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 epsilon(t) and/or permeability 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 omega(k) for the waves that can propagate in such a dynamic medium. This is a generalization of previous work [J. R. Zurita-Sanchez and P. Halevi, Phys. Rev. A 81, 053834 (2010)] thatwas restricted to the electric modulation epsilon(t). For our numericalwork (only) we assumed the harmonic modulations epsilon(t) = (epsilon) over bar [1 + m(epsilon) sin(Omega t)] and mu(t) = (mu) over bar [1 + m(mu) sin(Omega t + theta)], where Omega is the circular modulation frequency; m(epsilon) and m(mu) are, respectively, the strengths of the electric and magnetic modulations; and theta is the phase difference between these modulations. An analytic calculation for weak modulations (m(epsilon) << 1, m(mu) << 1) leads to two k bands, k(1)(omega) and k(2)(omega), that are separated by a k gap. If the modulations are in phase (theta = 0), this gap is proportional to vertical bar m(epsilon) - m(mu)vertical bar, while the gap is proportional to (m(epsilon) + m(mu)) if the modulations are out of phase (theta = pi). The gap thus disappears for equal, in-phase, modulations (m(epsilon) = m 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(epsilon,mu) less than or similar to 1). The photonic band structure k(omega) is periodic in omega, with period Omega, and there is an infinite number of bands k(1)(omega), k(2)(omega), ... Further, by allowing epsilon(t) and mu(t) to have imaginary parts, we examined the effects of damping [Im k(omega)] on the k bands. We also determined the optical response of a temporal photonic crystal slab, applying the above harmonic model for epsilon(t) and mu(t). The reflected and transmitted light represent a frequency comb of frequencies omega, vertical bar omega +/- Omega vertical bar, vertical bar omega +/- 2 Omega vertical bar, ... The transmission coefficients T-n (omega) for these harmonics n Omega of the modulation frequency strongly depend on the parameters m(epsilon), m(mu), and 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, T-n(omega) exhibits oscillations with peaks that resemble parametric resonances, rather than the usual Fabry-Perot resonances.