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*F*{

*n*(r),E(r). Here

*n*is the refractive index and

*F*is a known function. The equation can be solved by the method of successive approximations. We used only the first approximation. We supposed that permeability is equal to unity. A function describing the dependence of the refractive index on coordinates was selected. An example of the calculation is given in the paper. The solution may be generalized to the case when the refractive index depends on time.

_{2}=

*const*). The value of the component is linearly dependent on x

_{1}along the curves.

A procedure was set forth which allows to solve the problem when the beam propagates through the inhomogeneous gas. If the permittivity is close to unity and the permeability is equal to unity we get the explicit solution. The obtained solution may be generalized to the case when the permittivity depends on time.

_{3}= 0 ( x

_{3}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.

*MO*of a Point Spread Function (PSF)

*O*can be measured with the aid of special transparent image. PSF

*O*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

*O*and MTF

*MO*are connected by the eigen-values problem relative convolution and multiplication operations: O*Y=

*MO*Y. We may introduce MTF

*MR*of resolving function

*R*: R*Y=

*MR*Y and MTF

*MRO*of (R*O): R*O*Y=

*MRO*Y. We have [1] equality:

*MRO*=

*MR*

*MO*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

*O*, MTFs:

*MO*,

*MR*and

*MRO*, and numerous applications of the ultra-resolution method are considered.

*R*. Each linear device which forms image is characterized by two functions: PSF

*O*and MTF

*M*on observation area

*D*. The function

*R*resolves a function

*O*, if their cyclical product (or convolution) is equal (or near) to

*d*- delta function:

*R*O*=

*O*R*=

*d*on

*D*. The resolving image

*R*I*on

*D*can be presented as measured one by fine “registering system” with narrower PSF

*R*O*that cannot be achieved physically. If area

*D*is as small as one of

*O*, then the corresponding values of

*M*are relatively large ones and there is no problem for compensation of distortions. The special arrangement of local subareas of

*D*is obtained by solving the next multi sensors (rays) problem: to separate the image I and the distorting PSF

*O*on the observation area

*D*so that at small quantity of data the resolution problem with

*R*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

*O*(

*r*) of PSF and the set of resolving functions

*R*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.

*D*(

_{n}*r*) has the form:

*D*(

_{n}*r*)=

*c*(1-exp(-

*br*)sin(

*ar*)/(

*ar*)),

*a*,

*b*,

*c*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.

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