derived : with ω$_{s}$ = γ 4 πM$_{s}$ . The pole and zero frequency of bulk modes, ω$_{p}$ and ω$_{z}$ , is related to the zeroes of µ$_{y}$ and f ( µ ) = µ$_{y}$µ$_{z}$ − µ 2 yz . These give and where These three frequencies, ω$_{m}$ , ω$_{p}$ and ω$_{z}$ divide the bulk region into the shaded regions shown in Fig.2(b). Note that the lowest two bands overlap, and are deformed strongly near the resonance ω$_{m}$ . Surface modes, which is indicated by “SP” in Fig. 2(b) can exist in the gaps between ω$_{m}$ and ω$_{p}$ . The surface branches are nonreciprocal with respect to propagation, with ω ( ⃗ k ) = ω ( − ⃗ k ). The negative branch begins from the intersection of the lightline with the ω$_{m}$ resonance, and is indicated by “-L” . The branch terminates in the left middle bulk band. The positive branch begins at the lightline with frequency which is slightly higher than ω$_{m}$ , and then terminates at the upper bulk band. A method for detecting surface modes at THz frequencies is Attenuated Total Reflection (ATR)[1] and has been used to study surface modes on antiferromagnets[25]. In this method, a high optical index prism is used to couple electromagnetic radiation to surface excitations that lie off the vacuum light line. A sharp dip of reflectivity indicates excitation of surface modes. Using results from Ref. [8], reflectivity in ATR is given by R = ∣ ( 1 + re − 2 β$_{o}$$^{d}$) − iβ$_{o}$ ( 1 − re − 2 β$_{o}$$^{d}$) k$_{z}$ (1 + re − 2 β$_{o}$$^{d}$) + iβ$_{o}$ (1 re − 2 β$_{o}$$^{d}$) ∣ (38) ∣$^{k$_{z}$}$ ∣$^{2}$ where d represents the distance between prism and the sample, and r is defined as 10 r = β$_{o}$ − κ β$_{o}$ + κ (39) ω = [ ϵ$_{x}$ω 2 m − ω 2 afm cos 2 θ + Ω 2 me ϵ$_{x}$ − 1 ] 1 / 2 (37) Ω 2 mp = ω 2 m + ω 2 p + 4 ω 2 s sin 2 θ (36) ω$_{z}$ = 1 √$_{2}$ [ Ω 2 mp + ( Ω 4 mp − 4 ω 2 $_{m}$ω 2 p ) 1 / 2 ] 1 / 2 (35) ω$_{p}$ = [( ω 2 afm cos 2 θ + Ω 2 me ) + (2 ω$_{s}$ω$_{a}$ cos 2 θ ) ] 1 / 2 (34) ω$_{m}$ = [( ω 2 afm cos 2 θ + Ω 2 me ) + (2 ω$_{s}$ω$_{me}$ sin 2 θ ) ] 1 / 2 (33)