Parity-time ($\mathcal{PT}$)-symmetric wave devices, which exploit balanced interactions between material gain and loss, exhibit extraordinary properties, including lasing and flux-conserving scattering processes. In a seemingly different research field, periodically driven systems, also known as time-Floquet systems, have been widely studied as a relevant platform for reconfigurable active wave control and manipulation. In this article, we explore the connection between $\mathcal{PT}$-symmetry and parametric time-Floquet systems. Instead of relying on material gain, we use parametric amplification by considering a time-periodic modulation of the refractive index at a frequency equal to twice the incident signal frequency. We show that the scattering from a simple parametric slab, whose dynamics follows the Mathieu equation, can be described by a $\mathcal{PT}$-symmetric scattering matrix, whose $\mathcal{PT}$-breaking threshold corresponds to the Mathieu instability threshold. By combining different parametric slabs modulated out of phase, we create $\mathcal{PT}$-symmetric time-Floquet systems that feature exceptional scattering properties, such as coherent perfect absorption (CPA)-laser operation and bidirectional invisibility. These bidirectional properties, rare for regular $\mathcal{PT}$-symmetric systems, are related to a compensation of parametric amplification due to multiple scattering between two parametric systems modulated with a phase difference.

• Received 22 November 2017

DOI:https://doi.org/10.1103/PhysRevA.97.013839