Magnetostatic Echoes
Authors: Kaplan D. E. , Kooi C. F. 
Published: April 2005

Coherent excitation of an ensemble of higher-order magnetostatic modes in a ferromagnet by a sequence of two or three microwave pulses can result in the reradiation of echo pulses1 due to phase coherent states of the many excited modes ocurring at times subsequent to pulse application. The pulse spectrum breadth Δω must be appreciably greater than the frequency breadth of the individual modes. A model for the magnetostatic mode echo can be formulated in a manner similar to that of the spin echo.2 The salient features of the magnetostatic model echo differing from that of the spin echo are a consequence of a restriction to small angles of magnetization opening in the ferromagnetic experiment, nonuniform coupling to the participating magnetostatic modes, and the possibility of instability effects abstracting excited modes from the ensemble contributing to the mode echo. Excitation by two pulses of time separation τ should result in the appearance of a train of echoes spaced at time intervals τ following the second pulse. The decay of the primary two pulse echo as a function of increasing τ yields a relaxation time T2 characterizing an average rate of decay of the transverse energy of the excited mode ensemble. The decay of a three-pulse echo, occurring at time T+τ for excitation pulses applied at times 0, τ, and T will be governed by longitudinal relaxation if T is varied with τ held constant.
Magnetostatic mode echo phenomena have been observed in high-purity single-crystal YIG spheres, disks, and rods at microwave X-band frequencies. The samples are contained in low Q cavities or terminated waveguide. Pulse durations of 10−8 sec are used at peak powers ranging from 10−2 to 10 W.
Measurements of the two-pulse echo decay constant as a function of temperature over the range 1.6° to 300°K yield results compatible with the temperature dependence of the total linewidth ΔH for this material using the relationship T2=2/γΔH. A magnetization relaxation time at 300°K of T2=0.75 μsec has been observed in a [100] YIG rod of diameter 0.1 in. and length 0.3 in. with Hdc parallel to the rod axis. The existence of magnetostatic mode echoes appears to be experimentally favored by large samples with geometry other than spherical. This is a consequence of a higher density of modes in certain frequency intervals for nonspherical geometries, thus allowing more modes to be accomodated within the pulse spectrum breadth. Furthermore it is easier to excite higher order modes in such samples as evidenced, for example, by the experiments of Dillon.3 The echo appears most strongly with increasing Hdc in the magnetostatic mode region. A decrease in transverse relaxation time is noted as the high-field limit for the appearance of the echo is approached. This latter result appears to be associated with the density of degenerate spin wave states in this region. Multiple two-pulse echoes are easily observed in the YIG rod at 300°K. The amplitude of the secondary two-pulse echoes is found to be a critical function of the excitation pulse level and may exceed that of the primary echo under some conditions of excitation.
Measurements of T1 by means of the three-pulse echo technique, yield roughly the relationship T1∼T2/2. The observation is complicated by the existence of a large number of echoes arising from three closely spaced pulses.