# Ultra-Compact Digital Metasurface Polarization Beam Splitter via Physics-Constrained Inverse Design
###### Abstract  
Inverse design effectively promotes the miniaturization of integrated photonic devices through the modulation of subwavelength structures. Utilizing a theoretical prior based inverse design, we propose an ultra-compact integrated polarizing beam splitter consisting of a standard silicon-on-insulator (SOI) substrate and a tunable air-silicon column two-dimensional code metasurface, with a footprint of \(5\times 2.7\) um\({}^{2}\). The effective refractive index of the waveguide is modulated by adjusting the two-dimensional code morphology in the additional layer to achieve efficient polarization beam splitting. The simulation results demonstrate high performance, with a low insertion loss (\(<\)0.87 dB) and high extinction ratio (\(>\)10.76 dB) in a bandwidth of 80 nm covering the C-band. The device can withstand manufacturing errors up to \(\pm\)20 nm and is robust to process defects, such as the outer proximity effect, and thus is suitable for ultra-compact on-chip optical interconnects. silicon photonics; polarization beam splitter; inverse design; silicon-based metasurface +
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computing capabilities, the designed devices have been proven to have the advantages of small footprints, high performance, and multiple degrees of freedom in the design process [36, 37]. Utilizing a non-linear search algorithm based on a direct-binary search, [PERSON] and co-workers proposed an ultra-compact PBS with a footprint of 2.4 \(\times\) 2.4 \(\upmu\)m\({}^{2}\) and an extinction ratio greater than 10 dB within a bandwidth of 32 nm [10]. [PERSON] and colleagues proposed a 3D-printed PBS in the near-infrared with extinction ratios as much as 5 dB at the optimal wavelength [32]. However, we note that the existing research utilizing inverse design mostly considered the sophisticated optical response of nanostructured arrays as a mathematical optimization aiming at device output power under a two-dimensional approximation. Various algorithms (binary search algorithm (DBS), genetic algorithm, gradient descent, etc.) were applied to optimize a predefined figure of merit (FOM) and give a definite result, which cannot match the corresponding physical interpretation. Therefore, these methods have relatively low design diversity and high demand for computing power. The computational complexity increases exponentially with the increase in nanostructure size and the introduction of multi-parameter or multi-dimensional optimization objectives. Since the completely non-physical structure optimization process lacks upper-level guidance, some blind and inefficient operations need to be performed in the initial stage, leading to a waste of computing power. To solve these problems, a physical-model-guided inverse design method is needed, which can provide more constraints for the optimization process while retaining the advantages of the high degree of freedom in the design process and large optimization space of inversely designed components. In this work, we propose a digital two-dimensional code metasurface PBS based on a gradient index, which was inspired by the theoretical experience of our previous work [38]. The physical model of the gradient refractive index is determined by theoretical analysis, which effectively avoids the solutions being trapped in local optimums and gets rid of the uncertainty resulting from the sensitivity to stochastic initial values. Through further optimization, the topographic structure of the two-dimensional code array of the metasurface layer is adjusted to realize the local effective mode refractive index modulation of the waveguide layer within this limited range. This design method reflects a high degree of design freedom, a large optimization potential, and a small dependence on computing power. We demonstrate a novel PBS structure with a minimum 2.7 \(\times\) 5 \(\upmu\)m\({}^{2}\) footprint. The metasurface structure obtained by the inverse design is directly built on the 220 nm thick SOI waveguide, which acts as a 120 nm thick additional material layer.