  We present a method for stellarator coil design via gradient-based
optimization of the coil-winding surface. The REGCOIL (Landreman 2017 Nucl.
Fusion 57 046003) approach is used to obtain the coil shapes on the winding
surface using a continuous current potential. We apply the adjoint method to
calculate derivatives of the objective function, allowing for efficient
computation of analytic gradients while eliminating the numerical noise of
approximate derivatives. We are able to improve engineering properties of the
coils by targeting the root-mean-squared current density in the objective
function. We obtain winding surfaces for W7-X and HSX which simultaneously
decrease the normal magnetic field on the plasma surface and increase the
surface-averaged distance between the coils and the plasma in comparison with
the actual winding surfaces. The coils computed on the optimized surfaces
feature a smaller toroidal extent and curvature and increased inter-coil
spacing. A technique for visualization of the sensitivity of figures of merit
to normal surface displacement of the winding surface is presented, with
potential applications for understanding engineering tolerances.
