3 V. FREQUENCY DISPERSION When the dispersion of ε and µ is important the expression for the field energy density (14) has to be replaced by conventional material (in our case vacuum). In this situation the general boundary conditions (5) and (6) give (see, e.g., Ref. [17]) W = 1 8 π ( ∂ ( εω ) ∂ω ⃗ E 2 + ∂ ( µω ) ∂ω ⃗ H 2 ) . (15) ψ 1 = ψ 2 , 1 µ 1 ∂$_{ν}$ ψ 1 = 1 µ 2 ∂$_{ν}$ ψ 2 (TM) (12) ψ 1 = ψ 2 , 1 ε 1 ∂$_{ν}$ ψ 1 = 1 ε 2 ∂$_{ν}$ ψ 2 (TE) . (13) ∂ ( εω ) ∂ω > 0 , ∂ ( µω ) ∂ω > 0 (16) The energy density W defined by Eq. (15) is positive provided that instead of the special case in Eq. (8). For TE polarization we have for any values of ⃗ E 2 and ⃗ H $^{2}$. These inequalities imply lower bounds for the derivatives Using these boundary conditions together with the outgoing-wave conditions (10) at infinity we find that solu- tions of the mode equation (7) always have a negative Q -factor for 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 which will be discussed later in more detail we mention a mode in the limac¸on cavity with ε = − 9 / 4 and µ = − 1 ( | n | = 1 . 5 ). The normalized frequency is Ω 0 R/c = 45 . 1060 + i 0 . 1647 . The quality factor is therefore about Q 0 = − 137 . with ω > 0 . For conventional materials with positive ε and µ and nonresonant response to external fields the derivativesare small, and therefore often frequency dispersion can be safely ignored. For NIMs with negative ε and µ this is never possi- ble. In a transparency region, where we can neglect absorption in the material, causality requires the following additional in- equalities [30] Note that for a conventional material a similar effect may occur if a thin active layer with very strong gain is placed at the interface. In that case the jump of the derivative im- posed for negative index materials in Eq. (12) or (13) would be caused by an outflow of energy from that amplifying layer. It seems that for a passive NIM the exponential increase of the electromagnetic intensity contradicts the law of energy conservation. This is, however, not the case as can be seen by considering the electromagnetic field energy density For negative ε and µ these inequalities are stronger than the ones in Eq.(17). Note that there is a controversy in the NIM community concerning the existence of negative refraction in such a transparency region; see, e.g., Refs. [31–33]. This c on- troversy is, however, not settled yet; so in the following we assume that a transparency region with negative ε and µ ex- ists. ∂ε ∂ω > 2(1 − ε ) ω , ∂µ ∂ω > 2(1 − µ ) ω . (18) ∂ε ∂ω > − ε ω , ∂µ ∂ω > − µ ω , (17) W = 1 8 π ( ε ⃗ E 2 + µ ⃗ H 2 ) . (14) ε ( ω ) = 1 − ω 2 p ω 2 (19) ε ( ω ) = 1 − Ω 2 ω 2 − ω 2 0 + iγω . (20) It is worth mentioning that in (unrealistic) materials hav- ing no absorption at all in the whole frequency interval, the inequalities in Eq. (18) can turn into equalities. This is, for instance, the case for the nonlossy Drude model with the di- electric function and plasma frequency ω$_{p}$ . Another, more general model which is often used to describe NIMs locally in frequency space is the Drude-Lorentz system In the limiting case of no absorption γ → 0 the inequali- ties in Eq. (18) are fulfilled. For the subtle issue of causali ty in the nonlossy Drude-Lorentz model we refer the reader to Ref. [34]. It is worth mentioning that Dirac’s wave equation for rel- ativistic electrons possesses a similar “radiation catast rophe” which disappears in a proper quantum field theoretical treat - ment. For the NIM materials, the problem of negative field energy density can be, however, solved already on the wave equation level, namely by the inclusion of the frequency dis- persion of the electric permittivity and the magnetic perme- ability [4, 30]. In this equation and also in the following ones we suppress the dependency on the spatial coordinates for notational con- venience. As already pointed out by Veselago [4], as ε, µ < 0 the field energy (14) in a NIM would be negative and un- bounded from below. In our case of a quasi-bound cavity mode this results in an exponential decay of the field energy towards −∞ , as the cavity permanently looses (positive) en- ergy to the outside. Hence, the field intensity ⃗ E 2 > 0 inside the cavity increases exponentially in time. The imaginary part of the frequency Ω is therefore positive, and the quality factor negative, consistent with our numerical finding.