Dispersion relation of a plasmonic waveguide on substrate

Dispersion relation of the surface plasmon polariton (SPP) propagating along a subwavelength gold waveguide deposited on a substrate is studied theoretically. The eigenvalues of a 1D waveguide are calculated using the finite element method combined with the method of moments in the visible-NIR regime. The dispersion curve is obtained directly by calculating the complex wavenumber versus real frequency, rather than the complex frequency versus real wavenumber. This method particularly benefits the problem with frequency-dependent material property. The shape effect (square or rectangle) of a gold nanowire and the refractive index of substrate on the propagation length, phase velocity and group velocity are investigated. Moreover, the mode profile is analyzed to evaluate the local confinement of SPP's evanescent field. In addition, the analytical solution of a circular nanowire is provided for comparison. Our results show that a plasmonic waveguide is a lowpass filter with a cutoff frequency, which decreases as the substrate's refractive index increases.