( ω 2 m − ω $^{2}$) ( ω 2 p − ω $^{2}$) − [ 2 ω$_{s}$ω sin θ ∓ ( ω 2 r − ω $^{2}$)] 2 = 0 (44) A surface mode at k = −∞ may exist for some values of ω$_{r}$, ω$_{m}$ and ω$_{p}$ . For a solution to exist in this region, the function f ( µ ) of Eq. (44) should cross zero. This gives a condition 12 FIG. 4. The function in Eq. (43) represented by f ( µ ) is shown in (a) using parameters for BaMnF$_{4}$. The condition for surface modes in k >> ω/c region has no solution. In (b) a solution exists using parameters where λ/α = 1 . 5, where the surface begin to overcome the bulk as shown in (c). In (d) the surface branch is slightly higher than bulk when the ratio increase to λ/α = 2. where ω$_{r}$ = ( ω 2 afm cos 2 θ + Ω 2 me ) 1 / 2 represents the pole frequency of permeabilities given by Eq.(13-16) and is located slightly below ω$_{m}$ so that ω$_{r}$ < ω$_{m}$ < ω$_{p}$. Assume that the asymptotic frequency for the surface is located in the region above the pole frequency ω$_{p}$ . For k = + ∞ , the second term in Eq. (44) is always larger than the first term. Hence, Eq. (44) is never satisfied and the surface modes do not exist in this wavelength region. for k = ±∞ . In terms of ω$_{m}$ , ω$_{p}$ and ω$_{r}$ Eq. (43) becomes 2 4 6 8 10 −0.05 0 0.05 /2 c  (cm −1 ) $_{f(}$µ) +k  −k  $_{m}$  (a)The absence of surface modes at − k . 0.2 0.4 0.6 0.8 −6 −4 −2 0 2 /2 c  (cm −1 ) f ( ) $_{s}$  +k  −k  (b)The existence of surface modes at − k . −6 −4 −2 0 0.25 0.35 0.45 0.55 k  (cm −1 ) /2 c  (cm −1 ) SP −A  −L  (c)The parameters are λ/α =1.5. −6 −4 −2 0 0.15 0.25 0.35  k  (cm −1 ) /2 c  (cm −1 ) SP  −A −L  (d)The parameters are λ/α =2.