Thomas J. Karr

Temporal response of atmospheric turbulence compensation

The temporal kernel for the phase error variance due to atmospheric turbulence is calculated. This kernel, convolved with the servo transfer function, determines the temporal error of any turbulence compensation scheme.

Perturbation growth by thermal blooming in turbulence

The stability of a phase-compensated laser beam propagating in a turbulent absorbing fluid is considered. Small-scale transverse optical perturbations caused by turbulence and noise grow in thermal blooming by two instabilities: the uncompensated stimulated thermal Rayleigh scattering instability and the closed-loop instability. Linearized perturbation theory is used to calculate the electric field spectrum as a Taylor series in time and as a superposition of stable and unstable modes. The method is applicable to fluids with arbitrary parameter variations along the path. Compensated perturbations grow exponentially, and uncompensated ones grow quasi-exponentially. The instability growth rates and the turbulence and noise excitation strengths are derived for a simple fluid with homogeneous parameters. The linearized theory of perturbation growth is in good agreement with numerical simulations of full nonlinear thermal blooming. If the growth rate exceeds the damping rate from other phenomena, then the perturbations grow until they are limited by nonlinear saturation, at which point the beam is significantly degraded. At saturation the laser beam spontaneously breaks into small-scale transverse structures such as filaments or ribbons. The strongest damping mechanism in the open air is typically wind shear, which sets a threshold blooming rate and a threshold absorbed irradiance. Below threshold the perturbations grow linearly; above threshold they grow quasi-exponentially. Other atmospheric damping phenomena, such as diffusion and turbulent mixing, have a smaller effect.

Computation of small-scale velocity turbulence and its effect on optical scintillations and stimulated thermal Rayleigh scattering

Coherent high-power light beams propagating long distances through turbulent fluids are subject to many kinds of scattering effects; among these are small-scale thermal index instabilities, in which the fluid is heated by the small fraction of light that is absorbed, amplifying the pre-existing index fluctuations and producing small-angle stimulated thermal Rayleigh scattering. Turbulent velocity fluctuations can inhibit the rate of growth of these instabilities by dispersing the thermal perturbations created by the beam. Methods for computing the turbulent diffusion of the heating perturbations, compatible with fast-Fourier-transform beam propagation computations, are presented. Propagation calculations of scintillation coherence times and small-scale velocity turbulence thresholds for stimulated thermal Rayleigh scattering are included.

Turbulent mixing effects in models of thermal blooming

A two-dimensional hydrodynamic model incorporating turbulent mixing is derived by averaging the three-dimensional hydrodynamics equations over slabs of thickness L in the direction of propagation. The resulting equations are identical with those for two-dimensional, incompressibl flow with additional terms representing the average effect of velocity fluctuations within each slab. These equations depend only parametrically on the coordinate in the direction of propagation. The extra terms have a simple physical interpretation but must be modeled in order to close the equations. Correlation times of scintillations are computed from a numerical simulation of plane-wave blooming using this model. A comparison with the correlation times from simulations with only wind shear indicate that the effect of turbulent mixing is relatively small for typical values of the Kolmogorov inner scale.

Effect of a random wind field on thermal blooming instabilities

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.

Linear theory of thermal blooming in turbulence

In this paper we show how the electric field spectrum for a high energy laser beam propagating through a uniform atmosphere can be calculated using the linear theory of small fluctuations. The beam is modeled analytically as an infinite plane wave which propagates through a medium with constant absorption and no transverse wind. Return-wave phase compensation is modeled as a filter in the transverse Fourier domain. The linear theory describes the growth of small fluctuations on the beam and accurately predicts the evolution of the electric field spectrum until the magnitude of the fluctuations approach the original beam irradiance. The accuracy is tested by comparing with the spectrum calculated using ORACLE, a full wave optics thermal blooming code.