Gradient Based Optimization Methods for Metamaterial Design

The gradient descent/ascent method is a classical approach to find the minimum/maximum of an objective function or functional based on a first-order approximation. The method works in spaces of any number of dimensions, even in infinite-dimensional spaces. This method can converge more efficiently than methods which do not require derivative information; however, in certain circumstances the "cost function space" may become discontinuous and as a result, the derivatives may be difficult or impossible to determine. Here, we discuss both level set methods and eigenfunction optimization for representing the topography of a dielectric environment and efficient techniques for using gradient methods to solve different material design problems. Numerous results are shown to demonstrate the robustness of the gradient-based approach.