connection in optics, where the permittivity plays the role of the potential. We begin by deriving the relationship between the Helmholtz equation and the KdV equation in terms of the current induced in a material when a permittivity profile is changed slightly. It is then shown that the KdV equation can be used to design a plethora of bounded complex potentials that are relfectionless from both sides for all angles of incidence, and planar periodic media that exhibit a real Bloch vector for all angles of propagation. Finally we apply the KdV equation to reduce the reflection of a wave from an interface between two media of differing refractive indices