arX

iv:1

510.

0645

8v1

[ph

ysic

s.op

tics]

21

Oct

201

5

Electromagnetic-Magnetoelectric Duality for Waveguides

Nafiseh Sang-Nourpour,1, 2, 3 Benjamin R. Lavoie,4, 2 R. Kheradmand,1 M. Rezaei,5, 6 and Barry C. Sanders2, 7, 8, 9

1Photonics Group, Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz 51665-163, Iran2Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada3Photonics Group, Aras International Campus of University of Tabriz, Tabriz 51666-16471, Iran

4Department of Electrical and Computer Engineering,

Schulich School of Engineering, University of Calgary, Alberta T2N 1N4, Canada5Department of Theoretical Physics, Faculty of Physics, University of Tabriz, Tabriz 51664, Iran

6Department of Physics, Applied Science Faculty, Simon Fraser University, Burnaby, Canada7Hefei National Laboratory for Physics at Microscale and Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui, China8Shanghai Branch, CAS Center for Excellence and Synergetic

Innovation Center in Quantum Information and Quantum Physics,

University of Science and Technology of China, Shanghai 201315, China9Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

We develop a theory for waveguides that respects the duality of electromagnetism, namely thesymmetry of the equations arising through inclusion of magnetic monopoles in addition to includingelectrons (electric monopoles). The term magnetoelectric potential is sometimes used to signify themagnetic-monopole induced dual to the usual electromagnetic potential. To this end, we introducea general theory for describing modes and characteristics of waveguides based on mixed-monopolematerials, with both electric and magnetic responses. Our theory accommodates exotic mediasuch as double-negative, near-zero and zero-index materials, and we demonstrate that our generaltheory exhibits the electromagnetic duality that would arise if we were to incorporate magneticmonopoles into the media. We consider linear, homogeneous, isotropic waveguide materials withslab and cylindrical geometries. To ensure manifest electromagnetic duality, we construct genericelectromagnetic susceptibilities that are dual in both electric charges and magnetic monopoles usinga generalized Drude-Lorentz model that manifests this duality. Our model reduces to standardcases in appropriate limits. We consider metamaterials and metamaterial waveguides, as well asmetal guides, as examples of waveguides constructed of mixed-monopole materials, to show thegenerality of our waveguide theory. In particular, we show that, in slab and cylindrical waveguides,exchanging electric and magnetic material properties leads to the exchange of transverse magneticand transverse electric modes and dispersion equations, which suggests a good test of the potentialduality of waveguides. This theory has the capability to predict waveguide behavior under anexchange of electromagnetic parameters from the dual waveguide.

I. INTRODUCTION

Electromagnetic waveguides are physical devices thatconfine and control the energy of propagating waves, andare able to efficiently direct electromagnetic (EM) radi-ation [1]. The applications of these structures, such asin radars and optical fibers, provide motivation for fur-ther advances in this field. In recent years, metamateri-als [2] have enriched the capabilities of waveguides, forexample by tailoring properties such as EM susceptibil-ities as well as by generating new kinds of modes [3, 4].Metamaterials are designed to have both negative andpositive EM susceptibilities within a certain frequencywindow, and can even have both susceptibilities negativein some window, which is the so-called double-negativemedium [2, 5, 6].

Although breakthroughs are being made with custom-designed susceptibilities using metamaterials, which aresynthesized as some specific geometric arrangement ofmaterials to realize electric and magnetic properties thatmight not match naturally available materials [2], herewe treat metamaterials as materials in their own rightat both macroscopic and microscopic levels. Our mate-

rial description of metamaterials is natural at the macro-scopic level where the purpose of the metamaterial isto behave as an effective material on a sufficiently largescale. On the microscopic level, the material descrip-tion is advantageous, as we shall see, because we canemploy microscopic models such as the Drude-Lorentzmodel for calculating electric and magnetic responses ofthe medium [7].

In order to describe exotic metamaterials, for exam-ple with double-negative index properties, as materialsper se, our description of the material must manifestfull electromagnetic duality. This electromagnetic dual-ity arises by incorporating magnetic monopoles into theelectromagnetic equations. One way to admit monopolesis to construct two potentials, namely the usual elec-tromagnetic potential arising from Maxwell’s equations,and a second potential, introduced by Cabibbo and Fer-rari [8, 9] and sometimes referred to as the magnetoelec-tric potential [10, 11]. Whereas a metamaterial can besynthesized from regular materials to yield exotic prop-erties such as double-negative indices, introducing mag-netic monopoles into the theory enables a purely micro-scopic description of these media as materials rather than

2

metamaterials. We find the fully dual description of elec-tromagnetism conducive to building both the formalismand the intuition for a fully general theory of waveg-uides with utility for designing metamaterial waveguidesas well as material waveguides.

Using this convenient formal description of materialsin the framework of fully dualistic electromagnetic the-ory, we can generalize waveguide theories to accommo-date this fully general setting. Although impressive ef-forts have generalized waveguide descriptions for vari-ous geometries and constituent materials [1, 4, 12, 13],a fully comprehensive theory of waveguides for all ma-terials and metamaterials is lacking. Our aim is to con-struct a general waveguide theory incorporating full dual-ity and showing how this theory can convert descriptionsof ordinary waveguides into exotic ones simply using theprinciple of duality. We focus on duality for waveguidescomprising linear, homogeneous, isotropic materials (in-cluding exotic materials with magnetic monopoles) anddo not deal with varying the geometry at this time; a gen-eral theory incorporating full duality as well as generalgeometries is challenging but could be quite interestinggiven the importance and challenges of conformal trans-formation optics [14].

Our generalized waveguide theory features the full du-ality inherent in Maxwell’s equations and gives a quanti-tative description of waveguide phenomena, both macro-scopically and microscopically. The relevant backgroundof the theory is presented in Sec. II. We detail our ap-proach in Sec. III. The mathematical results and exam-ples are presented in Sec. IV. We conclude with a discus-sion of our theory in Sec. V.

II. BACKGROUND

This section provides the relevant background requiredto develop our generalized waveguide theory. We be-gin with a brief review of Maxwell’s equations and theconcept of electromagnetic duality. We then show howduality naturally leads to the idea of the general mixed-monopole material and discuss how metamaterials canbe thought of as an approximate mixed-monopole ma-terial. Finally, we review waveguide theory for the slaband cylindrical geometries.

A. Maxwell’s Equations and Electromagnetic

Duality

In source-free regions, Maxwell’s equations have a sym-metric form, and the duality of Maxwell’s equations be-comes apparent. These equations can also be written inregions containing charges and currents, but the presenceof electric charges and absence of magnetic ones leadsto an asymmetry in Maxwell’s equations [7]. Restor-ing the symmetry of Maxwell’s equations can be done

by mathematically allowing for the existence of magneticmonopoles [15, 16].Considering both electric and magnetic charges (by in-

serting terms for a magnetic charge density and currentdensity) allows Maxwell’s equations to be written as fol-lows [7]:

∇ ·D = ρe, ∇ ·B = ρm,

∇×E = −∂B

∂t+ Jm, ∇×H =

∂D

∂t+ Je,

(1)

where E is the electric field, D is the electric displace-ment field, B is the magnetic induction field and H isthe auxiliary magnetic field. The free magnetic and elec-tric current densities are represented by Jm and Je, re-spectively, and ρm and ρe are free magnetic and electriccharge densities, respectively [7]. To ensure symmetry,we require the magnetic monopole to behave in a man-ner analogous to the electric charge [15].In this paper, we assume linear, homogeneous and

isotropic materials to simplify the expressions, allowingus to focus on duality. The components of the auxiliaryfields in these types of materials are [7]:

D(ω)def= ε(ω)E(ω),

B(ω)def= µ(ω)H(ω),

(2)

with ε(ω) and µ(ω) the frequency-dependent electric per-mittivity and magnetic permeability of the material. Thepermeability of most materials is constant, but for cer-tain materials, such as magnetic materials, it can befrequency-dependent.Equations (1) and (2) reveal that for systems in which

the symmetric Maxwell’s equations are valid, electricparameters, including field, current density, charge orany parameter related to them, can be easily exchangedto magnetic parameters, without changing the form ofMaxwell’s equations, and vice versa. This EM dualitycan be manifest in any structure in which symmetricMaxwell’s equations exist. If a material were to featureelectromagnetic duality, we would expect to find withinit both electric and magnetic monopoles, as can be seenfrom Eq. (1).

B. Mixed-Monopole Materials

Mixed-monopole materials, materials that containboth electric and magnetic monopoles, can expand theparameter space for electromagnetic phenomena such asnegative, positive and zero permeability and permittivity.Although magnetic monopoles are not known to exist,they are theoretically permitted and serve as a valuablemathematical tool [17]. The electric and magnetic prop-erties of materials are encompassed in the permittivityand permeability of the medium, respectively.Different models are presented in the literature to

characterize frequency-dependent permittivity and per-meability. However,models should exhibit all the EM

3

features of materials, with refractive index ranging fromdouble-positive to double-negative values. One modelthat can be used for ε and µ in mixed-monopole ma-terials is the Drude-Lorentz model [7]. This model isa good candidate for describing EM susceptibilities ofmixed-monopole materials, as it is derived from a mi-croscopic description of the response of a material to anincident electromagnetic field [7].Electric permittivity is the result of a material polariz-

ing in response to an external electric field. The Drude-Lorentz form for the frequency-dependent permittivity ofa mixed-monopole material is [7]

ε(ω)

ε0= 1 +

Feω2e

ω20e

− ω2 + iΓeω, (3)

where Fe is electric oscillation strength, ω0e is electricresonance frequency, Γe is the electric damping constant,and all describe charge-Ion dipole oscillation (Drude-Lorentz model). The angular frequency of the incidentelectromagnetic field is ω. In this equation ωe is the elec-tric plasma frequency and is

ωedef=

√

Neq2emeε0

, (4)

where Ne is the number density of electrons, qe and me

are electron charge and mass and ε0 is vacuum permit-tivity [7].The magnetic susceptibility of a mixed-monopole ma-

terial is, also, described by a frequency-dependent func-tion, which is called the permeability. The frequency-dependent permeability is generated by magnetic dipoleoscillations in mixed-monopole materials. As we requiremagnetic monopole behavior to mirror that of electriccharges, the permeability, like the permittivity, is de-scribed by the Drude-Lorentz model:

µ(ω)

µ0

= 1 +Fmω

2m

ω20m

− ω2 + iΓmω(5)

with ω0m the magnetic resonance frequency [4, 18, 19],Fm is the magnetic oscillation strength and Γm the mag-netic damping constant, which are all parameters of mag-netic monopole-monopole coupling interaction in mixed-monopole materials. In Eq. (5) the magnetic plasma fre-quency ωm obeys

ωmdef=

√

Nmq2mmmµ0

(6)

with Nm the number density of magnetic charges, qm andmm the magnetic charge and mass, and µ0 the vacuumpermeability. The complex-valued permittivity and per-meability are denoted as ε = ε′ + iε′′ and µ = µ′ + iµ′′,respectively.The two expressions (3) and (5) are identical up to

converting between electric and magnetic symbols. This

similarity motivates us to introduce one symbol for bothelectric permittivity and magnetic permeability:

ζedef= ε, ζm

def= µ. (7)

We can use this convenient notation to refer to either εor µ as ζ, with the meaning made clear by context. Ac-cordingly, we can use the parameters

Fe,m, ωe,m, ω0e,m , Γe,m (8)

as the electric and magnetic components of oscillationstrength, plasma frequency, resonance frequency anddamping, respectively.The complex refractive index of the lossy mixed-

monopole material is

ndef=

√

εµ

ε0µ0

= nr + ini. (9)

As square roots have two possible solutions, one posi-tive and one negative, we must choose the appropriatecase. We are assuming passive materials and so choosethe sign that corresponds to ni > 0 to ensure that thematerial does not have gain. This choice leads to nr < 0at frequencies that have both ε′ < 0 and µ′ < 0 [4].Although materials containing both magnetic and elec-

tric monopoles are not known to exist in nature, we canapproximate the desired behavior with a material thatbehaves the same way on a large enough scale. For in-stance, artificially engineered metamaterials, which areexamples of mixed-monopole materials and exhibit bothelectric and magnetic effects, can be used to simulate thepresence of magnetic monopoles [18]. As metamateri-als approximate the response of mixed-monopole materi-als, their EM properties can be studied using the Drude-Lorentz model.

C. Metamaterials

Metamaterials are artificially engineered materials [2]that have properties and functionalities previously notattainable in natural materials [20–23]. Recently, meta-materials with different structures and properties areconstructed and investigated [19, 20, 24–26]. Develop-ments in metamaterial science have provided the possi-bility to realize materials with positive, zero and negativeeffective refractive indices [25, 27–30]. Zero-refractive-index metamaterials have infinite wavelength, quasi-infinite phase velocity and also quasi-uniform phase oflight in the material. The quasi-uniform phase of lightimplies that the dipoles in the metamaterial are oscillat-ing uniformly [29].Now we consider metamaterials constructed solely with

metal wires, meshes or plates [18]. For such metamateri-als, bulk permittivity is the dominant electric response.Therefore, the electric resonance frequency ω0e is zero so

4

the Drude-Lorentz model (3) reduces to

ε(ω)

ε0= 1−

ω2e

ω(ω + iΓe), (10)

which is the Drude model for electric susceptibility [4, 7].Whereas ω0e = 0 for the electric susceptibility, the

magnetic resonance frequency ω0m is not necessarily zeroin Eq. (5). The nonzero value of ω0m is due to the sub-wavelength structure of the metamaterial components re-sponsible for its magnetic susceptibility. For example,the fishnet metamaterial [18, 25], which we analyze here,the magnetic response is due to current loops in both theslabs and the necks of the structure.As metamaterials derive their magnetic response from

their structure, rather than from magnetic monopoles,the parameters in Eq. (5) are not functions of monopole-monopole interactions. Most notably, the plasma fre-quency in a metamaterial is simply the frequency of theincident field, ωm = ω. The monopoles in a materialwill naturally oscillate at the plasma frequency when per-turbed. A metamaterial does not have monopoles, butmagnetic dipoles created by current loops. As these cur-rent loops are driven by the incident field, they will tendto oscillate at the driving frequency, i.e. that of the inci-dent field. The remaining parameters in Eq. (5) reflectthe effects of the metamaterial geometry, as well as theresistance of the metal, and the capacitance and induc-tance created between the plates [18].

D. Waveguides

Waveguides are structures that have the ability to con-fine and direct the energy of EM waves. Waveguidescan be made of different materials and can have differentstructures, such as rectangular, slab and cylindrical ge-ometries. The characteristics and behavior of waveguidemodes depend on both the geometry and materials [1].In order to study the EM properties of waveguides

in this paper, slab and cylindrical geometries are con-sidered, as examples. Waveguides having a slab geom-etry can support two different mode types, transverseelectric (TE) and transverse magnetic (TM). Cylindricalguides support both TE and TM, as well as a third modetype, which is neither TE or TM, called a hybrid (HE)mode [1]. A schematic diagram of a slab and cylindricalwaveguide is illustrated in Fig. 1.To characterize different modal behavior in the waveg-

uides, the dispersion relation is needed, which is attainedby solving Maxwell’s equations in both the core andcladding, and then applying the appropriate boundaryconditions for the geometry [1, 7]. The dispersion rela-tions of TM and TE modes for the symmetric slab guideare [1, 4]

2ζ1ζ2γ1γ2

= −

(

ζ21γ21

+ζ22γ22

)

tanh(γ1w), (11)

a

(a) (b)

z

12

2

1

2

w

FIG. 1. Diagram of the slab and cylindrical waveguides. Re-gion 1 in both guides is for the core made of lossless dielectricand region 2 is cladding based on an mixed-monopole mate-rial, metamaterials in this case. The width of the slab guideis w and the radius of the cylindrical guide is a = w/2. Thepropagation direction of the fields is along the z axis.

with ζ in Eq. (11) referring to both the electric permit-tivity and to the magnetic permeability (7). In Eq. (11)

γj =

√

β̃2 − ω2εjµj (12)

is the complex wave number for the transverse compo-nents of the fields, j = 1, 2 (1 and 2 indicate the core and

cladding, respectively). Here, β̃ = β + iα is the complexpropagation constant (in the propagation direction) withβ and α the dispersion and attenuation, respectively, andw is the core width.For the HE modes in a cylindrical guide, the dispersion

relation is given by [1, 4]

(

1

κ21

+1

γ22

)2

=

(

µ1

κ21

J ′

m(aκ1)

Jm(aκ1)+

µ2

γ22

K ′

m(aγ2)

Km(aγ2)

)

×

(

ε1κ21

J ′

m(aκ1)

Jm(aκ1)+

ε2γ22

K ′

m(aγ2)

Km(aγ2)

)(

a2ω2

β̃2m2

)

,

(13)

with Jm and Km the Bessel and modified Bessel func-tions, respectively, and the prime representing the deriva-tive of Jm and Km with respect to r (radial coordinate).Here γ2 and κ1 = iγ1 are the wave numbers for thecladding and core, respectively, a is the core radius andm ≥ 0 is an integer characterizing azimuthal symmetry.We can recover the dispersion relations for the TE and

TM modes of a cylindrical waveguide by setting m = 0in Eq. (13), which reduces it to two separate dispersionequations, one for TE and the other for TM modes. Forthe m > 0 case, hybrid modes are needed to satisfy theboundary conditions [1]. As the TM and TEmodes of thecylindrical guide are similar to those of the slab guide,we focus on the HE modes. These dispersion relationsdescribe the modal dispersion and can be solved to obtainthe complex propagation constant.As metamaterials are examples of mixed-monopole

materials, metamaterial waveguides can be studied as

5

examples for more general waveguides based on mixed-monopole materials. Due to wide applicability of opticalwaveguides, metamaterial waveguides have been recentlystudied in order to extend waveguides’ range of applica-tion even further. The goal for metamaterial waveguidesis to realize optical properties significantly beyond con-ventional dielectric and metal waveguides [3, 4, 31, 32].In order to exploit fully the promise of metamaterial

waveguides, we need to understand their nature and, inparticular, the types of modes that are supported. Meta-material waveguides can support three distinct mode be-haviors: ordinary, surface plasmon polariton (SPP) andSPP-ordinary hybrid modes that are the combinationof SPP and ordinary modes. Ordinary modes are re-lated to travelling EM fields inside the core, whereas forSPP modes the energy is concentrated around the core-cladding interface [4]. To decrease the complexity of ourresults, we focus on ordinary modes in this paper.

III. APPROACH

We generalize the Drude-Lorentz model used for theresultant permittivity and permeability in order to ex-plore duality and its effects on the waveguide at themicroscopic level. Although we recognize that thewaveguides can comprise both material and metamate-rial components, our analysis at the microscopic leveltreats the magnetic properties as arising from magneticmonopoles in the medium, analogous to plasmonic ef-fects arising from electric monopoles (namely, electrons)in the medium. Our treatment of electric and magneticmonopoles is discuss in Subsec. III A, and we develop thegeneralized Drude-Lorentz model in Subsec. III B.Our incorporation of magnetic monopoles into the the-

ory is not meant to imply that we are aiming to usemagnetic monopoles in construction of waveguides butrather to exploit a manifestation of electromagnetic du-ality at the microscopic level by treating the medium ashaving magnetic monopoles. On the macroscopic level,this magnetic monopole description of permittivity andpermeability is valid as an effective theory.In Subsec. III D, we present our numerical method

to solve the dispersion equations of slab and cylindri-cal waveguides considerd in this paper. By numericallysolving the dispersion equations and obtaining the com-plex propagation constant of waveguides, we investigatethe EM characteristics of slab and cylindrical waveguides(Fig. 1) constructed from mixed-monopole materials us-ing the parameters introduced in Subsec. III C.

A. Mixed-Monopole Materials

In this subsection we explain the nature of mixed-monopole materials, which include both electric andmagnetic monopoles. In this subsection, we introduceand discuss the Drude-Lorentz model and describe the

electromagnetic response of mixed-monopole materials.Then we introduce certain materials as representative ex-amples of mixed-monopole materials.We consider mixed-monopole materials in this paper,

which comprise electric and magnetic monopoles withpositive and negative charges. Rather than refer tocharges, we generalize to the terms electric monopole andmagnetic monopole for the negative charges, and we referto the massive positive charges as electric and magneticions. Our treatment can be extended to dipoles withlow-mass negative charges ions and high-mass positivecharges. We recognize that in metamaterials the mag-netic dipoles are actually generated by electric currents.Nevertheless the notion of valence magnetic monopolesand oppositely charged massive magnetic nuclei is con-venient to build a fully dualistic description.We employ the Drude-Lorentz model for both electric

and magnetic susceptibilities [18]. This model can de-scribe both dielectric and metallic materials. In the limitof weak coupling between charges and ions, the Drude-Lorentz model reduces to the Drude model for metals [5]:negative charges are detached from the ions, allowingthem to move more freely within the material. The Drudeand Drude-Lorentz models assume that inter-electron in-teractions are negligible [7].We extend this conventional electrical approach to

incorporate magnetic monopole-monopole interactions.This is justified by the fact that the Drude-Lorentz modelassumes minimal inter-dipole interaction (i.e. there isenough separation between the atoms that nearest-neighbour interaction is negligible compared to the ef-fects of the incident field); thus, we also assume a largeinter-monopole separation consistent with the large inter-monopole separation for the electric case.In this extended Drude-Lorentz model, various param-

eters are included such as the mass of the monopoles. Theeffective masses of the electric and magnetic monopolesdepend, in part, on the band structure created by the lat-tices formed from the massive electric and magnetic ions.Determining the band structure and effective masseswould require a quantum treatment, but is not germaneto our work.In this subsection we have shown the properties of

mixed-monopole materials and the capability of Drude-Lorentz model for describing them. With this knowledge,we are now equipped to generalize the Drude-Lorentzmodel applicable to linear, homogeneous isotropic ma-terials for application to mixed-monopole materials.

B. Generalized Drude-Lorentz model

In this subsection, we construct the Drude-Lorentzmodel for a mixed-monopole material. Then we showhow this model can be used to determine the permittivityand permeability of the medium and furthermore showhow electromagnetic duality manifests in these expres-sions. We use the relations for the generalized Drude-

6

Lorentz model and refractive index to show differentkinds of materials with positive, negative, near-zero andzero refractive index.We introduce a modified form of the Drude-Lorentz

model, called the generalized Drude-Lorentz model, to fa-cilitate studying duality in waveguides. The generalizedmodel allows us to investigate EM symmetries of systemsunder the exchange of material properties from electric tomagnetic and vice versa. The generalized Drude-Lorentzmodel has the form

ζe,mζ0e,m

= ζbe,m+

Fe,mω2e,m

ω20e,m

− ω2 + iΓe,mω, (14)

where ζe,m is permittivity or permeability, ζ0e,m is vac-uum permittivity or permeability, ζbe,m

is the back-ground permittivity or permeability, and the other pa-rameters (8) have been explained earlier. Subscripts e

and m refer to electric and magnetic parameters, respec-tively.Our generalized model respects the duality of electro-

magnetism and describes the plasmonic effects of bothelectric and magnetic monopoles. We see this mathe-matical formalism as being inspired by treating formallya magnetic monopole plasma. To study EM duality inthe system, we replace ε (3) and µ (5) by ζe,m, respec-tively, in the generalized-model dispersion relations.By applying duality transformations (exchanging EM

parameters), we can realize the duality of electromag-netism in different systems. The Drude-Lorentz model(and possibly others) can be recovered from the general-ized Drude-Lorentz model through appropriate parame-ter choices. This model has the ability to describe mate-rials with all possible values for refractive index.The refractive index can be expressed in terms of the

generalized model as

n =

√

ζeζmζ0eζ0m

. (15)

This equation, along with Eq. (14), allows for positive,negative, near-zero and zero-refractive-index materials,given the appropriate parameter values. The conditionfor obtaining a refractive index of zero depends on whereζe and ζm become zero, so there are generally four fre-quencies at which zero-refractive-index occurs. There arealso regions with near-zero refractive index.In summary, expressions for generalized Drude-Lorentz

model (14) and for refractive index (15) manifest theduality of electromagnetism. The generalized Drude-Lorentz model can show exotic material properties witheven negative and zero refractive indices.

C. Characterization

In this subsection, we introduce dispersion equationsand suitable waveguide parameters that augment the ma-terial parameters introduced so far. Then we proceed to

construct equations in order to determine the values ofthese parameters. These values are obtained by solvingthe dispersion equations in two cases, namely, for slaband cylindrical waveguides as they are easily solved ex-amples and provide a clear foundation for studying morecomplicated cases.We can rewrite dispersion relations for TE and TM

modes in slab guide using the generalized model, asEq. (11), where we suppressed the subscript on ζe,m forreadability. In Eq. (11) ζj becomes εj for TM modesand µj for TE modes. As γj contains only the productof ε and µ, it is invariant under the duality transforma-tion. This unified equation makes the duality betweenTE and TM modes evident. Therefore, the generalizedmodel provides a unified relation that can be used to cal-culate both mode types. We see from Eq. (11) that thedispersion relation for the TE (TM) modes of the slabguide can be transformed into the dispersion relation forthe TM (TE) modes by replacing the magnetic versionof ζ with the electric one and vice versa.For HE modes the unified dispersion equation is:

(

1

κ21

+1

γ22

)2

=

(

ζm1

κ21

J ′

m(aκ1)

Jm(aκ1)+

ζm2

γ22

K ′

m(aγ2)

Km(aγ2)

)

×

(

ζe1κ21

J ′

m(aκ1)

Jm(aκ1)+

ζe2γ22

K ′

m(aγ2)

Km(aγ2)

)(

a2ω2

β̃2m2

)

,

(16)

where κ1 is the complex wave number in the core (κ1 =iγ1) and γ2 is the complex wave number for the cladding(subscripts 1 and 2 are for core and cladding regions ofthe cylindrical guide, respectively). Just like the slabguide, the dispersion relation for TE (TM) modes canbe transformed into that for the TM (TE) modes witha duality transformation. The dispersion relation for theHE modes, however, is left unchanged.We can now characterize the waveguide parameters for

various modes of the slab and cylindrical waveguides.The two waveguide parameters are relevant. One pa-rameter is the effective refractive index

neff = β/k0, k0 = ω/c = λ0/2π (17)

of a guided mode, which we see is directly proportionalto the real part β of the propagation constant [4]: Theeffective refractive index (17) is frequency dependent andaccounts for total modal dispersion (i.e., combined effectsof both chromatic and waveguide dispersion) with thevariations leading to different mode characteristics.We also need to consider mode attenuation in a waveg-

uide, which accounts for intensity reduction in the direc-tion of travel along the guide. Attenuation is mathemat-ically described by the imaginary part of the complexpropagation constant, which we denote as α. As withthe effective refractive index, the mode attenuation is fre-quency dependent and is a combination of both materiallosses and energy leaking from the waveguide. These twoparameters allow us to characterize the waveguide modesand how they are affected by duality. We consider waveg-uides with a metamaterial or metal cladding and an air

7

core. We then exploit the symmetry of the generalizedmodel to study EM duality in these structures.In this subsection, we have introduced two relevant

waveguide parameters. The two important parametersare effective refractive index of waveguide and waveg-uide attenuation that are related to waveguides’ propaga-tion constants. Studying these parameters enables us tosee waveguides behavior beside investigating EM dualityin guides. However, the propagation constant is attain-able by solving the transcendental dispersion equations ofwaveguides and requires numerical approaches for solvingthe equation. In the next subsection we briefly explainour numerical method for solving dispersion equations.

D. Numerical method

As our waveguides’ dispersion equations (11) and (13)are transcendental, they cannot be solved exactly and nu-merical techniques are required to solve the dispersion re-lations. Various numerical approaches are available, andwe choose Newton’s method because exact expressionsfor derivatives are available and the technique converges.Obtaining the complex propagation constant for slab

and cylindrical waveguides requires the dispersion equa-tions to be solved. As they are transcendental, we solvethem numerically using the numerical root-finding func-tion Mathematicar’s FindRoot function, which is basedon Newton’s method. Using this function, we find thecomplex propagation constant for the supported modesat different frequencies.As multiple modes can propagate in a waveguide at a

given frequency, different initial seed values are neededto obtain the propagation constant for different modes.Thus, we need to provide an initial seed value for eachmode at a frequency where the value of β̃ is known ap-proximately. We can then increment the frequency valueand solve for β̃ by using the previously obtained valueas a seed value for FindRoot. This procedure minimizesthe possibility of obtaining a β̃ value for a different modethan the one for which we are solving and is repeated forall desired modes.In this subsection, we explained the numerical method

we use to solve the transcendental dispersion equation ofslab and cylindrical waveguides to find the complex prop-agation constant for different modes in waveguides. Byfinding the propagation constant of waveguides, we cannow investigate the different parameters of waveguides.

E. Summary of Approach

In this section we presented the properties of mixed-monopole materials and introduced the Drude-Lorentzmodel as an appropriate model to describe these mate-rials. We then generalized the Drude-Lorentz model toa model that manifests the duality of electromagnetism

and describes materials with refractive indices rangingfrom positive to negative values.Next we presented the dispersion equations of waveg-

uides. The effective refractive index and attenuationare two relevant parameters that describe the waveg-uides and are attainable by numerically solving disper-sion equations of waveguides. Investigating these param-eters enable the investigation of EM duality in waveg-uides. We finally presented our numerical method forsolving the dispersion equations of slab and cylindricalwaveguides.

IV. RESULTS

In this section, we present our results for bulk metama-terial, which is described well by the behavior of mixed-monopole materials, and for metamaterial waveguides.This section’s results yield clear physical insight regard-ing waveguide behavior including metal and non-metalwaveguide media. We ensure that our results are con-sistent with existing theory. To verify that the gener-alized model is correct and reliable, we recreated somepreviously obtained numerical results using the appro-priate parameters [4]. To do so, we solved the modaldispersion relation of the symmetric slab and cylindricalguides and numerically determined the complex propa-gation constants. We compared our calculations and nu-merical results against previous work [4] using identicalparameters. Our results reproduce previous facts andsatisfy the expected duality relations.We first examine the effects of the duality transforma-

tion on the electromagnetic properties of a bulk meta-material. We then apply the results to metamaterial-dielectric waveguides, to determine the behavior of selectmodes under the duality transformation.

A. Metamaterials

To characterize the EM properties of metamaterialswe study the behavior of the permittivity, permeabilityand refractive index of a metamaterial and depict nu-merical results inFig. 2. These results are obtained bysolving Eqs. (14) and (15). The metamaterial parame-ters used are ζbe,m

:= 1, Fe = 1, Fm = 0.5, ω0e := 0,

ω0m := 0.2ωe, ωe = 1.37 × 1016 s−1, ωm = ω andΓm = Γe = 2.73× 1013 s−1 [4]. These parameters repre-sent a metamaterial with a metal-like permittivity, due tothe dominant response of the metal used to construct themetamaterial, and a permeability that is induced throughoscillating magnetic dipoles.Figure 2(a) shows that the permittivity and perme-

ability of metamaterials have positive, negative and zerovalues in different frequency regions. In the frequencyregions where both ε and µ are negative, the refractiveindex is also negative, which corresponds to a double-negative material as shown in Fig. 2(b). For our param-

8

FIG. 2. Plots of (a) ζ/ζ0 that reduce to electric and magneticresponses: ε/ε0 (black line) and µ/µ0 (grey line), respectively,and (b) Refractive index of metamaterial as a function offrequency, using the generalized model. Solid and dashed linesrepresent real and imaginary parts, respectively.

eter choice, the material is double-negative in the fre-quency range 0.2ωe . ω . 0.3ωe.

Therefore, from Figs. 2(a,b) we see that there are fre-quency regions in which the material has permittivityand permeability having positive, zero and negative val-ues that let the refractive index have all possible valuesranging from positive to negative values. Our model alsosolves ε > 0, µ < 0 and double-zero refractive-index ma-terials, however, because of the parameters choice in thispaper we can not see these regions in the graphs. Ourgeneralized model thus describes materials with all pos-sible values for refractive index.

By exchanging electric and magnetic properties inmetamaterials, the behavior of ǫ and µ, in Fig. 2(a), is re-versed, whereas the refractive index of the metamaterialremains unchanged. The reason for this behavior can beunderstood from the equation for the refractive index (9),which contains only the product of ε and µ. Therefore,by exchanging electric and magnetic properties, whichexchanges ε and µ, the behavior of refractive index doesnot change. These results show the EM duality of meta-materials by applying duality transformations.

FIG. 3. Plots of (a) effective refractive index, (b) at-tenuation, for different TE modes in the untransformedslab metamaterial-dielectric waveguide, using the generalizedmodel. The solid, dashed, dot-dashed and dotted lines are forTE0, TE1, TE2 and TE3 modes, respectively.

B. Slab metamaterial-dielectric waveguide

The first waveguide example is a symmetric slabmetamaterial-dielectric guide with hollow core and meta-material cladding. We study the effective refractive indexand attenuation of guided modes in this waveguide. Fig-ure 3 illustrates the behavior of the attenuation and neff

for the first few TE modes in the untransformed (beforeexchanging EM properties) slab metamaterial-dielectricwaveguide as a function of frequency.

To test the EM symmetries of this slab waveguide, weexchange the permittivity and permeability on the ma-terials. By applying this duality transformation in theslab metamaterial-dielectric waveguide, the behavior ofthe TE modes change in the transformed guide. Figure 4shows the behavior of the TE modes in the transformedguide, which is identical to the TM modes in the un-trasformed guide. This exchange of TE and TM modesunder the duality transformation shows the EM symme-try of the slab metamaterial-dielectric waveguide.

9

FIG. 4. Plots of (a) effective refractive index, (b) attenuationfor different TE modes of the transformed slab metamaterial-dielectric waveguide, using the generalized model. The solid,dashed, dot-dashed and dotted lines are for TE0, TE1, TE2

and TE3 modes of transformed guide, respectively.

C. Cylindrical metamaterial-dielectric waveguide

For the next example, we consider a cylindricalmetamaterial-dielectric waveguide with a hollow core andmetamaterial cladding. As with the slab guide, we in-vestigate the effect of a duality transformation on selectmodes, HE in this case. The behavior of neff and theattenuation of the modes is illustrated in Fig. 5.

These plots show that the HE mode behavior in acylindrical metamaterial-dielectric guide shows a com-plete EM symmetry under a duality transformation. Thisis in contrast to the TE and TM modes, which exchangeroles under a duality transformation, but expected giventhat the dispersion relation for these modes is invariant.However, the contributions of the electric and magneticfields do exchange roles, as they depend on the permit-tivity and permeability in an asymmetric way.

As for the slab guide, the dispersion relations for theTE and TM modes in the cylindrical guide convert toeach other under the duality transformation. This meansthe TE and TM modes in the cylindrical guide have thesame symmetric behavior as those in the slab guide whenapplying the duality transformation.

FIG. 5. Plots of (a) effective refractive index, (b) attenu-ation in both the untrasformed and trasformed cylindricalmetamaterial-dielectric waveguide. The thick solid, dashed,dot-dashed and dotted lines are for HE10, HE11, HE12 andHE13 modes, respectively. The effective refractive index andattenuation remain unchanged under a duality transforma-tion.

D. Slab metal-dielectric waveguide

We have demonstrated the fact that metamaterialshave a symmetric behavior under duality transforma-tions. We have also shown that the slab and cylindricalmetamaterial-dielectric waveguides respect EM duality.When we apply our generalized model to metal waveg-uides, we see consistent results with previous work [4],showing the generalized model works for metals, as wellas metamaterials.

We now perform the same procedure for the slab metal-dielectric waveguide, composed of a hollow core andmetal cladding, to gain better insight on the generalityof model. We first study the behavior of TE modes inthe guide with the plots of effective refractive index andattenuation in Figs. 6(a,b). We then perform the dualitytransformation on the waveguide.

The physical interpretation of this transformation isthat, by applying duality transformation, the metalcladding would have magnetic monopoles as free chargecarriers, rather than electrons. This hypothetical “mag-netic metal” would support the TM modes in trans-

10

FIG. 6. Plots of (a) effective refractive index, (b) attenua-tion, for different TE modes in the untransformed slab metal-dielectric hollow waveguide, using the generalized model. Thesolid, dashed, dotted-dashed and dotted lines are for TE0,TE1, TE2 and TE3 modes, respectively.

formed guide, which have the same behavior as TE modesin untransformed metal guide, as Figs. 7(a,b). These TEmodes are the modes that the TM modes of the metalguide are transformed into under the duality transforma-tion. For the HE modes in the metal-dielectric cylindri-cal guide, the duality transformation does not alter anyof the plots, which means that this waveguide has EMsymmetry under duality transformation.

E. Summary

In section IV, we investigated metamaterials, which areexamples of mixed-monopole materials and are a gener-alization of standard materials to include both electricand magnetic responses. Our results demonstrate theability of our general model to provide a full descriptionof materials, including double-positive, double-negative,positive-negative, negative-positive, permittivity-zero,permeability-zero and double-zero materials. We theninvestigated EM duality in waveguides constructed frommetamaterials. Upon performing the duality transforma-tion, the TE (TM) modes of a slab or cylindrical waveg-uide become TM (TE) modes, verifying the duality that

FIG. 7. Plots of (a) effective refractive index, (b) attenua-tion, for different TE modes in the transformed slab metal-dielectric hollow waveguide, using the generalized model. Thesolid, dashed, dotted-dashed and dotted lines are for TE0,TE1, TE2 and TE3 modes, respectively.

is inherent in the generalized model. The HE modes in acylindrical metamaterial-dielectric guide respect the EMduality by exchanging EM properties, though the effec-tive refractive index and attenuation do not change. Inother words, the TM modes of the untransformed guideare identical to the TE modes of the transformed guide,except that the E and B fields are exchanged, so we callthem TE, rather than TM. The metal guides, as well asmetamaterial waveguides, show EM symmetry under theduality transformations.

V. CONCLUSION

In this paper, we develop a generalized theory forwaveguides that respects the duality of electromag-netism. We demonstrate the usefulness of this theory byshowing two concepts. First, we show the ability of ourgeneralized Drude-Lorentz model to describe general ma-terials including mixed-monopole materials and metama-terials, along with waveguides based on these materials.Our generalized model underpins all physically accessiblevalues of both permittivity and permeability, and has theability to include double-positive, positive and negative,double-negative and even zero-index materials. By ex-

11

ploiting the concept of magnetic monopoles, our theorycan easily treat arbitrary combinations of general mixed-monopole materials, standard materials (such as metalsand dielectrics) and metamaterials for selected waveguidegeometries.Second, we show that if waveguides in one parameter

regime are constructed, we can also identify the electro-magnetically dual regime with the same properties andmake use of magnetic, rather than electric, properties.The study of EM symmetry in the systems is realizableby exchanging media from electric to magnetic material,by applying the duality transformations, and then study-ing the behavior of the refractive index of materials andthe modes in the waveguides. This symmetric behavior istestable experimentally by investigating and comparingthe EM properties of original and dual waveguides.Our theory gives us a good macroscopic description

of the waveguide behavior and simplifies the calculationsby using the duality of electromagnetism. We use theprinciple of duality in this paper to predict a waveguidethat has the same behavior with the electric and mag-netic fields swapped. In this case, we propose testing

two waveguides with reversed parameters.

Our results demonstrate that metamaterials exhibit asymmetric behavior under the duality transformation forall frequencies considered herein. This symmetry is dueto the magnetic response of these materials, as well astheir electric response, which simulates the existence ofmagnetic charges, and restores the duality of Maxwell’sequations. Our model enables us to explore the EM sym-metry of slab and cylindrical waveguide structures. Thisability of our model can be used to investigate EM sym-metries in other structures, as well. Our theory leads toa testable prediction for hollow-core metamaterial andmetal waveguides that allows swapping the electric andmagnetic properties.

VI. ACKNOWLEDGMENT

We acknowledge financial support from NSERC, AITFand the 1000 Talent Plan of China.

[1] C. Yeh and F. I. Shimabukuro, The Essence of Dielectric

Waveguides (Springer, New York, 2008).[2] V. M. Shalaev, Nature Photon. 1, 41 (2007).[3] I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar,

Phys. Rev. E 67, 057602 (2003).[4] B. R. Lavoie, P. M. Leung, and B. C. Sanders,

Phot. Nano. Fund. Appl. 10, 602 (2012).[5] N. Engheta and R. W. Ziolkowski, Metamaterials:

Physics and Engineering Explorations (Wiley, Piscat-away, 2006).

[6] S. A. Ramakrishna, Rep. Prog. Phys. 68, 449 (2005).[7] J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley,

New York, 1999).[8] N. Cabibbo and E. Ferrari, Nuovo Cim. 23, 1147 (1962).[9] R. Mignani and E. Recami,

Nuovo Cim. A 30, 533 (1975).[10] F. Moulin, Nuovo Cim. B 116, 869 (2001).[11] D. Fryberger, Found. Phys. 19, 125 (1989).[12] H. Choo, M.-K. Kim, M. Staffaroni, T. J. Seok, J. Bokor,

S. Cabrini, P. J. Schuck, M. C. Wu, and E. Yablonovitch,Nature Photon. 6, 838 (2012).

[13] K. Halterman, S. Feng, and P. L. Overfelt,Phys. Rev. A 76, 013834 (2007).

[14] L. Xu and H. Chen, Nature Photon. 9, 15 (2015).[15] A. Rajantie, Contemp. Phys. 53, 195 (2012).[16] L. J. LeBlanc, Nature 505, 627 (2014).[17] P. A. M. Dirac, Proc. R. Soc. A 133, 60 (1931).

[18] R. S. Penciu, M. Kafesaki, T. Koschny, E. N. Economou,and C. M. Soukoulis, Phys. Rev. B 81, 235111 (2010).

[19] J. Pendry, A. Holden, D. Robbins, and W. Stewart,IEEE Trans. Microw. Theory Tech. 47, 2075 (1999).

[20] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968), (In En-glish) Usp. Fiz. Nauk, 92, 517–526, 1964 (In Russian).

[21] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire,Science 305, 788 (2004).

[22] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).[23] W. J. Padilla, D. N. Basov, and D. R. Smith,

Mater. Today 9, 28 (2006).[24] A. Boltasseva and V. M. Shalaev,

Metamaterials 2, 1 (2008).[25] D. R. Smith, W. J. Padilla, D. C. Vier,

S. C. Nemat-Nasser, and S. Schultz,Phys. Rev. Lett. 84, 4184 (2000).

[26] S. P. Burgos, R. de Waele, A. Polman, and H. A. Atwa-ter, Nat. Mater. 9, 407 (2010).

[27] E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta,and A. Polman, Phys. Rev. Lett. 110, 013902 (2013).

[28] S. S. Islam, M. R. I. Faruque, and M. T. Islam,Materials 8, 4790 (2015).

[29] J. Schilling, Nature Photon. 5, 449 (2011).[30] R. Maas, J. Parsons, N. Engheta, and A. Polman,

Nature Photon. 7, 907 (2013).[31] K. Y. Kim, J. Opt. A: Pure Appl. Opt. 9, 1062 (2007).[32] G. D’Aguanno, N. Mattiucci, M. Scalora, and M. J.

Bloemer, Phys. Rev. E 71, 046603 (2005).