Temporal photonic (time) crystal with a square profile of both permittivity ɛ(t) and permeability μ(t)

We present a comprehensive study of a ``temporal photonic crystal'' or ``time crystal'' with a square profile of both its permittivity $\ensuremath{\varepsilon}(t)$ and permeability $\ensuremath{\mu}(t)$. The continuity of the displacement field $D(t){e}^{ikx}$ and magnetic field $B(t){e}^{ikx}$(where $k$ is the wave number) across the (simultaneous) discontinuities of $\ensuremath{\varepsilon}(t)$ and $\ensuremath{\mu}(t)$ facilitates the Kr\"onig-Penney methodology of Solid State physics, leading to an analytic photonic band structure (PBS) that relates the wave frequency $\ensuremath{\omega}$ to $k$. It is periodic in $\ensuremath{\omega}$, with the period given by the modulation frequency $\mathrm{\ensuremath{\Omega}}$ and exhibits $k$ bands separated by $k$ gaps. The PBS depends qualitatively on three parameters: the strengths of the electric and magnetic modulations ${m}_{\ensuremath{\varepsilon}}$ and ${m}_{\ensuremath{\mu}}$ and $\ensuremath{\tau}={t}_{1}/({t}_{1}+{t}_{2})$ associated with the intervals ${t}_{1}$ and ${t}_{2}$ that comprise a modulation period $T(=2\ensuremath{\pi}/\mathrm{\ensuremath{\Omega}})$. For equal electric and magnetic modulations the PBS is composed of straight lines and there are no $k$ gaps. This can be explained by the fact that for ${m}_{\ensuremath{\varepsilon}}={m}_{\ensuremath{\mu}}$ the wave impedance is continuous at the abrupt interfaces of $\ensuremath{\varepsilon}(t)$ and $\ensuremath{\mu}(t)$; hence no reflections occur. Conversely, the existence of gaps for ${m}_{\ensuremath{\varepsilon}}\ensuremath{\ne}{m}_{\ensuremath{\mu}}$ can then be associated with diffraction occurring for the frequencies $\ensuremath{\omega}=(n/2)\mathrm{\ensuremath{\Omega}}$ in the presence of discontinuities of the wave impedance. In comparison to harmonic modulation, where only the first gap is appreciable, for square modulation large gaps, that increase with the difference $|{m}_{\ensuremath{\varepsilon}}\ensuremath{-}{m}_{\ensuremath{\mu}}|$, exist even between distant bands. In the particular case ${m}_{\ensuremath{\mu}}=\ensuremath{-}{m}_{\ensuremath{\varepsilon}}$ [with $\ensuremath{\varepsilon}(t)$ and $\ensuremath{\mu}(t)$ oscillating out of phase], all the gaps have equal widths. The field $D(t)$ displays the Bloch-Floquet behavior, namely, oscillations of frequency $\mathrm{\ensuremath{\Omega}}$ being modulated by an envelope of frequency $\ensuremath{\omega}(<\mathrm{\ensuremath{\Omega}})$. For $\ensuremath{\omega}=(n/2)\mathrm{\ensuremath{\Omega}}$, the fields are standing waves. We also studied the optical response to a monochromatic wave incident at the modulated slab, namely, the spectral behavior of the frequency combs that are transmitted and reflected by the slab, as well as the field profile inside the slab. Especially interesting is the case of equal modulations ${m}_{\ensuremath{\varepsilon}}={m}_{\ensuremath{\mu}}$ where only the fundamental harmonic $n=0$ is transmitted if the modulations are out of phase, while no light at all is reflected if they are in phase.