arXiv 1502.072352015-02math-phmath.MP

利用半经典极限推导光子晶体中的光线光学方程

Derivation of Ray Optics Equations in Photonic Crystals Via a Semiclassical Limit

Giuseppe De Nittis, Max Lein

备注：: 49 pages, added Section 5.2 with a more detailed discussion of previous results, to appear in Annales of Henri Poincaré

摘要：

在这项工作中，我们提出了一种新的方法来射线光学极限：我们重写的动态麦克斯韦方程组薛定谔形式和证明叶戈罗夫型定理，一个强大的半经典技术。我们实现了这个计划的周期性光导体，光子晶体，从而使量子光之间的类比半经典的布洛赫电子和射线光学在光子晶体严格。我们的主要结果，定理3.3和4.1，给出了二次观测量的射线光学极限，其中，适用于局部平均的能量密度，坡印廷矢量和麦克斯韦应力张量。我们的是第一次严格推导的射线光学方程，其中包括所有的子领先阶项，其中一些也是新的物理文献。虽然射线光学极限，我们最初证明（定理3.3）适用于任何拓扑类的光子晶体，我们也考虑了真正的电磁场在非回旋光子晶体传播的射线光学极限。这样的扩展是不平凡的，因为实场的射线光学极限必然是一个多带问题。

英文摘要：

In this work we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. Our main results, Theorems 3.3 and 4.1, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all sub-leading order terms, some of which are also new to the physics literature. While the ray optics limit we prove initially (Theorem 3.3) applies to photonic crystals of any topological class, we also consider the ray optics limit for real electromagnetic fields propagating in non-gyrotropic photonic crystals. Such an extension is non-trivial, because the ray optic limit for real fields is necessarily a multiband problem.

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