Topology-Optimized Dielectric Cavities for Enhanced Excitonic Light Emission from $\rm WSe_{2}$

Photonic inverse design and, especially, topology optimization, enable dielectric cavities with deeply sub-diffraction mode volumes and high quality factors, thus offering a powerful platform for enhanced light-matter coupling. Here, we design and fabricate arrays of CMOS-compatible silicon cavities on sapphire with extreme subwavelength transverse mode sizes of only 30-40 nm ($\rm V\sim\lambda^3/2500$). These cavities are engineered for deterministic coupling to a monolayer (or few-layer) excitonic material, producing strong near-field localization directly beneath the 2D material. Photoluminescence (PL) measurements show reproducible tenfold enhancements relative to bare silicon, consistent with numerical simulations that account for material absorption and fabrication tolerances. Furthermore, time-resolved PL measurements reveal pronounced lifetime shortening and non-exponential dynamics, indicating cavity-mediated exciton-exciton interactions. The optimized cavity geometry enhances the far-field collection efficiency and supports scalable integration with van der Waals semiconductors. Our results show that the arrays of topology-optimized dielectric cavities are a versatile, scalable platform for controlling excitonic emission and interactions, which creates new opportunities in nonlinear optics, optoelectronics, and quantum photonics.