  We demonstrate a combined frequency and time domain investigation of
injection-locked, constriction-based spin Hall nano-oscillators by Brillouin
light scattering (BLS) and time-resolved magneto-optical Kerr effect (TR-MOKE).
This was achieved by applying an alternating current in the GHz regime in
addition to the direct current which drives auto-oscillations in the
constriction. In the frequency domain, we analyze the width of the locking
range, the increase in intensity and reduction in linewidth as a function of
the applied direct current. Then we show that the injection locking of the
auto-oscillation allows for its investigation by TR-MOKE measurements, a
stroboscopic technique that relies on a phase stable excitation, in this case
given by the synchronisation to the microwave current. Field sweeps at
different direct currents clearly demonstrate the impact of the spin current on
the Kerr amplitude. Two-dimensional TR-MOKE and BLS maps show a strong
localization of the auto-oscillation within the constriction, independent of
the external locking.
