Electromagnetic waves in parallel plate uniaxial anisotropic chiral waveguides

Abdul Ghaffar; Majeed A. S. Alkanhal

doi:10.1364/OME.4.001756

1. Introduction

Studies on propagation of electromagnetic waves in material waveguides are important in the current arena of advanced technologies for device design. Material waveguides have a lot of uses in many applications in microwave engineering and, hence, many types of waveguides with negative index metamaterials have been investigated for this purpose [1,2]. Many published researches on waveguides based on new materials show that chiral metamaterials have been of great research interest [3–5]. These metamaterials may exhibit special electromagnetic response that does not occur in natural materials. Such type of synthesized materials can be designed to have their permittivity and permeability artificially designed to have negative or positive values. Studies relevant to these materials that focused on isotropic chiral media have been carried out by several authors [6,7].

The uniaxial anisotropic chiral material (UACM) is a generalization of the well known isotropic and chiral media. Lindell et al. [8] studied the UACM using the vector transmission line approach. In these chiral materials, chirality appears only for a special direction. In this direction the permittivity and permeability have different values as compared to their values in the isotropic transverse plane. Usually, a UACM can be realized by doping small chiral items (such as wire spirals) into an anisotropic host medium. Cheng and Cui [9] investigated negative refractions in UACM. Many researchers studied the waveguides with UACM filling [10–15]. Within this context, parallel plate waveguides filled with chiral media have attracted noticeable research efforts and numerous researchers focused on the guides with isotropic chiral media [1–7].

The previous researches motivated adding this study of electromagnetic waves in UACM filled planar waveguides. In this study, we analytically explore electromagnetic fields, energy flux, and total power, propagating in a perfect electric conducting planar waveguide filled with UACM. The dispersion relation for the propagation of energy flux in the guide is obtained. The derived general formulas and the computed results in this work yield the results presented in [1], for the special case of material parameters for isotropic chiral filling, which verified the analytical formulations presented in this paper. The present analysis is the general for following planer waveguide problems under special conditions, which shows the novelty of our work i.e., from a simple formulation the subsequent waveguide problems can be formulated into anisotropic/ isotropic chiral filled parallel plate waveguide and anisotropic/ isotropic achiral filled parallel plate waveguide. The hybrid modes will propagate in UACM filled parallel plate waveguide support to the backward waves, which results into the formation of slow light in the guide. The proper selection of waveguide material may support the slow waves originate due to back ward wave to be strong in the guide. This analysis may also be helpful to find out the propagating region and the decaying region of guide which is prominent feature of the work. The harmonic (iωt) time dependence is adopted and suppressed in what follows.

2. Formulations

The uniaxial anisotropic chiral medium (UACM) parallel plate waveguide is shown in Fig. 1. It consists of two infinite perfect electric conductor (PEC) parallel planes in the y and z directions and is filled with UACM. The incident wave is considered to be propagating in the z direction. The constitutive relations for the UACM are given as [10–15].

\begin{array}{l} D = [ε_{t} {\bar{\bar{I}}}_{t} + ε_{z} {\hat{e}}_{z} {\hat{e}}_{z}] . E - j γ \sqrt{μ_{0} ε_{0}} {\hat{e}}_{z} {\hat{e}}_{z} . H \\ B = [μ_{t} {\bar{\bar{I}}}_{t} + μ_{z} {\hat{e}}_{z} {\hat{e}}_{z}] . H + j γ \sqrt{μ_{0} ε_{0}} {\hat{e}}_{z} {\hat{e}}_{z} . E \end{array}

Unit vectors for Cartesian coordinate system are denoted by

{\hat{e}}_{x}, {\hat{e}}_{y}

and

{\hat{e}}_{z}

. In the above relations

γ

is the chirality parameter which is the cause of electromagnetic coupling in the UACM and

{\bar{\bar{I}}}_{t} = {\hat{e}}_{x} {\hat{e}}_{x} + {\hat{e}}_{y} {\hat{e}}_{y} .

Parameters

μ_{t}

and

ε_{t}

are the transverse components of the permeability and permittivity and

μ_{z}

and

ε_{z}

are the permeability and permittivity of the longitudinal components, respectively. The symbols

μ_{0}

and

ε_{0}

are for the permeability and permittivity of free space, respectively.

Fig. 1 A parallel plate uniaxial anisotropic chirowaveguide.

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The excited fields in the uniaxial chiral medium propagate as two components with different wave numbers [9]. The two wave numbers in uniaxial chiral medium are defined as [12–15]

k_{\pm c}^{2} = λ^{2} / 2 [μ_{z} / μ_{t} + ε_{z} / ε_{t} \pm \sqrt{{(μ_{z} / μ_{t} - ε_{z} / ε_{t})}^{2} + 4 γ^{2} μ_{z} ε_{z} / μ_{t} ε_{t}}]

where

λ^{2} = ω^{2} μ_{t} ε_{t} - β^{2} .

With no variation along the y and x axes, we have

δ^{2} E_{\pm z} / δ x^{2} + k_{\pm c}^{2} E_{+ z} = 0

The solutions are found to be in the following form

E_{\pm z} = A_{\pm} \cos (k_{\pm c} x) + G_{\pm} \sin (k_{\pm c} x)

where

A_{\pm}

and

G_{\pm}

are the unknown constants to be obtained by using the appropriate boundary conditions. The corresponding eigen-functions can be written as follows [12]:

(E_{\pm z,} H_{\pm z}) = (E_{\pm z,} j α_{\pm} / η_{t} E_{\pm z})

with

α_{\pm} = (α_{\pm} / λ^{2} - ε_{z} / ε_{t}) \sqrt{ε_{t} μ_{t}} / γ \sqrt{μ_{z} ε_{z}}, η_{t} = \sqrt{ε_{t} / μ_{t}}

and

λ = \sqrt{k^{2} - β^{2}}

The total electric and magnetic fields in the UACM can be expressed as [14]

\begin{array}{l} E = E_{+} + E_{-} \\ H = j / η_{t} (H_{+} + H_{-}) \end{array}

The electromagnetic fields can be broken into transverse and longitudinal components as

\begin{array}{l} E = E_{t} + {\hat{e}}_{z} E_{z} \\ H = H_{t} + {\hat{e}}_{z} H_{z} \end{array}

The relationships between the transverse and the axial components are derived as [8,11]

E_{t} = (- j β / k_{+ c}^{2} \nabla_{t} - k_{t} α_{+} / k_{+ c}^{2} {\hat{e}}_{z} \times \nabla_{t}) E_{+ z} + (- j β / k_{- c}^{2} \nabla_{t} - k_{t} α_{-} / k_{- c}^{2} {\hat{e}}_{z} \times \nabla_{t}) E_{- z}

Using the PEC boundary conditions

E_{\pm z} = 0

and

E_{\pm y} = 0

at

x = \pm a / 2

we get

(\begin{matrix} \cos (k_{+ c} a / 2) & \sin (k_{+ c} a / 2) & \cos (k_{- c} a / 2) & \sin (k_{- c} a / 2) \\ \cos (k_{+ c} a / 2) & - \sin (k_{+ c} a / 2) & \cos (k_{- c} a / 2) & - \sin (k_{- c} a / 2) \\ \frac{α_{+}}{k_{+ c}} \sin (k_{+ c} a / 2) & - \frac{α_{+}}{k_{+ c}} \cos (k_{+ c} a / 2) & - \frac{α_{-}}{k_{- c}} \sin (k_{- c} a / 2) & \frac{α_{-}}{k_{- c}} \cos (k_{- c} a / 2) \\ \frac{α_{+}}{k_{+ c}} \sin (k_{+ c} a / 2) & \frac{α_{+}}{k_{+ c}} \cos (k_{+ c} a / 2) α_{+} / k_{+ c} & - \frac{α_{-}}{k_{- c}} \sin (k_{- c} a / 2) & \frac{α_{-}}{k_{- c}} \cos (k_{- c} a / 2) \end{matrix}) (\begin{matrix} A_{+} \\ G_{+} \\ A_{-} \\ G_{-} \end{matrix}) = 0

The determinant of the 4x4 matrix equal to zero leads to the following characteristics Eqs.

α_{-} \sin (k_{\pm c} a / 2) \cos (k_{\pm c} a / 2) / k_{- c} + α_{+} \sin (k_{\pm c} a / 2) \cos (k_{\pm c} a / 2) / k_{+ c} = 0

From the above Eqs., it can be shown that:

Ω_{1, 2} = (k_{+ c} + k_{- c}) \sin (k_{+ c} a / 2 + k_{- c} a / 2) \pm (k_{+ c} - k_{- c}) \sin (k_{+ c} a / 2 - k_{- c} a / 2) = 0

Equation (11) leads to a set of bifurcated modes starting from the cut-off frequency .Applying the boundary conditions, the coefficients

A_{\pm}

and

G_{\pm}

can be deduced [11]

A_{+} Ω_{2} (β) = G_{+} Ω_{1} (β)

with

A_{-} = - A_{+} \cos (k_{+ c} a / 2) / \cos (k_{- c} a / 2)

and

G_{+} = - G_{-} \sin (k_{+ c} a / 2) / \sin (k_{- c} a / 2)

. It is noted from these relations that for any propagating mode in parallel plate waveguide, either

Ω_{1} = 0

or

Ω_{2} = 0

which corresponds to an arm of bifurcated modes in the dispersion diagram. From (16), it is observed that when

Ω_{1} = 0

then

A_{+}

vanishes while

G_{+}

disappears when

Ω_{2} = 0

. As a result, the arm of modes corresponding to

Ω_{1} = 0

has

E_{z}

and

H_{z}

as odd functions of x coordinate and

E_{x},

E_{y},

H_{x}

and

H_{y}

as even functions of x. On the other hand, for the other arm of the bifurcated modes corresponding to

Ω_{2} = 0

,

E_{z}

and

H_{z}

are even functions of x coordinates and the transverse components

E_{x},

E_{y},

H_{x}

and

H_{y}

are now, odd function of x. The energy flow can be obtained

S_{z} = 1 / 2 Re (E \times H) . {\hat{e}}_{z}_{z} = 1 / 2 Re (E_{x} H_{y}^{*} - E_{y} H_{x}^{*})

3. Results and discussions

In this section, the above procedure is utilized to compute the dispersion curves, the energy flux, and the behavior of the electromagnetic field components in the parallel plate waveguide filled with UACM. Three different sets of electromagnetic constitutive parameters of the UACM: $ε_{t} > 0, ε_{z} > 0; ε_{t} < 0, ε_{z} < 0$ and $ε_{t} < 0, ε_{z} > 0$ are considered for further analysis. To check the accuracy of the formulation and the derived expressions, we compared numerical results obtained from Eq. (8) under special conditions that $ε_{t} = ε_{z} = ε_{0},$ and $μ_{t} = μ_{z} = μ_{0}$ at $γ = - 0.11, k a = 0.86,$ and $β a = 3$ with published results of [5,6]. Both results show similar behavior as depicted in Fig. 2.

Fig. 2 Comparison of intensity components of our work (dotted) and results in [1](solid line).

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It is assumed that $μ_{t} = μ_{z} = μ_{0}$ the operating wave length $λ = 1.55$ micron and $a = 0.11$ microns throughout the presented computations. Figures 3 and 4 are plots of Eq. (11), which show the magnitude variations of modes $Ω_{1}$ and $Ω_{2}$ against $β a$ corresponding to three material variations; type-i: $ε_{t} = 2 ε_{0,}$ $ε_{z} = 3 ε_{0,}$ type-ii: $ε_{t} = - 2 ε_{0,}$ $ε_{z} = 3 ε_{0,}$ and type-iii: $ε_{t} = 2 ε_{0,}$ $ε_{z} = - 3 ε_{0,}$ respectively. From these Figs., we observe that curves are bifurcated starting from the same cut off frequency and their variation may be noticed as $β a$ increases. From Fig. 3(a), it is observed that when $ω^{2} μ_{t} ε_{t} < β^{2},$ both waves, $k_{+ c}$ and $k_{- c}$ , are in the propagation region but when $ω^{2} μ_{t} ε_{t} > β^{2}$ both waves enter into the decaying region and will not propagate. Similarly both $Ω_{1}$ and $Ω_{2}$ modes will propagate till $ω^{2} μ_{t} ε_{t} < β^{2}$ and begin to decay at $ω^{2} μ_{t} ε_{t} > β^{2}$ For material type ii, Fig. 3(b) show that the $k_{+ c}$ wave propagates in the forward direction when $ω^{2} μ_{t} ε_{t} > β^{2}$ and attenuates when $ω^{2} μ_{t} ε_{t} < β^{2}$ whereas $k_{- c}$ wave attenuates when $ω^{2} μ_{t} ε_{t} > β^{2}$ and propagates in the backward direction when $ω^{2} μ_{t} ε_{t} < β^{2} .$ Due to these conditions mode $Ω_{1}$ will be in the propagation mode in the $ω^{2} μ_{t} ε_{t} > β^{2}$ region and when $ω^{2} μ_{t} ε_{t} < β^{2}$ , it will decay. Similarly for mode $Ω_{2}$ $ω^{2} μ_{t} ε_{t} > β^{2}$ is the evanescent region and $ω^{2} μ_{t} ε_{t} < β^{2}$ is the propagation region. Figure 3(c) show the situation for material type iii and it is noted that the $k_{+ c}$ wave is attenuated and $k_{- c}$ is propagating in both regions $ω^{2} μ_{t} ε_{t} > β^{2}$ and $ω^{2} μ_{t} ε_{t} < β^{2}$ . Both regions are evanescent regions for mode $Ω_{1}$ and propagation regions for $Ω_{2}$ mode.

Fig. 3 Characteristics curves for mode $Ω_{1}$ (thick solid line) and mode $Ω_{2}$ (thick dotted line) in (a) type-i material, (b) type-ii material, and (c) type-iii material.

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Fig. 4 The energy flux density in (a) type-i and type-ii and (b) type-iii materials for different values of chirality parameter.

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The behaviors of the energy flux densities are revealed in Figs. 4(a) and 4(b). These Figs. show the variation of the normalized energy flux density against $x / a$ , corresponding to the three material types at allowed values of the propagation constant obtained from the characteristics curves for different values of chirality parameters. The existence of backward wave propagation phenomenon is prominent in the planar waveguide filled with the UACM. It can be observed that in type-i and type-ii materials, the value of the chirality parameter changes the amplitude and the orientation of the flux density. The flux density appears in forward direction at higher values of the chirality parameter and in the backward direction at lower values of the chirality parameter. In case of type-iii material, the flux density propagates in the backward direction at all values of the chirality parameter.

4. Conclusions

Theoretical and numerical analysis of electromagnetic waves in parallel plate waveguides containing UACM are presented. The dispersion diagram, the cut-off frequencies, the propagating modes, and the evanescent modes in the UACM waveguide have been derived. Properties of the energy flux propagation in this waveguides have been investigated and some special properties are reported. It is observed that the UACM in parallel plate waveguide supports the propagation of backward waves. The values of the chirality parameter can influence the magnitudes of the energy flux propagating in the forward direction and could reverse the orientation of the flux density propagation at some chirality values as well.

Acknowledgment

The authors would like to thank the Deanship of the Scientific Research and the Research Center at the College of Engineering, King Saud University for their financial and administrative support.

References and links

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Electromagnetic waves in parallel plate uniaxial anisotropic chiral waveguides

Abstract

1. Introduction

2. Formulations

3. Results and discussions

4. Conclusions

Acknowledgment

References and links

Cited By

Figures (4)

Equations (13)

Optical Materials Express