Thin metallic ﬁlms bounded by two dielectric media can support long- and short-range surface plasmon polaritons. The dispersion relation associated with these hybrid modes is given by a complex transcendental equation that can only be solved numerically. In this contribution, a simple approximate analytical solution for the short-range modes is derived. To validate the proposed analytical solution, short-range surface plasmon polaritons in a two-dimensional periodic array of small holes are studied and compared to full-wave simulations. The results prove a spectral response in excellent agreement.
Lorente Crespo, M., & Mateo-Segura, C. (2016). Simple closed-form solution to predict short-range surface plasmons in thin ﬁlms bounded by asymmetric media and its application to hole arrays. IEEE Photonics Journal, 8(1), . https://doi.org/10.1109/JPHOT.2016.2521267