Electromagnetic modes in cavities made of negative-index metamaterials
Jan Wiersig, Julia Unterhinninghofen, Henning Schomerus, Ulf Peschel, Martina Hentschel
aa r X i v : . [ phy s i c s . op ti c s ] J a n Electromagnetic modes in cavities made of negative-index metamaterials
Jan Wiersig, Julia Unterhinninghofen
Institut f¨ur Theoretische Physik, Universit¨at Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
Henning Schomerus
Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
Ulf Peschel
Institute for Optics, Information and Photonics, University Erlangen-Nuremberg, D-91058, Erlangen, Germany
Martina Hentschel
Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, D-01187 Dresden, Germany (Dated: November 20, 2018)We discuss electromagnetic modes in cavities formed by metamaterials with negative refraction and demon-strate that the straightforward approach to substitute negative values of the electric permittivity and the magneticpermeability leads to quasi-bound states with a negative quality factor. To ensure positive quality factors and aconsistent physical interpretation of the quasi-bound states it is essential to include the frequency dispersion ofthe permittivity and the permeability, as required by positive field energy and causality. The basic mode equa-tion and the boundary conditions including linear frequency dispersion are derived. As an example we considera disk-like cavity with deformed cross sectional shape. The transition from the unphysical nondispersive casewith negative quality factors to the dispersive case with positive quality factors is demonstrated numerically andin an analytical perturbative treatment.
PACS numbers: 42.25.-p, 42.60.Da, 78.67.Pt
I. INTRODUCTION
Negative-index metamaterials (NIMs) are artificial com-posites characterized by simultaneously negative values of theelectric permittivity ε and the magnetic permeability µ [1–3].These materials were theoretically predicted already in 1968by Veselago [4]. In present days there is a strong interest insuch materials because of potential applications, such as elec-tromagnetic cloaking [5], subwavelength imaging [6] and fo-cussing of light [7].Cavities which confine electromagnetic waves in all threespatial dimensions have also attracted considerable attentionin the recent years, in particular in the optical regime [8]. Thestrong interest is partly due to the numerous future applica-tions, such as single-photon emitters [9] and ultralow thresh-old lasers [10–12], and partly due to the possibility to ad-dress fundamental questions of light-matter interaction [13]and ray-wave correspondence [14–16].The fabrication of electromagnetic cavities made of NIMsis rather challenging with nowadays technology. The lit-erature in this field is therefore limited to a few theoreti-cal studies, e.g., on one-dimensional cavities made of dis-tributed Bragg reflectors [17] and two-dimensional superscat-terers [18]. These works, however, consider scattering ofplane waves and do not treat the electromagnetic modes asquasi-bound states with finite lifetime. In this paper we fillthis gap and discover a subtle difficulty when defining quasi-bounded states in the canonical way. We show that for a con-sistent physical interpretation of modes in NIMs the frequencydispersion of ε and µ is crucial. Our consideration is gen-eral and applies to all sorts of NIM cavities. For illustrationwe present results for two-dimensional disk cavities with de- formed cross-sectional shape.The paper is organized as follows. Sections II and III pro-vide a brief review on Maxwell’s equations for monochro-matic fields and on electromagnetic cavities made of conven-tional materials. The boundary conditions for NIM cavitesand the appearance of negative quality factors are discussedin Sec. IV. Section V deals with the basic properties of thefrequency dispersion of ε and µ . A modified mode equationincluding linear dispersion and a numerical solution of thisequation is presented in Sec. VI. A discussion of the effectsof the dispersion is given in Sec. VII. Finally, Sec. VIII con-tains the conclusions. II. MAXWELL’S EQUATIONS
The source-free Maxwell’s equations in the frequency do-main are ∇ × ~E = i ωc ~B , ∇ × ~H = − i ωc ~D , (1) ∇ · ~D = 0 , ∇ · ~B = 0 , (2)where c is the speed of light in vacuum. The constitutive re-lations for monochromatic fields with frequency ω are in theisotropic and linear regime ~D ( ~r, ω ) = ε ( ~r, ω ) ~E ( ~r, ω ) , (3) ~B ( ~r, ω ) = µ ( ~r, ω ) ~H ( ~r, ω ) (4)with electric permittivity ε and magnetic permeability µ . Theboundary conditions at an interface between a material 1 anda material 2 are given by ~ν × ( ~E − ~E ) = 0 = ~ν × ( ~H − ~H ) , (5) ~ν · ( µ ~H − µ ~H ) = 0 = ~ν · ( ε ~E − ε ~E ) , (6)where ~ν is the local normal vector on the boundary. III. CONVENTIONAL CAVITIES
When discussing electromagnetic modes in cavities madeof conventional materials one usually takes advantage of twosimplifications: (i) the frequency dispersion of the permittiv-ity ε is ignored, assuming that the frequency interval of inter-est is sufficiently small. (ii) The permeability µ is assumed tobe constant and unity at all frequencies, throughout the wholespace.In the following, we focus on disk-like cavities mainly forillustration purposes. We emphasise here that our results canbe extended to arbitrarily shaped three-dimensional cavities ina straightforward manner. For the quasi-2D geometry of thedisk one separates the ( x, y ) -dynamics in the plane of the diskfrom the z -dynamics by expanding the electric field in termsof ~E ( x, y ) e ink z z , with k z = 0 or k z finite. In the latter casethe refractive index n = √ εµ in the mode equation can bereplaced by an effective index. Assuming a piecewise con-stant index of refraction one can derive the following modeequation for a quasi-two-dimensional disk [19, 20] − ∇ ψ = n Ω c ψ , (7)where Ω is the frequency of the mode. This Helmholtz equa-tion holds for both transverse magnetic (TM) and transverseelectric (TE) polarization. For TM polarization the elec-tric field is perpendicular to the cavity plane with E z = Re [ ψ ( x, y ) e − i Ω t ] . At the boundary between a material 1 anda material 2 the general boundary conditions (5) and (6) givethe following continuity relations for the wave function ψ andits normal derivative ∂ ν ψ along the normal ~ν , ψ = ψ , ∂ ν ψ = ∂ ν ψ (TM) , (8)assuming that µ = µ . For TE polarization, the mag-netic field is perpendicular to the cavity plane with H z = Re [ ψ ( x, y ) e − i Ω t ] . The boundary conditions are ψ = ψ , n ∂ ν ψ = 1 n ∂ ν ψ (TE) , (9)again assuming µ = µ . At infinity, outgoing wave condi-tions in the two-dimensional disk plane ψ ∼ ψ out = h ( θ, k ) exp ( ikr ) √ r (10)with wave number k = Ω /c are imposed for both polariza-tions, which results in quasi-bound states with frequencies Ω situated in the lower half of the complex plane. Whereas the real part is the usual frequency, the imaginary part is related tothe lifetime τ = − / [2 Im Ω] . The quality factor of a quasi-bound state is defined by Q = − Re Ω / [2 Im Ω] . These reso-nant states, first introduced by Gamow [21] and by Kapur andPeierls [22], are connected to the peak structure in scatteringspectra; see [23] for an introduction.As an example we choose a disk-like cavity with the bound-ary curve being the limac¸on of Pascal, which reads in polarcoordinates ρ ( φ ) = R (1 + e cos φ ) . (11)For vanishing deformation parameter e this gives the circu-lar disk with radius R . We choose a deformed disk with e = 0 . . Exactly this geometry has been studied for con-ventional materials in the context of directed light emissionfrom microlasers, theoretically [24] as well as experimen-tally [11, 25–27]. The value of R itself is not relevant, onlythe ratio R/λ is important, where λ = 2 π/k is the wave-length. Therefore, we consider in the following a normalizedfrequency Ω R/c = kR . Figure 1 shows a typical TM polar-ized mode in a limac¸on cavity with low index of refraction, n = 1 . , computed with the boundary element method [28].The mode is localized along an unstable periodic ray trajec-tory, i.e., it is a so-called scarred mode [29]. From the relativeintensity of the different segments we can assess the directionof energy flow as indicated by the arrows. The frequency is Ω R/c = 44 . − i . , the Q -factor is therefore about . Such a medium- Q , scarred mode is well suited to demon-strate how the light is (partially) refracted out. This will beuseful in the following comparison to the NIM cavities. FIG. 1: (color online). Near-field intensity pattern of an electromag-netic mode with normalized frequency Ω R/c = 44 . − i . in a conventional cavity with ε = 9 / and µ = 1 ( n = 1 . ) sur-rounded by vacuum with ε = µ = 1 . Arrows illustrate the directionof the energy flow. IV. NIM BOUNDARY CONDITIONS
In the case of a NIM cavity the permeability µ can no longerto be treated as spatially uniform as this quantity changes signat the interface between of the NIM cavity and the surroundingconventional material (in our case vacuum). In this situationthe general boundary conditions (5) and (6) give (see, e.g.,Ref. [17]) ψ = ψ , µ ∂ ν ψ = 1 µ ∂ ν ψ (TM) (12)instead of the special case in Eq. (8). For TE polarization wehave ψ = ψ , ε ∂ ν ψ = 1 ε ∂ ν ψ (TE) . (13)Using these boundary conditions together with theoutgoing-wave conditions (10) at infinity we find that solu-tions of the mode equation (7) always have a negative Q -factorfor NIMs, i.e., the intensity of such a solution ψ does not de-cay but instead increases exponentially in time, which is un-physical for a passive material. As a typical example whichwill be discussed later in more detail we mention a mode inthe limac¸on cavity with ε = − / and µ = − ( | n | = 1 . ).The normalized frequency is Ω R/c = 45 . i . .The quality factor is therefore about Q = − .Note that for a conventional material a similar effect mayoccur if a thin active layer with very strong gain is placedat the interface. In that case the jump of the derivative im-posed for negative index materials in Eq. (12) or (13) wouldbe caused by an outflow of energy from that amplifying layer.It seems that for a passive NIM the exponential increaseof the electromagnetic intensity contradicts the law of energyconservation. This is, however, not the case as can be seen byconsidering the electromagnetic field energy density W = 18 π (cid:16) ε ~E + µ ~H (cid:17) . (14)In this equation and also in the following ones we suppressthe dependency on the spatial coordinates for notational con-venience. As already pointed out by Veselago [4], as ε, µ < the field energy (14) in a NIM would be negative and un-bounded from below. In our case of a quasi-bound cavitymode this results in an exponential decay of the field energytowards −∞ , as the cavity permanently looses (positive) en-ergy to the outside. Hence, the field intensity ~E > insidethe cavity increases exponentially in time. The imaginary partof the frequency Ω is therefore positive, and the quality factornegative, consistent with our numerical finding.It is worth mentioning that Dirac’s wave equation for rel-ativistic electrons possesses a similar “radiation catastrophe”which disappears in a proper quantum field theoretical treat-ment. For the NIM materials, the problem of negative fieldenergy density can be, however, solved already on the waveequation level, namely by the inclusion of the frequency dis-persion of the electric permittivity and the magnetic perme-ability [4, 30]. V. FREQUENCY DISPERSION
When the dispersion of ε and µ is important the expressionfor the field energy density (14) has to be replaced by W = 18 π (cid:18) ∂ ( εω ) ∂ω ~E + ∂ ( µω ) ∂ω ~H (cid:19) . (15)The energy density W defined by Eq. (15) is positive providedthat ∂ ( εω ) ∂ω > , ∂ ( µω ) ∂ω > (16)for any values of ~E and ~H . These inequalities imply lowerbounds for the derivatives ∂ε∂ω > − εω , ∂µ∂ω > − µω , (17)with ω > . For conventional materials with positive ε and µ and nonresonant response to external fields the derivatives aresmall, and therefore often frequency dispersion can be safelyignored. For NIMs with negative ε and µ this is never possi-ble.In a transparency region, where we can neglect absorptionin the material, causality requires the following additional in-equalities [30] ∂ε∂ω > − ε ) ω , ∂µ∂ω > − µ ) ω . (18)For negative ε and µ these inequalities are stronger than theones in Eq.(17). Note that there is a controversy in the NIMcommunity concerning the existence of negative refraction insuch a transparency region; see, e.g., Refs. [31–33]. This con-troversy is, however, not settled yet; so in the following weassume that a transparency region with negative ε and µ ex-ists.It is worth mentioning that in (unrealistic) materials hav-ing no absorption at all in the whole frequency interval, theinequalities in Eq. (18) can turn into equalities. This is, forinstance, the case for the nonlossy Drude model with the di-electric function ε ( ω ) = 1 − ω p ω (19)and plasma frequency ω p . Another, more general modelwhich is often used to describe NIMs locally in frequencyspace is the Drude-Lorentz system ε ( ω ) = 1 − Ω ω − ω + iγω . (20)In the limiting case of no absorption γ → the inequali-ties in Eq. (18) are fulfilled. For the subtle issue of causalityin the nonlossy Drude-Lorentz model we refer the reader toRef. [34]. VI. MODIFIED MODE EQUATION
In the following, we show that when dealing with quasi-bound states not only Maxwell’s equations but also the con-stitutive relations including the frequency dispersion have tobe extended to the complex frequency plane. This fact di-rectly leads to a modified mode equation which is capable ofdescribing linear frequency dispersion. This mode equationpredicts positive quality factors in agreement with the require-ments of positive field energy and causality.The key observation to start with is that a quasi-bound state ~E Ω ( t ) = ~E e − i Ω t (21)with complex-valued frequency Ω is not a monochromaticwave as its Fourier decomposition ~E Ω ( t ) = Z ∞−∞ dω ~E ( ω ) e − iωt (22)gives nonvanishing ~E ( ω ) in a region around ω ≈ Re (Ω) . Withthis decomposition we can write the constitutive relation (3)as ~D Ω ( t ) = Z ∞−∞ dω ε ( ω ) ~E ( ω ) e − iωt . (23)To proceed further, let us fix a real-valued frequency ω r aroundwhich we study electromagnetic modes. We restrict ourselvesto modes with sufficiently small | Im (Ω) | , i.e., not too small Q -factor, and Re (Ω) ≈ ω r . This allows the expansion of thepermittivity ε ( ω ) ≈ ε ( ω r ) + ∂ε∂ω (cid:12)(cid:12)(cid:12) ω r ( ω − ω r ) . (24)Inserting this expansion into Eq. (23) gives ~D Ω ( t ) = ε ( ω r ) ~E Ω ( t ) + ∂ε∂ω (cid:12)(cid:12)(cid:12) ω r (cid:18) i ∂∂t − ω r (cid:19) ~E Ω ( t ) . (25)Now we exploit the property of quasi-bound states ∂∂t ~E Ω = − i Ω ~E Ω , which can be deduced from Eq. (21) and leads to ~D Ω ( t ) = ˜ ε (Ω) ~E Ω ( t ) , (26)with modified permittivity ˜ ε (Ω) = ε ( ω r ) + ∂ε∂ω (cid:12)(cid:12)(cid:12) ω r (Ω − ω r ) . (27)Equation (26) represents an analytic continuation of the per-mittivity ε ( ω ) to the complex-frequency plane Ω . For thiscontinuation the linearization in Eq. (24) is not needed. Infact, for the modified mode equation that we derive in the fol-lowing, the extension to any analytic function ε ( ω ) is straight-forward. Nevertheless, for clarity we restrict ourself to a lin-ear frequency dispersion. We would like to point out that theimaginary part of ˜ ε is related to the frequency dispersion andnot to optical absorption in the material, which is neglected here. Nevertheless, a realistic Ω = ω r with negative imag-inary part introduces a kind of loss in the originally losslessmedium which counteracts the nonphysical exponential in-crease of the intensity, thereby turning the negative qualityfactor into a positive one.In the following it will be convenient to express the valuesof the derivatives of ε and µ at the fixed frequency ω r by theirdimensionless linear dispersions α ε = − ∂ε∂ω (cid:12)(cid:12)(cid:12) ω r ω r ε , α µ = − ∂µ∂ω (cid:12)(cid:12)(cid:12) ω r ω r µ . (28)For NIMs ( ε < , µ < ) these quantities α ε , α µ have tobe chosen larger than 1 to satisfy the inequalities (17) (forconventional dielectrics α ε , α µ must be smaller than 1.). Tosatisfy the inequalities (18) for NIMs we must have α ε > − ε , α µ > − µ . (29)With the quantities in Eq. (28) we can rewrite Eq. (27) as ˜ ε (Ω) = ε ( ω r ) (cid:18) α ε ω r − Ω ω r (cid:19) . (30)We can derive a modified permeability in an analogue way ˜ µ (Ω) = µ ( ω r ) (cid:18) α µ ω r − Ω ω r (cid:19) . (31)As a result of our considerations we can use Maxwell’sequations (1)-(2) and the constitutive relations (3)-(4) formonochromatic waves with modified permittivity and perme-ability given by Eqs. (30) and (31). A direct consequence isthe modified mode equation − ∇ ψ = ˜ n (Ω) Ω c ψ , (32)with the modified refractive index ˜ n (Ω) = p ˜ ε (Ω)˜ µ (Ω) ≈ n ( ω r ) (cid:18) α n ω r − Ω ω r (cid:19) (33)and the dimensionless linear dispersion α n = − ∂n∂ω (cid:12)(cid:12)(cid:12) ω r ω r n = α ε + α µ . (34)In the derivation we have ignored terms of order (Ω − ω r ) ,which is consistent with Eq. (24). For the square root inEq. (33) we choose the positive branch. Note that the (modi-fied) refractive index can be defined negative or positive. Thisdoes not matter for our purpose, as the sign of the refractiveindex neither enters the mode equation (32) nor the modifiedboundary conditions ψ = ψ , µ ∂ ν ψ = 1˜ µ ∂ ν ψ (TM) (35) ψ = ψ , ε ∂ ν ψ = 1˜ ε ∂ ν ψ (TE) . (36)The phenomenon of negative refraction is here a result of therelative sign of the permittivity and the permeability in theboundary conditions (35) and (36).After the quantities ε , µ and their first derivative are spec-ified at a given reference frequency ω r , the modified modeequation (32), the modified permittivity (30), the modifiedpermeability (31), the modified refractive index (33), and themodified boundary conditions (35)-(36) have to be solvedself-consistently. This can be done with only slight modifi-cations of standard approaches, such as the boundary elementmethod [28].As an example we consider again a mode in the limac¸oncavity with normalized frequency Re (Ω) R/c around . . Wefix the reference frequency ω r R/c therefore to be . . Notethat the precise value of the reference frequency is not rele-vant as long as we choose the values for the permittivity, thepermeability and their derivatives accordingly, e.g. by readingoff their values from a material dispersion curve at the givenreference frequency. For the NIM we consider ε ( ω r ) = − / and µ ( ω r ) = − as in the previous section. To fulfill the con-straints from Eq. (29) α ε > , α µ > , (37)we first set α ε = α µ = 4 . and vary this value later. Accord-ing to Eq. (34) the quantity α n is then also . . Figure 2(a)shows as an example a TM polarized mode. The Q -factorturns out to be positive, Q = 438 , due to the inclusion of thedispersion. Interestingly, the spatial mode pattern is rather in-sensitive to the inclusion of frequency dispersion, as a closercomparison of the dispersive case in Fig. 2(a) and the (un-physical) nondispersive case in Fig. 2(b) shows. FIG. 2: (color online). (a) Near-field intensity pattern of an electro-magnetic mode with Ω R/c = 45 . − i . in a NIM cavitywith ε ( ω r ) = − / , µ ( ω r ) = − ( | n ( ω r ) | = 1 . ), ω r R/c = 45 . ,and linear frequency dispersion α ε = α µ = 4 . . Arrows illus-trate the direction of the energy flow. (b) Unphysical mode with Ω R/c = 45 . i . in a NIM cavity with ε = − / and µ = − ( | n | = 1 . ) calculated without frequency dispersion. Contrasting the mode in the NIM cavity in Fig. 2(a) witha corresponding mode in a conventional material in Fig. 1clearly reveals the negative refraction. To be more precise,one part of the beam is confined by total internal reflectionand another part is refracted out. Moreover, a careful inspec-tion shows a different Goos-H¨anchen shift (GHS) for the NIM and for the conventional material, as can be seen in Fig. 3. TheGHS is a lateral shift of totally reflected beams along the op-tical interface due to interference [35] (for GHS in cavitiessee Refs. [36, 37]). According to Ref. [38], a light beam in aconventional material reflected at the interface to a NIM ex-periences a negative GHS. In our case the light propagates inthe NIM and is reflected at the interface to a conventional ma-terial. We compute the GHS by reflecting a Gaussian beamat a planar dielectric interface, neglecting boundary curvatureeffects [39]. For the mode in the conventional cavity, seeFig. 3(a), we find a nearly perfect agreement between sucha beam reflection at a planar interface and the full mode cal-culation. For the mode in the NIM cavity, see Fig. 3(b), theagreement is somewhat reduced, but nevertheless the appear-ance of a negative GHS can be clearly seen.
FIG. 3: (color online). Goos-H¨anchen shift in (a) a conventional cav-ity (cf. Fig. 1), and (b) a NIM cavity [cf. Fig. 2(a)]. The dotted line isthe center of a Gaussian beam being reflected at the dielectric inter-face. The appearance of the Goos-H¨anchen shift along the boundaryis evident. The mode pattern is the full numerical solution of therespective mode equation.
VII. DISCUSSION
To quantify small differences in the spatial pattern of themode ψ without frequency dispersion [as, e.g., in Fig. 2(b)]and a mode ψ with dispersion [as, e.g., in Fig. 2(a)] we exam-ine the normalized spatial overlap S = | R C dxdy ψ ∗ ψ | qR C dxdy ψ ∗ ψ qR C dxdy ψ ∗ ψ . (38)We restrict the integrals to the interior of the cavity C as theexterior is influenced by the actual value of the quality factor.For the modes in Fig. 2 (a) and (b) we find − S ≈ . × − ,so indeed the overlap is nearly unity. The lower panel of Fig. 4shows − S as function of the linear frequency dispersion α n .Except near α n = 1 , the overlap − S is below . . Theupper panel of Fig. 4 shows Q/ ( − Q ) with Q = − asfunction of α n . It can be observed that for α n > , where thefield energy is positive, the Q -factor is also positive.Can we understand the basic features observed in Fig. 4?To do so, let us first note that the quantities | (˜ n − n ) /n | , | (˜ ε − ε ) /ε | , and | (˜ µ − µ ) /µ | are small, as required by the linearexpansions, e.g., in Eq. (24). The applicability of perturbation -101234 Q / (- Q ) α n = α ε = α µ - S positive field energy causality FIG. 4: (color online). Q/ ( − Q ) (upper panel) and overlap dif-ference − S [lower panel, cf. Eq.(38)] vs. linear frequency dis-persion α n = α ε = α µ . All quantities are dimensionless. Here, Q = − is the quality factor in the nondispersive case α n = 0 .In the regime α n > the electromagnetic field energy is positiveand for α n > also causality holds. The empty dots mark the re-sults of the full solution of the mode equation (32). The solid line isthe theoretical prediction in Eq. (39). theory to the Helmholtz equations (7) and (32) therefore im-plies that the spatial mode pattern does not depend much onthe frequency dispersion. But why is the Q -factor so stronglydependent on the frequency dispersion? This can be under-stood from the observation that for long-lived modes with high Q = − Re Ω / [2 Im Ω] , the imaginary part of the frequency Ω is small compared to the real part of Ω . Therefore, even asmall modification of Im (Ω) in absolute numbers can have arelatively large effect on the quality factor.To see how Q changes with the linear dispersion α n , con-sider the mode equation (7) with frequency Ω and the modi-fied mode equation (32) with frequency Ω . In both cases usethe same negative ε , µ . Both mode equations can give thesame spatial mode pattern provided that n Ω = ˜ n Ω . Fromthis relation we find for | ω r − Re (Ω) | ≪ ω r and | Im (Ω) | ≪ Re (Ω) that Q − Q ≈ α n − . (39)This shows that the modified refractive index turns the (non-physical) negative quality factor Q into a positive one, Q > as soon as α n > , i.e., exactly under the condition whichensures a positive field energy in the NIM. To illustrate thisrelation from another point of view, let us rewrite the righthand side of Eq. (39) as α n − − v p v g , (40)where v p = c/n is the phase velocity and v g = ∂ω/∂k isthe group velocity with wave number k = nω/c . The in-equalities (17) required by a positive field energy ensure that the phase velocity v p and the group velocity v g have a differ-ent sign in a NIM. The inequalities (18) required by causalitycarry over to | v g | < | v p | , i.e., superluminal energy propaga-tion is forbidden.As demonstrated in Fig. 4, the expression in Eq. (39) isin excellent agreement with the full solution of the modifiedmode equation (32). Note that near α n = 1 , where Q ≈ ,the absolute value of the imaginary part of Ω and the quantities | (˜ n − n ) /n | , | (˜ ε − ε ) /ε | , and | (˜ µ − µ ) /µ | are not small. Hence,the linear approximations in our theory are not justified, whichexplains why in this region − S is larger.From another point of view, the expression in Eq. (39) canalso be understood purely in real frequency space. Considera resonant structure of some spectrum, let’s say the Wignerdelay time or a scattering cross section. Note that such res-onances are determined only by the value of the product ζ = n ( ω ) ω of refractive index n and the frequency ω . In ourexample of the limac¸on cavity it is nkR rather than just kR ,with wave number k = ω/c . Let the full width at half max-imum (FWHM) of the resonant peak be δζ = nδω + ωδn .Comparing the nondispersive case ( δn = 0 ) with the disper-sive case we get nδω = nδω + ωδn , (41)again assuming that the change in the refractive index doesnot change the spatial mode structure. With Q = ω/δω and Q = ω/δω we arrive after a few algebraic manipulations atEq. (39).A related expression as in Eq. (39) exists also for conven-tional materials where it was used to predict enhancement ofquality factors in microcavities using highly dispersive mate-rials [40]. For small group velocity v g , corresponding to slowlight, the effect of the Q -factor enhancement in Eqs. (39) and(40) is strongest. This prediction has been confirmed recentlyin experiments on slow light in photonic crystals [41].Finally, we note that scattering of a monochromatic wavewith (real-valued) frequency ω at an obstacle made of a NIMis described by the ordinary mode equation (7) using incomingand outgoing wave conditions [17, 18]. However, when cal-culating spectra it is necessary to use the frequency-dependentpermittivity ε ( ω ) and permeability µ ( ω ) to be consistent withthe requirement of causality. This is usually ignored, perhapsbecause in a spectrum the FWHM δω cannot be easily distin-guished from − δω . VIII. CONCLUSIONS
We addressed quasi-bound electromagnetic modes innegative-index metamaterial cavities. The simple approachwhich substitutes negative values of the electric permittiv-ity and magnetic permeability into the boundary conditions(which suffices to obtain negative refraction) gives rise tomodes with negative Q -factor. This unphysical behavior canbe removed by including linear frequency dispersion in themode equation and in the boundary conditions, as is requiredby a positive field energy and causality. At complex reso-nance frequency, the effective permittivity then acquires a fi-nite imaginary part, which attenuates the mode even in ab-sence of physical absorption.As an example we studied a disk-like cavity with noncircu-lar cross-sectional shape. The modified mode equation resultsin modes with positive quality factor and clear signatures ofnegative refraction and negative Goos-H¨anchen shift. Acknowledgments
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