Parametric properties of fields in a slab of time-varying permittivity

Plane waves in a nonconducting charge-free medium with a permittivity given by can be written as products of Mathieu functions and exponentials . The behavior of the waves, which depends on the choice of the wave number , is deduced entirely from the properties of the Mathieu functions. In a periodically varying medium of infinite extent, one encounters exponentially increasing waves for certain bands of wave numbers. The existence of such waves in a permittivity-modulated slab immersed in a medium of arbitrary constant permittivity is investigated by attempting to satisfy the condition of continuity of the fields at the interfaces. It is found that the modulation index must exceed a critical value if exponentially increasing waves are to be produced. This critical value is shown to depend on the width of the slab and on the relation of the permittivities of the slab and the surrounding medium. For the case that is less than that critical value, the response of the slab to a normally incident plane wave is described. It is shown that the slab may be designed to act as a parametric oscillator, amplifier, filter, or frequency converter.