Controlling a bunch of multiple filaments by means of a beam diameter

Transcription

1 Appl. Phys. B 82, (2006) DOI: /s Applied Physics B Lasers and Optics o.g. kosareva 1, n.a. panov 1 n. akozbek 2 v.p. kandidov 1 q. luo 3 s.a. hosseini 3 w. liu 3 j.-f. gravel 3 g. roy 4 s.l. chin 3 Controlling a bunch of multiple filaments by means of a beam diameter 1 International Laser Center, Physics Department, M.V. Lomonosov Moscow State University, Moscow, , Russia 2 Time Domain Corporation, Cummings Research Park, 7057 Old Madison Pike, Huntsville, AL 35806, USA 3 Centre d Optique, Photonique et Laser (COPL) et Département de Physique, de Génie Physique et d Optique, UniversitéLaval,Québec, QC, G1K 7P4, Canada 4 Defense Research and Development Canada-Valcartier (DRDC-Valcartier), Val-Belair, QC, G3J 1X5, Canada Received: 22 April 2005/Revised:18July 2005 Published online: 19 November 2005 Springer-Verlag 2005 ABSTRACT We demonstrate a three orders of magnitude increase and stability in the backscattered fluorescence signal from nitrogen molecules by terawatt femtosecond laser pulse induced air filaments using a new method. The method is based on squeezing the initial beam diameter using a telescope. The effect of laser shot-to-shot fluctuations was included in numerical simulations by a random distribution of the initial intensity in both squeezed and non-squeezed beams. Statistical processing of the simulation results shows that the average diameter of plasma channels as well as the total amount of free electrons generated in a bunch of multiple filaments in air is larger in the squeezed beam. Shot-to-shot stability of the simulated plasma density increases in the squeezed beam. The change of this plasma density with propagation distance is in good qualitative agreement with the change of the range-corrected nitrogen fluorescence signal with distance. PACS Jx; Jf; Ay; Wt 1 Introduction Femtosecond laser pulses create dynamic light filamentation in bulk transparent materials [1 10]. By filaments we understand comparatively long regions of the spatially and temporally localized radiation zones, which generate free electrons in the medium. A typical diameter of a filament produced by a Ti:sapphire laser amplification system with the central wavelength at 800 nm is of the order of 100 µm in atmospheric air [1 4] and 2 10 µm in condensed matter [5 10]. Owing to the time-dependent self-focusing, the temporal pulse width inside the filament can be shortened [11, 12] and could be as short as a few femtoseconds resulting in only a few optical cycle pulse [12]. It was found experimentally [3] and confirmed in the simulations [13] that the energy flowing through a 500-µm pinhole centered on the longitudinal axis of a single filament is of the order of 5% 10% of the initial pulse energy. All the rest of the initial pulse energy is contained in a wide beam background [13 17], which feeds the filament throughout the propagation length. If the peak power of the pulse is several times higher than the critical power for self-focusing in the medium ( 5GW in Fax: (095) , kosareva@phys.msu.ru air, 1 5MWin condensed matter), then multiple filaments arise in the transverse beam profile [4, 17 20]. The so-called parent filaments born from the natural imperfections in the beam produce next generations of filaments. The bunch of filaments, similar to a single filament, is sustained by the energy contained in the beam background. Child and parent filaments compete for energy throughout the propagation [21]. The unique possibility of the existence of such confined propagation in free space is important for several applications. Lightning discharge control requires long and homogeneous plasma channels which are produced by the filaments [22]. Longitudinal homogeneity of the plasma channel is also important for waveguide writing in dielectric solids [7, 10]. Atmospheric remote sensing requires high conversion efficiency to both visible and infrared parts of the supercontinuum [23], as well as to the third harmonic in air [24, 25], when one uses the technique of absorption of the white light/third harmonic by pollutant molecules. New applications arise from the possibility of generating high-intensity few-cycle pulses after filamentation [12]. In comparison with sub-10- femtosecond pulse generation using a capillary tube or a hollow fiber [26, 27], filamentation allows one to obtain a fewcycle pulse in free space. In addition, this pulse is already selfcompressed and does not need to be sent into a dispersioncompensation device. Launching such a pulse into a medium of interest allows one to study ultra-fast molecular and atomic processes (see e.g. Refs. [28, 29]). To meet the demands of the applications, it is desirable to control the filamentation by adjusting the initial parameters of the laser pulse. There are different ways to control these parameters. At the beginning of femtosecond pulse propagation experiments in the atmosphere, initial pulse chirp and wavefront divergence were suggested as the tools for governing the starting position and the length of the filaments [30]. These methods were afterwards justified numerically [31] and successfully applied experimentally: in the experiment the conversion efficiency to the white light was six times larger when a 100-fs pulse was negatively chirped up to 600 fs [23]. The inevitable formation of multiple filaments in the pulse with peak power higher than the critical power for selffocusing in air requires small-scale control of the filamentation. The term small-scale control means that the transverse size of regular perturbations introduced into the beam

2 112 Applied Physics B Lasers and Optics is small in comparison with the initial beam diameter. The formation of multiple filaments in the locations predetermined by regular intensity and/or phase perturbations was described in Refs. [32, 33]. Control of multiple filaments with a mesh was performed both experimentally and numerically in Refs. [34, 35], where stochastic hot-spot formation was suppressed by periodic perturbations introduced by a metallic mesh. As an alternative to small-scale control of filamentation, one can use large-scale control and govern the propagation by changing the whole beam size or shape. For example, imposing ellipticity into the beam is one way to induce the filament formation at the predetermined positions in the transverse section of the pulse [36]. Another way of large-scale control of the filamentation is changing the initial beam diameter with a telescope at the beginning of the propagation. This method was experimentally observed in Ref. [37] and is further developed in this paper. In atmospheric remote sensing applications it is important to have a stable nonlinear nitrogen fluorescence signal, which is closely correlated with the distribution of plasma in the filaments. By shot-to shot stability we understand both the reproducibility of the signal intensity along the filament and the location of the filament s starting position. Further interest in the fluorescence registration is that many of the ionization and fragmentation products, including air pollutants, would fluoresce and this fluorescence can be detected using a LIDAR system [38]. Amplification of the clean fluorescence signal from nitrogen molecules in the direction opposite to femtosecond pulse propagation was first demonstrated by Luo et al. [39]. In the experiments with a long propagation path of 100 m the backscattered nitrogen fluorescence signal (BSF) was clearly detectable over the whole range [21]. However, the BSF signal varied randomly in terms of both the signal intensity distribution along the propagation path as well as the starting position of the filamentation. The physical origin of these fluctuations in the BSF signal was attributed to a competition among multiple filaments, arising from both the imperfections in the beam profile introduced by the laser system and propagation through atmospheric turbulence. In subsequent experiments it was demonstrated that by squeezing the initial beam diameter both the intensity of the fluorescence signal and the stability in the starting position of the filament can be increased essentially [37]. The purpose of this paper is to show both experimentally and by means of numerical simulations that the nonlinear fluorescence signal can be controlled by squeezing the initial beam diameter with a convex concave lens system. The result is that in the experiment we obtain a three orders of magnitude increase in the fluorescence signal and an essential increase in the signal stability from one laser shot to another. Numerical simulations are performed with the initial conditions, which are changing randomly from one realization to another. Each random realization simulates a single-shot pulse in the experiment. Statistical processing of the simulation results shows that squeezing of the initial beam diameter leads to the increase of the average amount of free electrons in the plasma channel and the increase in the stability of the fluence and plasma distribution from shot to shot. An explanation of this increase is in the more effective usage of the beam background energy in the case of initial beam squeezing. 2 Experiment and results The setup has been described elsewhere [37]. A linearly polarized femtosecond laser beam with a repetition rate of 10 Hz, 42-fs duration (FWHM), and a central wavelength of 800 nm with a bandwidth of 23 nm (FWHM) is used in this experiment. The pulse energy can vary from a few mj to 85 mj. The standard deviation of the fluctuations of the laser energy is 4.4%. It was shown in Ref. [21] that these energy fluctuations have a minor contribution to the fluorescence signal instability in comparison with the contribution from multiple-filament competition. The beam is guided from a vacuum compressor (10 3 Torr) to a corridor (length 101 m) through a 10-m vacuum pipe terminated with a 1.5-cm-thick CaF 2 window. In our experiments, a laser beam with two different diameters is used. The output beam diameter after the vacuum pipe is 25 mm (1/e 2 ). A telescope system consisting of a convex lens and a concave lens with 100 cm and 30 cm focal lengths, respectively, is introduced to decrease the beam diameter to 8mm(1/e 2 ). The divergence of the small beam can be changed by varying the position of the concave lens. The BSF signal of a nitrogen molecule is collected with a LIDAR system and detected by a photomultiplier tube (PMT, Hamamatsu R7400P, with 1-ns response time; gain = ). Considering that the lifetime of the N 2 fluorescence signal is around 1 2ns[40], the resolution of this detection system will be around cm. The field of view of the LI- DAR is equal to 16 mrad. In this experiment the LIDAR was put in an off-axis position. In this configuration, the first 6m of the propagation is blocked in order to avoid the scattering of the pump pulse from the last mirror in the setup. The whole range of the laser propagation after 6m and up to 100 m is covered by the field of view of the LIDAR system. The filamentation starts at 12 m in the large beam and 8min the small beam. Therefore, no important data was lost due to the blocking of the first 6mof the propagation path. Two broadband dielectric mirrors reflecting around 800 ± 50 nm are placed in front of the PMT to eliminate any backscattering of the fundamental laser light. This is followed by a band-pass filter (UG 11, 4-mm thick, band pass 200 to 400 nm) transmitting the major spectral lines from nitrogen molecules and ions in the nm region. In Fig. 1a, a three-dimensional (3D) graph of 300 shots at 30 mj/pulse with large beam diameter (25 mm) ispresented. In this graph the x axis is the direction of propagation of the beam that is calibrated from the time scale of the oscilloscope and the y axis represents the laser shot number. The peaks are BSF signals. One can notice strong fluctuations of the signal from shot to shot. These fluctuations are seen in both the starting position of the filament as well as the value of the signal intensity. A dramatic change is observed when the initial beam diameter is squeezed down to 8mm (Fig. 1b). The steep rise of the fluorescence signal is now at nearly the same position at each subsequent laser shot. Besides, the intensity of the signal is approximately 1000 times larger than in the large beam. Note that the fluorescence signals were calibrated by the different oscilloscope scales and different applied high voltages on the PMT. As a result of this calibration, the signal intensity in this paper for both large and small beams is presented in the same relative units. Fig-

3 KOSAREVA et al. Controlling a bunch of multiple filaments by means of a beam diameter 113 FIGURE 1 Experimentally registered 300 shots of backscattered fluorescence signal waveform detected by PMT. The laser energy was fixed at 30 mj/pulse. (a) The diameter of the beam is 25 mm. (b) By using the telescope the beam diameter is reduced to 8 mm ure 2a and b show the average and standard deviation of the waveforms from the plots in Fig. 1a and b, respectively. The peak of the average fluorescence signal in an 8-mm-diameter beam (Fig. 2b) is 923 times higher than the peak of the average signal in a 25-mm beam (Fig. 2a). The standard deviation related to the signal itself at a certain propagation distance is also much higher in the larger beam over the whole propaga- tion distance. For example, in the peak of the average signal the standard deviation in the large beam constitutes 200% of the signal value, while in the small beam it falls to 38% of the signal value. This is the quantitative evidence for the dramatic increase in the backscattered fluorescence signal stability when filamentation is controlled by the initial beam size. Our previous numerical simulations [21] have shown the physical reason for the fluorescence signal increase in the case of decreasing distance between the initial perturbations in the beam profile. Closer separation distance between the initial perturbations leads to the formation of a larger number of child filaments and, as a consequence, a more intense and longer plasma channel, which gives rise to a higher fluorescence signal. The problem of fluorescence signal increase was studied in the deterministic case. Numerical simulations presented in the following sections are aimed at statistical analysis of the pulse propagation with randomly changing transverse intensity distribution in both small- and large-beam cases. By means of statistical processing of the simulation results, we reveal the mechanisms that lead to either high or low fluorescence signal intensity and stability. 3 Numerical simulations FIGURE 2 The backscattered fluorescence signal averaged over 300 shots (see Fig. 1a and b). The standard deviation is shown for each 10th data point. (a) Initial beam diameter is 25 mm, (b) initial beam diameter is 8 mm Perturbations in the beam profile are initially created in the laser amplification system. The laser system ends with a vacuum compressor connected with the output to the corridor by means of a 10-m vacuum pipe with a 1.5-cmthick CaF 2 exit window. Initial perturbations are enhanced due to self-focusing in the CaF 2 window in such a way that the hot-spot zones are created with a typical size of the experimental beam radius a exp = 8.8mm (this corresponds to 25-mm beam diameter at 1/e 2 ). These hot-spot zones can be seen in Fig. 3a and b, which show the transverse distribution at the output of the exit window. The relative positions and amplitudes as well as transverse sizes of the hot-spot zones vary from one laser shot to another. Two randomly chosen laser shots can be compared in Fig. 3a and b. Further into the propagation, these zones seed random multiple filamentation in the experiment.

4 114 Applied Physics B Lasers and Optics FIGURE 3 Experimental transverse fluence distribution registered after the exit CaF 2 window of the vacuum system. White contours of equal fluence stress the perturbations induced by the window. Beam radius defined at 1/e level of fluence is a exp = 8.8 mm (corresponds to 25 mm at 1/e 2 ).(a)and(b) correspond to two different laser shots In order to qualitatively model a random beam distribution at the output of the CaF 2 window, we have represented the initial light field by a sum of a large Gaussian beam with the radius a 0 and five smaller Gaussian perturbations with the radius a 0 /2: Ẽ i (x, y, z = 0,τ)= E 0i e τ2 /2τ0 ( 2 exp x2 + y 2 ) + j=1 a 2 0 2a [ α j exp ( 2 (x x j) 2 + (y y j ) 2 )]. (1) The center of the large beam is located at the point (x = 0, y = 0). It is deterministic and does not change from one realization to another. One realization of a random field given by (1) corresponds to one laser shot in the experiment; i is the number of the realization. The coordinates of the centers of smaller perturbations (x = x j, y = y j ) are distributed randomly within the interval [ 2a 0, 2a 0 ]. The probability density function is equal to unity within this interval and zero outside this interval. Random amplitudes α j have the same probability density function within the interval [0, 1]. Each realization of the light field E has its own set of random values x j, y j, α j.thevalueτ = t z/v g is the time in the coordinate system moving with the pulse. Simulation of the shot-to-shot random distribution of light along the filament requires the solution of the 3D+ time statistical problem of the pulse propagation. In order to numerically solve this problem using a computer of a moderate size (the simulations were performed on one node of an IBM Cluster 1350 computer, so that two Intel Xeon TM 2.6 GHz processors with a total RAM of 1.5GBwere used), we have introduced a smaller beam size into the simulations than it was in the experiment. The case of a larger beam was modeled with the radius a 0 = 1.1mm in Eq. (1), while the case of a smaller beam was modeled with the value of a 0 = 0.7mm. In addition, we took the lower input pulse energy W 0 = 7mJinstead of 30 mj as used in the experiment. In all cases the input pulse energy was W 0 = 7mJand the half-pulse duration at e 1 intensity level was τ 0 = 25 fs. A decrease of initial pulse energy in comparison with the experimental one was taken in order to decrease high spatio-temporal gradients arising in the course of propagation. At the same time a comparatively high ratio of peak power to the critical power for self-focusing in air P/P cr = 10 allows us to perform the simulations in multiplefilamentation regime. We have studied 12 random realizations of a large beam and 12 random realizations of a small beam. The number of the realizations obtained for each beam size is not very large. Therefore, in order to check the convergence of statistical characteristics, we have performed successive averaging over 7, 8, 9, 10, 11, and 12 realizations of the obtained simulation results. Starting from the average value obtained from eight realizations and until the average value obtained from 12 realizations, the maximum relative change of these values was less than 8%. The input pulse energy is kept constant throughout all 24 realizations of a random field E given by Eq. (1). This is provided by the maximum intensity I 0i = (cn 0 /8π) E 0i 2 at z = 0 defined for each particular ith realization by I 0i = τ 0 π + 5 j=1 [ α j exp { exp W 0 ( x2 +y 2 2a 2 0 ) ( 2 (x x j ) 2 +(y y j ) 2 a 2 0 )] } 2 dx dy. (2) Assuming that the pulse propagates along the z axis with the group velocity v g, the equation for the slowly varying amplitude of the electric field E(x, y, z, t) is ( E 2ik z + 1 ) E v g t = 2 E x E y 2 + 2k2 n 0 ( nk + n p ) E ikαe, (3) where the first two terms on the right-hand side of Eq. (3) describe diffraction. In the third term we take into account the nonlinearity of the medium. Following Ref. [41], we represent the Kerr contribution n k in the form n k (t) =

5 KOSAREVA et al. Controlling a bunch of multiple filaments by means of a beam diameter 115 n 2eff (t) E(t) 2,where n 2eff (t) = 1 2 n E(t) 2 t H(t t ) E(t ) 2 dt. (4) The response function H(t) was defined in Ref. [42]. For the Gaussian pulse with 42-fs duration (at FWHM), n 2eff (0) = 0.57n 2 and the effective critical power for self-focusing P cr eff = 11.8GW. The plasma contribution to the refractive index n p is given by ω2 p n p = 2n 0 ω 2, (5) where ω p = 4πe 2 N e /m e is the plasma frequency and ω is the laser central frequency corresponding to λ = 800 nm.the free electron density N e (x, y, z, t) depends on the spatial coordinates and time according to the kinetic equation N e = R( E 2 )(N 0 N e ), (6) t where R denotes the ionization rate for oxygen and nitrogen molecules and N 0 is the density of neutral molecules (20% oxygen and 80% nitrogen). To calculate the ionization rate R( E 2 ) in Eq. (4) we used the model based on that of Perelomov, Popov, and Terent ev (PPT) [43] for the ionization of a hydrogen-like atom in the linearly polarized electric field E. The values of the effective charges of oxygen and nitrogen molecular ions are taken from Ref. [44] in order to fit the experimental data on ion yields. The last term on the right-hand side of Eq. (3) describes the energy losses due to the ionization, where the absorption coefficient α = I 1 mhω( N e / t), m is the order of the multiphoton/tunneling process, I = cn 0 E 2 /8π is the laser pulse intensity, and k = 2πn 0 /λ is the wavenumber. In Eq. (2) we have neglected the terms associated with self-steepening and material dispersion in air. The reason for this is that in both experiment and simulations we seed the filaments by quite strong initial intensity perturbations; therefore, we predetermine spatial break-up of the initial beam into multiple filaments. Indeed, an ideal Gaussian 42-fs pulse centered at 800 nm with a Gaussian spatial intensity distribution input energy of 30 mj and a beam diameter of 25 mm would self-focus in air at a distance of 32 m. For comparison, the characteristic dispersion length L disp = τ0 2/k ω for a 42-fs pulse in air is 39 m. Potentially, the two effects are comparable and material dispersion could affect the ideal pulse even at the initial stage of self-focusing. However, due to the perturbations in the beam profile, the large beam self-focuses at z f 14 m L disp (Fig. 2a). Therefore, spatial effects dominate temporal dispersive broadening from the beginning of the propagation. The next contribution of material dispersion can come into effect after the start of the filamentation. In our case of propagation in air the shortest duration τ f of the pulse reached in the course of non-stationary self-focusing is of the order of 3fs. The corresponding dispersion length is L disp = τf 2/k ω = 56 cm. At the same time, the filament radius a f in air is approximately 40 µm. The corresponding longitudinal scale of spatial transformation ka 2 f = 1.25 cm L disp. Thus, the spatial effects, which dominate the dynamics of multiple filamentation, develop faster than the temporal effects. Since the purpose of our current study is spatial control of multiple filamentation, we can neglect material dispersion at this stage. As far as self-steepening is concerned, it is shown in Refs. [45, 46] that this higher-order nonlinear effect mainly influences the temporal dynamics of the pulse as well as the white-light continuum conversion efficiency and to a less extent the spatial dynamics of the pulse intensity. The system of equations (2) (6) with the initial conditions (1) was solved on a grid with non-equidistant grid steps along both x and y coordinates. The size of the grid steps was dependent on the step number and was decreased in the region of inhomogeneities. The overall grid size was ( ) in the transverse plane and 512 equidistant steps in the time domain. The (x, y,τ) grid-size limitation is imposed by 1.5GB RAM of one IBM Cluster 1350 node with two processors. The run of one realization of the pulse given by Eq. (1) (corresponding to a single laser shot) took us around 35 h. We did not include other nodes in the computations in order to avoid time-consuming data exchange between the nodes, which would cause a several times increase in the computer time for the same number of grid steps. The number of z steps varied from 1000 to To check the validity of the simulations, we tested the phase growth between the current grid point and the neighboring points in the directions (x, y,τ) at each propagation step along the coordinate z. Throughout all the calculations this phase growth did not exceed π/10 rad. As the result of the simulations of Eqs. (1) (6) we obtain time-resolved distributions of the light field complex amplitude E(x, y, z,τ) and free electron density N e (x, y, z,τ),at each location z along the propagation direction. Usually, in experiments the fluence distribution is registered by means of a CCD camera, although femtosecond-resolved measurements can also be performed with special techniques (see e.g. Refs. [11, 12]). The fluence distribution is given by J (x, y, z) = cn 0 8π E (x, y, z,τ) 2 dτ. (7) The fluorescence signal registered in the experiment is associated with the overall amount of electrons generated at each location z at the end of the pulse τ end. This amount is characterized by the linear plasma density: D e (z,τ end ) = 4 Filament bunch energy N e (x, y, z,τ end ) dx dy. (8) We start from the analysis of the simulated fluence distribution J(x, y) and the electron density distribution N e (x, y) in successive transverse planes z along the direc-

6 116 Applied Physics B Lasers and Optics FIGURE 4 Simulated transverse fluence (the first and the third rows) and electron density distributions (the second and the fourth rows) at different propagation distances. (a) (g) are related to the small beam, a 0 = 0.7 mm, while (h) (n) are related to the large beam, a 0 = 1.1 mm. The initial conditions in (a) and (h) correspond to the two random laser shots, each given by Eq. (1), and white contours of equal fluence are shown to stress the initial beam shape. Maximum intensity in (a) is W/cm 2,andin(h) W/cm 2. The initial energy is the same for all panels and equal to 7 mj, pulse duration 42 fs (FWHM), and central wavelength 800 nm. The electron density in (e) (g), (l) (n) is obtained at the same propagation distance as the fluence in (b) (d), (i) (k), respectively. The contours of equal electron density are shown on the logarithmic scale; each contour value is defined through the equation N e contour = N emax 4 const, where the value const changes from 3 to zero with the step 1, N emax = cm 3 tion of propagation. In the first and the second rows of Fig. 4 (panels a to g) we show the development of transverse fluence patterns and electron density in the pulse with the smaller beam diameter. In the third and the fourth rows of Fig. 4 (panels i to n) the fluence and the electron density patterns are shown for the larger beam. Each of the initial distributions in Fig. 4a and h represents one random realization described by Eq. (1). The propagation distance increases from left to right; the fluence and electron density patterns are shown at the same positions z (if related to the same beam). Both absolute and diffraction length units are indicated. The diffraction length for the larger beam is z d = ka0 2 = 9.5m and the diffraction length for the smaller beam is z d = ka0 2 = 3.8m. According to the scenario of multiple-filament formation [21, 47], in both the small and the large beams one can see the formation of parent filaments from the initial pertur-

7 KOSAREVA et al. Controlling a bunch of multiple filaments by means of a beam diameter 117 bations, marked by the numbers 1, 2, and 3 in panels b, c and i, j of Fig. 4. Electron density distributions in panels e and l show that by z = 0.15z d two developed filaments are created. By a developed filament we assume a filament that forms a plasma burst with an electron density higher than 10 5 of atmospheric density N a = cm 3.Further on, from z = 0.15z d to z = 0.25z d (Fig. 4, panels c and j) child filaments are formed on the perturbations arising from the interference of rings produced by the parents. The childformation process differs in the small and in the large beams. In the small beam child filaments are created in both interference zones: between the filaments 1 and 2 as well as between the filaments 2 and 3 (fluence panel c and electron density panel f). However, in the large beam the zone between the 2nd and the 3rd parent filaments remains free of children up to z = 0.25z d (238 cm). This is seen in fluence panels j, k and electron density panels m, n. Finally, in the small beam we have a tight bunch of closely spaced plasma channels (panels d and g), while in the large beam we rather have a set of narrow electron density hot spots (panels k and n). This is in qualitative agreement with experimentally obtained multiple-filament distributions shown in Fig. 1 of Ref. [37]. In order to quantitatively demonstrate that squeezing of the initial beam leads to a higher growth rate of filaments with distance, we have averaged the number of the developed filaments N f born in successive transverse planes along the propagation direction. Figure 5 shows the growth of the average number of filaments with propagation distance. Averaging was performed over 12 realizations of an initially small beam (squares) or an initially large beam (circles). Based on the linear fits shown by solid lines, we can find the approximate filament growth rates, which are 10 filaments per 1mof propagation in the case of the small beam and seven filaments per 1min the case of the large beam. Thus, in the small-beam case the growth of filaments is 1.4 times faster than in the large-beam case. Also, the threshold distance at which the number of filaments starts to grow significantly is shorter in the case of the small beam. Comparing the standard deviations for the two curves in Fig. 5, we find the decrease in the maximum standard deviation from 2 in the large-beam case to 1.6 in the small-beam case. Faster and more predictable (because of less standard deviation) formation of filaments in the small beam is associated with more effective usage of the background energy source. To demonstrate this, we show in Fig. 6a c the dependence of the energy contained in multiple filaments on the propagation distance. The energy contained in the bunch of filaments is defined in the following way: after the formation of developed parent filaments (see e.g. filaments 1, 2 in panel b of Fig. 4) we register the half-maximum value of the fluence J FWHM, which is around 0.75 J/cm 2. The filament diameter at this fluence level is 190 µm and the diameter of the plasma channel at half-maximum is 100 µm. Afterwards, we integrate the light field intensity over the pulse time τ within the whole pulse and over the spatial (x, y) coordi- FIGURE 5 Growth of multiple filaments with propagation distance in the simulations. Average number of filaments in a set of transverse planes along the propagation direction z is shown by empty squares for the small beam, a 0 = 0.7 mm, or empty circles for the large beam, a 0 = 1.1 mm. Averaging was performed over 12 realizations. Solid curves represent linear fits to the average values FIGURE 6 Filament bunch energy W bunch extracted from the simulation results using Eq. (9). (a) Small beam, a 0 = 0.7 mm, integration is performed over the 2D region Σ, where the fluence J > J FWHM = 0.75 J/cm 2. (b) Large beam, a 0 = 1.1 mm, integration is performed over the 2D region Σ, where the fluence J > J FWHM = 0.75 J/cm 2. (c) Large beam, a 0 = 1.1 mm, integration is performed over the 2D region Σ, where the fluence J > J FWHM = 0.3J/cm 2. Initial pulse energy is the same for all plots and equal to W 0 = 7mJ

8 118 Applied Physics B Lasers and Optics nates within the region, where the fluence is higher than the value J FWHM : W bunch (z) = cn 0 8π dτ E (x, y, z,τ) 2 dx dy, (9) whereσ is the two-dimensional (2D) region with the fluence J > J FWHM.ThevalueW bunch given by Eq. (9) is averaged over 12 realizations of the small beam and plotted in Fig. 6a. The result of averaging over 12 realizations of the large beam is shown in Fig. 6b. The important result is that in the case of the small beam the fraction of the initial pulse energy equal to 10% 20% of W 0 is contained within the bunch of filaments (Fig. 6a). In the large beam just 2% 4% of the initial energy W 0 is higher than the chosen level J FWHM = 0.75 J/cm 2 (Fig. 6b). Besides, the standard deviation of the value W bunch in Fig. 6b is of the order of this value itself. Thus, one can see that the usage of the initial energy is more effective in the small-beam case. We note that in the simulations we controlled the total pulse energy at each position along the z axis. The energy loss by the end of the propagation path is nearly the same in the small beam and in the large beam ( 10% of the input pulse energy in the large beam and 15% in the small beam). The longitudinal extension of the bunch of filaments having the fluence higher than J FWHM = 0.75 J/cm 2 has approximately the same value of 100 cm in both Fig. 6a and b. For comparison, we present the percentage of energy contained within the region Σ with the fluence higher than 0.3J/cm 2 in the large beam (Fig. 6c). In this case around 10% 25% of the initial pulse energy is concentrated in the two-dimensional region Σ. The longitudinal extension of this region is more than 120 cm. Thus, depending on the fluence threshold that one needs to use for a particular purpose, either a large or a small beam can be employed in the experiment. Qualitative interpretation of the larger filament bunch energy in the case of the small beam can be given in the following way. Each filament is competing for the energy from the background source. If the two filaments are located closer to each other, the perturbation created by the rings diverging from these filaments in the plasma will most likely have enough peak power (energy) to develop into a child of the two parent filaments. If the initial parent filaments are further from each other, but have to share the same amount of energy as in the case of closer location, a new perturbation created between them might not be developed into a filament as soon as in the close-location case. This is because in the further-location case the new perturbation of a small size does not contain a critical power for self-focusing, i.e. P perturbation < P cr (P perturbation = πa 2 p I p,wherea p and I p are the characteristic radius and peak intensity of the perturbation, respectively). Larger energy concentration in the smaller beam results in higher peak intensity of the perturbation I p and, hence, the probability of having P perturbation > P cr in the perturbation of a certain size increases. Estimates of the further and closer locations of the neighboring parent filaments can be given on the basis of the analysis of the overlapping wavefront surfaces produced by the perturbations in the course of self-focusing. Our simulations in a single-filament regime show that the filament energy is mainly concentrated in the region where the transverse spatial phase ϕ changes in the range 0.36 ϕ max ϕ max.zero phase level corresponds to the surrounding beam background, where the self-focusing is not pronounced, while ϕ max is reached in the filament center. The power contained within the region of essential phase change is P cr and is almost independent of the initial pulse peak power. In the case of the two perturbations, the measure of the effective interaction can be defined in terms of the phase level at which the two wavefront surfaces belonging to different filaments overlap. If the overlap is at the level close to zero, the energy interchange between the perturbations is negligible and the situation can be classified as further location of filaments. If the overlap is at the level 0.36 ϕ max or higher then one can say that the two filaments interact intensively and the situation can be classified as closer location of filaments. The simulations in the two-perturbation regime have shown that the phase level of the overlap decreases from 0.82 ϕ max to 0.15 ϕ max if the distance between the maxima of the perturbations increases by 1.6 times from the small-beam radius a 0small = 0.7mmto the large-beam radius a 0large = 1.1mm. In our case the beam radius a 0 corresponds to the average distance between the initial perturbations and therefore is a good reference for measuring the relative locations of the two filaments originating from these perturbations. Finally, we note that according to the wavefront surface overlap criterion closer location between the filaments corresponds to the distance smaller than a 0small, while further location of filaments corresponds to the distance larger than a 0large. 5 Plasma channel diameter and nonlinear fluorescence A nonlinear fluorescence signal can be registered during the relaxation of the excited states of ionized and neutral molecules created during the ionization in the highintensity zones. Thus, the origin of the behavior of the fluorescence signal along the propagation distance should be examined in the plasma distribution. Earlier in this paper we have already used the transverse distribution of the plasma in order to count multiple filaments growing along the propagation direction, as shown in Fig. 5. Plasma hot spots formed at the end of the pulse by the simulated laser shots shown in Fig. 4c and j are plotted in Fig. 7a and b, respectively. One can see that the location and the number of filaments in each transverse plane can be calculated quite easily as soon as we define the filaments at a certain plasma level. For plasma channels shown in Fig. 7 we indicate the diameter at 1/e level of the maximum of each particular plasma spot. For example, in Fig. 7a in the small beam this electron density maximum varies from cm 3 to cm 3, while in Fig. 7b in the large beam this maximum value varies from cm 3 to cm 3. On average, the diameter of the plasma channels in the large beam is less than in the small beam. To make it obvious, we plot the average diameter of the plasma channels, which is calculated in each transverse beam section, as a function of the propagation distance. In Fig. 8 the average diameters

9 KOSAREVA et al. Controlling a bunch of multiple filaments by means of a beam diameter 119 FIGURE 7 Simulated plasma channels produced in a single realization of either the small (a) or the large(b) beam at the same propagation distance z = 0.25z d in diffraction lengths. The contours of equal electron density are shown on the logarithmic scale; each contour value is defined through the equation N e contour = N emax e const, where the value const changes from 3.5 to zero with the step 0.5, N emax = cm 3. The diameter of each plasma channel is defined at the level 1/e of the corresponding channel maximum density (and not of the overall maximum N emax ). Fluence distributions in the pulse that created these plasma channels are shown in (c) and(f) of Fig. 4, respectively are plotted versus distance shown both in diffraction length units in Fig. 8a and in absolute units in Fig. 8b. One can see in Fig. 8a that from the start of the filament bunch formation, the average diameter of the plasma channels in the small beam (the curve marked by empty squares) is larger than the diameter of the plasma channels in the large beam (the curve marked by empty circles). The overall tendency is the decrease in the average plasma channel diameter with distance in both the small and the large beams. Therefore, when the propagation distance is plotted versus distance in centimeters, the plasma diameter of the large beam appears to be larger than in the small beam in the vicinity of z 120 cm.thishappens because the distance z 120 cm is the starting point of filamentation for the large beam and corresponds to the peak of the filament bunch energy as shown in Fig. 6b. The decrease in the average plasma channel diameter is the consequence of the growing number of multiple filaments with distance (see Fig. 5). Indeed, at the beginning of the propagation the major part of the initial pulse energy converges towards several transverse perturbations, which form just a few parent filaments. The next generation of filaments comes mostly from the secondary perturbations arising from the interference of rings diverging from the parents. Multiplication of filaments leads to the decrease of the fraction of background energy from which they are seeded. Therefore, the corresponding plasma diameter decreases gradually with distance. The fluorescence signal depends on the total amount of electrons in the transverse section [48]. Therefore, we have calculated the linear density of the plasma D e, characterizing the total amount of free electrons in the transverse beam section created by the end of the pulse (see Eq. (8)). Two groups of curves shown in Fig. 9 correspond to the linear density calculated for the large and the small beams. Two bold curves marked by empty squares or circles show the average plasma density at a given propagation distance in the small and large beams, respectively. Curves marked by filled squares show the standard deviation from the average value in the small beam, while the two curves shown by filled circles show the standard deviation from the average value in the large beam. Thin dashed and solid curves show the plasma density in some chosen random realizations in the small and in the large beam, respectively. The dotted curve is the extrapolation of the linear density obtained in the small beam towards larger propagation distances. The total number of free electrons in each transverse beam section along the whole propagation distance studied is larger in the small beam (compare the two bold curves marked by empty circles and empty squares in Fig. 9). We extrapolated by the linear fit the dependence of the average density in the small beam on distance for z > 120 cm (dotted curve in Fig. 9). One can see that this extrapolation goes higher than nearly all random realizations of the plasma density in the large beam. The important issue is the stability of the generated plasma. The maximum relative standard deviation (the ratio of the standard deviation to the average value at the same z location) is 35% in the small beam and 63% in the large beam. Thus, by squeezing the beam diameter we not only obtained a higher amount of free electrons but also the stability of this amount from shot to shot is achieved. Both the increase in the total number of the generated electrons as well as the increase in the stability of the plasma density from shot to shot is in qualitative agreement with experimental results presented in Figs. 1 and 2. To make it more obvious, we plot in Fig. 10 the experimentally obtained average fluorescence signal from the small beam (empty triangles) together with the simulated average linear plasma density (empty squares). Both values are shown as functions of the propagation distance normalized to the corresponding diffraction length z d,whichis62 m for the experimental small beam and 3.8mfor the simulated small beam. Range correction of the signal is done through the multiplication of the signal obtained from a certain distance z by the square of this distance z. This procedure is needed since in the simulations we obtain the linear plasma density D e directly at the point of interest z, while in the simulations the range-resolved signal is always measured at z = 0.

10 120 Applied Physics B Lasers and Optics FIGURE 9 Simulated linear electron density calculated from Eq. (8) and averaged over 12 realizations for both small (empty squares)andlarge(empty circles) beams. For each average value the standard deviation is shown by the vertical solid line. Thin dashed lines for the small beam or thin solid lines for the large beam indicate the dependence of the linear electron density on distance in several random realizations of the initial laser pulse given by Eq. (1). The dotted curve is a linear fit to the average density in the small beam FIGURE 8 Average simulated plasma channel diameter and its standard deviation in the small (empty squares) orthelarge(empty circles) beam. Averaging is performed over all plasma channels found at a certain propagation distance throughout 12 realizations of the initial pulse given by Eq. (1). (a) Propagation distance is in diffraction lengths. (b) Propagation distance is in centimeters Good qualitative agreement between the calculated plasma density and the observed fluorescence signal is obtained, as seen in Fig. 10. Initial fast growth of both values after the start of the filamentation is replaced by a slower increase later in the propagation. The lack of fluctuations in the experimental signal in comparison with the simulation is explained by the much larger number of statistically processed random realizations (12 in the simulations and 200 in the experiment). Earlier start of the filamentation in the simulations than in the experiment is associated with the Gaussian transverse shape of random perturbations. At the same time, in the experimental beam a typical transverse shape of perturbations is far from Gaussian and, therefore, in spite of the same typical size and the ratio P perturbation /P cr, the beam self-focuses at a larger distance than in the simulations. We find that beam squeezing leads to the increase in the diameter of plasma channels and in the total number of electrons as well as the stability of this number from one laser shot to another. The result of this is the stronger and more stable fluorescence signal in comparison with the large beam. FIGURE 10 Simulated average linear electron density versus propagation distance in the small beam (empty squares, right vertical axis). Experimentally obtained backscattered nonlinear fluorescence signal in the small beam corrected by the factor of z 2 (empty triangles, left vertical axis). Diffraction length is 3.8 m for the simulated beam, a 0 = 0.7mm(1.16-mm diameter at FWHM) and 61.6 m for the experimental small beam (8-mm diameter at FWHM) In this work we have performed multiple-filamentation control with the beam diameter in order to increase the intensity and stability in the backscattered fluorescence signal registered from the plasma. The side effect associated with the use of the squeezed beam is the earlier onset of the filamentation (compare Fig. 2 a and b in the case of the experiment, and the two curves in Fig. 5 in the case of the simulations). However, if one needs to delay in the propagation the beginning of plasma formation, an initial negative chirp of the pulse can be employed [31]. 6 Summary and conclusion By varying the initial beam diameter, we demonstrated controlled multiple filaments in air. When the beam

11 KOSAREVA et al. Controlling a bunch of multiple filaments by means of a beam diameter 121 diameter is decreased from 25 mm to 8mmby keeping the input laser pulse energy constant, the measured backscattered nonlinear fluorescence signal in the experiment from nitrogen molecules is increased by three orders of magnitude. Additionally, the signal becomes much more stabilized: its standard deviation decreases from 300% of the average value in the case of the large beam to 50% of the average value in the case of the small beam. Numerical simulations performed for a set of random initial pulse distributions imitating laser shots in both the largeand the small-beam cases have revealed the following effects, which are associated with beam squeezing. In the small beam there is a faster growth of multiple filaments with propagation distance, a larger average diameter of plasma channels, and a larger overall amount of electrons in the transverse beam section. In addition, statistical processing of the simulation results shows that in the case of the small beam the number of filaments, the diameter of the plasma, and the amount of electrons are more predictable from one laser shot to another, since the ratio of the standard deviation to the corresponding average value is decreased in comparison with the large-beam case. Larger and more stable plasma density obtained in the small beam in the simulations is in good agreement with experimental results showing a larger and more stable fluorescence signal in the small beam. The reason for the increased value of the signal and for the stability increase is due to the more effective usage of the background energy in the small beam. Indeed, 1.4 times beam squeezing leads to the increase in the filament bunch energy from 4% to 20% of the initial pulse energy. Finally, we have shown the possibility of increasing the total number of electrons and the fluorescence signal by means of initial beam size. Delayed initiation of multiple filaments and plasma channels may be introduced with the help of the initial pulse chirp. Further experiments and numerical simulations should be performed on the optimization of the amount of plasma and the fluorescence signal by simultaneously varying initial beam size and initial pulse chirp. ACKNOWLEDGEMENTS We would like to acknowledge the technical support of M. Martin. This work was supported in part by NSERC, DRDC-Valcartier, Canada Research Chairs, CIPI, and FQRNT. O.G. Kosareva, N.A. Panov, and V.P. Kandidov are grateful for the support of the European Research Office of the US Army under Contract No. N P-6051 and the Russian Foundation for Basic Research (Grant No ). N.A. Panov is grateful for the support from the Dynasty Foundation. V.P. Kandidov, O.G. Kosareva, and S.L. Chin are grateful for the support of the NATO Linkage (Grant No. PST.CLG ). REFERENCES 1 A. Braun, G. Korn, X. Liu, D. Du, G. Mourou, Opt. Lett. 20, 73 (1995) 2 E.T.J. 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