IntroductionUntil quite recently, the study of electromagnetic scattering from metallic or dielectric gratings has been largely confined to specific planes of incidence, and a single polarization 11-61. Within the ast few years, application of the Method of Moments y71 has resulted in general solutions for scattering k o m metallic lattices and gratings 18 -10 1 This paper reports the development of a general numerical method [ 11 1 for calculating the scattering properties of dielectric gratings Figure 1 depicts an infinite dielectric grating of finite depth. An incident plane electromagnetic wave may impinge upon the structure from any arbitrary direction (B,cp), and may be of either polarization. The formulation calculates the reflected and transmitted components of both polarizations, including appropriate grating lobes.
The Numerical MethodFor simplicity, the method for a singleinterface problem (i.e. t = m) will be described. The two-interface problem ( j , finite) is reduced to a pair of single-interface formulations by even-odd excitation analysis By applying Floquet's theorem to the structure a typical cell may be isolated by planes at y = yo andThe electric fields in the exterior or freespace region are expressed as Eo= A m m 5 + X , R m T m m m where Am are incident wave coefficients Rm are reflected wave coefficients Y f are Floquet plane waves TM and TE to z. Similarly, within the grating structure, the electric fields are represented by where are interior wave coefficients Tn are two-media Floquet waves T M and TE to y Nicolaos G a Alexopoulos The yh set is a generalization of the modes used by Collin [4,5].These modes are not normal modes of the structure itself, but are dependent upon'the incidence direction (0,V) of the incident wave. Equations (1) and e) are complemented by similar expressions for the H fields. Tangential E and H fields are equated at the xy plane and a modified Galerkints method [12] is used to arrive at either of two equivalent matrix equations, rlN1 = CY,,,] X [Bn] (3) o r [Ih] = CY' 1 X [ R , ] M,m (4) If equation (3) is solved for the interior mode coefficients, Bn, the observable quantities, Q, must be calculated by an equation of the form Rm = C -Am (5)n where the Cnm are mode coupling coefficients. Both the Bn and Cnm vary over several orders of magnitude. This results in an error magnification effect which makes the calculated results for Rm numerically unstable. However, the direct solution of equation (4) for the Rm set has been found to be extremely stable.
VerificationThe formulation has been verified ! JY a combination of theoretical simulation and experiment. In the theoretical verification the formulation has been shown to be smoothly continuous between the two major planes of incidence, i.e. cp = 0 and cp = !No.
Theoretical SimulationsBy allowing dimension a to approach the period by or by letting €2 approach €3, the numerical method may simulate a sheet of homogeneous dielectric, whose scattering properties are well known. Convergence to these limiting cases is...
