# Dynamical Manipulation of Surface Plasmon Polaritons
## 2 Directional Excitation of Surface Plasmon Polaritons (SPP)  
### Polarization Controlled Excitation of SPP  
SPP source is the fundamental component of 2D plasmonic system. Slits and gratings are the commonly used structure to satisfy the wave vector matching condition and generate SPP [1; 2]. Conventionally, the excitation of SPP is bidirectional and symmetrical. Dynamic control of the propagation direction of SPP is desired for real applications. Figure 1a shows polarization controlled directional SPP coupler which consists of two parallel columns of orthogonal rectangle slits with separations [48]. SPP waves propagating to the left and right are the interference of the SPP launched by each column, which can be expressed as  
\[ \begin{array}{l}I_{r}\propto\left[(I_{1}+I_{2})+2\sqrt{I_{1}I_{2}}\cos(k_{ sp}s+\delta)\right]\\ I_{l}\propto\left[(I_{1}+I_{2})+2\sqrt{I_{1}I_{2}}\cos(k_{sp}s-\delta)\right] \end{array} , \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the intensities of SPP generated by the two columns of slits, \(k_{sp}\) is the wave vector of SPP. \(\delta\) is relative phase between the two orthogonal components of incident light, which determines the polarization state. When the separation \(s\) is adjusted to \(s=\lambda_{sp}/2\), Equation (1) can be simplified to  
\[ \begin{array}{l}I_{r}\propto\left[(I_{1}+I_{2})-2\sqrt{I_{1}I_{2}}\sin\delta \right]\\ I_{l}\propto\left[(I_{1}+I_{2})+2\sqrt{I_{1}I_{2}}\sin\delta\right]\end{array} . \tag{2}\]  
From Equation (2), it can be concluded that the overall intensity of excited SPP \(I=I_{1}+I_{2}\) is independent of relative phase, but the proportion of SPP propagating to the left or right can be tuned by changing polarization state. Particularly, with the illumination of circularly polarized light, \(I_{1}\) and \(I_{2}\) are equal and the relative phase is \(\delta=\sigma_{\pm}\frac{\pi}{2}\), where \(\sigma_{\pm}=\pm 1\) represents left circularly polarized (LCP) and right circularly polarized (RCP) light, respectively. Thus, SPP excited by LCP light will propagate to the left (\(I_{r}=0\)) and SPP excited by RCP light will propagate to the right (\(I_{l}=0\)). For linearly polarized light which can be regarded as the combination of LCP and RCP light, the excited SPPs propagate equally to the left and right side. The counterclockwise, clockwise, and linear arrows in Figure 1a indicate the polarization stats of the incident light. The experimental results further demonstrate that controlled directional SPP excitation is possible by changing the handedness of the incident light [48]. Based on anomalous diffraction, a similar functionality is realized with a metasurface consisting of spatial-variant subwavelength slits [49], as shown in Figure 1b. Light normally incident on the metasurface will experience both ordinary and anomalous diffraction. The orientation angle \(\alpha\) of the slits has no effect on the ordinary diffracted light and the metasurface acts as a grating, while the anomalous diffracted light is imprinted with a spin-dependent phase \(2\alpha\). As a result, the anomalous diffraction beam patterns are not symmetrical about the surface normal, which makes it possible to realize directional SPP excitation. To excite SPP, the in-plane wave vector of diffracted light should satisfy. \[k_{sp}=\frac{2m\pi}{s}+\sigma_{\pm}\frac{2\Delta\alpha}{s}. \tag{3}\]For LCP incident light with a frequency of \(\omega_{2}\), the SPPs generated by the \(m=1\) anomalous diffraction is shifted to a higher frequency \(\omega_{1}\) and will propagate to the left, while the SPPs generated by the \(m=-1\) anomalous diffraction are shifted to a lower frequency \(\omega_{3}\) and will propagate to the right. When the handedness of the incident light is switched, the propagation direction of SPPs will be reversed correspondingly. The phase gradient \(\Delta\alpha\) inducing extra in-plane momentum accounts for the polarization-controlled SPP excitation.