Abstract

Waves and their interactions with time-periodic media have recently been of theoretical interest for their unusual properties. It has been revealed that such interactions lead to momentum gaps at $\omega = N\Omega /2$, where $\omega$ is the wave frequency, $\Omega$ is the modulation frequency, and $N \in {\mathbb Z}$. While waves can propagate at the boundaries of the momentum gaps, it has been shown that waves exhibit parametric amplification in the momentum gaps due to the active nature of such temporally dynamic media. In this paper, the coupled waves theory is used for the modeling of the parametric amplification phenomena in time-periodic optical media. Simple formulas are derived using the 0th and first Fourier components of $\theta (t) = 1/\varepsilon (t)$, approximating the results of the cumbersome stability analysis of the associated hypergeometric equation of the electric displacement field and providing the conditions to define the wave parameters, which can fall in the momentum gap, resulting in parametric amplification or allow propagating modes. Numerical results are compared with the Hill’s determinant of the Floquet theory and FDTD simulations showing that the coupled waves theory can efficiently model the first and second momentum gaps. Further extensions can be made to higher-order gaps, by considering higher-order harmonics and following the same steps of this analysis.

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