Well-posed analysis for a class of Helmholtz equation with periodic coefficients

The scattering theory in periodic structures has many applications in modern microoptics and industry. Periodic structures are often referred to as diffraction gratings. In this paper we consider a planar dielectric-layer modulated grating problem. The diffraction problem may be modeled by a Helmholtz equation with periodic coefficients. Results on existence and uniqueness of the solution for the diffraction problem are obtained by variational method and integral equation method, respectively. At the end of the paper, we also discuss the Born approximation to the solution of an equivalent integral equation.