Growing fields in a temporal photonic (time) crystal with a square profile of the permittivity $ε(t)$

We investigate a band structure $ω(k)$ of a photonic time crystal with periodic square (step) modulation in time of its permittivity $ε(t)$⁠, oscillating between the value $ε1$ (sustained for a fraction of time τ of the period) and the value $ε2$ [fraction (1 − τ)]. The strength of modulation is $m=(ε1−ε2)/(ε1+ε2)$⁠. We find that $ω(k)$ can be periodic in a wave number k (in addition to the frequency ω), provided that a certain function $f(m,τ)$ of the parameters m and τ is an irreducible rational number. However, even for arbitrary values of m and τ, $f(m,τ)$ can be approximated by a fractional number to any desired degree of periodicity. Hence, for square modulation, a photonic band structure is necessarily periodic or quasi-periodic in the wave number. Moreover, for appropriate sets of the parameters m and τ, the modes associated with k values within the band gaps can have identical values of the imaginary part of ω. For simultaneous excitation of these modes, all the fields would grow in time at the same rate, resulting in powerful amplification.