Smooth-Curvature Bend Design Guided by Variational Analysis for Adiabatic Multimode Integrated Photonics
Multimode photonic integrated circuits enable ultralow-loss on-chip optical interconnects and microwave-photonic processing, yet waveguide bends dominate both chip footprint and excess loss. A high-performance multimode waveguide bend (MWB) must transmit the working mode with low loss while suppressing intermodal coupling, forcing a trade-off among bending radius, operating bandwidth, and fabrication tolerance. Here we formulate constant-width MWB design as a curvature-dependent variational problem. By constructing a figure of merit that incorporates higher-order-mode excitation, fundamental-mode mismatch, and sidewall field intensity, we derive a necessary condition for adiabatic optimality: the curvature profile must approach infinite differentiability throughout the bend, including its junctions with the input and output straight waveguides. This condition explains the limitations of circular and Euler bends and motivates a smooth polynomial curvature (SPC) family with a closed-form beta-function representation. We further introduce an optimized SPC hybrid (SPCh) bend that balances junction smoothness and the interior curvature gradient. On the 220 nm silicon-on-insulator platform, SPCh bends achieve a mode extinction ratio below $-37$ dB from 1500-1600 nm at an effective radius of $16\,\mu\mathrm{m}$, providing more than 22 dB stronger mode suppression than a representative Euler bend at the same radius. The simulated response remains robust to $\pm 60$ nm waveguide-width deviations. Fabricated SPCh-based microring resonators reach an intrinsic quality factor of up to $7.53 \times 10^6$ and a free spectral range of up to 100 GHz through a standard silicon foundry process. The resulting design strategy provides compact, broadband, and fabrication-tolerant multimode bends for high-density optical interconnects and microwave-photonic systems.