I had two guesses at the reasons that may cause the oscillation of light output of the laser diode, that happens at high currents.

1. excitation of next mode. At low power, the lasing will happen at the frequency where the conditions are the most favourable. However, as the power is increased, other modes of resonator may come into play. Sum of two sinusoids at close frequencies manifests as beating, which might be what I observe.

2. oscillation may be caused by the onset of catastrophic optical damage. Say, the thermal runaway starts. It causes increased absorption, which reduces light intensity in lasing cavity. Reduced intensity causes thermal runaway to abort, and then it happens over and over again.

I can relatively easily rule out possibility 1. I am observing "beating" at frequencies of a few MHz. I can easily calculate the first beat frequency.

Wavelength of emission is about 650 nm. The length of crystal is about 0.3 mm, so is the length of cavity. That means the optical path of one loop of light travel is 2*0.3mm*3.49 (3.49 is refraction index, and 2 is because a full loop is a travel back and forth). And I get that in the path length, 3200 full waves fit. So, the lowest beat frequency is the frequency of emission light divided by 3200. That yields 143 GHz. That's way higher than what I observe.

I haven't yet come up with a way to proof or disproof explanation 2....