Development of a Wideband Bi-layered Mantle-Cloak for Perfect Electric Conductor Cylindrical Objects under Obliquely Incident Plane Wave

This paper presents a mantle cloak structure to provide a wideband capability to hide a metallic cylinder over a broad span of angles of obliquely impinging plane waves. A bi-layered dielectric structure and a sheet of metasurface are used to enclose the object to be hidden. At first, the scattered field, including co-and cross-polarized components, is analytically formulated using circumferential waves for a multi-layer wrap around the cylinder. A tensor description is also applied to model the impedance of the metasurface for TE and TM waves. Then, a metasurface made of rectangular patches is considered


INTRODUCTION 1
New advances in metamaterial technology propose the feasibility of electromagnetics invisibility, which mainly relies on the complex interaction between waves and artificial materials [1][2][3].To this end, Pendry et al. [4] introduced the Coordinate Transform (CT) method, considered one of the most effective techniques to achieve invisibility [5], which relies on the concept of the curving electromagnetic waves around the object performed by employing a special coating to create an enclosed environment, which ensures that the illuminated waves move around the object to avoid reflecting, interpreting the interior space of the container as virtually invisible.By this method, an ideal conversion-based coating could completely separate the coated object, making it appear that the coating and inner area do not exist in terms of electromagnetic properties [6].
As of late, there have been significant improvements in making metamaterials.In this regard, experimental evidence has validated the application of the CT technique to microwave bands.Numerous efforts have been made to apply this method to visible light *Corresponding author email: neshat@um.ac.ir (M.H. Neshati) frequencies [1,7,8].Plasmonic cloaking is another helpful technique to achieve invisibility using the property of scattering elimination using metamaterials with a low electrical permeability coefficient [9][10][11][12][13] which creates an anti-phase scattering wave, reducing the scattered fields and eventually concealing the object.
Other unconventional techniques have been investigated in the literature in covering a sphere to be invisible.The limited size of the applied metamaterial coating results in an inherent need for a certain thickness.However, increasing the thickness of the layer is not suitable due to reducing bandwidth and increasing sensitivity.The plasmonic cloaking method requires thinner layers than the conversion-based approach and typically requires a finite thickness to function correctly [14][15][16].
A new masking technique based on the concept of mantle cloak has been proposed to address this issue by decreasing the Radar Cross Section (RCS) for various shaped objects, including planar, cylindrical, elliptical, and spheroidal objects at microwave and lower band of terahertz (THz) frequencies [14,15,[17][18][19][20].This technique relies on a patterned conductive thin surface coating, known as a metasurface, to diminish the scattered components appropriately.By calculating the parameters of the metasurface, an optimal surface reactance is obtained to effectively suppress the object's scattered fields, resulting in its invisibility.
Most published research on cylindrical objects focuses on obtaining cloaking for TE and TM plane waves illuminated perpendicular to the PEC cylinders.In contrast, electromagnetic cloaking in the case of obliquely illuminated waves is still a challenging topic, that demands further investigation [21].Most investigations have examined the performance of isotropic mantle cloaks, designed to be inductive or capacitive in one or two dimensions.Nevertheless, in practical applications, reducing the scattered field by cylindrical objects with thicknesses comparable to the wavelength of the incident wave is often preferable.This reduction should be satisfied over a wide range of polarization states.
When an obliquely incident electromagnetic wave illuminates a PEC cylinder, coupling polarization occurs, leading to an increase in the total RCS of the structure, which is generally undesirable.In addition, the cloaking frequency range is slightly shifted, which is also a challenging task.It should be pointed out that while an obliquely incident wave exposes a bare PEC cylinder, the scattered fields are not coupled.In contrast, the crosspolarized coupling is introduced for a dielectric or coated PEC cylinder using dielectric layers [22].As a result, scattering, and consequently acquiring invisibility is an exciting subject [23].Moreover, covering these objects over a wide range of angles of the incident wave is another exciting issue due to existing of the crosscoupling.
In a few studies, graphene has been applied as a mantle-cloaked material for dielectric cylinders with low permittivity at lower frequencies of the tera-hertz band [24,25].Nonetheless, the authors facilitated their analytical explanation by bypassing the cross-polarized components of the scattered fields.This simplification is necessary due to the cylinder's small size and low permittivity.A comparable level of simplification was considered by Bilotti et al. [26].Additionally, a singlelayered mantle cloak, possessing specific surface reactance, simply suppresses one or, at most, a limited number of orders of the scattered field, whereas using multi-layered coating introduces a few benefits, including the ability to suppress higher orders of the scattered fields, facilitating multi-band and wideband invisibility.Moreover, the range of invisibility angle is widened, and more importantly, cloaking feasibility for TM and TE polarized waves.
In our previous study [21], we thoroughly addressed and validated the analytical formulation of the scattering issue for a metallic cylinder enlightened obliquely by TM and TE polarized wave.A single-layered mantle cloak is also designed and optimized to suppress the scattered components by considering the co-and cross-polarized terms.
In this paper, the preliminary step involves expressing the scattered fields concerning co-and cross-polarized components originating from the PEC cylinder coated with multiple layers of dielectric material and subjected to the oblique incidence of a plane wave.Afterward, a numerical study is accomplished using CST EM simulator [27] to evaluate the accuracy of the derived results for a bi-layered structure.Moreover, further investigations are directed to develop upon the findings for a bi-layered mantel cloak to lessen the RCS of the metallic cylinders over a wide frequency range.

1. Single-Layered Mantel Cloak
Figure 1 shows a single-layered mantel cloak, in which a metallic cylinder with a radius of a is covered by a dielectric layer with a thickness of between a and ad.In addition, a metasurface is positioned directly on top of the spacer.The mantle cloak impedance is formulated by a tensor consisting of off-diagonal terms, in which the crosspolarized components are considered.These factors are mathematically represented by Equation ( 1) to model the metasurface impedance.
An oblique incident wave is illuminated on the cylinder by the angle of φᵢ and inclination angle of θᵢ as shown in Figure 1.For a TM wave, the cylindrical harmonics of the electric field are given by Equation (2) [28].
Using Jacobi-Anger expansion [28], electric fields inside the dielectric layer and its associated cross-polarized
1 cos Using impedance boundary conditions at the metasurface with a radius of d a  = , the unknown coefficients of the mentioned equations are calculated.This is involved to satisfy the continuity of the tangential components of the electric fields.In addition, the discontinuity of the tangential terms of the magnetic field by the total induced current on the metasurface is the other condition.In addition, the continuity of the tangential components of electric and magnetic fields have to be satisfied on the planar surface, and the absence of tangential electric fields on the surface of the cylinder.The TM

2D
 of the structure is obtained by summation of the co-and cross-polarized terms of the scatted fields, calculated by the asymptotic forms of Hankel functions and their derivatives [22].For a TM wave, TM 2D  is obtained using Equations (5a) to (5c).

Multi-Layered Mantel Cloak
To develop the concept of a mantle cloak using a multi-layered structure, the scattering formulation is extended for nonelectrically small objects over a wide frequency range.In this instance, the focus is directed towards an arrangement featuring an infinitely extended cylinder of a radius of a, which is covered by N conformal anisotropic metasurfaces, which exhibit an effective surface impedance tensor equivalent to The metasurfaces' radii are denoted by Equation (6b).
, , 1, 2, , In addition, the object under consideration is subjected to an external obliquely incident plane wave, with the electric field oriented parallel to the cylinder axis, known as TM, or perpendicular to it, designated as TE wave.The electromagnetic formulation conducted in the case of mentioned scenario is comparable to that of the singlelayered mantle cloak.The only difference is the existence of an additional spatial term (between two metasurfaces) in which the electric field d z E requires an accurate description.Equation (7a) is used for the co-polarized components.
Moreover, Equation (7b) is given for the crosspolarization terms.

A B Ci Di
Ei and cr n Fi .

3. Bi-Layered Mantel Cloak
As a specific example of a multi-layered mantel cloak, a bi-layered structure is considered in this section, which is shown in Figure 2. Considering a TM-polarized wave illuminates the object under investigation, the impinged, scattered, and transferred electric and magnetic fields are denoted by Equations (8a) to (8d).
The cross-polarized terms are also provided by Equations (9a) to (9c).
It should be noted that using the duality theorem, the scattered fields for a TE plane wave are derived using the same method as that of the TM one [21].Accordinghly, ten equations are obtained, which can be solved by applying boundary conditions, leading to finding scattering coefficients at different boundaries.Then,

1. Single-layered Mantel Cloak
To design a single-layered cloak, a metallic cylinder is considered in our investigation with a length and radius of 10

2. Bi-Layered Mantel Cloak
To evaluate the impact of the radius cylinder on the scattered fields, a TM  = fields, while for cylinders with a radius of nearly o, free space wavelength, at least up to the eight order of the scattering modes affect the co-polarized terms.Due to the ability of a metasurface to manipulate a single scattering mode, it is now possible to achieve cloaking for the objects that satisfy the quasi-static condition, in which the cylinder radius satisfies r  0 /10 [Ergin, #1].As can be appreciated from Figure 5, depending on the covering radius, there exists the possibility to obtain a minimum scattering width for a cylinder at a specific magnitude of the surface reactance of the cloak, denoted by Xs.The minimum scattering width is linked to significantly reducing the object's visibility.Nonetheless, dielectric objects require a positive surface reactance, which is an inductive surface, while in the case of metallic objects, they require a sheet of capacitive reactance.
It is worth saying that the application of the singlelayered mantle cloaking in reducing the total RCS is only limited to small radius cylinders relative to the operating free-space wavelength over a narrow bandwidth.However, for large objects multi-layered covering structure is proposed to widen available bandwidth [18].
Because the examination of metallic cylinders is significantly more pronounced when interacting with a TM wave in comparison with that of a TE one, in this paper, a TM wave at 3 GHz illuminates a limited length of a cylinder with a radius of 0. .To design the required metasurface providing the mentioned values of reactance, a quadrilateral capacitive patch is applied as shown in Figure 6.This structure provides two distinct periodicity parameters including Dz and Dφ, which are optimized to satisfy the required situation.Initially, Equation ( 10) is used to estimate TM S X , in which c and o  represent light speed and inherent impedance of free space, respectively.Table 1 summarizes the optimal parameters of the single-and bi-layered designed mantle cloaks.In this regard, in our optimization process, initially, we assume to have geometrical parameters obtained from analytical formula.Then, we run the optimizer in CST MWS to achieve a minimum total RCS.Notably, the frequency range in which the invisibility is achieved using the bi-layered structure is nearly doubled compared to that of the single-layered cloak.Figure 7 also confirms that, at 3 GHz, the bi-layered mantle cloak provides 1 GHz bandwidth, corresponding to a fractional bandwidth of 33%.So, the bi-layered metasurface can significantly reduce scattered fields over a wideband frequency range, interpreting the object as invisible.The photos of the electric field for the cloaked-and bared cylinder are shown in Figure 8, which confirms that the field distribution is strongly disturbed in the case of the cloaked cylinder.However, by covering the metallic cylinder using the designed metasurface, the wavefronts for the cloaked-and bared-cylinder is illustrated in Figure 8.It can be concluded that field distribution is powerfully disturbed in the case of the cloaked cylinder.However, by covering the metallic cylinder using the designed metasurface, the wavefronts rotate around the object leading to become invisible.The simulated patterns of the total RCS for the cloaked and uncloaked cylinder are depicted in Figure 9 versus φ in the observation angle of Table 2. summarizes a comparison between published works in the literature and the proposed technique in terms of cloaking performance and bandwidth.As can be seen, the introduced bi-layeedr rectangular patch metasurfaces with 33% bandwidth at 3 GHz provides better performance than the other published works wich represent the robustness of the  proposed methode to increase the cloaking bandwidth while an obliquely incident wave is impinged on a PEC cylinder.

CONCLUSION
This paper uses the established mantle cloak technique to study electromagnetic invisibility for a metallic cylinder illuminated by a broad range of obliquely incidence angles for both TM and TE waves.It is shown that the scattered fields provide co-and cross-polarized terms due to obliquely impinging waves.To settle the issue and consider these two components simultaneously, a metasurface sheet impedance is used, which is modeled by a tensor model for both waves.The metasurface is made by rectangular-patch, which provides capacitive reactance to diminish the scattered fields and significantly decrease the total RCS leading to hiding the object from the incoming wave.The systematic formulation is verified using numerical investigation by CST software.To widen the available bandwidth, a bilayered mantle cloak is also designed, resulting in nearly 33% cloaking bandwidth, which is twice that of the single-layered mantle clock at 3 GHz for a PEC cylinder with a radius of 10 mm.

Figure 1 .
Figure 1.Single-layered mantel cloak using metasurface to cover the PEC cylinder illuminated obliquely by a plane wave

Figure 2 .
(a) bi-layered mantel cloak to invisible a dielectriccoated cylinder under oblique incidence wave, (b) side view and the definition of the incident and scattered angle using the obtained coefficients, including co n A and cr n B , the total RCS of the object is calculated.


, respectively, and o is the free space wavelength at 3 GHz.The cylinder is illuminated by an obliquely oriented TM wave at 3 GHz with   = 60 ∘ and the polarization plane is perpendicular to the propagation direction.Figure3depicts the simulated scattering width contours versus chromaticity based on a relative radius of d aa and the vital metasurface impedance, TM S X , for the dielectric constant of 4 r  = .Moreover, it can be concluded that TM 50 on the covering radius, obtaining the minimum scattering width for a cylinder at a specific value of the surface reactance, denoted by Xs, is feasible.The aforementioned minimum scattering width significantly redusces the object's visibility, thereby establishing cloaking effectiveness.

Figure 4
shows the required impedance of the applied metasurface for the mentioned metallic cylinder, whereas the dielectric constant of the spacer is 4

=
at i = 60 and the required mantle cloak is designed to obtain low total RCS.As discussed in section II, using a single-or bilayered dielectric covering with a dielectric constant of 4, which provides capacitive reactance, the required reactance are

Figure 7 Figure 6 .
Figure7shows the total RCS of the single-and bi-layered mantel cloak.It can be seen that invisibility is obtained for a wide range of i using the optimized values of the relative thickness ad/a and spacer permittivity,

Figure 7 .
Figure 7.The total RCS for the single-and bi-layered mantel cloak versus frequency the designed bi-layered metasurface as mantle cloak significantly reduces the scattered fields.Also, it indicates that in all observation angles the cloak is efficient.

Figure 8 .Figure 9 .
Figure 8.The snapshots of the normalized electric fields of the cloaked and uncloaked cylinder for 60 i  =

TABLE 1 .
The calculated and refined optimal parameters for single-and bi-layered mantle cloaks

TABLE 2 .
The comparsion of cloaking and bandwidth performaance with published works