Study of Stacked High Tc Superconducting Circular Disk Microstrip Antenna in Multilayered Substrate Containing Isotropic and/or Uniaxial Anisotropic Materials

In this paper, we present a rigorous full-wave analysis able to estimate exactly the resonant characteristics of stacked high Tc superconducting circular disk microstrip antenna. The superconducting patches are assumed to be embedded in a multilayered substrate containing isotropic and/or uniaxial anisotropic materials (the analysis is valid for an arbitrary number of layers). London’s equations and the two-fluid model of Gorter and Casimir are used in the calculation of the complex surface impedance of the superconducting circular disks. Numerical results are presented for a single layer structure as well as for two stacked circular disks fabricated on a double-layered substrate.


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 millimeterwave antennas exhibiting high gain and wide bandwidth. Owing to their many attractive features and excellent advantages [1], microstrip antennas have attracted attention in both theoretical research and engineering applications over the past decades. Microstrip antennas are used in an increasing number of applications, ranging from biomedical diagnosis to wireless communication [2], [3]. However, as typical disadvantages are their capability to resonate at a single frequency, narrow bandwidth and low gain [4].
The discovery of high Tc superconductor oxides has made possible a new class of microwave and millimeterwave devices operating at temperatures considerably above liquid helium temperatures. Passive microwave components are among the first practical devices fabricated from these high Tc materials in which the surface resistance is a measure of the device performance at any frequency [5]. The extremely low surface resistance of superconductors facilitates the development of microwave devices with better performance than conventional devices [6]. The low surface resistance corresponds to a large quality factor and improved performance such as higher gain and lower insertion loss in passive microwave devices [7]. However, the narrow bandwidth that occurs with large quality factor is a major obstacle to the wider application of superconducting microstrip antennas [6], [7].
While maintaining the advantages of conventional single patch microstrip antennas, microstrip antennas of stacked configuration, consisting of one or more patches parasitically coupled to a driven patch, overcome the inherent narrow bandwidth limitation by introducing additional resonances in the frequency range of operation, achieving wide bandwidths [8], [9]. In addition, stacked microstrip configurations offer the possibility of dualfrequency operation [8], [10].
In this paper, we present a rigorous full-wave analysis of stacked high Tc superconducting circular disk microstrip antenna. We consider that the radiating elements are embedded in a multilayered structure containing isotropic and/or uniaxial anisotropic materials. Although the theoretical analysis presented in the current paper is valid for an arbitrary number of layers, we give here only numerical results for a single layer structure as well as for two-stacked circular disks fabricated on a double-layer substrate. The paper is organized as follows. In section 2, we develop new explicit expressions for the evaluation of the spectral dyadic Green's functions of two stacked circular disks embedded in a multilayered medium. These new expressions are also valid for structures involving more than two circular disks. London's equations and the twofluid model of Gorter and Casimir are used in the calculation of the complex surface impedance of the superconducting circular disks. In section 3, to validate the proposed approach, our numerical results are compared with theoretical and experimental data available in the literature for two different configurations. The first configuration consists on a single superconducting patch on a grounded dielectric substrate, while, the second configuration is composed of two stacked circular disks fabricated on a double-layer substrate. Finally concluding remarks are summarized in section 4.

Formulation
In equation (1), 0  is the free-space permittivity and diag stands for the diagonal matrix with the diagonal elements appearing between the brackets. Equation (1) can be specialized to the isotropic substrate by allowing be the value of the transverse electric field at the plane of the patch of radius i a . Owing to the revolution symmetry of the multilayered medium of figure 1 around the z-axis, when the Helmholtz equations for the longitudinal field components z E and z H are solved in cylindrical coordinates inside each of the layers of that medium, it turns out that the dependence of z E and z H on the  coordinate can be written as Following a mathematical reasoning similar to that shown in [1], we obtain a relation among ) , ( in the spectral domain given by In (7), j T is the matrix representation of the th j layer in the (TM, TE) representation [2]. It is to be noted that 11 G is related to the bottom disk and 22 G is related to the top disk. The dyadic Green's functions 12 G and 1 2 G characterize the coupling between the bottom patch and the top patch. Whatever the number of layers in the stacked configuration, the new explicit expression shown in equation (6) allows the computation of the dyadic Green's functions easily using simple matrix multiplications. Note also that the expression given in (6) is valid for stacked configurations containing more than two circular disks.
Unlike the case of perfectly conducting patches, the transverse electric field does not vanish on the area of high Tc superconducting patches. This is because the superconductor shows a resistance, although the value of this resistance is very small. The two-fluid model can be used to explain the general behavior of superconductors at microwave frequencies. The two-fluid model treats the current carriers in the superconductors as being of two distinct types: a supercarrier fraction, which carries current without any dissipation, and a normal fraction, which exhibit resistive scattering similar to the electrons in normal metals.
Considering the superconducting effects, we need simply to modify equations (4) and (5) The resistive part of the complex conductivity ) ( 1  may arise from normal electron conduction within nonsuperconducting grains and scattering from grain boundaries, flux vibration at pinning centers, and normal electron conduction due to thermal agitation in the superconducting state. The temperature dependence of 1  is as follows [7]: In equation (11),  is the angular frequency and ) (T  is the London penetration depth at temperature T . Now, that we have include the effect of the superconductivity of circular disks in the Green's functions formulation, the Galerkin procedure of the moment method can be easily

Validation of the proposed approach
For a circular radiating element the resonant modes are denoted by TMnm, where n and m denote variations in the azimuthal and radial directions, respectively. In the following only results for the TM11 mode are presented. This fundamental mode is characterized by the resonant frequency 11 r f , which is the lowest resonant frequency. Note that the proposed approach can give results not only for the fundamental mode but also for high order modes. If we want to calculate the resonant frequency for a specific mode, it is important to choose the basis functions such that they include qualitative natures of the true unknown current distributions of this mode. Also, the values of the three initial guesses used in the Muller's method for root seeking of the characteristic equation should be taken close to the resonant frequency of the considered mode. To this end, we can use the magnetic wall cavity model to obtain the values of the three guesses. In order to validate the proposed method for the case of a single circular disc printed on a one-layered substrate, we compare in Table 1 our numerical results with those obtained via the modified magnetic wall cavity model [11]. The circular disc is fabricated using a YBCO ( [11] despite its high permittivity because it allows the growth of high-quality (low surface resistance) YBCO superconducting films [11]. The operating temperature is K 77 

T
. Table 1   the calculated resonant frequencies and those obtained via the modified cavity model [11] for different radii of the superconducting circular disk and differences between these two results less than 2.5% are obtained. Note that the small differences between our numerical results and those obtained via the modified cavity model can be attributed to the fact that the cavity model do not account rigorously for the effect of fringing fields, especially for thick substrates.
A standard configuration for a microstrip antenna is a single patch supported above a ground plane by a simple dielectric substrate. This is a simple configuration that is relatively easy to fabricate, but it is limited in its functional capabilities. Dual-frequency structures are useful in situations where the antenna is required to operate efficiently at two distinct frequencies. Conventional microstrip antennas do not guarantee this type of operation. A more complex configuration, consisting of two microstrip patches in a stacked configuration, offers performance features that are not usually obtainable from the singlepatch single-dielectric configuration. These features include wider bandwidth and dual-frequency characteristics. In order to validate the proposed theory for the case of two stacked circular disks, numerical results are obtained for the parameters used in the experiment of Long and Walton [12].  Table 2, we have given at the same time the lower resonance and the upper resonance. It is observed from these comparisons that the agreement between theory and experiment is excellent. In section 2, we have shown that the new explicit expression for the evaluation of the dyadic Green's functions is valid for stacked configurations containing more than two circular disks. Hence, the proposed method can be easily applied to compute the complex resonant frequencies of stacked microstrip antennas with three radiating elements.

Conclusions
A rigorous full-wave analysis able to estimate exactly the resonant characteristics of stacked high Tc superconducting circular disk microstrip antenna has been presented. The superconducting circular disks are embedded in a multilayered structure containing isotropic and/or uniaxial anisotropic materials. London's equations and the two-fluid model of Gorter and Casimir have been used in the calculation of the complex surface impedance of the superconducting circular disks. New explicit expressions for the evaluation of the dyadic Green's functions of two stacked circular disks fabricated on a multilayered substrate have been derived. These expressions are also valid for stacked configurations containing more than two patches. Contrary to the equivalent boundary method [13], no transformation is required when the developed Green's functions are used in the moment method analysis of stacked high Tc superconducting rectangular microstrip patches. The numerical results obtained are compared with previously published numerical results and with measurements. Good agreement is found in all cases among all sets of results.