Integral equation analysis of a low-profile receiving planar microstrip antenna with a cloaking superstrate

Authors


Abstract

[1] Eliminating the electromagnetic interaction of a device with its background is a topic which attracts considerable attention both from a theoretical as well as from an experimental point of view. In this work, we analyze an infinite two-dimensional planar microstrip antenna, excited by an incident plane wave, and propose its potential operation as a low-profile receiving antenna, by suitably adjusting the parameters of its cloaking superstrate. We impose a semi-analytic integral equation method to determine the scattering characteristics of the microstrip antenna. The method utilizes the explicit expressions of the Green's function of the strip-free microstrip and yields the surface strip's current as the solution of a suitable linear system. Subsequently, the antenna's far-field response is obtained. Numerical results are presented for the achieved low profile of the receiving antenna, by choosing suitably the cloaking superstrate parameters. It is demonstrated that for specific cloaking parameters the scattered field by the antenna is considerably reduced, while the received signal from the antenna is maintained at sensible levels. We point out that the material values achieving this reduction correspond to a superstrate filled with anϵ-near-zero or a low-index metamaterial. Finally, the variations of the device reaction for various superstrates are depicted, concluding that for optimized superstrate's parameters, the reaction values are significantly reduced, while at distinct scattering angles even approach zero.

1. Introduction

[2] The relative benefits of conventional microstrip antennas, possessing a conducting strip printed on a grounded microwave substrate, as well as their attractive resonance and radiation characteristics have been meticulously documented in the literature [see, e.g., Pozar and Schaubert, 1995; Wong, 1999; Nasimuddin, 2011; Gao et al., 2001; Bzeih et al., 2007; Rubelj et al., 1997]. More precisely, the microstrip antennas offer light-weight, ease of fabrication, conformability to planar and non-planar surfaces, integrability with microwave and millimeter-wave circuits, as well as mechanical robustness. Microstrip antennas have been widely utilized in communication systems, radars, sensors, radio frequency identifications etc. The progress in wireless communications has further increased the demand for microstrip antennas, capable to be embedded in portable, hand-held, and wireless network devices [Nasimuddin, 2011]. Moreover, the remarkable electromagnetic properties of various types of metamaterials paved the way for the incorporation of similar substances materials as substrates or superstrates of microstrip antennas in order to achieve features, such as high directivity and increased gain [Xu et al., 2008; Attia et al., 2009].

[3] On the other hand, diminishing the electromagnetic interaction of a device or component is a topic that has attracted considerable attention from both a theoretical as well as from an experimental point of view. The general idea of cloaking a sensor, without affecting its capability to receive signals proportional to the measured quantity, is analyzed by Alù and Engheta [2009], where a system design is proposed whose presence is not perceived by the surrounding. An alternative materialization of the aforementioned concept, based on optics transformation, is presented by Greenleaf et al. [2011], where a sensor effect that breaks the strong connection between cloaking and shielding, allowing the former without the latter, is thoroughly described. Scattering cancelation mechanisms by means of isotropic and homogeneous thin covers in both electromagnetic and acoustic domains have been recently investigated [Guild et al., 2011], which could be employed in achieving low profile for receiving structures.

[4] In this work, motivated by the above considerations, we consider a two-dimensional (2-D) infinite planar microstrip antenna, excited by an incident plane wave, and propose its potential capability of operating as a low-profile receiving antenna, after incorporating a suitable cloaking superstrate. The aforementioned problem is indeed an inherently difficult and challenging problem to achieve a low profile, due to the facts that the microstrip structure is considered to be infinite, while the incident plane wave constitutes also an excitation with infinite wavefront. In particular, the under consideration 2-D planar microstrip antenna is comprised of a perfect electric conductor (PEC) plane, a slab substrate, a thin rectangular PEC strip, and a slab superstrate. The goal of the analysis lies in the determination of the influence of the superstrate on the scattering features of the microstrip. Additionally we aim at achieving a low profile for the receiving antenna by choosing appropriately the thicknessh and material parameters ϵ2 and μ2of the superstrate. We note that the utilization of the terminology “cloaking” in the present context does not concern that electromagnetic waves get around the object to be cloaked but more likely refers to a “carpet-cloaking” in the sense of concealing the presence of the strip from its environment; for carpet-cloaking seeLi and Pendry [2008].

[5] We implement a semi-analytic integral equation method to determine the scattering characteristics of the receiving microstrip antenna (for a discussion on analysis methodologies of microstrip antennas seeTsitsas and Valagiannopoulos [2011] and Valagiannopoulos and Tsitsas [2008] and of scattering by strips embedded in homogeneous or layered media see Gürel and Chew [1992], Cangellaris [1991], and Gürel and Aksun [1996]; aspects of optimal modeling of microstrip antennas are discussed by Kolundžija and Bajic [2002]). First, we determine analytically the Green's function as well as the primary induced fields on the homogeneous (strip-free) layered structure. Then, we apply a methodology based on the radiation integral to compute the scattered field by the microstrip antenna. In particular, we consider the integral representation of the scattered electric field due to the presence of the PEC strip, incorporating the unknown surface current induced on the strip. To this end, we decompose the PEC strip into a suitably large number of thin cylindrical wires, in each one of which flows a different but constant filamentary current. By imposing the boundary condition on the wires, we result to a linear system the solution of which provides the unknown current coefficients. Having determined these coefficients, we compute the total field in the vacuum region (above the antenna) and subsequently the antenna's far-field response. Furthermore, in order to investigate the regulating effect of the superstrate on the scattered field generated by the overall structure and then seek to reduce considerably the field generated by the actual antenna with respect to that by the superstrate-free antenna, we define two quantities of interest: thecloaking (or low-profile) factor, and the antenna's reaction.The former provides a measure of the reduction of the mean value of the far-field radiated over all directions under the presence of the superstrate compared to the superstrate-free case. The latter expresses the normalized far-field over the primary excitation for specific scattering angles. Also, in order to show that the possible reduction of the radiated field by the microstrip antenna is accompanied by a maintenance of the antenna's sensible received signal, we define the receivedequivalent current induced on the strip.

[6] Numerical simulations are presented dealing with the achieved low-profile features of the receiving microstrip antenna by choosing suitably the parametersϵ2, μ2, and h of the cloaking superstrate. It is demonstrated that for a specific neighborhood of material parameters ϵ2 and μ2the scattered field from the microstrip antenna is considerably reduced, while the received equivalent current on the strip is maintained at sensible levels. In fact, we show that for those parameter values the cloaking factors may be decreased to nearly 25%, meaning that the field generated by the microstrip is nearly one quarter of the field generated by the superstrate-free microstrip. Also, we indicate that the values ofϵ2 and μ2for which this far-field reduction is achieved, correspond to a superstrate composed of anϵ-near-zero (ENZ) material [Silveirinha and Engheta, 2006; Alù et al., 2007] or a low-index metamaterial (LIM) [Lovat et al., 2006; Valagiannopoulos, 2007a]. Numerical results are also presented concerning the cloaking factors reductions versus the superstrate thickness h, the operating frequency f, and the incidence angle ϕi. Finally, we depict the variations of the device reaction for various superstrate parameters and conclude that, for no superstrate and for an arbitrary superstrate, the values of the device reaction are very close, while in the case that the superstrate's parameters are optimized, then the device reaction values are significantly reduced, while at certain distinct scattering angles these values even approach zero.

2. Mathematical Formulation

2.1. Geometrical Configuration and Incident Field

[7] The under consideration two-dimensional geometrical configuration, as well as the respective coordinate systems (Cartesian (xyz) and cylindrical (ρϕz)), are depicted in Figure 1. The configuration is comprised of a perfect electric conductor (PEC) plane at y = −w covered by a slab substrate (region 1) of thickness 2w composed of a magnetically inert material with relative dielectric permittivity ϵ1. A thin rectangular PEC strip of thickness 2a and length 2b is located at y = w − a, namely slightly below the boundary y = w, and hence lying entirely inside region 1. The structure of the PEC plane, the substrate, and the PEC strip is additionally covered by a slab superstrate (region 2) of thickness h, composed of a material with relative dielectric permittivity ϵ2 and relative magnetic permeability μ2. The infinite plane region (vacuum region 0) above the superstrate is characterized by permittivity ϵ0 and permeability μ0. The entire structure is assumed uniform along the direction inline image.

Figure 1.

Geometrical configuration of the receiving microstrip antenna under investigation composed of a planar PEC strip, with thickness 2a and length 2b, lying between a substrate and a superstrate. The microstrip antenna is excited by a primary plane wave with angle of incidence ϕi. The implementation of our semi-analytical method requires that the PEC strip is decomposed into a suitably large number 2U + 1 ≡  inline image of thin wires with radius a. The surface current flowing on each wire is assumed to be constant.

[8] The device described above, which is assumed to operate as a receiving antenna, is excited by a unit amplitude primary plane wave traveling in region 0 along the ray ϕ = ϕi. Suppressing the time dependence exp(iωt), with ω = 2πf being the angular frequency, the z–polarized primary electric field is expressed by the following two equivalent expressions as

display math

where k0 = ω inline imageis the wave number of region 0. Note that, due to the two-dimensional nature of the configuration as well as the primary excitation, all the generated electric fields in each region will be alsoz–polarized, and hence will be described and referred to hereinafter by their z–components.

[9] The goal of the following analysis lies in the determination of the influence of the superstrate on the scattering features of the planar microstrip antenna and additionally in the investigation of the possibility of designing a low-profile receiving antenna, by choosing appropriately the thicknessh and material parameters ϵ2 and μ2 of the superstrate layer.

2.2. Green's Function of the Homogeneous Structure

[10] The formulation of an appropriate integral representation of the scattered electric field by the microstrip antenna requires a suitable analytic expression of the Green's function of the homogeneous (without the PEC strip) problem.

[11] To this end, we consider the respective homogeneous structure (where the PEC strip is now absent) which is excited by a two-dimensional infinite along thez-axis line source, located at an arbitrary point (x′, y′) inside the substrate region 1 (note that region 1 is where the strip is also located in the original physical problem depicted in Figure 1), possessing normalized current density

display math

where δ(·) is the Dirac function.

[12] The Green's function is defined as the electric field generated at the observation point (xy) by the aforementioned line source (see for example the discussions by Valagiannopoulos [2010] and Tsitsas et al. [2007]) and is computed here as an application of Sommerfeld's method [Sommerfeld, 1949; Stratton, 1941]. More precisely, the z–component of the primary field G1pr (singular term of the Green's function), generated by this line source under the assumptions that the entire space inline image2 is filled by the material (ϵ1ϵ0μ0) of the substrate region 1, is given by

display math

where k1 = k0 inline image and k2 = k0 inline image are the wave numbers of regions 1 and 2 respectively, while H0(2)denotes the second kind and zero-order cylindrical Hankel function.

[13] On the other hand, by imposing the relevant boundary conditions (PEC boundary condition on y = −w and transmission boundary conditions on y = w and y = w + h), we obtain the explicit Fourier integral expression of the secondary electric field induced by the line source in region 1 of the homogeneous structure (representing the smooth term of the Green's function)

display math

where the components of the integral kernel are given by

display math
display math

with the sign of the square roots gp(β) ≡ inline image (p = 0, 1, 2) corresponding to Re(gp(β)) > 0 and Im(gp(β)) > 0, while the remaining auxiliary functions are defined by

display math

[14] Furthermore, the secondary electric field, induced in the region 0 of the homogeneous structure, is expressed by the Fourier integral

display math

where

display math

2.3. Primary Induced Fields on the Homogeneous Structure

[15] The investigation of the scattering phenomena by the under consideration microstrip antenna, by applying an integral equation method, requires also appropriate expressions of the fields induced on the homogeneous structure, due to the primary plane wave (1).

[16] The induced transmitted secondary electric field in the substrate region 1 is explicitly given by

display math

where kr0 ≡ inline image (r = 1, 2), while the transmission coefficient T is defined by

display math

with

display math

[17] Moreover, the induced reflected secondary electric field in the air region 0 is explicitly given by

display math

where the reflection coefficient ρ is defined by

display math

[18] Note that the total electric field in region 0 is the sum of the primary field (1) and the secondary field (13).

3. Radiation Integral Method

3.1. Computation of the Surface Current on the PEC Strip

[19] The computation of the scattered field by the microstrip antenna of Figure 1 will be subsequently carried out by applying a radiation integral method. In particular, according to the radiation integral formula [see, e.g., Tai, 1994, p. 228], the generated scattered electric field in the substrate region 1, due to the presence of the PEC strip, has the following integral representation

display math

where K() is the surface current induced on the surface C of the PEC strip, while is the integration variable along this surface.

[20] Next, we decompose the PEC strip into a suitably large number 2U + 1 ≡  inline image of thin wires with radius a, as shown in Figure 1. These wires are posed adjacent to each other and their centers are located along the line {−b ≤ x ≤ by = w − a}. Furthermore, we assume that the surface current flowing on each wire is constant. Then, this decomposition of the PEC strip yields the following integral expression of the scattered field, resulting directly from (15):

display math

where κu corresponds to the constant z–polarized unknown electric current flowing on the surface Cu of the u–th cylinder (u = −U,…,U). Also, notice that the source coordinates in the Green's functions have been expressed in the local coordinate system, centered at the u–th cylinder by means of the polar variables ρ and ϕ. The integration length variable , included in ρ and ϕ, is utilized for the line integrations around the circular boundaries of the wires.

[21] Next, we impose the PEC boundary condition, namely the vanishing of the total tangential electric field, on the surfaces of the thin cylindrical wires, which we assume that coincide with their centers (xy) = (xvw − a) = (2avw − a) for v = − U,…,U. In this way, we obtain the following (2U + 1) × (2U + 1) linear system, with respect to the unknown current coefficients κu

display math

[22] Analytic progress can be made further in the elements of the system's matrix by considering separately the two terms, resulting from the smooth and the singular component of the Green's function. In particular, the term involving the smooth component, exhibits negligible variation with respect to the integration variable , due to the limited extent of each wire's cross section (thin-wire approximation), and hence can be simplified, by invoking the mean value theorem of integral Calculus, as:

display math

Note that in the right hand side of the above expression, we have utilized the Cartesian variables for the source coordinates in the Green's function component. Equation (18)may be also interpreted as the average-value approximation, typically considered in the Pocklington approximation for thin wires.

[23] Moreover, as far as the integral of the singular component of the Green's function is concerned, it can be explicitly evaluated by using standard analytical integration techniques [see, e.g., Valagiannopoulos, 2011a], as follows:

display math

where J0is the zero-order cylindrical Bessel function.

[24] Hence, we substitute the derived expressions (18) and (19) in the elements of the linear system's matrix (17) and then solve this system numerically to determine the current coefficients κu (u = −U,…,U). A basic characteristic of the system (17) is that as the strip's discretization variable a becomes smaller, the matrix of (17) becomes diagonally dominant [see, e.g., Varga, 1956, p. 22], since its diagonal elements, given by (19), include the factor H0(2)(k1a). Hence, the solution of (17) is assured to exhibit a robust behavior with absence of numerical instabilities as well as a fast convergence.

[25] We note that we have considered here a discretization of the strip into a suitable number of thin consecutive wires. On the other hand, one could consider a thin solid strip and model the current on it by means of pulse or triangular basis functions along x. However, in that case the basis functions should properly express the singularities along the edges of the strip, as, e.g., in the work by Valagiannopoulos [2007b] where the basis functions tend properly to infinity at the ends of a cylindrical patch. Here, the utilized procedure of decomposing the strip into thin wires is stable and robust since we try to mimic the behavior of the entire strip with a cluster of small wires touching one another; related aspects of this procedure are discussed by Valagiannopoulos [2011b]and C. A. Valagiannopoulos and A. Sihvola (On modeling perfectly conducting sharp corners with magnetically inert dielectrics of extreme complex permittivities, 2012, available at http://arxiv.org/abs/1201.5401), where it has been successfully applied for different problems of scattering by strips. Also, we remark that the obtained results prove to be meaningful also for modest values of the strip's thickness (and not necessarily only for very thin-strips); seeFigure 2 below as well as Valagiannopoulos [2011b, Figure 3].

Figure 2.

Absolute value of the scattered electric far-field |E0scat(ρ0ϕ)| for ρ0 = 5λ as function of the observation azimuthal angle ϕ for f = 2 GHz, ϵ1 = 4.2 − 0.084iϵ2 = 5.5, μ2 = 0.26, w = 5 mm, h = 1.5wb = wϕi = 45°, with a/b = 0.007 (dotted red line), 0.005 (dashed-dotted blue line), 0.004 (dashed green line), 0.0036 (solid black line).

3.2. Radiated Far-Field by the Microstrip Antenna

[26] The total reaction of the microstrip antenna, due to the excitation by the primary field E0pr of (1), is the total induced field in the air region 0, which by following the preceding analysis is expressed as the sum of the component E0sec induced in the homogeneous structure, given by (13), and the component E0scat due to the actual presence of the strip.

[27] The latter component admits the following integral representation:

display math

which, by considering again the decomposition of the strip in thin wires and following similar arguments with the previous section, takes the form:

display math

where now the surface current's coefficients κu have already been determined as the solution of the linear system (17).

[28] Finally, we will determine the far-field response of the microstrip antenna, by expressing the scattered field(21) in polar coordinates (ρϕ) and then taking the approximation as ρ → +. To this end, we will utilize the following stationary phase approximation formula [see, e.g., Erdelyi, 1956, pp. 50–52]:

display math

By combining (8), (21) and (22), we obtain the following far-field approximation

display math

where

display math

3.3. Cloaking Factor and Antenna's Reaction

[29] We aim at investigating numerically the effect of the superstrate on the total field generated by the receiving microstrip antenna of Figure 1and then at seeking to reduce considerably the field generated by this actual configuration with respect to that generated by the superstrate-free configuration. Therefore, we define the followingcloaking (or low-profile) factor:

display math

where in the denominator the electric far-fields inline image and inline image correspond to ϵ2 = μ2 = 1 (namely to the case that the superstrate is absent), while ρ0 is a suitably large radius such that the stationary phase (22)and far-field(23)approximations are valid. The numerator of this factor constitutes a measure of the mean value of the far-field radiated over all directions by the actual receiving microstrip antenna. On the other hand, its denominator measures the respective mean value of the radiated far-field by the corresponding microstrip without the superstrate. Hence, in order for the receiving microstrip antenna to be as low-profile as possible we will search, in the numerical simulations of the next section, for those superstrate parametersϵ2μ2, and h, which make the cloaking factor inline image as small as possible.

[30] Furthermore, in order to investigate the generated far-field for specific observation anglesϕ, and compare it with the primary excitation, given by (13), we define the microstrip antenna's far-fieldreaction as follows:

display math

3.4. Received Current by the Microstrip Antenna

[31] Finally, we need also to show that the possible reduction of the radiated field by the microstrip antenna is accompanied by a relative maintenance of the antenna's sensible field sensing (received signal). To this end, it is worthwhile to consider also the received signal of the microstrip antenna as a function of frequency and examine it in conjunction with the cloaking factor, in order to derive results in context with the ones of Alù and Engheta [2009] and Castaldi et al. [2010].

[32] For the under consideration microstrip antenna, the received signal is the total equivalent current inline image induced on the strip, which is defined as follows:

display math

This current is measured in Amperes, it is proportional to the received voltage, and it expresses the strength of the signal recorded by the receiving sensor.

[33] Here, we note that since we consider the strip to be the receiving element of the antenna, it is natural to expect that there must be a port leading to a conducting path (such as a coaxial cable [see, e.g., Gauthier et al., 1997]) through which electromagnetic energy is conveyed from the strip toward a receiver. However, as is usually the case in existing related investigations [see, e.g., Valagiannopoulos and Tsitsas, 2008; Castaldi et al., 2010], the contribution of this path is not taken into account for the present research purposes. This fact does not affect essentially the optimized design of the device, while still assures its proper functionality. Indeed, a small wire transmitting the signal from the metallic surface into the recording system is not expected to alter significantly the wave behavior of the considered “wire-free” structure. In fact, such receiving configurations have been successfully proposed and analyzed without taking into account wires, conducting paths or ports [Valagiannopoulos and Tsitsas, 2008; Castaldi et al., 2010].

4. Numerical Results

4.1. Convergence of the Method

[34] The cylinders radius aconstitutes actually the discretization parameter of the PEC strip; hence, it must be chosen suitably small in order to obtain convergent far-field results. Thus, prior to proceeding with the presentation and discussion of specific numerical results, we will first investigate the far-field convergence with respect to the ratioa/b, for some typical values for the parameters of the receiving microstrip antenna.

[35] To this end, Figure 2 depicts the fields convergence pattern with respect to the ratio a/b. More precisely, we plot the absolute value of the scattered electric far-field |E0scat(ρ0ϕ)| for ρ0 = 5λ as function of the observation azimuthal angle ϕ for f = 2 GHz, ϵ1 = 4.2 − 0.084iϵ2 = 5.5, μ2 = 0.26, w = 5 mm, h = 1.5wb = wϕi = 45°. Four different field curves are depicted corresponding to cylinders radii a = 0.007b, 0.005b, 0.004b, and 0.0036b. The method's convergence is evident from the curves of Figure 2. In the numerical results of the following section we will fix the discretization radius a at a constant value, which has been a priori checked to provide convergent field results for all the cases under examination.

4.2. Cloaking Features of the Microstrip

[36] We proceed to the investigation of the achieved low-profile features of the receiving microstrip antenna, by choosing suitably the parametersϵ2, μ2, and h of the cloaking superstrate.

[37] For all the numerical simulations, included hereafter, we consider that the substrate is composed of FR4 laminate [Nascimento and da Lacava, 2011], characterized by complex dielectric permittivity ϵ1 = 4.2 − 0.084i, and thickness w = 10 mm. Also, the constant length of the PEC strip is b = 1.5w, while the common radius of the cylinders, discretizing the strip, is a = b/30. The suitably electrically large radius, corresponding to the far-field computations, isρ0 = 5λ, where λ is the wavelength of the primary field, exciting the microstrip antenna.

[38] Figures 3a and 3b depict the cloaking factor inline image contour plots with respect to the superstrate's relative dielectric permittivity ϵ2 and relative magnetic permeability μ2 for f = 2 GHz, h = 2w, and angles of incidence ϕi = 30° and 60°, respectively. The motivation for examining such contour graphs is to find an operation point with respect to ϵ2 and μ2 for which the scattered field from the microstrip antenna is considerably reduced. Indeed, such an operating point exists, equalling approximately (ϵ2μ2) = (0.35, 16.5) for ϕi = 30° and (ϵ2μ2) = (0.03, 6.4) for ϕi = 60°.At these two operating points, the cloaking factors are decreased to nearly 25%, meaning that the field generated by this microstrip configuration is nearly the one quarter of the field generated by the corresponding superstrate-free configuration. Note that the corresponding power will decrease further significantly. Importantly, we observe fromFigure 3b that the operating point (ϵ2μ2) = (0.03, 6.4), for which the aforementioned far-field reduction is achieved, corresponds to the case of a superstrate filled with anϵ-near-zero (ENZ) material [Silveirinha and Engheta, 2006; Alù et al., 2007] or a low-index metamaterial (LIM) [Lovat et al., 2006; Valagiannopoulos, 2007a]. In fact, such materials are known to possess related remarkable electromagnetic properties, including controlling the radiation pattern, tunneling of electromagnetic energy etc [Silveirinha and Engheta, 2006; Alù et al., 2007].

Figure 3.

Cloaking factor contour plots as functions of the superstrate's relative dielectric permittivity ϵ2 and relative magnetic permeability μ2 for f = 2 GHz, h = 2w, and (a) ϕi = 30°, and (b) ϕi = 60°.

[39] Furthermore, Figures 4a and 4b show the cloaking factor inline image as function of the superstrate's relative dielectric permittivity ϵ2 for constant μ2 = 8.83, and relative magnetic permeability μ2 for constant ϵ2 = 0.23, respectively, for f = 2 GHz, ϕi = 45°, with superstrate thickness h = 1.5w, 2w, 2.5w. Also here, as in Figure 3, the cloaking factors attain considerably decreased values (for example reaching field reductions of the order of 20%). For each value of h the cloaking factor curve attains a minimum occurring each time for different value of ϵ2 or μ2. In fact, as h increases the values of ϵ2 and μ2 for which the minimum is attained is shifted to the ENZ or LIM zone, namely to values of ϵ2 and μ2 approaching 0. On the other hand, it is also interesting to note that as h decreases, the regions for which the cloaking factor values are much smaller than 1 become wider; in other words, all minima become sharper, as h increases. Besides, for ϵ2 and μ2 near zero, we observe that the variations of the cloaking factor curves seem to be actually independent of the superstrate thickness h.

Figure 4.

Cloaking factor as function of the superstrate's (a) relative dielectric permittivity ϵ2 for constant μ2 = 8.83, and (b) relative magnetic permeability μ2 for constant ϵ2 = 0.23, with f = 2 GHz, ϕi = 45°, and h = 1.5w(dashed-dotted red line),h = 2w (dashed green line), h = 2.5w (solid blue line).

[40] Figures 5a and 5b represent the cloaking factor inline image as function of the operating frequency f for ϕi = 45°, and h = 1.5w, ϕi = 45°, with (a) ϵ2 = 0.5, 1, 1.5, 2 for constant μ2 = 8.5, and (b) μ2 = 1, 2, 3, 4 for constant ϵ2 = 0.5. From Figure 5a we observe that as ϵ2 decreases, the value of the attained minimum also decreases, while simultaneously the minimum becomes sharper. The same conclusions hold from the statements of Figure 5b but in the case that μ2increases. We point out that indeed for the selected values of parameters the frequency response of the microstrip is not so wide-band, in the sense of achieving relatively small values of the cloaking factors for a narrow frequency interval. However, we note that the presented scenario inFigure 5corresponds to the case where the superstrate parameters have been chosen arbitrarily and they have not been forced to achieve an optimal low-profile behavior. The reason for the selection of this scenario was to demonstrate that reduced values of the cloaking factors may indeed be obtained without necessarily optimizing all the superstrate parameters. Of course, a systematic treatment leading to the optimal values ofϵ2 and μ2would offer a more wide-band behavior of the microstrip and would reduce further the cloaking factor values.

Figure 5.

Cloaking factor as function of the device's operating frequency f for ϕi = 45°, and h = 1.5w with (a) ϵ2= 0.5 (dotted red line), 1 (dashed-dotted green line), 1.5 (dashed blue line), 2 (solid black line) for constantμ2 = 8.5, and (b) μ2= 1 (dotted red line), 2 (dashed-dotted green line), 3 (dashed blue line), 4 (solid black line) for constantϵ2 = 0.5.

[41] Furthermore, the small attained values of the cloaking factor versus frequency, which were observed in Figure 5, need to be tested with respect to the maintenance of the antenna's sensible received signal. To this end, Figures 6a and b depict the received equivalent current inline image as function of the antenna's operating frequency f for (a) ϵ2 = 0.5, μ2 = 8.5 and (b) ϵ2 = 0.5, μ2 = 4. In other words, we represent the received equivalent currents in those cases that substantial cloaking effect was observed (namely the optimized cases with respect to ϵ2 and μ2 of Figure 5). Also, in both Figures 6a and 6bthe superstrate-free case (ϵ2 = μ2 = 1) is depicted in order to compare the received “cloaked” signal to that of an antenna with absent superstrate which operates normally as a receiver. As is shown in Figure 6, the equivalent current on the strip is, as expected, smaller in the cloaked case due to the presence of the optimized superstrate. Nevertheless, the received signal is sufficiently sensible to be detected and subsequently recorded by a measuring equipment. Consequently, the utilized cloaking superstrate layer aids in designing a low-profile sensor, relatively “invisible” to the surrounding space, which is still capable of “observing” and “sensoring” efficiently. In addition, and as far as the frequency response is concerned, the equivalent current of the cloaked antenna has a similar waveform to that of the uncloaked one. In particular, the variation of the equivalent current for the cloaked antenna follows the corresponding one of the superstrate-free antenna with a small frequency delay and without the rapid fluctuations.

Figure 6.

Received equivalent current as function of the device's operating frequency f for ϕi = 45°, and h = 1.5w with (a) ϵ2 = 0.5, μ2 = 8.5 (solid red line), and (b) ϵ2 = 0.5, μ2= 4 (solid red line). In both Figures 6a and 6b the superstrate-free caseϵ2 = μ2 = 1 (dashed blue lines) is depicted for comparison.

[42] In Figure 5the frequency dispersion of the ENZ material was not taken into account. Hence, it is now interesting to investigate how its presence affects the bandwidth of the cloaking effect. To this end, we assume a Drude model around the design frequency, namely consider that the frequency-dependent permittivity of the superstrate is expressed as [see, e.g.,Valagiannopoulos, 2011c]:

display math

where fPis the so-called plasma frequency. FromFigure 5 substantial cloaking effect is observed when ϵ2 = 0.5. Therefore, we select the plasma frequency fP = 1.41 GHz in order to obtain values of ϵ2 close to 0.5 around f = 2 GHz.

[43] Figures 7a and 7b depict the cloaking factor as function of the operating frequency f for (a) ϵ2 = 0.5, and ϵ2 = ϵ2(f) with constant μ2 = 8.5, and (b) ϵ2 = 0.5, and ϵ2 = ϵ2(f), with constant μ2 = 4. The permittivity function ϵ2(f) follows the Drude model (28). Notice that (cf. Figure 5) the specific fixed utilized values of ϵ2 and μ2 correspond to the minimum cloaking factor values around f = 2 GHz. From Figure 7 it is evident that the frequency dispersion of the ENZ material does not affect essentially the cloaking operation of the antenna.

Figure 7.

Cloaking factor as function of the frequency f for ϕi = 45°, and h = 1.5w with (a) ϵ2 = 0.5, μ2 = 8.5 (blue line with squares), ϵ2 = ϵ2(f), μ2 = 8.5 (red line with circles), and (b) ϵ2 = 0.5, μ2 = 4 (blue line with squares), ϵ2 = ϵ2(f), μ2 = 4 (red line with circles). In both Figures 7a and 7b, the inset shows the permittivity function ϵ2(f), following the Drude model (28).

[44] Moreover, Figures 8a and 8b depict the cloaking factor inline image as function of the angle of incidence ϕi for f = 2 GHz, h = 2w, and for superstrate permittivity and permeability varying as in Figure 5. The conclusions derived from Figure 8 are quite similar with those of Figure 5 in the sense that the cloaking factor values generally decrease for decreasing ϵ2 and increasing μ2.

Figure 8.

Cloaking factor as function of the angle of incidence ϕi for f = 2 GHz, h = 2w, with (a) ϵ2= 0.5 (dotted red line), 1 (dashed-dotted green line), 1.5 (dashed blue line), 2 (solid black line) for constantμ2 = 8.5, and (b) μ2= 1 (dotted red line), 2 (dashed-dotted green line), 3 (dashed blue line), 4 (solid black line) for constantϵ2 = 0.5.

[45] Finally, Figures 9a and 9b depict the device reaction as function of the observation azimuthal angle ϕ for ϕi = 45°, with (a) f = 2 GHz, h = 2w, and no superstrate (ϵ2 = μ2 = 1), arbitrary superstrate (ϵ2 = 12.56, μ2 = 13.78), optimal superstrate (ϵ2 = 0.26, μ2 = 8.35), (b) f = 3 GHz, h = 3w, and no superstrate, arbitrary superstrate, optimal superstrate (ϵ2 = 0.01, μ2 = 0.28). Note that in the cases of no superstrate and of an arbitrary superstrate the values of the device reaction are very close and its variations (oscillations) are very similar. However, in the case that the parameters of the superstrate are optimized, by following the aforementioned procedure based on contour plots as those of Figure 3, then the device reaction values are significantly reduced, while at certain distinct scattering angles these values even approach zero.

Figure 9.

Device reaction versus the angle ϕ for ϕi = 45°, with (a) f = 2 GHz, h = 2w, and no superstrate (ϵ2 = μ2 = 1) (dashed black line), arbitrary superstrate (ϵ2 = 12.56, μ2 = 13.78) (dashed-dotted blue line), optimal superstrate (ϵ2 = 0.26, μ2 = 8.35) (solid red line), (b) f = 3 GHz, h = 3w, and no superstrate (dashed black line), arbitrary superstrate (dashed-dotted blue line), optimal superstrate (ϵ2 = 0.01, μ2 = 0.28) (solid red line).

5. Conclusions and Discussion

[46] We utilized a semi-analytical integral equation method to analyze the scattering characteristics of an infinite two-dimensional planar microstrip antenna, excited by an incident plane wave. The main focus was given at the investigation of the device's potential operation as a low-profile receiving antenna, by suitably altering the physical and geometrical parameters of the cloaking superstrate. Numerical simulations were presented concerning the achieved low-profile features of the receiving antenna. It was demonstrated that for specific cloaking material parameters a significant scattered field reduction of the order of 25% is achieved, while simultaneously the received signal of the antenna (equivalent current on the strip) still remains sensible and detectable. Importantly, we note that the material values achieving this field reduction correspond to a superstrate filled with anϵ-near-zero material or a low-index metamaterial. Optimized superstrate's parameters were also reported for which the device reaction values are considerably reduced, even approaching zero at distinct scattering angles.

[47] Interesting future work directions concern the conduction of a systematic investigation of the optimization of the cloaking superstrate's parameters in order to achieve the smallest possible cloaking factor and device reaction values for a given microstrip antenna.

[48] Since it is found that very low values of permittivity or refractive index are favorable to low cloaking factors, it is worth to discuss how can materials with such parameters be realized. Indeed, the experimental fabrication of metamaterials with low index for a specific band is a topic of high impact which has attracted recently a growing scientific interest. For example, in the work by Huang et al. [2011]it is demonstrated that, in the microwave regime, certain dielectric photonic crystals with reasonable dielectric constants manipulate waves as if they had near-zero refractive indices. On the other hand, in the work byAdams et al. [2011]an inclusion-free ENZ material layer, operating at optical frequencies, is experimentally realized and its performance is characterized. Finally, in the work byRizza et al. [2011] the homogenization of a nanolaminate with metal and dielectric constituents is experimentally investigated and it is reported that the permittivity is described by an improved approach, which allows tailoring the geometry to achieve the ENZ condition at a visible wavelength.

Acknowledgments

[49] The authors would like to thank sincerely the Associate Editor as well as both reviewers for their constructive comments.

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