Fig. 2: Distance-dependent transmission spectra and spin-wave group velocities.

a–e Transmission spectra \({S}_{21}\) measured as a function of field angle \({{{{{\rm{\theta }}}}}}\) from 0 to180 degree on Samples Z01, Z02, Z07, Z08, and Z09 with spin-wave propagation distances \(d\) of 2, 6, 12, 20, and 24 µm, respectively. The spin-wave wavevector \(k\) is along the LSMO [110] direction. The data displayed in the map is the imaginary part of the transmission spectra, and the color bar denotes the amplitude of the imaginary part, which is in arbitrary units. f–h Simulation results of angle-dependent spin-wave transmission over 2, 6, and 12 µm. The color code is taken from the real part of the total transmission signal \(s(d)\) picked by the NSL or CPW in the simulations (see Supplementary Sec. V). i Peak-to-peak frequency span \(\triangle f\) extracted from signals attributed to mode Y in a–e at \({{{{{\rm{\theta }}}}}}=90^\circ\) and plotted as a function of \(1/d\). The linear fit (red line) yields an average group velocity of 0.8 km s⁻¹. j The dispersion relation obtained by frequencies \(f\) of mode X extracted from the field-dependent transmission spectra at 54° (Supplementary Fig. 11) and wavevectors \(k\) from the Fourier transformation of CPWs (Supplementary Fig. 2).