Temporal photonic crystals with modulations of both permittivity and permeability

We present an in-depth study of electromagnetic wave propagation in a temporal photonic crystal, namely, a nonconducting medium whose permittivity $\ensuremath{\varepsilon}(t)$ and/or permeability $\ensuremath{\mu}(t)$ are modulated periodically by unspecified agents (these modulations not necessarily being in phase). Maxwell's equations lead to an eigenvalue problem whose solution provides the dispersion relation $\ensuremath{\omega}(k)$ for the waves that can propagate in such a dynamic medium. This is a generalization of previous work [J. R. Zurita-S\'anchez and P. Halevi, Phys. Rev. A 81, 053834 (2010)] that was restricted to the electric modulation $\ensuremath{\varepsilon}(t)$. For our numerical work (only) we assumed the harmonic modulations $\ensuremath{\varepsilon}(t)=\overline{\ensuremath{\varepsilon}}[1+{m}_{\ensuremath{\varepsilon}}sin(\mathrm{\ensuremath{\Omega}}t)]$ and $\ensuremath{\mu}(t)=\overline{\ensuremath{\mu}}[1+{m}_{\ensuremath{\mu}}sin(\mathrm{\ensuremath{\Omega}}t+\ensuremath{\theta})]$, where $\mathrm{\ensuremath{\Omega}}$ is the circular modulation frequency; ${m}_{\ensuremath{\varepsilon}}$ and ${m}_{\ensuremath{\mu}}$ are, respectively, the strengths of the electric and magnetic modulations; and $\ensuremath{\theta}$ is the phase difference between these modulations. An analytic calculation for weak modulations $({m}_{\ensuremath{\varepsilon}}\ensuremath{\ll}1,{m}_{\ensuremath{\mu}}\ensuremath{\ll}1)$ leads to two $k$ bands, ${k}_{1}(\ensuremath{\omega})$ and ${k}_{2}(\ensuremath{\omega})$, that are separated by a $k$ gap. If the modulations are in phase $(\ensuremath{\theta}=0)$, this gap is proportional to $|{m}_{\ensuremath{\varepsilon}}\ensuremath{-}{m}_{\ensuremath{\mu}}|$, while the gap is proportional to $({m}_{\ensuremath{\varepsilon}}+{m}_{\ensuremath{\mu}})$ if the modulations are out of phase $(\ensuremath{\theta}=\ensuremath{\pi})$. The gap thus disappears for equal, in-phase, modulations $({m}_{\ensuremath{\varepsilon}}={m}_{\ensuremath{\mu}})$. An exact solution of the eigenvalue equation confirms that these approximations hold reasonably well even for moderate modulations. In fact, there are no $k$ gaps for equal modulations even if these are very strong $({m}_{\ensuremath{\varepsilon},\ensuremath{\mu}}\ensuremath{\lesssim}1)$. The photonic band structure $k(\ensuremath{\omega})$ is periodic in $\ensuremath{\omega}$, with period $\mathrm{\ensuremath{\Omega}}$, and there is an infinite number of bands ${k}_{1}(\ensuremath{\omega})$, ${k}_{2}(\ensuremath{\omega}),...$ Further, by allowing $\ensuremath{\varepsilon}(t)$ and $\ensuremath{\mu}(t)$ to have imaginary parts, we examined the effects of damping $[\mathrm{Im}\phantom{\rule{4pt}{0ex}}k(\ensuremath{\omega})]$ on the $k$ bands. We also determined the optical response of a temporal photonic crystal slab, applying the above harmonic model for $\ensuremath{\varepsilon}(t)$ and $\ensuremath{\mu}(t)$. The reflected and transmitted light represent a frequency comb of frequencies $\ensuremath{\omega}$, $|\ensuremath{\omega}\ifmmode\pm\else\textpm\fi{}\mathrm{\ensuremath{\Omega}}|$, $|\ensuremath{\omega}\ifmmode\pm\else\textpm\fi{}2\mathrm{\ensuremath{\Omega}}|,...$ The transmission coefficients ${\mathcal{T}}_{n}(\ensuremath{\omega})$ for these harmonics $n\mathrm{\ensuremath{\Omega}}$ of the modulation frequency strongly depend on the parameters ${m}_{\ensuremath{\varepsilon}}$, ${m}_{\ensuremath{\mu}}$, and $\ensuremath{\theta}$, as well as on the thickness of the slab. Moreover, they can much exceed unity, as a result of energy transfer from the source of modulation. In a particularly interesting case, ${\mathcal{T}}_{n}(\ensuremath{\omega})$ exhibits oscillations with peaks that resemble parametric resonances, rather than the usual Fabry-Perot resonances.