Structure-Adaptive Topology Optimization Framework for Photonic Band Gaps with TM-Polarized Sources
We present a structure-adaptive topology optimization framework for engineering photonic band gaps with TM-polarized sources based on computation of the photonic density of states with a uniform source substituting for the standard Dirac delta function sources in formalisms analogous to $Γ$-point integration and to integration over a full Brillouin zone. We generalize the limiting uniform and Dirac delta function sources to more general collections of sources, such that the union of the sources in a given collection is hyperuniform. The uniform-source approach necessarily leads to the fastest computations. We also demonstrate how our approach can be generalized to the treatment of the frequency-dependent optical response of materials. Finally, we show that we can recover known two-dimensional photonic crystals for the TM polarization. A key advantage of our work is its ability to optimize for a specific midgap frequency and band gap in a structure-adaptive manner. Our work leverages the insight that the determination of the minimum supercell size and the minimum precision to which the frequencies within the photonic band gap must be sampled will lead to the observation of photonic-crystal structures when the $Γ$-point formalism for the uniform-source approach is employed. Additionally, our $Γ$-point and full Brillouin zone formalisms for the uniform-source approach inherently encourage binarized designs even in gradient descent.