Figure 1

(a) An SPP-emitting dipole is situated on the $z$ axis at a distance $d$ from a metal/dielectric interface in the $x,y$ plane. (b) Left: The $z$ component of the instantaneous SPP electric field in volts per meter generated by a dipole in air that oscillates out of plane with respect to a gold surface (${\epsilon }_1=1,{\epsilon }_2=-9.9+1.0i$) at a frequency corresponding to a vacuum wavelength of 652 nm. The dipole is situated a distance of $d=50\mathrm{nm}$ above the interface and has a strength of $p_0=1\mathrm{D}(1\mathrm{D}=3.3356\times {10}^{-30}\mathrm{C}\mathrm{m})$. The field is evaluated on the air side ($n=1$) at a distance of $z=5\mathrm{nm}$ above the interface. Right: Plot of the time-averaged electric-field intensity of the SPP field at $z=5\mathrm{nm}$ above the interface ($\langle |\vec{E}{{|}^2\rangle }_t$) in ${\mathrm{V}}^2/{\mathrm{m}}^2$. The color scale is in powers of 10 to enhance visibility. (c) Left: The $z$ component of the instantaneous SPP electric field (in V/m) at $z=5\mathrm{nm}$ created by a dipole as in (a), oscillating in plane with the metal surface. Right: plot of the time-averaged electric-field intensity at $z=5\mathrm{nm}$ for the parallel dipole in ${\mathrm{V}}^2/{\mathrm{m}}^2$. The plots take near-field effects into account, which add small perturbations in the immediate vicinity of the source.[26]Reuse & Permissions

Figure 2

(a) Top: the time evolution of a dipole (blue arrows) with $\alpha =0.44\pi ,\beta =-0.15\pi ,\delta =-0.83\pi$, and $\gamma =0$ in arbitrary units. The dipole rotates, tracing out an ellipsis (blue) in three-dimensional space. The projections of the trajectory of the dipole on the $x,z,z,y$, and $x,y$ planes are also shown (red). Center: The $z$ component of the SPP electric field launched by the dipole at an air/gold interface, evaluated at $z=5\mathrm{nm}$ above the gold interface (in volts per meter). The dipole emits at a frequency corresponding to a vacuum wavelength of 652 nm. Bottom: plot of the time-averaged electric-field intensity at $z=5\mathrm{nm}$. In contrast to the fields plotted in Figs. 1b and 1c, the radiation field is clearly asymmetric about the origin. (b) Top: the time evolution of a dipole (blue) that rotates out of plane at the same frequency, with parameters chosen as $\alpha =\arctan \left|\Gamma \right|\approx 0.402\pi ,\beta =0$, and $\delta =\gamma =\arctan \left(\mathrm{Im}\right\{\Gamma \}/\mathrm{Re}\{\Gamma \left\}\right)\approx -0.517\pi$. This choice ensures that the dipole launches SPPs unidirectionally in the $+x$ direction when placed in air above the metal surface. Center: the $z$ component of the SPP electric field at $z=5\mathrm{nm}$ generated by the unidirectionally emitting dipole (in V/m). Bottom: plot of the electric-field intensity at $z=5\mathrm{nm}$ (in ${\mathrm{V}}^2/{\mathrm{m}}^2)$. The plots take the near field of the dipole into account,[26] hence the extinction can be seen to be imperfect directly to the left of the dipole. The values used to specify geometry, material properties, and dipole strength in (a) and (b) are the same as used in Figs. 1b and 1c.Reuse & Permissions

Figure 3

(a) The SPP radiation pattern of a dipole in air above a gold surface with $\alpha ={\alpha }_{\mathrm{uni}}=\arctan \left|\Gamma \right|\approx 0.402\pi$ and $\beta =0$ for various values of the phase $\delta$, showing continuous tuning of the emission from right unidirectionality to left unidirectionality via intermediate extinction ratios (in arbitrary units). The phases are (1) $\delta ={\delta }_{\mathrm{uni}}=\arctan (\mathrm{Im}\left\{\Gamma \right\}/\mathrm{Re}\left\{\Gamma \right\})\approx -0.517\pi$ (right unidirectional), (2) $\delta ={\delta }_{\mathrm{uni}}+0.33\pi$, (3) $\delta =0$, and (4) $\delta ={\delta }_{\mathrm{uni}}+\pi$ (left unidirectional). Whereas out-of-plane rotating dipoles lead to a highly asymmetric emission of SPPs, a tilted linear dipole will also create an asymmetric pattern as shown in curve (3) due to optical losses in the gold. (b) The SPP radiation patterns of a unidirectionally emitting dipole ($\alpha ={\alpha }_{\mathrm{uni}},\delta =\gamma ={\delta }_{\mathrm{uni}}$) for several values of $\beta$ (in arbitrary units), showing how the direction of unidirectional emission can be moved in plane by rotating the axis of rotation of the dipole about the $z$ axis. The values of $\beta$ are: (1) $\beta =0$, (2) $\beta =\pi$, and (6) $\beta =\pi /2$. The dielectric constants, dipole position, and frequency are the same as in Figs. 1 and 2. The motion of the dipoles is plotted in the center and bottom parts of the figure in diagrams labeled from (1) to (6). The motion of the dipole is shown in blue with the projections of the trajectory on the $x,z,z,y$, and $x,y$ planes shown in red (arbitrary units).Reuse & Permissions