Scattering of electromagnetic waves from a chiral coated nihility cylinder hosted by isotropic plasma medium

M. Z. Yaqoob; A. Ghaffar; Majeed A.S. Alkanhal; I. Shakir; Q. A. Naqvi

doi:10.1364/OME.5.001224

1. Introduction

The electromagnetic properties of the artificially designed materials that are often known as metamaterials, have magnetize many researchers, opticians, and engineers due to their tremendous uses as wave guiders, microwave controller, electromagnetic invisibility cloak, perfect reflectors, phase shifters and filters [1–3]. Metamaterials have purposed new degrees of freedom regarding the realization of unusual electromagnetic properties at different ranges of frequencies, which are beyond the natural materials. The split ring resonator (SRR), plasma, chiral, chiral nihility and perfect electromagnetic conductor are some typical metamaterials, which are being studied extensively in literature [4–6]. The nihility is the electromagnetic nilpotent, and has the most astonishing impact in the field of optics and electromagnetics, in this medium both the relative permeability and permittivity have null magnitude [7].

Many researchers have paid the attention to this electromagnetic trinity and performed lot of work on the scattering of electromagnetic radiation from nihility material/objects i.e., Lakhtakia discussed the scattering from nihility sphere and analyze that the Extinction efficiency of the nihility sphere is more than the perfect electric conductor sphere [8]. In further addition, the electromagnetic scattering from infinite nihility cylinder is also studied by Lakhtakia and Geddes [9]. Ahmad et al. discussed the electromagnetic scattering from metamaterial coated nihility circular cylinder [10]. To get more control on electromagnetic scattering, the coating of material i.e., chiral metamaterial is applied on nihility cylinder, and problem transformed into the electromagnetic scattering from chiral coated nihility cylinder [11]. Sobia et al., discussed the more generalized problem i.e., scattering of electromagnetic radiation from chiral coated nihility cylinder placed in the chiral metamaterial [12].

Lot of research work regarding the electromagnetic scattering from plasma coated perfect conducting objects has been done in literature. Plasma recently realized as metamaterial and has found applications in negative refractive index materials, photonic crystals, target protection and communication [13–15]. Plasma based composites metamaterials have the recent interest of researchers due to their numerous application in the communication, defense technology, rocket science and space sciences [16]. To meet the recent interest, we have presented the electromagnetic scattering from chiral coated nihility cylinder hosted by isotropic plasma medium. The extended classical scattering theory is used to accomplish the scattering problem. The influence of plasma density, effective collision frequency, plasma oscillation and chirality on the scattering amplitude is analyzed. This work generalizes the subsequent scattering problems i.e., scattering from chiral/achiral coated nihility/ chiral nihility/PEC/PMC circular cylinder placed in dielectric or free space, which shows the novelty of our work. Moreover, present work has practical importance in target protection and microwave controlling devices. The time harmonic dependence is taken $e^{- j ω t} .$

2. Formulations

In this section, the analytical formulation and geometry of the scattering problem is presented. The whole space is divided into the three regions with respect to the medium i.e., I, II and III that represents plasma, chiral metamaterial and nihility/ chiral nihility medium respectively, as shown in Fig. 1.The plasma is a host medium in which infinitely long chiral coated nihility circular cylinder is placed, and considered to be isotropic and homogenous. In addition to this, the chiral coating and nihility cylinder are also considered as homogenous, linear and isotropic. The thickness of the coating is measured by the parameter $Δ ρ = b - a,$ where $ρ_{2} = b$ is the radius of coated chiral cylinder and $ρ_{1} = a$ is the radius of inner nihility core. To retain the homogeneity and uniformity in the thickness of chiral coating, the concentrically coated chiral nihility cylinder is taken in this scattering problem.

Fig. 1 Concentric coated chiral coated nihility cylinder placed in plasma medium.

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The incident scattered and transmitted fields are expanded in terms of the cylindrical vector wave functions (CVWFs) to secure the symmetry in the coordinate systems [9, 10].

E^{i} = E_{0} \sum_{n = - \infty}^{\infty} j^{n} [M_{n}^{(1)} (k_{1} ρ) + N_{n}^{(1)} (k_{1} ρ)]

H^{i} = \frac{j E_{0}}{η_{1}} \sum_{n = - \infty}^{\infty} j^{n} [M_{n}^{(1)} (k_{1} ρ) + N_{n}^{(1)} (k_{1} ρ)]

E^{s} = E_{0} \sum_{n = - \infty}^{\infty} j^{n} [a_{n} N_{n}^{(2)} (k_{1} ρ) + b_{n} M_{n}^{(2)} (k_{1} ρ)]

H^{s} = \frac{j E_{0}}{η_{1}} \sum_{n = - \infty}^{\infty} j^{n} [a_{n} M_{n}^{(2)} (k_{1} ρ) + b_{n} N_{n}^{(2)} (k_{1} ρ)]

where

a_{n}

and

b_{n}

are the Co and Cross-polarized coefficients respectively, and

η_{1} = \sqrt{μ_{1} / ε_{1}}

represents the intrinsic impedance of the plasma medium. The

k_{1} = ω \sqrt{ε_{1} μ_{1}}

is the wavenumber in the plasma medium, which is the function of (electron) plasma frequency

(n),

effective collision frequency

(v)

and the incident field frequency

(ω)

. The following equations represent the transmitted fields in the coating as well as the chiral cylinder

\begin{array}{l} E^{c} = E_{0} \sum_{n = - \infty}^{\infty} j^{n} [c_{n} [M_{n}^{(3)} (k_{+} ρ) + N_{n}^{(3)} (k_{+} ρ)] + d_{n} [M_{n}^{(3)} (k_{+} ρ) + N_{n}^{(3)} (k_{+} ρ)]] \\ - E_{0} \sum_{n = - \infty}^{\infty} j^{n} [e_{n} [M_{n}^{(2)} (k_{+} ρ) + N_{n}^{(2)} (k_{+} ρ)] + f_{n} [M_{n}^{(2)} (k_{+} ρ) + N_{n}^{(2)} (k_{+} ρ)]] \end{array}

\begin{array}{l} H^{c} = \frac{j E_{0}}{η_{2}} \sum_{n = - \infty}^{n = \infty} j^{n} [c_{n} [M_{n}^{(3)} (k_{+} ρ) + N_{n}^{(3)} (k_{+} ρ)] + d_{n} [M_{n}^{(3)} (k_{+} ρ) + N_{n}^{(3)} (k_{+} ρ)]] \\ - \frac{j E_{0}}{η_{2}} \sum_{n = - \infty}^{n = \infty} j^{n} [e_{n} [M_{n}^{(2)} (k_{+} ρ) + N_{n}^{(2)} (k_{+} ρ)] + f_{n} [M_{n}^{(2)} (k_{+} ρ) + N_{n}^{(2)} (k_{+} ρ)]] \end{array}

E^{t} = E_{0} \sum_{n = - \infty}^{\infty} j^{n} [g_{n} [M_{n}^{(1)} (k_{+} ρ) + N_{n}^{(1)} (k_{+} ρ)] + h_{n} [M_{n}^{(1)} (k_{-} ρ) - N_{n}^{(1)} (k_{-} ρ)]]

H^{t} = \frac{j E_{0}}{η_{3}} \sum_{n = - \infty}^{\infty} j^{n} [g_{n} [M_{n}^{(1)} (k_{+} ρ) + N_{n}^{(1)} (k_{+} ρ)] + h_{n} [M_{n}^{(1)} (k_{-} ρ) - N_{n}^{(1)} (k_{-} ρ)]]

where the

η_{2} = \sqrt{μ_{2} / ε_{2}}

and

η_{3} = \sqrt{μ_{3} / ε_{3}}

are the impedance of the region II and III respectively, while the

k_{\pm 2} = ω (\sqrt{ε_{2} μ_{2}} \pm β_{2})

and

k_{\pm 3} = ω (\sqrt{ε_{3} μ_{3}} \pm β_{3})

are the wavenumbers in coating and chiral cylinder respectively. The appropriate boundary conditions are applied to calculate the Co and Cross polarized scattering coefficients i.e.,

a_{n}

and

b_{n}

respectively, and further utilized to calculate the bistatic echo widths as in [11,12].

4. Numerical results and discussions

The numerical approach is used, to get more physical understanding and insight physics of the electromagnetic scattering from chiral coated nihility cylinder placed in plasma medium. Further, the scattering coefficients, bistatic echo widths and plots against different parameters, are obtained numerically. The MATHEMATICA professional software package is used to simulate the scattering problem through programing. The infinite series solution is approximated by the mesh loop value from −5 to 5, while, in the whole results, the incident frequency, size of the coated and inner core is taken as 1GHz, 10cm and 5cm respectively. To check the functionality of software package and accuracy in our work, some of the results are compared with already published literature under special conditions and good agreement is found, as shown in Fig. 2(a).When the host medium plasma is replaced by the free space i.e., $ε_{1} = ε_{o}$ & $μ_{1} = μ_{o}$ and chiral coating is replaced by dielectric coating i.e., $ε_{r 3} = 9.8, μ_{r 3} = 1 & β_{2} = 0.0$ then the present scattering problem is transformed into the scattering from dielectric coated nihility cylinder [10]. For further reliability and accuracy, under second special condition, i.e., by replacing the inner nihility core $(ε_{r 3} \to 0.0, μ_{r 3} \to 0.0, β_{3} = 0.0)$ by PEC cylinder $(ε_{r 3} \to \infty, μ_{r 3} = 1, β_{3} = 0.0),$ the problem transforms into scattering of electromagnetic radiation from PEC coated cylinder [13], as given in Fig. 2(a).. While the Fig. 2(a). gives the comparison between the bistatic echo widths of chiral coated nihility cylinder placed in free space and isotropic plasma. It is obvious from this comparison that the plasma environment can be used to increase or tune the scattering amplitude.

Fig. 2 Comparison of (a) Present work with published literature [10,13] (b) Co-polarized bistatic echo widths of chiral coated nihility cylinder placed in free space and isotropic plasma medium at $(b = 10 c m, a = 5 c m,$ $f = 1 G H z, n = 1.0 \times 10^{16} m^{- 3} and v = 1.0 \times 10^{10} H z, ε_{r 2} = 2.5 a n d β_{2} = 0.002)$ .

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The comparison between the bistatic echo widths of chiral coated cylinders (nihility, chiral nihility, PEC and PMC) placed in isotropic plasma medium is presented in Fig. 3.The different inner cores of cylinders i.e, Nihility, Chiral Nihility, PEC and PMC in region III are realized by the constitutive parameters $(ε_{r 3}, μ_{r 3}, β_{3})$ i.e., $(ε_{r 3} \to 0.0, μ_{r 3} = \to 0.0, β_{3} = 0.0),$ $(ε_{r 3} \to \infty, μ_{r 3} = 1, β_{3} = 0.0)$ and $(ε_{r 3} = 1, μ_{r 3} \to \infty, β_{3} = 0.0)$ respectively. Figure 3(a). presents the comparison between the Co polarized scattering coefficients of chiral coated nihility/ chiral nihility/PEC and PMC cylinder placed in the plasma medium while the comparison between the cross polarized scattering coefficients of these cases is shown in Fig. 3(b). It is obvious from Fig. 3(a). and Fig. 3(b). that the chiral coated nihility core has more stealth capability as compared to other cores and also the behavior of Co and Cross polarized fields is opposite to each other. In Fig. 4, the influence of plasma on the Co and Cross polarized scattering echo widths is presented. It is clear from the Fig. 4(a). that with the increase of plasma density the Co polarized scattering echo width’s amplitude also increases, while the cross polarized scattering echo width’s amplitude is decreasing with the increase in the plasma density as depicted in Fig. 4(b).

Fig. 3 Comparison of (a) Co-polarized bistatic echo widths of chiral coated cylinders placed in plasma medium. (b) Cross-polarized bistatic echo widths of chiral coated cylinders placed in plasma medium at $(b = 10 c m, a = 5 c m, f = 1 G H z, n = 1.0 \times 10^{15} m^{- 3} and v = 1.0 \times 10^{10} H z, ε_{r 2} = 2.5, β_{2} = 0.002 a n d β_{3} = 0.001)$ .

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Fig. 4 Influence of electron density on (a) Co-polarized bistatic echo widths. (b)) Cross-polarized bistatic echo widths at $(b = 10 c m, a = 5 c m, f = 1 G H z, v = 1.0 \times 10^{10} H z, ε_{r 2} = 2.5 a n d β_{2} = 0.0002) .$

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In Fig. 5 the effect of effective collision frequency on the scattering bistatic echo width of Co and Cross polarized scattering coefficient is presented. The relative permittivity of the plasma is consist of two parts i.e., real part (energy stored) and imaginary part (energy dissipation), the effective collision frequency is inversely proportional to the imaginary part. Therefore by increasing the effective collision frequency the scattering width increases. The Fig. 6 depicts the effect of chirality on the bistatic echo widths of the Co and Cross polarized scattering coefficients. The Fig. 6(a). shows that the influence of chirality parameter on the Co polarized bistatic scattering echo width is almost negligible while, the cross polarized bistatic echo width is strongly influenced by the chirality parameter as given in Fig. 6(b).

Fig. 5 Influence of effective collision frequency on (a) Co-polarized bistatic echo widths (b) Cross-polarized bistatic echo widths at $(b = 10 c m, a = 5 c m, f = 1 G H z, n = 1.0 \times 10^{15} m^{- 3}, ε_{r 2} = 2.5 a n d β_{2} = 0.002) .$

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Fig. 6 Influence of chirality parameter on (a) Co-polarized bistatic echo widths (b) Cross-polarized bistatic echo widths $(n = 2.0 \times 10^{15} m^{- 3}, v = 1 G H z a n d ε_{r 2} = 2.2)$ .

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5. Conclusion

The canonical boundary value scattering problem is studied to analyze the scattering of cylindrical waves from chiral coated nihility infinite circular cylinder placed in isotropic plasma medium. The influence of plasma parameters (plasma density and effective collision frequency) on the scattering echo width is analyzed and concluded that scattering amplitude can be controlled and tuned by tuning the plasma parameters. Furthermore, the effect of chirality on the bistatic echo width is also reported. It is concluded that the present work will be help full for target protection and microwave controlling devices.

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research (DSR) at King Saud University for its funding of this research through the Research Group no RG-1436-001.

References and links

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Scattering of electromagnetic waves from a chiral coated nihility cylinder hosted by isotropic plasma medium

Abstract

1. Introduction

2. Formulations

4. Numerical results and discussions

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (6)

Equations (8)

Optical Materials Express