# Ultra-Compact Digital Metasurface Polarization Beam Splitter via Physics-Constrained Inverse Design
## 2 Structure and Method  
Figure 1a,b shows the proposed PBS, which is realized by utilizing a 120 nm metasurface structure formed on a standard commercial SOI platform. Incident light with both TE and TM polarization enters the metasurface PBS from the input port, which is a silicon waveguide of 220 nm thickness, while the effective refractive index of the waveguide is modulated by adjusting the distribution of the subwavelength structure. Then, TE and TM modes are separated due to the different responses to the metasurface and then separate into different output ports. The footprint of the device is as compact as 2.7 \(\times\) 5 \(\upmu\)m\({}^{2}\). The metasurface layer, as shown in Figure 1c, is a two-dimensional code structure composed of a cylindrical silicon layer with a diameter of 5 \(\upmu\)m and a number of unevenly distributed square air holes etched inside. The square air holes have a side length of 80 nm, which is far below the working wavelength. According to the law of medium equivalence, a  
certain uniformly-changed, isotropic refractive index profile can be achieved by changing the topographic structure of the metasurface layer [39]. Owing to the strong birefringence of silicon, TE and TM polarized light reflect different effective indices when transmitted in the same waveguide. The propagation of light from different polarizations in the waveguide follows for the TE and TM modes, respectively:  
\[ \begin{split}\left(\mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{ \mathrm{TM}}^{2}\right)^{1/2}&\mathrm{k}_{0}\mathrm{h}=\mathrm{m }\pi+\tan^{-1}\left(\frac{\mathrm{n}_{\mathrm{silicon}}^{2}}{\mathrm{n}_{ \mathrm{silicon}}^{2}}\right)\left(\frac{\mathrm{N}_{\mathrm{TM}}^{2}-\mathrm{ n}_{\mathrm{silicon}}^{2}}{\mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{\mathrm{TM}}^{2}} \right)+\tan^{-1}\left(\frac{\mathrm{n}_{\mathrm{silicon}}^{2}}{\mathrm{n}_{ \mathrm{silicon}}^{2}}\right)\left(\frac{\mathrm{N}_{\mathrm{TM}}^{2}-\mathrm{ n}_{\mathrm{nit}}^{2}}{\mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{\mathrm{TM}}^{2}}\right)\\ &\left(\mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{\mathrm{TE}}^{2 }\right)^{1/2}&\mathrm{k}_{0}\mathrm{h}=\mathrm{m}\pi+\tan^{-1} \left(\frac{\mathrm{N}_{\mathrm{TE}}^{2}-\mathrm{n}_{\mathrm{silica}}^{2}}{ \mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{\mathrm{TE}}^{2}}\right)+\tan^{- 1}\left(\frac{\mathrm{N}_{\mathrm{TE}}^{2}-\mathrm{n}_{\mathrm{nit}}^{2}}{ \mathrm{n}_{\mathrm{silicon}}^{2}-\mathrm{N}_{\mathrm{TE}}^{2}}\right)\end{split}  \tag{1}\]  
Here, \(\mathrm{N}_{\mathrm{TE}}\) and \(\mathrm{N}_{\mathrm{TM}}\) is the effective refractive index of the two polarization modes, \(\mathrm{k}_{0}\) represents the wavenumber when light propagates in a vacuum, and \(\mathrm{h}\) indicates the thickness of the waveguide. Therefore, the effective refractive indices of different modes of light have different responses to the equivalent thickness h of the waveguide layer, which is determined by the thickness of the additional metasurface layer in the considered PBS. Figure 2a shows the variation curve of the effective refractive index with the thickness of the silicon-based metasurface layer, where a relationship of positive correlation can be observed. As the  
Figure 1: Schematic of the metasurface polarization beam splitter. (**a**) Top view of the proposed PBS. Inset: the zoom-in top view of a single pixel in the metasurface layer. (**b**) Cross-section of the area marked by the red line, which indicates the structural form of metasurface built on the standard SOI. The topmost metasurface layer can be divided into a series of small pixels of 80 \(\times\) 80 nm\({}^{2}\), each of which has two optional modes: etched air hole or silicon. (**c**) The metasurface structure, in which the light area represents the doped air, and the dark area represents the 120 nm thick silicon structure.