6 FIG. 4: (color online). Q/ ( − Q 0 ) (upper panel) and overlap dif- ference 1 − S [lower panel, cf. Eq.(38)] vs. linear frequency dis- persion α$_{n}$ = α$_{ε}$ = α$_{µ}$ . All quantities are dimensionless. Here, Q 0 = − 137 is the quality factor in the nondispersive case α$_{n}$ = 0 . In the regime α$_{n}$ > 1 the electromagnetic field energy is positive and for α$_{n}$ > 4 also causality holds. The empty dots mark the re- sults of the full solution of the mode equation (32). The solid line is the 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 on the frequency dispersion. But why is the Q -factor so strongly dependent 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 a small modification of Im (Ω) in absolute numbers can have a relatively 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 0 and the modi- fied mode equation (32) with frequency Ω . In both cases use the same negative ε , µ . Both mode equations can give the same spatial mode pattern provided that n Ω 0 = ˜ n Ω . From this relation we find for | ω$_{r}$ − Re (Ω) | ≪ ω$_{r}$ and | Im (Ω) | ≪ Re (Ω) that Q − Q 0 ≈ α$_{n}$ − 1 . (39) This shows that the modified refractive index turns the (non- physical) negative quality factor Q 0 into a positive one, Q > 0 as soon as α$_{n}$ > 1 , i.e., exactly under the condition which ensures a positive field energy in the NIM. To illustrate this relation from another point of view, let us rewrite the right hand side of Eq. (39) as α$_{n}$ − 1 = − v$_{p}$ v$_{g}$ , (40) where v$_{p}$ = c/n is the phase velocity and v$_{g}$ = ∂ω/∂k is the group velocity with wave number k = nω/c . The in- equalities (17) required by a positive field energy ensure th at We addressed quasi-bound electromagnetic modes in negative-index metamaterial cavities. The simple approach which substitutes negative values of the electric permittiv- ity and magnetic permeability into the boundary conditions (which suffices to obtain negative refraction) gives rise to modes with negative Q -factor. This unphysical behavior can be removed by including linear frequency dispersion in the mode equation and in the boundary conditions, as is required by a positive field energy and causality. At complex reso- nance frequency, the effective permittivity then acquires a fi- VIII. CONCLUSIONS Finally, we note that scattering of a monochromatic wave with (real-valued) frequency ω at an obstacle made of a NIM is described by the ordinary mode equation (7) using incoming and outgoing wave conditions [17, 18]. However, when cal- culating spectra it is necessary to use the frequency-dependent permittivity ε ( ω ) and permeability µ ( ω ) to be consistent with the requirement of causality. This is usually ignored, perhaps because in a spectrum the FWHM δω cannot be easily distin- guished from − δω . A related expression as in Eq. (39) exists also for conven- tional materials where it was used to predict enhancement of quality factors in microcavities using highly dispersive mate- rials [40]. For small group velocity v$_{g}$ , corresponding to slow light, the effect of the Q -factor enhancement in Eqs. (39) and (40) is strongest. This prediction has been confirmed recent ly in experiments on slow light in photonic crystals [41]. again assuming that the change in the refractive index does not change the spatial mode structure. With Q = ω/δω and Q 0 = ω/δω 0 we arrive after a few algebraic manipulations at Eq. (39). nδω 0 = nδω + ωδn , (41) explains why in this region 1 − S is larger. From another point of view, the expression in Eq. (39) can also be understood purely in real frequency space. Consider a resonant structure of some spectrum, let’s say the Wigner delay 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 our example 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 As demonstrated in Fig. 4, the expression in Eq. (39) is in excellent agreement with the full solution of the modified mode equation (32). Note that near α$_{n}$ = 1 , where Q ≈ 0 , 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, wh ich the phase velocity v$_{p}$ and the group velocity v$_{g}$ have a differ- ent sign in a NIM. The inequalities (18) required by causality carry over to | v$_{g}$ | < | v$_{p}$ | , i.e., superluminal energy propaga- tion is forbidden. -1 0 1 2 3 4 Q/(-Q 0 ) 0 1 2 3 4 5 $_{n}$ =  0.0001 0.001 1-S positive field energy causality