Employing Complex Analysis to Engineer Finite Photonic Crystals

We perform topology optimization for photonic crystal structures by minimizing the integrated photonic local density of states weighted by a window function having the form of a generalization of the Lorentzian¹. In two dimensions, we find phase transitions for the TM polarization as a function of the number of poles in the window function, with the phase following the ultimate transition agreeing with known crystal structures. Our method has advantages over conventional approaches involving the computation of photonic bandstructures in that it allows for arbitrary crystal symmetry and can treat finite structures. This work has potential applications in the design of mirrors, waveguides, and cavities, and in the discovery of novel photonic crystals in three dimensions.<br/><br/>1. H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, Fundamental limits to near-field optical response over any bandwidth, Phys. Rev. X 9, 011043 (2019).