The frequencies ω$_{m}$ and pole frequency ω$_{p}$ also increase under an external magnetic field. The difference between ω$_{m}$ and ω$_{p}$ is given approximately by As a consequence, the middle bulk band narrows in frequency with increasing magnetic field. At a field of 15 T, the BaMnF$_{4}$ middle bulk band is sufficiently narrow that a surface mode rises above the middle band and extends to small wavelengths, as illustrated in Fig. 5(b). Lastly, we consider the case where the dielectric constant depends on frequency. We suppose that the frequency dependence is given by the soft phonon mode according to with the dielectric function Here ω 2 L = ω 2 T + f ϵ$_{o}$ϵ$_{b}$ and ϵ$_{b}$ = 8 . 3 is dielectric constant background. FIG. 6. Dispersion relation for a frequency dependent dielectric function. The frequency is in- troduced according to Eq. (35). In (a), the case when of a magnetic resonance frequency lower than the electric resonance frequency is shown. In (b), the case of a magnetic resonance frequency higher than the electric resonance frequency is shown. Since there is no dynamic electric and magnetic coupling for the TE modes, there is no effect on the surface mode frequencies even though the electric resonance is near the magnetic resonance in frequency (this can be achieved by modifying the phonon mass). Effects on 14 ω 2 T = ( β$_{1}$ + P 2 o β$_{2}$ ) f (47) ϵ$_{x}$ = ϵ$_{b}$ ω 2 L − ω 2 ω 2 T ω 2 (48) Δ 2 ≈ ( ω 2 p − ω 2 m ) ∝ (cos 2 θ − sin 2 θ ) (46) −10 0 10 2.9 3.1 3.3 3.5 3.7 3.9  k  (cm −1 ) /2 c  (cm −1 ) −A −L L  A  SP  e $_{m}$  (a)The case where ω e > ω m . −15 −5 5 15 2.5 2.7 2.9 3.1 3.3  k  (cm −1 ) / 2 c  (cm −1 ) $_{m}$  −A  L A  e SP  −L  (b)The case where ω m > ω e .