Spin-Wave Instability in Nonuniform Fields and the Generation of Amplified Echoes

Amplified echoes can be described as a manifestation of nonlinear spin-wave interaction in an inhomogeneous magnetic field. In a typical experiment a very small pulse is incident on a ferrimagnetic sample at t=-τ and a large pulse is incident at t=0. An echo is thereupon emitted at t=τ with an amplitude greatly exceeding that of pulse 1. The disturbances associated with the two incident pulses and with the echo can be represented by ``quasi-stationary'' spin waves whose wave vectors vary with time on account of the field inhomgeneity. The wave vectors of the three waves are matched throughout and nonlinear coupling under proper conditions gives rise to unstable solutions in which the pair of spin waves associated with the pulse 1 and the echo grow exponentially at the expense of the spin wave generated by the large second pulse. At t=τ the echo spin wave evolves into a uniform precession and at that time radiates a highly amplified echo. The process is generically related to the Suhl instability in a uniform ferrimagnet in which pairs of opposite spin waves grow exponentially at the expense of the uniform precession mode. A simple mathematical model is introduced to illustrate this behavior.