光波包动力学中的几何问题
Geometrical Aspects in Optical Wavepacket Dynamics
摘要：
我们建立了光波包在介电常数和磁导率具有周期性结构的非导电介质中传播的半经典理论，即，非导电光子晶体。我们采用量子力学的形式主义，以澄清其联系的电子系统。它涉及几何相位，即，Berry相位，以一种自然的方式，描述了轨道运动和内部旋转之间的相互作用。在此基础上，我们讨论了光学霍尔效应的几何特性。我们还考虑了一个减少的理论，没有周期性结构的系统，并将其应用到在界面反射/折射的横向位移。对于椭圆偏振的一般入射光束，每个反射/透射光束的横向位移的相同结果由以下不同的方法给出：（i）波包动力学的分析评估，（ii）总角动量（TAM）守恒{ \it为单个光子}，和（iii）波包动力学的数值模拟。这与经典电动力学的一个结果是一致的。这意味着单个光子的TAM守恒已经在波动光学，即经典电动力学中考虑了。最后，我们展示了我们的理论的应用到一个二维光子晶体，并提出了一个优化设计的光学霍尔效应的增强光子晶体。
英文摘要：
We construct a semiclassical theory for propagation of an optical wavepacket in non-conducting media with periodic structures of dielectric permittivity and magnetic permeability, i.e., non-conducting photonic crystals. We employ a quantum-mechanical formalism in order to clarify its link to those of electronic systems. It involves the geometrical phase, i.e., Berry phase, in a natural way, and describes an interplay between orbital motion and the internal rotation. Based on the above theory, we discuss the geometrical aspects of the optical Hall effect. We also consider a reduction of the theory to a system without periodic structure and apply it to the transverse shift at an interface reflection/refraction. For generic incident beams with elliptic polarizations, an identical result for the transverse shift of each reflected/transmitted beam is given by the following different approaches; (i) analytic evaluation of wavepacket dynamics, (ii) total angular momentum (TAM) conservation {\it for individual photons}, and (iii) numerical simulation of wavepacket dynamics. It is consistent with a result by classical electrodynamics. This means that the TAM conservation for individual photons is already taken into account in wave optics, i.e, classical electrodynamics. Finally, we show an application of our theory to a two-dimensional photonic crystal, and propose an optimal design for the enhancement of the optical Hall effect in photonic crystals.
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