Analytical investigation of higher-order plasmonic modes on a metal-dielectric interface

Based on the plane-wave expansion method, we have analytically obtained a set of higher-order transverse magnetic (TM) modes that can exist on a metal-dielectric interface. Any linear combination of these modes can be supported on the interface. Our analysis is based on the assumption of nonuniformity of the electromagnetic fields in all directions. We have also shown that no higher-order transverse electric (TE) mode can propagate on the interface. Furthermore, the orientation of Poynting vector and energy flow of the TM modes are described. Our results show that the Poynting vector of a higher-order TM mode always makes an angle with the interface, which means TM modes are not pure surface waves.

Figure 1

Schematic of a metal-dielectric interface.

Figure 2

$z$ component of electric field corresponding to (a) the fundamental mode and (b) a higher-order TM mode on the silver-vacuum interface.

Figure 3

Poynting vector and the related angles. The dashed line shows the trajectory of $\mathit{P}$ on the $yz$ plane.

Figure 4

Values of the angles (a) $\theta$ and (b) $\varphi$ corresponding to the Poynting vectors of ${\text{TM}}_{\text{x}}$ modes. These values correspond to medium 1. The values are given in degrees.

Figure 5

Values of the angles (a) $\theta$ and (b) $\varphi$ corresponding to the Poynting vectors of ${\text{TM}}_{\text{x}}$ modes. These values are calculated in medium 2. The values are given in degrees.