We formulate and exploit a computational inverse-design method based on topology optimization to demonstrate photonic crystal structures supporting complex spectral degeneracies. In particular, we discover photonic crystals exhibiting third-order Dirac points formed by the accidental degeneracy of monopolar, dipolar, and quadrupolar modes. We show that, under suitable conditions, these modes can coalesce and form a third-order exceptional point, leading to strong modifications in the spontaneous emission (SE) of emitters, related to the local density of states. We find that SE can be enhanced by a factor of 8 in passive structures, with larger enhancements ∼√n³ possible at exceptional points of higher order n.United States. Air Force Office of Scientific Research (FA9550-14-1-0389)National Science Foundation (U.S.) (DMR-1454836)National Science Foundation (U.S.) (DGE1144152
