A design method for planar waveguide concave diffraction gratings is presented in the paper

which is named Modified Recursive Definition of Facet Positions. The method is based on a modification of Recursive Definition of Facet Positions. We can design the planar waveguide concave diffraction gratings accurately by using this method

especially for some special grating structures. The central position of grating facets can be presented by two constraint functions

and its numerical solution is derived from a recursive formula approach. To improve the accuracy of computation

a Newton iteration procedure is introduced in the recursive approach while computing each facet position. Two typical design examples

a flat-filed mount and a flat-top mount

are given to assess the necessity of adopting Newton iteration procedure subsequently. The simulative results prove the modification is crucial for the accurate definition of grating facet position.