# Ultra-Compact Digital Metasurface Polarization Beam Splitter via Physics-Constrained Inverse Design
## 2 Structure and Method
thickness increases, the difference between the effective refractive indices in different polarization modes gradually increases, which is beneficial for the output of the TE and TM modes to different ports. With a reasonable manufacturing difficulty, the variation ranges of the effective indices are considered for the selection of as many possibilities for the optical path modulation by the additional layer with a structure of two-dimensional code. Therefore, the thickness of the additional silicon layer is set to 120 nm, where the maximum relative effective refractive index of the TE mode is 1.08 and that of the TM mode is 1.27. The digital two-dimensional code metasurface structure is used to set the refractive index with a high degree of freedom and large parameter space. The length of the minimum etched cell in the two-dimensional code is set to 80 nm, which is far below the operating wavelength. Therefore, the considered device is discretized into thousands of 80 nm \(\times\) 80 nm silicon/air pillars, which are defined as \"pixels\". Each such pixel can occupy two states: silicon or air. Here, by setting the limit of the gradient refractive index, we realize the optimization of the metasurface structure without high computing power while retaining high manufacturing tolerance. The dielectric constants of the different polarized lights in the doped medium are [40]:  
\[ \begin{array}{c}\varepsilon_{\mathrm{TE}}=\mathrm{p}\varepsilon_{\mathrm{ air}}+(1-\mathrm{p})\varepsilon_{\mathrm{silicon}}\\ \frac{1}{\varepsilon_{\mathrm{TM}}}=\frac{\mathrm{p}}{\varepsilon_{\mathrm{ air}}}+\frac{1-\mathrm{p}}{\varepsilon_{\mathrm{silicon}}}\end{array}  \tag{2}\]  
where p is the volume fraction, which is defined as the area ratio of the air holes in the specified unit area in the two-dimensional case. For optical materials, the effective refractive index N is numerically equal to the square root of the dielectric constant \(\varepsilon\). The regulation of the air volume fraction of the surface area of the metamaterial is realized by nanostructure arrays that can be regarded as a two-dimensional code distribution. The metasurface silicon layer with a thickness of 120 nm is divided into a series of square units with a side  
Figure 2: (**a**) The normalized effective refractive indices for the TE and TM mode as a function of the thickness of the metasurface layer. (**b**) Relationship between the number of etched grids and the normalized effective refractive indices for the TE and TM mode, thereby matching the typical cell structures of each density mode. (**c**) Index profile of the TM mode at the initial time and (**d**) after optimization. length of 240 nm, and each unit is further divided into a nine-square lattice structure of 3 \(\times\) 3 small pixels. Then, the local approximate thickness can be adjusted by determining the number of etched pixels in each unit, thereby realizing the different local efficient refractive indices in the corresponding area. The number of etched pixels in each unit ranges from 0 to 9. It can be seen from the figure that as the number of etched cells in each unit decreases, the corresponding local effective refractive index increases, and the index difference between the TE and TM modes in the unit gradually increases. As a result, light of different polarizations can be deflected by different degrees in the waveguide layer. Therefore, the local effective refractive indices of each cell can be divided into 10 density modes, and the corresponding effective refractive index is shown in Figure 2b. Except for mode 0 (none of the pixels are etched) and mode 9 (all the pixels are etched), each mode has several possible code arrangements with an equivalent local refractive index. As can be observed in natural phenomena such as mirages, light travels along a curved path in a variable index medium, and the optical path depends on the gradient index distribution. For the same subwavelength structure, light with different polarization states will show different gradient indices.