Thermal blooming compensation instabilities

A general model is developed for the time-dependent growth of small perturbations in thermally bloomed beams with and without correction. Intensity and phase everywhere along the path of an intense forward beam and a weak backward beam are determined from the initial beam and path conditions by five time-dependent Green functions. The Green functions are exact solutions of the combined linearized blooming and turbulence problems and are given in closed form for an arbitrary path. Any correction method is a boundary condition connecting the forward and backward fields. Time-dependent instabilities correspond to singularities in the appropriate combination of Green functions. Perfect field conjugation gives perfect correction and is stable at all spatial frequencies. Perfect phase-reversal correction is unstable at all spatial frequencies. The instability growth rate is proportional to absorbed irradiance, and the total gain in a convection clearing time is proportional to the dimensionless blooming number. High gain is predicted in moderate blooming. Convection shear reduces the gain for irradiances with an instability growth rate much smaller than the shear rate and suppresses the gain for irradiances with a Rayleigh-range optical-path-difference growth rate much smaller than the shear rate.