The intensity saturation during self-focusing of high-power ultrashort laser radiation in an optical medium is considered. The propagation of a femtosecond laser pulse in air from the collapse stop point to a nonlinear focus are considered in detail. The criterion for stopping the intensity collapse is established in the approximation of the nonlinear Schrödinger equation. This criterion makes it possible to take into account the nonlinear activity of various optical media during the propagation of high-power femtosecond laser pulses. The parameters of the effective permittivity model that satisfy this criterion are determined.
Numerical simulation of high-power femtosecond laser pulse propagation in air under conditions of amplitude modulation by metal wire-mesh masks are carried out. The use of such modulation masks leads to splitting of initial laser beam into separate parts (subbeams) of lower energy. Such spatial structuring of optical radiation causes large-scale diffraction interaction between subbeams, which leads to regularization of the diffraction pattern formed at the stage of pulse self-focusing. It is shown that the use of modulation meshes generally reduces the total length of the beam filamentation, but significantly increases the longitudinal continuity of the laser plasma density distribution in laser filaments. For full-scale meshes providing modulation of the entire beam area the formed filamentation region demonstrates larger extent than by using meshes with partial beam covering.
The numerical simulations of propagation of high-power femtosecond laser pulses in air under conditions of superposed spatial phase modulation are carried out on the basis of the reduced form of nonlinear Schrödinger equation for time-averaged electric field envelope. Initial spatial modulations are applied to pulse wavefront profiling by a staggered phase mask with variable phase jumps between adjacent elements. It is shown that with specific phase modulations, the pulse filamentation region in air can be markedly shifted further and elongated compared to a non-modulated pulse.
The propagation of high-power ultrashort laser pulses on aerospace paths when laser radiation is sent from the Earth into space is analyzed theoretically. Titanium-sapphire and CO2 lasers with femtosecond and picosecond pulse durations, respectively, are considered as radiation sources. The effective intensity and effective radius of curvature of the wavefront of high-power laser beams are estimated on the basis of the nonlinear layer model. It is shown that, on the whole, the use of a CO2 laser makes it possible to achieve more precise focusing of radiation on an object during energy transfer under the conditions of power losses caused by strong molecular absorption. Also, more precise focusing on aerospace paths up to 400 km long is ensured by the use of a laser beam with a lower peak power and a larger initial radius.
Experimental data on self-focusing and filamentation of high-power ultrashort laser pulses for wide-aperture (centimeter) beams of sub-terawatt power are interpreted on the basis of the diffraction-ray model of single filamentation. The intensity inhomogeneities in the initial transverse profile of the laser beam, the Kerr self-focusing of which leads to the formation of high-intensity light channels, are considered. The parameters of these inhomogeneities are estimated, which is important for numerical simulation of multiple filamentation of ultrashort laser pulses along atmospheric paths.
The main processes during the propagation of high-power femtosecond laser pulses in air are analyzed. The theoretical description of self-focusing and filamentation of laser radiation was performed on the basis of the diffraction-ray tubes method. Using the method of characteristics, the equation for the intensity was modified, and the equation for the power in the diffraction-ray tube was obtained. A relationship for the power balance of various processes in the filamentation domain was established. The mechanisms for replenishing energy losses in the filamentation domain caused by plasma formation was determined. The first of them is the transverse flux from the adjacent tube (so-called energy-replenishing diffraction-ray tube). The second mechanism is associated with the inflow of energy stored in the previous time slices of the laser pulse due to dispersion.
The propagation of high-power femtosecond laser radiation through an aerosol is considered. The propagation of high-power femtosecond laser radiation is simulated by passing through flat phase screens that simulate the passage through an aerosol. They have effective optical parameters corresponding to the properties of aerosol, which are found, respectively, for different values of intensity levels in the case of nonlinear interaction from the solution of the direct problem.
An efficient model of multiple filamentation is described, which makes it possible to carry out estimated calculations and numerical experiments that simulate the effect of a turbulent atmosphere on the characteristics of femtosecond laser beams. The results of numerical experiments on the formation of a beam of optical filaments during the propagation of terawatt pulses of a titanium-sapphire laser under conditions of weak and moderate turbulence are presented. It is shown that, at moderate turbulence, the channeling mode for large (with a radius of 0.8 to 1 cm) intensity inhomogeneities in the initial profile of the laser beam is retained at distances of hundreds of meters. This occurs even when the value of the relative peak power of laser pulses exceeds the value of the critical self-focusing power no more than 1.5 times. It is found that the dependence of the relative radius of the laser beam on the propagation distance, normalized to the corresponding scale, remains constant on average.
The propagation of ultrashort laser pulses in the atmosphere is accompanied by nonlinear effects. The most low-threshold of them is the effect of cubic nonlinearity along with nonlinear absorption manifesting in aerosol. This effect should lead to the transformation of the scattering phase function formed in a liquid droplet aerosol. To study this effect, numerical and experimental studies on droplets of various sizes and geometries were carried out. As expected, the cubic nonlinearity inclusion should lead to an increase in the effect of backward scattering.
The evolution of the small-scale transverse structure of high-power femtosecond laser radiation propagating in air in the filamentation mode has been experimentally and theoretically studied. Experimental results were obtained using wideaperture (centimeter) collimated beams of femtosecond pulses of a titanium-sapphire laser. As a result, the features of propagation of spatially isolated high-intensity light channels, the diameter of which is several millimeters, were determined. They are formed because of Kerr self-focusing of intensity inhomogeneities in the initial transverse profile of the laser beam. It is shown that the formation of a filament (a localized light structure, the existence of which is associated with the plasma formation and the generation of conical emission) does not occur in each of these channels. The theoretical evaluation of this light channels characteristics was carried out. It is based on the diffraction-ray model of single filamentation of femtosecond laser pulses. Studying of the evolution transverse profile of a laser beam with a centimeter radius and subterawatt power along air path showed that the initial radius of intensity inhomogeneities, for which laser filamentation occur, is equal several (2.5-3.5) millimeters. The power in these inhomogeneities varies from 19 to 26 GW. Differences in the values of the radius and power of these inhomogeneities are the cause of different distances from the laser pulse source at which their self-focusing occurs.
The results of studies of the femtosecond laser pulses propagation in air in the mode of self-focusing and filamentation are presented. The analysis of the numerical simulation results is carried out based on the method of diffraction-ray tubes. This approach made it possible to evaluate the characteristics of the energy-replenishing diffraction-ray tube, which forms a nonlinear focus, contains energy sufficient for compensation of the beam energy loss throughout the entire life cycle of the filament, and after its completion determines the boundaries of the postfilamentation channel (PFC). It is found that the average power in PFC formed by beams of sub- and millimeter radius is about 0.6-0.9 critical power for self-focusing, and it weakly depends on the initial parameters of the femtosecond laser pulse. In general, the energy consumption of radiation on filamentation decreases with increasing the initial beam radius. The angular divergence of PFC decreases by an order when the initial radius of the laser beam increases from submillimeter to millimeter values. For the last class of beams, the angular divergence of PFC remains approximately constant (20-25 microradians). Dependence of the angular divergence of post-filamentation light channels on the peak pulse power is less pronounced.
The effect of the group velocity dispersion of a pulse on the propagation of pulses of a titanium-sapphire laser in air is considered. Numerical simulation is performed for powerful femtosecond laser pulses with a duration of 100 and 20 fs, while laser beams with different initial radius and peak powers is considered. The study is based on the analysis of the results of numerical solutions of the nonlinear Schrödinger equation in a Kerr-plasma dissipative dispersion medium, carried out in the framework of the diffraction-ray tube method. The manifestation of dispersion is detected in the case when the dispersion length is not the smallest scale of all the characteristic longitudinal scales of the problem. The relation for the effective dispersion length is obtained in order to evaluate the effect of normal dispersion on the propagation of powerful femtosecond laser pulses. It has been established that in a nonlinear focusing medium, an increase in the initial radius of the laser beam can cause a disruption of filamentation even at high levels of supercritical power.
The results of natural experiments of the propagation of powerful femtosecond laser radiation in glass and water, accompanied by multiple filamentation, and the results of numerical simulation of the process are presented. Based on the diffusive equations for density of the number of filament estimates of the positions of maxima in the number of filaments were obtained. Sufficient criterion of macroscopic refocusing for density of the number of filaments was established.
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