Olga I. Dokukina

On the problem of beam focusing in the turbulent atmosphere

Two parts of the problem were analyzed. The first one is the adequate description of turbulence. The result is the simulation of the evolution of the refractive index due to turbulence. The second one is the beam focusing on condition that the refractive index is subject to spatial and temporal variations. The turbulence was simulated with the aid of the solution to the Navier-Stokes equations. Three kinds of initial conditions were used: (i) the vortical field was given, the velocity divergence (dilatation) being zero and the temperature being constant; (ii) the velocity divergence was given, the vorticity being zero and the temperature being constant; (iii) there is a temperature distribution, the vorticity and the dilatation being zero. In all cases, the initial values of the density are constant. The problem is set in the infinite space, the initial data being random functions. The solution of the Navier- Stokes equations was reduced to the solution of integral equations of the Volterra type. The iterative procedure was used. The comparison of the subsequent iterations allows to conclude that the convergence takes place. The problem of compensation for turbulent distortions of a laser beam was solved. The resolving function determines the necessary deformation of the mirror. The knowledge of the resolving function indicates the way to the beam focusing in the turbulent atmosphere.

Local-linear method of super-resolution for compensation of image distortions using new model of turbulence

Atmospheric turbulence is one of the important factors that influence on scene spatial resolution. In order to restore an image with minimum distortions one must know the correlation function for fluctuations of refractive index and the distorting PSF as a result. Grid-generated turbulence is a classic example of homogeneous and isotropic turbulence. Statistical properties of this flow have been investigated experimentally. In our case of grid-generated turbulence the statistical properties are distinct from the Kolmogorovs two-thirds law. Calculations performed on the basis of the linearized three-dimensional unsteady Navier-Stokes equations gave similar results. We modelled laser beam propagation through turbulent atmosphere and obtained numerical data for the distortion of images. The distortion of PSF and the set of resolving functions were found according to the structure function. The problem of compensation of distortions caused by turbulence was solved with the aid of a new local-linear super-resolution method. This method allows to resolve turbulent distortions of PSF at low signal-to-noise ratio.

Modelling of propagation and scintillation of a laser beam through atmospheric turbulence

The investigation was fulfilled on the basis of the Navier-Stokes equations for viscous heat-conducting gas. The Helmholtz decomposition of the velocity field into a potential part and a solenoidal one was used. We considered initial vorticity to be small. So the results refer only to weak turbulence. The solution has been represented in the form of power series over the initial vorticity, the coefficients being multiple integrals. In such a manner the system of the Navier- Stokes equations was reduced to a parabolic system with constant coefficients at high derivatives. The first terms of the series are the main ones that determine the properties of acoustic radiation at small vorticity. We modelled turbulence with the aid of an ensemble of vortical structures (vortical rings). Two problems have been considered : (i) density oscillations (and therefore the oscillations of the refractive index) in the case of a single vortex ring; (ii) oscillations in the case of an ensemble of vortex rings (ten in number). We considered vortex rings with helicity, too. The calculations were fulfilled for a wide range of vortex sizes (radii from 0.1 mm to several cm). As shown, density oscillations arise. High-frequency oscillations are modulated by a low-frequency signal. The value of the high frequency remains constant during the whole process excluding its final stage. The amplitude of the low-frequency oscillations grows with time as compared to the high-frequency ones. The low frequency lies within the spectrum of atmospheric turbulent fluctuations, if the radius of the vortex ring is equal to several cm. The value of the high frequency oscillations corresponds satisfactorily to experimental data. The results of the calculations may be used for the modelling of the Gaussian beam propagation through turbulence (including beam distortion, scintillation, beam wandering). A method is set forth which describes the propagation of non-paraxial beams. The method admits generalization to the case of inhomogeneous medium.

Grid turbulence and its interaction with a shock wave

Turbulent fluctuations of density and pressure in air and argon in a shock tube have been investigated as well as their interaction with a shock wave reflected from a perforated plate at the end of a shock tube. Air and argon were used as test gases. The Mach number of the incident shock was 1.9–3.9, that one of the reflected shock was 1.4–2.4. The turbulent length scale behind the incident shock was measured as well as that one behind the reflected shock. The last value is a few times less than the former. It was established that there is overpressure in the turbulent flow behind the reflected shock. The value of the overpressure is 12% in argon and 9% in air.

Pressure fluctuations within a turbulent gas flow and their interaction with a shock wave

The interaction of a shock wave with a turbulent air flow is investigated experimentally. The turbulence was created with the aid of a grid. On its reflection from a perforated disc the wave propagated through a turbulent flow. The Mach number of the incident shock was equal to 1.9–4, the Mach number of the reflected wave was equal to 1.6–2.5. We found the autocorrelation functions of pressure fluctuations and their phase diagrams. The turbulent length scale of pressure fluctuations behind the incident shock was determined. The appropriate quantity behind the reflected wave is less of an order as compared with the previous case. It is established that the pressure behind the reflected wave in the turbulent flow is 7–8% higher as compared with the pressure in the laminar flow, if other conditions are the same.

Statistical properties of density fluctuations in the atmosphere

The improvement of the parametrix method for solving the full system of the Navier-Stokes is presented. As known the fundamental solution equation is an oscillatory one. These oscillations are observed while analyzing the density evolution. Their frequency diminishes as the time grows. The approximate expression is presented for density in the neighborhood of a vortical structure. The laser beam propagation has been analyzed. The method will enable to find time average quantities. We considered the mathematical theory of the laser-schlieren technique. Experimental data on grid-generated turbulence are presented.

Problems related to the beam propagation in the tubulent atmosphere

Related problems are as follows: (i) evolution of the vortical structures which play an important role in turbulence; (ii) laser beam propagation through turbulence; (iii) object-targeting problem. The parametrix method was used. The convergence of the coupled iterative procedure was discussed. We investigated the influence of a point thermal source on the vorticity of a cylindrical vortex. We revised the 3D object-targeting problem.

Characterization of a laser beam propagating through the turbulent atmosphere

The method is proposed in order to model the propagation of a laser beam through turbulence. Two kinds of beams are under consideration: (i) beams of small power; (ii) high-power beams with effect of self-focusing (including ultra-short laser pulses). Turbulent atmosphere can be modeled with the aid of the full Navier-Stokes equations. As known, turbulent fluctuations decay in isolated system due to dissipation. To prevent decay, energy transfer from outside must exist. So we introduced an additional term into the equation of energy. The amplitude of arising disturbances has been investigated. The Navier-Stokes equations were reduced to integral ones and solved by iterative procedure. The comparison of the subsequent iterations demonstrates rapid convergence. The nonlinear solution has an important feature: the dispersion law depends on coordinates and time. The range of applicability of the numerical method has no restrictions on the value of the Reynolds number. Turbulent properties can be found by averaging over initial data. Some theoretical results were confirmed by experiments relating to grid turbulence. We have also considered the 3D objecttargeting problem. Some examples are given.

The effect of the strong turbulence regime on the laser beam propagation and focusing

The influence of the strong turbulence regime on the laser beam propagation and focusing has been investigated. Turbulence was modeled on the base of the full Navier-Stokes equations which were transformed into the system of the Volterra type. The influence of a source-like term in the energy equation was analyzed. Statistical properties of turbulence were calculated. Computations were compared with experiments. We investigated the propagation of a laser beam with the aid of parabolic equation method. Focusing for linearly and radially polarized beam was considered. The problem of object-targeting system and the propagation of ultra-short laser pulses were also discussed.

Modelling of laser beam propagation through the whirlwind

We modelled a whirlwind in the atmosphere with the aid of system of the linear [1] and weakly nonlinear 3D system of the Navier-Stokes equations. We investigated Gaussian beam propagation through the modelled whirlwind in the atmosphere. Parabolic equation method has been applied for study of the intensity variations of the beam. We investigated the evolution of the whirlwind velocity field and laser beam propagation through it. The examples of the distorted laser beam are presented as the images in 2D plane. A new method was set forth to the compensation of random distortions. The method was applied for compensation of distortions of a laser beam propagating through the whirlwind in atmosphere. Results may be applied for the compensation distortions of images from modern optical telescopes, in targeting problem and even for control of laser beam focusing on object-target in the case of the random turbulent medium.