Abstract <p>We consider the calculation of a diffractive optical element (DOE) generating a required intensity distribution in a given plane. For the calculation, we propose a version of the supporting quadric method widely used in similar geometrical optics problems. The method is based on minimizing the root-mean-square error in a certain class of functions using a gradient method and leads to a DOE with a quasi-regular surface relief. We derive explicit expressions for the gradient and Hessian of the root-mean-square error, which makes it possible to efficiently apply the gradient method and provides the theoretical basis for using the methods of the second order. The obtained theoretical results are illustrated by a DOE design example.</p>