Curvature-induced radiation of surface plasmon polaritons propagating around bends

We present a theoretical study of the curvature-induced radiation of surface plasmon polaritons propagating around bends at metal-dielectric interfaces. We explain qualitatively how the curvature leads to distortion of the phase front, causing the fields to radiate energy away from the metal-dielectric interface. We then quantify, both analytically and numerically, radiation losses and energy transmission efficiencies of surface plasmon polaritons propagating around bends with varying radii as well as sign of curvature.

Figure 1

(a) The geometry of the curved metal-dielectric interface. A rounded edge, characterized by a permittivity ${ϵ}_i$, has a bend angle $\theta$ and a finite bend radius $R$. The bend is confined to the region of space shown, with center of curvature at the origin. The rest of the space is occupied by a medium with ${ϵ}_o$. Axes $x_1$ and $x_2$ extend along the boundaries between regions I and II and regions II and III, respectively. In the $x_1\text{−}{x}_2$ plane, regions I and III are semi-infinite. The system is also infinite in extent along the entire $z$ axis. Arrows indicate incident and reflected fields in region I, and transmitted field in region III. In addition, (a) shows the intensity of SPPs in gray scale. The intensity distribution is calculated for SPPs traveling around a metallic corner (b), using the single-mode approximation developed in the text. We also consider the complementary configuration, in which SPPs propagate around a dielectric void (c).

Figure 2

(Color online) Dispersion relations of various transverse-magnetic modes on a metal cylinder with $R=2\mu \mathrm{m}$. The dispersion relation for the fundamental mode (solid line) closely follows that of SPPs on a flat metal-dielectric interface (dashed-dotted line). The horizontal scale of the latter is normalized by the factor $R$, so that the curve is a plot of $\omega$ as a function of $kR$, instead of $k$. Dispersion relations of modes with higher radial excitations are also shown (dashed lines). These modes are less confined to the interface and do not approach ${\omega }_p∕\sqrt{2}$ asymptotically. Higher radial excitation modes with even weaker surface confinement exist (not shown), all lying to the right-hand side of the three dashed lines shown.

Figure 3

(Color online) Plot of $\mathrm{Im}\left[m\right]$ as a function of $G$ for $\omega R∕c=1000$ and ${ϵ}_o=1$. The premittivity ${ϵ}_i$ is real and varies from ${ϵ}_i=-30$ to $-200$.

Figure 4

Comparison of intensity profiles of SPPs in region I (solid line) and the fundamental mode in region II (dashed line), calculated for $\omega R∕c=800$. The profile mismatch is barely visible, indicating that SPPs in region I couple predominately to the fundamental mode in region II.

Figure 5

(Color online) (a) The upper bound for the transmittance, ${\mathsf{T}}_u$, plotted for a silver-air interface with bend angle $\theta =90°$, as a function of bend radius $R$ for wavelengths $\lambda =500\mathrm{nm}$ (dashed-dotted line), $\lambda =600\mathrm{nm}$ (dashed line), and $\lambda =700\mathrm{nm}$ (solid line). (b) Magnified view of the upper bound in the diffraction-dominated regime.

Figure 6

(Color online) (a) FDTD simulation results showing the magnitude of the magnetic field for $R=\lambda =630\mathrm{nm}$. The SPPs are incident from the bottom and propagate counterclockwise around the bend. (b) Analytical result for the transmittance upper bound (black trace) compared to numerical simulations for various values of $R$ (solid circles). The upper bound values are clearly higher than the simulation results, confirming the consistency of our analytical method.

Figure 7

(Color online) FDTD simulation results showing the magnitude of the magnetic field for (a) $R=\lambda =630\mathrm{nm}$ and (b) $R=0$ when SPPs, incident from the bottom, propagate around a circular dielectric void. The radiation loss is significantly reduced compared to the propagation around a metallic bend.

Figure 8

(Color online) (a) Schematic diagram of a SPP microresonator embedded near a sharp edge. SPPs are incident from the bottom along the direction of the arrow, coupled to the resonator and coupled out again. (b) Numerical FDTD simulation showing the magnitude of the magnetic field for $d=8.4\mathrm{nm}$ and $r=378\mathrm{nm}$. A significant portion of the field is seen to couple into the resonator.