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International Journal of Antennas and Propagation

Volume 2012 (2012), Article ID 170963, 16 pages

http://dx.doi.org/10.1155/2012/170963

## A Coupled-Field Expansion Method for Single-Layer and Multilayer Planar Periodic Structures

^{1}Center for Electromagnetic and Antenna Engineering, Department of Electronic Engineering, Macquarie University, Sydney, NSW 2109, Australia^{2}China Aerospace Science and Technology Corporation, Number 5 Academy, Number 513 Institute, Yantai 264000, China

Received 10 July 2011; Accepted 28 December 2011

Academic Editor: Wei Hong

Copyright © 2012 Yading Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new, rigorous, field-based, seminumerical analysis method is presented to obtain the reflection and transmission coefficients of 2D planar periodic structures with arbitrarily shaped metallization patterns for both normal and oblique incidence conditions. It is useful for the analysis, design, and optimization of many single-layer and multilayer planar structures, such as frequency-selective surfaces (FSSs), artificial magnetic conductor (AMC) surfaces, electromagnetic bandgap (EBG) structures, some metamaterials and high-impedance surfaces. In this coupled-field expansion method (CFEM), the *x*- and *y*-components of the vector magnetic potential in each homogeneous region in a unit cell are expanded in terms of Bloch-Floquet modes and the solution to the coupled-field problem is formulated. The unique, analytical formulation presented here leads to a linear system with reasonably simple matrix elements. By cascading the matrices representing each interface, multilayer periodic structures are analyzed in a very
flexible way. Being field based, CFEM does not require substrate Green's functions to analyze surfaces printed on dielectric substrates. The method was validated by analyzing one single-layer periodic surface (a printed AMC on a dielectric substrate) and one multilayer periodic surface (a circular polarizer) and comparing CFEM results with HFSS results.

#### 1. Introduction

Periodic two-dimensional arrays of thin conducting patterns have been widely investigated as frequency-selective surfaces (FSS), artificial magnetic conductors (AMC), electromagnetic bandgap (EBG) structures, planar metamaterials, circular polarizers, and high-impedance surfaces. Fully numerical methods, such as the finite-element method (FEM) [1] and the finite-difference time-domain (FDTD) [2] method compute the field distribution in a unit cell completely numerically. The direct output of these methods, that is, numerical values of the fields at sampled points, does not indicate interesting information, such as which modes are propagating and which modes are evanescent, unless further processing is carried out.

The Bloch-Floquet theorem, which describes the periodic electromagnetic wave distribution in periodic structures, is an important starting point for many semianalytical methods. Periodic Green’s functions based on TE and TM polarized Bloch-Floquet expansions have been proposed and have been applied with the Method of Moments (MoM) to obtain the full-wave solution for 2D periodic arrays [3–8]. The electric or magnetic current distribution is the direct output of these methods. A single-layer FSS in free space was analyzed in [3] by combining TE and TM polarized Bloch-Floquet modes with the MoM. In [4], the MoM was extended for an FSS printed on a dielectric substrate. A set of entire-domain basis functions numerically calculated using the boundary integral resonant mode expansion (BI-RME) method was proposed for multilayer FSSs in [5]. In [6, 7], a Green’s function that takes into account the singularity at metal patch edges was exploited with entire-domain basis functions, and the singularity was treated analytically. A cascaded generalized scattering matrix was presented in [8] to efficiently solve multilayer problems. All these semianalytical methods solve an integral equation and compute the current distribution first. For substrate-based and printed structures, they require Green’s functions.

In this paper, we present a new, rigorous, field-based, seminumerical, full-wave analysis method for analyzing single-layer/multilayer planar periodic structures by analytically expanding the fields in each homogeneous region between interfaces. In the development of this method, the concepts were taken from established methods such as the MoM, Mode Matching Method, and Spectral Domain method. In this coupled-field expansion method (CFEM), the problem space is divided into two/several homogeneous regions which are coupled. The coupling occurs through aperture areas (e.g., apertures) in the planar interface between regions. The and components of the vector magnetic potential in each region is expanded using Bloch-Floquet modes. The unknown coefficients in the expansion are determined by analytically enforcing interface conditions over the entire area they are applicable. By characterizing each interface using a matrix and cascading the matrices, multilayer periodic structures are analyzed in a very flexible way. As we solve for fields instead of currents, substrate Green’s functions are not required to analyze multilayer periodic structures on dielectric substrates.

The theory for single-layer conducting patterns is described in Section 2.2. Then, CFEM is extended for multilayer case in Section 2.3. To verify the accuracy of this method, we present numerical results for a printed AMC on a dielectric substrate and a multilayer circular polarizer, and compare CFEM results with HFSS results in Section 3. The CFEM results agree very well with HFSS simulation results for normal and oblique incidence. The surface designed by this method has been successfully employed in the small EBG-resonator antennas.

#### 2. Thoery

##### 2.1. Formulation of the Problem

Due to translational periodicity [9], we only need to analyze a unit cell of a 2D periodic structure. The cross-section of a generic multilayer structure, which is periodic in the and directions, is displayed in Figure 1. We denote each homogeneous layer by , , where is the number of layers. , are the upper and lower half spaces, respectively. The interface between the and regions, denoted by , is located at . All 2D periodic thin conducting patterns are located at these interfaces (although some interfaces can be simple dielectric interfaces without any conducting patterns). Each homogeneous layer has a thickness of . The theory presented in this paper is applicable to two types of periodic structures:Type A: has a conducting pattern and/or a dielectric interface at ;Type B: has a PEC ground plane at .

An FSS with multiple printed metal patterns is an example of a Type A structure. A multilayer AMC surface is a Type B structure.

The following notations are adopted throughout this paper.(a): the incident electric field in Region .(b): the reflected electric field in Region if the top conducting pattern at were a complete PEC without any apertures. Note that, at , and .(c): the component of the equivalent magnetic currents at the top interface of region .(d): the component of the equivalent magnetic currents at the bottom interface of region .(e): the electric field created by and in Region .(f): the electric field in Region due to all equivalent magnetic currents.(g): the total electric field in Region .(h): is the component of the total magnetic vector potential in Region . and are the components of the magnetic vector potential created by the top and bottom equivalent magnetic currents, respectively, in Region .(i), : the permittivity and permeability of the dielectric in region .

The total field in region is given by where the scattered field is totally due to the magnetic currents at the bottom interface, since there is no top interface for region . In other regions, In Type B structures, .

In general, the conducting patterns on each interface have different periodicity from each other. In such cases, a larger unit cell with an effective periodicity is defined. The unit cell of a multilayer structure is considered as a group of coupled regions. Each region is bounded by Bloch-Floquet periodic boundaries on four sides. Let us take Region , , for example, as shown in Figure 2. The range of is . In Figure 2, without any loss of generality, we have assumed that and to simplify the explanation. The apertures in the top and bottom interfaces ( and ) of Region are replaced by equivalent magnetic currents on PECs. The top equivalent magnetic currents now appear in the aperture areas of Surface , just below the PEC at , while the bottom equivalent magnetic currents (, ) exist in the apertures of Surface , just above the bottom PEC at .

##### 2.2. Expansion of the Coupled Fields in a Unit Cell of a Single-Layer Periodic Structure

In order to develop CFEM step by step, this subsection illustrates how to expand fields in each homogeneous region of a unit cell in a single-layer structure. For this purpose, we first transform the unit cell to two waveguides. The unit cell, shown in Figure 4(a), is separated into two regions, Region and Region , as shown in Figure 4(b). The metal pattern at is replaced with a PEC and equivalent magnetic currents in the aperture areas of both regions. Each region has Bloch-Floquet periodic boundaries on four sides.

As shown in Figure 4(b), the range of Region is . Its top is open and its bottom has equivalent magnetic currents, and , just above a PEC. In Type A structures, the range of Region is with an open boundary at the bottom. In Type B structures, the range of Region is with a PEC at the bottom as shown in Figure 4(b). In both cases, the top boundary of Region 1 is a PEC at . There exist equivalent magnetic currents, , , just below this PEC.

Next we apply image theory to Region and remove the PEC surface at , by doubling the magnetic currents. This transforms Region to the “waveguide” shown in Figure 3(a), which has Bloch-Floquet periodic boundaries on four sides, opened at two ends and excited by at . When image theory is applied to Region of a Type A structure, it is transformed to a similar waveguide, but excited by at . Region of a Type B structure is transformed to the waveguide shown in Figure 3(c), which is excited by at and their images at , . In our theory, we exploit the fact that the excitation in Figure 3(c) can be realized by superposing the excitation in Figure 3(c) at an infinite series of coordinates.

To find the electromagnetic fields produced by magnetic currents in Region and Region [10], we solve the following wave equations for the magnetic vector potential () in the two corresponding waveguides: where and represent the coordinate of each layer of equivalent magnetic currents (i.e., excitations in Figures 3(a) and 3(c)). is infinite at , otherwise it is zero.

The general solutions of (3) and (4), the inhomogeneous, second-order partial differential equations, are: Here, we select as eigenmodes of the waveguide bounded by Bloch-Floquet periodic boundary conditions. The sum of these modal functions, weighted with coefficients , gives the full-wave solution including all propagating and evanescent waves. The subscript represents the sequence of modal functions given below. These modal functions are the same in both regions: where , are the and components of the incident wave vector ; is the lattice constant along the direction; is the lattice constant along the direction; and are integers from to . In the special case of normal incidence, .

By substituting in the wave equation, one can prove that the in (7) is given by where For a Type A structure, is given by For a Type B structure, is given by To shorten the mathematical expressions, let us define two new functions, and , to represent the 3D field modes in (7) and their derivatives: where At , The electric and magnetic fields produced by the magnetic currents in each region are obtained by substituting (7) and (8) into (5) and (6):

##### 2.3. Coupled-Field Expansion for a Unit Cell of a Multilayer Periodic Structure

Region , shown in Figure 2, is the generalized case of Region and Region in Section 2.2. This subsection extends the field expansion in Section 2.2 to the multilayer case. Let us first apply image theory to Region . Its top equivalent magnetic currents , are doubled when the top PEC at is removed. The same applies to its bottom equivalent magnetic currents , when the bottom PEC at is removed. In addition, multiple images of these currents appear in an infinite number of surfaces. This transforms Region to a waveguide with Bloch-Floquet periodic boundaries at the four sides, excited by and at and a series of their images at , as well as , at and a series of their images at , where . This excitation scheme for a typical Region is shown in Figure 3(d).

As the top and bottom regions are different from a typical region, shown in Figure 2, we need to consider four types of “waveguides”, as follows.(1)For , Region leads to the “waveguide” in Figure 3(a) because the top of this region is open and hence there are no magnetic currents on the top. This has only one layer of magnetic currents.(2)In Type A structures, for , Region leads to the “waveguide” in Figure 3(b) because the bottom of this region is open. In Type B structures, there is no field in this region as Surface is a PEC.(3)In Type B structures, for , Region leads to the “waveguide” in Figure 3(c) because Surface is a complete PEC without apertures and hence there are no equivalent magnetic currents on it.(4)All other typical regions have metallic patterns at both top and bottom, which leads to the generic “waveguide” in Figure 3(d).

Note that the last type of “waveguide” is required only for multilayer structures. We exploit the fact that the excitations in Figure 3(d) can be realized by shifting and superposing the excitations in Figures 3(a) and 3(b) at an infinite periodic series of coordinates.

To find the electromagnetic fields in Region produced by magnetic currents on Surfaces and , we solve the following wave equations for the magnetic vector potential () in the corresponding waveguides: where and , . is infinite at , otherwise, it is zero. is infinite at , otherwise it is zero. The series of and represents the coordinate of each layer of equivalent magnetic currents. (i.e., excitations in Figures 3(a), 3(b), 3(c), and 3(d)).

The magnetic potentials produced by the equivalent magnetic currents in the typical waveguide in Figure 3(d) can be expressed as a linear superposition of the potentials created by the equivalent magnetic currents at the two interfaces ( and ), and their corresponding series of images. So, let us separate them as Now, we expand these potentials as where and are defined in (9).

In (23)–(26), and are given by where To shorten the mathematical expressions, let us define four new functions, , , , and to represent the 3D field modes in (23) and (26), and their derivatives: where It can be shown that, at , Similarly, at ,

For further compactness, let us define three operators: where is an arbitrary function. , , and represent the integration over the aperture areas, metal patch areas, and the whole unit cell area at the interface , respectively. The operator satisfies the following orthogonality: Because the integration area of a unit cell is equal to the sum of its aperture area and patch area, In the following subsections, various interface continuities are enforced, in order to find the unknown coefficients and in (23)–(26).

Since Region is the generalized case of Region and in Section 2.2, we will enforce the boundary conditions on Surface .

##### 2.4. Enforcing the Tangential Electric Field to Be Continuous across Apertures in Surface

Let us consider Region , and apply operator (35) to both sides of (48) at . The result is Due to the orthogonality in (36), only the , term is nonzero on the right-hand side. Then, considering that and as given in (32), we have Therefore, Similarly, we can derive the following expressions by applying the operator (35) to (48) for Region and applying (35) to (53) for Region and Region : Let us consider the interface in a typical region (). The component of the total electric field on this plane is given by . Then, as directed magnetic currents do not produce an directed electric field on the same plane (see (26)), , and, therefore, at this plane at . Similarly, we have for Region at the same plane at . Since the tangential components of the total electric field are zero on the metal areas of the interface, and they are continuous in the aperture areas of the interface, we find that In region , the component of the total electric field is the sum of the components of the incident field (), the reflected field (), and the scattered field. However, at the interface, and cancel out due to the PEC, and hence the above relationships are valid also for region at interface .

Let us substitute (44) into (40)-(41) and make a simple rearrangement which leads to the following relationship between the coefficients for the two coupled regions and : where is a new unknown coefficient, which will replace both and eventually.

Similarly, considering in (45) and (42)-(43), we have Note that the above two equations establish a simple relationship between the coefficients in Region and those in Region at . We have exploited this relationship to introduce a new set of unknown coefficients and , which are common to both regions. Note that the coefficients and are associated with the th interface whereas the former coefficients are associated with both the regions and interfaces.

Let us substitute (46) and (47) in (23)–(26). The electric and magnetic fields produced by the magnetic currents in each region are obtained by substituting these new expressions for the magnetic vector potentials in (20) and (21):

##### 2.5. Enforcing the Tangential Electric Field to Be Zero on the Metal Patch Areas in Surface

As the tangential electric field is zero on the metal patch areas of the interface, and . Hence, we can replace in (40) by as we only consider at . In other words, the integration area in (40) can be reduced to the aperture area: Similarly, one may prove that Let us replace in (56) using and expand in (56) using (48). We obtain Note that (57) leads to the same equation (60), when the above steps are followed.

Similarly, substitution of (47) and (53) in (58)-(59) leads to where Note that the same integral (62) appears in both (60)-(61).

##### 2.6. Enforcing the Tangential Magnetic Field to Be Continuous across the Aperture Areas in Surface

The total in region is where and are the components of the magnetic field of incident and reflected waves in region . At the interface , due to the PEC, and hence The total in a typical region, , is Now let us apply operator to (64): Next, we consider the interface and Region below it. So, we set in (65) and apply the same operator : Since the magnetic field is continuous across the aperture at interface , . So, we subtract (67) from (66). After simply rearranging the sequence of terms, we obtain Note that due to (37). Similarly the continuity of the tangential component of the magnetic field across the apertures at the other typical interfaces (, ) leads to

Let us repeat this process with . At interface , the relationship is For any other typical interface , ,

Next, we express all terms inside the operator in the right-hand side of (68)–(71) in terms of unknowns . Let us start from in (69). Replacing in (55) with and then applying the operator, we obtain

Due to the orthogonality in (36), only the , term is nonzero on the right-hand side. Hence, where

Next and are expanded using (61). Similarly, we apply operator to in Region . The result is where when , , otherwise .

We similarly expand the other terms in (68)–(71) and express the right-hand side of all these equations in terms of .

##### 2.7. Building a Linear System of Equations

The linear system will be based on the expanded versions of (68)–(71).

###### 2.7.1. Intermediate Interfaces

The matrix relationship developed in this subsection applies to all intermediate, typical surfaces . For Type A structures, they range from and, for Type B structures, .

Let us define three column vectors to represent unknowns in (69) and (71): where and , are from .

By substituting expansions (73)-(75), and so forth in (69) and (71), we obtain the linear subsystem for each intermediate layer: where The above expression describes the relationship between the unknowns for three adjacent surfaces. In this expression, , , , , , , , , , , , and are twelve submatrices. Their elements are where , and , , are given by

###### 2.7.2. Last Interface ()

When the derivation in the above subsection is repeated for the last interface of a Type A structure, that is, , the result is This involves only two unknown vectors because there are no magnetic currents below this surface.

By manipulating the above equation, can be expressed in terms of : Let us now define then (83) becomes Note that now we have a relationship connecting unknowns for two adjacent interfaces.

In a Type structure, the last surface with magnetic currents is . The above derivation is valid for this case too. In this case, ; the relationship between and is similar to (85).

###### 2.7.3. Second-Last Interface

As an example, let us now consider the next interface, . The objective here is to derive a relationship between and , by exploiting (78) and (85). Let us first substitute in (78) and rewrite it as Now, we substitute (85) to replace . The result is After rearrangement, we get the following relationship between