  Topological photonic crystals (PCs) can support robust edge modes to
transport electromagnetic energy in an efficient manner. Such edge modes are
the eigenmodes of the PDE operator for a joint optical structure formed by
connecting together two photonic crystals with distinct topological invariants,
and the corresponding eigenfrequencies are located in the shared band gap of
two individual photonic crystals. This work is concerned with maximizing the
shared band gap of two photonic crystals with different topological features in
order to increase the bandwidth of the edge modes. We develop a semi-definite
optimization framework for the underlying optimal design problem, which enables
efficient update of dielectric functions at each time step while respecting
symmetry constraints and, when necessary, the constraints on topological
invariants. At each iteration, we perform sensitivity analysis of the band gap
function and the topological invariant constraint function to linearize the
optimization problem and solve a convex semi-definite programming (SDP) problem
efficiently. Numerical examples show that the proposed algorithm is superior in
generating optimized optical structures with robust edge modes.
