The propagation length of surface plasmon polaritons (SPP) is limited by the internal losses in the metal and the radiation losses due to the structure imperfections at the interface. These losses can be compensated by introducing material with optical gain (active medium) in the system. The required gain for complete loss compensation depends not only on the dissipative properties of the metal but also on the geometry of the structure. The geometry factor can weaken the gain requirements and one promising system, in this respect, is a metal layer embedded in an active medium. Here we present an analytical solution for this system, which allows to find the curves of exact loss compensation in the complex plane of the dielectric function of the gain medium. These curves determine the minimum required gain of the active medium for lossless SPP propagation. The solution is given for both modes of SPP propagation. This approach is then illustrated and discussed for silver and gold layers embedded in an active media.

中文翻译：

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被有源电介质包围的金属层中表面等离子体激元的损耗补偿条件：解析解

表面等离子体激元 (SPP) 的传播长度受到金属内部损耗和由于界面结构缺陷造成的辐射损耗的限制。这些损耗可以通过在系统中引入具有光学增益的材料（活性介质）来补偿。完全损耗补偿所需的增益不仅取决于金属的耗散特性，还取决于结构的几何形状。几何因素会削弱增益要求，在这方面，一个有前途的系统是嵌入有源介质中的金属层。在这里，我们提出了该系统的解析解，它允许在增益介质的介电函数的复平面​​中找到精确的损耗补偿曲线。这些曲线确定了无损 SPP 传播所需的最小活性介质增益。对 SPP 传播的两种模式都给出了解决方案。然后针对嵌入在活性介质中的银和金层来说明和讨论这种方法。