Reciprocity theorem and accuracy evaluation on scattering fields by dielectric gratings

In the theory of gratings, a diffraction solution must satisfy the energy balance and the reciprocity. This paper examines the diffraction by multilayered dielectric gratings, including lossy layers of which the solutions do not satisfy the energy balance. Applying the shadow theory to the matrix eigenvalues method, the symmetry of scattering factors in the theory and that of diffraction efficiencies are shown in some formulae and then are validated numerically. In addition, the reciprocity errors for diffraction efficiencies and scattering factors are newly considered as an accuracy criterion. It is shown that, on numerical calculations, the reciprocity errors for diffraction efficiencies have only to be checked.