Electrically tunable mantle cloaking utilizing graphene metasurface for oblique incidence

Journal Pre-proofs Regular paper Electrically Tunable Mantle Cloaking Utilizing Graphene Metasurface for Oblique Incidence Zahra Hamzavi-Zarghani, Alireza Y. ahaghi, Ladislau M. atekovits PII: DOI: Reference:

S1434-8411(19)32222-8 https://doi.org/10.1016/j.aeue.2020.153080 AEUE 153080

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International Journal of Electronics and Communications

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2 September 2019 6 January 2020 8 January 2020

Please cite this article as: Z. Hamzavi-Zarghani, A.Y. ahaghi, L.M. atekovits, Electrically Tunable Mantle Cloaking Utilizing Graphene Metasurface for Oblique Incidence, International Journal of Electronics and Communications (2020), doi: https://doi.org/10.1016/j.aeue.2020.153080

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Electrically Tunable Mantle Cloaking Utilizing Graphene Metasurface for Oblique Incidence Zahra Hamzavi-Zarghani1,2 , Alireza Y ahaghi1 , Ladislau M atekovits2 1. School of Electrical and Computer Engineering, Shiraz University, Shiraz 71946, Iran, 2. Dipartimento di Elettronica e Telecomunicazioni, Politecnico di Torino, 10129 Torino

Abstract An analytic formulation to achieve mantle cloaking of a dielectric cylinder using scattering cancellation method for TMz polarized oblique incidence is presented. To obtain this goal, a closed form expression for the required surface impedance of the covering metasurface is derived. By properly tuning the chemical potential of graphene, its surface impedance is adjusted for the invisibility of the considered cylinder for arbitrary angles of the impinging plane wave. The scattering width reduction of 15.7dB and 21.27dB is obtained with 3-dB bandwidth of 18.7% and 10.6% at 1 THz and 1.5 THz, respectively for the incident angle of 60 degrees. Furthermore, by optimizing the surface impedance of graphene, finite cylinders can also be cloaked. The results indicate 11.4dB and 13.5dB radar cross section (RCS) reduction with 30% and 22% 3-dB bandwidth for the incident angles of 45 and 80 degrees, respectively. Full-wave simulations verify the analytical results. The effect of the cloaking is examined by comparing the input impedance of a pyramidal horn antenna when it is loaded with an array of bear and cloaked cylinders positioned in the aperture. Keywords: Graphene, Invisibility, Mantle Cloaking, Oblique Incidence. 1. Introduction Recently, researchers have shown great interest in cloaking and scattering manipulation. Several techniques have been considered to achieve invisibility (1)-(5). Transformation optics which was for the first time introduced by Pendry et al in 2006 (6) is one of the most known methods for cloaking. In

Preprint submitted to International Journal of Electronics and CommunicationsJanuary 9, 2020

this method, a metamaterial layer covers the object and guides the impinging wave around it avoiding the wave from having interaction with the object. Because of needing thick inhomogeneous and anisotropic material, this method is not very practical (7). Another technique is plasmonic cloaking based on the scattering cancellation method on which a layer with negative or near-zero relative permittivity covers the object. The layer produces an anti-phase scattered field relative to the field scattered by the object. These scattered fields cancel each other making the object invisible (8). Another technique based on the scattering cancellation method is mantle cloaking. In this method, contrary to the plasmonic cloaking which needs a thick layer, an ultra-thin metasurface covers the object which produces anti-phase scattered field and leads to cloak the object (9). Several researchers have investigated mantle cloaking in antenna applications (10)-(15). The majority of the previously reported results on cloaking were achieved with the assumption of normally incident plane wave. Indeed no closed form expression for the required surface impedance of the covering metasurface to reduce scattering of an object under oblique illumination of a plane wave has been introduced. However, in several applications objects are exposed to oblique impinging waves. In this paper, we derive an analytic formula for the required surface impedance of coating metasurface for oblique illumination for the first time. By considering the tunability characteristic of graphene (16)-(25), proper surface impedance for cloaking a dielectric cylinder under different arbitrary angles of wave illumination is achievable by adjusting the chemical potential of graphene. We obtain a high reduction of scattering for both finite and infinite dielectric cylinders which are under an oblique incidence by TMz polarized plane wave relative to the longitudinal axis. Also, the reflection coefficient of a horn antenna in the presence of some cloaked and uncloaked inclined dielectric cylinders in its aperture is presented. This paper is structured as follows: Section 2 reports the analytical formulation for investigation of a dielectric cylinder under oblique incidence and the derivation of closed form expression for the required surface impedance to obtain invisibility for a given angle of impinging plane wave. In section 3, analytical results of total scattering width (SW) of infinite cylinders are compared with results obtained by full-wave simulations. Furthermore, the scattering reduction of a finite cylinder for different incident angles by comparing the total RCS of bare and coated objects is studied. Besides, reflection coefficient of a horn antenna in presence of some cloaked and uncloaked in3

Figure 1: Dielectric cylinder under oblique incidence.

clined dielectric cylinders in its aperture is presented. In the last section, some conclusions are drawn. 2. Analytical Investigation of Oblique Incidence The structure of a dielectric cylinder coated by a graphene monolayer is shown in Fig. 1. For the sake of simplicity, in our formulation, the height of the cylinder is considered infinite. It is illuminated by a TMz polarized plane wave which is obliquely incident on the cylinder with the angle of θi relative to the z axes. The incident electric field (E i ) is described as: E i = E0 (ˆ x cos θi + zˆ sin θi )e−jβ0 (x sin θi −z cos θi )

(1)

The incident electric field (E i ), the scattered field (E s ) and the field inside the object (E in ) can be written as a summation of Bessel and Hankel functions in the cylindrical coordinate. Their z components are as follows (26): Ezi = E0 sin θi e(jβ0 z cos θi ) ∞ X j −n Jn (β0 r sin θi )ejnφ n=−∞

4

(2)

Ezs = E0 sin θs e(−jβ0 z cos θs ) ∞ X j −n Cn Hn(2) (β0 r sin θs )ejnφ

(3)

n=−∞

Ezin = E0 sin θ2 e(jβz cos θ2 ) ∞ X j −n An Jn (βr sin θ2 )ejnφ

(4)

n=−∞

where β0 and β are the wavenumbers in the free space and in the cylinder (2) and Jn and Hn are the Bessel function of the first kind and Hankel function of the second kind, respectively. θs is the scattering angle and θ2 is the angle that wave enters into the cylinder. An and Cn are the coefficients of the electric fields inside the object and that of the scattered wave, respectively which should be determined by applying boundary conditions on the surface of the cylinder at r = a. The first boundary condition enforces the continuity of tangential electric field which leads to:

sin θi

∞ X

Jn (β0 a sin θi )+sin θi

n=−∞

∞ X

Cn Hn(2) (β0 a sin θi )

n=−∞ ∞ X

= sin θ2

An Jn (βa sin θ2 )

(5)

n=−∞

The second boundary condition which relates the tangential electric and magnetic fields to the surface impedance of the covering metasurface (which is a graphene monolayer here) is written in a tensorial form, as follows: i s in in ~ ~ ~ ~ ~ (6) r × (H + H − H ) = Z s .Js rˆ × E = Z s .ˆ r=a

~ denotes the magnetic field, J~s is the current density at the surface where H and Z s is the surface impedance tensor of the metasurface (27). According to (28), cross-polarization mutual coupling for a dielectric cylinder with a small radius relative to the wavelength is very low and therefore can be neglected. Moreover, by properly designed covering metasurface, one can reduce the unwanted coupling (28) and therefore neglect the offdiagonal elements of the surface impedance matrix, i.e. Zszφ = Zsφz = 0. 5

Zszz and Zsφφ should be determined in order to be able to design the proper metasurface for the purpose of mantle cloaking of a dielectric cylinder under oblique incidence. The equation (6) transforms to two scalar equations. The first one is: Ezin = Zszz (Hφi + Hφs − Hφin )

(7)

By considering the relation between Hφ and Ez (26), Eq. (7) leads to:

sin θ2 =

∞ X

An Jn (βa sin θ2 )

n=−∞ ∞ X

Zszz [β0 J 0 (β0 a sin θi ) jwµ n=−∞ n

+β0

∞ X

Cn Hn0(2) (β0 a sin θs )

n=−∞ ∞ X

−β

An Jn0 (βa sin θ2 )]

(8)

n=−∞

Moreover, one has a second equation as in Eq. (9): Eφin = Zsφφ (Hzi + Hzs − Hzin )

(9)

Due to considering TMz polarization and neglecting cross-polarization coupling (28), the z component of the magnetic field is zero (Hz =0). On the other hand, the φ component of the electric field inside the cylinder is (26): (in)

Eφ ∞ X

= jE0

cot θ2 jβz cos θ2 e βr

nj −n+1 An Jn (βr sin θ2 )ejnφ

(10)

n=−∞

Since we are investigating mantle cloaking of a cylinder with small radius (βa  λ0 ), the first harmonic (n = 0) has a much higher contribution in the scattered field of the cylinder than the others; because of this, higherorder harmonics of scattering coefficients can be neglected (29). For n = 0, 6

it results that Eφin = 0, so Eq. (9) is true for every Zsφφ . For simplicity, we can consider Zsφφ equal to Zszz that corresponds to an isotropic metasurface, hence: Zsφφ = Zszz = Zs

(11)

By solving Eqs. (5) and (8) the scattering coefficient (Cn ) can be achieved. As already mentioned, we consider a cylinder with a small radius and therefore the first harmonic of the scattering coefficient plays the most important role in the scattered field. Therefore, by equating C0 to 0, one can significantly reduce the scattering of the object. By equating C0 to 0, we can determine Zs , the required surface impedance for cloaking a dielectric cylinder under the illumination of an oblique incident plane wave with an angle of θi . It is achieved as follows: jη0 sin θi sin θ2 J0 (β0 a sin θi )J0 (βa sin θ2 ) r sin θi J0 (β0 a sin θi )J1 (βa sin θ2 ) − sin θ2 J0 (βa sin θ2 )J1 (β0 a sin θi ) (12) Here, r is the relative permittivity of the cylinder, η is the intrinsic 0 p impedance of the air and sin θ2 = 1 − (cos θi )2 /r . The next step after finding the required surface impedance is designing a metasurface according to the Eq. (12), to cover the cylinder in order to reduce its scattering. We consider graphene monolayer for this purpose because of its simplicity (it is homogeneous and isotropic leading to less cross-polarization) and also its ability to make the structure tunable. The conductivity of graphene is modeled with Kubo formula which is the sum of interband and intraband parts (30)-(34): Zs = √

σintra

KB e2 T µc − KµcT B = −j 2 [ + 2 ln(e + 1)] π~ (w − 2jτ −1 ) KB T   je2 2|µc | − (w − jτ −1 )~ σinter = ln 4π~ 2|µc | + (w − jτ −1 )~

(13) (14)

where KB is the Boltzmann’s constant, µc is the chemical potential of graphene, e is the electron charge, τ is the relaxation time of graphene, ~ is the reduced Plank0 s constant and T is the temperature.

7

3. Results and Discussions The total scattering width (SW) of infinite bare and cloaked cylinders are investigated. Scattering width can be expressed as (26): ∞ X 4 SW = |Cn |2 β0 sin θi n=−∞

(15)

Our goal is to cloak a dielectric cylinder with the relative permittivity of r = 4 and radius of a = 15µm at the two frequencies of 1 and 1.5 THz when it is illuminated by TMz polarized plane wave with the incident angle of 60◦ . The radius of the considered cylinder at 1 THz is λ200 . According to Eq. (12) the required surface impedance is 755.6j Ω at 1 THz and 466.1j Ω at 1.5 THz. It is noted that both are positive reactances. The characteristics of the graphene are chosen to fulfill these values of the surface impedances. They are as follows: T = 300 K, τ = 1 ps, µc is 0.0625 eV at 1 THz and 0.153 eV at 1.5 THz. Figure 2 shows the numerical and analytical results for the SW of the bare and coated cylinders under the illumination of 60◦ oblique incidence. Numerical results are obtained by simulation of the cylinders with CST Microwave Studio (35), and applying (15) leads to analytic curves. It illustrates the suppression of scattering by the cloaked cylinders using the derived closed-form expression for the required surface impedance. Furthermore, it can be observed from the figure that by adjusting the chemical potential of the graphene, scattering reduction can be achieved for different frequencies. The scattering reduction at 1 THz is 15.7dB and at 1.5 THz is 21.27dB. 3-dB bandwidth is 18.7% and 10.6% at 1 THz and 1.5 THz, respectively. The figure also shows good agreement between simulation results and the ones obtained analytically. Figure 3 shows the variation of the surface reactance of the covering graphene layer for three illumination angles of 45, 60, 80 degrees and Fig. 4 shows the scattering width reduction for different incident angles. The electric field distribution for the cloaked and uncloaked cylinders with a radius of 30 µm (λ0 /10) is shown in Fig. 5. Warmer colors correspond to the higher E field values. Figures 5 (a) and (c) represent perturbation of the electric field when it exposes to the bare cylinder with the incident angle of 30◦ and 45◦ , respectively. Figures 5(b) and (d) illustrate that covering the cylinder with the designed graphene monolayer leads to a significant reduction of the scattering and therefore the illuminated plane wave passes the cylinder without high perturbation. 8

Figure 2: Scattering width of the bare and cloaked cylinders for the incident angle of 60◦ . Red: Uncloaked, Blue and Green: cloaked for 1THz and 1.5THz with different µc . Solid line: Analytical, Dashed line: Simulation. Blue: chemical potential=0.0625 eV, Green: chemical potential=0.153 eV

In the next step, we consider a finite cylinder and we prove that in spite of assuming infinite structure for our analytical formulation, the achieved closed-form expression can also be used for finite one, with further optimization. A dielectric cylinder with the relative permittivity of r = 4 and radius of a = 30µm at height of h = 240 µm (0.8 λ0 ) at 1 THz under illumination of oblique incidence with the incident angles of 45◦ and 80◦ is studied. In the literature, there is an approximate formula which relates the 3-D RCS to the 2-D one as follows (26), (36): σ3−D ' σ2−D

2h2 sin2 θi λ0

(16)

Using Eq. (16), we are able to calculate RCS of the finite cylinder and the curves for the two oblique incidences of 45◦ and 80◦ are plotted in Fig. 6. The results achieving from the simulation by the commercial software (CST) are also plotted to validate the analytical ones. The results illustrate 11.4dB and 13.5dB reduction in total RCS of the cylinder under oblique illumination with 30% and 22% 3-dB bandwidth for the incident angles of 45◦ and 80◦ , respectively. Moreover, the comparison between analytical and simulation results shows acceptable agreement. However, by increasing the height of the cylinder, the two curves will be more similar. Figure 7 represents the far-field results for the RCS of the cylinder under these two mentioned angles of incidence in polar system in 7(a) and (c) φ = 0◦ and 7(b) θ = 45◦ 9

Figure 3: Variation of surface reactance for three illumination angles of 45, 60, 80 degrees

planes and in 7(d) θ = 80◦ plane. The polar plots demonstrate that at every observation angle, the RCS of the coated cylinder is smaller with respect to the uncoated one and comparison between CST and HFSS results shows very good agreement. As the final investigation for 3-D structures, we consider an array of inclined cylinders with a radius of 30 µm placed in the aperture of a pyramidal horn antenna. The scattering cylinders produce a high reflection at the antenna input port which can have a bad effect on the horn radiation (37). In (38), it has been shown that the presence of the cylinders in front of the horn antenna weakens the radiation operation of the antenna in gain, sidelobe level and cross-polarization. In order to make this antenna0 s blockage as small as possible, we cover the cylinders with the graphene monolayer which was optimized for cloaking purposes. Figure 8 shows the numerical results of the amplitude of the reflection coefficient from the horn antenna in the presence of the bare and coated array of inclined cylinders in its aperture. The cylinders are placed with 60◦ slop in order to meet oblique incidence conditions. It can be noticed that covering the cylinders with the properly designed graphene monolayer, effectively decreases their scattering resulting in a significant reduction in the input reflection coefficient of the antenna. For biasing the graphene layer, the structure in Fig. 9 is suggested. One metallic contact is put on graphene and the other one is put on top of the cylinder acting as a ground and a voltage supply is connected to the contacts. By changing the biasing voltage, the chemical potential of the graphene can 10

Figure 4: Scattering width reduction for different incident angles

(a)

(b)

(c)

(d)

Figure 5: Electric field distribution for the (a) uncloaked and (b) cloaked cylinders with the incident angle of 30◦ and (c) uncloaked and (d) cloaked cylinders with the incident angle of 45◦

be adjusted. Moreover, fabrication of the graphene layer cloak around a cylinder is possible. Since graphene is mechanically flexible it is feasible to wrap the graphene based metasurface around each arbitrary object (39),(40). Table. 1 compares some cloaking structures in the literature. It indicates that by considering the size of the cylinder, RCS reduction and tunability feature of cloaking, our work has an acceptable performance. 4. Conclusion In this paper, we have presented an analytic approach to design a metasurface to cloak a dielectric cylinder under the oblique incidence of a TMz polar11

(a)

(b) Figure 6: Total RCS of the bare and cloaked cylinders for the incident angle of (a) 45◦ , µc = 0.165 eV and (b) 80◦ , µc = 0.175 eV. Blue: Cloaked, Red: Uncloaked, Solid line: Analytical, Dashed line: Simulation

12

(a)

(b)

(c)

(d)

Figure 7: Far field results for the RCS of the cylinder with the incident angle of (a) and (b) 45◦ and (c) and (d) 80◦ in polar system for the observation angle of (a) and (c) constant φ = 0◦ and (b) constant θ = 45◦ and (d) constant θ = 80◦ . Blue: Cloaked, Red: Uncloaked. Dashed line: CST, Solid line: HFSS

Figure 8: Reflection amplitude of the horn in presence of the uncloaked and cloaked cylinders with the slope of 60◦ .

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Figure 9: The suggested structure for biasing the cloaked cylinder.

l References RCS reduction (36) 10dB (8) 13dB (41) 18dB (42) 10dB (43) 8dB (44) 12dB our work 13.5dB

Size of the cylinder 0.1λ 0.13λ 0.08λ 0.13λ 0.13λ 0.12λ 0.1λ

Cloaking technique plasmonic cloaking adding metastrips mantle cloaking plasmonic cloaking core-shell nanoparticles graphene metasurface graphene monolayer

Table 1: Comparison of the cloaking structures

ized plane wave. A closed-form expression for the required surface impedance of the considered metasurface has been derived. A graphene monolayer was our choice as the metasurface due to its tunability characteristics. By properly adjusting the chemical potential of the graphene, the required surface impedance for cloaking the cylinder with different arbitrary incident angles of the illuminated wave can be achieved. Several scenarios have been investigated to illustrate the correct performance of our design. The results show an effective reduction of scattering by the cloaked cylinders (both finite and infinite ones). The results obtained from analytical calculations were compared to the simulation results using CST Microwave Studio. The agreement between the two results validates our approach.

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