Electromagnetic wave propagation in media whose permittivity varies periodically in time

We have developed a general theory for propagation of plane electromagnetic waves in a medium with permittivity that is varying periodically in time. The Bloch-Floquet theorem dictates that these are a superposition of harmonic modes whose frequencies differ by $2\pi /T$, where $T$ is the period of $\varepsilon (t)$. For arbitrary periodicity, the dispersion relation $\omega (t)$ for the ``Bloch frequency'' is given in terms of the roots of an infinite determinant whose elements depend on the Fourier coefficients of $\varepsilon (t)$. For small variation of $\varepsilon (t)$ around an average $\varepsilon _0 $,$\omega (t)$ is characterized by regions of the wave vector $k$ that are forbidden for propagation. These are centered at \textit{$\omega $} and $k$ values that are, respectively, integer multiples of $\pi /T$ and of $\pi \varepsilon _0^{1/2} /cT$. The widths of the gaps are proportional to the corresponding Fourier coefficients of $\varepsilon (t)$. In the special case of square-periodic variation of $\varepsilon (t)$, there is no need to recur to a perturbational calculation, because the dispersion relation can be derived analytically, with no approximations. Again, we find wave vectors gaps whose edges are located at the frequencies $\omega =0,\pi /T,2\pi /T,...$ .