This paper presents an efficient full-wave analysis of an open-ended waveguide radiating through a two-layer superstrate. The spectral dyadic Green's function for a multilayer medium is formulated through cascaded matrices, each of which corresponds to one layer of the superstrate independently. The reflection coefficient matrix is derived by enforcing the field matching condition at the opening of the waveguide. The generalized pencil of function (GPOF) method and Gaussian quadrature are adopted to accelerate the computation. Once the excitation and the Green's function are known, the far-field radiation pattern and directivity are obtained directly from the spectra of the electric and magnetic fields in the half free space. An open-ended WR-90 waveguide radiating through a two-layer superstrate is considered as an example to verify the proposed efficient full-wave analysis. The effect of the superstrate on the directivity of open-ended waveguides is also examined and discussed.
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 In this paper, an efficient full-wave analysis of the radiation from an open-ended waveguide through a two-layer superstrate is presented. Contributions are made in the following aspects. Firstly, the spectral dyadic Green's function in a multilayer superstrate is formulated so that it can be derived through cascaded matrices, each of which represents one layer of the superstrate independently. Therefore, it is very convenient to be modified for arbitrary superstrate configurations. Secondly, the generalized pencil of function (GPOF) [Hua and Sarkar, 1989] method and Gaussian quadrature are employed to accelerate the computation of reflection coefficient matrix. The convergence behavior of this efficient method is also investigated. Thirdly, the influence of the superstrate dimensions on the high antenna directivity are examined to show the properties of radiation through the two-layer medium.
 The rest of the paper is organized as follows. Formulations of the efficient full-wave analysis method are given in section 2. In section 3, numerical results of an open-ended WR-90 waveguide radiating through a two-layer superstrate are provided. The efficiency and convergence behavior of our method as well as the high directive radiation from open-ended waveguides are also examined. Conclusions are given in section 4.
2.1. Spectral Dyadic Green's Function in a Multilayer Superstrate
 The model of an open-ended waveguide radiating through a multilayer superstrate is illustrated in Figure 1. The superstrate consists of N dielectric layers which are infinitely large in the horizontal direction. The permittivity, permeability and thickness of the ith layer are εi, μi and hi, respectively. The interface between the ith layer and (i + 1)th layer is at z = di. Above the superstrate is the half free space, which is indexed with N + 1 for convenience. A waveguide opening on an infinitely large ground plane is mounted at the bottom of the superstrate.
 In the N-layered superstrate, the spectral transverse electric and magnetic fields in the ith layer can be given by
xi and yi are the spectral functions of the x and y components of the incident wave (propagating in the +z direction), respectively. xi and yi are the corresponding functions for the reflected wave (propagating in the −z direction).
 Considering the continuity of transverse electric and magnetic fields at z = di (i = 1, 2, …, N) and N+1 = 0, we find the following relationship between the first layer and the half free space,
 It is seen that the effect of the ith layer is represented by matrix Ti in (2). The benefit brought by this concise expression is that if a layer is changed or added or removed, only the corresponding matrix Ti is required to be modified or inserted or deleted from (2) with other parts unchanged. This feature of independence is very useful since no sophisticated modification of the formulation is needed when the superstrate configuration varies from one to another. Letting
we can find the expression of 1∣z=0 in terms of 1∣z=0
where G is the spectral dyadic Green's function given by
 A simple model shown in Figure 2, where the superstrate consists of two layers (N = 2), is considered here to show more details of our formulation. The following assumptions are made for this model for simplicity: (1) μ1 = μ2 = μ3 = μ and μ is the permeability of free space; (2) ε = ε = ε = 1, ε = εr ≫ 1; (3) the excitation is provided by an open-ended rectangular waveguide.
2.2.1. Spectral Green's Function
 With the assumptions made above, we have k3 = k1, kz3 = kz1, B3 = B1. The matrices C and D in (6) can be explicitly shown as
We define the 2 × 2 matrix G as follows,
Substituting (7) into (5) and comparing the result with (8), all elements of G can be explicitly expressed as
and J1 = L1−1C1, J2 = L2−1C2, K1 = L1−1S1, K2 = L2−1S2, F1 = K1/k, F2 = K2/k. It is not difficult to note that g′22 can be derived from g′11 by exchanging kx and ky and g′12 = g′21.
2.2.2. Spectrum of the Excitation
 The spatial expressions of the transverse electric fields of a particular mode in a rectangular waveguide of cross-section a × b (shown in Figure 2) are given as follows,
where Nmn is the normalization coefficient. Sx and Sy are the mode functions of the transverse electric fields in and directions, respectively,
It is seen that all TE or TM modes can be numbered with an integer pair (m, n). In order to simplify the notation, we reindex each mode with only one integer l (l = 1, 2, …, L). Therefore, all considered modes in the waveguide are rearranged and the electric and magnetic fields in the waveguide are given by
where Al+ and Al− are the magnitude of mode l propagating in the + and − directions, respectively. Yl is the admittance of the lth mode. At the opening of the waveguide (z = 0), (13) can be expressed by the following matrix equations
 Taking the double Fourier transform of both sides of (14), we have
where = (1, 2, …, L) is the double Fourier transform of and
2.2.3. Reflection Coefficient Matrix Γ
 Considering the continuity of the transverse fields along the waveguide opening, substituting (16) into (4), and taking the double inverse Fourier transform of both sides of the obtained equation, we have
 Left-multiplying both sides of (19) by T and integrating on the waveguide opening S and considering (17) over the aperture, the following equation is derived,
Since ∫∫STdS results in a unit matrix, it can be removed from the left-hand side of (20). Then we exchange the order of the integrations on the right-hand side of (20) and arrive at the matrix equation
 Therefore, the reflection coefficient matrix Γ is of the form
 Since Y has been well defined in (15), the next step is to calculate the matrix U. According to (22), the element in the lth row and l'th column of U is
and the expressions of qs(s = 1,2,3) are listed in Table 1. The parity of the integrand Z(kx, ky) is determined by the combination of m, m′, n, n′, which is illustrated in Table 2. The result of integration in (27) is 0 if either m + m′ or n + n′ is odd. This indicates that the lth mode and l'th mode cannot be coupled when either m + m′ or n + n′ is odd. Therefore, if a mode cannot be coupled from the incident mode, this mode can be removed from matrix U, which decreases the dimension of U without affecting the accuracy. Moreover, when both m + m′ and n + n′ are even, Z is even with respect to both kx and ky and the integration domain can be reduced to only one quadrant, as shown in (29). Furthermore, it can be concluded from (28) that the integral domain can always be reduced to one quadrant as long as the spectra of both the lth and l'th mode in the waveguide are symmetric or antisymmetric with respect to kx and ky.
Table 1. Expressions of q1, q2, and q3 for Different Cases of Coupling
TE Coupled to TE
TE Coupled to TM
TM Coupled to TE
TM Coupled to TM
a2b2(n2m′2 + n′2m2)
abmn(b2m′2 − a2n′2)
abm′n′(b2m2 − a2n2)
Table 2. Parity of Z With Respect to kx and ky
m + m′ Is Odd
m + m′ Is Even
The Parity of Z With Respect to kx
n + n′ is odd
n + n′ is even
The parity of Z with respect to ky
 After some observations on the spectral dyadic Green's function, it is found that G has poles on the (kx, ky) plane, which corresponds to surface waves in the superstrate. The positions of the poles depend on the value of kx2 + ky2, therefore it is more convenient to extract the poles from the integrand in the system of cylindrical coordinates. Using the equations kx = kρ cos ψ, ky = kρ sin ψ, (29) is transformed from Cartesian coordinates to cylindrical coordinates.
where Q(kρ, ψ) = kρZ(kρ cos ψ, kρ sin ψ). Now the positions of the poles are only related to kρ and integration of the poles can be obtained by the theorem of residue. Usually the number and position of poles on the integration path of kρ can be obtained by solving a transcendental equation numerically. If we assume that there are J poles (kp1, kp2, …, kpJ) for a given ψ(ψ = ψt), the integration of kρ in (30) can be decomposed as
where Q*(kρ, ψt) = Q(kρ, ψt) − Σs=1J is a function of kρ without poles, Rs(ψt) is residues of Q(kρ, ψ) at (kps, ψt). Therefore, when both m + m′ and n + n′ are even, the expression of ull′ in (30) becomes
 When the integration in (32) is numerically solved, the reflection coefficient matrix Γ is then obtained from (24).
2.2.4. Accelerating Technique
 It is noticed that computing the integration in (32) directly by summing up a lot of sample points may be time-consuming since the integrand Q*(kρ, ψ) is slowly converging and oscillating. Therefore, an accelerating technique is proposed here to improve the computational efficiency. Firstly, it is noticed that the wavenumber k1 corresponds to the half free space since k1 = k3. As has been pointed out by Chew , kρ = k1 is a branch point and the integrand Q*(kρ, ψt) with respect to kρ is not smooth there. The integration interval of kρ is segmented into (0, k1) and (k1, +∞) so that the curve is smooth within each range and less sample points are required later. Secondly, the integrals in the two intervals are calculated separately. Gaussian quadrature are adopted to evaluate the integral in (0, k1).
where K1 is the number of abscissas with respect to kρ from 0 to k1, ws and kρs are the weights and abscissas, respectively. The generalized pencil of function (GPOF) method [Hua and Sarkar, 1989] is employed to fit the integrand Q*(kρ, ψt) over (k1, +∞) with the summation of a series of exponential functions through some sample points. Therefore, the integration of kρ in (k1, +∞) is converted to a simple summation as shown in (34).
where Me is the number of terms in the exponential series, cs and ds are the resultant coefficients obtained with GPOF, which are dependent on ψt implicitly. Since Q*(kρ, ψ) diminishes as kρ approaches to +∞, the real part of ds is always negative for s = 1, 2,…, Me.
 It is also found that the integration in (32) with respect to ψ can also be calculated by Gaussian quadrature. Therefore, the double integral of ull′ has now been converted to a summation as
where K2 is the number of abscissas with respect to ψ and ψt represents the abscissas in interval (0, π/2) and
It will be shown later that the computation efficiency has been greatly improved after using this accelerating technique.
2.2.5. Radiation Pattern
 According to (14) and (24), when A+ and Γ are known, the spectrum of the transverse electric field at the opening w∣z=0 can be obtained. Substituting (2) into (1) when N = 2, the spectra of the electric and magnetic fields in the half free space (layer 3) are derived.
Therefore, the far-field radiation pattern in terms of Eθ and Eϕ are given as follows.
r is the distance from the point of observation to the origin and η = 120πΩ. When the radiation pattern is known, the directivity of the aperture antenna can be readily calculated.
3. Results and Discussions
 The radiation of an open-ended rectangular waveguide through a two-layer superstrate illustrated in Figure 2 is analyzed using the proposed method. The model of the rectangular waveguide is WR-90, which is opened on an infinitely large ground plane. The thicknesses of the first and second layers of the superstrate are 14.993mm and 2.3473mm, respectively. The relative permittivity of the second layer is εr = 10.2.
Figure 3 presents the reflection coefficient results in the rectangular waveguide obtained using our method and compared with those by Ansoft High-Frequency Structure Simulator (HFSS). The number of modes L considered is 11. The number of abscissas (K1) with respect to kρ is 9. The number of abscissas (K2) with respect to ψ is 9. The numbers of sample points (Ms) and exponential terms (Me) are 300 and 7, respectively. It is seen in the figure that the results agree very well with each other, which therefore verifies the accuracy of our method in calculating the reflection coefficient of an open-ended waveguide radiating through the two-layer superstrate.
Figure 4 presents the comparison of radiation pattern results at 10GHz obtained by (38) and those by Ansoft Designer. It is seen that the results obtained by both methods are in excellent agreement, which proves the accuracy of our method in calculating the radiation pattern. Since the radiation pattern is known, the directivity is found to be 15.9dB, which is about 10dB higher than that of a simple open-ended WR-90 waveguide. The gain-enhancement result obtained using our full-wave analysis is in good agreement with that experimentally studied by Tan et al. , which therefore demonstrates the validity of the proposed full-wave method in calculating the radiation through the two-layer superstrate.
 In order to show the efficiency of our method, the CPU time of our full-wave analysis is presented and the normalized computation time results are given in Table 3. It is seen that though GPOF contains operations such as singular value decomposition and pseudo-inversion of matrices, the computation time of our method is much less than that of direct integration. More time can be saved using our method when higher accuracy is expected.
Table 3. Comparison of Normalized Computation Time
3.2. Convergence Study
 The convergence of reflection coefficient results with respect to the parameters adopted in the proposed numerical method is examined in this section. Here we define Δ to be the variation of the reflection coefficient when the parameters change. The model considered is the same as that described in section 3.1 and the operating frequency is 10GHz.
Table 4 presents the reflection coefficient results with respect to L, which is the number of modes considered in the waveguide. It is seen that a very accurate result can be obtained with a relative small mode number in this case. Table 5 examines the convergence of our method with respect to K1, the number of abscissas in calculating the integral of kρ from 0 to k1. It is seen that the change of the reflection coefficient result is very small when the number of abscissas is greater than 9. Table 6 shows the convergence behavior of our method with respect to K2, the number of abscissas in calculating the integral of ψ from 0 to π/2. It is found that the convergence performance is quite good when K2 ≥ 9. The convergence behavior of our method with respect to Ms, the number of sample points employed by GPOF to fit the integration tail, is shown in Table 7. As more sample points of the integrand are used, the variation of the reflection coefficient becomes smaller, which leads to a more accurate result. Table 8 shows the convergence behavior of our method with respect to Me, which is the number of exponential terms generated by GPOF. It is found that 7 exponential terms can provide quite accurate result.
Table 4. Reflection Coefficient With Respect to the Mode Number L (K1 = 9, K2 = 9, Ms = 300, Me = 7)
Table 5. Reflection Coefficient With Respect to the Number of Abscissas K1 (L = 19, K2 = 9, Ms = 300, Me = 7)
Table 6. Reflection Coefficient With Respect to the Number of Abscissas K2 (L = 19, K1 = 9, Ms = 300, Me = 7)
Table 7. Reflection Coefficient With Respect to the Number of Sample Points Ms (L = 19, K1 = 9, K2 = 9, Me = 7)
Table 8. Reflection Coefficient With Respect to the Number of Exponential Terms Me (L = 19, K1 = 9, K2 = 9, Ms = 300)
3.3. High Directivity
 In section 3.1, it has been shown that the directivity of the open-ended WR-90 waveguide radiating through a specific two-layer superstrate is 15.9dB, which is about 10dB higher than that of the same open-ended waveguide without superstrate. In this part, the relationship between the high directivity and the superstrate dimensions in terms of h1 and h2 is examined. The waveguide model is WR-90; the operating frequency is 10GHz and the relative permittivity of layer 2 is εr = 10.2.
Figure 5 provides the directivity contour of the open-ended waveguide radiating through a two-layer superstrate. High directivity, which is represented by the bright areas, appears periodically with respect to h1 and h2. The periods of h1 and h2 are λ1/2 and λ2/2, respectively, where λi(i = 1, 2) represents the wavelength in the ith layer. This periodicity of high directivity is in agreement with the experimental results of Tan et al. . It is noticed that the nearest bright area to the bottom-left corner of the figure is located around the point where h1 = λ1/2 and h2 = λ2/4, which represents the thinnest two-layer superstrate that can achieve a high directivity.
 An efficient full-wave analysis of an open-ended waveguide radiating through a two-layer superstrate has been presented in this paper. The field equations have been set up in the form of cascaded matrices, through which the spectral dyadic Green's function for the multilayer superstrate has been derived. The generalized pencil of function method and Gaussian quadrature have been employed to improve the computation efficiency for the reflection coefficient matrix. After that, the radiation pattern and the antenna directivity have been calculated directly using the spectra of transverse electric and magnetic fields in the half free space. An open-ended WR-90 waveguide with a two-layer superstrate has been studied to verify the efficiency of our method. Numerical results have been presented to show that the proposed method is very accurate and efficient. The convergence behaviors of our method and the influence of the superstrate dimensions on the directivity have also been examined.