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Parity-Time (PT) symmetric wave devices, which exploit balanced interactionsbetween material gain and loss, exhibit extraordinary properties, includinglasing and flux-conserving scattering processes. In a seemingly differentresearch field, periodically driven systems, also known as time-Floquetsystems, have been widely studied as a relevant platform for reconfigurableactive wave control and manipulation. In this article, we explore theconnection between PT-symmetry and parametric time-Floquet systems. Instead ofrelying on material gain, we use parametric amplification by considering atime-periodic modulation of the refractive index at a frequency equal to twicethe incident signal frequency. We show that the scattering from a simpleparametric slab, whose dynamics follow Mathieu equation, can be described by aPT-symmetric scattering matrix, whose PT-breaking threshold corresponds to theMathieu instability threshold. By combining different parametric slabsmodulated out-of-phase, we create PT-symmetric time-Floquet systems thatfeature exceptional scattering properties, such as CPA/Laser operation andbidirectional invisibility. These bidirectional properties, rare for regularPT-symmetric systems, are related to a compensation of parametric amplificationdue to multiple scattering between two parametric systems modulated with aphase difference.

Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems