Photonic time crystals (PTC) arise in time-modulated media. PTCs are manifested by the generation and amplification of so-called “time reversed” waves propagating in the direction opposite the incoming light. Superficially, the observed phenomenon bears a resemblance to the widely known phenomena of optical parametric generation (OPG) and amplification (OPA) using second- or third-order optical nonlinearities. I show that while indeed the same physical mechanism underpins both PTC and OPA, the difference arises from the boundary conditions. Thus, while dispersion for both PTC and OPA exhibits the same bandgap in momentum space, only in the case of PTC can one have propagation in that bandgap with exponential amplification. I also show that PTC can be engineered with both second- and third-order nonlinearities, and, rather unexpectedly, modulating permittivity in the ultrafast (few femtoseconds) rate is not a necessity. Also, one can emulate all the PTC features using materials with a few picoseconds response time commensurate with the propagation time through the medium.

中文翻译：

---

光子时间晶体和参量放大：相似性和区别


光子时间晶体（PTC）出现在时间调制介质中。 PTC 的表现是产生并放大所谓的“时间反转”波，该波以与入射光相反的方向传播。从表面上看，观察到的现象与众所周知的使用二阶或三阶光学非线性的光学参量生成（OPG）和放大（OPA）现象相似。我表明，虽然 PTC 和 OPA 的物理机制确实相同，但差异是由边界条件引起的。因此，虽然 PTC 和 OPA 的色散在动量空间中表现出相同的带隙，但只有在 PTC 的情况下，才能在该带隙中以指数放大的方式进行传播。我还表明，PTC 可以设计为具有二阶和三阶非线性，而且出人意料的是，在超快（几飞秒）速率下调制介电常数并不是必需的。此外，还可以使用具有几皮秒响应时间（与通过介质的传播时间相当）的材料来模拟所有 PTC 功能。