# Self-Action of Laser Beams in the Atmosphere

Nonlinear effects in gases changing the medium permittivity lead to self-action of laser beams. Insignificant changes of the wave phase due to a change in the refractive index in the elementary volume are accumulated at large distances into significant distortions of the wave phase and amplitude. At the beam self-action, its angular spectrum transforms, which leads to a change in the propagation trajectory, self- defocusing, and self-focusing of the radiation. The self-action of spatially modulated waves in the atmosphere causes thermal action of the laser radiation (heating, kinetic cooling) and effects of changes in the medium polarizability [25—31].

In the nonturbid atmosphere, the main factor decreasing the efficiency of laser energy transfer to long distances is the effect of thermal blooming having the lowest energy thresholds. In what follows, we restrict our consideration to only this effect.

There are numerous works devoted to the problem of thermal blooming of laser radiation [3,32—34]. These works systematize theoretical investigations and present experimental data on the thermal distortion of beams in model media. The current experimental study of this effect in the actual atmosphere is discussed in Reference 35.

It should be noted that laser experiments in the actual atmosphere are extremely expensive. In this connection, for the correct prediction of the atmospheric propagation of high-intensity laser beams, high requirements are imposed on the accuracy and reliability of analytical and numerical calculations of various parameters. This circumstance has led to development of numerous approaches and methods for theoretical study of the discussed effect and numerical algorithms for their implementation [3,33,34,36—44].

In the Institute of Atmospheric Optics (Tomsk, Russia), this problem was solved within the framework based on the comprehensive consideration of the atmospheric effect on the optical wave parameters. This formulation of the problem indicated that, along with consideration of the effect of self-action on the beam parameters, it is also necessary to take into account the influence of atmospheric turbulence distorting the beam coherence and making stochastization of the temperature field in the beam channel. It is also important to take into account the nonideal character of laser sources and some other features dictated by practical issues of application of high-power lasers in the actual atmosphere.

A characteristic feature of self-action of laser beams in the atmosphere is the mutual influence of different types of transformation of beam parameters (spatial, amplitude, frequency, temporal). This is caused by the participation of both linear (speckle structure of the beam due to scattering at turbulent and discrete inhomogeneities of the atmosphere) and nonlinear effects in this mutual influence of transformations. Thus, the amplitude nonlinear conversions of the beam lead to a change in the diffraction characteristics of the channel [39], while the stimulated Raman scattering influences the radiation divergence [45].

To study theoretically the combined influence of the atmosphere on the laser beam characteristics, we used two approaches. One of them is based on the field description of effects in the atmosphere. In this case, we speak about development and implementation of numerical methods for solution of the parabolic equation. Another approach assumes the development of the high-efficiency method for solution of similar diffraction problems, which is based on the method of splitting by physical factors in combination with the fast Fourier transform (FFT) method [36]. The results of investigation of the nonlinear propagation of beams with this method are reported in Reference 46.

Additionally to the approach regarding the propagation of high-power laser beams in the atmosphere based on the parabolic equation, the researchers have developed the original theory of the method of radiation transfer equation as a ray method of the wave theory. In the following, we generally characterize this method and illustrate the results of its application with particular examples.

In the nonlinear optics of the atmosphere, a lot of practically important problems require the study of self-action of wide-aperture laser beams under conditions of significant nonlinear distortions. This indicates that the main interaction in the nonlinear medium occurs in the zone of geometric shadow of the beam. The study of this problems based on the apparatus of quasi-optical equation appears to be quite difficult. Therefore, it is natural to refer to new approaches.

One of them in the methodology of the theory of wave processes is the ray approximation, which is treated in the theory of wave propagation in inhomogeneous media as the method of construction of shortwave asymptotic of the wave equation providing solution of diffraction problems based on the geometric optics equations [38].

The method of radiation brightness transfer equations allows one to extend significantly the domain of application of the ray methods and to study the problem of self-action at the wide range of process parameters.

For smoothly inhomogeneous low-attenuated media, the system of equations for the determination of intensity of pencil laser beams by the method of the radiation brightness transfer equation is formulated as follows:

where *J* is the radiation brightness; R is the transverse vector of a point in the beam; *z* is the coordinate of propagation direction; n is the transverse vector of the tangent line to the trajectory of the geometric-optics ray; e is the perturbation of the medium permittivity caused by nonlinear effects (see also Chapter 2).

In every particular case, the system is supplemented with the equation determining the form of the dependence of permittivity perturbation e on the intensity e = e( *I*). For integration of transfer Equation 4.27 with initial condition (4.28), the classical method is the method of characteristics. In this method, the intensity is connected with the brightness at the entrance aperture (4.28) by the integral form

with the initial conditions in the observation plane

The method of characteristics consists essentially in the fact that the integral for intensity determination (4.30) is written at the characteristics emitted from the spatial point (R, z) to the initial plane *z =* 0 in the direction *—n ^{r}* and intersecting the initial plane at the point R

^{0}= R

^{,}(z

^{/}= 0) in the direction n

^{0}= n

^{,}(z

^{/}= 0). The characteristics obey the geometric-optics equations.

The behavior of the ray characteristics demonstrates clearly peculiarities of the integrated solution. Ray trajectories become concentrated at the focus points and rarefied at the beam blooming. If in geometric optics the beam intensity is determined from the law of energy preservation in an elementary ray tube, the area of whose cross section is calculated with participation of one central ray, then in the method of transfer equation many rays are involved in the intensity calculation. This fact resolves the problem of caustic and allows the diffraction at the beam aperture to be taken into account.

The method of characteristics in different modifications was applied in References 37, 39, 41, 42, 44 to problems of propagation of coherent and partially coherent laser beams in the atmosphere for the wide range of the process parameters and paths of different lengths. Different modifications of the theory were used for investigation of self-action problems, whose solutions were *a priori *unknown, in particular, problems of self-induced caustic [37], transformation of coherent properties of radiation in a nonlinear medium [41], nonlinear refraction of wide-aperture beams [41], and fluctuation phenomena in a nonlinear medium [44]. Below we illustrate the use of the method of transfer equation for problems of propagation of high-power laser radiation in the atmosphere.

The results obtained in Reference 41 illustrate peculiarities of self-action of the partially coherent radiation at a vertical atmospheric path under conditions of kinetic cooling of the medium. Compare the coherent beam with the partially coherent beam having the same energy parameters and dimensions. For the coherent beam the effect of self-focusing is observed at the vertical path, while for the partially coherent beam, due to its higher initial diffraction divergence, the beam defocusing takes place (Figure 4.2).

This figure shows the dependence of the effective relative beam radius on the dimensionless distance for laser beams with different degrees of spatial coherence for the time *t = t _{p}* and

*6 = t*

_{V}*7*

*it*0,5

_{p}=*(t*is the pulse duration;

_{p}*t*is the time of vibrational-translational relaxation of nitrogen molecules);

_{VT}*1*is the coherent radiation with the initial divergence ©0 = 1.69 X 10

^{-}, the nonlinearity parameter

Figure 4.2 Relative effective beam radius *versus* the dimensionless distance. Atmospheric model—midlatitude summer.

*P = L _{d}IL_{n}* = 123; 2 is for the partially coherent radiation,

*©0p*= 1.69x10

^{c}^{5},

*P = Lp/Ln =*12,3.

Along with the mentioned approaches for studying the self-action of high- intensity laser radiation in homogeneous and inhomogeneous nonlinear refractive media, the methods were developed. They allow the *apriori* estimation of the influence of nonlinear effects for beams with different profiles and different mechanisms of radiation interaction with the medium [43] and, for some cases, provide exact solutions for effective (integral) parameters of the beam. Here, the criterion of energy transfer is chosen to be the effective beam intensity, whose initial value determines the degree of nonlinear distortions for beams of different profile

The parameters entering into the definition of the effective intensity, the effective beam radius *R,* and the vector of displacement of the beam centroid R_{c}, obey the following equations [3,43]:

where *P* is the beam power, *p* is the wave phase, *k* is wavenumber, ©_{e} is the effective width of the angular spectrum (directional pattern):

The scale *p _{lk}* characterizes the diffraction properties of the beam;

*F*and

_{e}*F*are the effective curvature radii of the beam phase front.

_{A}The parameters *p _{de}, F _{e}, F_{A}* have the following form:

where *A* is the wave amplitude;

The similarity was found in the behavior of effective parameters of collimated beams of different classes at the self-action under conditions of strong nonlinear distortions for different mechanisms of radiation interaction with the medium.

Figure 4.3 Relative effective beam intensity near the centroid as a function of the parameter *z/L _{n}* for the nonlinear medium with stationary wind nonlinearity. Different dots correspond to beam with different initial profiles. (Zemlyanov, A. A., and Martynko, A. V.,

*Atmos. Ocean. Optics,*vol. 4, No. 11, pp. 834-838, 1991.)

This effect consists essentially in the fact that the form of dependence of the relative effective intensity *I _{e}(z)/I_{e}(z* = 0) on the generalized distortion parameter (z/L

_{n})

^{2}is approximately identical for different beam classes (Figure 4.3). It can be explained by formation of the nonlinear layer near the radiator, in which the limiting directional pattern of the beam is formed. In this case, the nonlinear component of the limiting angular divergence of the beam is determined as

*©*=

_{nlXl}*R*

_{e0}/L_{n}.The universal form of the longitudinal scale of nonlinearity was found

the use of which for estimation solves automatically the well-known problem of underestimation of the thresholds of nonlinear effects in methods based on the “aberrationless” approximation or approximation of geometric optics.

Conditions for the formation of the limiting angular divergence in the initially homogeneous refractive medium have been studied. The solution was obtained for the effective parameters of the beam after the nonlinear layer

is the squared nonlinear divergence of the beam;

where *в = (F _{e}* )

^{—2}— (Fi)

^{-2}is the factor of aberration distortions of the beam; (

*F*

_{i})

^{—1}=

*(F*)

^{—1}+

*(F*)

_{n}^{—i},

*F*is the nonlinear component of the effective wavefront curvature radius

_{n}*F*

_{e}*1*

*;*asterisked parameters are calculated at the boundary of the zone, where nonlinear effects manifest themselves (nonlinear layer). This zone can be found from the condition of saturation of the angular divergence

In the more general case, *F _{e} < F_{A}* from definitions of the scales

*F*and

_{e}*F*Consequently, the structure of the solution for the effective beam radius after the nonlinear layer is different for the aberration and aberrationless cases. The scale

_{A}.*F*can be both positive (self-defocusing) and negative (self-focusing). It is the most sensitive indicator of properties of a refractive medium. Therefore, it makes sense to determine the thresholds of nonlinear effects at an inhomogeneous path from the analysis of the scale

_{n}*F*. The condition for significant self-defocusing (self-focusing) of the beam at the distance

_{n}*z*is inequality |F

_{n}| <

*z*. If the beam is focused on the detection plane

*F = z,*then the nonlinear effects show themselves against the background of the diffraction effects at |f;| <

*L*or if the characteristic refraction angle 0

_{d}*0/*

_{n}= R_{e}*F*I exceeds the diffraction divergence of the beam:

_{n}©n >00.

For the case of weak nonlinear distortions at an inhomogeneous path, the equations for *F _{n}* and the nonlinear component of the limiting divergence

*©*can be written in the form ( = e

_{n(xi}_{max}(z

*)s*(R)):

where *L _{e}ff* is the scale characterizing inhomogeneity of the atmospheric path parameters. If the condition

*L*fulfilled, the situation of strong nonlinear distortions of the beam takes place at the inhomogeneous path. It is obvious that there is an intermediate region, where

_{n}< Lffis*L*

*=*

_{n}*L*

*f*. In this case, the beam parameters determining the effective intensity after the nonlinear layer can be obtained only numerically.

Figure 4.4 illustrates the dependence of the limiting angular divergence of the beam on the scale of nonlinearity. The data were obtained from the numerical s olution of the problem of self-action of the pulsed radiation at a vertical atmospheric path by the method of brightness transfer equation.

This dependence has two asymptotics: strong *© _{nixi}* = R

_{0}/L

_{n}and weak

*©*

_{n}*x,*= RoLf /Ln nonlinear distortions. Since the solutions exactly corresponding to the case of strong distortions

*L*were not obtained in Reference 41, the data of numerical calculations were extrapolated from the adjacent region to the region of strong distortions.

_{n}< L_{e}ffIt turned out that the calculated dependence *© _{mXi}(L_{r})* can be well described by the approximation equation

Figure 4.4 Nonlinear component of the limiting beam divergence as a function of the nonlinearity length (self-action of the long pulse at the vertical atmospheric path): (1) calculation by the method of transfer equation (solid curve) and extrapolation dependence (dashed line), asymptotic of strong (2) and weak (3) nonlinear distortions, triangles are for the approximation dependence.

The developed technique for estimating the efficiency of laser radiation energy transfer in nonlinear refractive media has allowed the qualitative and quantitative analysis of some practically important multiparameter problems of atmospheric nonlinear optics with the use of the equations for the integral beam parameters.