content_image/352632.jpg content_image/352630.jpg content_image/352631.jpg content_image/352629.jpg content_image/352628.jpg content_image/352627.jpg content_image/352626.jpg
where ω$_{L}$ is the frequency of the longitudinal phonon mode, and ω$_{T}$ is the frequency of the transverse phonon mode. 

## IV. RESULTS AND DISCUSSION 

Since there is no dynamic ME coupling, the dispersion relation for bulk polaritons can be calculated from the macroscopic electromagnetic wave equation 

<img src='content_image/352632.jpg'>

For the TE modes, the constitutive connecting ⃗ B with ⃗ H are defined by ⃗ µ = 1 + 4 π⃗ χ$_{m}$ . Using these relations, the solution of the wave equations for the bulk modes is of the form 

and an implicit expression for the bulk mode frequency is: 

<img src='content_image/352630.jpg'>

<img src='content_image/352631.jpg'>

Equation (26) has two zeros, one is for ϵ$_{x}$ = 0 and the other for f ( µ ) = µ$_{y}$µ$_{z}$ − µ 2 yz = 0. It also has two resonance poles from ϵ$_{x}$ and another from the condition µ$_{y}$ = 0. As a result, there are three bands of bulk polariton modes. 

The dispersion relation for surface modes is obtained by assuming solutions in the form: 

and 

<img src='content_image/352629.jpg'>

<img src='content_image/352628.jpg'>

where β and β$_{o}$ are positive real attenuation constants for the sample and vacuum, re- spectively. An implicit relation for the attenuation factor β of the medium is derived by substituting Eq. (27) into the wave equation Eq. (24): 

<img src='content_image/352627.jpg'>

An explicit relation for the attenuation constant β$_{o}$ (in vacuum) is given by 

<img src='content_image/352626.jpg'>

