content_image/332368.jpg content_image/332367.jpg content_image/332366.jpg content_image/332365.jpg content_image/332364.jpg content_image/332360.jpg content_image/332363.jpg
derived : 

with ω$_{s}$ = γ 4 πM$_{s}$ . 

<img src='content_image/332368.jpg'>

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 

<img src='content_image/332367.jpg'>

<img src='content_image/332366.jpg'>

<img src='content_image/332365.jpg'>

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 

<img src='content_image/332364.jpg'>

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 

∣$^{k$_{z}$}$ − − ∣$^{2}$ where d represents the distance between prism and the sample, and r is defined as 

<img src='content_image/332360.jpg'>

<img src='content_image/332363.jpg'>

