Integral equation solver for conical diffraction of multi-profile gratings


Dr. G. Schmidt


Dr. L. Goray - St Petersburg Academic University, Nanotechnology Research and Education Centre RAS


International Intellectual Group, Inc.


February 2009 - January 2011


A diffraction grating is an optical component with a periodic pattern, which splits and diffracts light into a finite number of beams traveling in different directions. Since their invention about two centuries ago diffraction gratings have attracted great attention especially for spectroscopic applications. Nowadays modern semiconductor technologies as for example ion-etching technique paved the way for fabricating optical devices with complicated structural features within the length-scale of optical waves. Such diffractive elements have many technological advantages and can be designed to perform functions unattainable with traditional optical devices. The directions, in which a grating diffracts light, are known from simple grating equations. From a grating user point of view it is more important to know how much of the light goes into specific directions and how this depends on the grating structure. It is widely accepted that the prediction of these efficiencies or even the design of new structures has to rely on accurate mathematical models and numerical codes for solving the full electromagnetic vector-field equations.

Numerical codes based on integral equations are rather popular for solving classical diffraction problems, where the grating grooves/lines are oriented perpendicular to the plane of incidence. It turned out that integral equation methods are very efficient in certain scenarios, which combine several difficulties (non-smooth profile curves, high wave-numbers and thin coating layers) that make them incapable of being treated by other methods. However, gratings are frequently used with obliquely incident angles. The resulting ''conical'' diffraction by an arbitrarily oriented planar grating are often treated by three-dimensional methods. An elegant analytic two-dimensional approach has been developed in [1] and later on implemented in the FEM package DiPoG for solving direct and optimal design problems for diffraction gratings. Recently an integral formulation of conical diffraction was studied in [2], [3]. It has been shown that these problems can be formulated as a system of singular integral equations.

Goals of Project

  • Design of a robust and reliable numerical method for solving the system of singular integral equations of conical diffraction
  • Implementation of an efficient solver for gratings with one profile
  • Implementation of two different approaches to treat conical diffraction by multi-profile gratings
  • Extension to photonic crystal diffraction gratings


  1. J. Elschner, R. Hinder, F. Penzel, G. Schmidt, Existence, uniqueness and regularity for solutions of the conical diffraction problem. Math. Mod. Meth. Appl. Sci., 10 (2000), 317--341.
  2. G. Schmidt, Integral equations for conical diffraction by coated gratings. WIAS Preprint No. 1296 2008, to appear in J. Int. Equ. Appl.
  3. G. Schmidt, Boundary Integral Methods for Periodic Scattering Problems, in: Around the Research of Vladimir Maz'ya II. Partial Differential Equations, pp. 337-364, Springer (2010), cf. WIAS Preprint no. 1435 , 2009.
  4. L. I. Goray, G. Schmidt, Solving conical diffraction grating problems with integral equations. J. Opt. Soc. Amer. A, 27 (2010), 585--597, cf. WIAS Preprint No. 1469, 2009.