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coupling is large, then the ME susceptibility will also be large and thereby increase the imaginary part of β . 

In the case where k >> ω $_{c}$, the attenuation constants for the material and vacuum regions can be approximated by ϵ$_{z}$β 2 ≈ ϵ$_{y}$k 2 y and β$_{o}$ ≈ k$_{y}$ . The dispersion relation Eq.(36) then reduces to: 

<img src='content_image/343170.jpg'>

The surface modes require the permitivity ϵ$_{y}$ to be negative, and the longitudinal phonon frequency polarized along z direction, ω$_{oz}$ , is lower than ω ′ $_{ey}$. This means that the value of ϵ$_{z}$ is positive in regions where surface modes exist, and the requirement in Eq.(45) is never satisfied. Therefore, surface modes do not exist in the limit k >> ω $_{c}$. 

We now study the influence of external fields on the band structures. Results are shown in Figs.4(b) and 4(d). Results for an electric field value of 5 × 10 8 V/m along the direction of spontaneous polarisation (but with zero magnetic field) are shown in Fig.4(b). The effect of the electric field is to shift the window where surface modes exist to higher frequencies. The upward frequency shift is due to the increase of spontaneous polarisation, which directly increases the phonon frequencies ω ′ ey and ω$_{Ly}$ . However, the change in ω$_{Ly}$ is smaller than the change in ω ′ ey (which is due to the third term in Eq.(43)) and so the surface mode window is narrowed. The electric field increases the canting angle slightly, as shown in Fig.4(a), and the effect on magnetic resonance is negligible. 

Results for a magnetic field of 10 T (with zero electric field) are shown in Fig.4(d). The magnetic field increases the canting angle (see Fig.4(c)) and shifts ´ ω$_{m}$ to a lower frequency (as shown in Fig.4(d)) but the effects on the bulk bands are negligible. This shift can be understood if we consider the case where the ME coupling is neglected. Then the frequency ω$_{me}$ −→ 0, and also Ω$_{me}$ −→ 0. In this case, the magnetic resonance frequency will take the form ω 2 m = ˜ ω 2 afm + Ω 2 0 −→ ω$_{a}$ ( ω$_{a}$ − 2 ω$_{e}$x ) cos 2 θ . It can be seen from this expression that a magnetic field reduces the magnetic resonance frequency. We note that this effect is mentioned in Ref. $^{22}$. 

The surface modes can be modified if the magnetic resonance frequency can be shifted into a surface wave “window”, as shown in Fig.5(b) and 6(b). In BaMnF$_{4}$ where the electric and magnetic resonance frequency are well separated around 37cm − $^{1}$, it is very difficult to arrange the magnetic resonance frequency inside this window. This may be possible for a suitably prepared material (or artificially constructed material) whose frequency separation 

