4 VI. MODIFIED MODE EQUATION here. Nevertheless, a realistic Ω = ω$_{r}$ with negative imag- inary part introduces a kind of loss in the originally lossless medium which counteracts the nonphysical exponential in- crease of the intensity, thereby turning the negative quality factor into a positive one. In the following it will be convenient to express the values of the derivatives of ε and µ at the fixed frequency ω$_{r}$ by their dimensionless linear dispersions 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 to be extended to the complex frequency plane. This fact di- rectly leads to a modified mode equation which is capable of describing linear frequency dispersion. This mode equation predicts 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 with complex-valued frequency Ω is not a monochromatic wave as its Fourier decomposition α$_{ε}$ = − ∂ε ∂ω ∣ ω r ω$_{r}$ ε , α$_{µ}$ = − ∂µ ∂ω ∣ ω r ω$_{r}$ µ . (28) ⃗ E Ω ( t ) = ⃗ E 0 e − i Ω t (21) ⃗ E Ω ( t ) = ∫ ∞ −∞ dω ⃗ E ( ω ) e − iωt (22) α$_{ε}$ > 2 − 2 ε , α$_{µ}$ > 2 − 2 µ . (29) ˜ ε (Ω) = ε ( ω$_{r}$ ) ( 1 + α$_{ε}$ ω$_{r}$ − Ω ω$_{r}$ ) . (30) ⃗ D Ω ( t ) = ∫ ∞ −∞ dω ε ( ω ) ⃗ E ( ω ) e − iωt . (23) ˜ µ (Ω) = µ ( ω$_{r}$ ) ( 1 + α$_{µ}$ ω$_{r}$ − Ω ω$_{r}$ ) . (31) ε ( ω ) ≈ ε ( ω$_{r}$ ) + ∂ε ∂ω ∣ ω r ( ω − ω$_{r}$ ) . (24) − ∇ $^{2}$ψ = ˜ n $^{2}$(Ω) Ω 2 c 2 ψ , (32) ⃗ D Ω ( t ) = ε ( ω$_{r}$ ) ⃗ E Ω ( t ) + ∂ε ∂ω ∣ ω r ( i ∂ ∂t − ω$_{r}$ ) ⃗ E Ω ( t ) . (25) ⃗ D Ω ( t ) = ˜ ε (Ω) ⃗ E Ω ( t ) , (26) ˜ ε (Ω) = ε ( ω$_{r}$ ) + ∂ε ∂ω ∣ ω r (Ω − ω$_{r}$ ) . (27) ˜ n (Ω) = √ ˜ ε (Ω)˜ µ (Ω) ≈ n ( ω$_{r}$ ) ( 1 + α$_{n}$ ω$_{r}$ − Ω ω$_{r}$ ) (33) α$_{n}$ = − ∂n ∂ω ∣ ω r ω$_{r}$ n = α$_{ε}$ + α$_{µ}$ 2 . (34) with the modified refractive index and the dimensionless linear dispersion As a result of our considerations we can use Maxwell’s equations (1)-(2) and the constitutive relations (3)-(4) for monochromatic waves with modified permittivity and perme- ability given by Eqs. (30) and (31). A direct consequence is the modified mode equation We can derive a modified permeability in an analogue way With the quantities in Eq. (28) we can rewrite Eq. (27) as For NIMs ( ε < 0 , µ < 0 ) these quantities α$_{ε}$ , α$_{µ}$ have to be chosen larger than 1 to satisfy the inequalities (17) (for conventional dielectrics α$_{ε}$ , α$_{µ}$ must be smaller than 1.). To satisfy the inequalities (18) for NIMs we must have In the derivation we have ignored terms of order (Ω − ω$_{r}$ ) $^{2}$, which is consistent with Eq. (24). For the square root in Eq. (33) we choose the positive branch. Note that the (modi- fied) refractive index can be defined negative or positive. Th is does not matter for our purpose, as the sign of the refractive index neither enters the mode equation (32) nor the modified boundary conditions ψ 1 = ψ 2 , 1 ˜ µ 1 ∂$_{ν}$ ψ 1 = 1 ˜ µ 2 ∂$_{ν}$ ψ 2 (TM) (35) ψ 1 = ψ 2 , 1 ˜ ε 1 ∂$_{ν}$ ψ 1 = 1 ˜ ε 2 ∂$_{ν}$ ψ 2 (TE) . (36) Equation (26) represents an analytic continuation of the per- mittivity ε ( ω ) to the complex-frequency plane Ω . For this continuation the linearization in Eq. (24) is not needed. In fact, 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 the imaginary part of ˜ ε is related to the frequency dispersion and not to optical absorption in the material, which is neglected 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 with modified permittivity Inserting this expansion into Eq. (23) gives To proceed further, let us fix a real-valued frequency ω$_{r}$ around which we study electromagnetic modes. We restrict ourselves to modes with sufficiently small | Im (Ω) | , i.e., not too small Q -factor, and Re (Ω) ≈ ω$_{r}$ . This allows the expansion of the permittivity gives nonvanishing ⃗ E ( ω ) in a region around ω ≈ Re (Ω) . With this decomposition we can write the constitutive relation (3) as