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PHYSICAL REVIEW A 92, 033857 (2015)

Filamentation and supercontinuum generation in solid-state dielectric media with picosecond laser pulses J. Galinis,1 G. Tamoˇsauskas,1 I. Graˇzuleviˇci¯ut˙e,1 E. Keblyt˙e,1 V. Jukna,2,3 and A. Dubietis1,* 1

Department of Quantum Electronics, Vilnius University, Saul˙etekio Avenue 9, Building 3, LT-10222 Vilnius, Lithuania 2 Centre de Physique Théorique, CNRS, Ecole Polytechnique, F-91128 Palaiseau, France 3 Laboratoire d’Optique Appliquée, ENSTA ParisTech, Ecole Polytechnique, CNRS, F-91762 Palaiseau, France (Received 6 May 2015; published 30 September 2015) Filamentation and supercontinuum generation with 1.3-ps, 1055-nm laser pulses in YAG crystal is investigated numerically and experimentally. Numerical simulations based on solving the unidirectional nonparaxial propagation equation uncover that the self-focusing dynamics of a picosecond laser pulse markedly differs from that observed in a femtosecond filamentation regime. We show that spatiotemporal transformation of the picosecond pulse is governed by the free electron plasma, which defocuses and absorbs its rear part, resulting in the formation of several subpulses of femtosecond duration, which thereafter undergo peculiar spatiotemporal dynamics and have different contributions to spectral superbroadening. The numerical findings are confirmed experimentally by measuring the spatiotemporal intensity profiles of the wave packet at various stages of propagation where relevant events of the spectral broadening occur. DOI: 10.1103/PhysRevA.92.033857

PACS number(s): 42.65.Jx, 42.65.Sf

I. INTRODUCTION

Self-focusing of ultrashort laser pulses in a transparent dielectric medium gives rise to a phenomenon, termed filamentation, which dramatically modifies the spatial, temporal, and spectral properties of the incident radiation. The physics of filamentation involves a wealth of nonlinear effects which simultaneously take place in space and time: self-focusing, self-phase modulation, pulse splitting and/or compression, pulse-front steepening and generation of optical shocks, multiphoton absorption, and excitation of free electron plasma. Complex interplay between these nonlinear effects along with diffraction and group velocity dispersion (GVD) leads to the occurrence of a “dynamic structure with an intense core, that is able to propagate over extended distances much larger than the typical diffraction length while keeping a narrow beam size without the help of any external guiding mechanism,” termed a light filament [1]. One of the most spectacular and visually perceptible effects produced by the filamentation is white light or, more generally, supercontinuum (SC) generation, offering a simple and very efficient method for production of broadband radiation with high spatial and temporal coherence, which serves as a useful tool for diverse applications in ultrafast spectroscopy and contemporary nonlinear optics. In bulk solid-state dielectric media the practical scheme for SC generation is as compact as it is efficient: under suitable low-numerical-aperture focusing conditions filamentation and spectral superbroadening are achieved in a few millimeters of propagation without the onset of optical damage of material (see, e.g., [2]). Thanks to advances in our understanding of filamentation phenomena, the emerging dynamical picture of SC generation with femtosecond laser pulses is fairly well understood. In the regime of a normal GVD, which spans from the ultraviolet to the near-infrared in most solid-state dielectric media, the SC generation arises from the pulse splitting at the nonlinear

*

[email protected]

1050-2947/2015/92(3)/033857(5)

focus [3–5]. Blue-shifted spectral broadening is associated with a steep front, which develops at the tail of the trailing subpulse, whereas red-shifted spectral broadening is favored from the leading front of the leading subpulse [6] (see also [7] for experimental proof). In the entire spatiotemporal domain, pulse splitting and SC generation are described in terms of the formation of two X waves with different carrier frequencies and traveling at different group velocities [8]. While blue-shifted spectral broadening is generally defined by the bandgap and dispersive properties of a nonlinear medium [9–11] and the X-wave phase-matching condition, in particular [12,13], redshifted spectral broadening could be enhanced by choosing a looser focusing of the input beam [14]. Recent studies uncovered the whole evolution cycle of femtosecond filaments, which fundamentally connects self-focusing and defocusing cycles with spatiotemporal dynamics and relevant events in spectral broadening [15]. It is important to mention that at typical filamentation intensities in the normal-GVD regime, spatiotemporal transformations occurring within a femtosecond filament are driven by multiphoton absorption and the GVD, and the effect of free electron plasma is negligible, as verified by the numerical simulations and experiments [15–18]. Recent progress on novel solid-state laser systems operating around 1 μm and development of compact ultrabroadband tabletop optical parametric chirped pulse amplification (OPCPA)–based systems has prompted experimental investigations of SC generation in solid-state dielectric media using longer, subpicosecond and picosecond pump pulses [19–24]. However, despite the current interest, filamentation and SC generation with picosecond laser pulses in solid-state media are still very poorly investigated topics, especially concerning the spatiotemporal evolution and related dynamics of spectral superbroadening. In this paper we present an extensive numerical and experimental study on filamentation and SC generation in YAG crystal using 1.3-ps, 1055-nm input pulses. Our results demonstrate that a self-focusing picosecond laser pulse undergoes peculiar spatiotemporal dynamics, which is governed by the free electron plasma and therefore is markedly

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different from the femtosecond laser pulse dynamics, which is driven by multiphoton absorption. II. NUMERICAL MODEL

The spatiotemporal and spectral evolution of the picosecond laser pulse in YAG crystal, which combines a high nonlinearity and a high optical damage threshold [25], was numerically simulated by the unidirectional nonparaxial propagation equation for the spectral components of the laser pulse envelope ˆ E(ω,k ⊥ ,z) [26], ∂ Eˆ 2 = i( k 2 (ω) − k⊥ − k0 − k0 ω)Eˆ ∂z Jˆ ω −1 ˆ P +i , + i 2n(ω)c 0 ω

(1) (2)

where k(ω) denotes the dispersion relation of the medium, k⊥ is the transverse wave vector, and n(ω) is the refractive index of the medium calculated from a Sellmeier equation [27], for which k0 ≡ k(ω0 ) and k0 ≡ dk/dω|ω0 were evaluated at the central laser frequency. The nonlinear polarization and current source terms P (t,r,z) and J (t,r,z) were computed in the space-time domain, assuming revolution symmetry, (3) 0−1 P = 2n0 n2 |E|2 E, W Ug ρ 1 − E, (4) 0−1 J = n0 c σ (1 + iω0 τc )ρ + |E|2 ρnt where n0 = 1.74 and n2 = 6.5 × 10−16 cm2 /W are the linear and nonlinear refractive indexes, respectively, σ = 5.6 × 10−22 m2 is the cross section for inverse Bremsstrahlung, τc = 0.1 fs is the effective collision time, ρ is the density of electrons generated during propagation, and ρnt = 2.1 × 1022 cm−3 is the density of neutral molecules. W is the rate of optical-field ionization calculated from Keldysh’s theory [28] with electronhole mass ratio m∗ = 1 and bandgap Ug = 6.5 eV. The density of electrons in the conduction band was obtained by a simple rate equation in the form ρ ∂ρ σ ρ 2 − = W (ρnt − ρ) + |E| ρ 1 − , (5) ∂t Ug ρnt τrec where the terms on the right-hand side describe photoionization, ionization by avalanche and recombination, respectively, where τrec = 1 ps is the free electron recombination time. The pulse parameters, such as the input beam diameter, pulse width, and energy, were taken to match the experimental ones, as described below. III. EXPERIMENTAL SETUP

The experiment was performed using an amplified Nd:glass laser system (Twinkle; Light Conversion Ltd.), which delivered 1.3-ps, 1055-nm pulses with an energy of up to 5 mJ at a 10-Hz repetition rate. The laser output beam was suitably attenuated, reduced in size, spatially filtered, and thereafter focused with an f = +42 mm lens onto a 25-μm-FWHM spot size located at the input face of the YAG crystal. The spatiotemporal intensity distributions of the emerging wave packet

were measured using a three-dimensional imaging technique. More specifically, the spatially resolved cross-correlation function was recorded by the sum-frequency generation in thin (200-μm) BBO crystal using a short (165-fs) gating pulse at 1055 nm, which was produced by the nonlinear secondharmonic compression and its subsequent down-conversion back to the fundamental wavelength (see [29] for details). The cross-correlation signal was imaged onto the CCD camera (Grasshopper 2; Point Grey), and the entire spatiotemporal distribution of the resulting wave packet was reconstructed by merging a series of consecutive cross-correlation images recorded by changing the time delay between the object and the gating pulses in a 65-fs step. High dynamic range spectral measurements were performed by imaging the output face of the crystal onto the fiber tip with 600 μm core diameter, in succession connected to two calibrated fiber spectrometers (AvaSpec-2048 and AvaSpecNIR256-2.5, both from Avantes), altogether covering 0.3 − 2.5 μm effective spectral range and operating in a single-shot regime. The SC spectra were averaged over 100 laser shots and corrected according to sensitivity functions of the spectrometers and transmission of the color filters and partially reflecting mirror used for the suppression of the pump wavelength. The spatiotemporal distributions of the wave packet and corresponding spectra were measured at relevant stages of its nonlinear dynamics, using three YAG crystals of 3, 6 and 10 mm length. The input energy of 8.8μJ was chosen so as to excite stable and reproducible SC spectrum at the output of the longest (10 mm) crystal, without of the onset of the optical damage under either single or multiple shot exposure. IV. RESULTS AND DISCUSSION

Figure 1 summarizes the numerically simulated propagation dynamics of a 1.3-ps laser pulse at 1055 nm with a 25-μm FWHM diameter and 8.8 μJ energy (input intensity, 600 GW/cm2 ) in a YAG crystal. The temporal evolution of the pulse shown in Fig. 1(a), which demonstrates that the input picosecond pulse transforms into a sequence of intense femtosecond fragments (subpulses), which exhibit different propagation properties and make different contributions to spectral broadening and SC generation, as shown in Fig. 1(b). During the first millimeter of propagation, the input Gaussian wave packet, as shown in Fig. 2(a), quickly shrinks in dimensions due to self-focusing and reaches a nonlinear focus at approximately z = 1 mm, where the maximum density of free electron plasma (ρ = 7.3 × 1020 cm−3 ) is generated. The plasma channel, whose highest density peaks on the propagation axis, defocuses and absorbs the rear part of the pulse, inducing a considerable pulse shortening and shifting the intensity peak to the front of the pulse, as shown in Fig. 2(b). As a large fraction of the pulse energy is completely pushed out of the propagation axis, a peculiar ring-shaped intensity distribution forms at the tail, as shown in Fig. 2(c). At the same time the leading part of the pulse experiences considerable plasma-induced self-shortening, down to 70 fs at z = 1.2 mm, and reshapes into a distinct subpulse (the first sub-pulse), which continues generation of the plasma and which, in turn, defocuses and absorbs its trailing edge, rapidly pushing the intensity

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FIG. 3. (Color online) Numerically simulated spatiotemporal intensity distributions of the wave packet at (a) z = 3 mm, (b) z = 6 mm, and (c) z = 10 mm. Subpulses are indicated by arrows and labels. (d–f) Numerically simulated respective cross-correlation functions generated with a 165-fs Gaussian pulse. (g–i) Experimentally measured spatiotemporal intensity distributions of the wave packet at the output of 3-, 6-, and 10-mm-long YAG samples. FIG. 1. (Color online) Numerically simulated results of propagation of a 1.3-ps, 1055-nm, 8.8-μJ laser pulse in a YAG crystal: (a) temporal evolution, (b) spectral broadening, (c) plasma density, and (d) beam radius (FWHM). z is the propagation distance.

peak toward the forefront. Interestingly, despite this dramatic reshaping, only a moderate spectral broadening around the carrier wavelength at this propagation stage is observed. As soon as the intensity of the first subpulse starts to decrease, plasma generation decreases as well, and the ring-shaped rear part of the wave packet rebuilds a temporal structure on the beam axis, which experiences conditional

FIG. 2. (Color online) Normalized spatiotemporal intensity distributions of the wave packet illustrating the early stage of selffocusing: (a) at the input, z = 0 mm; (b) z = 1.0 mm; (c) z = 1.2 mm. Note the logarithmic intensity scale used to reveal fine spatiotemporal features.

splitting into the second and third subpulses at approximately z = 2.5 mm, as shown in Fig. 3(a). Please also refer to [30], which provides a complete set of spatiotemporal dynamics versus propagation length. Note that conditional pulse splitting at this stage does not induce any (at least, detectable) spectral broadening, as shown in Fig. 1(b). The second and third subpulses afterward experience rather different propagation dynamics: as the second subpulse (the leading fragment) never gains an intensity sufficient to induce an apparent spectral broadening; in contrast, the third subpulse (the trailing fragment) self-focuses at z = 5.6 mm, as evident from the occurrence of a distinct secondary plasma peak with ρ = 4.7 × 1019 cm−3 , which correlates with the reduction of the beam diameter, shown in Figs. 1(c) and (d), respectively. At its nonlinear focus the third subpulse splits into two fragments, labeled the fourth (leading) and fifth (trailing) subpulses, as shown in Fig. 3(b). The pulse splitting gives rise to a considerable blue-shifted spectral broadening with the occurrence of a distinct blue peak centered at 540 nm and a cutoff at 450 nm (at the 10−7 intensity level). With further propagation the blue peak broadens toward the carrier wavelength, yet maintaining a constant cutoff wavelength of 450 nm, which is defined by the bandgap and chromatic dispersion of the nonlinear medium [9–11]. Interestingly, the splitting event induces only a weak red-shifted spectral broadening due to the fast decrease in the intensity of the leading fragment (fourth subpulse), as the associated wave packet takes a bow-shaped spatiotemporal intensity distribution, which experiences rapid

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FIG. 5. (Color online) Temporal structure of the on-axis SC radiation and its spectral analysis, showing the contributions of subpulses to the blue-shifted and red-shifted spectral broadenings: (a) numerically computed temporal profile; (b) experimentally measured cross-correlation function with a 165-fs pulse. Dashed black curves show the multipeaked temporal structure that corresponds to the full SC spectrum, while right-hand solid (blue) and left-hand solid (red) curves show the peaks after spectral filtering in the visible (λ < 0.7 μm) and in the infrared (λ > 1.4 μm) spectral ranges, respectively. See text for details. FIG. 4. (Color online) SC spectra after propagation in 3-, 6-, and 10-mm-long YAG samples: (a) numerically simulated, as-taken lineups from Fig. 1(b); (b) experimentally measured.

diffraction spreading. Note that the first, second, and fifth subpulses exhibit quasistationary propagation, maintaining a short (few tens of femtoseconds) pulse duration on the beam axis as they develop peculiar, X-shaped spatiotemporal profiles [see Fig. 3(c)]. Moreover, the first and second subpulses experience gradual self-compression; self-compression of the first subpulse results in progressive red-shifted spectral broadening, whereas the second subpulse undergoes yet another, more subtle conditional splitting at z = 7.2 mm, producing a weak subpulse (not labeled) located at the center of the overall temporal structure, which is not visible in Fig. 1(a) due to its very low intensity. Figures 3(d)–3(f) present the corresponding numerically simulated spatiotemporal crosscorrelation functions computed with a 165-fs Gaussian pulse, to be compared with Figs. 3(g)–3(i), which present the results of experimental measurements. The experimental data captured the relevant events of spatiotemporal reshaping, as well as the complex spatiotemporal shapes of the subpulses, in great detail. The only difference is the temporal spread of the split subpulses, which is larger in the numerical data and could be attributed to limitations of our numerical model, which does not account for transient plasma-generation-related effects (e.g., exciton generation) and their impact on the complex propagation dynamics, which may be of importance in the case of long (picosecond) input pulses. The relationship between spatiotemporal transformations of the wave packet and spectral broadening was verified also experimentally by spectral measurements at the output of 3-, 6-, and 10-mm-long YAG crystals. A comparison between the simulated and the measured SC spectra is provided in Fig. 4, showing a fair agreement between the numerical and the experimental data. In particular, experimental measurements capture the relevant events in spectral broadening: spectral broadening around the carrier wavelength at z = 3 mm due

to plasma-induced self-shortening of the first subpulse, the occurrence of the isolated blue peak at z = 6 mm due to splitting of the third subpulse, and thereafter broadening of the blue peak toward the carrier wavelength, along with strong red-shifted spectral broadening to above 2 μm at z = 10 mm. Finally, concerning the multipeaked temporal structure of the pulse, we performed a quantitative test, which allowed us to identify the temporal sources contributing to the blue-shifted and red-shifted spectral broadenings, i.e., to associate the particular subpulses with the respective spectral content of the SC. Figure 5(a) shows the numerical data which compare the onaxis temporal structure of the pulse at z = 10 mm, as computed for the entire spectral content and for the numerically filtered spectra in the visible (λ < 0.7 μm) and in the infrared (λ > 1.4 μm) spectral ranges. Figure 5(b) shows the experimental data, as obtained from the cross-correlation measurements, applying appropriate spectral filtering of the sum frequency signal using short- and long-pass color glass filters. These results suggest that the first subpulse is responsible for the generation of a strongly red-shifted SC spectrum, whose progressive broadening with propagation is favored from continuous self-compression of the pulse, as could be verified from the numerical data in Figs. 3(a)–3(c) and, in more detail, in [30]. More specifically, the numerical simulation yields that the first subpulse becomes as short as 15 fs at z = 10 mm; its extremely short duration is conveniently confirmed by the experimentally measured width of the cross-correlation function (170 fs), which almost equals the duration of the gating pulse itself. Note that the expected contribution of the fourth subpulse (the leading fragment after splitting of the third subpulse) to the red-shifted spectral broadening is apparently weak. Although its peak at positive times could be detected at shorter propagation distances, e.g., at z = 6 mm (not shown here), its intensity rapidly decreases with propagation due to fast diffraction spreading in space, and at z = 10 mm the fourth subpulse becomes virtually absent in the numerical and experimental data. The blue-shifted portion of the SC spectrum

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is clearly associated with the intense fifth subpulse (the trailing fragment after splitting of the third subpulse). Finally, these results also show that the second subpulse does not contribute to large either red-shifted or blue-shifted spectral broadenings, and its spectral components are located in the vicinity of the carrier wavelength (1055 nm).


In conclusion, we have presented a numerical and experimental study of filamentation and SC generation in YAG crystal with a 1.3-ps, 1055-nm laser pulses. Our results highlight a good agreement between the numerical and the experimental data and validate a peculiar filamentation and SC generation scenario of picosecond laser pulses, which is markedly different compared to the case of femtosecond-pulse filamentation. While the femtosecond wave packet collapses as a whole, its spatiotemporal transformation is mainly driven by multiphoton absorption, and SC generation occurs due to pulse splitting at the nonlinear focus (see, e.g., [6,14,15]); in contrast, the picosecond wave packet undergoes a series of free electron plasma-driven transformations in space and

time, which occur at various stages of propagation, resulting in multipeaked temporal and, more generally, spatiotemporal intensity distributions at the output of the nonlinear medium. The uncovered peculiar filamentation scenario of picosecond laser pulses was fairly confirmed by the experimental measurements, capturing the occurrence and evolution of a complex multipulse spatiotemporal structure at the output of YAG crystal at relevant stages of nonlinear propagation, as recorded by a three-dimensional imaging technique. Despite the multipulse temporal structure of the SC radiation, which covers the wavelength range from 480 nm to more than 2 μm, our findings suggest that the red-shifted and blue-shifted spectral broadenings are associated just with the propagation of two distinct subpulses, which are well separated in time. In particular, our results could be readily applicable to a variety of newly developed picosecond laser sources, operating in the near-infrared spectral range and dedicated to the development of simple, efficient, and inexpensive tabletop OPCPA-based laser systems (see, e.g., [31]) providing a broadband seeding source, which markedly simplifies the overall setup, excluding the need for employing an optically synchronized broadband laser oscillator source.

[1] A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47 (2007). [2] J. B. Ashcom, R. R. Gattass, C. B. Schaffer, and E. Mazur, J. Opt. Soc. Am. B 23, 2317 (2006). [3] J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996). [4] S. A. Diddams, H. K. Eaton, A. A. Zozulya, and T. S. Clement, Opt. Lett. 23, 379 (1998). [5] A. A. Zozulya, S. A. Diddams, A. G. Van Engen, and T. S. Clement, Phys. Rev. Lett. 82, 1430 (1999). [6] A. L. Gaeta, Phys. Rev. Lett. 84, 3582 (2000). [7] A. L. Gaeta, Top. Appl. Phys. 114, 399 (2009). [8] D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, Phys. Rev. Lett. 96, 193901 (2006). [9] A. Brodeur and S. L. Chin, J. Opt. Soc. Am. B 16, 637 (1999). [10] M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, Phys. Rev. Lett. 91, 043905 (2003). [11] M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, Appl. Phys. B 77, 185 (2003). [12] D. Faccio, A. Averchi, A. Couairon, A. Dubietis, R. Piskarskas, A. Matijosius, F. Bragheri, M. A. Porras, A. Piskarskas, and P. Di Trapani, Phys. Rev. E 74, 047603 (2006). [13] D. Faccio, A. Averchi, A. Couairon, M. Kolesik, J. V. Moloney, A. Dubietis, G. Tamosauskas, P. Polesana, A. Piskarskas, and P. Di Trapani, Opt. Express 15, 13077 (2007). [14] V. Jukna, J. Galinis, G. Tamosauskas, D. Majus, and A. Dubietis, Appl. Phys. B 116, 477 (2014). [15] A. Jarnac, G. Tamosauskas, D. Majus, A. Houard, A. Mysyrowicz, A. Couairon, and A. Dubietis, Phys. Rev. A 89, 033809 (2014). [16] A. Dubietis, E. Gaiˇzauskas, G. Tamoˇsauskas, and P. Di Trapani, Phys. Rev. Lett. 92, 253903 (2004). [17] S. Skupin and L. Bergé, Physica D 220, 14 (2006). [18] E. Gaiˇzauskas, A. Dubietis, V. Kudriaˇsov, V. Sirutkaitis, A. Couairon, D. Faccio, and P. Di Trapani, Top. Appl. Phys. 114, 457 (2009).

[19] M. Bradler, P. Baum, and E. Riedle, Presented at the International Conference on Ultrafast Phenomena (Optical Society of America, Washington, D.C., 2010), paper ME25. [20] M. Bradler and E. Riedle, Presented at Advanced Solid-State Photonics (Optical Society of America, Washington, D.C., 2011), paper AMD4. [21] M. Schulz, R. Riedel, A. Willner, T. Mans, C. Schnitzler, P. Russbueldt, J. Dolkemeyer, E. Seise, T. Gottschall, S. Hdrich, S. Duesterer, H. Schlarb, J. Feldhaus, J. Limpert, B. Faatz, A. Tünnermann, J. Rossbach, M. Drescher, and F. Tavella, Opt. Lett. 36, 2456 (2011). [22] R. Riedel, A. Stephanides, M. J. Prandolini, B. Gronloh, B. Jungbluth, T. Mans, and F. Tavella, Opt. Lett. 39, 1422 (2014). [23] A.-L. Calendron, H. C ¸ ankaya, G. Cirmi, and F. X. Kärtner, Opt. Express 23, 13866 (2015). [24] I. Graˇzuleviˇci¯ut˙e, M. Skeivyt˙e, E. Keblyt˙e, J. Galinis, G. Tamoˇsauskas, and A. Dubietis, Lith. J. Phys. 55, 110 (2015). [25] S. Xu, J. Qiu, T. Jia, C. Li, H. Sun, and Z. Xu, Opt. Commun. 274, 163 (2007). [26] A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. RamirezGongora, and M. Kolesik, Eur. Phys. J. Special Topics 199, 5 (2011). [27] M. J. Weber, Handbook of Optical Materials (CRC Press, Boca Raton, FL, 2003). [28] L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965). [29] D. Majus, O. Jedrkiewicz, M. Molteni, P. Ragazzi, P. Di Trapani, and G. Tamoˇsauskas, Eur. Phys. J. Special Topics 199, 77 (2011). [30] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevA.92.033857 for detailed numerically simulated spatiotemporal dynamics of self-focusing picosecond laser pulse in YAG crystal. [31] A. Vaupel, N. Bodner, B. Webb, L. Shah, and M. Richardson, Opt. Eng. 53, 051507 (2014).

V. CONCLUSIONS

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