Closed-Form Approximations to Solutions of Plasmon Dispersion at a Dielectric/Conductor Interface

The dispersion equation for surface plasmons (SPs) at a dielectric/conductor interface has been studied extensively with respect to the design of plasmonic devices. A key design requirement is the reduction of damping in the propagating SP polariton wave. Satisfaction of this constraint requires that an “electron gas” in a conducting medium, such as a doped semiconductor, move along the interface at speeds that approximate the polariton wave speed. At these low relative speeds, the efficient exchange of energy between the drifting electrons and a traveling SP polariton may be enabled. The ill-conditioned, eighth-order, complex coefficients dispersion equation derived earlier is dependent upon six parameters that vary over many orders of magnitude. The dispersion equation is also found to be singular in regions of practical interest, and, taken together, these properties have hampered the success of numerical investigations. Therefore, the dispersion equation is analytically investigated here, and closed-form results are found for the parametric dependence of the surface polariton's propagation constant on five dimensionless groups. These new solutions show how compensation of propagation losses without the use of structures can be achieved and provide avenues that guide device design. The closed-form results are used to initialize numerical optimization by providing sufficiently accurate starting points within the parameter space that avoid numerical ill-conditioning.