Conditions for loss compensation of surface plasmon polaritons in a metal layer surrounded by an active dielectric: an analytic solution

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.