Showing papers on "Femtosecond pulse shaping published in 2013"
TL;DR: In this article, a review of the evolution of the self-focus phenomenon in light beams is presented, and the current status of this rapidly growing area of nonlinear optics and laser physics is discussed.
Abstract: 2012 marked the 50th anniversary of the first published prediction of the self-focusing phenomenon in light beams. The recent revived interest in the subject is due to advances in high-power femtosecond laser technology and due to the possibility they provided of creating extended filaments of high light field intensity in gases and condensed media. This review shows in retrospect how our understanding of the self-action of light evolved from the self-focusing of laser beams in the 1960s to the filamentation of femtosecond laser pulses at present. We also describe the current status of this rapidly growing area of nonlinear optics and laser physics. Finally, we discuss, in general terms, what the phenomena of laser beam self-focusing and laser pulse filamentation have in common and how they differ.
145 citations
TL;DR: A 2.0 PW femtosecond laser system at 800 nm based on the scheme of chirped pulse amplification using Ti:sapphire crystals is reported on, which is the highest peak power ever achieved from a femTosecond Laser system.
Abstract: We report on a 2.0 PW femtosecond laser system at 800 nm based on the scheme of chirped pulse amplification using Ti:sapphire crystals, which is the highest peak power ever achieved from a femtosecond laser system. Combining the index-matching cladding technique and the precise control of the time delay between the input seed pulse and pump pulses, the parasitic lasing in the final booster amplifier is effectively suppressed at the pump energy of 140 J at 527 nm. The maximum output energy from the final amplifier is 72.6 J, corresponding to a conversion efficiency of 47.2% from the pump energy to the output laser energy. The measured spectral width of the amplified output pulse from the final amplifier is 60.8 nm for the full width at half-maximum (FWHM) by controlling the spectral evolution in the amplifier chain, and the recompressed pulse duration is 26.0 fs. The technology of cross-polarized wave (XPW) is applied in a broadband front-end, and the pulse contrast is improved to ~1.5 × 10−11 (−100 ps before the main pulse) which is measured at 83 TW power level with a repetition rate of 5 HZ.
120 citations
TL;DR: Two experiments confirming that hypocycloid Kagome-type hollow-core photonic crystal fibers (HC-PCFs) are excellent candidates for beam delivery of MW peak powers and pulse compression down to the sub-50 fs regime are presented.
Abstract: We present two experiments confirming that hypocycloid Kagome-type hollow-core photonic crystal fibers (HC-PCFs) are excellent candidates for beam delivery of MW peak powers and pulse compression down to the sub-50 fs regime. We demonstrate temporal pulse compression of a 1030-nm Yb:YAG thin disk laser providing 860 fs, 1.9 µJ pulses at 3.9 MHz. Using a single-pass grating pulse compressor, we obtained a pulse duration of 48 fs (FWHM), a spectral bandwidth of 58 nm, and an average output power of 4.2 W with an overall power efficiency into the final polarized compressed pulse of 56%. The pulse energy was 1.1 µJ. This corresponds to a peak power of more than 10 MW and a compression factor of 18 taking into account the exact temporal pulse profile measured with a SHG FROG. The compressed pulses were close to the transform limit of 44 fs. Moreover, we present transmission of up to 97 µJ pulses at 10.5 ps through 10-cm long fiber, corresponding to more than twice the critical peak power for self-focusing in silica.
102 citations
TL;DR: In this paper, a spectral domain modulation (SDFM) based angle-to-wavelength pulse shaper was proposed to overcome the unwanted background arising from other pump-probe modalities.
Abstract: The stimulated Raman scattering signal is often accompanied by unwanted background arising from other pump-probe modalities. We demonstrate an approach to overcome this challenge based on spectral domain modulation, enabled by a compact, cost-effective angle-to-wavelength pulse shaper. The pulse shaper switches between two spectrally narrow windows, which are cut out of a broadband femtosecond pulse and selected for on- and off- Raman resonance excitation, at 2.1 MHz frequency for detection of stimulated Raman scattering signal. Such spectral modulation reduced the unwanted pump-probe signals by up to 20 times and enabled stimulated Raman scattering imaging of molecules in a pigmented environment.
95 citations
TL;DR: In this paper, a nano-scale photoelectron source optimized towards ultrashort pulse durations and well-suited for time-resolved diffraction experiments is presented.
Abstract: We present a nano-scale photoelectron source, optimized towards ultrashort pulse durations and well-suited for time-resolved diffraction experiments. A tungsten tip, mounted in a suppressor-extractor electrode configuration, allows the generation of 30 keV electron pulses with an estimated pulse duration of 37 fs at the gun exit. We infer the pulse duration from particle tracking simulations, which are in excellent agreement with experimental measurements of the electron-optical properties of the source. We furthermore demonstrate femtosecond laser-triggered operation. Besides the short electron pulse duration, a tip-based source is expected to feature a large transverse coherence as well as a nanometric emittance.
92 citations
TL;DR: This work implements both chirped pulse amplification and divided pulse amplification in the same femtosecond fiber amplifier setup, which allows an equivalent stretched pulse duration of 2.4 ns in a compact tabletop system.
Abstract: We implement both chirped pulse amplification and divided pulse amplification in the same femtosecond fiber amplifier setup. This scheme allows an equivalent stretched pulse duration of 2.4 ns in a compact tabletop system. The generation of 77 W of compressed average power at 4.8 MHz, together with 320 fs and 430 μJ pulses at a repetition rate of 96 kHz, is demonstrated using a distributed mode-filtering, rod-type, ytterbium-doped fiber. Limitations in the temporal recombining efficiency due to gain saturation inside the fiber amplifier are identified.
83 citations
TL;DR: In this article, the structure and evolution of ultrashort shock waves generated by femtosecond laser pulses in single-crystal nickel films are investigated by molecular dynamics simulations, and an elastic response is maintained for shock amplitudes exceeding the Hugoniot elastic limit determined from simulations of steady shock waves.
Abstract: The structure and evolution of ultrashort shock waves generated by femtosecond laser pulses in single-crystal nickel films are investigated by molecular dynamics simulations. Ultrafast laser heating is isochoric, leading to pressurization of a 100-nm-thick layer below the irradiated surface. For low-intensity laser pulses, the highly pressurized subsurface layer breaks into a single elastic shock wave having a combined loading and unloading time $\ensuremath{\approx}$10--20 ps. Owing to the time-dependent nature of elastic-plastic transformations, an elastic response is maintained for shock amplitudes exceeding the Hugoniot elastic limit determined from simulations of steady shock waves. However, for high-intensity laser pulses (absorbed laser fluence $g$$0.6\phantom{\rule{0.28em}{0ex}}\mathrm{J}/{\mathrm{cm}}^{2}),$ both elastic and plastic shock waves are formed independently from the initial high-pressure state. Acoustic pulses emitted by the plastic front support the motion of the elastic precursor resulting in a fluence-independent elastic amplitude; whereas the unsupported plastic front undergoes significant attenuation during propagation and may fully decay within the metal film.
75 citations
TL;DR: It is demonstrated that a given field envelope produces a unique and unequivocal chirp-scan map and that, under some asymptotic assumptions, both the spectral amplitude and phase of the measured pulse can be retrieved analytically from only two measurements.
Abstract: We investigate a variant of the d-scan technique, an intuitive pulse characterization method for retrieving the spectral phase of ultrashort laser pulses. In this variant a ramp of quadratic spectral phases is applied to the input pulses and the second harmonic spectra of the resulting pulses are measured for each chirp value. We demonstrate that a given field envelope produces a unique and unequivocal chirp-scan map and that, under some asymptotic assumptions, both the spectral amplitude and phase of the measured pulse can be retrieved analytically from only two measurements. An iterative algorithm can exploit the redundancy of the information contained in the chirp-scan map to discard experimental noise, artifacts, calibration errors and improve the reconstruction of both the spectral intensity and phase. This technique is compared to two reference characterization techniques (FROG and SRSI). Finally, we perform d-scan measurements with a simple grating-pair compressor.
75 citations
TL;DR: In this article, the authors demonstrate temporal pulse compression in gas-filled kagome hollow-core photonic crystal fiber (PCF) using two different approaches: fiber-mirror compression based on self-phase modulation under normal dispersion, and soliton effect self-compression under anomalous dispersion with a decreasing pressure gradient.
Abstract: We demonstrate temporal pulse compression in gas-filled kagome hollow-core photonic crystal fiber (PCF) using two different approaches: fiber-mirror compression based on self-phase modulation under normal dispersion, and soliton effect self-compression under anomalous dispersion with a decreasing pressure gradient. In the first, efficient compression to near-transform-limited pulses from 103 to 10.6 fs was achieved at output energies of 10.3 μJ. In the second, compression from 24 to 6.8 fs was achieved at output energies of 6.6 μJ, also with near-transform-limited pulse shapes. The results illustrate the potential of kagome-PCF for postprocessing the output of fiber lasers. We also show that, using a negative pressure gradient, ultrashort pulses can be delivered directly into vacuum.
73 citations
TL;DR: The results demonstrated that the DSR phenomenon could exist in Yb-doped fiber lasers, which could be used to achieve wave-breaking-free, ultrahigh-energy pulse.
Abstract: We reported on the dissipative soliton resonance (DSR) phenomenon in a mode-locked Yb-doped fiber laser by using the nonlinear polarization rotation technique. It was found that the multi-pulse oscillation under high pump power could be circumvented by properly adjusting the polarization controllers, namely, the wave-breaking-free rectangular pulse in DSR region was achieved. As the DSR signature, the pulse duration varied from 8.8 ps to 22.92 ns with the increasing pump power. Correspondingly, the maximum pulse energy was 3.24 nJ. The results demonstrated that the DSR phenomenon could exist in Yb-doped fiber lasers, which could be used to achieve wave-breaking-free, ultrahigh-energy pulse.
69 citations
TL;DR: In this article, the formation of laser-induced periodic surface structures upon irradiation of titanium, silicon, and fused silica with multiple irradiation sequences consisting of parallel polarized Ti:sapphire femtosecond laser pulse pairs (pulse duration 50-150fs, central wavelength ∼800-nm) was studied experimentally.
TL;DR: In this article, the authors presented a numerical model with which they have obtained excellent quantitative agreement with two recent experiments in the femtosecond regime, and they have been able to correctly predict both the observed pulse duration and the output power for the first time.
Abstract: Optically pumped vertical-external-cavity surface-emitting lasers (OP-VECSELs), passively modelocked with a semiconductor saturable absorber mirror (SESAM), have generated the highest average output power from any sub-picosecond semiconductor laser. Many applications, including frequency comb synthesis and coherent supercontinuum generation, require pulses in the sub-300-fs regime. A quantitative understanding of the pulse formation mechanism is required in order to reach this regime while maintaining stable, high-average-power performance. We present a numerical model with which we have obtained excellent quantitative agreement with two recent experiments in the femtosecond regime, and we have been able to correctly predict both the observed pulse duration and the output power for the first time. Our numerical model not only confirms the soliton-like pulse formation in the femtosecond regime, but also allows us to develop several clear guidelines to scale the performance toward shorter pulses and higher average output power. In particular, we show that a key VECSEL design parameter is a high gain saturation fluence. By optimizing this parameter, 200-fs pulses with an average output power of more than 1 W should be possible.
TL;DR: In this article, an all-normal-dispersion passively mode-locked Yb-doped fiber laser using a graphene oxide/polyvinyl alcohol (GO-PVA) saturable absorber without surfactant was demonstrated.
Abstract: We have demonstrated an all-normal-dispersion passively mode-locked Yb-doped fiber laser using a graphene oxide/polyvinyl alcohol (GO–PVA) saturable absorber without surfactant, for the first time to the best of our knowledge. The experimental results show that the pulse duration of the mode-locked lasers varies from 191 ps to 1.68 ns, while the cavity round trip time changes from 24 to 458 ns, through the variation of the cavity length. In addition, the proposed passively mode-locked fiber laser demonstrates a maximum average output power of 539 mW with a laser cavity length of 94 m, and the corresponding single pulse energy reaches 0.429 μJ. The proposed mode-locked fiber lasers with large chirp pulses may find potential applications in fiber chirped pulse amplification systems for micromachining, material processing and diagnostic applications.
TL;DR: Two-stage fiber pre-amplifiers and a high energy large mode area (LMA) fiber amplifier were used to boost pulse energy up to 54 µJ before pulse compressor with chirped pulse amplification technique and pulse energy and duration of 910 fs were obtained.
Abstract: In the paper, a 2 µm high energy fs fiber laser and amplification system is presented based on Tm doped fibers. The seed laser was designed to generate pulse train at 2024 nm at a repetition rate of 2.5 MHz. An AOM was used as a pulse picker to further lower the repetition rate down to 100 kHz. Two-stage fiber pre-amplifiers and a high energy large mode area (LMA) fiber amplifier were used to boost pulse energy up to 54 µJ before pulse compressor with chirped pulse amplification technique. After compressor, pulse energy of 36.7µJ and pulse duration of 910 fs and were obtained.
TL;DR: In this paper, a self-induced narrow-band laser at 428 nm has a pulse duration of 2.6 ps with perfect linear polarization property, which opens new possibilities for remote detection in the atmosphere.
Abstract: We report, for what we believe to be the first time, on the generation of remote self-seeding laser amplification by using only one 800 nm Ti:Sapphire femtosecond laser pulse. The laser pulse (~ 40 fs) is first used to generate a filament either in pure nitrogen or in ambient air in which population inversion between ground and excited states of nitrogen molecular ions is realized. Self-induced white light inside the filament is then serving as the seed to be amplified. The self-induced narrow-band laser at 428 nm has a pulse duration of ~2.6 ps with perfect linear polarization property. This finding opens new possibilities for remote detection in the atmosphere.
TL;DR: Two all fiber-based laser systems are demonstrated to achieve high energy and high average power femtosecond pulsed outputs at wavelength of 1 µm.
Abstract: Two all fiber-based laser systems are demonstrated to achieve high energy and high average power femtosecond pulsed outputs at wavelength of 1 µm. In the high energy laser system, a pulse energy of 1.05 mJ (0.85 mJ after pulse compressor) at 100 kHz repetition rate has been realized by a Yb-doped ultra large-core single-mode photonic crystal fiber (PCF) rod amplifier, seeded with a 50 µJ fiber laser. The pulse duration is 705 fs. In the high average power experiment, a large mode area (LMA) fiber has been used in the final stage amplifier, seeded with a 50 W mode locked fiber laser. The system is running at a repetition rate of 69 MHz producing 1052 W of average power before compressor. After pulse compression, a pulse duration of 800 fs was measured.
TL;DR: The results indicate that temporal pulse shaping can be advantageously used as a mean to control the periodic nanoripples' formation and thus the outcome of laser assisted nanofabrication process, which is desirable for the applications of nanopatterned transparent semiconductors.
Abstract: We demonstrate the capability to control the ripple periodicity on polycrystalline ZnO films by applying temporally delayed femtosecond double pulses. It is shown that there is a characteristic pulse separation time for which one can switch from low- to high- spatial-frequency ripple formation. Results are interpreted based on the relation of the characteristic delay time with the electron-phonon relaxation time of the material. Our results indicate that temporal pulse shaping can be advantageously used as a mean to control the periodic nanoripples’ formation and thus the outcome of laser assisted nanofabrication process, which is desirable for the applications of nanopatterned transparent semiconductors.
TL;DR: In this paper, a chirped pulse amplification (CPA) setup utilizing a graphene mode-locked femtosecond fiber laser as a seed source was presented, which allowed the amplification of the seed up to 1?W of average power with linearly polarized 810 fs pulses and 20?nJ pulse energy at a 55?MHz repetition rate.
Abstract: In this work we demonstrate for the first time, to our knowledge, a chirped pulse amplification (CPA) setup utilizing a graphene mode-locked femtosecond fiber laser as a seed source. The system consists of a mode-locked Er-fiber oscillator operating at 1560?nm wavelength, a grating-based pulse stretcher, two-stage amplifier and a grating compressor. The presented setup allows the amplification of the seed up to 1?W of average power (1000 times amplification) with linearly polarized 810 fs pulses and 20?nJ pulse energy at a 55?MHz repetition rate. The whole design is based on single-mode fibers, which allows one to maintain excellent beam quality, with M2 less than 1.17.
TL;DR: It is shown that drilling with radially polarized pulses produces holes with smoother and better-delineated walls compared with the other polarizations used, whereas linearly polarized pulses can machine 20-nm wide single grooves in fused silica when the electric field of the pulse is aligned perpendicular to the cutting direction.
Abstract: In this article we compare the results of micromachining of fused silica and silicon with tightly focused scalar (viz., circularly and linearly polarized) and vector (viz., azimuthally and radially polarized) femtosecond laser pulses. We show that drilling with radially polarized pulses produces holes with smoother and better-delineated walls compared with the other polarizations used, whereas linearly polarized pulses can machine 20-nm wide single grooves in fused silica when the electric field of the pulse is aligned perpendicular to the cutting direction. The observed polarization-controlled micromachining is due to the formation of sub-diffraction-limited nanostructures that are optically produced in the multi-pulse irradiation regime.
TL;DR: In this article, the effect of ultrashort laser-induced morphological changes upon irradiation of silicon with double pulse sequences is investigated under conditions that lead to mass removal, and the proposed underlying mechanism is based on the combination of carrier excitation and energy thermalization and capillary wave solidification.
Abstract: The effect of ultrashort laser-induced morphological changes upon irradiation of silicon with double pulse sequences is investigated under conditions that lead to mass removal. The temporal delay between twelve double and equal-energy pulses (Ep=0.24J/cm2 each, with pulse duration tp=430fs, 800nm laser wavelength) was varied between 0 and 14ps and a decrease of the damaged area, crater depth size and periodicity of the induced subwavelength ripples (by 3-4%) was observed with increasing pulse delay. The proposed underlying mechanism is based on the combination of carrier excitation and energy thermalization and capillary wave solidification and aims to provide an alternative explanation to the control of ripple periodicity by temporal pulse tailoring. This work demonstrates the potential of pulse shaping technology to improve nano/micro processing.
TL;DR: All four 5S-7S two-photon transitions in a rubidium vapor are determined with both statistical and systematic uncertainties below 10(-11), which is an order of magnitude better than previous experiments on these transitions.
Abstract: An experimental realization of high-precision direct frequency comb spectroscopy using counterpropagating femtosecond pulses on two-photon atomic transitions is presented. The Doppler broadened background signal, hampering precision spectroscopy with ultrashort pulses, is effectively eliminated with a simple pulse shaping method. As a result, all four 5S-7S two-photon transitions in a rubidium vapor are determined with both statistical and systematic uncertainties below 10(-11), which is an order of magnitude better than previous experiments on these transitions.
TL;DR: A high power polarization maintaining femtosecond Tm-doped fiber laser system is demonstrated and mode locked pulses with a pulse repetition rate of 30.84 MHz are demonstrated.
Abstract: A high power polarization maintaining femtosecond Tm-doped fiber laser system is demonstrated. A chirped fiber Bragg grating with normal dispersion was used to compensate the anomalous dispersion from the regular fiber in the 2 µm seed oscillator to generate mode locked pulses with a pulse repetition rate of 30.84 MHz. After chirped pulse amplification, an amplified power of 78 W was obtained. The pulse was compressed by a chirped volume Bragg grating based pulse compressor. A pulse duration of 760 fs and an average power of 36 W were obtained after compressor.
TL;DR: The rapid development of commercial multi-kW continuous-wave fibre lasers and high-peak power fibre lasers leads to a natural convergence toward high-average power high- peak power pulsed fibre lasers whose applications include marking, trimming, micromachining, precision drilling, welding and cutting while retaining the characteristics advantage of the fibre laser.
Abstract: We present a pulsed fiber laser system with average power up to 265 W, pulse energy up to 10.6 mJ, pulse duration adjustable in the range 500 ps–500 ns, repetition rate fully controllable from single-shot operation up to 1 MHz, and the ability to control peak power independently of pulse energy. The system has a compact, all-spliced construction. Such a versatile laser will have wide applications in materials processing.
TL;DR: The unusual dependence of femtosecond laser writing on the light polarization and direction of raster scanning is demonstrated in silica and chalcogenide glasses and two different mechanisms contributing to the observed anisotropy are identified.
Abstract: The unusual dependence of femtosecond laser writing on the light polarization and direction of raster scanning is demonstrated in silica and chalcogenide glasses. Two different mechanisms contributing to the observed anisotropy are identified: the chevron-shaped stress induced by the sample movement and the pulse front tilt of ultrashort light pulse. Control of anisotropies associated with the spatio-temporal asymmetry of an ultrashort pulse beam and scanning geometry is crucial in the ultrafast laser machining of transparent materials.
TL;DR: In this paper, the role of MI and self-Raman self-scattering has been discussed and a new method for increasing the pulse train repetition rate through frequency modulation of the seed wave has been proposed.
Abstract: Optical pulse generation and compression have been numerically studied in anomalous dispersion decreasing fibers. We show that evolution of modulation instability (MI) observed with chirped wave packets in tapered fibers produces the mechanism for generation of ultrashort pulses with high repetition rates. The role of MI and Raman self-scattering has been also discussed. The simulations show that pulse chirping enhances self-Raman scattering at early stages of pulse propagation and improves compression of the generated pulses. It is also shown that the presence of amplitude and frequency modulation of the seed wave provide essential impact on the pulse train formation. The new method for increasing the pulse train repetition rate through frequency modulation of the seed wave has been proposed.
TL;DR: The nonlinear pulse compression of temporally divided pulses, which is presented in a proof-of-principle experiment, holds promise for overcoming fundamental limitations of the pulse peak power that lead to destruction of the fiber or ionization limitations in high-energy hollow-core compression.
Abstract: We report on the nonlinear pulse compression of temporally divided pulses, which is presented in a proof-of-principle experiment. A single 320 fs pulse is divided into four replicas, spectrally broadened in a solid-core fiber, and subsequently recombined. This approach makes it possible to reduce the nonlinearities in the fiber and therefore to use total input peak power of about 13.3 MW, which is more than three times higher than the self-focusing threshold. Finally, the combined output pulse could be compressed to sub-100 fs pulse duration. This general and universal approach holds promise for overcoming fundamental limitations of the pulse peak power that lead to destruction of the fiber or ionization limitations in high-energy hollow-core compression.
TL;DR: The carrier-envelope phase (CEP) stability of hollow-fiber compression for high-energy few-cycle pulse generation and the onset of an ionization-induced CEP instability is observed, which saturates beyond input pulse energies of 1.25 mJ.
Abstract: We investigated the carrier-envelope phase (CEP) stability of hollow-fiber compression for high-energy few-cycle pulse generation. Saturation of the output pulse energy is observed at 0.6 mJ for a 260 μm inner-diameter, 1 m long fiber, statically filled with neon. The pressure is adjusted to achieve output spectra supporting sub-4-fs pulses. The maximum output pulse energy can be increased to 0.8 mJ by either differential pumping (DP) or circularly polarized input pulses. We observe the onset of an ionization-induced CEP instability, which saturates beyond input pulse energies of 1.25 mJ. There is no significant difference in the CEP stability with DP compared to static-fill.
TL;DR: Recommendation for shortest parabolic pulse formation is made based on the analysis presented, which found that both approaches could produce parabolic pulses as short as few hundred femtoseconds applying commercially available fibers, specially designed all-normal dispersion photonic crystal fiber and modern femTosecond lasers for pumping.
Abstract: Formation of parabolic pulses at femtosecond time scale by means of passive nonlinear reshaping in normally dispersive optical fibers is analyzed. Two approaches are examined and compared: the parabolic waveform formation in transient propagation regime and parabolic waveform formation in the steady-state propagation regime. It is found that both approaches could produce parabolic pulses as short as few hundred femtoseconds applying commercially available fibers, specially designed all-normal dispersion photonic crystal fiber and modern femtosecond lasers for pumping. The ranges of parameters providing parabolic pulse formation at the femtosecond time scale are found depending on the initial pulse duration, chirp and energy. Applicability of different fibers for femtosecond pulse shaping is analyzed. Recommendation for shortest parabolic pulse formation is made based on the analysis presented.
TL;DR: A theoretical and experimental comparison with conventionally derived BEBOP, BIBOP, and BURBOP-180° pulses is given, finding that while the overall transfer efficiency of a single pulse pair is reduced, resulting transfer to undesired coherences is reduced by several percent.
TL;DR: In this article, a deformable mirror was used to control the plasma channel length by introducing a spherical aberration into the initial transverse spatial distribution of a femtosecond laser pulse.
Abstract: Filamentation of focused UV and IR femtosecond laser pulses and plasma channel formation governed by variable wavefront distortions was experimentally and numerically studied. A deformable mirror was used to control the plasma channel length by introducing a spherical aberration into the initial transverse spatial distribution of a femtosecond laser pulse. An at least double increase of the plasma channel length was observed with increasing deformation of the mirror. Numerical calculations show that the hat-like phase shape of the aberration ensures that the energy of the initial laser pulse remains confined for a longer distance within the limited transverse size of the filament.