content_image/372265.jpg content_image/372264.jpg
regions are determined by the attenuation constant. 

Using the attenuation constants, we obtain a narrow window between transversal and longitudinal phonon frequencies, ω ′ ey and ω$_{Ly}$ , associated with the pole and zero value of ϵ$_{y}$ (see Fig.2(b)). In this figure, since the ME coupling is weak, hence the frequency ω$_{Ly}$ is very slightly below the induced frequency ω$_{i}$ . Since the value of ϵ$_{y}$ between these two frequencies is negative, surface modes can be obtained inside this narrow window. The surface modes start from the crossing between the lightline ( ω = ck ) and the resonance frequency ´ ω$_{ey}$ and terminate at the longitudinal phonon frequency ω$_{Ly}$ . The frequency ω$_{Ly}$ can be approximated as 

<img src='content_image/372265.jpg'>

and is indicated in Fig.2(d). Since the surface modes terminate at the longitudinal phonon frequency, the gap around ω$_{i}$ does not influence the surface modes. 

Interestingly, from the expression for the attenuation constant of Eq.(39), and the dis- persion relation of surface modes in Eq.(41), it was only possible to satisfy the boundary conditions with a complex β . The resulting mode is not a true surface mode but is instead a pseudo-surface wave $^{21}$. Because the solution for the attenuation constant in Eq.(39) is complex, the attenuation constant for the material sample consists of real and imaginary part as illustrated in Fig.3(a) and 3(b). 

The imaginary part of β is 

<img src='content_image/372264.jpg'>

The positive value of the real part defines regions where the surface modes can exist. The existence of an imaginary part indicates that the solution in Eq.(37) is a psuedosurface mode, and not purely localized to the surface. Instead, energy ”leaks” into the bulk. The wave is comprised of a localised component, that travels along the surface and decays into the material according to the real part of β , and a component that travels into the material with wave number equal to the imaginary part of β . 

Values for the imaginary parts of β are plotted as a function of frequency in Fig.3(b) for the case of no applied fields. Imaginary β depends linearly on the magnetoelectric susceptibility and becomes large near the electric resonance frequency ´ ω$_{ey}$ . One can also see from Fig.3(b), that the coupling directly influences the magnitude of ”leakage”. If the 

