Electromagnetic cloak to restore the antenna radiation patterns affected by nearby scatter Open Access

Very high frequency (VHF) radio wave communications used in variety of applications such as railways, maritime and air traffic control, space research, as well as for VHF radio and television could be strongly perturbed by complicated environmental conditions. The objects located near an electromagnetic source that could not be moved away due to technical restrictions can further damage the signal as a results of undesirable scattering. Even a subwavelength scatter located in a close proximity of the source could strongly affect its far-field radiation patterns. Therefore, in order to remediate the far-field radiation of electromagnetic source one needs “to hide” the obstacle from the source. The concept of electromagnetic invisibility cloak recently developed in metamaterial science^1–5 has trigged a great amount of research aimed to reduce or eliminate an unwanted scattering from an obstacle. Actually, one can distinguish four principal types of electromagnetic cloak. First, the transformation optics technique allows to reroute the incident field around an obstacle making it invisible.^1,6 Such an elegant theoretical finding is working for any monochromatic incident plane wave in any frequency range. However, it often requires extensive engineering to approximate the exotic parameters necessary for anisotropic and inhomogeneous metamaterial shell. Additional drawback of this approach is the dispersion of material parameters that renders the “transformation optics” cloak non ideal for the pulse propagation.⁷ 

An additional approach “to hide” an object in microwave frequency range is based on transmission-line network.⁸ It does not require the exotic material parameters. Instead, the cloak itself is a carefully designed three-dimensional mesh of metal or other materials that mimics the wave propagation in free space. Here, the transmission-line network should be impedance-matched to free space in order to efficiently couple the incident electromagnetic wave to the cloak. Ideally, the fields are confined inside the transmission lines allowing to hide any object placed in the space between adjacent sections of transmission lines.

In the third approach, inspired by the cloak for acoustic waves,⁹ the active electromagnetic cloak is based on introducing the discontinuity in the field at the boundary of an obstacle to cancel its scattering field.¹⁰ The field discontinuity is achieved by locating the active sources at the object boundary. The collective source radiation mimics the electromagnetic field opposite to the scattering one as a result of constructive and destructive interference. Here, the knowledge about the scattering field is required for engineering the electromagnetic contribution (phase and amplitude) of each source. A priory, such type of cloak is not limited to a monochromatic plane wave excitation and has been recently realized for a point-source-like excitation.¹⁰ 

Usually, for each electromagnetic invisibility concept mentioned above the practical implementation is rather cumbersome^8,10,11 that hampers its application. The forth alternative approach to achieve electromagnetic invisibility is the concept based on scattering cancelation firstly developed in optics^2,12 and experimentally realized in the microwave region¹³ for plane wave excitation. It has been recently shown that the scattering from spherical and cylindrical objects in microwave can be significantly reduced with use of ultrathin metasurface covering the obstacle.^14–16 For such type of cloak, called mantle cloak, the incident field excites the conduction currents, which, in turn, reradiate field in antiphase with the field scattered from the obstacle. Therefore, the scattering cancelation is the result of destructive interference. Such a metasurface can be described by a low-profile surface with thoroughly designed surface impedance. Based on scattering cancelation the cloak can be easily scaled to any, a priory, frequency range, but remains limited to a moderately sized object compared to electromagnetic radiation wavelength. Despite disadvantage mentioned above, the possibility to design a cloak of ultra-thin thickness could be attractive for practical use.

One of the promising application of electromagnetic cloak is the reduction of antenna’s blockage caused by the scattering of antenna radiation from the nearby obstacles. Recently in Ref. 17 the transmission-line cloak has been proposed in order to significantly reduce the scattering from metallic cylinders in the vicinity of a directive horn antenna. In Ref. 18 it has been demonstrated theoretically that the radiation patterns of infinitely long electric source perturbed by an infinitely long cylindrical obstacle can be completely restored with use of mantle cloak. In Refs. 19 and 20 the concept of mantle cloak has be successfully applied to simulate the reduction of mutual blockage between neighboring dipole antennas. However, there is no studies, to the best of our knowledge, that demonstrate that electromagnetic cloak can be applied for the monopole antennas affected by a nearby scatter on a finite ground plane. This type of antennas is widely used in VHF frequency range for variety of applications in public and military services. However, its performance is often perturbed in complicate environment. The use of cloak for such type of antennas is not straightforward since the ground plane, that is inevitable part in antenna design, can produces the secondary scattering that should be canceled together with the wave initially scattered by the obstacle. Moreover, in particular applications sometimes it is required that the antenna should be tilted from its vertical.

The goal of this paper is to verify the applicability of the mantle cloak concept for VHF monopole antenna. First, we demonstrate how a subwavelength scatter located in a close proximity of antenna could block its radiation. Then, we redo the simplified model of the plane wave scattering on an infinite cylindrical scatter^14,15 in order to unreveal the resonant nature of the invisibility provided by the mantle cloak that we aim to employ. Next, inspired by the idea of practical use we simulate a real scenario taking into account a finite size of an antenna and scatter as well as the presence of finite ground plane. We also consider the situation when the antenna is inclined in respect to the ground plane. Finally, we studied the problem of impedance matching for complete radiating system (i.e. antenna, obstacle, and the cloak) that is one of the fundamental tasks in getting an antenna to radiate. Actually, the cloaked obstacle can modify the impedance of an antenna as well as its frequency bandwidth. Therefore, we complete our study by analyzing how the cloak affects these antenna characteristics.

Let us first model the situation that is close to realistic antenna radiation experiment where the cloaking could be highly beneficial: a scatter is placed in the immediate vicinity of an antenna. The scenario we simulate is as following. A monopole antenna with the length l_a such as l_a ≃ λ/4 is mounted on a finite ground plan with the radius R together with a subwavelength size obstacle that displaced from the antenna by the subwavelength distance t < λ/5 (see Fig. 1). The obstacle is a perfect electric conductor (PEC) cylinder with radius r ≪ λ and finite length l < λ/2. The cylinder is located at a ground plane and affects the radiation patterns of antenna. To stress the influence of a ground plane we simulate the antenna emission, when the antenna is rotated at an angle θ from the z axis.

First, let us analyze the far-field radiation patterns of antenna for the different antenna’s rotation angle θ (see Fig. 1). The calculations have been performed with use of the commercial software COMSOL Multiphysics²¹ based on the finite element method. The entire modeling domain is bounded by a perfectly matched layer (PML) which acts like an anechoic chamber absorbing all radiated energy and guarantees the outgoing wave boundary conditions. In our calculations the monopole antenna is the classical quarter-wave whip, i.e. l_a ≃ λ/4, modeled by a thin metallic cylinder with a lump port excitation. The antenna is matched to 25 Ohm.

For the reference we first calculate the radiation patterns of a single antenna on a finite-sized ground plane for θ = 0 (see Fig. 2(a). The radiation patterns of such antenna is omnidirectional with a peak of radiation slightly moved from xy plane due to the size effect of the ground plane.²² Once a PEC cylindrical scatter is present (Fig. 2(b)) we observe that the far-field radiation patterns of antenna are strongly damaged in the x direction, since the electromagnetic field is scattered by a cylinder. The ground plane also modifies the antenna radiation pattern especially if antenna is inclined (Fig. 2(c)) since the fields are scattered by the both obstacle and ground plane. For example, if the antenna is tilted towards the cylinder (θ = 30^o) the radiation patterns are damaged the most significantly: from the E- and H-planes one can see that the antenna’s gain along x direction is reduced by more then 10dBi. Note, that without scatter the radiation patterns of vertical as well inclined antennas are nearly identical (cf. also Figs. 7).

From Fig. 2 one can deduce that it is highly desirable to put the invisibility concept to practical use especially for antennas applications. Before applying the cloak that should hide a cylinder from the antenna let us first consider a model when the infinite PEC cylinder is cloaked from the incident plane wave. Such a strongly simplified model allows us to obtain the analytical solution of the problem and demonstrate the resonant nature of invisibility. While the model derivation can be found in literature^14,15 we decided to redo it in the next section for the reader convenience: the results obtained in the framework of this simplified model we use further for simulation of monopole antenna radiation in the presence of finite cloaked cylinder.

We consider the scattering of plane electromagnetic wave by PEC cylinder infinite in z-direction. The geometry of the problem is depicted in Fig. 3. We introduce a cloak by an impedance surface with effective surface impedance Z_s located at thin dielectric coverture with dielectric permittivity ε. With the knowledge of the main technical requirements on the cloak size and weight, we consider a coverture that has ultra-thin thickness such as b − a < a/4, where a and b are the inner and outer radius of the cloak cylinder. We exploit the concept of the cloak based on scattering cancelation with an objective to hide the cylinder and simultaneously keep the ultra-thin size of the cloak. In our model the cloak size is 25% of an obstacle, which we used for a sake of graphic representation of results.

The structure is illuminated by a TM plane wave with electric field parallel to z axis. We consider here only such type of polarization since it’s the closest to our final scenario aimed for antenna application. Benefiting from cylindrical symmetry of an obstacle the incident electric and magnetic field can be expressed as in terms of an infinite Fourier-Bessel series²³ as:

while the scattered fields consisting of out-going waves can be expanded in terms of of Hankel functions as

Here in all these expressions J_n(x), $H n ( 1 ) ( x )$ and $H n ( 2 ) ( x )$ are the Bessel function of the first kind, and the Hankel functions of the first and second kinds, respectively. The prime denotes the derivative with respect to the argument. E₀ is the amplitude of incident field, k₀ = 2πf/c and $k=2πf ε /c$ are wavevectors in a vacuum and substrate, c is the speed of light, f is the frequency, η₀ is the free space impedance. The field in a dielectric coverture can be represented as superposition of in-coming and out-going waves

where η is the wave impedance of the dielectric coverture with permittivity ε. Within the PEC cylinder the electric field is zero.

The unknown coefficients can be determined by using appropriate boundary conditions at the interfaces ρ = a and ρ = b. Since the substrate is covered by an impedance surface with effective surface impedance Z_s we apply the impedance boundary conditions at the interface ρ = b:

while at the interface ρ = a the total field is zero:

With Eqs. (1)-(6) the boundary conditions (7)-(9) result in a system of linear equations. By solving it we obtain the expression for scattering coefficient A_n in the form:

where $α= H n ( 2 ) ( kb ) P n + H n ( 1 ) ( kb )$⁠, $β= [ H n ( 2 ) ( kb ) ] ′ P n + [ H n ( 1 ) ( kb ) ] ′$⁠, and $P n =− H n ( 1 ) ( k a ) / H n ( 2 ) ( k a )$⁠. The effective surface impedance Z_s can be written as Z_s = R_s + iX_s, where R_s is the effective surface resistance and X_s is the effective surface reactance that could be either inductive (X_s > 0) or capacitive (X_s < 0).

The scattering coefficient A_n determines the the total scattering cross section σ that is defined as a ration of a total power scattered by a cloaked cylinder to the incident power per unit length on the scatter:

For the plane wave incidence the overall visibility of a cloaked cylinder can be judged from the the quantity of total scattering cross section. Thus, if the scattering coefficient A_n is zero, then σ = 0 that corresponds to a total invisibility of an obstacle.

An additional characteristic to analyze an object scattering is the backscattering cross section σ_B, which is defined as the ratio of the power scattered in the direction toward the source to the incident power per unit length on the scatter. When the cylindrical wave decomposition is used for a scattered field (see Eq. (3),(4)) we can obtain

Obviously, that in case of the invisibility the backscattering cross section is zero since the invisibility presumes the zero total scattering cross section σ. We decided to add σ_B to our consideration since it is directly related to radar cross section widely used in antenna’s community.

In Fig. 4 we present the scattering and backscattering cross sections calculated for a PEC infinite cylinder as well as PEC cylinder cloaked by an impedance surface mounted on a dielectric coverture with dielectric function ε. In our calculations we took ε = 10 that is typical for aluminum oxide – the material that is one of the most cost effective and widely used as electrical isolator for integrated circuits. Note, that varying the dielectric permittivity of the substrate the cloak properties discussed below can be shifted to another frequency band while keeping the main physical conclusions in this article still plausible. We consider the lossless case, i.e. R_s = 0, that is reasonable for the metal-based metamaterials in the frequency range under consideration. Therefore, the scattering and backscattering cross sections we plot as a function of effective surface reactance X_s = Im(Z_s). We accompany these results by the scattered electric field amplitudes calculated for each cylindrical harmonic. One can see that the scattering from a thin uncloaked cylinder is dominated by a zero harmonic (cf. Fig. 4), hence, in order to cloak a cylinder it is enough to minimize its amplitude. The impedance surface mounted on the subwavelength coverture above a cylinder modifies significantly the electromagnetic response of the structure. First, for a effective surface reactance around X_s = − 50 Ohm, the scattering as well as backscattering cross section vanishes manifesting nearly the total invisibility of subwavelength PEC cylinder. Note also, that for X_s < − 50 Ohm the scattering cross section of cloaked cylinder is still less then the scattering cross section of a bear PEC cylinder. Even without the total invisibility one can observe the scattering reduction that could be useful to restore the antenna radiation patterns. However, for X_s > − 50 Ohm the cloak acts in a different way: it resonantly enhances the scattering from a coated cylinder due to the resonant enhancement of higher order harmonics amplitudes at X_s = − 40.5 Ohm and X_s = − 33.4 Ohm. Similar behavior has been observed in optics for a coated plasmonic cylinder.²⁴ 

It would be also interesting to analyze the results as a function of frequency. Fig. 5 shows that for a given effective surface impedance the reduction of scattering occurs in a rather broad frequency range. Obviously, that the situation can be complicated for the lager cylinder due to non negligible contribution from the next higher-order scattering harmonics, thus the multiresonant or multilayered metasurfaces could be required.

Analyzing the scattering cross section spectra as well as the evaluation of the scattering cross section as a function of effective surface reactance X_s we notice the classical form of Fano resonance.²⁵ The asymmetric line-shape of the resonance is produced by the interference between a background state associated here with the scattering from uncloaked PEC cylinder and the resonant state associated with the resonance behavior of the zero cylindrical harmonic for a cloak cylinder. Near the resonant value of X_s (see in Fig. 4) or resonant frequency (see in Fig. 5(b)) the background scattering amplitude of the zero cylindrical harmonic varies slowly but rest significate, while the resonant scattering amplitude changes both in magnitude and phase quickly. To illustrate the properties of Fano resonance we plot in the inset to Figs.4 and 5 the phase and its derivative of the zero cylindrical harmonic. Close to the resonance we observe the phase jump and the resonant profile of its derivative. For the next cylindrical harmonic the asymmetric profile is not so pronounced due to the weak value of background scattering amplitude.

Let us come back to initial problem and analyze the radiation of monopole antenna mounted on a ground plan and perturbed by a scattering from PEC cylinder located nearby. This time we cloak the cylinder by an effective impedance surface. Note, that as compared to the problem of plane wave scattering by an infinite cylinder considered in a previous section, the problem of antenna radiation that we tackle here differs significantly. The cylinder has a finite length and the both, monopole antenna and obstacle are mounted on a ground plane. Let us analyze if the resonances of impedance surface discussed in previous section can be of practical use. In particular, it has been recently proposed to exploit the higher order resonances of the metasurface to booster the radiation from the source.¹⁸ In Fig. 6 we plot the reflection coefficient at the input port of antenna S11 as a function of effective surface reactance X_s. First, we check that the unperturbed monopole antenna is perfectly adapted and demonstrates the reflection less then -30dB at the resonance frequency. The presence of the obstacle slightly disturbs the antenna matching and increases the reflection up to -17dB. In case of the cloak we observe a series of resonances, namely at X_s = − 38 Ohm and X_s = − 31 Ohm, the maximums of which correspond with an accuracy about 6% and 8% to the resonances of the first and the second cylindrical harmonics obtained from analytical model (cf. Fig. 4). At that the antenna is adapted only at X_s < − 45 Ohm with reflection coefficient S11 less then -10dB. Therefore, we conclude that the resonances of the higher order harmonics can be hardly of use for a monopole antenna under the study. While for the problem of the plane-wave scattering on the infinite cloaked cylinder the resonances of the higher order cylindrical harmonics provides the enhanced scattering, in case of antenna radiation they also increase the reflection to the input port making the radiating system ill-adapted.

Let us now analyze the case when the impedance surface reduces strongly the scattering from the cylinder. From the analytical study presented in pervious section we know that the invisibility is expected at X_s = − 50 Ohm that corresponds to the zero of the fundamental cylindrical mode. We also checked (cf. Fig. 6) that at X_s = − 50 Ohm our radiating system is still adapted with the reflection coefficient S11 less then -15dB. Now, let us analyze the radiation patterns and check if the mantle cloak could restore the monopole antenna radiation patterns. In Fig. 7 we plot radiation patterns in the both E- and H-planes for different inclinations of antenna. We observe that the radiation of inclined unperturbed antenna experiences only the subtle changes. However, the obstacle strongly damages the antenna radiations and this phenomenon is the most pronounced when the antenna is inclined towards the cylinder (cf. Fig. 7 θ = 30^o). We “hide” the obstacle by the impedance surface with effective surface impedance X_s = − 50 Ohm. We observe that for each angle of inclination the radiation patterns of the antenna in H-plane are mostly restored while they remains slightly perturbed in E-plane. The most striking results demonstrating the cloak efficiency is obtained for θ = 30^o.

In order to emphasis the practical applicability of the discussed phenomena we replace in our analysis the hypothetical impedance surface by the real metamaterial accessible with the standard printed circuit board (PCB) lithography. To design the cloak we chose a metasurface composed of 2D arrays of metallic rods arranged in a form of Jerusalem crosses (JCs) (see Fig. 8). The metallic rods have been model as a PEC that is a reasonable approximation in VHF frequency range. In order to simulate the required effective impedance Z_s as a first approximation we used the analytical expression for surface impedance given in the appendix to Ref. 16. Then, we performed a numerical optimization procedure to obtain the final parameters of the structure given in the caption to Fig. 8. Please note that Z_s is highly sensitive to the effective dielectric function (1 + ε)/2 (cf. Ref. 16), nevertheless we decided to keep the permittivity constant in our study (ε = 10) giving preference to existing materials. In Fig. 8 we demonstrate that the radiation patterns of a monopole antenna perturbed by a cylindrical obstacle located in its near-field can be strongly improved. However, the antenna radiation is not completely restored since the cylinder rests slightly visible. Here, the metasurface reproduce a required effective surface impedance owing to electric currents induced in metallic rods, that in turn re-emit the electromagnetic wave with phase adjusted in a way to reduce the scattering. In order to visualize further the performance of the cloak we present in Fig. 9 the near-field calculated for the case of the monopole antenna along, for the antenna perturbed by an obstacle and the antenna with a cloaked obstacle. We observe that at the distances from the antenna that are larger then λ/2 the near-fields of antenna in H-plane is completely restored. We should notice here, that in order to simulate the required effective impedance we tested a rather complicated structure based on JCs, while the other type of array can be found in literature.¹⁶ The reason is that such array is much more technologically feasible in VHF range in comparison to other simple configuration like an arrays of metallic patches or cross dipoles.¹⁶ For example, for an array of metallic patches that is much easier to design,¹⁶ the required effective reactance can be obtained only for the extremely short patch-to-patch distances making them too challenging for the fabrication and computation. In particular, for an array that consists of six metallic patches along the circumference of the cylinder of radius r = 20 cm the patch-to-patch distance should be about 0.1 mm that is 2.7 × 10⁴ times less then wavelength for f = 108 MHz.

Finally, we consider the problem of impedance matching for monopole antenna. Actually, the cloaked obstacle can modify the impedance of the antenna as well as its frequency bandwidth.¹⁹ The impedance matching is one of the fundamental tasks in getting an antenna to radiate. In Fig. 10 we plot the the reflection coefficient S11 at the input port and the input impedance Z as a function of frequency. First, we observe that the unperturbed monopole antenna is perfectly matched to the impedance of 25 Ohm and demonstrates the reflection less then -10 dB in a frequency band from 104.5 MHz to 113 MHz. Once the antenna is perturbed by a PEC cylinder the adaptation becomes slightly worse due to the undesired parasitic scattering from the obstacle. We observe that the resonance of antenna as well as its frequency band shift to the lower frequency. At that, the bandwidth of perturbed antenna is increased by 10%. Once the obstacle is cloked by the surface with required effective impedance this shift is larger while the reflection is stronger. Nevertheless, even in the case of the cloak based on JC metasurface the antenna rests adapted providing the reflection coefficient at the input port S11 less then -10dB in bandwidth from 102 MHz to 109.5 MHz. Therefore, the complete system (antenna, obstacle and cloak) is still operating in the bandwidth that is narrowed only by 11%. It is interesting to note that the input impedance is perturbed only slightly in the presence of the obstacle and impedance surface. However, for real metasurface based on periodic arrangement of metallic rods the impedance strongly changes for the frequencies that are above the frequency corresponding to invisibility. Obviously, that the effect of invisibility is resonant effect and nearly total invisibility is observed only for the particular frequency (see Sec. II). Nevertheless, we found that the use of the cloak is always beneficial in comparison with uncloaked obstacle within the bandwidth of the radiating system. To demonstrate this we plotted the radiation patterns that corresponds to the bandwidth limits 102 MHz and 109 MHz (Fig. 9(d), 9(e)).

In this work we have used an analytical scattering approach as well as the rigorous numerical solution of Maxwell equations to demonstrate how the invisibility concept can be applied to the practical VHF antenna communications. As an electromagnetic source we chose a monopole VHF antenna that is widely used for variety of applications in public and military services. For an obstacle we considered a PEC cylinder. The both monopole antenna and cylinder were located at the finite ground plane and had a finite length. We have shown that a PEC cylinder can be shielded from the antenna radiation by cloaking it with impedance surface. The value of the effective impendence necessary to reach nearly total invisibility is close to the model value obtained from analytical approach considering the plane wave scattering from a infinite cloaked cylinder. We have also checked the case when monopole antenna is inclined in respect to the cylindrical scatter and the ground plane. Our results show that owing to impedance surface one can “hide” the obstacle from the antenna radiation and almost completely restore its radiation. Next, we replaced in our simulation the hypothetical impedance surface by a realistic metal-based metasurface in order to be as close as possible to realistic experimental situation. Finally, we analyzed how the cloak can modify the input impedance of antenna and its frequency band. We found that in spite of the fact that the invisibility is the resonant effect, the use of the cloak is always beneficial in the frequency band of radiating system. The obtained results could find its applications to ensure the steady VHF radio wave communications in complicated environmental conditions.

This work has been supported by the European Defence Agency (“Mimicra”) and AIRBUS Innovation Works (“Eurocopter”). The authors thank Alexandre Sellier and Gérard-Pascal Piau for stimulating discussions.