PUBLICATIONS. Radio Science. Power flux distribution in chiroplasma-filled perfect electromagnetic conductor circular waveguides

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: Energy flux distribution in chiroplasma PEMC waveguide may be obtained The dispersion diagram, cutoff frequencies, and the propagating modes The chirality, plasma, and cyclotron frequency effects on the energy flux Correspondence to: A. Ghaffar, Citation: Ghaffar, A., and M. A. S. Alkanhal (2016), Power flux distribution in chiroplasmafilled perfect electromagnetic conductor circular waveguides, Radio Sci., 51, , doi:. Received 30 APR 2015 Accepted 9 MAR 2015 Accepted article online 16 MAR 2016 Published online 30 MAR American Geophysical Union. All Rights Reserved. Power flux distribution in chiroplasma-filled perfect electromagnetic conductor circular waveguides A. Ghaffar 1,2 and Majeed A. S. Alkanhal 1 1 Department of Electrical Engineering, King Saud University, Saudi Arabia, 2 Department of Physics, University of Agriculture, Faisalabad, Pakistan Abstract Theoretical analysis of the electromagnetic wave propagation in circular waveguides with a chiroplasma core coated with a perfect electromagnetic conductor (PEMC) is presented. The presented formulations and analysis are general for any perfect electric, perfect magnetic, or perfect electromagnetic conductor (PEMC) waveguides filled with any general anisotropic/isotropic metamaterial including plasma. The characteristic equation for the modes in this waveguide is obtained, and the behavior of the dispersion curves and the power flux are examined and evaluated numerically. The results demonstrate that the behavior of the power flux transported in the guide, in magnitude and orientation, is highly determined by the chirality parameter and the plasma and cyclotron frequencies. The mode cutoff frequencies are sensitive to the variations in the filling material and chirality parameters and likewise affected by the variations in the PEMC admittance parameter. 1. Introduction The knowledge of electromagnetic wave behavior in diverse materials is important for novel material-waveguide designs with conceivable optical and electromagnetic device and system applications. Waveguides of different materials have various uses in several applications in microwaves, millimeter waves, terahertz, and optical wave regimes. Waveguide structures of negative index metamaterials have been examined [Pelet and Engheta, 1990], and it is noticed that chiral metamaterials are of great scientific researchmerits[oussaid and Haraoubia, 2004; Gulistan et al., 2012; Ali et al., 2012]. The chiral materials have distinct electromagnetic properties that are not found in naturally occurring materials [Lindell et al., 1994; Ghaffar and Alkanhal, 2014]. The research work related to the chiral metamaterials that were devoted to anisotropic chiral media has been reported by different authors [Baqir and Choudhury, 2014; Dong and Li, 2012a; Dong and Li, 2012b; Baqir and Choudhury, 2012]. The invention of microwave devices fabricated using ferrites, semiconductors, and bianisotropic chiral and biisotropic chiral and plasma metamaterials have opened many problems of electromagnetic (EM) wave propagation in anisotropic complex materials and inspired theoretical and experimental research [Hu and Ruan, 1998; Abdoli-Arani, 2014; Uhm et al., 1988; Shen, 1991]. The waveguide with perfect electromagnetic conductor (PEMC) boundary [Baqir and Choudhury, 2014; Hussain and Naqvi, 2007] or chiroplasma core is of particular interest in the field of wave propagation characterization. The realization and applications of PEMC medium have been studied by many researchers [Lindell and Sihvola, 2005; Lindell and Sihvola, 2006b; Lindell and Sihvola, 2006a]. Anisotropic-filled waveguides have many distinct characteristics over bi-isotropic chiral guiding media [Eftimiu and Pearson, 1989]. Plasma is an attractive medium because it is an anisotropic medium due to the presence of the biasing magnetic field [Torres-Silva et al., 2000; Marakassov and Fisanov, 2001], which allows a form of control over the chirality of the medium. Conceiving chiroplasma as a guiding mean in waveguide structures can alter the features of the electromagnetic waves propagating modes [Guo, 2013; Gao et al., 2014; Gong, 1999]. The analysis of such type of research is very attractive because applications of chiroplasma waveguide in engineering electromagnetics will undertake a widespread knowledge of the characteristics of its eigenmodes. It is confirmed theoretically that the perfect electromagnetic (PEMC) material behaves like a reflector of electromagnetic (EM) waves; however, it is different from the perfect magnetic (PMC) and the perfect electric (PEC) [Ruppin, 2006]. The PEMC is a general form of both the PMC and the PEC. The boundary conditions defined for PEMC interfaces are [Lindell and Sihvola, 2005] as follows: nðh þ MEÞ ¼ 0 (1) n ðd MBÞ ¼ 0 where n is the unit vector normal, and M is the scalar admittance parameter describing the PEMC. The values M = 0 and M ± equal to PMC and PEC, respectively. The EM power cannot enter into the PEMC medium GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 231

2 because the complex Poynting vector is imaginary for the real values of M [Ghaffar et al., 2014a; Ghaffar et al., 2014b]. The presented work in this paper represents a waveguide made of a chiroplasma material core and an outer perfect electromagnetic conductor (PEMC) boundary. In this paper, we present analysis and computations of the electromagnetic field propagation and the power flux density in the Figure 1. A PEMC chiroplasma waveguide. chiroplasma-filled cylindrical waveguide. The presented formulations and analysis are generalized for any waveguide core filled with a general anisotropic/isotropic material including plasma and metamaterials with (PEC, PMC, or PEMC) coating materials. The dispersion relations for the guide are developed and used to get the propagating modes and their power flux. Results obtained show that the chirality values and plasma and cyclotron frequencies can retain noteworthy effect on the power values of the propagating modes. The time-harmonic (jωt) dependence is assumed and omitted in what follows. The paper is organized as follows. Basic formulas are derived in section 2. The results and discussions are presented in section 3. Finally, conclusions are succeeding in section Formulations Consider a circular waveguide filled with a chiroplasma material core of radius b and bounded by a PEMC. The geometry of the problem is depicted in Figure 1. Unit vectors are denoted in cylindrical coordinate system as bρ; ϕ, b and bz. The constitutive expressions for the chiroplasma material are defined below [Torres-Silva et al., 2000; Marakassov and Fisanov, 2001; Guo, 2013; Gao et al., 2014; Gong, 1999] D ¼ ε 0 ε:e þ jγh (2) H ¼ jγe þ 1 μ B (3) where γ is the chirality parameter, ε 0 is the permittivity of free space, and μ is the permeability of free space. The permittivity tensor can be written as ε 1 jε 2 0 ε ¼¼ jε 2 ε 1 0 (4) 0 0 ε 3 qffiffiffiffiffiffiffi where ε 1 ¼ ε 0 1 ω2 p ω ; ε ω 2 ω 2 2 ¼ ε cω 2 p 0 p ωω ð 2 ω 2 c Þ ; ε 3 ¼ ε 0 1 ω2 p Ne ω, with ω 2 p ¼ 2 m eε 0 and ω c ¼ eb0 m eε 0. The terms N,m e, B o, and e are the plasma density (electron density), electron mass, magnitude of the magnetic field, and the electron s charge, respectively. The plasma frequency is ω p, and the cyclotron frequency is ω c. The wave equation obtained from Maxwell s equations for the longitudinal field components E z and H z in the anisotropic chiroplasma material [Hu and Ruan, 1998] can be expressed as 2 t E z 2 t H z þ p 1 jp 2 jp 3 p 4 Ez H z ¼ 0 (5) where p 1 ¼ β2 þ ω 2 με 1 ε3 þ ωμγ βε 2 þ 2ωμγ ; ε 1 ε 1 p 2 ¼ ω p μ βε 2 þ 2ωμγ ; ε 1 p 3 ¼ βωε 2ε 3 β2 γε 3 þ βγ2 ε 2 ωμ þ ω2 2 μγε 2 ω ωμγ ε 1 þ ε 3 þ 2μγ 2 þ β 2 γ; ε 1 ε 1 ε 1 ε 1 GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 232

3 and p 4 ¼ ω2 με 2 2 ð 1 ε 2 Þ ε 1 β 2 þ 2ω 2 μ 2 γ 2 ε 2 βωμγ ε 1 where β is the propagation wave number along the guide direction, γ is the chirality parameter, and ω is the angular frequency of the wave. One can obtain the modal field structure by finding the eigenvectors and eigenvalues of the matrix in (5). Denoting eigenvalues as k 2 1;2, we get qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 1;2 ¼ p 1 þ p 4 ± ðp 1 p 4 Þ 2 4p 3 p 2 =2 (6) The corresponding eigenfunctions define the hybrid nature of the propagating modes as ðe z ; H z Þ ¼ E z ; jh 1;2 E z (7) where h 1;2 ¼ p 1 k 2 1;2 =p 2 is the mode hybrid factor. The electromagnetic fields can be broken into transverse and longitudinal components as E ¼ E t þ be z E z (8) H ¼ H t þ be z H z (9) In the core region, the longitudinal electromagnetic field components can be written as E z ¼ ½A m J m ðk 1 ρþþ B m J m ðk 2 ρþšexpðjm φþexpð jβzþ (10) H z ¼ j½a m h 1 J m ðk 1 ρþþb m h 2 J m ðk 2 ρþšexpðjm φþexpð jβzþ (11) The transverse field components in the waveguide core are derived as E ρ ¼ j A m k 1 J mð k 1ρÞðu þ wh 1 Þ m ρ J mðk 1 ρþðu v þ ðw þ xþh 1 Þ þ B m k 2 J mð k 2ρÞðu þ wh 2 Þ m ρ J mðk 2 ρþðu v þ ðw þ xþh 2 Þ exp½jmφ ð βzþš E ϕ ¼ ½A m k 1 J mð k 1ρÞ ð v þ xh 1 Þ m ρ J mðk 1 ρþðu v þ wh 1 þ xh 1 Þ þ B m k 2 J mð k 2ρÞ ð v þ xh 2 Þ m ρ J mðk 2 ρþðu v þ wh 2 þ xh 2 Þ exp½jmφ ð βzþš H ρ ¼ A m k 1 J mð k 1ρÞðy uh 1 Þ m ρ J mðk 1 ρþðy þ z uh 1 þ vh 1 Þ þ k 2 J mð k 2ρÞðy uh 2 Þ m ρ J mðk 2 ρþðy þ z uh 2 þ vh 2 Þ exp½jmφ ð βzþš B m H φ ¼ j A m k 1 J mð k 1ρÞðz þ vh 1 Þ m ρ J mðk 1 ρþðy þ z uh 1 þ vh 1 Þ þ k 2 J mð k 2ρÞðz þ vh 2 Þ m ρ J mðk 2 ρþðy þ z uh 2 þ vh 2 Þ exp½jmφ ð βzþš B m (12) (13) (14) (15) where A m and B m are constants, J m () is the Bessel function of the first kind, J mðþis its differentiation with respect to the argument, and m is a positive or negative integer specifying the azimuthal field dependence. Furthermore, u ¼ p 3ωμε 2 p 4 ðβε 3 ωμε 2 γþ ; ε 1 ðp 1 p 4 þ p 2 p 3 Þ v ¼ ωμð ð p 3 þ γp 4 Þ ; p 1 p 4 þ p 2 p 3 w ¼ p 1ωμε 2 þ p 2 ðβε 3 ωμε 2 γþ ; ε 1 ðp 1 p 4 þ p 2 p 3 Þ x ¼ ωμ ð p 1 þ γp 2 Þ ; p 1 p 4 þ p 2 p 3 GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 233

4 Figure 2. Comparison of normalized relative electric field intensity under special conditions of our work (solid line) and results obtained by Baqir and Choudhury [2012] (dotted line). y ¼ p 3ðβε þ ωμε 2 γþþγp 4 ½ðβε ð 1 ε 3 Þ ωμε 2 γþš ; ε 1 ðp 1 p 4 þ p 2 p 3 Þ and z ¼ ω ½ p 3μγ þ p 4 ðε 2 þ μγ 2 ÞŠ : p 1 p 4 þ p 2 p 3 Figure 3. Plot of f(β) with βb (a) EH 01 mode (b) EH 11 mode. GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 234

5 Figure 4. Plot of f(β) with βb for different values of the PEMC admittance parameter M (a) EH 01 mode (b) EH 11 mode. The dispersion equation of the guide is obtainable by using the correct boundary conditions at the interface of the chiroplasma material and the PEMC. Using the perfect electromagnetic conductor boundary conditions [Lindell and Sihvola, 2005; Ruppin, 2006] H z þ ME z ¼ 0 at ρ ¼ b (16) H φ þ ME φ ¼ 0 at ρ ¼ b (17) A set of two homogeneous equations is obtained by using the above boundary conditions, and the determinant made by the coefficients of those equations would be equal to zero in order to obtained nontrivial solution, which yields A m J m ðk 1 ρþðjh 1 þ MÞþB m J m ðk 2 ρþðjh 2 þ MÞ ¼ 0 (18) A m ðjzþ ð vh 1 ÞþMðv xh 1 ÞÞk 1 ρj mð k 1ρÞ mð jðyþ z uh 1 þ vh 1 ÞþMðu v þ ðw þ xþh 1 ÞÞmJ m ðk 1 ρþ þ B m ðjzþ ð vh 2 ÞþMðv xh 2 ÞÞk 1 ρj mð k 2ρÞ mð jðyþ z uh 2 þ vh 2 ÞþMðu v þ ðw þ xþh 2 ÞÞmJ m ðk 2 ρþ ¼ 0 (19) The dispersion equation obtained from above equations can be written as fðβþ ¼ k 1 ðm þ jh 2 Þðjzþ ð vh 1 ÞþMðv xh 1 ÞÞρ J mð k 1ρÞJ m ðk 2 ρþþðk 2 ðz þ vh 2 jmðv xh 2 ÞÞ ð jm þ h 1 Þρ J m k ð 2ρÞþm M 2 ðw þ xþþy þ z ð h2 h 1 ÞJ m ðk 2 ρþþj m ðk 1 ρþ ¼ 0 (20) The above expression (20) leads to the characteristic curve from which the propagation constant and the cutoff frequencies are obtained. The power density using equations (12) to (15) can be obtained from S z ¼ 1 ReðE HÞbe z ¼ Re E ρh φ E φh ρ (21) GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 235

6 Figure 5. Plot of f(β) with βb for different values of chirality parameter (a) EH 01 mode (b) EH 11 mode. 3. Results and Discussions In this section, the above described technique is used to find the dispersion curves and the power flux in the PEMC chiroplasma-filled waveguide. The derived analytical formulation is verified by comparing results obtained in this work, under the special conditions γ =0, M = 0, and ω p = 0, with published results of Baqir and Choudhury [2012] at ε 1 = ε 2 = ε and γ = 0. Both results coincide, and the accuracy of the developed expressions is confirmed as shown in Figure 2. The propagation constant f(β)can be obtained numerically from the characteristic equation (20). By applying the proper boundary conditions, a normalized constant A m or B m is obtainable from either equation (18) or (19). Throughout this analysis, we consider pure proper propagation situation at ω > ω p, which ensures that both permittivity and permeability of the plasma media are positive. The cutoff frequencies are obtainable at values of β when f(β) in equation (20) becomes identically zero. The effects of the admittance parameter M of the PEMC on the characteristic curves are examined firstly as depicted in Figures 3 and 4. Theoretically, the parameter M extends over an infinite range [Ruppin, 2006] in which M = 0 corresponds to the PMC case, M =± corresponds to the PEC case, and the intermediate value of M in between 0 and ± corresponds to the general PEMC. In the presented illustrations, it is found that a PEMC with value of M = ±0.9 or higher behaves similarly as a PEC. The computed results at M =± using the presented formulation precisely approach the results of the waveguide filled with an anisotropic plasma chiral medium under PEC boundary conditions. The values of β and hence the cutoff frequencies of the EH 01 and EH 11 modes can be read from Figures 3a and 3b, respectively, when f(β) approaches zero along the vertical axis with the corresponding βb values of the horizontal axis. Figure 3 represents the variation of f(β) against βb for the case with M = 0 (dotted line), M ± (dashed line), and M = ± 0.5 (solid line). When the admittance parameter M ±, the dispersion curve corresponds to a chiroplasma-filled waveguide coated with a PEC, when M = 0 then the dispersion curve corresponds to a chiroplasma-filled waveguide coated with a PMC, and when the admittance GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 236

7 Figure 6. The power flux density for different chirality parameter values (a) EH 01 mode (b) EH 11 mode. parameter M has some intermediate value, then the dispersion curve corresponds to a chiroplasma-filled waveguide coated with a PEMC. The propagation modes in the core are hybrid modes, and the plots in Figures 3a and 3b provide the values of the propagation constants corresponding to EH 01 and EH 11 guided modes, respectively, at γ = 0.4, ω c = rad/sec, ω c = rad/sec and kb = The cutoff frequency of the PMC coated chiroplasma-filled waveguide is at a higher value than that of the PEC coated chiroplasmafilled waveguide, whereas cutoff frequency of the PEMC coated chiroplasma-filled waveguide varies for different values of M in between 0 and ±. It is reported by Lindell et al. that when the parameter M has a nonzero finite value, then the PEMC medium behaves as a perfect reflector for electromagnetic waves in a different manner from that of the PEC and the PMC medium [Lindell and Sihvola, 2005; Lindell and Sihvola, 2006b; Lindell and Sihvola, 2006a]. The PEMC medium does not allow power to enter and is considered as a boundary with nonreciprocity in its boundary as a notable feature for the PEMC of finite M value. The finite value of M may be chosen between 0 and ± utilizing the formula M = tan α. When α =0 or α =90, then M = 0 and M =±, respectively. We assumed a moderate finite value of M with α between 0 and 90 for analysis that is more sensible. M = 0.5 is selected for the PEMC for the sake of comparison with the PEC and the PMC chiroplasma-filled waveguides. The effect of the PEMC admittance parameter M on the characteristic curves is depicted in Figures 4a to 4b for the values M = ± 0.7 (dotted line), M = ± 0.5 (solid line), and M = ± 0.05 (dashed line), with other parameters specified as γ = 0.4, ω c = rad/sec, ω c = rad/sec, and kb = Figure (4) indicates that higher values of the PEMC admittance parameter M yield higher cutoff frequencies of the EH 01 and EH 11 modes. The effects of the core-material chirality γ on the characteristic curve and the power flux are explained in the following illustrations. The effects of chirality on the characteristic curves are depicted in Figures 5a and 5b for the EH 01 and the EH 11 guided modes, respectively, with γ = 0.5 (dashed line), γ = 0.6 (solid line), and γ = 0.7 (dotted line), M = 0.5, γ = 0.4, ω c = rad/sec, ω c = rad/sec, and kb = Higher values of the chirality parameter have lower cutoff frequencies for the EH 01 mode and higher cutoff frequencies for the EH 11 mode. GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 237

8 Figure 7. The power flux density for the plasma frequencies (a) EH 01 mode (b) EH 11 mode. To examine the propagation of electromagnetic waves in plasma chirowaveguide, the cross-sectional distribution of the power flux density S z of the guided modes is considered. The behavior of the normalized power flux densities for the EH 01 and the EH 11 guided modes are revealed in Figures 6 8. All power density values are normalized to the highest value of the in the illustrated results. Figures 6a and 6(b) show the variation of the normalized power flux density against ρ/b for the EH 01 and the EH 11 guided modes for different values of the chirality parameter. These figures show variation of the normalized power flux density when kb = 1.55, γ = 0.01 (thick solid line), γ = (dashed line), γ = (dotted line), and γ = (solid line) and M = 0.5. The effect of the values of the chirality parameter on the levels of the power flux density of the EH 01 guided mode and the EH 11 guided mode is dissimilar as inferred from Figure (6). The power flux tends to concentrate in the core region in the forward direction for the EH 01 guided mode and supports power flow in the backward direction for the EH 11 guided mode. The power flux density of the EH 11 guided mode at γ = becomes positive which shows that the increased value of γ causes the guide to support power flow in the backward direction. The backward waves are concentrated in the central core and become almost zero beyond the core. The effects of the plasma frequency and the cyclotron frequency on the power flux are revealed in Figures 7 and 8. Figures 7a and 7b express the variations of the power flux densities with respect to ρ/b for the EH 01 and the EH 11 guided modes for different values of the plasma frequency at M = 0.5. The magnitude of the power flux density increases with the increase of the plasma frequency in the forward direction for the EH 01 guided mode, while it marginally increases and supports power flow in the backward direction for the EH 11 mode. The power flux S z is focused in the middle of the core in both modes. Figures 8a and 8b show the variation of the power flux density with respect to ρ/b for different values of the cyclotron frequency for the EH 01 and the EH 11 guided modes at M = 0.5. The magnitude of the power flux density decreases with the increase of the cyclotron frequency in the forward direction for the EH 01 guided mode and in the backward direction for the EH 11 guided mode. The power flux S z is focused in the middle of the core. GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 238

9 Figure 8. The power flux density for cyclotron frequencies (a) EH 01 mode (b) EH 11 mode. 4. Conclusions The electromagnetic wave propagation in a circular perfect electromagnetic conductor waveguide filled with a chiroplasma material has been analyzed in this paper. Expressions for the electromagnetic field components in the chiroplasma circular waveguide have been established. The cutoff frequencies, dispersion curves, and the propagation modes in the chiroplasma-filled waveguide have been examined. The chirality value and the cyclotron and the plasma frequencies have an obvious control on the magnitude of the power flow propagating along the guide. The higher modes tend to support backward propagating waves with power flux in the reverse orientation along the guide. Higher values of the chirality parameter cause the power flux to decrease for the EH 01 mode in the forward direction and to increase for the EH 11 mode in the backward direction and tend to focus in the middle of the waveguide core in both modes. For the EH 11 mode, the power flux flow is positive at reduced chirality. Increasing the value of the chirality causes the wave number to be negative and hence causes backward propagation. Higher values of the plasma frequency cause the power flux to increase, while higher cyclotron frequency values causes the power flux to decrease in the waveguide core material. Acknowledgments 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 Project RG Simulations data supporting the Figures 2 8 are available from the authors. References Abdoli-Arani, A. (2014), Modification of density profile at interaction of three superposing fundamental modes with plasma in a cylindrical waveguide, Waves Random Complex Media, 24, Ali, M. M., M. J. Mughal, A. A. Rahim, and Q. A. Naqvi (2012), The guided waves in planar waveguide partially filled with strong chiral material, Int. J. Appl. Electromagn. Mech., 38, Baqir, M., and P. Choudhury (2012), On the energy flux through a uniaxial chiral metamaterial made circular waveguide under PMC boundary, J. Electromagn. Waves Appl., 26, Baqir, M., and P. Choudhury (2014), Dispersion characteristics of optical fibers under PEMC twists, J. Electromagn. Waves Appl., 28, Dong, J.-F., and J. Li (2012a), Characteristics of guided modes in uniaxial chiral circular waveguides, Prog. Electromagn. Res., 124, Dong, J.-F., and J. Li (2012b), Guided modes in the circular waveguide filled with uniaxial chiral medium, Int. J. Appl. Electromagn. Mech., 40, GHAFFAR AND ALKANHAL CHIROPLASMA WAVEGUIDE 239

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