Electromagnetic waves in parallel plate uniaxial ... - OSA Publishing

Electromagnetic waves in parallel plate uniaxial anisotropic chiral waveguides Abdul Ghaffar1,2,* and Majeed A. S. Alkanhal 1 1

Dept. of Electrical Engineering, King Saud University, Riyadh, Saudi Arabia 2 Dept. of Physics, University of Agriculture, Faisalabad, Pakistan * [email protected]

Abstract: Theoretical analysis of electromagnetic wave propagation in planar waveguides filled with uniaxial chiral anisotropic media is presented. The guide parallel plates are assumed to be perfect electric conductors. The material filling the waveguide is a generalized medium that incorporates chiral and metamaterials. The behavior of the field intensities, the dispersion curves, and the energy flux for three varieties of uniaxial chiral media are examined numerically. The results demonstrate the phenomena of backward waves in uniaxial anisotropic chiral media. The comparisons of the computed results of the presented general formulations with published results for some material cases confirm the accuracy of the presented analysis. OCIS codes: (230.0230) Optical devices; (230.7390) Waveguides, planar.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

P. Pelet and N. Engheta, “The theory of chirowaveguides,” IEEE Trans. Antenn. Propag. 38(1), 90–98 (1990). I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, Incorporated, 1994). R. Oussaid and B. Haraoubia, “Behavior of chiral material in terms of a guided wave propagation,” I. J.of Appl. Electromagnetics and Mechanics 19(1–4), 631–635 (2004). R. Oussaid and B. Haraoubia, “Longitudinal and transverse operators formalism for chiral media: Application to guided structures filled with chiral material,” Can. J. Phys. 82(5), 367–378 (2004). S. Gulistan, A. A. Syed, and Q. A. Naqvi, “Fields in fractional dual DB waveguides containing chiral nihility metamaterials,” J. Electromagn. Waves Appl. 26(16), 2130–2141 (2012). M. M. Ali, M. J. Mughal, A. A. Rahim, and Q. A. Naqvi, “The Guided waves in planar waveguide partially filled with strong chiral material,” I. J.of Appl. Electromagnetics and Mechanics 38(2–3), 139–149 (2012). A. Viitanen and I. V. Lindell, “Plane wave propagation in uniaxial bianisotropic medium with an application to a polarization transformer,” Int. J. Infrared Millim. Waves 14(10), 1993–20010 (1993). Q. Cheng and T. J. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B 73(11), 113– 104 (2006). S. F. Mahmoud and A. J. Viittanen, “Modes in a hard surface waveguide with uniaxially anisotropic chiral material filling,” Prog. in Electromag. Res. 39, 265–279 (2003). J. F. Dong and J. Li, “Characteristics of guided modes in uniaxial chiral circular waveguide,” Prog. in Electromag. Res. 124, 331–345 (2012). J. F. Dong and J. Li, “Guided modes in circular waveguide filled uniaxial chiral medium,” I. J. of Applied Electromagnetics and Mech. 40(4), 283–292 (2012). M. A. Baqir and P. K. Choudhury, “On energy Flux through a uniaxial chiral metamaterials made circular waveguide under PMC boundary,” J. Electromagn. Waves Appl. 26(16), 2165–2175 (2012). M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral circular waveguide under DB boundary,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013). M. A. Baqir and P. K. Choudhury, “Flux through guides with microstructured twisted clad DB medium,” J. Nanomater. 2014, 1–6 (2014). M. A. Baqir and P. K. Choudhury, “Propagation through uniaxial anisotropic chiral circular waveguide under DB boundary,” J. Electromagn. Waves Appl. 27(6), 783–793 (2013).

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

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Received 12 Jun 2014; revised 19 Jul 2014; accepted 21 Jul 2014; published 4 Aug 2014

(C) 2014 OSA 1 September 2014 | Vol. 4, No. 9 | DOI:10.1364/OME.4.001756 | OPTICAL MATERIALS EXPRESS 1756

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].

D = ε t It + ε z eˆ z eˆ z  .E − jγ μ 0ε 0 eˆ z eˆ z .H   B = μ t It + µ z eˆ z eˆ z  .H + jγ μ 0ε 0 eˆ z eˆ z .E  

(1)

Unit vectors for Cartesian coordinate system are denoted by eˆ x , eˆ y and eˆ z . In the above relations γ is the chirality parameter which is the cause of electromagnetic coupling in the UACM and It = eˆ x eˆ x + eˆ y eˆ 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.

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Fig. 1. A parallel plate uniaxial anisotropic chirowaveguide.

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±2c λ 2 / 2  µ z / µt + ε z / ε t ± 

( µ z / µt − ε z / ε t )

2

+ 4γ 2 µ z ε z / µt ε t  

(2)

where λ 2 ω 2 µt ε t − β 2 . With no variation along the y and x axes, we have =

0 δ 2 E± z / δ x 2 + k±2c E+ z =

(3)

The solutions are found to be in the following form

= E± z A± cos(k ± c x) + G± sin(k ± c x)

(4)

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

(E

± z,

H ± z ) = ( E± z , jα ± / ηt E± z )

(5)

with α ± = ε t / µt and= λ k 2 − β 2 The total electric (α ± / λ 2 − ε z / ε t ) ε t µt / γ µ z ε z ,ηt = and magnetic fields in the UACM can be expressed as [14]

= E E+ + E−

= H j / ηt ( H + + H − )

(6)

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

E = Et + eˆ z Ez H = H t + eˆ z H z

(7)

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

Et =

( − jβ / k

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2 +c

∇t − ktα + / k+2c eˆ z × ∇t ) E+ z + ( − j β / k−2c ∇t − ktα − / k−2c eˆ z × ∇t ) E− z (8)



Using the PEC boundary conditions E± z = 0 and E± y = 0 at x = ± a / 2 we get

 cos ( k+ c a / 2 )   cos ( k+ c a / 2 ) α  + sin ( k+ c a / 2 )  k+ c α  + sin ( k+ c a / 2 )  k+ c

sin ( k+ c a / 2 )

cos ( k− c a / 2 )

− sin ( k+ c a / 2 ) −

α+ k+ c

α+ k+ c

cos ( k− c a / 2 )

cos ( k+ c a / 2 )

cos ( k+ c a / 2 ) α + / k+ c

− −

α− k− c

α−

k− c (9)

sin ( k− c a / 2 ) sin ( k− c a / 2 )

sin ( k− c a / 2 )   − sin ( k− c a / 2 )   A+  G  α− cos ( k− c a / 2 )   +  = 0 k− c   A−  G  α− cos ( k− c a / 2 )   −  k− c 

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

α − sin ( k± c a / 2 ) cos ( k± c a / 2 ) / k− c + α + sin ( k± c a / 2 ) cos ( k± c a / 2 ) / k+ c = 0 (10) From the above Eqs., it can be shown that:

Ω1,2=

( k+ c + k− c ) sin ( k+ c a / 2 + k− c a / 2 ) ± ( k+ c − k− c ) sin ( k+ c a / 2 − k− c a / 2 )=

0 (11)

Equation (11) leads to a set of bifurcated modes starting from the cut-off frequency .Applying the boundary conditions, the coefficients A± and G± can be deduced [11]

A+ Ω 2 ( β ) = G+ Ω1 ( β )

(12)

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 Ω1 =0 or Ω 2 =0 which corresponds to an arm of bifurcated modes in the dispersion diagram. From (16), it is observed that when Ω1 =0 then A+ vanishes while G+ disappears when Ω 2 =0 . As a result, the arm of modes corresponding to Ω1 =0 has Ez and H z as odd functions of x coordinate and Ex , 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 , Ez and H z are even functions of x coordinates and the transverse components Ex , E y , H x and H y are now, odd function of x. The energy flow can be obtained

S= 1/ 2 Re ( E × H ) .eˆ= 1/ 2 Re ( Ex H y∗ − E y H x∗ ) z z

(13)

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= ε= ε 0 , and µ= µ= µ0 at t z t z

γ= −0.11, ka = 0.86, and β a = 3 with published results of [5,6]. Both results show similar behavior as depicted in Fig. 2.

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

It is assumed that µ= µ= µ0 the operating wave length λ = 1.55 micron t z 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 Ω1 and Ω 2 against β a corresponding to three material variations; type-i: ε t = 2ε 0, ε z = 3ε 0, type-ii: ε t = −2ε 0, ε z = 3ε 0, and typeiii: ε t = 2ε 0, ε z = −3ε 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 µt ε t < β 2 , both waves, k+ c and k− c , are in the propagation region but when ω 2 µt ε t > β 2 both waves enter into the decaying region and will not propagate. Similarly both Ω1 and Ω 2 modes will propagate till ω 2 µt ε t < β 2 and begin to decay at ω 2 µt 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 µt ε t > β 2 and attenuates when ω 2 µt ε t < β 2 whereas k− c wave propagates in the backward direction attenuates when ω 2 µt ε t > β 2 and 2 2 when ω µt ε t < β . Due to these conditions mode Ω1 will be in the propagation mode in the ω 2 µt ε t > β 2 region and when ω 2 µt ε t < β 2 , it will decay. Similarly for mode Ω 2 ω 2 µt ε t > β 2 is the evanescent region and ω 2 µt 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 µt ε t > β 2 and ω 2 µt ε t < β 2 . Both regions are evanescent regions for mode Ω1 and propagation regions for Ω 2 mode. 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.

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Fig. 3. Characteristics curves for mode Ω1 (thick solid line) and mode Ω 2 (thick dotted line) in (a) type-i material, (b) type-ii material, and (c) type-iii material.

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

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