content_image/380135.jpg content_image/380145.jpg content_image/380146.jpg content_image/380147.jpg content_image/380139.jpg content_image/380134.jpg content_image/380140.jpg
a material 2 are given by 

<img src='content_image/380135.jpg'>

where ⃗ ν is the local normal vector on the boundary. 

When discussing electromagnetic modes in cavities made of conventional materials one usually takes advantage of two simplifications: (i) the frequency dispersion of the permit tiv- ity ε is ignored, assuming that the frequency interval of inter- est is sufficiently small. (ii) The permeability µ is assumed to be constant and unity at all frequencies, throughout the whole space. 

In the following, we focus on disk-like cavities mainly for illustration purposes. We emphasise here that our results can be extended to arbitrarily shaped three-dimensional cavities in a straightforward manner. For the quasi-2D geometry of the disk one separates the ( x, y ) -dynamics in the plane of the disk from the z -dynamics by expanding the electric field in terms of ⃗ E ( x, y ) e ink z $^{z}$, with k$_{z}$ = 0 or k$_{z}$ finite. In the latter case the refractive index n = √ εµ in the mode equation can be replaced by an effective index. Assuming a piecewise con- stant index of refraction one can derive the following mode equation for a quasi-two-dimensional disk [19, 20] 

where Ω is the frequency of the mode. This Helmholtz equa- tion holds for both transverse magnetic (TM) and transverse electric (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 and a material 2 the general boundary conditions (5) and (6) give the following continuity relations for the wave function ψ and its normal derivative ∂$_{ν}$ ψ along the normal ⃗ ν , 

assuming that µ 1 = µ 2 . 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 

again assuming µ 1 = µ 2 . At infinity, outgoing wave condi- tions in the two-dimensional disk plane 

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 

## III. CONVENTIONAL CAVITIES 

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real part is the usual frequency, the imaginary part is related to the lifetime τ = − 1 / [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 and Peierls [22], are connected to the peak structure in scattering spectra; 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 polar coordinates 

For vanishing deformation parameter e this gives the circu- lar disk with radius R . We choose a deformed disk with e = 0 . 43 . Exactly this geometry has been studied for con- ventional materials in the context of directed light emission from microlasers, theoretically [24] as well as experimen- tally [11, 25–27]. The value of R itself is not relevant, only the ratio R/λ is important, where λ = 2 π/k is the wave- length. Therefore, we consider in the following a normalized frequency 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 . 5 , 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 relative intensity of the different segments we can assess the direction of energy flow as indicated by the arrows. The frequency is Ω R/c = 44 . 9376 − i 0 . 0622 , the Q -factor is therefore about 361 . Such a medium- Q , scarred mode is well suited to demon- strate how the light is (partially) refracted out. This will be useful in the following comparison to the NIM cavities. 

In the case of a NIM cavity the permeability µ can no longer to be treated as spatially uniform as this quantity changes sign at the interface between of the NIM cavity and the surrounding 

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## IV. NIM BOUNDARY CONDITIONS 

