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Journal of Computational Electronics

, Volume 17, Issue 3, pp 949–958 | Cite as

Scattering by a magnetized plasma-coated topological insulator cylinder

  • Majeed A. S. Alkanhal
  • A. Ghaffar
  • M. M. Hussan
  • Y. Khan
  • I. Ahmad
  • Q. A. Naqvi
Article

Abstract

In the present paper, the scattering characteristics of a magnetized plasma-coated topological insulator cylinder are formulated and analysed graphically. Field equations at each interface are expanded in terms of cylindrical wave vector functions by imposing extended classical wave-scattering theory. By applying the boundary conditions, scattering matrices are obtained in terms of scattered and transmitted coefficients. The obtained results are also compared with published results to display the accuracy of the present formulation under some special conditions. Changes in the bistatic echo widths of the topological insulator cylinder are also recorded by varying the anisotropic plasma parameters (i.e., the applied magnetic field, plasma density and effective collisional frequency).

Keywords

Scattering Topological insulator Cylinder Plasma 

1 Introduction

Topological insulators represent a newly discovered state of quantum matter defined by quantum field and quantum band theory [1]. Basically, topological insulators are phases of matter characterised by a new order that does not fit into the symmetry-breaking paradigm. Global quantities are used to describe these new phases, which do not depend on the entire system. More generally, the order of valence bands contains nonstandard topological symmetry. Topological insulators are just like insulators with a bulk energy gap; moreover, their surface behaves as a metal (i.e., it possesses gapless edges (surface states) with time-reversal symmetry). After the discovery of topological insulators by Kane and Mele [2], to explore their different aspects and for a more physical insight, scientists and researchers worldwide have performed theoretical investigation [3, 4, 5, 6, 7, 8]. In the presence of an applied magnetic field (external), a topological insulator behaves as a pure insulator because its bulk energy gaps open up, which also breaks time-reversal symmetry [9]. It has also been reported that in response to induced external/magnetic-field topologically quantised effects, were called the magneto-electric effect (TME) [9, 10, 11, 12, 13, 14, 15, 16], the Lagrangian function has been employed to describe TME induced due to an external electric/magnetic field and given as \(L=L_0 +L_\theta \) [17].
$$\begin{aligned} L=\frac{\epsilon {{\varvec{E}}}^{\mathbf{2}}}{8}-\frac{1}{8\mu }{{\varvec{B}}}^{\mathbf{2}}+\frac{{\uptheta }\alpha }{4\pi ^{2}}\left( {{{\varvec{E.B}}}} \right) \end{aligned}$$
(1)
Fig. 1

Geometry of the presented scattering problem

Here \(\epsilon \), \(\mu \), \({{\varvec{E}}},\,{{\varvec{B}}},\,\theta \) and \(\alpha \) represent the relative permittivity, relative permeability of the topological insulator, electric field, magnetic flux densities, magnetoelectric polarisability and fine structure constant, respectively. The numeric value of the fine structure constant is \(\alpha =\frac{\mathrm{e}^{2}}{\hbar \pi }=\frac{1}{137}\) [18]. According to the time reversal transformation, \({{\varvec{E}}}\rightarrow {{\varvec{E}}}\) and \({{\varvec{B}}}\rightarrow -{{\varvec{B}}}\). Therefore, if we consider only a periodic system, there exist only two values of \(\theta \) , i.e., \(\theta =\pi \) (topological insulator) and \(\theta =0\) (vacuum). When we consider the time-reversal symmetry broken, then the topological magneto-electric polarisability is quantised in odd integer values of \(\pi \) , i.e., \(\theta =\left( {2n+1} \right) \pi \). The value of integer \(n\in {{\varvec{Z}}}\) can be determined by the nature of time-reversal symmetry breaking (the value of n can be controlled experimentally). In the present paper, we used \(\theta =\pi \) and \(41\pi \) [17, 18]. Because of the exceptional properties of the topological insulator, this medium has been used as a cylinder, as well as a sphere to observe the scattering characteristics for the enhancement of different optic applications (i.e., target protection, clocking, perfect reflectors and perfect observers). Furthermore, the effects on scattering widths were also investigated when the topological insulator was embedded in a chiral medium [19]. Abbas et al. [20] extended this work and studied the transmission characteristics by coating the topological insulator cylinder with metamaterials. Plasma has been considered the most astonishing material as a coating agent for studying the scattering behaviour of cylinders, spheres and waveguides. In previous literature, it was observed that by using an anisotropic plasma layer on a perfect electromagnetic conductor (PEMC), nihility spheres and cylinders, the scattering widths can be controlled more effectively by just varying the collisional frequency and plasma density [21, 22, 23]. However, until now, no one has investigated the problem of a topological insulator cylinder coated with some anisotropic medium. Thus, in the present paper, to extend the work already done, an anisotropic plasma has been used as a coating layer to investigate the bistatic echo width by varying the anisotropic plasma parameters.

Stealth or low invisibility does not mean completely vanishing from radar sources. Basically, it means low radar signals or signatures (i.e., the target can be observed by radar within a short distance) [24]. Radar cross section has a great importance in defence technology because it directly affects the detection range of a radar [25]. The invisibility of aircrafts from radar and the purpose of stealth can be achieved by changing the shape of the target and introducing coating or plasma. Plasma shielding has attracted many researchers due to its dissipative nature. Plasma, being a dissipative medium, absorbs a greater part of the incident wave and reduces radar cross section due to low scattering [26]. The present problem is physically regarded as a space ship covered with a magnetized plasma cloud.
Table 1

Conversion of present formulation to other materials

Material

\(\theta \)

\(\epsilon _{2r} \)

\(\mu _{2r} \)

PEC

0

\(10^{5}\)

1

PMC

0

1

\(10^{5}\)

Nihility

0

\(10^{-5}\)

\(10^{-5}\)

TIs

\(\pi \)

100

1

Fig. 2

a, b Bistatic echo widths of Topological insulator, PMC, PEC and nihility cylinder when \(a=5\,{\hbox {cm}},\, b=10\,{\hbox {cm}},\, f=1\,{\hbox {GHz}},\, \epsilon _1 =9.8\) and \(\mu _1 =1\) with already published literature (a = TE case, b = TM case) (Color figure online)

Fig. 3

a, b comparison of forward and backscattered echo width of PEC, nihility, PMC and topological insulator cylinder when \(a=5\,{\hbox {cm}},\, b=10\,{\hbox {cm}},\, f=1\,{\hbox {GHz}},\, \epsilon _1 =9.8\) and \(\mu _1 =1\) (Color figure online)

Fig. 4

a, b represents the forward scattering and backscattering echo widths of anisotropic plasma-coated topological insulator cylinder when \(a=0.5\lambda , b=1.5\,a, f=3\,{\hbox {GHz}},\, n=1\times 10^{15}{\hbox {m}}^{-3}\) and \(v=1\times 10^{10}\) Hz. (Red\(\,=\,\theta =\pi \) and Black\(\,= \theta =41\pi )\) TE Case (Color figure online)

Fig. 5

a, b represents the forward scattering and backscattering echo widths of anisotropic plasma-coated topological insulator cylinder when \(a=0.5\lambda ,\, b=1.5\,a,\, f=3\,{\hbox {GHz}},\, n=1\times 10^{15}{\hbox {m}}^{-3}\) and \(v=1\times 10^{10}\) Hz. (Red \(= \theta =\pi \) and Black= \(\theta =41\pi )\) TM Case (Color figure online)

Fig. 6

Forward scattering (a, c) and backscattering echo widths (b, d) of anisotropic plasma-coated topological insulator cylinder at different values of plasma density when \(a=0.5\lambda ,\, b=1.5\,a,\, f=3\,{\hbox {GHz}}\) and \(v=1\times 10^{10}\,\)Hz. (Red \(= n=1\times 10^{15}\), Black=\(\, n=5\times 10^{15}\), Blue=\(\, n=10\times 10^{15}\) and Green=\(\, n=15\times 10^{15})\) TE Case (Color figure online)

Fig. 7

a, b Forward scattering and backscattering echo widths of anisotropic plasma-coated topological insulator cylinder at different values of collisional frequency when \(a=0.5\lambda ,\, b=1.5\,a,\, f=3\,{\hbox {GHz}}\) and \(n=1\times 10^{15}{\hbox {m}}^{-3}\). (Red \(= v=1\times 10^{10}\) Hz and Blue=\(\, v=10\times 10^{10}\) Hz), respectively (Color figure online)

Fig. 8

Forward scattering (a, b) and backscattering echo widths (c, d) of anisotropic plasma-coated topological insulator cylinder at different values of collisional frequency when \(a=0.5\lambda ,\, b=1.5\, a,\, f=3\) GHz and \(v=1\times 10^{10}\) Hz. (Red \(= n=1\times 10^{15}\), Black=\(\, n=5\times 10^{15}\), Blue=\(\, n=10\times 10^{15}\) and Green=\(\, n=15\times 10^{15})\)TM (Color figure online)

Fig. 9

Forward scattering (a, c) and backscattering echo widths (b, d) of anisotropic plasma-coated topological insulator cylinder at different values of collisional frequency when \(a=0.5\lambda ,\, b=1.5\, a,\, f=3\,{\hbox {GHz}}\) and \(n=1\times 10^{15}{\hbox {m}}^{-3}\). (Red\(\,= v=1\times 10^{10}\), Black=\(\, v=5\times 10^{10}\), Blue=\(\, v=10\times 10^{10}\) and Green=\(\, v=15\times 10^{10})\)TM (Color figure online)

The geometry of the present problem is depicted in Fig. 1, where the whole space is sliced into three regions. The outer space (free space) is denoted by region I, the coating layer (anisotropic plasma layer) is indicated by region II and the core of the cylinder (topological insulator) is represented by region III. The time dependence of \(\mathrm{e}^{-j\omega t}\) has been considered throughout the mathematical formulation, where \(\omega =2\pi f\) and j is a complex number.

2 Formulations and methodology

Figure 1 shows the proposed geometry of the presented problem. In order to obtain the scattering characteristics, a topological insulator cylinder of infinite extent has been considered with radius a, as shown in Fig. 1. An anisotropic plasma has been considered as the coating layer on the topological insulator cylinder with radius b. For the sake of mathematical simplicity, the coating layer has been considered of uniform thickness. The whole geometry is sliced into three regions for more of a physical understanding as follows: Region 1 demonstrates the external medium, i.e., free space with \(\epsilon _{0} \) and \(\mu _{0} \) as permittivity and permeability with wave number \(k_0 =\omega \sqrt{\epsilon _{0} \mu _{0}}\). The coating layer of magnetized plasma has been regarded as Region 2. The plasma in magnetized form is considered to be anisotropic, cold, non-reciprocal, uniform, incompressible and homogeneous. This type of plasma is characterised by the permittivity tensor as [27, 28, 29]
$$\begin{aligned} \bar{\varepsilon }=\left| {{\begin{array}{ccc} {\epsilon _1 }&{} {-j\epsilon _2 }&{} 0 \\ {j\epsilon _2 }&{} {\epsilon _1 }&{} 0 \\ 0&{} 0&{} {\epsilon _3 } \\ \end{array} }} \right| \end{aligned}$$
(2)
The exact expression of \(\epsilon _{1} , \quad \epsilon _{2} \) and \(\epsilon _{3}\) can be written as
$$\begin{aligned} \epsilon _1= & {} 1-\frac{\omega _\mathrm{p}^2 \left( {\omega -jv} \right) }{\omega \left( {\left( {\omega -jv} \right) ^{2}-\omega _\mathrm{c}^2 } \right) }\\ \epsilon _2= & {} \,\frac{\omega c\, \left( {\omega _\mathrm{c} } \right) }{\omega \left( {\,\left( {\omega -jv} \right) ^{2}-\omega _\mathrm{c}^2 } \right) }\\ \epsilon _3= & {} \, 1\, -\, \frac{\omega _\mathrm{p}^2 }{\omega \,\left( {\omega -jv} \right) } \end{aligned}$$
where \(\omega _\mathrm{p} =\sqrt{\frac{n\mathrm{e}^{2}}{m\epsilon _{0} }}\) and \(\omega _\mathrm{c} =\frac{e\, B_0 }{m}\) with v, n, m and \(B_0 \) as the collisional frequency, plasma density, mass of the electron and magnetic field, respectively. Because of the anisotropy of plasma, when we solved Maxwell’s equation, two different waves were found to travel in different directions with corresponding wavenumbers \(k_1 =\frac{k_{0} }{\sqrt{m_{1} }}\) and \(k_2 =\frac{k_0}{\sqrt{m_{3} }}\), respectively, with \(m_{1} =\frac{\epsilon _{1} }{\epsilon _{1} ^{2}-\epsilon _{2}^{2}}\) and \(m_{3} =\frac{1}{\epsilon _{3}}\). Region 3 represents the topological insulator cylinder core with relative permittivity \(\epsilon _{2}\), relative permeability \(\mu _{2} \) and \(k_{3} =\omega \sqrt{\epsilon _2 \mu _2 }\) as the wavenumber.
The parallel polarised incident field on the anisotropic plasma-coated topological insulator cylinder travelling along the negative x-axis in terms of the cylindrical coordinates (\(\rho ,\varphi )\) can be written as
$$\begin{aligned} E_{0z}^\mathrm{inc} \left( {\rho ,\varphi } \right) =E_0 \mathrm{e}^{jk_0 \rho \mathrm{cos}\varphi } \end{aligned}$$
(3)
With the help of the Fourier transformation method, the incident field in terms of the Fourier–Bessel series can be written as
$$\begin{aligned} E_{0z}^\mathrm{inc} \left( {\rho ,\varphi } \right) =E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}J_n \left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(4)
Its corresponding magnetic flux density in the direction of \(\varphi \) can be obtained by using Maxwell’s equation.
$$\begin{aligned} B_{0\varphi }^\mathrm{inc} \left( {\rho ,\varphi } \right)= & {} \frac{1}{-j\omega }\frac{\partial E_{0z}^\mathrm{inc} \left( {\rho ,\varphi } \right) }{\partial \rho } \end{aligned}$$
(5)
$$\begin{aligned} B_{0\varphi }^\mathrm{inc} \left( {\rho ,\varphi } \right)= & {} j\sqrt{\epsilon _0 \mu _0 }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}J^{\prime }_n \left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(6)
Here, the Bessel function of the first kind and its derivative with respect to the whole argument are represented by \(J_n \left( {k_0 \rho } \right) \) and \(J^{\prime }_n \left( {k_0 \rho } \right) \), respectively. The scattering field from the topological insulator cylinder contains a cross-polarised component because of the magnetoelectric effect. Thus, the scattering field in region \(\rho >b\) by using Maxwell’s equations can be written as
$$\begin{aligned} E_{0z}^s \left( {\rho ,\varphi } \right)= & {} E_0 \mathop \sum \limits _{n=-\infty }^{\infty } j^{n}a_{n}^{e} \,H_n^1 \left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(7)
$$\begin{aligned} B_{0\varphi }^s \left( {\rho ,\varphi } \right)= & {} j\sqrt{\epsilon _0 \mu _0 }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}a_n^e \, H_n^{1 {\prime }}\left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(8)
$$\begin{aligned} H_{0z}^s \left( {\rho ,\varphi } \right)= & {} -j\sqrt{\epsilon _0 \mu _0 }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}b_n^e \, H_n^1 \left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(9)
$$\begin{aligned} E_{0\varphi }^s \left( {\rho ,\varphi } \right)= & {} -E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}b_n^e \, H_n^{1^{{\prime }}}\left( {k_0 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(10)
The total field in the region bounded by two interfaces \(\rho =a\) and \(\rho =b\) in terms of oppositely travelling cylindrical waves can be written as
$$\begin{aligned}&E_{1z} \left( {\rho ,\varphi } \right) =E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}[c_n^e \, H_n^2 \left( {k_1 \rho } \right) \nonumber \\&\qquad +d_n^e \, H_n^1 \left( {k_1 \rho } \right) ]\mathrm{e}^{jn\varphi } \end{aligned}$$
(11)
$$\begin{aligned}&B_{1\varphi } \left( {\rho ,\varphi } \right) =\,j\sqrt{\bar{\varepsilon } \mu }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}[c_n^e \, H_n^2 {\prime }\left( {k_1 \rho } \right) \nonumber \\&\qquad +d_n^e \, H_n^1 {\prime }\left( {k_1 \rho } \right) ]\mathrm{e}^{jn\varphi }\end{aligned}$$
(12)
$$\begin{aligned}&B_{1z} \left( {\rho ,\varphi } \right) =\,-\,j\sqrt{\bar{\varepsilon } \mu }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}[e_n^e \, H_n^2 \left( {k_2 \rho } \right) \nonumber \\&\qquad +f_n^e \, H_n^1 \left( {k_2 \rho } \right) ]\mathrm{e}^{jn\varphi } \end{aligned}$$
(13)
$$\begin{aligned}&E_{1\varphi } \left( {\rho ,\varphi } \right) =-\,E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}[e_n^e \, H_n^2 {\prime }\left( {k_2 \rho } \right) \nonumber \\&\qquad +f_n^e \, H_n^1 {\prime }\left( {k_2 \rho } \right) ]\mathrm{e}^{jn\varphi } \end{aligned}$$
(14)
The total field in Region 3
$$\begin{aligned}&E_{2z} \left( {\rho ,\varphi } \right) =E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}g_{n\, } J_n \left( {k_3 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(15)
$$\begin{aligned}&E_{2\varphi } \left( {\rho ,\varphi } \right) =-E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}h_n \, J_{n}^{{\prime }}\left( {k_3 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(16)
$$\begin{aligned}&B_{2\varphi } \left( {\rho ,\varphi } \right) =j\sqrt{\epsilon _2 \mu _2 }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}g_n \, J_{n}^{{\prime }}\left( {k_3 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(17)
$$\begin{aligned}&B_{2z} \left( {\rho ,\varphi } \right) =-j\sqrt{\epsilon _2 \mu _2 }E_0 \mathop \sum \limits _{n=-\infty }^\infty j^{n}h_n \, J_n \left( {k_3 \rho } \right) \mathrm{e}^{jn\varphi } \end{aligned}$$
(18)
Equations 718 contain unknown coefficients, which can be obtained by implementing the below boundary conditions.
$$\begin{aligned}&E_{1\varphi } \left( {\rho ,\varphi } \right) |_{r=a}=E_{2\varphi } \left( {\rho ,\varphi } \right) \, |_{r=a} \end{aligned}$$
(19)
$$\begin{aligned}&E_{1z} \left( {\rho ,\varphi } \right) |_{r=a}=E_{2z} \left( {\rho ,\varphi } \right) \, |_{r=a} \end{aligned}$$
(20)
$$\begin{aligned}&\frac{1}{\mu _1 }B_{1\varphi } \left( {\rho ,\varphi } \right) |_{r=a} \nonumber \\&\quad =\frac{1}{\mu _2 }B_{2\varphi } \left( {\rho ,\varphi } \right) |_{r=a} -\alpha \frac{\theta }{\pi }E_{2\varphi } \left( {\rho ,\varphi } \right) \, |_{r=a} \end{aligned}$$
(21)
$$\begin{aligned}&\frac{1}{\mu _1 }B_{1z} \left( {\rho ,\varphi } \right) |_{r=a} \nonumber \\&\quad =\frac{1}{\mu _2 }B_{2z} \left( {\rho ,\varphi } \right) |_{r=a} -\alpha \frac{\theta }{\pi }E_{2z} \left( {\rho ,\varphi } \right) \, |_{r=a} \end{aligned}$$
(22)
$$\begin{aligned}&E_{0z} \left( {\rho ,\varphi } \right) |_{r=b} \nonumber \\&\quad =E_{1z} \left( {\rho ,\varphi } \right) \, |_{r=b} \, \, \, \, \, 0\le \varphi \le 2\pi \end{aligned}$$
(23)
$$\begin{aligned}&E_{0\varphi } \left( {\rho ,\varphi } \right) |_{r=b} \nonumber \\&\quad =E_{1\varphi } \left( {\rho ,\varphi } \right) \, |_{r=b} \, \, \, \, \, 0\le \varphi \le 2\pi \end{aligned}$$
(24)
$$\begin{aligned}&\frac{1}{\mu _0 }B_{0\varphi } \left( {\rho ,\varphi } \right) |_{r=b} \nonumber \\&\quad =\frac{1}{\mu _1 }B_{1\varphi } \left( {\rho ,\varphi } \right) \, |_{r=b} \, \, \, \, \, 0\le \varphi \le 2\pi \end{aligned}$$
(25)
$$\begin{aligned}&\frac{1}{\mu _0 }B_{0z} \left( {\rho ,\varphi } \right) |_{r=b} \nonumber \\&\quad =\frac{1}{\mu _1 }B_{1z} \left( {\rho ,\varphi } \right) \, |_{r=b} \, \, \, \, \, 0\le \varphi \le 2\pi \end{aligned}$$
(26)
where
$$\begin{aligned} E_{0z} \left( {\rho ,\varphi } \right) =E_{0z}^\mathrm{inc} \left( {\rho ,\varphi } \right) +E_{0z}^s \left( {\rho ,\varphi } \right) \end{aligned}$$
(27)
and
$$\begin{aligned} B_{0\varphi } \left( {\rho ,\varphi } \right) =B_{0\varphi }^\mathrm{inc} \left( {\rho ,\varphi } \right) +B_{0\varphi }^s \left( {\rho ,\varphi } \right) \end{aligned}$$
(28)
By using Eqs. 418 in the above-mentioned boundary conditions (Eqs. 1928), eight sets of equations were obtained, which can be written in matrix form as below, where \(\eta _n =\sqrt{\frac{\mu _n }{\epsilon _n }}\), \(r_0 =k_0 b, \, r_1 =k_1 a,\, r_2 =k_2 a,\, r_3 =k_3 a,\, r_4 =k_1 b\) and \(r_5 =k_2 b\).
$$\begin{aligned}&\left| {\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ {J_{n} \left( {r_{0} } \right) } \\ {J_{n}^{\prime } \left( {r_{0} } \right) } \\ 0 \\ 0 \\ \end{array}} \right| = \left| \begin{array}{cccccccc} 0 &{} 0 &{} 0 &{} { - J_{n}^{'} (r_{3})} &{} 0 &{} 0 &{} {H_{n} ^{{2'}} (r_{2} )} &{} {H_{n} ^{{1'}} (r_{2} )} \\ 0 &{} 0 &{} { - J_{n} (r_{3} )} &{} 0 &{} {H_{n} ^{2} (r_{1} )} &{} {H_{n}^{1} (r_{1} )} &{} 0 &{} 0 \\ 0 &{} 0 &{} {\frac{{\eta _{1} }}{{\eta _{2} }}J_{n} ^{'} (r_{3} )} &{} { - i\,\eta _{1} \alpha \frac{\theta }{\pi }J_{n} ^{'} (r_{3} )} &{} {-H_{n} ^{{2'}} (r_{1} )} &{} {-H_{n}^{{1'}} (r_{1} )} &{} 0 &{} 0 \\ 0 &{} 0 &{} { - i\,\eta _{1} \alpha \frac{\theta }{\pi }J_{n} (r_{3} )} &{} {\frac{{\eta _{1} }}{{\eta _{2} }}J_{n} (r_{3} )} &{} 0 &{} 0 &{} { - H_{n} ^{2} (r_{2} )} &{} { - H_{n} ^{1} (r_{2} )} \\ { - H_{n} ^{1} (r_{0} )} &{} 0 &{} 0 &{} 0 &{} {H_{n} ^{2} (r_{4} )} &{} {H_{n} ^{1} (r_{4} )} &{} 0 &{} 0 \\ { - H_{n} ^{{1'}} (r_{0} )} &{} 0 &{} 0 &{} 0 &{} {\frac{{\eta _{0} }}{{\eta _{1} }}H_{n} ^{{2'}} (r_{4} )} &{} {\frac{{\eta _{0} }}{{\eta _{1}}}H_{n} ^{{2'}} (r_{4} )} &{} 0 &{} 0 \\ 0 &{} { - H_{n} ^{1} (r_{0})} &{} 0 &{} 0 &{} 0 &{} 0 &{} {\frac{{\eta _{0} }}{{\eta _{1} }}H_{n} ^{2} (r_{5} )} &{} {\frac{{\eta _{0} }}{{\eta _{1} }}H_{n} ^{1} (r_{5} )}\\ 0 &{} { - H_{n} ^{{1'}} (r_{0} )} &{} 0 &{} 0 &{} 0 &{} 0 &{} {H_{n}^{{2'}} (r_{5} )} &{} {H_{n} ^{{1'}} (r_{5} )} \\ \end{array} \right| \left| {\begin{array}{c} {a_{n}^{e} } \\ {b_{n}^{e} } \\ {c_{n}^{e} } \\ {d_{n}^{e} } \\ {e_{n}^{e} } \\ {f_{n}^{e} } \\ {g_{{n~}} } \\ {h_{{n~}} } \\ \end{array}} \right| \end{aligned}$$
Co- and cross-polarised coefficients \(a_n^e \, \hbox {and}\, b_n^e \), respectively, can be deduced by solving the above matrixes. By using these coefficients in the below equations, the co-polarised and cross-polarised scattering width can be obtained.
$$\begin{aligned} \sigma _{2D} =\mathop {\lim }\nolimits _{\rho \rightarrow \infty } \left[ {2\pi \rho \frac{\left| {E_s } \right| ^{2}}{\left| {E_i } \right| ^{2}}} \right] \end{aligned}$$
(29)
After some simple mathematical steps, we obtained the scattering echo widths in terms of co- and cross-polarisation_ENREF_30 [30].
$$\begin{aligned} \frac{\sigma _\mathrm{co} }{\lambda _0 }= & {} \frac{2}{\pi }\, \left| {\mathop \sum \limits _{n=-\infty }^\infty a_n^e \, \mathrm{e}^{in\varphi }} \right| ^{2} \end{aligned}$$
(30)
$$\begin{aligned} \frac{\sigma _\mathrm{cross} }{\lambda _0 }= & {} \frac{2}{\pi }\, \left| {\mathop \sum \limits _{n=-\infty }^\infty b_n^e \, \mathrm{e}^{in\varphi }} \right| ^{2} \end{aligned}$$
(31)

3 Results and discussion

In this section, the obtained mathematical equations were solved numerically and the obtained results are presented in graphical form. By applying specific boundary conditions, the obtained results were compared with already published literature to show the accuracy of the present analytical solution. The obtained results of the topological insulator cylinder can be transformed into a perfect electric conductor (PEC), perfect magnetic conductor (PMC) and nihility by simply choosing the values of \(\theta , \quad \epsilon _{2r} \) and \(\mu _{2r} \), as shown in Table 1.

When we remove the applied magnetic field, the anisotropic plasma will be converted into isotropic plasma. The results were obtained and compared with available literature [10, 11, 12] by considering \(a=5\,{\hbox {cm}},\, b=10\,{\hbox {cm}},\, f=1\,{\hbox {GHz}},\, \epsilon _1 =9.8\) and \(\mu _1 =1\), as shown in Fig. 2a, b. The comparison of both forward and backscattered echo widths of different cores (PEC, PMC, nihility and the topological insulator cylinder) are presented in Fig. 3a, b. the obtained results are quite interesting because the backscattered echo width of the topological insulator exist while for PEC, PMC and nihility its value is exactly zero.

Figure 4a, b represent the forward scattering and backscattering echo widths of the topological insulator cylinder coated with anisotropic plasma, respectively, under transverse magnetic (TM) polarisation when \(\theta =\pi \) and \(\theta =41\, \pi \). It has been analysed that the backscattering echo width when \(\theta =\pi \) is spread at more angles and shows higher magnitude than the other case (i.e., \(\theta =41\pi )\), as shown in Fig. 4 b. Meanwhile, the magnitude of forward scattering when \(\theta =41\pi \) shows higher magnitude behaviour compared to the \(\theta =\pi \) case. The forward scattering and backscattering echo widths of the topological insulator cylinder covered with an anisotropic plasma layer when transverse electric (TE) polarisation has been considered are shown in Fig. 5a, b. It has been observed that when we consider \(\theta =\pi \), the forward scattering echo width of the topological insulator cylinder shows spreading behaviour instead of \(\theta =41\pi \). The effects on the scattering width of the topological insulator cylinder by varying the plasma density were recorded and are displayed graphically in Fig 6. By analysing the results, it has been reported that the forward scattering echo width shows decreasing magnitude towards \(\varphi =0^{\circ }\) as we increase the plasma density, which can be confirmed from Fig. 6a, c. On the other hand, by increasing the plasma density, the backscattering echo width starts spreading and covers more angles and its magnitude shows increasing behaviour, which can be confirmed from Fig. 6b, d.

Figure 7a, b represents the forward scattering and backscattering widths at different values of effective collisional frequency. It has been observed that by increasing the collisional frequency, no variation in the forward and backscattering echo widths was found whether or not we chose \(\theta =\pi \) or \(\theta =41\pi \), as shown in Fig. 7a, b. For analysis, the effects of plasma density on the scattering echo widths of the topological insulator cylinder when TE polarisation has been considered are shown in Fig. 8. The forward scattering echo widths of the topological insulator cylinder covered with anisotropic plasma show increasing behaviour and also start spreading at wide angles when we increase the numeric value of plasma density, as shown in Fig. 8a, b. Similar to forward scattering, the backscattering echo width also shows the same behaviour, which can be verified from Fig. 8c, d. It has been observed that by increasing the effective collisional frequency, the forward scattering echo widths of both cases (i.e., \(\theta =\pi \) and \(41\pi )\) show increasing behaviour, as shown in Fig. 9a, c. On the other hand, the backscattering echo widths show constant behaviour no matter how much we increase the collisional frequency, as shown in Fig. 9b, d.

4 Conclusion

The problem of scattering widths of the topological insulator cylinder when covered with an anisotropic plasma layer has been solved analytically by considering magneto electric polarization (\(\theta \)) \(\leftharpoondown \) \(\psi \) equal to \(\pi \) \(\psi \) and 41\(\pi \). Variations in the scattering echo widths of the topological insulator cylinder were recorded by varying the anisotropic plasma parameters (i.e., the plasma density and effective collisional frequency) by considering transverse electric and transverse magnetic polarisation. It has been observed that by increasing the plasma density in the TM case, the forward scattering echo width shows decreasing behaviour at lower angles, while the backscattering echo width follows increasing behaviour at both lower and higher angles. It has also been noticed that by increasing the collisional frequency, both forward scattering and backscattering show constant behaviour. In TE polarisation, both plasma density and effective collisional frequency affect the scattering echo widths more effectively than the TM case. It has been noted that by increasing the plasma density, both the forward and backscattering echo widths start increasing. However, when we increase the collisional frequency, the forward scattering echo width show increasing behaviour, while the backscattering remains the same no matter how much we increase the value of the collisional frequency.

Meanwhile, by changing plasma parameters we can easily tune scattering efficiencies i.e., radar cross section can be increased/decreased. In conclusion, the obtained findings will be helpful for defence and stealth technologies.

Notes

Acknowledgements

The authors would like to extend their sincere appreciation to The Deanship of Scientific Research (DSR) at King Saud University, Riyadh, Saudi Arabia for their financial support through the Research Group Project No. RG-1438-12.

Author Contributions

MASA, AG and MMH derived analytical expressions and numerical analysis. They wrote the main manuscript text. YK developed methodology in the given study. MASA, and YK also secured the research grant for this project and will be paying the publication fee from the grant approved on their name. This project was accomplished under the supervision of MASA. MYN and QAN conducted computational calculations of Figs. 2 and 3. All authors reviewed the manuscript before submitting it to ‘Journal of Computational Electronics’.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Majeed A. S. Alkanhal
    • 1
  • A. Ghaffar
    • 2
  • M. M. Hussan
    • 2
  • Y. Khan
    • 1
  • I. Ahmad
    • 1
  • Q. A. Naqvi
    • 3
  1. 1.Department of Electrical EngineeringKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Department of PhysicsUniversity of AgricultureFaisalabadPakistan
  3. 3.Department of ElectronicsQuaid-i-AzamIslamabadPakistan

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