Wave propagation and dispersion in space-time periodic media

The dispersion characteristics of electromagnetic fields are examined for waves propagating in a medium with a permittivity which is modulated periodically with respect to time and one spatial co-ordinate. By taking a guided-wave approach, which expresses the fields of arbitrary sources in terms of a superposition of modal solutions, it is shown that many properties of the individual modes may be inferred by means of wavenumber diagrams. These diagrams are easily constructed for the special case of a vanishingly small periodic modulation. By using coupled-mode consideration, this special case is extended to serve as a good approximation for finite modulation amplitudes. In addition to yielding information on the dispersion character of modal fields, it is shown that the wavenumber diagrams may be employed to extract the modal constituents of a plane wave scattered by a layer containing a space-time periodic medium. The principal far-field components of waves excited by a localised arbitrary source embedded in such a medium may also easily be obtained from the wavenumber diagrams. The guided-wave approach, together with its associated wavenumber diagrams, is thus shown to serve as a powerful tool in analysing and understanding a large class of phenomena, which includes the diffraction of light by sound and parametric effects in nonlinear media, as well as other aspects of wave interactions in material bodies.