Parameterizing arbitrary shapes for optimization of optomechanical cavities

Optomechanical crystals (OMCs) exploit the coupling between optical and mechanical modes for applications in nonlinear optics, precision sensing, and quantum information processing. Conventional OMC designs typically rely on simple shapes such as ellipses to define photonic crystal lattice holes, as smooth transitions are crucial for maintaining high optical quality factors. These constraints, however, limit the design space and hinder the discovery of novel mechanical modes. We introduce a Fourier descriptor (FD)-based approach for generating and optimizing complex shapes in OMCs. FDs compactly represent arbitrary closed curves as sums of sinusoidal components, allowing smooth shape control with few parameters. By interpolating FD coefficients, our method enables continuous transitions between shapes without introducing extra variables, while inherently enforcing smoothness and avoiding self-intersections through boundary extraction. As a demonstration, we present the first OMC nanobeam design in hexagonal boron nitride (hBN), a promising material for integrated quantum photonics. Optimizing 23 design parameters, we obtain a highly non-intuitive mechanical mode with small mode volume and strong optomechanical coupling, illustrating the power of FD-based optimization for next-generation photonic devices.