Journal of Infrared, Millimeter, and Terahertz Waves

, Volume 32, Issue 11, pp 1350–1366

Dual-Frequency Behavior of Stacked High Tc Superconducting Microstrip Patches

Authors

  • Siham Benkouda
    • Electronics DepartmentUniversity of Batna
  • Mounir Amir
    • Electronics DepartmentUniversity of Batna
    • Electronics DepartmentUniversity of Batna
  • Abdelmadjid Benghalia
    • Electronics DepartmentUniversity of Constantine
Article

DOI: 10.1007/s10762-011-9842-1

Cite this article as:
Benkouda, S., Amir, M., Fortaki, T. et al. J Infrared Milli Terahz Waves (2011) 32: 1350. doi:10.1007/s10762-011-9842-1
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Abstract

The dual-frequency behavior of stacked high Tc superconducting rectangular microstrip patches fabricated on a two-layered substrate is investigated using a full-wave spectral analysis in conjunction with the complex resistive boundary condition. Using a matrix representation of each layer, the dyadic Green’s functions of the problem are efficiently determined in the vector Fourier transform domain. The stationary phase method is used for computing the radiation electric field of the antenna. The proposed approach is validated by comparison of the computed results with previously published data. Variations of the lower and upper resonant frequencies, bandwidth and quality factor with the operating temperature are given. Results showing the effects of the bottom patch thickness as well as the top patch thickness on the dual-frequency behavior of the stacked configuration are also presented and discussed. Finally, for a better comprehension of the dual-frequency operation, a comparison between the characteristics of the lower and upper resonances is given.

Keywords

Superconducting microstrip patchesStacked patchesDual-frequency operation

1 Introduction

The need for high data transmission rate coupled with ever increasing demand for mobile devices has generated a great interest in low cost, compact microwave and millimeter-wave antennas exhibiting high gain and wide bandwidth [1]. Microstrip patch antennas have demonstrated to be one of the most versatile antennas in recent years [2]. Nowadays, patches are probably the most used antennas in compact commercial designs [2]. The main advantages of these antennas are low profile, light weight, easy fabrication, integrability with microwave and millimeter-wave integrated circuits and conformability to curved surfaces [36]. However, as typical disadvantages are their capability to resonate at a single frequency, narrow bandwidth and low gain [7].

The extremely low surface resistance in high Tc superconducting thin films facilitates the development of microwave and millimeter-wave devices with better performance than conventional devices. High Tc superconducting microstrip patch antennas have higher gain than their normal counterparts [812], but they suffer from the extremely narrow bandwidth, which severely limits their application [9, 10].

Stacked patches are one of the most common solutions adopted to increase the bandwidth of patch antennas [1315], since the structure is as compact as the original (given that the thickness of the substrate is usually thin, and this is the only dimension that is increased) and the radiation pattern can keep its characteristics over all of the working frequency band [2]. Moreover, one can vary the basic configuration of this type of antenna by adding spacers, superstrates or multiple layers providing even more versatility to it [2]. Furthermore, stacked patches are useful in situations where the antenna is required to operate efficiently at two distinct frequencies (dual-frequency operation) [1619].

In the current paper, we investigate the dual-frequency behavior of stacked high High Tc superconducting microstrip patch antennas using a full-wave spectral domain technique in conjunction with the complex resistive boundary condition [1012]. As in [18], we consider both the case where the top rectangular patch is longer than the bottom one and the opposite case. Note that for the case of stacked high High Tc superconducting patches, other degrees of freedom contribute in the control of the dual-frequency behavior, which are the top and bottom patch thicknesses. Also in this paper, the influence of the temperature on the lower and upper resonances, bandwidth and quality factor of stacked high High Tc superconducting rectangular patches is examined.

2 Formulation

Figure 1 shows the geometry of two stacked high Tc superconducting rectangular microstrip patches fabricated with the same superconducting material. The sizes of the bottom patch of thickness e1 and the top one of thickness e2 are, respectively, a1 × b1 and a2 × b2. There is no offset between the two patches. The substrate is assumed to be made of two layers. The first (second) layer of thickness d1 (d2) is characterized by the free-space permeability μ0 and the permittivity ε0εr10εr2). In the theoretical formulation, we have considered that the two layers are not identical, but in Section 3, we consider that he substrate is assumed to be made of two layers of the same material (εr1 = εr2 = εr) and identical thickness (d1 = d2 = d). All fields and currents are time harmonic with the e+iωt time dependence suppressed. The transverse fields inside the jth layer (j = 1, 2) can be obtained via the inverse vector Fourier transform as [20]
https://static-content.springer.com/image/art%3A10.1007%2Fs10762-011-9842-1/MediaObjects/10762_2011_9842_Fig1_HTML.gif
Fig. 1

Geometry of two stacked high Tc superconducting rectangular microstrip patches.

$$ {\mathbf{E}}({{\mathbf{r}}_s}, z) = \left[ {\begin{array}{*{20}{c}} {{E_x}({{\mathbf{r}}_s}, z)} \\ {{E_y}({{\mathbf{r}}_s}, z)} \\ \end{array} } \right] = { }\frac{1}{{4{\pi^2}}}\int\limits_{{ - \infty }}^{{ + \infty }} {\int\limits_{{ - \infty }}^{{ + \infty }} {{ }{\mathbf{\bar{F}}}({{\mathbf{k}}_s}, {{\mathbf{r}}_s}){ }{.}} } \,{\mathbf{e}} ({{\mathbf{k}}_s}, z)\,d{k_x}\,d{k_y} $$
(1)
$$ {\mathbf{H}}({{\mathbf{r}}_s}, z) = \left[ {\begin{array}{*{20}{c}} {{H_y}({{\mathbf{r}}_s}, z)} \\ { - {H_x}({{\mathbf{r}}_s}, z)} \\ \end{array} } \right] = { }\frac{1}{{4{\pi^2}}}\int\limits_{{ - \infty }}^{{ + \infty }} {\int\limits_{{ - \infty }}^{{ + \infty }} {{ }{\mathbf{\bar{F}}}({{\mathbf{k}}_s}, {{\mathbf{r}}_s}){ }{.}} } \,{\mathbf{h}} ({{\mathbf{k}}_s}, z)\,d{k_x}\,d{k_y} $$
(2)
where \( {\mathbf{\bar{F}}}({{\mathbf{k}}_s}, {{\mathbf{r}}_s}) \) is the kernel of the vector Fourier transform [20], and
$$ {\mathbf{e}}(\,{{\mathbf{k}}_s},z) = \left[ {\begin{array}{*{20}{c}} {\frac{\text{i}}{{{k_s}}}\frac{{\partial {{\tilde{E}}_z}(\,{{\mathbf{k}}_s},z)}}{{\partial z}}} \\ {\frac{{\omega {\mu_0}}}{{{k_s}}}{{\tilde{H}}_z}(\,{{\mathbf{k}}_s},z)} \\ \end{array} } \right] = {{\mathbf{A}}_j}(\,{{\mathbf{k}}_s}){e^{{ - {\text{i}} \,{k_{{z j}}} z}}} + {{\mathbf{B}}_j}( {{\mathbf{k}}_s}) {e^{{{\text{i}} {k_{{z j}}} z}}} $$
(3)
$$ {\mathbf{h}}(\,{{\mathbf{k}}_s},z) = \left[ {\begin{array}{*{20}{c}} {\frac{{\omega {\varepsilon_0}{\varepsilon_{{rj}}}}}{{{k_s}}}{{\tilde{E}}_z}(\,{{\mathbf{k}}_s},z)} \\ {\frac{\text{i}}{{{k_s}}}\frac{{\partial {{\tilde{H}}_z}(\,{{\mathbf{k}}_s},z)}}{{\partial z}}} \\ \end{array} } \right] = {{\mathbf{\bar{g}}}_j}(\,{{\mathbf{k}}_s})\, \cdot \,\left[ {{{\mathbf{A}}_j}(\,{{\mathbf{k}}_s}){e^{{ - {\text{i}} \,{k_{{z j}}} z}}} - {{\mathbf{B}}_j}( {{\mathbf{k}}_s}) {e^{{{\text{i}} {k_{{z j}}} z}}}} \right] $$
(4)
In Eqs. 3 and 4, \( {\tilde{E}_z}(\,{{\mathbf{k}}_s},z) \) and \( {\tilde{H}_z}(\,{{\mathbf{k}}_s},z) \) are the scalar Fourier transforms of Ez(rs, z) and Hz(rs, z), respectively. Aj and Bj are two-component unknown vectors and
$$ {{\mathbf{\bar{g}}}_j}( {{\mathbf{k}}_s}){ } = { }\,diag\,\left[ {\frac{{\omega {\varepsilon_0}{\varepsilon_{{rj\,}}}}}{{{k_{{zj}}}}},\;\frac{{{k_{{zj}}}}}{{\omega {\mu_0}}}} \right],{k_{{zj}}} = {\left( {{\varepsilon_{{rj}}} k_0^2 - k_s^2} \right)^{{\,\tfrac{1}{2}}}} $$
(5)
with \( k_0^2 = {\omega^2} {\varepsilon_0}\, {\mu_0} \) and kzj is the propagation constant in the jth layer. Writing Eqs. 3 and 4 in the planes z = zj−1 and z = zj, and by eliminating the unknowns Aj and Bj, we obtain the matrix form
$$ \quad \quad \left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},z_j^{ - })} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},z_j^{ - })} \\ \end{array} } \right] = {{\mathbf{\bar{T}}}_j} \cdot \left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},z_{{j - 1}}^{ + })} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},z_{{j - 1}}^{ + })} \\ \end{array} } \right]\quad $$
(6)
with
$$ {{\mathbf{\bar{T}}}_j}{ } = { }\left[ {\begin{array}{*{20}{c}} {{{{\mathbf{\bar{T}}}}_j}^{{11}}} & {{{{\mathbf{\bar{T}}}}_j}^{{12}}} \\ {{{{\mathbf{\bar{T}}}}_j}^{{21}}} & {{{{\mathbf{\bar{T}}}}_j}^{{22}}} \\ \end{array} } \right]{ } = { }\left[ {\begin{array}{*{20}{c}} {{\mathbf{\bar{I}}} cos{\theta_j}} & { - {\text{i }}{{{\mathbf{\bar{g}}}}_j}^{{ - 1}}sin{\theta_j}} \\ {\quad - {\text{i }}{{{\mathbf{\bar{g}}}}_j}sin{\theta_j}} & {{\mathbf{\bar{I}}}\cos {\theta_j}} \\ \end{array} } \right] $$
(7)
which combines e and h on both sides of the jth layer as input and output quantities. In Eq. 7, θj = kzjdj and \( {\mathbf{\bar{I}}} \) stands for the 2 × 2 unit matrix. The matrix \( {{\mathbf{\bar{T}}}_j} \)is the matrix representation of the jth layer in the (TM, TE) representation. The continuity equations for the tangential field components are
$$ {\mathbf{e}}( {{\mathbf{k}}_s},z_j^{ - }) = { }{\mathbf{e}}( {{\mathbf{k}}_s},z_j^{ + }) = { }{\mathbf{e}}( {{\mathbf{k}}_s},{z_j}){,}\;j{ } = { }1,\;2 $$
(8)
$$ {\mathbf{h}}( {{\mathbf{k}}_s},z_j^{ - }) - { }{\mathbf{h}}( {{\mathbf{k}}_s},z_j^{ + }){ } = { }{\delta_{{j\,1}}}{ }{{\mathbf{j}}^1}( {{\mathbf{k}}_s}) + { }{\delta_{{j\,2}}}{ }{{\mathbf{j}}^2}( {{\mathbf{k}}_s}){, }j{ } = { }1,\;2 $$
(9)
In Eq. 9, j1(ks) (j2(ks)) is the vector Fourier transform of the current J1(rs) (J2(rs)) on the bottom (top) patch [21], and δji (i = 1, 2) is the Kronecker symbol. Using Eqs. 6, 8 and 9 yields
$$ \quad \quad \left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},d_1^{ + })} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},d_1^{ + })} \\ \end{array} } \right] = {{\mathbf{\bar{T}}}_1} \cdot \;\left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},{0^{ + }})} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},{0^{ + }})} \\ \end{array} } \right] - \quad \left[ {\begin{array}{*{20}{c}} {\mathbf{0}} \\ {{{\mathbf{j}}^1}( {{\mathbf{k}}_s})} \\ \end{array} } \right]\quad $$
(10)
$$ \quad \quad \left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},{{({d_1} + {d_2})}^{ + }} )} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},{{({d_1} + {d_2})}^{ + }} )} \\ \end{array} } \right] = {{\mathbf{\bar{T}}}_2} \cdot \;\left[ {\begin{array}{*{20}{c}} {{\mathbf{e}}( {{\mathbf{k}}_s},d_1^{ + })} \\ {{\mathbf{h}}( {{\mathbf{k}}_s},d_1^{ + })} \\ \end{array} } \right] - \quad \left[ {\begin{array}{*{20}{c}} {\mathbf{0}} \\ {{{\mathbf{j}}^2}( {{\mathbf{k}}_s})} \\ \end{array} } \right]\quad $$
(11)
In the unbounded air region above the top patch of the stacked structure \( (\,{d_1} + {d_2}\,\langle \,z\,\langle \infty { } {\text{and}} {\varepsilon_r} = 1\,) \) the electromagnetic field given by Eqs. 3 and 4 should vanish at z → + ∞ according to Sommerfeld’s condition of radiation, this yields
$$ {\mathbf{h}}( {{\mathbf{k}}_s},{({d_1} + {d_2})^{ + }}) = {{\mathbf{\bar{g}}}_0}( {{\mathbf{k}}_s}){ } \cdot { }{\mathbf{e}}( {{\mathbf{k}}_s},{({d_1} + {d_2})^{ + }}) $$
(12)
where \( {{\mathbf{\bar{g}}}_0}( {{\mathbf{k}}_s}) \) can be easily obtained form the expression of \( {{\mathbf{\bar{g}}}_j}( {{\mathbf{k}}_s}) \) given in Eq. 5 by allowing εrj = 1. The transverse electric field must necessarily be zero at the plane z = 0, so that we have
$$ {\mathbf{e}}( {{\mathbf{k}}_s},{0^{ - }}) = {\mathbf{e}}( {{\mathbf{k}}_s},{0^{ + }}) = {\mathbf{e}}( {{\mathbf{k}}_s},0) = {\mathbf{0}} $$
(13)
From Eqs. 1013, we obtain a relation among j1(ks), j2(ks), e(ks, d1) and e(ks, d1 + d2) given by
$$ {\mathbf{e}}\,(\,{{\mathbf{k}}_s},{d_1} ){ } = {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 1}}}(\,{{\mathbf{k}}_s}) + {{\mathbf{\bar{G}}}^{{12}}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 2}}}(\,{{\mathbf{k}}_s}) $$
(14)
$$ {\mathbf{e}}\,(\,{{\mathbf{k}}_s}, \,{d_1} + {d_2} ){ } = {{\mathbf{\bar{G}}}^{{21}}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 1}}}(\,{{\mathbf{k}}_s}) + {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 2}}}(\,{{\mathbf{k}}_s}) $$
(15)
The four 2 × 2 diagonal matrices \( {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}) \), \( {{\mathbf{\bar{G}}}^{{12}}}(\,{{\mathbf{k}}_s}) \), \( {{\mathbf{\bar{G}}}^{{21}}}(\,{{\mathbf{k}}_s}) \) and \( {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}) \) are the dyadic Green’s functions of the stacked configuration in the vector Fourier transform domain. It is to be noted that \( {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}) \) is related to the bottom patch and \( {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}) \) is related to the top patch. \( {{\mathbf{\bar{G}}}^{{12}}}(\,{{\mathbf{k}}_s}) \) and \( {{\mathbf{\bar{G}}}^{{21}}}(\,{{\mathbf{k}}_s}) \) characterize the interactions between the bottom patch and the top patch. These dyadic Green’s functions can be computed via the following explicit equations:
$$ {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}){ } = {\mathbf{\bar{T}}}_1^{{12}}{ }{.}\,{ }\left[ {{{{\mathbf{\bar{g}}}}_0}{ }{. }{\mathbf{\bar{T}}}_2^{{12}} - {\mathbf{\bar{T}}}_2^{{22}}} \right]\,{ }{. }\,{\left[ {{{{\mathbf{\bar{g}}}}_0}{ }{. }{{{\mathbf{\bar{\Lambda }}}}^{{12}}} - {{{\mathbf{\bar{\Lambda }}}}^{{22}}}} \right]^{{ - 1}}},{\mathbf{\bar{\Lambda }}}{ } = {{\mathbf{\bar{T}}}_2}{ }{.}\,{ }{{\mathbf{\bar{T}}}_1} $$
(16)
$$ {{\mathbf{\bar{G}}}^{{12}}}(\,{{\mathbf{k}}_s}){ } = {{\mathbf{\bar{G}}}^{{21}}}(\,{{\mathbf{k}}_s}){ } = - {\mathbf{\bar{T}}}_1^{{12}}{ }{.}\,{ }{\left[ {{{{\mathbf{\bar{g}}}}_0}{ }{. }{{{\mathbf{\bar{\Lambda }}}}^{{12}}} - {{{\mathbf{\bar{\Lambda }}}}^{{22}}}} \right]^{{ - 1}}} $$
(17)
$$ {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}){ } = - {{\mathbf{\bar{\Lambda }}}^{{12}}}{ }{.}\,{ }{\left[ {{{{\mathbf{\bar{g}}}}_0}{ }{. }{{{\mathbf{\bar{\Lambda }}}}^{{12}}} - {{{\mathbf{\bar{\Lambda }}}}^{{22}}}} \right]^{{ - 1}}} $$
(18)
It is worth noting that the new explicit expressions shown in Eqs. 1618 allow the computation of the dyadic Green’s functions easily using simple matrix multiplication. Another advantage is that no transformation is required when the Green’s functions computed from Eqs. 1618 are used in the vector Hankel transform analysis of stacked circular microstrip patches. Considering the superconducting effects, we need simply to modify Eqs. 14 and 15 by replacing \( {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}) \) by \( {\mathbf{\bar{G}}}_s^{{11}}(\,{{\mathbf{k}}_s}) = {{\mathbf{\bar{G}}}^{{11}}}(\,{{\mathbf{k}}_s}) - {Z_{{s1}}} \cdot {\mathbf{\bar{I}}} \) and \( {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}) \) by \( {\mathbf{\bar{G}}}_s^{{22}}(\,{{\mathbf{k}}_s}) = {{\mathbf{\bar{G}}}^{{22}}}(\,{{\mathbf{k}}_s}) - {Z_{{s2}}} \cdot {\mathbf{\bar{I}}} \), where Zs1 and Zs2 are, respectively, the surface impedance of the bottom and top superconducting patches. When the thicknesses of the superconducting patches are less than three times the zero-temperature penetration depth (λ0), Zs1 and Zs2 can be expressed as follows [1012]:
$$ {Z_{{s1}}} = \frac{1}{{\sigma \,{e_1}}},{Z_{{s2}}} = \frac{1}{{\sigma \,{e_2}}} $$
(19)
where σ is the complex conductivity of the superconducting films. It is determined by using London’s equation and the Gorter-Casimir two-fluid model as [1012]
$$ \sigma = {\sigma_n}\,{({{T} \left/ {{{T_c}}} \right.})^4} - {\text{i}}\,{{{(\,\,1 - {{({{T} \left/ {{{T_c}}} \right.})}^4}\,)}} \left/ {{( \omega {\mu_0}\lambda_0^2 )}} \right.} $$
(20)
where T is the temperature, Tc is the transition temperature, σn is the normal state conductivity at T = Tc and ω is the angular frequency.
Using the technique known as the moment method, with weighting modes chosen identical to the expansion modes, Eqs. 14 and 15 are reduced to a system of linear equations which can be written compactly in matrix form as [22]
$$ {\mathbf{\bar{Z}}}\, \cdot \,{\mathbf{C}} = {\mathbf{0}} $$
(21)
where \( {\mathbf{\bar{Z}}} \) is the impedance matrix and the elements of the vector C are the mode expansion coefficients to be sought [22]. Note that each element of the impedance matrix \( {\mathbf{\bar{Z}}} \) is expressed in terms of a doubly infinite integral [22]. The system of linear equations given in Eq. 21 has non-trivial solutions when
$$ \det [{\mathbf{\bar{Z}}}(\omega )] = 0 $$
(22)

Equation 22 is an eigenequation for ω, from which the characteristics of the stacked structure of Fig. 1 can be obtained. In fact, let \( \omega = 2\pi ( {f_r} + {\text{i}} {f_i} ) \) be the complex root of Eq. 22. In that case, the quantity fr stands for the resonant frequency, the quantity \( BW = {{{{2} {f_i}}} \left/ {{{f_r}}} \right.} \) stands for the bandwidth and the quantity \( Q = {{{ {f_r}}} \left/ {{(2{f_i}}} \right.}) \) stands for the quality factor. Once the complex resonant frequency is determined, the eigen vector corresponding to the minimal eigenvalue of the impedance matrix gives the coefficients of the currents on the patches. The currents are thus obtained in numerical form. These currents can be used for determining the electromagnetic field in each layer of Fig. 1 using Eqs. 14 and 1315.

Note that for the case of electrically thin substrates, only the z component of the electric field and the transverse component of the magnetic field (TM propagating waves) exist in each dielectric layer of the two-layered substrate.

The polarization of an antenna refers to the polarization of the electric field vector of the radiated wave [23]. Consequently, before studying the polarization, it is necessary to first calculate the radiated field. Using the stationary phase method [24, 25], we can obtain the radiation electric field of the structure shown in Fig. 1 in terms of the transverse electric field at the plane z = d1 + d2 as follows:
$$ \left[ {\begin{array}{*{20}{c}} {{E_{{\theta \prime}}}(r \prime,\theta \prime,\varphi \prime)} \\ {{E_{{\varphi \prime}}}(r \prime,\theta \prime,\varphi \prime)} \\ \end{array} } \right] = {\text{i}}{k_0}\frac{{{e^{{ - {\text{i}} {k_0} r\prime\quad \quad }}}}}{{2 \pi r\prime}}\left[ {\begin{array}{*{20}{c}} { - 1} & 0 \\ 0 & {\cos \theta \prime} \\ \end{array} } \right]\,\, \cdot \,\,{\mathbf{e}}\,(\,{{\mathbf{k}}_s}, \,{d_1} + {d_2} ) $$
(23)
where {r’, θ’, φ’} is a local set of spherical coordinates defined with respect to the Cartesian system \( \left\{ {x \prime \equiv x,y\prime \equiv y,z\prime \equiv z} \right\} \) with an origin placed at the plane z = d1 + d2 of Fig. 1. Substituting Eq. 15 into Eq. 23 yields
$$ \left[ {\begin{array}{*{20}{c}} {{E_{{\theta \prime}}}(r \prime,\theta \prime,\varphi \prime)} \\ {{E_{{\varphi \prime}}}(r \prime,\theta \prime,\varphi \prime)} \\ \end{array} } \right] = {\text{i}}{k_0}\frac{{{e^{{ - {\text{i}} {k_0} r\prime\quad \quad }}}}}{{2 \pi r\prime}}\left[ {\begin{array}{*{20}{c}} { - 1} & 0 \\ 0 & {\cos \theta \prime} \\ \end{array} } \right]\,\, \cdot \,\,\left\{ {\,{{{\mathbf{\bar{G}}}}^{{21}}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 1}}}(\,{{\mathbf{k}}_s}) + {\mathbf{\bar{G}}}_s^{{22}}(\,{{\mathbf{k}}_s}){ }{. }{{\mathbf{j}}^{{ 2}}}(\,{{\mathbf{k}}_s})\,} \right\} $$
(24)
In Eqs. 23 and 24, the stationary values of kx and ky are given by
$$ {k_x} = - {k_0}\sin \theta \prime\,\cos \varphi \prime $$
(25)
$$ {k_y} = - {k_0}\sin \theta \prime\,\sin \varphi \prime $$
(26)

It is clear from Eq. 24 that the radiation electric field depends on the bottom patch current as well as on the top patch current. To conclude this section, we point out that the stacked patches with a single feed produces linearly polarized radiation. If circular polarization is desired, the most directed approach is to use two feeds located geometrically 90° apart and with a relative phase shift of 90° [26]. This arrangement excites two orthogonal modes, each providing a linearly polarized wave at right angles to each other and at phase quadrature [26]. Circular polarization can also be produced by altering the shape of patches (geometrical deformation) [26, 27].

3 Results and discussion

3.1 Validation of the proposed approach

Although the full-wave analysis can give results for several resonant modes [28], only results for the TM01 mode are presented in this study. The choice of the basis functions is of prime importance as it conditions the stability and convergence of the moment method. This choice is dictated by physical considerations such as entire domain and edge condition of the patch current [4]. The space-domain basis functions for approximating the bottom patch current and the top patch current are chosen to be the TM current modes of two magnetic-wall rectangular cavities with dimensions a1 × b1 and a2 × b2, respectively. Through numerical convergence checks, it is found that only one mode per patch suffices to obtain convergent results. Unlike single-layer microstrip antennas, some spurious resonances are found in the stacked configuration. This result is in accordance with that discovered by Fan and Lee [29]. In effect, Fan and Lee have shown that stacked circular-disk and annular-ring microstrip antennas have some spurious resonances [29]. However, there are no spurious resonances for the case of single-layer circular-disk and annular-ring microstrip antennas [29]. When evaluating the elements of the impedance matrix \( {\mathbf{\bar{Z}}} \), the integrands for the integrals are singular [30]. The integrals are evaluated numerically along an integration path deformed above the real axis (in the complex ks plane) to avoid the singularities (see Fig. 2) [30]. The resonant frequencies are complex and have a small positive imaginary part [30]. The spurious resonances are characterized by a negative imaginary part and consequently one cannot confuse them with the resonant frequencies of the stacked structure. Alternatively, we can also use the magnetic wall cavity model [8], which is a simple analytical model, together with the work in [18], for the distinction between the spurious resonances and the resonant frequencies of the stacked patches.
https://static-content.springer.com/image/art%3A10.1007%2Fs10762-011-9842-1/MediaObjects/10762_2011_9842_Fig2_HTML.gif
Fig. 2

The integration path in the complex ks plane.

In order to confirm the computation accuracy, our numerical results are compared with those obtained from the magnetic-wall cavity model [8] for the patches shown in Fig. 1 when they are not in the presence of each other. The two patches are fabricated with a YBCO superconducting thin film with parameters σn = 106 S/m, λ0 = 140 nm, Tc = 89 K and e1 = e2 = 350 nm. The operating temperature is T = 50 K. Table 1 summarizes our computed resonant frequencies and those obtained via the magnetic-wall cavity model [8] for three different cases and differences between these two results of less than 3% are obtained. As a consequence, very good agreement between our results and those of the literature is achieved. This validates the theory proposed in this paper.
Table 1

Comparison of our calculated resonant frequencies with those obtained via the magnetic wall cavity model [8] for the patches shown in Fig. 1 when they are not in the presence of each other; a1 = a2 = 2900 μm, εr = 1, d = 20.5 μm, e1 = e2 = 350 nm, σn = 106 S/m, λ0 = 140 nm, Tc = 89 K and T = 50 K.

\( {b_1}\,(\,\mu {\text{m}}\,{)}\,{\text{x}}\,{b_2}\,(\,\mu {\text{m}}\,{)} \)

Resonant frequencies (GHz)

Magnetic wall cavity model [8]

Results of the present work

Top patch absent

Bottom patch absent

Top patch absent

Bottom patch absent

2500 x 3250

59.537

45.781

58.651

44.769

3420 x 2675

43.569

55.547

43.042

54.121

3930 x 3000

37.930

49.570

37.509

48.405

3.2 Influence of the temperature on the lower and upper resonant frequencies

Numerical results are obtained for the stacked structure shown in Fig. 1. The patches are fabricated using a YBCO superconducting thin film with parameters σn = 106 S/m, λ0 = 140 nm, Tc = 89 K and e1 = e2 = 35 nm. The rectangular patches of identical size (a1 × b1 = a2 × b2 = 1560 μm × 702 μm) are printed on a lanthanum aluminate substrate of thickness 2 d = 174 μm. The variation of the permittivity of the lanthanum aluminate substrate with the variation of the temperature, as indicated by the experiment of Richard’s et al. [8], is taken into account in the present subsection. Due to the presence of the top patch in the stacked configuration, two resonances associated with the two constitutive resonators of the stacked structure are obtained. The lower resonance is noted fl while the upper resonance is noted fu. Figure 3 shows the lower and upper resonant frequencies against the operating temperature. From the results of Fig. 3, it can be observed that increasing the temperature will decrease the lower and upper resonant frequencies. This decrease is significant for temperatures near the transition temperature. Also, another point of particular interest is the possibility of widening the separation between fl and fu by decreasing the operating temperature. As an example, in the case of T = 87 K, the separation between the upper and lower resonances is 7.17% of the lower resonant frequency. For T = 70 K, there is now a 11.36% separation between the upper and lower resonant frequencies.
https://static-content.springer.com/image/art%3A10.1007%2Fs10762-011-9842-1/MediaObjects/10762_2011_9842_Fig3_HTML.gif
Fig. 3

Lower and upper resonant frequencies of the high Tc superconducting microstrip stacked patches against operating temperature; \( a_{1} \times b_{1} = a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), d = 87 μm, e1 = e2 = 35 nm, σn = 106 S/m, λ0 = 140 nm and Tc = 89 K.

3.3 Effects of patches thicknesses on the dual-frequency behavior

3.3.1 Effects of patches thicknesses on the dual-frequency behavior when b2< b1

In Fig. 4a, b, results are presented for the lower and upper resonant frequencies of stacked high Tc superconducting rectangular patches against normalized thickness of bottom (top) patch. The size of the bottom patch is a1 × b1 = 1560 μm × 702 μm, the top patch has the same length and its width is slightly smaller (700 μm). The superconducting material characteristics are: \( {\sigma_n} = 9.83\,\,{10^5}\,{\text{S/m}} \), λ0 = 100 nm, Tc = 89 K. The operating temperature is T = 50 K. The substrate is composed of two layers with parameters εr = 23.81 and d = 87 μm. In Fig. 4a, b, the top (bottom) patch thickness is maintained constant, whereas, the bottom (top) patch thickness is varied. It is observed from Fig. 4a that when the bottom patch thickness grows, the lower resonant frequency increases, whereas, this last is not affected by the variation of the top patch thickness as indicated in Fig. 4b. This can be explained by the fact that when b2 < b1, the lower resonance is associated with the resonator formed by the bottom patch and the ground plane [18]. It is also observed from Fig. 4a and b that the upper resonant frequency is affected by the variation of the bottom patch thickness as well as by the variation of the top patch thickness. This is expected since when b2 < b1, the upper resonance is associated with the resonator formed by the two patches [18].
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Fig. 4

a Variations of resonant frequencies of the high Tc superconducting microstrip stacked patches with bottom patch thickness when b2 < b1; \( a_{1} \times b_{1} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 700\,\mu {\text{m}} \), εr = 23.81, d = 87 μm, e2 = 1 nm, \( {\sigma_n} = 9.83\,\,{10^5}\,{\text{S/m}} \), λ0 = 100 nm, Tc = 89 K and T = 50 K. b Variations of resonant frequencies of the high Tc superconducting microstrip stacked patches with top patch thickness when b2 < b1; \( a_{1} \times b_{1} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 700\,\mu {\text{m}} \), εr = 23.81, d = 87 μm, e1 = 1 nm, \( {\sigma_n} = 9.83\,\,{10^5}\,{\text{S/m}} \), λ0 = 100 nm, Tc = 89 K and T = 50 K.

3.3.2 Effects of patches thicknesses on the dual-frequency behavior when b2> b1

In Fig. 5a and b, the effects of patches thicknesses on the lower and upper resonant frequencies of stacked high Tc superconducting rectangular patches are also investigated. In these Figs, the size of the bottom patch is \( a_{1} \times b_{1} = 1560\;\mu {\text{ m}} \times 702\;\mu {\text{m}} \), the top patch has the same length and its width is slightly larger (704 μm). In Fig. 5a, b, the top (bottom) patch thickness is maintained constant, whereas, the bottom (top) patch thickness is varied. It is seen from Fig. 5a that when the bottom patch thickness increases, the lower resonant frequency increases, whereas, it remains almost constant when the top patch thickness grows as shown in Fig. 5b. Although when b2 > b1, the lower resonance is associated with the resonator formed by the top patch and the ground plane [18], it is important to note that the electromagnetic field configuration in this resonator is strongly influenced by the bottom patch, which is located inside this resonator. It is also seen from Fig. 5a and b that the upper resonant frequency is affected by the variation of the bottom patch thickness as well as by the variation of the top patch thickness. This last result can be explained by the fact that when b2 > b1, the upper resonance is related to the bottom patch and it is significantly perturbed by the fringing fields of the top patch [18].
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Fig. 5

a Variations of resonant frequencies of the high Tc superconducting microstrip stacked patches with bottom patch thickness when b2 > b1; \( a_{1} \times b_{1} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 704\,\mu {\text{m}} \), εr = 23.81, d = 87 μm, e2 = 1 nm, \( {\sigma_n} = 9.83\,\,{10^5}\,{\text{S/m}} \), λ0 = 100 nm, Tc = 89 K and T = 50 K. b Variations of resonant frequencies of the high Tc superconducting microstrip stacked patches with top patch thickness when b2 > b1; \( a_{1} \times b_{1} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 704\,\mu {\text{m}} \), εr = 23.81, d = 87 μm, e1 = 1 nm, \( {\sigma_n} = 9.83\,\,{10^5}\,{\text{S/m}} \), λ0 = 100 nm, Tc = 89 K and T = 50 K.

3.4 Influence of the temperature on the bandwidth and quality factor

The influence of the operating temperature on the bandwidth and quality factor of the stacked high Tc superconducting rectangular microstrip patches shown in Fig. 1 is investigated in Fig. 6. The parameters of the antenna are identical to those used in Fig. 3. Note that the variation of the permittivity of the lanthanum aluminate substrate with the variation of the temperature, as indicated by the experiment of Richard’s et al. [8], is taken into account in our calculations. The superconducting antenna operates at the lower resonant frequency. It is observed from Fig. 6a that the effect of varying the temperature on the bandwidth is significant only for temperatures near the transition temperature. This behavior agrees with that discovered experimentally by Richard’s et al. [8] for single-layer superconducting microstrip antennas. As an example, when the operating temperature is varied from T = 84 K to T = 87 K, the bandwidth increases from 1.379% to 3.182% for a large fractional change of 130.75%. In Fig. 6b, we study the effect of the temperature on the quality factor of the stacked high Tc superconducting rectangular microstrip patches. It can be observed that increasing the temperature will decrease the quality factor. This decrease is significant for temperatures near the transition temperature.
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Fig. 6

Bandwidth and quality factor of the high Tc superconducting microstrip stacked patches against operating temperature. The antenna operates at the lower resonant frequency; \( a_{1} \times b_{1} = a_{2} \times b_{2} = 1560\,\mu {\text{m}} \times 702\,\mu {\text{m}} \), d = 87 μm, \( {e_1} = {e_2} = 35\,{\text{nm}} \), \( {\sigma_n} = {10^6}\,{\text{S/m}} \), λ0 = 140 nm and Tc = 89 K.

3.5 Characteristics of the lower and upper resonances

In this subsection, for a better comprehension of the dual-frequency operation, we compare between the characteristics of the lower and upper resonances. We consider the third structure studied in Table 1, i.e., \( a_{1} \times b_{1} = 2900\;\mu {\text{m}} \times 3930\;mu {\text{m}} \), \( a_{2} \times b_{2} = 2900\;\mu {\text{m}} \times 3000\;\mu {\text{m}} \), εr = 1, d = 20.5 μm, e1 = e2 = 350 nm, σn = 106 S/m, λ0 = 140 nm, Tc = 89 K and T = 50 K. For the case of the lower resonance, the complex resonant frequency and the surface current densities on the bottom and top patches are
$$ f_l^{{ c}} = 37.503 + {\text{i}}\, {5}{.825}\; \times \;{1}{{0}^{{{ - 2}}}} \left[ {\text{GHz}} \right] $$
(27)
$$ { }J_y^{{1}}( x,y) = 0.994\cos \left( {254.453\; \pi \;y} \right) \left[ {{\text{A}}/{\text{m}}} \right] $$
(28a)
$$ { }J_x^{{1}}( x,y) = 0 \left[ {{\text{A}}/{\text{m}}} \right] $$
(28b)
$$ { }J_y^{{2}}( x,y) = 0.016\cos \left( {333.333\; \pi \;y} \right) \left[ {{\text{A}}/{\text{m}}} \right] $$
(29a)
$$ { }J_x^{{2}}( x,y) = 0 \left[ {{\text{A}}/{\text{m}}} \right] $$
(29b)
For the case of the upper resonance, the complex resonant frequency and the surface current densities on the bottom and top patches are
$$ f_u^{{ c}} = 55.617 + {\text{i}} \,{1}8.{246}\; \times \;{1}{{0}^{{{ - 2}}}} \left[ {\text{GHz}} \right] $$
(30)
$$ { }J_y^{{1}}( x,y) = 0.527\cos \left( {254.453\; \pi \;y} \right) \left[ {{\text{A}}/{\text{m}}} \right] $$
(31a)
$$ { }J_x^{{1}}( x,y) = 0 \left[ {{\text{A}}/{\text{m}}} \right] $$
(31b)
$$ { }J_y^{{2}}( x,y) = 0.849\cos \left( {333.333\; \pi \;y} \right) \left[ {{\text{A}}/{\text{m}}} \right] $$
(32a)
$$ { }J_x^{{2}}( x,y) = 0 \left[ {{\text{A}}/{\text{m}}} \right] $$
(32b)
From the results of Table 1 and Eq. 27, it is clear that the lower resonant frequency (fl = 37.503 GHz) is very close to the resonant frequency of the isolated bottom patch (37.509 GHz). This means that the lower resonant frequency is associated with the resonator formed by the bottom patch and the ground plane. We have also investigated the field structure in this resonator. Figure 7 shows the normalized longitudinal electric field in the resonator formed by the bottom patch and the ground plane when the antenna operates at the lower resonance. It is observed from Fig. 7 that the structure of the electric field is similar to that of a conventional microstrip antenna (microstrip antenna with single patch) [26]. This is expected since the current on the top patch being very weak does not affect significantly the fields inside the resonator formed by the bottom patch and the ground plane.
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Fig. 7

Normalized longitudinal electric field in the resonator formed by the bottom patch and the ground plane when the antenna operates at the lower resonance fl = 37.503 GHz; \( a_{1} \times b_{1} = 2900\,\mu {\text{m}} \times 3930\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 2900\,\mu {\text{m}} \times 3000\,\mu {\text{m}} \), εr = 1, d = 20.5 μm, \( {e_1} = {e_2} = 350\,{\text{nm}} \), \( {\sigma_n} = {10^6}\,{\text{S/m}} \), λ0 = 140 nm, z = 10.25 μm, Tc = 89 K and T = 50 K.

Concerning the upper resonant frequency, it is associated with the resonator formed by the two patches. Although the resonant length of this last resonator is b2, the value of the upper resonant frequency (fu = 55.617 GHz) is different from that of the isolated top patch (48.405 GHz). This can be explained by the fact that when the antenna operates at the upper resonant frequency, the configuration of the fields inside the resonator formed by the two patches is significantly perturbed by the fringing fields at the perimeter of the bottom patch (see Fig. 8).
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Fig. 8

Normalized longitudinal electric field in the resonator formed by the two patches when the antenna operates at the upper resonance fu = 55.617 GHz; \( a_{1} \times b_{1} = 2900\,\mu {\text{m}} \times 3930\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 2900\,\mu {\text{m}} \times 3000\,\mu {\text{m}} \), εr = 1, d = 20.5 μm, \( {e_1} = {e_2} = 350\,{\text{nm}} \), \( {\sigma_n} = {10^6}\,{\text{S/m}} \), λ0 = 140 nm, z = 30.75 μm, Tc = 89 K and T = 50 K.

We finish this section by a comparison between the radiated electric field at the lower resonant frequency (fl = 37.503 GHz) and the one obtained when the antenna operates at the upper resonant frequency (fu = 55.617 GHz). The considered plane is the principal plane \( \varphi \prime = \frac{\pi }{2} \). From the results of Fig. 9, it is observed that the radiated electric field remains important along the horizon \( ( \theta \to \frac{\pi }{2} ) \) for the lower resonant frequency as well as for the upper resonant frequency. It is also observed from Fig. 9 that the antenna radiates more efficiently when it operates at the upper resonant frequency. This result agrees with that discovered theoretically for stacked perfectly conducting rectangular microstrip patches [31].
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Fig. 9

Comparison between the radiated electric field at the lower resonant frequency \( ( \,{f_l} = 37.503\;{\text{GHz}} {)} \) and the one obtained when the antenna operates at the upper resonant frequency \( ( \,{f_u} = 55.617\;{\text{GHz}} {)} \); \( a_{1} \times b_{1} = 2900\,\mu {\text{m}} \times 3930\,\mu {\text{m}} \), \( a_{2} \times b_{2} = 2900\,\mu {\text{m}} \times 3000\,\mu {\text{m}} \), εr = 1, d = 20.5 μm, \( {e_1} = {e_2} = 350\,{\text{nm}} \), \( {\sigma_n} = {10^6}\,{\text{S/m}} \), λ0 = 140 nm, Tc = 89 K,T = 50 K and \( \varphi ' = \frac{\pi }{2} \).

4 Conclusion

The dual-frequency behavior of two stacked high Tc superconducting patches has been investigated. The analysis has been based on a full electromagnetic wave model with London’s equations and the Gorter-Casimir two-fluid model. New explicit expressions have been developed allowing the computation of the dyadic Green’s functions of the stacked configuration easily by using simple matrix multiplications. These new expressions can be easily generalized to the case of stacked multilayered structures involving more than two radiating patches. The stationary phase technique has been used for computing the radiation electric field. Our numerical results obtained via Galerkin’s method in the vector Fourier transform domain have been compared with those obtained from the magnetic-wall cavity model and very good agreement has been found. Numerical results have shown that the influence of the operating temperature on the lower and upper resonant frequencies, bandwidth and quality factor of the stacked high Tc superconducting patches is especially significant for temperatures near the transition temperature. Other results also have indicated that the decrease of the temperature constitutes a very efficient tool for widening the separation between the upper and lower resonances. It has been also found that whatever the relative sizes of the superconducting patches in the stacked configuration, the upper resonant frequency is affected by the variation of the bottom patch thickness as well as by the variation of the top patch thickness. On the other hand, the lower resonant frequency is sensitive only to the variation of the bottom patch thickness when both the thicknesses of the two stacked high Tc superconducting patches vary. Finally, we have shown that the superconducting antenna radiates more efficiently when it operates at the upper resonant frequency. This last result agrees with that discovered theoretically for stacked perfectly conducting rectangular microstrip patches [31].

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© Springer Science+Business Media, LLC 2011