content_image/431536.jpg content_image/431523.jpg content_image/431525.jpg content_image/431532.jpg
<img src='content_image/431536.jpg'>

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 

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 

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 

<img src='content_image/431523.jpg'>

<img src='content_image/431525.jpg'>

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. 

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 

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 

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). 

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]. 

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 . 

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- 

<img src='content_image/431532.jpg'>

## VIII. CONCLUSIONS 

