Filamentation of femtosecond laser pulses in turbulent air

Transcription

1 Appl. Phys. B 74, (2002) DOI: /s Applied Physics B Lasers and Optics s.l. chin 1 a. talebpour 1 j. yang 1 s. petit 1 v.p. kandidov 2 o.g. kosareva 2, m.p. tamarov 2 Filamentation of femtosecond laser pulses in turbulent air 1 Center for Optics, Photonics and Lasers, Department of Physics, Laval University, Quebec, G1K 7P4, Canada 2 International Laser Center, Physics Department, Moscow State University, , Moscow, Russia Received: 23 May 2001/Revised version:26 September 2001 Published online: 29 November 2001 Springer-Verlag 2001 ABSTRACT Formation and wandering of filaments in air are studied both experimentally and numerically. Filament-center deflections are collected from 1100 shots of 190-fs and 800-nm pulses in the plane perpendicular to the propagation direction. To calculate the filament wandering in air we have developed a model of powerful femtosecond laser pulse filamentation in the Kolmogorov atmospheric turbulence and employed the Monte Carlo method to model the propagation of several hundred laser pulses. Statistical processing of experimental and numerical data shows that filament-center displacements in the transverse plane obey the Rayleigh-distribution law. Parameters of the Rayleigh distribution obtained for numerical and experimental data are close to each other. PACS Bz; Jx; Uu 1 Introduction First experiments on long-range filamentation of powerful femtosecond laser pulses in air were performed in the middle-1990s in several laboratories [1 3]. In these experiments fs and 5 50 GW laser pulses from Ti:sapphire laser amplification systems operating at nm produced light filaments with the length of tens of meters. The remarkable feature of these filaments is that along most of the propagation distance more than 10% of the pulse energy is localized in the near-axis area with the diameter of about 100 microns. From the point of view of nonlinear optics the filamentation phenomenon is a small-scale transient self-focusing of laser radiation in air [4]. Self-focusing originates from the Kerr effect, which provides the increase in the nonlinear refractive index with the increase in the light-field intensity. If the pulse peak power is not larger than approximately eight threshold powers for self-focusing in air, only one filament is created [3]. With increasing peak power two or more filaments can be observed [5]. For an initially focused beam the filamentation starts right beyond the lens focal point [6]. Fil- Fax: / , kosareva@msuilc.ilc.msu.su amentation of focused subpicosecond laser pulses with central wavelength 248 nm was experimentally and numerically studied [7]. For the laser pulse used in this experiment 50% of the input energy was lost in the nonlinear focus and 40% was contained in the divergent beam surrounding the filament. The filament itself contained 10% of the input laser energy. Filamentation with various temporal lengths and various laser wavelengths is discussed in [8]. Experiments were performed using two laser systems: a Nd:YLF laser system with 525-fs pulses and a wavelength of 1053 nm and a Ti:sapphire laser system with 60-fs pulses and a wavelength of 795 nm. The length of the filament produced by 60-fs pulses with peak power of the order of 300 GW was more than 200 m. Atthe same time the length of the filament produced by 525-fs pulses with peak power of the order of 40 GW was approximately 50 m. In the case of 525-fs pulses and a peak power larger than 30 GW several (3 4) filaments were observed at the distance of filament formation. Small-scale self-focusing of laser pulses in condensed media was intensively discussed in the 1960s and 1970s in the context of laser thermonuclear fusion studies (see e.g. [9 12]). Observation of self-focusing in air has become available with the advent of laser systems that produce powerful femtosecond and picosecond laser pulses. For longer pulse duration self-focusing in air is suppressed by the nonlinear effects with lower threshold powers: heat defocusing and optical breakdown [12]. For femtosecond and subpicosecond pulses these effects do not develop due to a long response time. Filamentation of laser pulses in atmospheric air can be observed only if two conditions are simultaneously satisfied: ultrashort duration and high power of the radiation. The mechanisms that stop self-focusing in gases and condensed media are different. In optical glass two-photon absorption stops the growth of intensity in the nonlinear focus [13]. In gases self-focusing is stopped by defocusing of laser radiation in the plasma arising from photoionization of molecules in the nonlinear focus. As a result, the maximum light intensity in the filament does not exceed W/cm 2 for infrared pulses at nm [2 4] and W/cm 2 for ultraviolet pulses at 248 nm [7]. Nonlinear interaction of laser pulses with neutral molecules of air and self-produced laser plasma results in strong spatiotemporal gradients of light-field intensity and phase. The

2 68 Applied Physics B Lasers and Optics intensity peak produced by Kerr self-focusing is followed by a dynamic ring structure arising from strong nonlinear refraction in the plasma. The rings surrounding the filament converge and diverge alternately. As a result, the energy contained in the filament changes nonmonotonically with propagation distance. This effect was called the refocusing phenomenon. It was studied in [3, 4]. Filamentation is accompanied by the conical emission wide-band coherent radiation in the visible range. This radiation propagates forward in the narrow cone surrounding the filament [2, 14]. Experiments on self-focusing in water, methanol and CCl 4 [5, 15] demonstrated that the conical emission in the range nm has the same length of coherence as the input laser pulse with central wavelength of 800 nm. Nowadays wide-band radiation is regarded as a prospective coherent white-light source for sounding the atmosphere. The first experiments with femtosecond terawatt lidar were performed in Jena [16, 17]. Laser pulses with 100-fs duration, 2-TW peak power and 790-nm central wavelength were sent to a vertical atmospheric path. The back-scattered signal was registered from the height of 12 km. The absorption spectra of oxygen in the range nm and water vapor in the range nm were obtained at the height of m. In recent experiments [18] with terawatt laser pulses, infrared spectral broadening up to 4.5 microns was registered. This opens a new possibility for the application of laser-induced supercontinuum to the registration of pollutants in the atmosphere, because the absorption lines of most pollutants are in the range microns. Theoretical studies of long-distance propagation in air are based on the nonlinear Schrödinger equation for the complex amplitude of the electric field. The propagation equation includes diffraction, group-velocity dispersion, nonlinear refraction and absorption in the neutral gas and the plasma. The propagation equation is solved together with the equations describing transient nonlinear responses of the air and the plasma produced in the course of photoionization of air components in a strong laser field. Usually numerical simulations are applied to the solution of this three-dimensional nonstationary problem [3, 4, 19, 20]. In the case of the singlefilament regime, a cylindrical geometry of the experimental setup is used to reduce one spatial coordinate in the transverse direction and to save computational resources. For the interpretation of the simulation results several models have been applied: the self-channeling model [2, 6], the movingfocus model modified by the contribution of the self-produced laser plasma [3, 4] and the dynamic spatial replenishment model [20]. Numerical simulations allowed us to recapture the physical picture of the filamentation and accompanying effects of conical emission [4, 14] and refocusing [3, 20]. In [21] variational analysis has been applied to the study of the filamentation phenomenon. The equations for the beam width and the curvature radius of the laser-beam wavefront were derived and solved under the assumption of a Gaussian distribution of spatial intensity in the transverse direction. Filamentation is a stochastic process. This fact was first mentioned in [3] where random deflections of the filament center from the propagation axis were observed in the transverse plane perpendicular to the propagation direction. La Fontaine et al. [8] reported that not every perturbation in the beam profile formed earlier in the propagation was developed into a filament. In the femtosecond lidar experiments [16, 17] the pulse peak power was several hundred times larger than the threshold power for self-focusing in air. Therefore a bunch of filaments was created. In the conditions of natural atmospheric turbulence the formation of multiple filaments was random along the propagation distance. In order to regularize the process of the filament formation Woste et al. [16, 17] used a focusing lens with a 30-m focal length. In the focal point of the lens a long light filament was formed from which whitelight conical emission was generated. Theoretically, formation of filaments is the result of spatial instability of the light field in the strong optical nonlinearity. In the single-filament regime in a regular (nonstochastic) medium the instability is located at the point with the highest intensity, i.e. on the beam axis. However, in the real conditions the place of the filament formation depends on the transverse intensity fluctuations caused by limited spatial coherence of the laser radiation and natural perturbations of the refractive index in atmospheric air. The effect of atmospheric turbulence on filamentation of powerful femtosecond laser pulses in air was numerically studied [22]. Here the stochastic model of phase screens was employed to model refractive-index fluctuations in air. Kandidov et al. [22] showed that filaments were created in the places of random focusing of the radiation. Since in the conditions of self-focusing the light field is unstable relative to the local intensity fluctuations, the distance between the laser system output and the place of the filament formation is, on average, shorter in turbulence. The simulated picture of the filament wandering in the transverse plane is in qualitative agreement with the one obtained in the experiment. The Monte Carlo method was used to find the change of the transverse root-mean-square deviation of the filament from the propagation axis with distance. Numerical simulations of filamentation of a femtosecond pulse with peak power many times exceeding the threshold power for self-focusing in air were performed [23]. Dynamics of multiple filaments produced by a pulse with initially regularly positioned intensity perturbations was studied. In this paper we present experimental and theoretical studies of filament wandering caused by natural fluctuations of the refractive index in air. On the basis of statistical processing of the filament parameters obtained in the experiment and in the simulations, we develop a theory of filament formation in the conditions of natural atmospheric turbulence. 2 Experiment Our laser system consists of a Ti:sapphire oscillator followed by a regenerative and two multiple-pass Ti:sapphire amplifiers. The pulse duration is 190 fs (FWHM) and the central wavelength is λ = 800 nm. The laser pulse with 9.6-mm (at e 2 intensity level) beam diameter and energy of 4.6mJpropagates in the air of the laboratory building. The beam starts to self-focus at a distance of 30 m from the output of the compressor. Pulse peak power is 21 GW, i.e. 3.7 times larger than the threshold power for self-focusing in air, P cr, the value of which was estimated as 6.1GW[3].

3 CHIN et al. Filamentation of femtosecond laser pulses in turbulent air 69 To characterize the spatial profile of the filament its energy is attenuated by consecutive reflections from the surface of two prisms and, after further attenuation by neutral-density filters, the beam profile is captured by a CCD camera with a dimension of 6 6mm 2 ( pixels). The output of the CCD camera is a two-dimensional matrix F(x, y), the element of which at each point is proportional to the time-integrated value of the intensity of the filament at that point. A sample output of the fluence distribution in a single shot is shown in Fig. 1. A clearly pronounced maximum of the fluence distribution corresponds to the filament-center position at this propagation distance. From shot to shot the filament-center position changes randomly in the transverse plane XOY. A good fit to the fluence distribution is a Gaussian function of the form: { [ ( ) A x 2 ( ) ]} Xc y Yc F(x, y) = exp 2 + wx wy where X c, Y c are the coordinates of the filament center and w x, w y are the filament radii in x and y directions, respectively. To calculate the values of these parameters we proceed as follows. First, we integrate the function F(x, y) over the + area y =±60 pixels to find a function f(x) = F(x, y)dy. At the left-hand and right-hand sides of this area the values of F(x, y) are negligibly small. After calculating f(x) wefitit ( ) 2 to a Gaussian function of the form A wx ( 2 π/2 exp x Xc w x ), where A is a constant, and obtain parameters X c and w x.repeating this procedure, we integrate F(x, y) in the x direction and then we find Y c and w y. The diameter of the randomly displaced filament, d, is calculated as d = w 2 x + w2 y. (2) w x w y (1) Diameter of the filament d (pixel) propagation distance (m) FIGURE 2 The filament diameter d as a function of the propagation distance. Each point corresponds to a 20-shot average standard deviation were calculated according to the formulas of mathematical statistics [24]. At a distance of 30 m (Fig. 2) the filament diameter rapidly decreases. This indicates the start of the filamentation process. After the filament is created, the pulse energy remains localized in the narrow near-axis area with a diameter of about 1.5mm. This localization of energy takes place at least up to 105 m along the propagation direction. The large standard deviation of the diameter d at a distance of 30 m demonstrates that the filament formation is an irregular process. For example, for the data shown in Fig. 2 approximately one-third of all the pulses are far from being focused since the beam diameter is too large around 6mm. For statistical analysis of the filament parameters we collected 1100 shots at z = 30 m and another 1100 shots at z = 105 m. For each shot the position of the filament center In Fig. 2 we present the dependence of the diameter of the laser beam d on the propagation distance, z, that has been measured from the output of the compressor. Each point corresponds to a 20-shot average. The average value of d and its Y (pixels) Y, mm Y, mm a X, mm X, mm X (pixels) FIGURE 1 Fluence distribution in a single pulse measured by a CCD camera in the plane perpendicular to the propagation direction b X, mm X, mm FIGURE 3 The filament-center positions in the transverse plane at distances z = 30 m and z = 105 m from the output of the compressor: a experiment, 1100 laser shots; b simulations, 200 laser shots

4 70 Applied Physics B Lasers and Optics (X c, Y c ) and the diameter of the filament at 1/e 2 of fluence, d, were determined. In Fig. 3a we demonstrate the ensemble of the filament-center positions at these two distances. The average values of filament displacements in x and y directions are zero. The density of the filament-center positions is axially symmetric. From these observations two conclusions can be derived. First, there is no correlation between the filament displacement in x and y directions. Second, random displacements of the filament are statistically isotropic in the transverse plane XOY. Thus, statistical characteristics of the filament wandering depend only on the displacement of the filament center R c from the origin of the coordinate system located on the CCD camera. The distance R c is defined as R c = X 2 c + Y c 2. (3) This result means that random factors that lead to filament wandering are statistically isotropic in the plane perpendicular to the propagation direction. Among these factors are perturbations of the intensity and the phase caused by limited spatial coherence of the laser radiation and by refractive-index fluctuations in the propagation path. In Sect. 3 we develop a model of filamentation of powerful femtosecond laser pulses propagating in the conditions of refractive-index fluctuations in turbulent air. 3 Filamentation model A powerful femtosecond laser pulse propagates in the conditions of diffraction, group-velocity dispersion, Kerr self-focusing, plasma production and refractive-index fluctuations in atmospheric turbulence. The propagation equation for the slowly varying amplitude of the electric field E(x, y, z, t) is given by ( 2ik 0 z + 1 ) 2 k 2 E E = E k 0 v g t ω 2 t 2 + 2k 0 2 n 2 E 2 E + 2k 0 2 n pl (I)E (4) + 2k 02ñ(x, y, z)e ik 0 αe, where ñ(x, y, z) describes random fluctuations of the refractive index in turbulence and k 0 is a wavenumber. The ionization-energy loss is described by the coefficient α. The scattering and absorption of the radiation in the atmospheric aerosol are not taken into account. Nonlinear change of the refractive index in a strong laser field is defined by the Kerr effect and the response of the laser-induced plasma by n pl (I),whereI is the laser intensity. Kerr nonlinearity leads to self-focusing, while plasma causes defocusing of the radiation. The plasma contribution to the refractive index is proportional to the electron density. Free electrons are produced due to multiphoton or tunneling ionization of the air components, mainly from the molecules of oxygen and nitrogen. Thus, the dependence n pl (I) characterizes the process of plasma production from the oxygen and nitrogen molecules of air in the course of multiphoton and tunneling ionization. According to recent studies [2, 3, 14, 19, 20] the filamentation of a powerful femtosecond laser pulse consists of the following processes. The pulse with peak power larger than the threshold power for self-focusing in air undergoes selfcontraction during the propagation. In the nonlinear focus located in the central or leading part of the pulse the intensity reaches W/cm 2 and the ionization probability rapidly increases. The electron density accumulates during the pulse. The growth of the electron density is described by kinetic equations where the ionization rate is calculated according to models [25 27]. For a 800-nm laser wavelength the ionization rate is roughly proportional to the 8 th power of the laser intensity. Therefore, the plasma contribution is highly nonlinear. The laser-produced plasma leads to the strong defocusing of the radiation and limits the intensity growth caused by self-focusing. At the trailing edge of the pulse (beyond the nonlinear focus) a complex dynamic structure of aberrational rings surrounding the filament center is created. Filamentation in the conditions of atmospheric turbulence should be described in terms of a 3D + time stochastic equation with strong nonlinearity. Numerical simulation of this equation is not available with our computational facilities. However, the physical mechanism of filamentation allows us to build up a simpler model of formation and wandering of filaments in a randomly inhomogeneous medium. This model allows us to consider the initial stage of filamentation. By this initial stage we mean the nonlinear growth of the intensity up to the ionization threshold and formation of the nonlinear focus. At this stage of investigation we do not consider complex dynamics of the light field in the pulse along the whole propagation path. Below we present the justification of this assumption. According to the moving-focus model [3, 4] the most powerful slice of the pulse focuses the closest from the laser system output and defines the start of the filament. The slices at the front of the pulse contain less power and focus further along the propagation direction. The succession of the nonlinear foci formed by slices from central and leading parts of the pulse form the filament. Slices at the trailing part of the pulse defocus in the plasma. In the physical mechanism of filamentation it is essential that the ionization starts only after the intensity reaches the ionization threshold, which is about three orders of magnitude larger than the peak intensity of the input pulse. In the conditions of our experiment in air the threshold intensity for the ionization is estimated as W/cm 2 while the peak intensity in the input pulse is W/cm 2. Therefore, the laser-produced plasma does not affect the initial stage of the filament formation. The dispersion length for a 190-fs and 800-nm laser pulse propagating in air is of the order of 1km. Therefore, the group-velocity dispersion becomes important only after a fs contracted pulse [28] is created in the leading part of the pulse due to self-steepening. This peak is created in the nonlinear focus. Earlier in the propagation, group-velocity dispersion does not affect the filament formation. Stochastic simulations of powerful femtosecond laser pulse propagation in the turbulent atmosphere along the whole propagation path will require taking into account the effect of the ionization and group-velocity dispersion on the spatiotemporal transformation of the pulse in the filament.

5 CHIN et al. Filamentation of femtosecond laser pulses in turbulent air 71 FIGURE 4 Schematic picture of filament wandering in atmospheric turbulence. S turbj 1, S turb j, S turb j+1 are phase fluctuations on the screens that imitate atmospheric turbulence, z turb is the distance between the phase screens, 1, 2 are temporal slices of the pulse and z nf1, z nf2 are the positions of the nonlinear foci created by the slices 1, 2 At the initial stage of filamentation only Kerr self-focusing in randomly inhomogeneous media defines the position of the filament formation in a 3D space. Schematically this process is shown in Fig. 4. Before the filament is formed, refractiveindex fluctuations in air cause phase perturbations in the central slice of the pulse. Random small-scale perturbations near the maximum intensity location can initiate the formation of the nonlinear focus. The position of this focus is the place of the filament formation in the transverse plane. The place of the filament formation in the transverse plane changes randomly from shot to shot. Phase fluctuations, the size of which is larger than the transverse size of the laser pulse, are responsible for the wavefront tilt of the slice. The joint effect of large-scale and small-scale refractive-index fluctuations causes a random deviation of the nonlinear focus and the filament center from the propagation axis. Each slice in the leading edge of the pulse covers its own distance in randomly inhomogeneous air until it is self-focused. Therefore, random distortions of the wavefront (mainly tilts and focusing) formed in the course of this initial stage of propagation are different in each slice. As a result, the position of the filament center changes randomly with increasing propagation distance. According to our considerations the initial stage of the filament formation can be described by the following equation: 2ik 0 ( z + 1 v g t ) E = E + 2k 0 2 n 2 E 2 E + 2k 02ñ(x, y, z)e (5) The model described by (5) does not take into account inertial response of the Kerr nonlinearity [19, 20], since the latter does not influence the location of the nonlinear focus in space. The coefficient n 2 is chosen to be 1, cm 2 /W. The corresponding critical power for self-focusing is P cr = W [3]. Statistical characteristics of a three-dimensional field of refractive-index fluctuations ñ(x, y, z) are given by the model of atmospheric turbulence. For subpicosecond laser pulses the field ñ(x, y, z) may be regarded as stationary. Statistical isotropy of the filament-center wandering (Fig. 3a) allows us to assume that the experiment was performed in the conditions of the developed turbulence, which obeys the Kolmogorov 11/3 law [29]. To describe a wide range of refractive-index fluctuations we use the von Karman spectrum: Φ n (κ) = 0, 033C n 2 (κ 2 + κ 0 2 ) 11/6 exp{ κ 2 /κ m 2 } (6) where C n 2 is a structure constant of atmospheric turbulence, index n shows that the values Φ n and C n 2 refer to the refractive index n. Parameters κ 0 and κ m in (6) are given by: κ 0 = 2π/L 0 and κ m = 5.92/l 0 (7) where L 0 and l 0 are outer and inner scales of turbulence, respectively. The direct measurement of atmospheric turbulence parameters C 2 n, L 0 and l 0 has not been performed in the framework of our experiment. Therefore, to find the quantitative values of these parameters we will use the estimates from [30]. If the height of the atmospheric path is H = 1 2m then the outer scale of turbulence, calculated from the expression L 0 = 0.4H,isL 0 1m. The inner scale of turbulence is l 0 = 1mm. The structure constant C 2 n, which shows the intensity of atmospheric fluctuations, is a free parameter. We varied its value in the range C 2 n = cm 2/3 in order to fit the experimental results. The initial distribution of the electric field E(x, y, z = 0, t) is close to Gaussian in space and time [2, 3]. For coherent radiation the field E(x, y, z = 0, t) is given by E(x, y, z = 0, t) = E 0 exp { x2 + y 2 } } exp { t2 (8) 2τ 2 0 2a 2 0 where a 0 is the beam radius and 2τ 0 is the pulse duration. The peak power P 0 in the pulse exceeds the critical power for selffocusing P cr : P 0 > P cr, P 0 = πa 0 2 I 0, I 0 = cn 8π E 0 2, (9) where c is the speed of light. In this formulation the problem of filamentation in the turbulent atmosphere is reduced to the calculation of the nonlinear focus position in (x, y, z) space for the successive infinitely thin slices of the pulse. The nonlinear focus position of the central slice of the pulse is the beginning of the filament, and the nonlinear focus position of the slice with P P cr is the end of the filament. By the nonlinear focus

6 72 Applied Physics B Lasers and Optics position of any slice of the pulse we mean the point in (x, y, z) space at which the intensity reaches the ionization threshold. The solution to (5) for a random realization of refractiveindex fluctuations ñ(x, y, z) corresponds to the experimental measurements performed for one laser shot. A set of solutions to (5) obtained for statistically independent realizations of refractive-index fluctuations ñ(x, y, z) forms an ensemble that we used to calculate statistical characteristics of the filament in turbulent air. 4 Numerical simulation technique Numerical simulations of (5) and (8) are performed on the basis of the phase-screen model [31]. The calculations are performed on an (x, y, z) grid. In the simulations a randomly inhomogeneous medium is represented by a chain of phase screens located along the propagation axis. This chain made up of a finite number of scattering screens reproduces adequately the properties of a continuous medium, provided that the distance between the screens z is small compared with the characteristic scales of the field variation along the propagation coordinate z. These scales include the length of nonlinearity L nl, the diffraction length L d and the length of turbulence L turb [31]: z min{l ln, L d, L turb }. (10) The length of nonlinearity is defined as a distance along which the maximum phase growth due to self-focusing ϕ nl = n 2 k 0 L nl max{ E 2 } does not exceed 1rad.Then L nl = (n 2 k 0 max{ E 2 }) 1. (11) The length of turbulence L turb is defined as a distance along which the mean-square deviation of the phase due to refractive-index fluctuations does not exceed 1rad.Forthe von Karman model of atmospheric turbulence the length of turbulence is given by [ ( ) ] 2π 5/3 1 L turb = 2, 4π 2 k 2 0 0, 033C 2 n, (12) The diffraction length L d is given by the expression L d = k 0 a(z) 2.Thevaluea(z) coincides with the initial beam radius a 0 at z = 0. Through the value a(z) for z > 0 we denote the spatial scale of the nonlinear focal region in the beam cross section. In the course of Kerr self-focusing the intensity increases sharply in the region where a nonlinear focus is formed. Simultaneously, the lengths L nl and L d decrease. In order to satisfy the inequality (10) we decreased the interval between the phase screens as the plane of the nonlinear focus formation was approached. Since the simulation of phase screens that adequately reproduce atmospheric turbulence requires a lot of calculations we used two systems of phase screens. The first system imitates random phase fluctuations S turb (x, y, z) caused by refractive-index perturbations in the atmosphere. In this system of phase screens the turbulent phase screens are placed equidistantly along the propagation direction with the interval z turb. The interval z turb is selected from both the L 0 inequality (10) and the condition governing the applicability of the δ-correlated phase screens for the turbulent atmosphere given by L 0 z turb min{l turb, L d }. (13) Here L 0 is the outer scale of atmospheric turbulence defined earlier. The second system of phase screens reproduces the nonlinear phase growth ϕ nl (x, y, z) arising due to self-focusing. The distance z nl between nonlinear phase screens decreases with increasing intensity in accordance with the conditions (10), (11). In the initial stage of propagation, where the nonlinear phase growth is small and z nl > z turb, nonlinear phase screens are located in the same plane as the turbulent phase screens. Between both nonlinear and turbulent phase screens the light field undergoes only linear diffraction. In order to obtain a path of the nonlinear focus from one laser shot, we formed the chain of phase screens to simulate the atmospheric turbulence over the entire length of the filament. Then we considered the self-focusing of the succession of pulse slices passing through the same chain of the turbulent phase screens. The position of the nonlinear focus for a certain slice was defined as a point in (x, y, z) space, where the slice intensity reaches the ionization threshold. A set of such points calculated for all slices of the pulse, from the central slice at t = 0 to the slices in the leading front of the pulse, creates the trajectory of the filament for one laser shot. Statistical processing of the ensemble of such trajectories calculated for statistically independent chains of phase screens allows us to find the average distance from the output of the compressor to the beginning of the filament and the variance of the transverse deviation of the filament in the turbulent atmosphere. For the simulation of the turbulent phase screens we employ the modified method of subharmonics [32 34]. This method increases essentially the range of spatial scales of random phase fluctuations reproduced on the grid. As demonstrated in [35], the modified method of subharmonics with four iterations of phase-screen generation makes it possible to obtain a random field of phase fluctuations (6), the outer scale L 0 of which is two orders of magnitude larger than the transverse size of the grid in the plane XOY. In the simulations of filamentation in turbulence we used a square grid ( ) with the step h = 0.08 mm in the transverse section. This grid reproduces adequately phase fluctuations with the spatial scale ranging from L 0 = 1m to l 0 = 1mm. The distance between the turbulent phase screens was z turb = 1m. The distance between the nonlinear phase screens was decreased to z nl = 1cm as we approached the nonlinear focus. The diffraction length was L d (z = 0) 100 m for the input beam and L d (z = z f ) = 2.5cmfor the beam near the start of the filament. 5 Numerical simulation results Spatial distributions of the light field obtained from the numerical solution of the stochastic equation (5) demonstrate a random process of the nonlinear focus formation in

7 CHIN et al. Filamentation of femtosecond laser pulses in turbulent air 73 FIGURE 5 Spatial intensity distribution in the slice with 8.5-GW power (P = 1.4P cr ) at several distances from the output of the compressor. The structure constant of atmospheric turbulence is C n 2 = cm 2/3, the outer scale L 0 = 1 m and the inner scale l 0 = 1 mm. Intensity distribution at each distance is normalized to the maximum intensity I max at this distance. At z = 20 m I max = 1.4I 0, at z = 40 m I max = 2.5I 0, at z = 60 m I max = 3.6I 0,atz = 80 m I max = 7.3I 0, at z = 90 m I max = 16.2I 0 and at z = 100 m I max = 554I 0.Here I 0 = W/cm 2 is the maximum intensity at z = 0 the course of laser pulse filamentation in the turbulent atmosphere. Figure 5 shows the intensity distribution in the slice with P = 8.5GW= 1.4P cr at several distances z from the output of the compressor. In air the turbulent lenses induce distortions in the initially smooth beam profile. The strongest random focusing in the paraxial region becomes the nucleus of a nonlinear focus. At the beginning of propagation the intensity growth in the vicinity of the nucleus for the nonlinear focus is relatively slow. Near the distance of the nonlinear focus formation the intensity growth is very fast. One can see that in this random shot the nonlinear focus is displaced from the center of the input beam. The intensity in the nonlinear focus is much higher than the maximum values of intensity in all FIGURE 6 Trajectories of the filament center in the plane perpendicular to the propagation direction. The solid curve tracks the filament-center positions for the pulse. The random field ñ(x, y, z) is the same that was used to calculate the spatial intensity distribution shown in Fig. 5. The dashed curve is obtained for another realization of the random field ñ(x, y, z) the other slices of the pulse that do not focus at this distance z. Therefore, the position of the nonlinear focus nearly coincides with the maximum of the fluence at the same distance z. As a result, the filament center, defined as the position of the maximum of the fluence, is randomly displaced in the transverse section of the pulse. In the simulations we can follow the displacements of the filament center for a single pulse. For this purpose we calculated the nonlinear focus position for a succession of the pulse slices starting from the central slice towards the slices in the leading front of the pulse. The chain of the turbulent phase screens was the same for each slice. Figure 6 shows two trajectories of the nonlinear foci (and, consequently, the filament center) in the transverse plane XOY. The trajectories are obtained for two different realizations of the random field ñ(x, y, z) that describes refractive-index fluctuations in the atmosphere. Two random realizations of the field ñ(x, y, z) correspond to the propagation of two pulses in the laboratory experiment. Figure 6 demonstrates that for the pulses with 21-GW peak power the filament starts at z = m. This is in agreement with experimental data shown in Fig. 2. With increasing propagation distance the transverse deflection of the filament-center position from the position of the input beam center increases. See, for example, the trajectory (solid curve) in Fig. 6 that is obtained for the same random realization of refractive-index fluctuations as a set of intensity distributions in Fig. 5. At a distance z = 100 m the nonlinear focus position is shifted by 1.3mm along the x-axis and by 0.5mm along the y-axis (compare Figs. 5 and 6). The second trajectory in Fig. 6 (dashed curve) is obtained for another realization of the random field ñ(x, y, z).atz = 100 m the filament center is shifted down and to the left from the input beam center. 6 Statistical processing of experimental and numerical simulation results For statistical analysis of simulation results we have numerically solved equation (5) with 200 statistically independent chains of random phase screens. This corres-

8 74 Applied Physics B Lasers and Optics given by: Ψ lab/comp (R m ) = m P lab/comp (R i ) (14) i=1 The distance of the filament center R c = X c2 + Y c2 from the origin of the coordinate system XOY depends on the distances X c and Y c, which are statistically independent. Displacements X c and Y c are due to uncorrelated fluctuations of the refractive index in air on the path of the pulse propagation and, consequently, they obey the normal-distribution law. Therefore, it is reasonable to assume that the filament-center displacements R c obey the Rayleigh-distribution law: { ( ) } 2 Rc Ψ(R c ) = 1 exp, (15) w FIGURE 7 The normalized bar charts for the filament-center displacements R at distances z = 30 m and z = 105 m from the output of the compressor: solid curve corresponds to the experiment (1100 shots) and dashed curve to the simulations (200 shots) ponds to 200 laser shots in the experiment. The filamentcenter positions in simulations were measured at z = 30 m and z = 105 m as was done in the experiment. The value of the structure constant C n 2 was a fitting parameter in the simulations. The filament-center positions were calculated for several series of 200 shots and for each series we used its own C n 2 constant in order to find the best fit to the observed data. Figure 3b shows the simulated filament-center positions for C n 2 = 1, cm 2/3. There is an obvious qualitative agreement between the simulated and experimentally obtained filament-center positions (compare Figs. 3a and 3b). For the quantitative analysis of the results we have plotted bar charts for the samples of experimental {R lab } and simulated {R comp } values for the filament-center displacements. The sample size in the experiment is N lab = 1100 laser shots and in the simulations N comp = 200 laser shots. Figure 7 shows the normalized bar charts P lab (R m ) and P comp (R m ), where m is the number of the interval. The area of each bar chart is equal to unity. The sample size N comp in the simulations is much less than the sample size N lab in the experiment. Therefore, in order to obtain a valid estimate of the probability P comp (R m ), the width of the intervals in the bar chart plotted from the simulated data was chosen to be larger than the width of the intervals in the bar chart plotted from the experimental data. The empirical distribution functions of the filamentcenter displacements corresponding to these bar charts are where w is the parameter that characterizes the mean value of the filament-center displacements. The normalized probability function for the Rayleigh distribution has the form: P(R c ) = 2R { ( ) } 2 c w exp Rc 2 (16) w Experimentally obtained and simulated bar charts P lab (R m ) and P comp (R m ) correspond to the Rayleigh probability function (16) with the parameters w lab and w comp, respectively. The parameters w lab and w comp are calculated as follows. For an interval m parameter w m can be expressed using (14), (15): R m w m =, m = 1, 2,...M, (17) ln(1 Ψ(Rm )) where M is the number of intervals in the bar chart. Themeanvalueofwand its root-mean-square deviation are given by: w = 1 M M w m σ w = m=1 ( 1 M 1 ) 1/2 M (w m w) 2. (18) m=1 The absolute values of w and σ w obtained from statistical processing of experimental and numerical data are shown in Table 1. The comparison of the values of w and σ w shows that the structure constant of turbulence C n 2 = 1, cm 2/3 better corresponds to the experimental conditions than the structure constant C n 2 = cm 2/3.However,forz = 30 m the parameter w comp calculated with C n 2 = 1, cm 2/3 is less than w lab. At the same time for Propagation distance Experiment, Numerical simulations, 200 shots 1100 laser shots C n 2 = 1, cm 2/3 C n 2 = cm 2/3 30 m ± mm ± mm ± mm 105 m ± mm ± mm ± mm ( w m and σ w = M TABLE 1 The absolute values of w = M 1 m=1 data. The values σ w and w are shown in the form w ± σ w 1 M 1 ) 1/2 M (w m w) 2 obtained from statistical processing of experimental and numerical m=1

9 CHIN et al. Filamentation of femtosecond laser pulses in turbulent air 75 FIGURE 8 The distribution function of the filament-center displacements Ψ(R) in the probability scale of the Rayleigh law with parameter w lab : solid curve corresponds to the experiment (1100 shots) and dashed curve to the simulations (200 shots) z = 105 m the value of the parameter w comp is larger than w lab. The reason for this discrepancy may be the finite spatial coherence of the laser pulse that has not been taken into account in the simulations. Random fluctuations in the laser beam may be responsible for essential deflections of the filament center at the initial stage of propagation, where the contribution of turbulence to the nonlinear focus formation is not very large. To analyze the applicability of the Rayleigh law to the description of random displacements of the filament center, let us plot the distribution function Ψ(R m ) in the Rayleigh probability scale. If we define the probability scale from the Rayleigh distribution with the parameter w lab then the analytical dependence for the distribution function (15) will be plotted as a straight line with an angle of 45 degrees between this straight line and each of the coordinate axes (Fig. 8). The values of the empirical distribution function Ψ lab (R m ) calculated for m intervals are shown by dots. The analytical dependence of the Rayleigh-distribution function on the parameter w lab fits the distribution function obtained from the experimental data very well. In this probability scale the straight line calculated for the distribution function Ψ comp (R m ) with the parameter w comp makes an angle with the analytical dependence Ψ(R) with the parameter w lab. This is the consequence of the discrepancy between the Rayleigh parameters w comp and w lab obtained from the simulations and from the experiment. The distribution functions Ψ lab (R m ) and Ψ comp (R m ) plotted in the Rayleigh probability scale fit the straight lines. This proves the validity of our assumption that the filament-center displacements obey the Rayleigh-distribution law. 7 Conclusions We presented the results of an experimental and theoretical study of filament wandering in the propagation of powerful femtosecond laser pulses in a randomly inhomogeneous medium. We have experimentally obtained the ensemble of filamentcenter displacements in the plane perpendicular to the propagation direction. The observed picture of filament displacements obtained for 1100 laser shots proves the isotropic character of random perturbations that cause these displacements. For the perturbations associated with refractive-index fluctuations in air the isotropic character of the perturbations corresponds to the Kolmogorov atmospheric turbulence. Statistical processing of experimental data shows that the distribution function of filament-center displacements obeys the Rayleigh law. The parameter of the Rayleigh distribution characterizes the average value of displacements and increases with filament length up to 1.5mmat z = 100 m. We have developed a model of filament formation and wandering in the course of propagation of a powerful femtosecond laser pulse through atmospheric turbulence. In this model the filament wandering is associated with refractiveindex fluctuations in air. These fluctuations disturb the wavefront of the laser radiation and cause random formation and displacements of the filament. For numerical simulations of a femtosecond pulse self-focusing in atmospheric turbulence we have developed the phase-screen model that describes both turbulent and nonlinear perturbations of the wavefront of the laser radiation. The developed model allows us to simulate the initial stage of the powerful femtosecond laser pulse propagation in atmospheric turbulence. Using the Monte Carlo method and the Kolmogorov model of atmospheric turbulence we have found that the simulated filament displacements as well as experimentally obtained filament displacements obey the Rayleigh-distribution law. The parameter of the Rayleigh distribution calculated in the simulations is close to the one obtained from the experimental data. The discrepancy can be associated with the limited spatial coherence and random angular deflections of the laser radiation. The effect of these factors on the filament wandering in the atmosphere demands further study. Further study of stochastic filamentation, including multifilamentation and white-light generation, demands a more advanced model in order to perform the simulations of the pulse transformation along the whole propagation path in the turbulent atmosphere. 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