Turbulent fluctuations in the atmosphere distort a laser beam during its propagation. There exist two problems : (i) adequate description of the atmospheric turbulence and (ii) analysis of the propagation of the beam through turbulence, investigation of the beam spreading and distribution of its intensity. Unfortunately, only the scalar case of beam propagation has been considered often in the literature. Most part of authors studied only paraxial beams. Non-paraxial beams are considered in [1, 2]. Below we consider the propagation of a non-paraxial laser beam. The analysis has been made on the basis of the Maxwell equations. Two cases have been considered: (i) the permittivity and permeability are constant (the homogeneous atmosphere); (ii) the case when the permeability is equal to unity, the permittivity being dependent on coordinates. We assume that the permittivity is close to unity. Let us consider the first case. Some details of the solution were recently published in . We have a linear system of ordinary differential equations with constant coefficients due to the Fourier transform. The unknown functions are determined from the condition at the plain x<sub>3</sub> = 0 ( x<sub>3</sub> being the coordinate along the axis of the beam). In the second case (the permittivity is the function of coordinates being close to unity) we have a system of linear ordinary differential equations after the Fourier transform, too. The right-hand terms depend on the previous solution which was obtained for the homogeneous atmosphere. The solution is the sum of that one for the homogeneous atmosphere and that one for the variable part of the permittivity. Thus we have the solution which describes the propagation of the non-paraxial beam through the inhomogeneous atmosphere on condition that the variation of the refractive index is small. Numerical calculations were fulfilled for the components of the electric field.
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
Our approach for modeling laser beam propagation through turbulence involves parabolic equation method and results of experimental investigation in laboratory. The analytic solution to the problem of the Gaussian beam propagation through non-uniform gas has been derived. The solution depends on the refracted index, i.e. on the gas density. The density distribution can be found from the Navier-Stokes system. The appropriate solution may be constructed by two ways : (i) as a series in powers of vorticity which is supposed to be small; (ii) with the aid of the parametrix method which includes an iterative procedure. It follows from the solution that acoustic radiation of vortex rings arises. Statistical properties of the propagating beam were found from the solution to the parabolic equation as average over time. In experiments the propagation path was equal to 7 m. The laser beam propagation was accompanied by convection and lateral wind. The frequency of turbulent fluctuations was equal to 2-10 Hz. Phase trajectories were found as well as statistical properties of the beam intensity in turbulent gas flow. The conclusion is as follows. Statistical characteristics traditionally used for the estimation of the laser beam special distortions in the open space transmission channels are to be complemented by the dynamic parameters such as the space of embeddings dimension, characteristic frequencies for the phase trajectories and so on.
Density oscillations in the vicinity of vortex rings in air have been investigated. The calculations were fulfilled on the
basis of the Navier-Stokes equations. We used series expansions of unknown functions in powers of a parameter which
characterizes vorticity. As a result, we got a non-uniform system of parabolic differential equations with constant
coefficients. The frequency of oscillations depends only on the dimensions and the shape of the ring in the case of small
vorticity (weak turbulence). We analyzed oscillations generated by rings with circular cross-section. The size of the rings
varied in a wide range. It includes inertial range and dissipation range. It is interesting to note that first of all the
amplitude of oscillations increases, reaches its maximum and then decreases up to zero. These data can be used for
modeling the propagation of a Gaussian beam through the turbulent atmosphere. We analyzed intensity fluctuations
(scintillations) of the beam after the passage through the non-uniform region which contains vortex rings.
We considered an ill-posed problem (that of super-resolution) connected with image restoration. In such cases if the
input data are slightly changed, the solution may vary considerably. The proposed procedure is as follows. We change
the instrument function in such a manner that it will be reversible one within the limits of accuracy. The procedure
enables to solve some problems referring to the turbulent atmosphere.
The problem of the evolution of an ensemble of vortex rings in air has been solved. The full system of the Navier -Stokes equations was used. The parametrix method was applied. The calculations were performed for a wide range of the ring parameters (circular and elliptic cross-sections, various diameters of the rings, their different orientation in the space etc.). The initial value problem is as follows. The vorticity has non-zero value only inside the rings at initial instant, the density and the temperature being constant everywhere at t=0. If the density is known, then it is possible to find the refractive index. The solution to the Navier-Stokes equations is an oscillating one. Thus the refractive index is an oscillating function with respect to time. These results enable to model turbulence in an adequate way without using the Taylor frozen turbulence hypothesis. The evolution of the frequency spectrum of the density fluctuations was obtained. These results were compared with Tatarskii's data.
The intensity of a laser beam propagating through the ensemble of vortex rings in air was found with the aid of the parabolic equation method. A numerical procedure is set forth which allows to solve the problems of superresolution without using regularization methods. The task is as follows. There is a set of experimental data and an instrument function (with some error). We change the domain in such a manner that the corresponding MTF has nowhere zero values. The procedure enables to solve problems of focusing in the turbulent 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
We considered the mathematical theory of the laser-schlieren technique. Experimental data on grid-generated turbulence
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.
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 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.
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
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.
The problem of compensation for turbulent distortions of images is investigated.The full system of the Navier-Stokes equations was used for modeling of turbulent fluctuations. The problem of the convergence of the proposed mathematical procedure and the focusing of the laser beam in non-uniform atmosphere are also discussed.
Results may be applied to the compensation for distortions of images from modern optical telescopes without using deformable mirrors, to indication targeting problems and even to the control of laser beam focusing in the case of the turbulent atmosphere.
We modelled a whirlwind in the atmosphere with the aid of system of the linear  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.
We investigated Gaussian beam propagation through the turbulent atmosphere. The solutions of the linear and nonlinear unsteady 3D Navier -- Stokes equations have been used for modeling of turbulent fluctuations. Parabolic equation method has been applied for study of the intensity variations of the beam. Numerous examples of modeling are presented. Results may be applied to the study of turbulent fluctuations as well as of distortions of object images.
There are well known factors which provoke image distortions during the propagation of light in the atmosphere. We used the solution of the linear unsteady 3D Navier - Stokes equations in order to model turbulent fluctuations of the refractive index. Parabolic equation method has been used for the solution of the problem of laser propagation. Density distribution and structure function were determined as well as their temporal evolution. Distortions of the Gaussian beam due to the fluctuations of the refractive index are presented.
There are well-known factors which provoke image distortions during the propagation of light in the atmosphere. We used the solution of the linearized unsteady 3D Navier-Stokes equations in order to model turbulent fluctuations of the refractive index.This fact may be used for the construction of point spread function (PSF) for turbulent distortions. It was found that there exists a range of natural frequency for turbulent fluctuations. Structure function and correlation function were determined as well as their temporal evolution. A large amount of numerical results is presented.
The modern digital image forming systems are multi sensors or multi rays as a rule. The Modulation Transfer Function (MTF) <i>MO</i> of a Point Spread Function (PSF) <i>O</i> can be measured with the aid of special transparent image. PSF <i>O</i> of passive radio vision system can be measured for a point source. If Y is the ortho-normal system of Fourier harmonics in a small domain, then PSF <i>O</i> and MTF <i>MO</i> are connected by the eigen-values problem relative convolution and multiplication operations: O*Y=<i>MO</i> Y. We may introduce MTF <i>MR</i> of resolving function <i>R</i>: R*Y=<i>MR</i> Y and MTF <i>MRO</i> of (R*O): R*O*Y= <i>MRO</i> Y. We have  equality: <i>MRO</i> = <i>MR</i> <i>MO</i> in the frequency small domain. Ultra-resolution method gives point results of resolution and it is the most effective and stable method in order to increase resolution at present time. Examples of PSFs <i>O</i>, MTFs: <i>MO</i>, <i>MR</i> and <i>MRO</i>, and numerous applications of the ultra-resolution method are considered.
At Faculty of Physics, Moscow State University, the new image processing methods for different physical measuring systems are created. The main feature of the proposed super-, ultra-resolution methods consists in the diminishing of the dimensions of problems under consideration. In super- resolution method every actual (or virtual) ray has its own local vision domain. The local-linear super- resolution problem was solved on the special arranged set of actual (or virtual) rays. The introduced resolving function R  was not used. Point Spread Function (PSF) O and resolved O: R*O were needed for the illustration of results of the local-linear super-resolution method . In ultra-resolution (point) method, the resolving function R is directly used on small size vision domains ,and so is PSF O. The ultra-resolution method gives point results. In the super-resolution method each pixel was divided onto 2x2 and 4x4. The method of ultra-resolution gives us practically unlimited capability for "interpolation of pixels". "The pixel interpolation" certainly increases the dimensions of problem, but it enables us to perform a better presentation of the PSF O of the image measuring system. From the point of view of super- resolution method, the number of virtual rays in ultra-resolution method corresponds to the number of the small "interpolated pixels". The new ultra-resolution method is more effective and stable in comparing with the super-resolution method . Numerous applications are considered, too.
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 Kolmogorov's 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.
As known, the problem of compensation of turbulent distortions is an ill-posed one. Regularization methods are found to be inefficient. The problem can be solved by using local-linear super-resolution method. Its main feature consists in diminishing of problem dimensions. We use the terminology of multi sensors (rays) systems for this purpose and a new concept of resolving function <i>R</i>. Each linear device which forms image is characterized by two functions: PSF <i>O</i> and MTF <i>M</i> on observation area <i>D</i>. The function <i>R</i> resolves a function <i>O</i>, if their cyclical product (or convolution) is equal (or near) to <i>d</i> - delta function: <i>R*O </i>= <i>O*R </i>= <i>d</i> on <i>D</i>. The resolving image <i>R*I </i>on <i>D</i> can be presented as measured one by fine “registering system” with narrower PSF <i>R*O </i>that cannot be achieved physically. If area <i>D</i> is as small as one of <i>O</i>, then the corresponding values of <i>M</i> are relatively large ones and there is no problem for compensation of distortions. The special arrangement of local subareas of <i>D</i> is obtained by solving the next multi sensors (rays) problem: to separate the image I and the distorting PSF <i>O</i> on the observation area <i>D</i> so that at small quantity of data the resolution problem with <i>R</i> could be solved strictly and with the minimum border effect. Grid-generated turbulence in a shock tube was chosen as an example of homogeneous and isotropic turbulence. Statistical properties of this flow have been investigated experimentally. We found the correlation function and structure function for the fluctuations of refractive index. In our case of grid-generated turbulence the statistical properties are distinct from the Kolmogorov's two-thirds law. We modeled laser beam propagation through turbulent atmosphere and obtained the numerical results for the distortions of images. The distortion <i>O</i> (<i>r</i>) of PSF and the set of resolving functions <i>R</i> 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.
Statistical properties of a turbulent flow have been investigated experimentally in a shock tube with a turbulence grid. The fluctuations of the refractive index were detected with the aid of a laser-schlieren technique. It was found that the structure function <i>D<sub>n</sub></i>(<i>r</i>) has the form: <i>D<sub>n</sub></i>(<i>r</i>)=<i>c</i>(1-exp(-<i>br</i>)sin(<i>ar</i>)/(<i>ar</i>)),<i>a</i>,<i>b</i>,<i>c</i> being constants, in contrast with Kolmogorov's two-thirds law. These data were used in order to determine the PSF and MTF. Numerical results for the image of an object in a turbulent flow are also presented.
At Faculty of Physics, Moscow State University, the new super-resolution methods for different physical measuring systems are created. Special super-resolution methods for actual multi-rays systems of a millimeter wave range were developed. Until recent time the data obtained from an actual one-ray radio-vision system, were represented in such aspect: as though they were obtained from a virtual multi-rays system. Such an approach appears to be very successful: the problem of super-resolution has been solved at low values of a signal/noise ratio with the paralleled solutions on all the virtual rays. The paper is devoted to the problem of super-resolution of the data from actual compact 6-rays system of radio-vision. The experimental device of a 6-rays radio vision system was created without the mathematical modeling of its operation. It is possible even to tell that at the output of this system the poor results were gained. Therefore there was a problem to improve considerably the poor results: to increase the resolution of each ray. We suppose that obtained experience to be useful while constructing the modern all-weather multi rays real-time radio-vision systems. The methods of super-resolution of actual multi-rays microwave vision system is considered in the paper.
Targets usually have relatively steady geometrical forms. Backgrounds are characterized by uncontrollable variations of brightness. Those characterizations are taken into account in the model of the form of targets and in the model of background. If all the models are randomized, then we can distinguish the targets by an effective principle of the maximum likelihood of distinction. If the model of the geometrical form of the targets is deterministic and the background is an additive gaussian noise, then the morphological principle of distinction of the targets is effective. The distinction of targets is not effective in Cartesian space, for example at inverting brightness of the images of targets on a uniform background. In systems of radio-vision it is necessary to account into consideration the antenna pattern for effective distinction of targets. The local-linear method of super-resolution is included in the distinction and indication problems. The main idea of the new super-resolution method is in the resolving function. The local resolving functions are steady calculated in the local domains where antenna pattern is defined. The local resolving functions save the mean brightness' in the local domains. This property can be used in order to suppress the uncontrollable background. These and other questions are considered in the report by using examples of the model images of targets and backgrounds.
The local-linear method of super-resolution was developed for the solution of problems of compensation (resolving) of PSF distortions at low values of a signal/noise ratio. The resolving function, noise factor and value of increased resolution are introduced into consideration. The mathematical questions of applications of the local-linear method of super-resolution and problem of parallel calculations are considered.