## 1 Introduction

[2] Periodic structures can find a variety of applications in the area of the solid state physics and microwave engineering [*Wu*, 1995; *Mittra et al*., 1988; *Yablonovitch*, 1987; *Smith et al*., 2000]. The EM scattering from periodic structures has been studied using many methods. They include the mode-matching method [*Chen*, 1970a; *Chen*, 1970b; *Chen*, 1971], the finite element method (FEM) [*Bardi et al*., 2002; *Pelosi et al*., 2000], the boundary integral-modal (BI-modal) method [*Bozzi et al*., 2001], the finite element-boundary integral (FE-BI) method [*Gedney et al*., 1992; *Eibert et al*., 1999], and the integral-equation (IE) method [*Mittra et al*., 1988], [*Pous et al*., 1991; *Wan et al*., 1995; *Mathis et al*., 1998; *Stevanović et al*., 2006; *Dardenne et al*., 2008; *Hu and Song*, 2010, 2011; *Hu et al*., 2011; *Peterson et al*., 1998; *Trintinalia et al*., 2004; *Jordan et al*., 1986; *Silveirinha et al*., 2005].

[3] In this paper, an IE approach in spatial domain is developed to analyze the scattering from a doubly periodic array of three-dimensional (3-D) general objects. The periodic array can be filled with different media in different layers. The 3-D perfect electric conductor (PEC) or dielectric objects in each layer can have arbitrary shapes. Actually, Wang and Ling in [*Wang et al*., 1991] proposed the idea of using the equivalence principle and connection scheme (EPACS) to efficiently characterize a deep cavity structure. Trintinalia and Ling in [*Trintinalia et al*., 2004] extended the IE approach with the EPACS to handle the multilayered doubly periodic array without PEC or dielectric inclusions. Hu and Song in [*Hu and Song*, 2011] delineates the IE approach using the EPACS to calculate electromagnetic (EM) scattering from multilayered and semi-infinitely layered arrays of two-dimensional (2-D) PEC objects. The authors in [*Hu et al*., 2011] extended the approach in [*Hu and Song*, 2010] to handle a doubly periodic array of three-dimensional (3-D) PEC objects. However, the formulation in [*Hu et al*., 2011] is applicable to multilayered arrays, rather than semi-infinitely layered arrays. In this work, the IE approach with the EPACS is further extended to handle the doubly periodic array of dielectric objects. Furthermore, this paper also formulates the EM problem from a semi-infinitely layered array of 3-D PEC or dielectric objects. The formulation of the IE approach with the EPACS is compactly unified for treating several cases, including multilayered or half-space media and a multilayered or semi-infinitely layered array of PEC and dielectric objects.

[4] As shown in [*Hu and Song*, 2010; *Hu et al*., 2011 and *Trintinalia et al.*, 2004], the application of the EPACS can avoid the use of multilayered doubly periodic Green's functions [*Park et al*.,1998]. Actually, due to the periodicity of the array, the computational domain is first restricted to one period. In one period, each layer is treated as an individual cell [*Hu and Song*, 2010, 2011; *Hu et al*., 2011; *Peterson et al*., 1998; *Trintinalia et al*., 2004]. Then, the equivalence principle can be applied separately to each individual cell to obtain the surface integral equations for equivalent currents on the outer boundary of the cell and the inner boundary (i.e., the general object's surface). Specifically, the electric field integral equations (EFIE) and the magnetic field integral equations (MFIE) [*Balanis*, 1989; *Harrington*, 1968] are applied on the outer boundary [*Trintinalia et al*., 2004]. The Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation [*Mautz et al*., 1979] or the EFIE of the impedance boundary condition (IBC) [*Cwik et al*., 1987] is applied on the surfaces of general objects. When the surface impedance of objects approaches zero, the EFIE of the IBC can be reduced to that of PEC. After discretizing the integral equation, the unknowns on the object's surface are eliminated. Then, the periodic boundary condition (PBC) is applied to eliminate the unknowns on the left and right sides of the outer boundary. Thus, the relationship between the top and bottom surfaces can be established. Next, the boundary condition of tangential continuity is used to connect two adjacent cells and eliminate the unknowns on the interface between two cells. For an *N*-layer array, (*N-1*)-time connection procedure is applied. For identical layers, the logarithm algorithm can be applied to reduce the times of the connection procedure from *N-1* to *L*_{0} [*Hu and Song*, 2010; *Hu et al*., 2011]. The logarithm algorithm makes use of the fact that the relationship between the topmost and bottommost surfaces of Layers 1–2* ^{n}* is identical to that of Layers 2

^{n + 1}− 2

^{2n}. Then, one can obtain the relationship between the topmost and bottommost surfaces of the array. In addition, two more integral equations on the topmost and bottommost surfaces should be adopted to establish the complete system of equations. These integral equations involve the doubly periodic Green's functions (DPGF) in free space. The fast convergence for the DPGF in free space can be achieved by using some acceleration techniques [

*Jordan et al*., 1986;

*Silveirinha et al*., 2005;

*Hu and Song*, 2011].

[5] Moreover, based on the 3-D EPACS, an effective approach is proposed to handle the semi-infinitely layered case in which one unit consisting of several layers is repeated infinitely along one direction. The basic idea for the 2-D case in [*Hu and Song*, 2010] has been extended to the 3-D case. Finally, the numerical results will be given to validate the developed approach.