Green’s functions, propagation invariants, reciprocity theorems, wave-field representations and propagator matrices in two-dimensional time-dependent materials

The study of wave propagation and scattering in time-dependent materials is a rapidly growing field of research. Whereas for one-dimensional applications, there is a simple relation between the wave equations for space-dependent and time-dependent materials, this relation is less straightforward for multi-dimensional materials. This article discusses fundamental aspects of two-dimensional electromagnetic and acoustic wave propagation and scattering in homogeneous, time-dependent materials. This encompasses a review of transmission and reflection at a single time boundary, a discussion of the Green’s function and its symmetry properties in a piecewise continuous time-dependent material, a discussion of propagation invariants (including the net field-momentum density), general reciprocity theorems and wave field representations. Analogous to the well-known expression for Green’s function retrieval by time-correlation of passive measurements in a space-dependent material, an expression is derived for Green’s function retrieval by space-correlation of passive measurements in a time-dependent material. This article concludes with the discussion of the propagator matrix for a piecewise continuous time-dependent material, its symmetry properties and its relation with the Green’s function.