# Valley-Selective Response of Nanostructures Coupled to 2D Transition-Metal Dichalcogenides
## 3 Nanostructures for Valley-Selective Directional Emission  
### Single Nanoantennas  
As mentioned in the introduction, the proposed nanostructures for valley-selective response can perform spatial separation of the valley degree of freedom with surface waves or separation in K-space when photons with opposite helicity are emitted in different directions. Moreover, spatial separation of the valley degree of freedom may be accompanied by real dividing of valley excitons [24] or the transformation of valley exciton pseudospin to photonic degrees of freedom via optical spin-orbit coupling [21; 23; 89; 90; 91; 92] (or both these processes). Below, we describe all these possibilities with various examples from the recent literature.  
Photons as information carriers are more robust and controllable than excitons, especially at room temperature. Thus, nanostructures supporting the coupling of valley pseudospin to photonic degrees of freedom (e.g., transverse optical spin angular momentum, t-OSAM) are desirable for on-chip integrated valley devices. The optical spin-orbit coupling effect provides a robust one-to-one transformation of the valley degree of freedom into t-OSAM because of spin-momentum locking. One realization of this idea was implemented in Reference [23] (Figure 2a).  
The structure consists of a single silver nanowire placed on top of a TMD flake (WS\({}_{2}\)), consisting of five layers, which is an indirect semiconductor, but still preserves valley-polarization properties [93]. The silver nanowire supports guided modes possessing t-OSAM in the out-plane direction. Spin-momentum locking of these plasmonic modes allows the transformation of valley polarization into directional emission: the system, excited by circularly polarized light in the center of the nanowire, launches plasmonic modes carrying t-OSAM in the left or right direction as a function of their valley polarization. Note that, because of the structure symmetry, this operation can be achieved under asymmetric-laser-spot excitation in the transverse direction (y). This effect can be described by the coupling efficiency, \(K_{\text{exp}}=(I_{L}-I_{R})/(I_{L}+I_{R})\), where \(I_{L}\) and \(I_{R}\) are the light intensities directed to the left and right side, respectively. Thus, when this value is zero, the structure shows the same coupling on both sides. The measurement results of the coupling efficiency for 594-nm \(\sigma^{+}\) and \(\sigma^{-}\) light excitation as a function of the laser-spot coordinate in the transverse direction are shown in Figure 2b,c. The purple dotted lines show the total PL intensity measured from the ends of the nanowire as a function of the excitation position.  
Plasmonic (Au) nanoantennas for valley-dependent far-field emission (separation in K-space) when photons with opposite helicity are emitted in different directions were theoretically proposed in Reference [22] (Figure 2d). The antenna consists of two closely arranged plasmonic bars different in their lengths. The short bar possesses electric-dipole resonance, while the longer one has quadrupole resonance at the same operation wavelength. The entire nanoantenna being excited by a circularly polarized dipole (modeled as two orthogonal short dipoles with a relative phase of \(\Delta\varphi=\pm 90^{\circ}\)) emits in different directions. The asymmetry originates from the constructive or destructive interference of fields from dipole and quadrupole resonances. Thus, when the coupled 1L-TMDC-nanoantenna system is excited by light of different circular polarizations, the emission from the two valleys (+K and \(-\)K) is emitted in opposite directions (Figure 2e). The nanoantennas in Figure 2d are assumed to be located far enough from each other such that the response of the system is defined with good approximation by the response of the isolated nanoantenna.