Wei,

the dispersion can also be different (omega vs k)

Thank you very much Stuart! I will look up for more references! 

Originally the distinction between DE-modes and MSBV-modes was found by Damon and Eshbach [1]. They solved the Landau-Lifshitz Equation in combination with the (magneto static) Maxwell equations for an infinitely extended ferromagnetic film taking into account the dipolar fields generated by the spin waves during the precession of each magnetic moment. But they did not include exchange interaction and this is why the result, that different modes appear with different dispersions, is an effect related purely to the dipolar interaction.

The main conclusion f the paper is, that there are two independent (actually orthogonal) solutions to the problem of spin waves in in-plane magnetized film:

- the first solution is a spin wave mode where the precession amplitude is constant over the film thickness. These modes can travel in ANY direction in the film - perpendicular and parallel to the magnetization. However, when propagating parallel to the magnetization the frequency is decreasing with increasing wavenumber (negative dispersion, negative group velocity) which gave them the name: Backward Volume Wave. These waves can propagate in any direction within the film plane, but when the wave vector is perpendicular to the magnetization, the dispersion is constant.

- the second solution - the Magneto-Static Surface-Wave (or Damon Eshabch wave) has a maximum of the precession amplitude at the films surface and the amplitude decays exponentially when going into the film. Hence, they are called surface waves. These waves are non-reciprocal. This means, that the wave can propagate only in one direction at a given surface. Waves with opposite wavenumber travel at the other film surface. These modes show an increasing frequency for increasing wave numbers (positive dispersion and positive group velocity). The exist for propagation perpendicular to the magnetization and even for directions a little off the perpendicular direction up to a critical angle (details in Ref [1] see Fig.3).

NOTE: The distinction between the two different classes of modes is usually only good for thick films where the definition of a surface wave makes sense (typically above thicknesses of 100nm) AND if exchange energy does not play an important role. If one considers films with thicknesses below 100nm and with a higher exchange content, e.g., Permalloy or Cobalt, then it does not make sense anymore to call waves DE-waves or BVMS-waves. In this case there is on general solution to the problem, which describe nearly all angles of propagation. The difference the is simply: Wavenumber perpendicular to M: positive dispersion, high and positive group velocity. Wavenumber parallel to M: negative dispersion, small and negative group velocity. Calculations can be found in [2,3].

[1] R. W. Damon and J. R. Eshbach, J Phys Chem Solids 19, 308 (1961)

[2] B. Kalinikos and A. Slavin, J Phys-Condens Mat 19, 7013 (1986).

[3] S. O. Demokritov, B. hillebrands, and A. Slavin, Physics Reports 348, 441 (2001).

Thanks Dr. Schultheiss for such a thorough explanation!