An optimum design of concave gratings is proposed based on Newton recursive algorithm,which differs from the traditional design based on a power-series expansion of the light-path function.The structure of a concave grating is represented by two constraint equations that may be chosen to meet some specific demands.From the grating pole,the facet positions are determined from the numerical solutions for the roots of two constraint equations in sequence.Anastigmatic mounts of concave gratings based on this algorithm are optimized,and numerical results are simulated.Further applications of this algorithm are also discussed,including the one that can not be designed with the theory of the power-series expansion of light-path function.