  Photonic inverse design typically seeks designs parameterized by binary
arrays, where the values of each element correspond to the presence or absence
of material at a particular point in space. Gradient-based approaches to
photonic inverse design often include thresholding of non-binary optimization
arrays; when the thresholding is sufficiently sharp, binary designs are
obtained. However, difficulty can arise due to vanishing gradients for sharp
thresholds, which cause optimization to stall. Here, we present hypersphere
optimization, a new method of carrying out binarized photonic optimization on a
high dimensional manifold. We numerically show that, on the manifold, upstream
gradients from the objective function can smoothly transition a design from one
nearly-binary array to another with minimal pixel flips. Our method is an
approximate gradient-based optimization method for binary arrays that can be
applied to various photonic inverse design problems and beyond.
