Topology-optimization framework for photonic band gaps with TM- and TE-polarized sources based on the photonic density of states

We present a topology-optimization framework for engineering photonic band gaps with TM- and TE-polarized sources based on computation of the photonic density of states with a uniform source substituting for the standard Dirac delta function sources. Optimization is performed in a manner that allows for targeting of a given midgap frequency and minimum band gap. We apply this framework in formalisms analogous to Γ-point integration and to integration over a full Brillouin zone. We also demonstrate how our approach can be generalized to the treatment of the frequency-dependent optical response of materials and could potentially be employed to invert photonic and phononic bandstructures. Finally, we show that we can recover known two-dimensional photonic crystals for the TM and TE polarizations, and we demonstrate waveguiding applications of our designs. Additionally, our full Brillouin zone formalism for the uniform-source approach inherently encourages binarized designs even in gradient descent.