We develop a modal method that solves Maxwell's equations in the presence of the linearized hydrodynamic correction. Using this approach, it is now possible to calculate the full diffraction for structures with a period of the order of the plasma wavelength, including not only the transverse but also the longitudinal modes appearing above the plasma frequency. As an example for using this method we solve the diffraction of a plane wave near the plasma frequency from a bimetallic layer, modeled as a continuous variation of the plasma frequency. We observe absorption oscillations around the plasma frequency. The lower frequency absorption peaks and dips correspond to the lowest longitudinal modes concentrated in the lower plasma frequency region. As the frequency is increased, higher order longitudinal modes are excited and extend to the region of higher plasma frequency. Moreover, an examination of the propagation constants of these modes reveals that the absorption peaks and dips are directly related to the direction of phase propagation of the longitudinal modes. Furthermore, we formulate a variant of the plane wave expansion method, and use it to calculate the dispersion diagram of such longitudinal modes in a periodically modulated plasma frequency layer.
Yanai, A., Mortensen, N. A., & Levy, U. (2013). Absorption and eigenmode calculation for one-dimensional periodic metallic structures using the hydrodynamic approximation. Physical Review B Condensed Matter, 88(20), [205120 ]. https://doi.org/10.1103/PhysRevB.88.205120