Propagation of the non-paraxial Gaussian beam through the inhomogeneous atmosphere

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 [3]. 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 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.