Coupling, resonance transmission, and tunneling of surface-plasmon polaritons through metallic gratings of finite length

The Green's-theorem integral equation formulation is employed to study numerically the coupling into surface plasmon polaritons by illuminating a finite metal grating with a Gaussian beam from the vacuum half-space above the metal surface. A flat surface impedance boundary condition is used to simplify the scattering integral equations. The grating coupler period is chosen so that the first diffracted order excites a surface plasmon polariton and the zeroth diffracted order is the only radiating order. In particular, the surface magnetic field and the angular distribution of scattered intensity are calculated. These functions provide in turn the total intensity of the radiation scattered into the vacuum and the total power flow carried by the surface plasmon polariton, from which the energy balance is monitored. In this way, the zeroth order and coupling efficiencies are studied as functions of the angle of incidence ${\theta }_0$ and the grating coupler height $s_c$, with the aim of analyzing the influence of the length of the illuminated coupler. Our results show that, when the illuminated coupler length is decreased, the photon-surface plasmon polariton coupling resonance broadens as a function of both ${\theta }_0$ and $s_c$, and that larger values of $s_c$ are required to optimize this coupling resonance. In addition, the coupling geometry is exploited to obtain the reflection and transmission coefficients, and the intensity of the scattered volume waves, of a surface plasmon polariton thus excited that impinges on another finite metal grating, called a grating scatterer. Two frequencies of the incoming surface plasmon polariton are considered that lie very close to the lower band edge of the gap in the surface plasmon polariton dispersion relation for the infinite grating scatterer. If the frequency is in the band, a strongly oscillating, resonant behavior of the transmission coefficient as a function of the grating length is obtained. For a frequency in the gap, transmission is negligible unless the grating is short enough that the surface plasmon polariton can tunnel through it.