Numerical integral methods to study plasmonic modes in a photonic crystal waveguide with circular inclusions that involve a metamaterial

We present several numerical integral methods for the study of a photonic crystal waveguide, formed by two parallel conducting plates and an array of circular inclusions involving a conducting material and a metamaterial. Band structures and reflectance were calculated, for infinite and finite photonic crystal waveguides, respectively. The numerical results obtained show that the numerical methods applied provide good accuracy and efficiency. An interesting detail that resulted from this study was the appearance of a propagating mode in a band gap due to defects in the middle of the photonic crystal waveguide. This is equivalent to dope a semiconductor to introduce allowed energy states within a band gap. Our main interest in this work is to model photonic crystal waveguides that involve left-handed materials (LHMs). For the specific LHM considered, a surface plasmon mode on the vacuum–LHM interface was found.