## 1. Introduction

[2] Extensive recent interest in manipulation and localization of the light in novel applications of plasmonic resonance such as near-field microscopy, surface defect detection, subwavelength lithography, and developing tunable optical filters has renewed interest in modeling wave scattering from grating surfaces. When solving the problem of scattering from large periodic array of identical cavities or holes, it is useful to approximate the structure as an infinite array and take advantage of the periodicity of the electromagnetic fields. Several methods were reported in the literature to solve the problem of scattering from an infinite array of cavities or holes engraved in metallic or dielectric screens. Among these methods is the coupled-wave method (CWM) where the permittivity of the grating is expanded in the Fourier series and the component of the wave number along the grating is determined by the Floquet condition as given by *Moharam and Gaylord* [1986]. Theoretically, an infinite number of the space harmonics is needed to expand the fields. In practice, a large number of space harmonics must be considered to achieve accurate results. However, the convergence rate, especially for the TE polarization case (where the magnetic field has a component only along the axis of the cavities), is very slow as given by *Li and Haggans* [1993].

[3] The problem of diffraction by an infinite periodic conducting grating was investigated using a hybrid method combining the finite element method (FEM) and the method of moments (MoM) by *Gedney and Mittra* [1991]. Using the equivalence theorem, the fields inside of the cavities or holes were decoupled from the outside region by closing the apertures with a perfect electric conductor (PEC) surface and introducing the equivalent magnetic currents over the openings. Using Floquet theorem, the scattered fields outside the cavities or holes are produced by the periodic equivalent magnetic current and the fields inside the single cavity or hole were calculated using FEM. Imposing field continuity as the boundary condition at the aperture of the single cavity or hole results in a set of equations describing the unknown equivalent magnetic current. The MoM is employed to determine the unknown magnetic current coefficients. In the works of *Delort and Maystre* [1993] and *Pelosi et al.* [1993], a hybrid FEM and Floquet mode expansion of the scattered fields was used to analyze the problem of scattering from periodic structures where the FEM is applied inside the unit cell which encloses all inhomogeneities and the periodic boundary condition is applied on the lateral boundaries of the unit cell. In addition, field continuity is forced on the upper and lower boundaries of the unit cell. In this method the upper and lower boundaries must be chosen in a homogenous medium (i.e., in upper half-space above the cavity) to achieve fast convergence that results in a prohibitive increase of the solution domain.

[4] In the work of *Ichikawa* [1998], the finite difference time domain method (FDTD) is used to analyze the problem of electromagnetic scattering from a dielectric gratings. The FDTD algorithm is applied in the unit cell of the grating, including two different dielectric materials. The periodic boundary condition and the absorbing boundary condition (ABC) are applied on the lateral walls of the unit cell and top and bottom boundaries of the unit cell, respectively, to truncate the mesh region. Placing the ABC close to the scatterer results in nonphysical reflections into the solution region yielding numerical errors. To minimize these errors, the domain truncating boundaries (i.e., top and bottom boundaries) should be chosen far enough from the cavity. However, placing the boundaries of the computational domain far from the cavity leads to a prohibitive increase in the computational cost. The infinite array of bottle-shaped cavities is investigated by *Depine and Skigin* [2000] using the mode-matching technique (MMT). Although the mode-matching technique is a highly accurate and efficient, it cannot be used for cavities having inhomogeneous or anisotropic fillings.

[5] Recently, the method of moment is applied to solve the problem of scattering from an infinite array of rectangular cavities in an impedance screen as given by *Baranchugov et al.* [2008]. The image theory is applied to reduce the infinite array to a single cell of one period. Due to the periodicity, the scattered fields are expanded in terms of Floquet series. Also the fields inside the cavities are expressed in terms of the eigenfunction series of the parallel plate waveguide with impedance walls. By forcing the continuity of the fields at the aperture of the cavity, the integral equation is derived and solved using the MoM. Another method using the overlapping T-block method (OTM) and the Floquet theorem was reported by *Cho* [2008]. The array is divided into the infinite T blocks associated with each cavity. The fields inside each T block are calculated using the Green's function and the mode-matching techniques. By superposing the fields in overlapping T blocks and using the Floquet theorem, the total fields are expressed in the closed form. However, this method is limited to cavities with canonical shape and homogenous fillings.

[6] In this paper we develop a new FEM-based method to solve the problem of scattering from an infinite periodic array of identical cavities engraved in an infinite perfect electric conductor screen. Using the two-boundary formulation that was initially reported by *McDonald and Wexler* [1972] and then applied to the problem of finite array of cavities by *Alavikia and Ramahi* [2009], the solution domain is confined to *only* one cavity that is divided into interior and exterior regions. The finite element formulation is applied inside the interior regions to obtain the solution of the Helmholtz's equation. The surface integral equation using the free-space Green's function, which was reported by *Alavikia and Ramahi* [2009], is applied at the openings of all cavities as a global boundary condition. Taking advantage of the field's periodicity at the apertures of the cavities, the free-space Green's function is replaced by the quasi-periodic Green's function, thus limiting the surface integral equation to the aperture of *only* one cavity. The Neumann or Dirichlet boundary condition is applied on the wall of the cavity for the TE or TM polarization, respectively. The advantage of this method is that no periodic boundary condition that would require a constrained mesh scheme is used in this formulation. Also, we emphasize that in the method presented here, the surface integral equation using the quasi-periodic Green's function is used to derive a linear system of equation as a constraint that connects the field values on the boundary to the field values on the apertures of the single cavity in the array by considering the coupling between all cavities. The attractive feature of two-boundary formulation combined with quasi-periodic Green's function is that no singularities in Green's function arises while applying the surface integral as a boundary constraint.