Modal properties of parallel plate waveguide partially loaded with double negative materials

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Modal properties of parallel plate waveguide

partially loaded with double negative materials

Ayşegül Pekmezci & Ercan Topuz

To cite this article: Ayşegül Pekmezci & Ercan Topuz (2019) Modal properties of parallel plate waveguide partially loaded with double negative materials, Journal of Electromagnetic Waves and Applications, 33:11, 1443-1452, DOI: 10.1080/09205071.2019.1614096

To link to this article: https://doi.org/10.1080/09205071.2019.1614096

Published online: 09 May 2019.

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JOURNAL OF ELECTROMAGNETIC WAVES AND APPLICATIONS 2019, VOL. 33, NO. 11, 1443–1452

https://doi.org/10.1080/09205071.2019.1614096

Modal properties of parallel plate waveguide partially loaded

with double negative materials

Ayşegül Pekmezci and Ercan Topuz

Electronics and Communications Engineering Department, Doğuş University, Istanbul, Turkey

ABSTRACT

Properties of eigen-solutions in parallel plate waveguides partially loaded with Lorentz type Double Negative Materials (DNG), mod-elled by lossless/lossy, identical/non-identical electric, magnetic parameters are presented. Modal cut-off phenomena associated with surface waves, transitions between evanescent/propagating, forward/backward waves and their dependence on frequency, fill-ing factor and dispersive material parameters are investigated. Novel conditions are given for the existence of surface waves and for the emergence of complex eigenvalues in the absence of losses, and supported with numerical results of exact frequency domain analysis.

ARTICLE HISTORY Received 4 October 2018 Accepted 24 April 2019 KEYWORDS

Backward waves; complex waves; double negative material (DNG); eigen-solutions; parallel plate waveguide; surface waves

Introduction

Investigations on propagation in dispersive materials having simultaneously negative per-mittivity and permeability date back to the pioneering theoretical work of Veselago in 1968 [1]. It was pointed out that these kinds of materials would have unusual characteristics, such as the negative index of refraction and backward wave propagation. These materials which are not observed in nature were coined as Metamaterial (MTM) or Double Negative Mate-rial (DNG). At the end of the twentieth century, Pendry et al. experimentally demonstrated that a composite medium of periodically placed thin metallic wire structures exhibits nega-tive permittivity (ε) and periodic arrangement of metallic split ring resonators (SRR) exhibit negative permeability (μ) over certain frequency bands [2,3]. In 2000, Smith et al. combined these two structures in a single configuration to create a material possessing a negative index of refraction for a frequency band in the GHz range [4]. More recently, various shapes of split ring resonators (SRRs) have been studied to generate dispersive permeability, and new double-negative (DNG) structures are realized from the arrangement of these SRRs and metallic wires or SRRs alone printed on dielectric substrates [5–7]. The unusual propagation characteristics of DNG media has led to new applications in diverse areas such as imaging, cloaking, antennas and guiding structures which are covered in numerous papers, special issues and several books [8–15].

Reports investigating the propagation characteristics and potential applications of metamaterial loaded planar and rectangular waveguides focus on problems dealing with CONTACT Ayşegül Pekmezci apekmezci@dogus.edu.tr

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Figure 1.Conﬁguration of parallel plate waveguide partially loaded with a pair of parallel layers con-sisting of DPS (layer 1) and DNG (layer 2) materials.

specific applications or propagation characteristics, often using the rather restrictive assumption of nondispersive DNG media [16–18]. Alu and Engheta [16] presented a com-prehensive analytical analysis and numerical results for the modes propagating in bilayer parallel plate waveguide loaded with pairwise combinations of lossless single negative (ε > 0, μ < 0 or ε < 0, μ > 0), DPS (double positive) and DNG slabs of arbitrary thick-nesses at fixed frequency (i.e. for constant values of constitutive parameters). Nefedov and Tretyakov [17] investigated DPS-DNG slab combinations with several layer thicknesses and presented numerical results for dispersion characteristics of propagating modes over wide frequency bands assigning constant values to material parameters. Cory and Shtrom [18] addressed DPS-DNG slab configuration in a parallel plate waveguide of given plate separa-tion and presented numerical results of modal dispersion for different values of geometrical loading factor (t/a in Figure1) over wide frequency bands, also using constant values for material parameters. In this paper, we properly take dispersive effects into account and investigate the characteristics of modal fields supported by a parallel plate waveguide par-tially loaded with a homogeneous DNG slab characterized by Lorentz model in the config-uration shown in Figure1. Considering the canonical problem of propagation in DPS/DNG layered planar waveguides, the existence conditions of and the transitions between the evanescent, forward/backward propagating, surface and complex waves in the absence of losses are investigated via exact frequency domain analysis. The results presented in this paper may provide potential applications in the design of ever increasing MTM devices such as multiband and evanescent wave filters, phase shifters and antennas [11–15].

It should be noted that this paper does not refer to a particular realization format of the DNG medium, rather to a general homogeneous 2-D media with dispersive characteristics represented by relation (2) at microwave frequencies within the X-band. To investigate all wave types supported by the structure, the frequency region is chosen to extend a few GHz on both sides of the frequency where the refractive index of the DNG layer becomes−1. Eigen-solutions in DNG loaded regions

For TEztype modes, the non-vanishing field components are Ey, Hx, Hzand the eigenvalue

equation can be expressed as in [19]

F(ω, β) = μr1 k1 tan(k1t) + μr2 k2 tan[k2(a − t)] = 0, (1) where ki= k0

εriμri− b2 with i= 1, 2; k0= ω√ε0μ0; b= β/k0, and ejωt time

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JOURNAL OF ELECTROMAGNETIC WAVES AND APPLICATIONS 1445

Table 1.Lorentz model parameters.

Parameter ωpe ω0e e ωpm ω0m m

Scenario A _ω_se√48_/5 _ω_se_/5 0 ;_ω_se_/200 _ω_sm√48_/5 _ω_sm_/5 0 ;_ω_sm_/200 Scenario B ω_se√48/5 ω_se/5 0 ;ω_se/200 ω_sm4/3 ω_sm/3 0 ;ω_sm/200

which for the lossless case are: real, imaginary or complex for propagating, evanescent and complex modes, respectively. In Figure1, layer 1 is a nondispersive DPS medium with a thickness of t and assumed to be air (εr1= μr1= 1), layer 2 is a dispersive DNG medium

whose electric (χe) and magnetic (χm) susceptibilities are modelled with the single pole

Lorentz model: χe,m= ω2 pe,pm ω2 oe,om− ω2 + jωe,m ; εr2= ε0(1 + χe) μr2 = μ0(1 + χm) , (2)

whereωpe,pmdenotes the plasma,ωoe,omthe resonance frequency, ande,mthe damping

coefficients. In this model, permittivity and permeability have negative real parts if the oper-ating frequency is betweenωoe,om,

ω2

oe,om+ ω2pe,pm

. When the resonance frequency is chosen as zero, the generic function given in relation (2) reduces to Drude model, and yields negative values for frequencies below the plasma frequency [ω < ωpe,pm]. Hence, beside

being more general, a double Lorentz medium also has the potential of supporting broader negative refractive index bandwidth than the Drude–Lorentz medium [11]. The normalized Lorentz parameters used in numerical calculations for two scenarios are listed in Table1, whereωse,smdenote reference frequencies yieldingεr2(ωse)= −1 and μr2(ωsm)= −1 in

the lossless case, respectively.

In the dispersion diagrams shown in Figure2positive values are assigned to real, and negative values to imaginary values of b= β/k0, representing phase progression of

prop-agating modes and decay of the evanescent modes in the increasing z-direction. It has to be noted that the dispersion diagrams given for the lossless Lorentz model are only slightly modified by the inclusion of small losses ( = ωs/1000 and = ωs/200) to the

model. Indeed, the differences between dispersion diagrams are not discernable in the scale of the figures, except for the reference frequency (fs) and vicinities of cut-off

transi-tions (fc) as shown in Figure2(b) and Figure2(c). Figure2(a) provides a convenient means

for visually assessing the dependence of phase velocity (νp) on frequency, since at any

fre-quency its relative value is just the inverse of the value read from the dispersion curve,

c/νp = b and the dependence of group velocity (νg) on frequency where the slope of the

curve at any point is equal to the value of relative group velocity at the corresponding fre-quency, dko= dβ = νg/c. Thus, phase and group velocities are in opposite directions for

both propagating modes over the plotted frequency ranges for t= a/4. We will denote this behaviour, typical in DNG media, as “contra-directional” flow (backward mode) as opposed to forward mode or “co-directional” flow typical in DPS media, where phase and group velocities are in the same directions.

Clearly, the frequency domain analysis of this paper is applicable for any given data set for DNG constitutive parameters. To demonstrate this and provide a test case for assessing the accuracy of our codes, the dispersion diagram given in [18, Figure3] was calculated with the same parameters, a= 2.286 cm, d = (a − t) = a/4, εr1= μr1= 1, εr2= −9, μr2= −1,

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Figure 2.Dispersion diagram for Scenario A (fse= fsm= fs= 9 GHz) where t = a/4 for (a) lossless case

and lossy cases around (b) fs= 9 GHz and (c) fc = 8.26 GHz.

Figure 3.Dispersion diagram of propagating and one of the complex modes calculated with the same parameters used in [18], together with superimposed data points from Figure3in [18].

with the data reported in [18] for comparison purposes. One observes that the dispersion curves of both propagating and complex modes with real (βr) and imaginary (βi) parts are in

perfect agreement over the frequency range considered, kod∈ 0.3 − 3, f (GHz) ∈ 2.5 − 25.

Modal cut-off phenomena

A salient feature of wave propagation in a parallel plate waveguide partially loaded with lossless DNG slab is the existence of modal cut-off frequencies characterized by vanish-ing of the group velocity, arisvanish-ing in three distinctly different ways. One can differentiate these by the values of phase velocity at cut-off frequencies, as; (A) evanescent – propagat-ing wave transitions wherein the phase velocity increases to infinity, (B) co-directional – contra-directional flow transitions wherein phase velocity has a finite value, and (C) cut-off transitions around the reference frequency fswherein the phase velocity tends to zero. In

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Figure 4.Variation of cut-oﬀ frequencies atβ = 0 versus t/a for f_se= f_sm= f_s= 13 GHz.

Figure 5.Plot of (a) dispersion diagram, (b) phase/group velocities and (c) E-ﬁeld distribution in the cross-section of waveguide at f = 7.5 GHz for Scenario A when t = 3a/4 and fse= fsm= fs= 13 GHz.

Evanescent – propagating wave transitions

Cut-off frequencies (fc), defined byβ (fc)= 0 are shown in Figure4as a function of the

geo-metrical filling factor parameter (t/a), for Scenario A and reference frequency fs = 13 GHz

(horizontal line). For instance, fc1, fc2and fc3indicated in figure for t/a= 0.75 correspond

to the transition frequencies from evanescent to propagating modes in the dispersion diagram given in Figure5(a).

Co-directional – contra-directional flow transitions and the emergence of complex eigenvalues

For certain parameter ranges the dispersion diagrams exhibit interesting behaviours, an example of which is depicted in Figure5. Let us focus on the propagating mode solutions in frequency region 6.5–8.5 GHz in Figure5(a) and the variation of phase and group velocities in Figure5(b). We see that this mode undergoes a cut-off transition of the type considered in

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Figure 6.Variation of the real (dash) and imaginary (dot) parts for one of the complexβ values. The numbers on the propagating mode dispersion curve (solid) correspond to those indicated in Figure5.

(A) above, at the frequency of 6.96 GHz (koc= 1.46) where the phase velocity is infinite and

the group velocity is zero. If the frequency is increased up to the “turning point” at about 8.41 GHz (kot= 1.76) the solution ceases to exist at higher frequencies, but the dispersion

curve smoothly connects to another branch extending towards lower frequencies. Thus, there are two propagating wave solutions at all frequencies between 6.96 and 8.41 GHz, one co-directional and the other contra-directional, and at the cut-off frequency, the common phase velocities remain finite while the group velocity becomes zero. The dominant part of the transmitted power resides in the DPS (air) medium for the co-directional case and in the DNG medium for the contra-directional case, as demonstrated in Figure5(c). These two solutions approach each other as the frequency is increased, merge at the cut-off frequency 8.41 GHz and disappear beyond it. This is the frequency at which complex conjugate pairs of solutions forβ emerge, and one of the complex conjugate pairs with positive real and negative imaginary parts are plotted in Figure6.

Surface waves and cut-off transition around fs

Surface wave type modal fields which cling to the interface of DPS/DNG media exist in parameter regions wherein one has solutions for real values ofβ, and imaginary values of both k1and k2. This leads to the condition, b = β/k0 > max (εr1μr1, εr2μr2). Writing

imaginary values of k1and k2as, ¯k1, ¯k2> 0 the eigenvalue equation is expressed as,

¯k1t = tanh−1[A tanh(¯k2d)] where A = |μr2|

μr1

¯k1

¯k2

, d= a − t. (3)

Using relation (3), the conditions for the existence of surface wave type solutions in the partially loaded parallel plate waveguide can then be stated as:

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Figure 7.Dispersion diagrams of propagating modes for lossless non-identical parameters given in Sce-nario B: (solid) fse= 9 GHz, fsm= 10 GHz, t = 3a/4 and (dot) fse= 10 GHz, fsm= 9 GHz, t = a/4. For

both cases refractive index n = −1 at about f = 9.45 GHz indicated with dashed vertical line.

(1) In the frequency range considered the two media should be of the types DPS – DNG, and f has to be in the vicinity of fs.

(2) If fs < fcthen A< 1, a solution exists for d > t and disappears for f < fs.

(3) If fs > fcthen A> 1, a solution exists for d < t and disappears for f > fs.

(4) For any configuration, there will be at most a single surface wave type solution. It has to be noted that conditions (2) and (3) impose constraints on the relative magni-tudes of both widths and constitutive parameters of DNG, DPS slabs, thereby augmenting similar conditions reported in the literature, for supporting surface waves when t and d are free parameters [16, Equation (24)] and for suppressing them when t→ ∞, i.e. for grounded slab [20, Equation (13)].

Dispersion diagrams are given in Figure7for two lossless DNG slabs with non-identical parameters and different filling factors t/a of 1/4 and 3/4. It is clearly seen from this figure that only one of the two propagating modes supported in 7–12 GHz band, yields surface waves with asymptotically decreasing phase and group velocities as the frequencyωsm

whereμr2(ωsm)= −1, is approached. It can easily be shown that the same considerations

also apply for the dual case of TMzmodes in the vicinity ofωsewhereεr2(ωse)= −1. Thus, for

surface wave type slow wave solutions, fsis identified as a cut-off frequency, at which both

phase and group velocities tend to zero. Evidently, surface wave type modes will not be supported in the special case t= d = a/2. The eigenvalue equation in relation (1) becomes an identity at f= fsbut has no solution in this vicinity, thus, the cut-off frequency at f= fs

and t= d indicated in Figure4is a numerical artefact and should be discarded. In this case, propagating fields are obtained only for modes with cut-off frequencies away from fs, as

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Figure 8.Dispersion diagram for Scenario A and t = d = a/2 for fs= 13 GHz.

Conclusion

Properties of the modal fields supported by a parallel plate waveguide partially loaded with a DNG slab are investigated. It is shown that the group velocities of propagating modes vanish at frequencies of the following three distinctly different cut-off transitions:

(A) Evanescent – propagating wave transitions wherein the phase velocity increases to infinity.

(B) Forward – backward propagating wave (co-directional – contra-directional flow) tran-sitions wherein phase velocity has a finite value.

(C) Surface wave transitions at reference frequencyωsm(for TEzmodes) or its immediate

vicinity wherein phase and group velocity tends to zero.

Of special interest are the cut-off transitions (B) and (C) mentioned above. It is shown that these lead to novel conditions for the existence of surface waves and of complex eigenval-ues in the absence of losses, which can be phrased as: excluding the case of t/a ∼= 1/2, a single mode will become of surface wave type as the reference frequency is approached, other propagating modes of the structure may undergo through a type (B) cut-off transition in the frequency region considered resulting in the emergence of complex eigenvalues.

The results presented in this paper may provide potential applications in the design of conductor backed MTM structures widely used in designing planar antennas with improved/tailored performances, and also in the design of MTM devices such as multiband and evanescent wave filters, phase shifters, cavity resonators, couplers and diplexers.

Disclosure statement

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Notes on contributors

Ayşegül Pekmezciwas born in Zonguldak, Turkey, in February 1984. She received her BSc and MSc degrees in Electrical and Electronics Engineering from University of Gaziantep, and PhD degree in Electronics and Communication Engineering from Doğuş University in 2007, 2010 and 2018, respec-tively. Currently, she is with the Department of Electronics and Communications Engineering, Dogus University in Istanbul, Turkey. Her research interests include theoretical and computational aspects of scattering and interaction with dispersive media, numerical solutions of electromagnetic wave problems.

Ercan Topuzreceived his MSc and PhD degrees in electronics and communication from Istanbul Technical University, Istanbul, Turkey, in 1965 and 1973, respectively. Currently, he is with the Depart-ment of Electronics and Communications Engineering, Dogus University in Istanbul, Turkey. He has authored or co-authored over 70 technical papers in books, journals, and conferences. His research interests include theoretical and computational electromagnetics, microwave/optical devices and systems. Dr. Topuz is a member of the Electromagnetics Academy.

ORCID

Ayşegül Pekmezci http://orcid.org/0000-0003-0893-1056

References

[1] Veselago VG. The electrodynamics of substances with simultaneously negative values ofε and μ. Sov Phys Usp.1968;10(4):509–514.

[2] Pendry JB, Holden AJ, Robbins DJ, et al. Low frequency plasmons in thin-wire structures. J Phys Condens Matter.1998;10:4785–4809.

[3] Pendry JB, Holden AJ, Robbins DJ, et al. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans Microw Theory Tech.1999;47:2075–2084.

[4] Smith DR, Schultz S, Padilla WJ, et al. Composite media with simultaneously negative permeabil-ity and permittivpermeabil-ity. Phys Rev Lett.2000;84(18):4184–4187.

[5] Chen H, Ran L, Huangfu J, et al. Left-handed materials composed of only S-shaped resonators. Am Phys Soc Phys Rev E.2004;70(057605):1–4.

[6] Li Z, Aydin K, Ozbay E. Transmission spectra and the effective parameters for planar metamate-rials with omega shaped metallic inclusions. Opt Commun.2010;283:2547–2551.

[7] Kishor K, Baitha MN, Sinha RK, et al. Tunable negative refractive index metamaterial from V-shaped SRR structure: fabrication and characterization. Opt Society Am B. 2014;31(7): 1410–1414.

[8] Pendry JB. Negative refraction makes a perfect lens. Phys Rev Lett.2000;85(18):3966–3969. [9] Schurig D, et al. Metamaterial electromagnetic cloak at microwave frequencies. Science.2006;

314(5801):977–980.

[10] Engheta N. An idea for thin sub-wavelength cavity resonators using metamaterials with negative permittivity and permeability. IEEE Antennas Wireless Prop Lett.2002;1:10–13.

[11] Rennings A, et al. Double-Lorentz transmission line metamaterial and its application to tri-band devices. IEEE MTT-S Int. Microwave Symp Dig.2007;1:1427–1430.

[12] Eleftheriades GV, Engheta N. Metamaterials: Fundamental, and applications in the microwave and optical regimes. Proc IEEE.2011;99(10):1618–1621. Special Issue.

[13] Tretyakov S, et al. The century of metamaterials. J Opt.2017;19(8):1–3. Special Issue.

[14] Engheta N, Ziolkowski RW. Metamaterials: physics and engineering explorations. Hoboken (NJ): John Wiley & Sons;2006.

[15] Capolino F. Applications of metamaterials. Boca Raton (FL): CRC Press;2009; Sect. III-IV. [16] Alù A, Engheta N. Guided modes in a waveguide filled with a pair of single-negative (SNG),

double-negative (DNG), and/or double-positive (DPS) layers. IEEE Trans Microw Theory Tech.

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