Unidirectional and diffractionless surface plasmon polaritons on three-dimensional nonreciprocal plasmonic platforms

Light-matter interactions in conventional nanophotonic structures typically lack directionality. For example, differently from microwave antenna systems, most optical emitters (e.g., excited atoms/molecules and simple nanoantennas) exhibit quasi-isotropic dipolar radiation patterns with low directivity. Furthermore, surface waves supported by conventional material substrates do not usually have a preferential direction of propagation, and their wavefront tends to spread as it propagates along the surface, unless the surface or the excitation is properly engineered and structured. In this article, we theoretically demonstrate the possibility of realizing unidirectional and diffractionless surface plasmon polariton modes on a nonreciprocal platform, namely, a gyrotropic magnetized plasma. Based on a rigorous Green's function approach, we provide a comprehensive and systematic analysis of all the available physical mechanisms that may bestow the system with directionality, both in the sense of one-way excitation of surface waves and in the sense of directive diffractionless propagation along the surface. The considered mechanisms include (i) the effect of strong and weak forms of nonreciprocity, (ii) the elliptic-like or hyperbolic-like topology of the modal dispersion surfaces, and (iii) the source polarization state, with the associated possibility of chiral surface-wave excitation governed by angular-momentum matching. We find that three-dimensional gyrotropic plasmonic platforms support a previously unnoticed wave-propagation regime that exhibit several of these physical mechanisms simultaneously, allowing us to theoretically demonstrate unidirectional surface plasmon polariton modes that propagate as a single ultranarrow diffractionless beam. We also assess the impact of dissipation and nonlocal effects. Our theoretical findings may enable a new generation of plasmonic structures and devices with highly directional response.

Figure 1

Different physical mechanisms that influence the directionality of the excitation and propagation of surface plasmon polaritons (SPPs). Panels (a)–(h) show qualitative sketches of the typical in-plane SPP pattern (surface-wave intensity in different directions on an interface) that may be obtained with a specific combination of source polarization (linear or circular), SPP modal dispersion (elliptic or hyperbolic), and medium properties (reciprocal or strongly nonreciprocal). The inset shows the system configuration under study: a generic plasmonic material occupying the lower half space ($z<0$) forms an interface with a different medium in the upper half space (e.g., free space), where an electric-dipole source is located. The panels corresponding to linear source polarization assume a $z$-oriented dipole, whereas for circularly polarized emitters the plane of circular polarization is indicated by dashed green lines (the dipole rotates in the plane containing the green line and the $z$ axis). In this work, particularly attention is devoted to panels shaded in blue and red.

Figure 2

Band diagram (density plots) for the bulk modes of a magnetized plasmonic material with bias along the $y$ axis, for different propagation directions defined by the angle $\psi$ with respect to the $+x$ axis (see Fig. 1): (a) $\psi =0^{\circ }$, (b) $\psi ={30}^{\circ }$, (c) $\psi ={60}^{\circ }$, and (d) $\psi ={90}^{\circ }$. The cyclotron frequency is set to ${\omega }_c/{\omega }_p=0.22$, where ${\omega }_p$ is the plasma frequency. $k_p$ is the free-space wave number at ${\omega }_p$. The white horizontal strip indicates the band gap between TM-like modes. For each panel, the inset provides a zoom around the band gap.

Figure 3

Surface modes on an interface between a magnetized plasmonic material and an opaque (metallic) isotropic material, as sketched in the inset on top. Left column: Equifrequency contours in $k_x{k}_y$ space (red dashed lines) for the dispersion function of the TM SPP modes supported by the structure in the inset. The magnetized plasma has cyclotron frequency ${\omega }_c/{\omega }_p=-0.22$, and the isotropic material has permittivity ${\epsilon }_m=-2$. Three different frequencies $\omega /{\omega }_p$ have been considered, within [(a)] and above [(c), (e)] the bulk-mode band gap. The colors of the density plots correspond to the magnitude of the integrand in Eq. (6) (brighter colors mean higher intensity), indicating which portion of the equifrequency contour contributes more strongly to SPP excitation for the chosen source: a dipolar emitter linearly polarized along the $z$ axis and located a distance $d=0.5c/{\omega }_p$ above the surface. The red arrows indicate the main directions of energy flow (i.e., SPP group velocity). Right column: SPP patterns in the $xy$ plane, corresponding to each equifrequency contour on the left, for the same linearly polarized dipolar emitter. The SPP patterns represent the amplitude of the field $|{\mathbf{E}}_z^s|$, calculated exactly with Eq. (6), at a fixed radial distance $1.2{\lambda }_0$ from the source (where ${\lambda }_0$ is the free-space wavelength for each panel). In each panel, the fields are normalized to their maximum value.

Figure 4

Surface modes on an interface between a magnetized plasmonic material and a transparent isotropic material (free space), as sketched in the inset on top. The figure is similar to Fig. 3, but for two frequencies below the bulk-mode band gap: (a), (b) $\omega /{\omega }_p=0.47$, and (c), (d) $\omega /{\omega }_p=0.75$. The magnetized plasma has cyclotron frequency ${\omega }_c/{\omega }_p=0.9$. The linearly polarized dipolar emitter is located a distance $d=0.05c/{\omega }_p$ above the surface. Left column: Equifrequency contours for the SPP modes (red dashed lines), overlapped to density plots indicating which portion of the equifrequency contour contributes more strongly to SPP excitation (brighter colors) for the chosen source, as in Fig. 3. Right columns: Corresponding SPP patterns on the $xy$ plane.

Figure 5

SPP patterns in the $xy$ plane on the surface of a nonmagnetized plasma, excited by a dipolar emitter with (a) linear polarization $\mathit{\gamma }=\widehat{z}$, and (b), (c) circular polarization, $\mathit{\gamma }=\widehat{x}+i\widehat{z}$ and $\mathit{\gamma }=\widehat{x}-i\widehat{z}$, respectively. The SPP patterns represent the amplitude of the field $|{\mathbf{E}}_z^s|$, calculated exactly with Eq. (6), at a fixed radial distance ${\lambda }_0$ from the source (where ${\lambda }_0$ is the free-space wavelength). Intensities are normalized to the linear case in (a). The emitter is located a distance $d={\lambda }_0/20$ above the interface and oscillates at frequency $\omega /{\omega }_p=0.55$.

Figure 6

Schematic of the relevant quantities involved in SPP excitation/propagation: group velocity ${\mathit{v}}_g$ (red arrows; direction of SPP energy flow), linear momentum $\mathbf{k}$ (blue arrows; direction of SPP phase flow if real, and of evanescent decay if imaginary), and transverse spin angular momentum $\mathbf{S}$ (green arrows; normal to the plane of rotation of the electric field) for a plasmonic surface mode. Two cases have been considered: (a) nonmagnetized reciprocal plasma (isotropic bidirectional surface waves), and (b) magnetized nonreciprocal plasma (semihyperbolic unidirectional surface waves, i.e., below-the-gap surface modes), interfaced with vacuum. The vectors in panel (b) refer to the dominant large-$k$ asymptotic region of the hyperbolic equifrequency contour in Fig. 4.

Figure 7

Elliptic-like SPP patterns in the $xy$ plane on the surface of a magnetized plasma, excited by a dipolar emitter, with linear or circular polarization: $\mathit{\gamma }=\widehat{z}$ (blue line), $\mathit{\gamma }=+\widehat{x}+i\widehat{z}$ (black line), and $\mathit{\gamma }=-\widehat{x}+i\widehat{z}$ (red line). All other parameters are the same as in Fig. 3.

Figure 8

Semihyperbolic SPP patterns in the $xy$ plane on the surface of a magnetized plasmonic material, excited by a dipolar emitter at frequency $\omega =0.7{\omega }_p$, with different circular-polarization states: (a) $\mathit{\gamma }=-\widehat{x}+i\widehat{z}$, (b) $\mathit{\gamma }=\widehat{x}+i\widehat{z}$, (c) $\mathit{\gamma }=\cos \left(\varphi \right)\widehat{x}+\sin \left(\varphi \right)\widehat{y}+i\widehat{z}$, and (d) $\mathit{\gamma }=\cos \left(\varphi \right)\widehat{x}-\sin \left(\varphi \right)\widehat{y}+i\widehat{z}$. The SPP patterns represent the amplitude of the field $|{\mathbf{E}}_z^s|$, calculated exactly with Eq. (6), at a fixed radial distance $0.7{\lambda }_0$ from the source (where ${\lambda }_0$ is the free-space wavelength). The emitter is located in vacuum, at a distance $d=0.05c/{\omega }_p$ above the magnetized plasma. The plasma cyclotron frequency is ${\omega }_c/{\omega }_p=0.9$. The angle considered in panels (c) and (d) is $\varphi ={60}^{\circ }$, for this specific example. The intensities of the SPP patterns are all normalized by the same value. Panels (e) and (f) show the electric-field intensity distributions of the launched semihyperbolic SPP beams, corresponding to the cases in panels (a) and (d), respectively, obtained via full-wave simulations performed with CST Microwave Studio [54]. We considered the same parameters as in our exact Green's function calculations, except for the inclusion of moderate dissipation in the plasmonic material, defined by a collision frequency $\mathrm{\Gamma }/{\omega }_p=0.003$. An animation of the simulated time-harmonic electric field, for the case in panel (f), is included in the Supplemental Material [16].