IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.XX, NO. XX, XXX XXXX 1

Energy Coupled Mode Theory for ElectromagneticResonators

Sameh Y Elnaggar, Richard J. Tervo, and Saba M. Mattar

Abstract—There is recent interest in the inter/intra-elementinteractions of metamaterial unit cells. To calculate the effectsof these interactions which can be substantial, an”ab-initio”general coupled mode equation, in the form of an eigenvalueproblem, is derived. The solution of the master equation givesaccurate estimates of the coupled frequencies and fields in termsof an arbitrary number of uncoupled modes, having equal ordifferent frequency values. By doing so, the problem size islimited to the number of modes rather than the, usually large,discretized spatial and temporal domains obtained by rigorousfull-wave solvers. Therefore, the method can be consideredas anapproximate numerical recipewhich determines the behavior ofa complex system once its simpleringredientsare known. Besidesquantitative analysis, the coupled mode equation proposesa pic-torial view of the split rings’ hybridization. It can be rega rded asthe electromagnetic analog of molecular orbital theory. Althoughbeing an approximate method, the solution of the eigenvalueproblem for different configurations gives valued informationand insight about the coupling of metamaterials unit cells.Unlikerigorous numerical techniques, closed form expressions can bederived using the coupled mode equation. For instance, it isshown that the behavior of split rings as a function of the relativeposition and orientation can be systematically explained.This isdone by singling out the effect of each relevant parameter suchas the coupling coefficient and coupling induced frequency shiftcoefficients.

Index Terms—Metamaterials, stereometamaterials, split ringresonators, coupled mode theory, coupled resonators, hybridiza-tion.

I. I NTRODUCTION

M ETAMATERIALS (MMs) are artificial media that ex-hibit fascinating electromagnetic properties not present

in nature [1]. They are usually manufactured from arrays ofsub-wavelength resonators. Therefore, effective media theoryis used to study their behavior [1]. By analogy with naturalmaterials, the sub-wavelength resonators form the basic unitcells (atoms) of the MM. Each unit cell has electric andmagnetic multipoles. If the cells inter-spacing is small, theycan strongly interact and hence substantially change the mediaproperties [2]–[4]. Inspired by stereochemistry, the interactioncan be controlled by changing the spatial arrangement of the

Manuscript received September 30, 2014; revised May 7, 2015. SMMacknowledges the Natural Sciences and Engineering Research Council ofCanada for financial support in the form of a discovery (operating) grant. SYNis grateful for the financial assistance from the Universityof New Brunswickin the form of pre-doctoral teaching and research assistantships.S Y. Elnaggar is currently with the School of Engineering andInformationTechnology, University of New South Wales, Canberra, ACT, Australia.Richard J. Tervo is with the Department of Electrical and Computer Engi-neering, University of New Brunswick, Fredericton, NB, Canada.Saba M. Mattar is with the Chemistry Department, Universityof NewBrunswick, NB, Canada.Corresponding author: Saba M. Mattar (email: mattar@unb.ca ).

resonators. This concept, widely known asstereometamateri-als, was applied to different MM configurations [2], [4]–[8].Similarly, the media properties can change by tuning the unitcell’s resonant frequency. This can be done for example, byincorporating a photo-conductive semiconductor [6], [9],[10].

The analogy between the MM unit cells and atoms isutilized to visualize the interaction between MM unit cellsas thehybridization of unit cells’ modes. This concept wasfirst applied to the study of nano-shells and nano-spheres [11],[12]. Later, it was adopted to qualitatively and experimentallyanalyze the interaction between either split ring resonators(SRRs), which are the building blocks of MM [13]–[16]. Theinteraction between the mutual magnetic and electric energiesdetermines the nature and strength of coupling. Due to thebi(iso/aniso)tropic property of split rings, both electric andmagnetic couplings play roles in the process of interaction[17].

Due to the sub-wavelength nature of MM unit cells, thecoupling between them was studied based on quasi-staticapproaches. For example, the current and charge densitiesare used to determine the Lagrangian (L). The Lagrangianequation of motion yields a system of coupled differentialequations which determines the interaction terms [2], [3],[18].Due to their importance, the analysis is usually carried outfor meta-dimers. It was shown that the net effect of boththe in-plane electrical and the out-of-plane magnetic dipolesdetermines the interaction behavior [4], [18]. Circuit modelsare also developed to model and to quantify split rings andtheir coupling [19]–[24].

Coupled Mode Theory (CMT), an approximate analyticaltechnique, proves to be very successful when applied toweakly coupled systems [25]. It was used in MM [26], [27],microwave filters [28]–[31], wireless power transfer [32]–[35] and magnetic resonance [36]–[39]. Unlike finite element,finite difference and method of moments, CMT reduces thecomputational domain to the number of modes. Other thanreducing the computational complexity, CMT provides anintuitive picture of how a complex system behaves in terms ofthe interaction of its relatively simpler subsystems. ThismakesCMT a very useful and powerful tool for studying MM.

With the aim of qualitatively and quantitatively analyzingthe hybridization of coupled systems including MMs, an”ab-initio” coupled mode formalism is developed in the currentarticle. The formalism decomposes the modes of a complexsystem to the modes of simpler subsystems. A similar analysiswas developed to study an electron paramagnetic probe con-sisting of a dielectric resonator and a cavity. However, it waslimited to the studied case and bounded to TE modes [37].

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Nevertheless, it was shown that a coupled mode formalism isstill capable of describing the system behavior even thoughthecoupling coefficient can be substantially large (≈ 0.4). Besidesthe ability to accurately estimate the eigen-frequencies,fielddependent parameters such as the quality factor and resonatorefficiency were accurately determined [38], [39]. The currentarticle provides a systematic derivation of the general coupledmode equation in the form of an eigenvalue problem. Oncesolved, the eigenvalues determine the resonant frequencies(eigen-frequencies), while the eigenvectors are used to findthe fields. The eigenvalues and eigenvectors are proven tosatisfy the resonance condition, where the average electricenergy is equal to the average magnetic energy and henceis named energy coupled mode theory (ECMT). It reducesto well known formulae when applied to special cases, suchas the hybridization of meta-dimers [3]. Moreover, ECMTprovides a complementary and generalized pictorial view tothe dipoles coupling widely used in examining the interactionof SRRs. It also gives a numerical procedure for calculatingfrequencies and fields. A meta-dimer and two double SRRs(DSRRs) configurations are numerically studied. The resultsare explained by assessing the effect of system parameters onthe coupling coefficientκ.

The paper is partitioned as follows: Section II presentsthe theoretical background with emphasis on field expansionsand notations used. Section III is devoted to the theoreticalderivation. Section IV presents the results and discusses thehybridization of split-ring resonators. Finally, the conclusionfollows in Section V.

II. T HEORETICAL BACKGROUND

As the resonators approach each other their modes overlapand couple. In the current article the number of coupled modes,N , is taken to be arbitrary. The coupled fields are expressedas the linear combination of the uncoupled ones. Therefore

E =

N∑

i=1

aiEi (1)

and

H =

N∑

i=1

biHi. (2)

Hereai andbi are the expansion coefficients of the electricand magnetic components respectively. TheN modes can beequal to or greater than the number of resonators. So a validcoupled mode can be the linear combination of the first modeof resonator one and the first mode of resonator two, or itcan be the first mode of resonator one, the second mode ofresonator one and the first mode of resonator two, etc. Thisprocedure is very helpful, for instance, if the frequenciesoftwo modes of one resonator are very close (or degenerate),so they both couple with a third nearby mode of a differentresonator. For example, a dielectric resonator’sTE01δ modemay couple with two the two degenerateTE011 andTM111

cylindrical cavity modes. Expansions (1) and (2) are equiv-alent to the linear combination of atomic orbitals (LCAO)in molecular orbital theory [40]. In general, the uncoupled

modes are independent and can be arbitrary. Thus, the basisset in expansions (1) and (2) is not necessarily complete.Moreover, the expansions do not guarantee that the boundaryconditions are satisfied at each point on the interfaces of theresonators. However for most practical cases, the field valuesof one resonator’s mode at the vicinity of another resonatorare sufficiently small and thus the boundary conditions areapproximately satisfied. For a more accurate treatment, thescattering fieldsEscat andHscat should be included (1) and(2). In this work, regardless of the fact that the scattered fieldsare ignored, the estimated frequencies and fields are accurate.

At the conductors’ surfaces, the surface current densityJs

is equal ton ×H wheren is the unit normal. Therefore, thecurrent densityJs over any conductor is expanded in termsof the uncoupled fields as

Js =N∑

i=1

bin ×Hi. (3)

Each ith uncoupled mode satisfies the sinusoidal time-varying Maxwells equations [41], therefore the curl of thefields can be written as:

∇×Ei = −γiµi(r)Hi, (4)

∇×Hi = γiǫi(r)Ei + Ji. (5)

Here γi, µi, ǫi and Ji are the complex frequencyγi =αi+jωi, permeability, permittivity and volume current densityof the ith uncoupled mode respectively. They are the systemparameters when the other resonators are not present orare infinitely away. In general,ǫi(r) and µi(r) change withposition r . Generally,ǫi(r) 6= ǫk(r ). For example, ifi andkrepresent theTE01δ modes of two dielectric resonators, then,in this case, it is clear thatǫi(r) 6= ǫk(r). Similarly, for thecoupled system

∇×E = −γµ(r)H (6)

∇×H = γǫ(r)E + J (7)

where γ, µ, ǫ and J are the corresponding symbols for thecoupled system. Generally speaking,ǫ(r) 6= ǫi(r). As anexample, consider the two dielectric resonators given above.In this case,ǫi(r) is equal toǫriǫ0 inside theith dielectricresonator andǫ0 elsewhere. However,ǫ(r) is equal toǫriǫ0inside theith resonator,ǫrkǫ0 inside thekth resonator andǫ0elsewhere.

In the current article, the Dirac Bra-ket notation is used torepresent the inner product. For example, the inner productof two vector fieldsA and B is denoted by〈A|B〉 and bydefinition it is equal to

〈A|B〉 ≡∫

V

A∗ ·B dv,

whereV is the total volume.

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III. A NALYTICAL DERIVATION

A. Resonance Condition

In this subsection, the resonance condition of the coupledsystem is derived in terms of the uncoupled parameters. Thetime-averaged stored magnetic (WM ) and electrical (WE )energy can be written as

WM =1

4〈H |µH〉, (8)

WE =1

4〈ǫE|E〉. (9)

At resonance and when the medium losses are negligibly small

WM = WE . (10)

Expanding the energy expressions (8) and (9) inE and H

according to (1) and (2). After some algebraic manipulationone can find that,

WE =1

4a†Aa, WM =

1

4b†Gb. (11)

whereAik = 〈ǫEi|Ek〉 is the(i, k) coupled electrical energyandGik = 〈Hi|µHk〉 is the (i, k) coupled magnetic energy.

B. Cross Relations of the uncoupled fields

In this subsection, a useful relation which describes theinteraction between different resonators’ modes is derived. Itwill be used in the next section to simplify the eigenvalueproblem. Expanding∇ · (Ei ×H

∗k ), using the identity,

∇ · (A×B) = (∇×A) ·B − (∇×B) ·A (12)

applying (4), (5) and integrating over the total volumeV∮

∂VEi×H

∗

k ·dS= −γi〈Hk|µiHi〉−γ∗

k〈ǫkEk|Ei〉−〈Ei|Jk〉

(13)

Relation (13) can be re-written as

Mik +Nik = −γ∗i Bik − γkD∗

ki, (14)

where Bik ≡ 〈µiHi|Hk〉 and Dik ≡ 〈ǫiEi|Ek〉. Mik =∫

∂V E∗i ×Hk ·dS andNik = 〈Ei|Jk〉 = −

∫

∂Ck

E∗i ×Hk ·dS,

where ∂Ck is the PEC surface of thekth resonator. TheBik andDik terms depend on the uncoupled parameters, they are calledthe uncoupled magnetic and electric energy, respectively.Thefull matrix expression of (14) is

M+N = −Γ†B −D†Γ, (15)

hereD† is the Hermitian transpose ofD and

Γ =

γ1 0 · · · · · · 00 γ2 ... 0...

.... . .

...0 0 · · · γn

(16)

γi is the complex angular frequency of theith uncoupled mode.Equations (14) and (15) relate the complex conjugate of

the reactive power-like componentsMik andNik to the bulkenergy componentsBik andDik.

C. Eigenvalue Problem

To find the eigenvalue equation, the expressions∇ · (E∗i ×

H) and∇ · (E × H∗i ) are used. Expanding∇ · (E∗

i × H)using (12), one can write

∇ · (E∗i ×H) = (∇×E

∗i ) ·H − (∇×H) ·E∗

i .

(4) and (7) simplify the above equation to

∇ · (E∗i ×H) = −γ∗

i µiH∗i ·H − γǫE ·E∗

i − J ·E∗i . (17)

ExpandingE andH in terms of the fieldsEi andHi ofthe uncoupled systems, using (1), (3), and (2), and integratingover the whole volume,

∑

k

(Mik + Fik + γ∗i Bik)bk + γAikak = 0 (18)

Fik = −∫

∀ ∂Ck

E∗i ×Hk · dS

Equation (18) relates theak and bk coefficients to oneanother. To derive the eigenvalue equation, one needs to findanother relation between the two coefficients and substitutebk in term ofak. This can be achieved by applying the aboveprocedure to∇· (E×H∗

i ). Noting that the electric fieldE isperpendicular to the conductors and hence to surface currentdensity(Js)|k, then

∫

E · J∗k dv = 0. Thus one can arrive at,

∑

i

γGkibi + [γ∗kDki +M∗

ik] ai = 0. (19)

The two relations (18) and (19) can be written in matrixform as

[

Γ†B +M+ F]

b+ γAa = 0, (20)(

M† + Γ†D)

a+ γGb = 0. (21)

The eigenvalue equation can be derived by noting that from(21)

b = − 1

γG−1

(

M† + Γ†D)

a. (22)

Substituting (22) back in (20) one arrives at,

A−1(

Γ†B +M+ F)

G−1(

M† + Γ†D)

a = γ2a. (23)

Using (15), (23) can be re-written as

A−1(

F −N − D†Γ)

G−1(

M† + Γ†D)

a = γ2a. (24)

For thin conductors, which is usually the case for metamate-rials cells, (15) shows that(F − N ) → 0. Accordingly, (24)simplifies to

−(

A−1D†Γ)

G−1(

M† + Γ†D)

a = γ2a. (25)

Equation (25) is the required eigenvalue equation, where theeigenvalues are the square of the complex angular frequencyand the eigenvectors are the coefficients of the fields given by(1). It represents anumerical recipewhich mixes theingre-dients (uncoupled modes) in specificquantities (determinedby the strength of overlap between the fields) to obtain theapproximated coupled frequencies and fields. For relativelyhigh Q resonators (Q > 15), such as the ones studied here,losses are considered to be small. Thus,Mik ≈ 0, γi ≈ jωi

(equivalently,Γ ≈ jΩ) andγ ≈ jω. Hence, (25) simplifies to(

A−1D†Ω) (

G−1ΩD)

a = ω2a. (26)

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.XX, NO. XX, XXX XXXX 4

It is worth noticing that the eigenvalue problem (26) doesnot depend on the modes amplitudes. Similarly if the frequen-ciesωi are scaled up or down,ω is changed by the same factor,something to be expected when losses are negligibly small.Noting that the modes amplitudes can always be selected suchthat the self terms (Gii andAii) are normalized to unity andthe off-diagonal terms (Gij andAij ) are usually much smallerin value, leads to the conclusion that the inversesA−1 andG−1

exist.

IV. RESULTS AND DISCUSSION

A. Proof of The Resonance Condition

It is interesting to verify that when losses are sufficientlysmall, the eigenvalue problem (26) satisfies the resonancecondition of the coupled system given by (10) or (11). Using(22)

WM =1

4b†Gb = 1

4

(

1

ωG−1ΩDa

)†

G(

1

ωG−1ΩDa

)

=1

4a†(

1

ω2D†ΩG−1ΩD

)

a. (27)

The eigenvalue problem (26) can be rewritten as

Aa =1

ω2D†ΩG−1ΩDa

Therefore,

WM =1

4a†Aa = WE . (28)

Relation (28) is identical to the resonance condition in (10).

B. Hybridization of Double split ring resonators

In this subsection, the eigenvalue problem (26) is solvedto find the modes of a meta-dimer, a circular and a rectan-gular DSRRs based on the hybridization of the two rings’fundamental modes. It is worth mentioning that the analysisis applicable to a broad class of interacting resonators such asloop-gap resonators [42], degenerate meta-dimers [3], asym-metric meta-dimers [6] and DSRRs. Since the rings are thin,all the permittivity functions(ǫ1(r), ǫ2(r), ǫ(r), ) can beapproximated byǫ0. Thus,A = D and (23) reduces to

ΩG−1ΩAa = ω2a (29)

If the integrations’ volume is large and contains the nearfields, thenGii = Aii. In fact, this is the resonance conditionfor the ith mode. Moreover, the coupling is weak such thatthe on-diagonal terms ofA and G dominate and the off-diagonal terms represent the perturbation due to coupling(A2

ii ≫ A12A21, i = 1, 2). Therefore, the eigenvalue problem(29) is simplified to(

ω21 ω1

ω1A12−ω2G12

A11

ω2ω2A21−ω1G21

A22

ω22

)

(

a1a2

)

= ω2

(

a1a2

)

.

(30)Furthermore, the operatorA and G can be symmetrized

by setting the complex amplitudes of the eigen-fieldsE1 and

E2 (H1 and H2) to be real (imaginary). Using (13), (30)simplifies to(

ω21 jω1

N21

A11

−jω2N12

A22

ω22

)

(

a1a2

)

= ω2

(

a1a2

)

. (31)

Solving (31), the coupled frequencies are found to be

ω2++ =

ω21 + ω2

2

2−√

(

ω21− ω2

2

2

)2

− ω1ω2(N12N21)

A11A22

(32)

and

ω2+− =

ω21 + ω2

2

2+

√

(

ω21 − ω2

2

2

)2

− ω1ω2(N12N21)

A11A22

, (33)

whereω++ andω+− are the angular frequencies of the sym-metric (bonding) and anti-symmetric (anti-bonding) modesrespectively. Thestrengthof coupling, which is proportionalto the shift of the frequencies from the uncoupled modes, canbe quantified by defining the coupling coefficientκ as

κ(ω1, ω2) =

√

− N12N21

ω1ω2A11A22

. (34)

Due to the interaction (off-diagonal terms) between theuncoupled modes, the coupled frequenciesω++ andω+− aredifferent fromω1 andω2. The interaction is due to the〈Ei|Jk〉term. This power interaction can be also explained in termsof the electric and magnetic field overlaps as given by (13)and (30). An important special case is whenω1 = ω2 = ω0.This, for example, represents the meta-dimer studied in [2],[3] using magnetic and electric interactions. For meta-dimers,(30) simplifies to(

ω20 ω2

0A12−G12

A11

ω20A21−G21

A22

ω20

)

(

a1a2

)

= ω2

(

a1a2

)

(35)

Equation (35) says thatκ can be regarded as the differencebetween a magnetic (κM ) and an electric (κE) components.This result was previously derived using the Lagrangian equa-tion of motion [3] and the perturbation method [43]. It isalso consistent with lumped circuit models, where couplingis modelled by a mutual inductance

(

Lm = G12/√G11G22

)

and a mutual capacitance(

Cm = A12/√A11A22

)

[44]. WhenκM = κE , the modes decouple. This reinforces the findingsof [3] that there is no avoided crossing wheneverκM = κE

as was before attributed to the higher order electric multipo-lar interactions [2]. The decoupling of modes even thoughκM and κE are not negligibly small may seem counterintuitive. Visualizing the coupling as a hybridization of twoatomic structures at which the electric-electric dipole and themagnetic-magnetic dipole interactions counteract, partially al-leviates the confusion but, does not remove it completely. Theeigenvalue problem (31) gives a general physical explanationby taking advantage of (34). The conditionκM = κE isequivalent toNik = 0 or equivalently

∫

VEi · J∗

k dv = 0.This means that there is no interaction between the uncoupledmodes, whenever the relative position and orientation of themeta-dimer atoms were meticulously tuned such thatEi is

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orthogonal toJk. In another words, there is no energy transfer,or a pathway, between the two uncoupled modes and henceno split in frequency.

The three different configurations analyzed in this sectionare quantitatively studied by solving the eigenvalue problem(29). As a first step and due to the lack of analytical ex-pressions, the fields and frequencies of the single uncoupledSRRs are computed using HFSSR© eigenmode solver (AnsysCorporation, Pittsburgh, PA, USA). The fields are exportedto a MATLAB R© code where the matricesG and A arecalculated and hence (29) is solved to determine the coupledfrequencies. Finally, the frequency values are compared tothe ones obtained by another HFSS eigenmode simulationof the complete DSRR systems. For the HFSS calculations,the conductivity of the SRR was assumed to be infinite. Thesolution domain was enclosed in an airbox which is 7 timeslarger than the SRR width. The structure is considered to beembedded in open space. Therefore, the airbox was subjectedto a Perfectly Matched Layer (PML) boundary condition [45].

1) Meta-dimer configuration:The coupling between therings forming a meta-dimer, as shown in Fig. 1, is usuallystrong. To validate the current approach,κ due to the in-teraction of the two fundamental modes on the rings of themeta-dimer shown in Fig. 1 and discussed in Ref. [46], iscalculated as a function of the twisting angleθ. κ, κM , κE andthe lower frequency are determined using (29). The frequencyand the net coupling are compared, as shown in Fig. 2 toHFSS eigenmode solver whereκω, the coupling coefficient,is calculated using the frequency splitting formula

κω =f2+− − f2

++

f2+− + f2

++

. (36)

g

Fig. 1. (a) A meta-dimer,rin = 2.5 mm, rout = 4 mm,Wg = 1 mm andseparation between the two rings= 2 mm. (b) The surface current distributionof the fundamental mode with a resonant frequency≈ 7 GHz. (c) The surfacecurrent distribution of the third mode with a resonant frequency≈ 22 GHz.

Even though the values ofκ is substantially large (approach-ing ≈ 0.45) and the split rings are radiative (Qrad ≈ 20),ECMT still gives accurate values over the whole twisting anglerange (θ ∈ [0, 180]).

The effect of the third order mode was included in (29)to yield a 4 × 4 matrix. The frequency was calculated overthe full rangeθ ∈ [0, 180]. It was found that the calculatedfrequency is more accurate. Nevertheless, unlike Ref. [46]theimprovement is insignificant (16,957 ppm using two modesonly and 16,738 ppm, using the four modes). The second modeis of an opposite symmetry to both the first and third modesand hence theE − J interactions are very small [46].

Fig. 2. The coupling coefficientsκ, κM , κE andκω for a tightly broadsidecoupled split rings forming a meta-dimer calculated over a range of twistinganglesθ. The inset shows the lower frequency found by applying ECMT andHFSS.

In the following, two non-degenerate configurations,A andB, are treated separately. ConfigurationA consists of twocoaxial circular SRR which have resonant frequencies of10.3 GHz and 15 GHz (Fig.3(a)). The net coupling strength(reflected in the magnitude of the frequency split) is calculatedas a function of the angleθ between the two gaps.

Wg

rin out

L

LWg

out

out

Lin

r

(a)

Fig. 3. The modes of the DSRR as a hybridization between the tworings’ modes. (a) ConfigurationA: Circular; The frequency of the smallring (resonator 1):f1 = 15 GHz, rout = 2.5 mm, rin = 2 mm andWg = 0.8 mm. The frequency of the large ring (resonator 2):f2 = 10.3GHz, rout = 1.5 mm, rin = 1 mm andWg = 0.1 mm. The right handside in (a) shows the coupled modes and frequencies; Symmetric (Bonding):f++ and Anti-symmetric (Anti-bonding):f+−. (b) ConfigurationB: Rect-angular; The frequency of the outer ring (resonator 2) isf2 = 1.35 THz,Lout = 36 µm, Lin = 30 µm andWg = 20 µm. The capacitive gap ofthe inner ringWg = 2− 16 µm, Lout = 24 µm andLin = 20 µm. Theright hand side in (b) shows the coupled modes and frequencies; Symmetric(Bonding): f++ and Anti-symmetric (Anti-bonding):f+−.

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TABLE ICIRCULAR DSRR: THE FREQUENCIES OF THE SYMMETRIC(f++) AND

ANTI -SYMMETRIC (f+−) MODES FOR DIFFERENTθ VALUES. THE

FREQUENCY OF THE INNER RING MODE(f1)= 15 GHz. THE FREQUENCYOF THE OUTER RING MODE(f2) = 10.3 GHz.θ IS THE ANGLE BETWEEN

THE TWO GAPS.

θ (deg) Symmetric Anti-symmetric

ECMT HFSS ECMT HFSS

0 10.29 10.26 15.46 15.6

90 9.9 9.56 15.95 16.3

180 9.55 9.24 16.93 16.6

Fig. 3(b) shows ConfigurationB. It consists of two co-axialsquare split rings. The outer ring has a fixed capacitive gap andhence a fixed resonant frequency off2 = 1.35 THz. The innerring’s gap is allowed to change from 2µm to 16µm whichis translated to a frequency range off1 = 1.64− 2.07 THz.

2) Analysis of ConfigurationA: Table. I shows the calcu-lated frequency of configurationA using both (29) and HFSSeigenmode solver. It is clear from the table that as the angleθ increases, the coupled frequencies deviate more from theuncoupled ones (f1 andf2). This can be explained by referringto the coupling coefficient expression (34). Both uncoupledangular frequenciesω1 andω2 are constant. The interactionsN12 and N21 are the only terms that change. Thus,κ isalways proportional to the reactive powersNik. The electricfield of the uncoupled modes is concentrated in the gap of theSRR. At the same time the conduction currentJk attains itsmaximum at the farthest side [23]. Therefore whenθ increases,κ increases.

3) Analysis of ConfigurationB: ConfigurationB is moreinteresting. Not only are the resonant frequencies in the farinfrared, but also the uncoupled frequencyω1 does change.Accordingly and from (34),κ is now a function of both theinteraction termsNik and the frequencyω1. The calculatedfrequencies are presented in Fig. 4 where again the valuescomputed using (29) are compared to those obtained by HFSSeigenmode solver. The results confirm the applicability of (29).It is also observed that asω1 increases, the shift in frequencyof the anti-symmetric mode| ω+−−ω1 | decreases. From (33),the frequency shift is a function of the productω1|N12|, which,as estimated in theAppendixbased on a lumped circuit modelapproach, decreases wheneverWg increases. To clarify why|N12| decreases whenWg increases, one refers to Fig. 5 whichillustrates how the hybridization can be visualized in terms ofthe interaction betweenE1 andJ2 of the uncoupled modes.From the figure the current density valuesJ2 are maximumnear the inner ring’s gap. Therefore whenWg increases (ω1

increases),E1 is distributed over a larger width and thusreducesN12. It is worth noticing that the frequency shift| (ω+− − ω1) /ω1 | is significantly large (≈ 15− 25%).

Unlike ω+−, ω++ does not significantly change withWg.This cannot be explained by simply referring to (32) whichwas derived based on the assumption that higher order termsare negligibly small. In fact, (32) and (33) predict that

ω2+− − ω2

1 = ω22 − ω2

++, (37)

which does not comply with the curves depicted in Fig. (4).

2 4 6 8 10 12 14 161

1.5

2

2.5

Wg (µm)

Fre

quen

cy (

TH

z)

Symmetric HFSSAntisymmetric HFSSSymmetric ECMTAntisymmetric ECMTf

2

f1

Fig. 4. The frequencies of the symmetric (f++) and anti-symmetric (f+−)modes of the rectangular double split-ring resonator.f1 is the frequency ofthe inner ring mode.f2 is the frequency of the outer ring mode.

1

8

6

4

2

0

a.u

Fig. 5. Rectangular DSRR: The plot of the magnitude and vector fields of theElectric field of the inner ring (E1) superimposed on the plot of the magnitudeand field vectors of the surface current density (J2). When the capacitive gapWg of the inner ring is extended,E1 spreads across a larger area.

To better understand whyω++ behaves as shown in Fig. (4),the higher order terms in the on-diagonal elements are re-tained. The on-diagonal terms are modified by subtractingχii = ω1ω2 (GikAki) / (A11A22) from ω2

i . For the DSRRshown in Fig. 3(b),Aki < 0 and henceχii < 0 . Thisis because the angle betweenE1 and E2 is 180. χii, thecoupling induced frequency shift coefficient, was theoreticallydescribed for coupled optical cavities [47]. The expressions(32) and (33) for the coupled frequencies are then modifiedby replacing eachω2

i with ω2i −χii. Accordingly, (37) becomes

ω22 − ω2

++ = ω2+− − ω2

1 + χ11 + χ22,

(i.e.,the shift betweenω++ andω2 is smaller than that betweenω+− andω1.) Thus, the effect ofχii is to pull ω++ up towardω2 and counteracts the influence of the off-diagonal crosscoupling term. With a similar argument to the one presentedin the Appendix, it can be shown that|χii| decreases asWg

increases, which keepsω++ curve approximately flat as Fig. 4

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.XX, NO. XX, XXX XXXX 7

shows.To determine the approximated fields using the coupled

mode formalism, the eigenvectors of (29) are computed andthe expansion (1) is used. Fig. 6 shows the electric field ofconfigurationB when Wg = 10 µm, f1 ≈ 1.93 THz. It isclear from the Fig. that the approximated total electric fielddoes not satisfy the boundary conditions at the rings’ surfaces.It also fringes inside the conductors. This limitation of ECMTcan be explained by noticing that, in general, the modesof a given uncoupled ring are independent and might notsatisfy the boundary conditions at other rings. Nevertheless,the eigenvalue problem (29) still gives very reasonable resultsas it shows the contribution of each of the uncoupled modesto the total DSRR fields.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

a.u

a.u

HFSS ECMT

Anti-symmetric

9

8

7

6

5

4

3

2

SymSymmetric

Fig. 6. The magnitude of the electric field,|E|, of the coupled modes of therectangular double split-ring resonator whenWg = 10 µm (f1 ≈ 1.93 THz).

It is worth to briefly discuss the similarities and differencesbetween the current approach and other very recent approachesbased on circuit models and electric field integral equation(EFIE) based scatterers’ models [22], [23], [46]. The circuitmodel approach relies on the calculation of the couplingcoefficientsκM andκE after the interaction energy betweenthe two rings’ degenerate modes is determined. Retardationis taken into account and hence the model is expected to beaccurate when the distance between the resonators is large[22], [23]. ECMT is an approach which only considers thenear field regime. Thus, it is expected to be accurate forcalculating tight and quasi tight coupled systems. Moreover,as demonstrated, ECMT can be used for degenerate and non-degenerate modes.

On the other hand, the EFIE-based approach relies on thesolution of Maxwell’s equation in an integral form. UnlikeECMT, radiation is included in the model and is automaticallycalculated by extending the frequency to the complex plane.Both ECMT and the EFIE-based approach are general and can

be applied to a system of an arbitrary number of conductingresonators and modes. ECMT, however, can be also appliedto dielectric resonators and cavities as well [37]–[39].

V. CONCLUSION

A general coupled mode equation in the form of aneigenvalue problem is derived. The eigen-frequencies aredetermined after finding the eigenvalues. The eigenvectorsare used to find the electromagnetic fields. If resonators arecompared to atoms, the eigenvalue problem can be consideredas the electromagnetic analog of molecular orbital theory.This conceptual view agrees with the way meta-materials unitcells are treated. It is shown that, for an arbitrary numberof resonators and/or modes, the solutions of the eigenvalueproblem guarantee that the resonance condition of the coupledmodes is always satisfied. As an immediate application, thebehavior of meta-dimers and DSRR was explained using theinteraction betweenEi andJk. Thus, the eigenvalue problemprovides an intuitive view to how resonators interact. The(Ei, Jk) interaction picture reduces the electric and magneticmutual coupling to a single interaction term. Two configura-tions were formulated and numerically solved and the resultswere compared to finite element simulations. To illustratethe versatility of the coupled mode formalism, the numericalfindings were explained using the(Ei, Jk) interaction picture.It was shown that for tight coupled resonators, the couplinginduced frequency shifts termsχii is very essential to correctlyexplain and quantify the DSRR behavior.

APPENDIX

Consider a simple LC circuit, with a capacitive gapWg,resonating at angular frequencyω1. In terms of the voltage onthe capacitorC, the average power is

P =1

2

√

C

LV 2,

where the relationω1 = 1/√LC was used. The capacitance

C ∝ 1/Wg, V ∝ |E|Wg. Thus, for a fixed powerP ,

ω1|E| ∝ 14

√

Wg

.

N12 is the integral ofE1 and J2. Thereforeω1|N12| ∼1/ 4

√

Wg.

ACKNOWLEDGMENT

We would also like to acknowledge CMC Microsystemsfor providing HFSSR© that facilitated this research. We wouldlike to thank Dr. Brent Petersen, Electrical and ComputerEngineering, University of New Brunswick, for his valuablefeedback on the manuscript.

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Sameh Y. Elnaggar has a B.Sc (Electrical Engi-neering) and M.Sc. (Electrical Engineering) fromAlexandria University and a Ph.D. (Electrical andComputer Engineering) from the University of NewBrunswick. His research interests are in compu-tational electromagnetic, interferometry, magneticresonance, quasi-optics, metamaterials and wirelesspower transfer. Currently, he is with School ofEngineering and Information Technology, Universityof New South Wales, Canberra.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL.XX, NO. XX, XXX XXXX 9

Richard J. Tervo has a B.Sc. (Physics) and M.Sc.(Experimental Nuclear Physics) from McMasterUniversity and a Ph.D. (EE) from Universit Laval.He joined the Department of Electrical and Com-puter Engineering in 1986.

Saba M. Mattar has a B.Sc. (Chemistry andPhysics) from Alexandria University, M.Sc. (Solidstate Science) from The American University inCairo and a Ph.D. (Chemistry) from McGill Univer-sity. He joined the Chemistry Department, Univer-sity of New Brunswick in 1986. He is presently anHonorary Research Professor at UNB. His interestsare theoretical and experimental magnetic resonance,trapping of reactive intermediates, electronic struc-ture of molecules, chemical physics, new metamate-rials, spectroscopy and instrumentation.