Radio Science

IE analysis of multilayered doubly periodic array of 3-D general objects using equivalence principle and connection scheme

Authors


Corresponding author: Fu-Gang Hu, Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, 50011, USA. (hufugang@sina.com)

Abstract

[1] An integral equation (IE) approach in the spatial domain is developed to analyze the electromagnetic (EM) scattering from the multilayered doubly periodic array of three-dimensional (3-D) general objects. This approach applies the equivalence principle to the interior problem of each layer. Then, the tangential continuity condition is applied to build up the connection between layers. The application of the equivalence principle and connection scheme (EPACS) allows us to avoid of the use of multilayered double periodic Green's functions. For 2N identical layers, the elimination of the unknowns between top and bottom surfaces can be accelerated using the logarithm algorithm. In addition, this approach is extended to handle the semi-infinitely layered array in which a unit consisting of several layers is repeated infinitely in one direction. The main contribution of this paper is to extend previous work on the EPACS to handle the 3-D perfect electric conductor or dielectric objects with arbitrary shapes. Numerical results are provided to validate the present approach.

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 inline image identical layers, the logarithm algorithm can be applied to reduce the times of the connection procedure from N-1 to L0 [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 1–2n is identical to that of Layers 2n + 1 − 22n. 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.

2 Formulation

2.1 Array with Finite Layers

[6] Figure 1 shows one period of a multilayered array of general objects impinged by a plane wave. The multilayered structure consists of laterally unbounded dielectric layers stacked in the z-direction, and each layer contains a doubly periodic array of 3-D metallic or dielectric inclusions. The multilayered structures can be planar or non-planar structures. Let a1 and a2 be the primitive lattice vectors in the xy-plane for the periodic array. Figure 2 shows the unknowns distribution on the boundaries of Cell 1. Using the equivalence principle, one can get the electric field integral equations (EFIE) and magnetic field integral equations (MFIE) [Trintinalia et al., 2004; Hu et al., 2011; Balanis, 1989; Harrington, 1968] on So

display math(1)
display math(2)

where inline image and inline image.

Figure 1.

Cells in each layer.

Figure 2.

Unknown currents on the interior side of So and Si of Cell 1.

[7] On Si, if the PMCHWT formulation [Mautz et al., 1979) is applied, one has

display math(3)
display math(4)

[8] The operators inline image and inline image for Cell i are given by

display math(5)

where inline image, inline image, Gi(,d)(R) = exp (−jki(,d)R)/(4πR), and inline image. inline image is the principal value integral of inline image. The subscript i(,d) represents i or i,d.

[9] If the impedance boundary condition (IBC) [Cwik et al., 1987] is applied, one has

display math(6)

where

display math(7)

[10] When inline image, the IBC boundary is reduced to the PEC boundary. Thus, this formulation is applicable to the PEC case.

[11] Discretizing equations ((1))–((4)) through applying the zero-order divergence-conforming vector basis functions [Graglia et al., 1997; Rao et al., 1982] on triangular elements yields

display math(8)
display math(9)
display math(10)
display math(11)

[12] If the IBC is applied, one can obtain from equations ((1))–((2)), ((6))

display math(12)
display math(13)
display math(14)

[13] The expressions of the coefficient matrix's elements are given in the Appendix.

[14] Eliminating the unknowns on the objects in terms of ((8))–((11)) or based on ((12))–((14)) yields

display math(15)

[15] Eliminating the unknowns on four sides with the aid of the periodic boundary condition (PBC), one can obtain

display math(16)

where

display math

[16] Similarly, one can apply the equivalence principle to obtain the relationship for Layer i

display math(17)

[17] In the next step, the connection scheme is applied to eliminate the unknowns between Layers 1 and N. From ((16)), one gets the initial relationship

display math(18)

where

display math

[18] Assume the following relationship has been found between Layers 1 and i (1 ≤ i ≤ N − 1)

display math(19)

[19] Applying ((17)) and the condition of tangential continuity of fields to the interface between Layers i and i + 1, one can obtain

display math(20)

[20] Rewriting A(i) and V(i + 1) as

display math(21)

[21] Combining ((19)) and ((20)) gives

display math(22)

where

display math(23)

[22] Thus, one can recursively find the relationship between Layer 1 and Layer N by using ((23)) [Hu and Song, 2010]

display math(24)

[23] It should be noted that if each layer is identical, the logarithm algorithm can be applied to speed up the procedure for finding A(N). For inline image, the N-time process of applying the connection scheme can be reduced to that of L0 times by replacing V(i + 1) with A(i) in ((23)).

[24] In addition to ((24)), two more conditions are required to solve the scattering problem. The following equations hold

display math(25a)
display math(25b)

on the top surface of Layer 1, and

display math(26a)
display math(26b)

on the bottom surface of Layer N. The subscript t indicates the tangential components of vectors. inline imagep and inline imagep are obtained from inline imagei and inline imagei by replacing the free-space Green's function with the doubly periodic Green's function [Jordan et al., 1986; Silveirinha et al., 2005; Hu et al., 2011]. Actually, ((25a))–((26b)) are obtained by setting up the problems equivalent to the original problem above the topmost surface and below the bottommost surface, respectively. Because there is no source exciting the incident field in the region below the bottom surface of Layer N, no incident field contributes to the total field in ((26a)) and ((26b)).

[25] Applying the weighted residue method to ((25a))–((26b)) [Hu and Song, 2011], one obtains

display math(27)

where

display math(28)

[26] Combining ((24)) and ((27)) gives

display math(29)

[27] Moreover, the formulations are applicable to the case in which there are no general objects inside the cell by changing ((15)) with

display math(30)

[28] The electric field Es, excited by the equivalent currents, can be given by

display math(31)

[29] The subscript 0 indicates background media. Assume the incident wave is the J0th-order Floquet's mode with coefficient inline image. For the Jth-order Floquet's mode [Hu and Song, 2011], the reflection and transmission coefficients can be given by

display math(32)

where

display math(33)

[30] The subscripts t and b in St,b indicate the topmost and bottommost surfaces, respectively. In the sign ∓, − and + correspond to t and b, respectively. The definition of inline image, ZJ, and γJ can be found in [Hu and Song, 2011].

2.2 Array with Semi-infinite Layers

[31] Assume a unit consisting of N0 (N0 ≥ 1) layers is repeated infinitely along inline image direction. Each unit must be identical to the other units. However, it is unnecessary for each layer to be the same in one unit. Each unit can be regarded as a two-port network and the semi-infinitely layered array is equivalent to the connection of an infinite number of two-port networks. For this semi-infinitely layered array, the impedance matrix representing the relationship between the equivalent magnetic and electric currents on the top surface of each unit should be identical because the network is infinitely extended when observed from the top surface of any unit. Let P0 denote each of these impedance matrices. Thus, by using the tangential continuity condition, one obtains

display math(34)

[32] On the other hand, by using EPACS, one gets

display math(35)

[33] Using the second set of equations from both ((34)) and ((35)), one can get

display math(36)

[34] Substitution of ((36)) into the first set of equations from ((35)) gives

display math(37)

[35] Comparing ((37)) with the first set of equations from ((34)), one can achieve

display math(38)

[36] ((38)) is an equation for the unknown matrix P0. It is impossible to find the explicit solution for P0. However, it can be solved by using the iterative procedure

display math(39)

where α is a relaxation factor, and X is a matrix to be determined, which satisfies X = f(X). Combining ((27)) and the first set of ((34)) gives the complete system of equations

display math(40)

where

display math(41)

[37] Similarly, this approach is applicable to the semi-infinitely layered array without objects inside each layer by means of ((30)).

3 Numerical Results

[38] To validate the developed approach, several periodic structures are simulated. The implemented Fortran program is run on a PC machine with an Intel CPU of 2.27 GHz. The double precision architecture is applied in the computation.

[39] The first example is a doubly periodic array of two PEC cylinders, which is shown in Figure 3 [Stevanović et al., 2006; Loutrioz et al., 1999]. The axis of each cylinder is along the x-direction. The artificial box enclosed by the outer boundary So is filled with the free space. The height of the box along z-axis is 6 mm. The other two dimensions of the box are determined by the lattice vectors. The lattice vectors are inline image mm and inline image mm. Each cylinder has the radius r = 0.75 mm and length L = 5 mm. The space between two cylinders is d = 6 mm. The structure also can be simulated by directly applying EFIE with the free-space DPGF on the PEC surface of two cylinders. This direct IE approach does not apply the connection scheme. Table 1 shows the computational information of the examples, including the numbers of elements on objects and artificial box, the total number of unknowns, memory requirements, and the CPU time per frequency point. Figure 4 shows the scattering parameters of the 0th-order TMz mode as a function of frequency. The agreements between the EPACS and direct IE results are very good. The direct IE results obtained using our in-house code [Hu et al., 2011] agree very well with those in [Stevanović et al., 2006], which have been validated by measurement results.

Figure 3.

Two cells in one period, including two meshed circular cylinders.

Table 1. Computational Information for Periodic Structures
ProblemsElements (Objects, Box)UnknownsMemory (MB)CPU Time at Each Frequency or Angle Point (s)
Figure 4232, 3001248808.9
Figure 661, 560175937021
Figure 7632, 2202556310165
Figure 80, 1805404153
Figure 9 (semi-inf.-layer)61, 1806194323
Figure 9 (4-layer)61, 180619412.8
Figure 9 (8-layer)61, 180619412.8
Figure 9 (32-layer)61, 180619413.1
Figure 12320, 30013809113
Figure 4.

Transmission coefficients of the 0th-order TMz mode. The incident angles are θi = 0 and ϕi = 0.

[40] The second example is a microstrip dipole in a dielectric medium shown in Figure 5. Figure 6 shows the reflection coefficients of 0th-order TEz mode. The numerical results agree very well with those from the direct IE approach.

Figure 5.

Doubly periodic array of microstrip dipole in a dielectric slab (Left: side view; Right: top view). εr1 = 2, μr1 = 1, and t1 = t2 = 1 mm. W = 2.5 mm and L = 5 mm. The lattice vectors are inline image mm and inline image mm.

Figure 6.

Reflection coefficients of the 0th-order TEz mode of a periodic array of microstrip dipoles. The incident angles are θi = 0 and ϕi = 0. The results of the direct IE approach are taken from [Eibert et al., 1999].

[41] For the dielectric case, a one-layer doubly periodic array of dielectric sphere is considered. The PMCHWT formulation is applied on the sphere's surface. Figure 7 shows the transmitted power coefficient of the array.

Figure 7.

Transmitted power coefficient of the 0th-order TMz mode of a one-layer doubly periodic array of dielectric spheres in free space. The relative permittivity and permeability of the sphere are εr = 8 and μr = 1, respectively. The radius of the sphere is R = 0.2 m. The lattice vectors are inline image m and inline image m. The incident angles are θi = 0 and ϕi = 0. The mode matching results are taken from [Matsushima et al., 2007].

[42] Figure 8 shows the reflection coefficient of a semi-infinitely layered array without PEC objects. There is a good agreement between the results from the proposed methods and the analytical solution. Then, the reflection coefficient of the semi-infinitely layered array of PEC patches is calculated and compared with that of a 4-layer, 8-layer, and 32-layer PEC patch array, which is shown in Figure 9. Because the media are lossy, when the number of layers increase, the solution for the multilayered array converges to that of the semi-infinitely layered array.

Figure 8.

Reflection coefficient the 0th-order TMz mode of semi-infinitely layered array without PEC object. εr = 2 − j0.1 and μr = 1. The lattice vectors are set to be inline image mm and inline image mm. The layer thickness is set to be 2 mm. The incident angles are θi = 0 and ϕi = 0.

Figure 9.

Reflection coefficient of the 0th-order TMz mode of the semi-infinitely layered and multilayered array of PEC patches. The medium parameters of each layer are εr = 4 − j0.2 and μr = 2 − j0.1. The lattice vectors are set to be inline image mm and inline image mm. The dimension of the patch and layer thickness are the same as in Figure 5. The incident angles are θi = 0 and ϕi = 0.

[43] The last example is a dielectric slab embedded with a four-layer array of PEC spheres impinged by a normally incident plane wave, which is shown in Figure 10. The reflection and transmission coefficients of the periodic array can be calculated using the developed approach. When the periodic unit size is much smaller than the wavelength, the effective permittivity and permeability of this slab with PEC sphere inclusions can be extracted in terms of the calculated scattering parameters (S-para.) [Smith et al., 2002; Hu et al., 2010]. Figure 11 shows the extracted effective medium parameters as a function of volume fraction p of inclusions. Figure 12 shows the effective medium parameters as a function of incident angles. The effective medium parameters are extracted from S-parameters obtained using the developed approach. They are compared with those from Maxwll-Garnell (MG) formula [Merrill et al., 1999]. It is worth noting that the extracted permittivity and permeability only slightly change with the incident angles because of the geometrically isotropic property of spheres.

Figure 10.

Four-layer array of PEC spheres embedded in a slab. inline image µm, inline image µm, and t = 6 µm. εri = 2 and μri = 1 (i = 1, ⋯, 4).

Figure 11.

Effective medium parameters of a slab as a function of volume fractions. The slab is embedded with a four-layer array of PEC spheres shown in Figure 10. θi = 0° and ϕi = 0°. The operating frequency of the incident wave is f = 90 GHz.

Figure 12.

Effective medium parameters of a slab as a function of incident angles. The slab is embedded with a four-layer array of PEC spheres shown in Figure 10. The volume fraction is p = 15%. α = 90°, and ϕi = 0°. The operating frequency is f = 100 GHz.

4 Conclusions and Discussions

[44] This paper presents an IE approach in the spatial domain to analyze a multilayered doubly periodic array of 3-D general objects. This approach applies the EPACS, and thus can avoid using of multilayered DPGF. In addition, based on EPACS, an effective approach is proposed to handle semi-infinitely layered periodic arrays of 3-D general objects. The developed approaches are validated by comparing the numerical results with reference results.

[45] The direct solver without fast algorithms is applied in this work, so the computational complexity of this method is the same as the direct solver. However, for many applications of periodic structures, the size of the unit cell is not too large, so the developed method is still applicable to the simulation of many periodic structures. Furthermore, how to apply the iterative solver with fast algorithms is an open problem and may be considered in our future work.

[46] Compared to the direct IE approach in free space, the present approach will take more computational resources. However, the present approach can handle a multilayered array. Compared to the direct IE approach in a multilayered medium, the present approach avoids applying multilayered periodic Green's functions. Furthermore, when an array has large numbers of identical layers, the present approach can apply the logarithm algorithm to reduce the computation. This advantage cannot be found in the FE-BI approach with iterative solvers. In addition, the present approach can handle semi-infinitely layered periodic arrays of 3-D objects. To the best of our knowledge, there are no published reports on this function.

Appendix A: Matrix Elements

[47] When the PMCHWT formulation is applied, the coefficient matrix elements are given by

display math(A-1)
display math(A-2)

[48] When the IBC formulation is applied, the coefficient matrix elements are given by

display math(A-3)
display math(A-4)

Acknowledgments

[49] This work was supported in part by Intel Corporation and by the National Science Foundation CAREER Grant ECS-0547161.

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