From a three-dimensional boundary value problem for the time harmonic classical Maxwell equations, we derive the dispersion relation for a surface wave, the edge plasmon-polariton (EP), that is localized near and propagates along the straight edge of a planar, semi-infinite sheet with a spatially homogeneous, scalar conductivity. The sheet lies in a uniform and isotropic medium; and serves as a model for some two-dimensional (2D) conducting materials such as the doped monolayer graphene. We formulate a homogeneous system of integral equations for the electric field tangential to the plane of the sheet. By the Wiener-Hopf method, we convert this system to coupled functional equations on the real line for the Fourier transforms of the fields in the surface coordinate normal to the edge, and solve these equations exactly. The derived EP dispersion relation smoothly connects two regimes: a low-frequency regime, where the EP wave number, $q$, can be comparable to the propagation constant, $k_0$, of the ambient medium; and the nonretarded frequency regime in which $|q|\gg |k_0|$. Our analysis indicates two types of 2D surface plasmon-polaritons on the sheet away from the edge. We extend the formalism to the geometry of two coplanar sheets.

中文翻译：

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各向同性半无限导电片上的边缘等离子体激元

从时间谐波经典麦克斯韦方程的三维边界值问题中，我们推导出表面波的色散关系，即边缘等离子体极化子 (EP)，它位于平面半平面的直边附近并沿其直边传播。 -具有空间均匀、标量电导率的无限片。薄片位于均匀且各向同性的介质中；并用作一些二维 (2D) 导电材料的模型，例如掺杂的单层石墨烯。我们为与片材平面相切的电场制定了齐次积分方程组。通过Wiener-Hopf方法，我们将该系统转化为实线上的耦合函数方程，用于表面坐标垂直于边缘的场的傅立叶变换，并精确求解这些方程。导出的 EP 色散关系平滑地连接了两个区域：低频区域，其中 EP 波数 $q$ 可以与环境介质的传播常数 $k_0$ 相媲美；以及 $|q|\gg |k_0|$ 的非延迟频率状态。我们的分析表明远离边缘的片材上有两种类型的 2D 表面等离子体激元。我们将形式主义扩展到两个共面片的几何形状。