  Photonic topology optimization is a technique used to find the electric
permittivity distribution of a device that optimizes an electromagnetic
figure-of-merit. Two common techniques are used: continuous density-based
optimizations that optimize a grey-scale permittivity defined over a grid, and
discrete level-set optimizations that optimize the shape of the material
boundary of a device. More recently, continuous optimizations have been used to
find an initial seed for a concluding level-set optimization since level-set
techniques tend to benefit from a well-performing initial structure. However,
continuous optimizations are not guaranteed to yield sufficient initial seeds
for subsequent level-set optimizations, particularly for high-contrast
structures, since they are not guaranteed to converge to solutions that
resemble only two discrete materials. In this work, we present a method for
constraining a continuous optimization such that it converges to a discrete
solution. This is done by inserting a constrained sub-optimization at each
iteration of an overall gradient-based optimization. This technique can be used
purely on its own to optimize a device, or it can be used to provide a nearly
discrete starting point for a level-set optimization.
