  In this paper, boundary conditions for Landau-Lifshitz equations at the
interface between two ferromagnets with different Dzyaloshinskii-Moriya
interactions are derived. We calculated and verified the analytical expression
for the energy flux density continuity for the spin-wave propagation through
the interface between two ferromagnets with/without Dzyaloshinskii-Moriya
interactions considering the boundary conditions mentioned. Analytical
expressions for reflection, transmission coefficients and the nonreciprocity
factor are calculated in the case of spin-wave propagation through a double
layer ferromagnet with/without Dzyaloshinskii-Moriya interactions in the
first/second layer. Two principally different types of nonreciprocity effects
for spin waves are revealed in such a double layer system. The material
parameters of a double layer ferromagnet with/without Dzyaloshinskii-Moriya
interactions in the first/second layer are found for which the extremely high
nonreciprocity factor (>10) is expected according to the results of
calculations. The results of the paper deepen the knowledge about the spin-wave
propagation control in magnonic devices.
