## 1. Introduction

[2] Specific wave propagation and radiation properties of certain diffraction phenomena by periodic structures motivate the use of such structures in several areas of optics, acoustics, electromagnetics and integrated optoelectronics. A dielectric waveguide with periodically varying refractive index is mainly utilized to obtain a frequency selective surface. In the microwave region, dielectric waveguide gratings are used as dichroic subreflectors in large reflecting antennas [*Agrawal and Imbriale*, 1979] and reflection and transmission filters [*Tibuleac et al.*, 2000]. Thin-film periodic dielectric structures are important in many integrated optics devices such as beam-to-surface-wave couplers, distributed feedback amplifiers and lasers, multiplexers, and demultiplexers [*Wang and Magnusson*, 1994]. Also, dielectric gratings play a significant role in resonant narrow-band plane-wave reflection filters. These devices operate by exciting a resonance (leaky wave mode) in a grating-waveguide structure and yield highly efficient filtering at the desired frequency with relatively small sidebands. Therefore, resonant grating filters are suitable for many applications such as laser-cavity mode selectors [*Norton et al.*, 1997] as well as security and anti-counterfeit devices, used in credit and identification cards [*Turunen and Wyrowski*, 1997].

[3] The diffraction phenomena by dielectric grating waveguides have already been analyzed by perturbation techniques and rigorous differential-equation methods. Specifically, the perturbation techniques provide physically intuitive but approximate results (valid only for weak grating perturbations, e.g. small grating thickness) [*Miyanaga and Akasura*, 1982]. The most widely used rigorous differential-equation methods are the rigorous coupled-wave analysis [*Moharam et al.*, 1995] and the modal method [*Chu and Kong*, 1977]. Both are vector diffraction methods that provide reflected and transmitted diffraction efficiencies for given sets of structural and incident field parameters. Their disadvantages consist in the facts that they are computationally intensive, require time-consuming root searches, large numerical matrices, and multiple time-intensive steps to determine the resonance's width [*Norton et al.*, 1997]. Besides, the finite difference time domain method [*Zunoubi and Kalhor*, 2006] and the boundary element method [*Nakata and Koshiba*, 1990] have also been applied for the investigation of the diffraction by grating waveguides. Subdomain methods, like for example the boundary element method, possess drawbacks concerning the creation of fictitious charges and currents on the boundaries of the subdomains due to the discontinuous variation of the electromagnetic field as well as the large number of unknowns, leading to large required memory and CPU time and possible numerical instabilities [*Athanasoulias and Uzunoglu*, 1995]. Almost all the above mentioned algorithms solve the diffraction problem following the three basic steps: (i) the unknown fields outside the grating region are expressed by appropriate series of plane waves with under determination coefficients (ii) a general solution of Maxwell's equations in the grating region is determined by using an appropriate method (iii) the boundary conditions at the diffraction grating and homogeneous grating interfaces are matched (see the related detailed discussion in *Jarem and Banerjee* [2000]). In addition, it is worth noting that layered periodic waveguides and antennas have also been analyzed by applying two-dimensional fast spectral domain algorithms for volume integral equations and using layered media Green's functions [*Eibert and Volakis*, 2000; *Eibert et al.*, 2003]. Besides, the propagation characteristics of coupled nonsymmetric grating slab waveguides have been investigated in [*Tsitsas et al.*, 2006].

[4] In this paper, we investigate the diffraction behavior of an infinite periodic grating slab waveguide subject to plane wave incidence with arbitrary incidence angle. The grating region consists of a periodic layer containing rectangular grooves. The developed methodology combines semianalytical integral equation techniques and is essentially based on a Method of Moments technique using entire domain basis functions. The standard electric field integral equation is employed for the electric field on the grooves. By applying Sommerfeld's method, a Fourier integral representation of the Green's function of the nongrating problem is derived. The integral equation is subsequently solved by applying an entire domain Galerkin's technique, based on a Fourier series expansion of the electric field on the grooves, leading to a nonhomogeneous linear system. The solution of this system provides the field on the grooves, by means of which the reflected and transmitted field's distributions are determined.

[5] The proposed method provides semianalytic solutions with high numerical stability, controllable accuracy, and high efficiency, since accurate results are obtained by using only a few expansion terms [*Jones*, 1964]. Therefore, this method achieves also economy in computer memory and CPU time. Convergence of the reflected and transmitted fields, which is checked by increasing successively the number of expansion functions, demonstrates the superior numerical efficiency with respect to the methods of [*Pai and Awada*, 1991; *Morf*, 1995]. This efficiency is mainly justified by the fact that both the unknown electric field and the entire domain expansion functions satisfy Helmholtz equation. A detailed analysis concerning convergence and numerical efficiency aspects is included in the numerical results and discussion section below. Furthermore, the Green's function is analytically expressed and all the involved integrations are analytically carried out. Thus, the computational cost is reduced, the accuracy increases and the sole approximation in the solution is the truncation of the expansion functions sets. Besides, the present formulation requires no discretization of the integral equation involved, while the boundary element method requires and depends strongly on the discretization in boundary elements [*Nakata and Koshiba*, 1990]. In addition, the use of the Green's function of the slab geometry provides a more compact formulation inherently satisfying the boundary conditions in the nongrating structure. This situation does not occur either in the above mentioned rigorous differential-equation methods (where the respective boundary conditions are imposed at the final step (iii)) or in the boundary element method (utilizing the free space Green's function).

[6] The numerical results of this paper, apart from the convergence mentioned above, mainly exhibit the optimal incident field's and grating's characteristics for the structure's efficient operation as a narrow band reflection or transmission filter. Specifically, Wood's anomalies are investigated by applying the developed integral equation method and its results are compared with those of previously established techniques. Moreover, the characteristics of the reflection and transmission resonances in the case of plane wave diffraction by arrays of dielectric rectangular cylinders are discussed. Finally, we investigate the dependence of the grating waveguide's reflectance spectrum from the grooves characteristics inside each grating's unit cell. It is demonstrated that the grooves modulation provides flexibility in the design of reflection grating filters and leads to very narrow resonances for specific choices of the grooves parameters.

[7] In the following analysis an exp(*jωt*) time dependence of the field quantities is assumed and suppressed.