Photonic crystals have revolutionized the field of optics with their unique dispersion and energy band gap engineering capabilities, such as the demonstration of extreme group and phase velocities, topologically protected photonic edge states, and control of spontaneous emission of photons. Time-variant media have also shown distinct functionalities, including nonreciprocal propagation, frequency conversion, and amplification of light. However, spatiotemporal modulation has mostly been studied as a simple harmonic wave function. Here, we analyze time-variant and spatially discrete photonic crystal structures, referred to as spatiotemporal crystals. The design of spatiotemporal crystals allows engineering of the momentum band gap within which parametric amplification can occur. As a potential platform for the construction of a parametric oscillator, a finite-sized spatiotemporal crystal is proposed and analyzed. Parametric oscillation is initiated by the energy and momentum conversion of an incident wave and the subsequent amplification by parametric gain within the momentum band gap. The oscillation process dominates over frequency mixing interactions above a transition threshold determined by the balance between gain and loss. Furthermore, the asymmetric formation of momentum band gaps can be realized by spatial phase control of the temporal modulation, which leads to directional radiation of oscillations at distinct frequencies. The proposed structure would enable simultaneous engineering of energy and momentum band gaps and provide a guideline for implementation of advanced dispersion-engineered parametric oscillators.

中文翻译：

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时空晶体动量带隙中电磁波的参数振荡

光子晶体凭借其独特的色散和能带隙工程设计能力，彻底改变了光学领域，例如演示了极高的群和相速度，受拓扑保护的光子边缘态以及对光子自发发射的控制。时变媒体还显示出独特的功能，包括不可逆传播，频率转换和光放大。但是，时空调制主要作为简单的谐波函数进行研究。在这里，我们分析时变和空间离散的光子晶体结构，称为时空晶体。时空晶体的设计允许对动量能带隙进行工程设计，在其中可以进行参数放大。作为构建参数振荡器的潜在平台，提出并分析了时空有限晶体。参数振荡由入射波的能量和动量转换以及随后由动量带隙内的参数增益进行的放大引发。在高于由增益和损耗之间的平衡确定的过渡阈值的混频相互作用中，振荡过程占主导地位。此外，动量带隙的不对称形成可以通过时间调制的空间相位控制来实现，这导致了不同频率的振荡的定向辐射。所提出的结构将能够同时设计能量和动量带隙，并为实施先进的色散工程参数振荡器提供指导。参数振荡由入射波的能量和动量转换以及随后由动量带隙内的参数增益进行的放大引发。在高于由增益和损耗之间的平衡所确定的过渡阈值之上的混频相互作用中，振荡过程占主导地位。此外，动量带隙的不对称形成可以通过时间调制的空间相位控制来实现，这导致了不同频率的振荡的定向辐射。所提出的结构将能够同时设计能量和动量带隙，并为实施先进的色散工程参数振荡器提供指导。参数振荡由入射波的能量和动量转换以及随后由动量带隙内的参数增益进行的放大引发。在高于由增益和损耗之间的平衡所确定的过渡阈值之上的混频相互作用中，振荡过程占主导地位。此外，动量带隙的不对称形成可以通过时间调制的空间相位控制来实现，这导致了不同频率的振荡的定向辐射。所提出的结构将能够同时设计能量和动量带隙，并为实施先进的色散工程参数振荡器提供指导。