# Dynamical Manipulation of Surface Plasmon Polaritons
## 3 Dynamical Focusing of SPP  
The initial studies about SPP focusing are done with circular or semicircular slits [13; 14]. In 2008, [PERSON] et al. found an interesting phenomenon that the SPP focused by semicircular slits would experience a spin-dependent transverse shift [57; 58], as shown in Figure 3a. The subwavelength scale displacement can be estimated with \(\lambda_{sp}/2\pi\). In analogy with the spin Hall effect in the free space, this effect is attributed to a geometric spiral phase which unifies the spin redirection Berry phase and the PB phase. In 2015, we experimentally observed the spiral phase in the terahertz region and proposed an intuitive model based on Fourier transformation to illustrate this effect [60]. It should be noted that the transverse shift is smaller than the wavelength of SPP because of the limited spiral phase ranging from 0 to \(\pi\). The circular or semicircular slits can be regarded as the constitution of subwavelength slits which are extensively utilized to design the optical metasurface. For a rectangle slit illuminated by circularly polarized light, the excited SPP field can be expressed as  
\[E_{sp}(r)=\sin\alpha\exp(i\sigma_{\mp}\alpha)\frac{\exp(ik_{sp}r+i\sigma_{\pm }\pi/2)}{\sqrt{r}}, \tag{6}\]  
where \(\alpha\) represent the orientation angle of the slit and \(r\) is the distance from the slit [62]. Thus, the phase and amplitude of SPP can be modulated by changing the orientation angle of the slits. Moreover, the phase modulation depends on incident polarization. As shown in Figure 3b, two columns of slits with a distance \(d\) are arranged according to the Fresnel zone plate to focus SPP [46] and the orientation angles are \(\alpha_{0}\) and \(-\alpha_{0}\), respectively. The phase of SPP propagating to the left and right can be written as  
\[\varphi_{sp}^{l}=2\sigma_{\pm}\alpha_{0}-k_{sp}d,\;\varphi_{sp}^{r}=2\sigma_{ \pm}\alpha_{0}+k_{sp}d. \tag{7}\]  
When \(\alpha_{0}=\pi/4\) and \(d=\lambda_{sp}/4\), SPP will be solely focused on the left for LCP incident light (\(\varphi_{sp}^{l}=0\), \(\varphi_{sp}^{r}=\pi\)) and will be focused on the right for RCP light (\(\varphi_{sp}^{l}=\pi\),\(\varphi_{sp}^{r}=0\)). In the experiment, the focal length is 24 \(\upmu\)m and incident wavelength is 980 nm, which indicates that the separation between the SPP focuses generated by LCP and RCP light is much larger than the wavelength. Besides, considering that the spiral phase is the origin of spin-dependent SPP focusing for the semicircular lens in Figure 3a, introducing additional spiral phase should be able to amplify the transverse displacement. In Figure 3c, orthogonal slit pairs with spatial variant orientation angles are positioned along the semicircular circle [62]. The SPP field generated by an orthogonal slit pair is given by \(E_{sp}(r)=-\exp(i\sigma_{\mp}2\alpha)\frac{\exp(ik_{sp}r)}{\sqrt{r}}\). The overall spiral phase contains two parts: the intrinsic spiral phase dependent on the handedness of incident light and the additional spiral phase caused by the rotation of slit pair, which can be written as \(\varphi_{sp}=\sigma_{\mp}(\theta+2\alpha)\). The separation between the two focuses generated by LCP and RCP light reaches 1500 nm for 632.8 nm incident light, which could reduce the crosstalk in on-chip communications. For the semicircular SPP lens, SPPs excited by different wavelength incident light are always focused in the center, because the optical paths of SPPs are always equal to radius of the semicircle. A multiple-wavelength SPP lens is desirable to enhance the capacity of on-chip SPP systems. As shown in Figure 3d, the positions of the nanoslits are optimized with an iterative algorithm to design a compact triplexer [81]. The different positions of the slits correspond to the phase modulation of SPP. In the experiment, \(5\times 6\) nanoslits are adopted and the positions of the SPP focuses generated by 820 nm, 850 nm, and 880 nm incident light are (0, \(-\)3 \(\upmu\)m), (0, 0), and (0, 3 \(\upmu\)m), respectively.