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We have numerically investigated the effect of random fluctuations on uncompensated (open-loop) and phase-compensated (closed-loop) small -scale thermal blooming instabilities of a collimated beam propagating through refractive index turbulence. We used the ORACLE time-dependent, three space-dimensional, wave-optics code on a Cray X-MP. The Monte Carlo random wind fields, v(z), were exponentially correlated along the propagation direction, z. The small scale instabilities are present up to a threshold value of the rms random wind, beyond which beam propagation appears to be stable; the threshold value is nearly independent of the correlation length as long as the latter is much shorter than both the length of the thermal blooming region and the Rayleigh range of dominant perturbations. We describe our results with a dimensionless shear parameter, S, that is directly proportional to the ratio of the turbulence scintillation rate to the thermal blooming rate. S is defined as: (formula available on paper) is the one-axis variance of the random wind and N is the time derivative of the thermal blooming optical path difference (OPD) in waves/sec. The open-loop calculations use a plane wave “beam” and a uniform medium. The Strehl ration at 50 waves of thermal blooming OPD remains approximately constant while S ≥ 2.3 and then it decreases rapidly as S decreases. The closed-loop calculations use a large Fresnel number finite beam, a non-uniform medium of length sL, absorption - exp(-z/L), and a Hufnagel-Valley typ On2 profile whose r0 Fresnel number was 24. For 10 is less than or equal to Np is less than or equal to 40 we see no evidence of the closed-loop instability at a wind clearing time (20 waves) for S ≥ (54 ± 2)/Np, where (formula available on paper) is twice the actuator spacing simulated by a Fourier filter. This rresult suggests that the ration of the turbulence scintillation rate to the closed-loop gain for a uniform wind determines the threshold.
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