August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4659
Post-filament self-trapping of ultrashort laser pulses A. V. Mitrofanov,1,2,3 A. A. Voronin,1,2 D. A. Sidorov-Biryukov,1,2 G. Andriukaitis,4 T. Flöry,4 A. Pugžlys,4 A. B. Fedotov,1,2 J. M. Mikhailova,1,6 V. Ya. Panchenko,3 A. Baltuška,4 and A. M. Zheltikov1,2,5,* 1 2 3
Russian Quantum Center, ul. Novaya 100, Skolkovo, Moscow Region 1430125, Russia
Physics Department, International Laser Center, M.V. Lomonosov Moscow State University, Moscow 119992, Russia
Institute of Laser and Information Technologies, Russian Academy of Sciences, Shatura, Moscow Region 140700, Russia 4 Photonics Institute, Vienna University of Technology, Gusshausstrasse 27-387, 1040 Vienna, Austria 5 6
Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, USA *Corresponding author: [email protected] Received May 20, 2014; revised June 18, 2014; accepted June 23, 2014; posted June 24, 2014 (Doc. ID 212277); published August 5, 2014 Laser filamentation is understood to be self-channeling of intense ultrashort laser pulses achieved when the selffocusing because of the Kerr nonlinearity is balanced by ionization-induced defocusing. Here, we show that, right behind the ionized region of a laser filament, ultrashort laser pulses can couple into a much longer light channel, where a stable self-guiding spatial mode is sustained by the saturable self-focusing nonlinearity. In the limiting regime of negligibly low ionization, this post-filamentation beam dynamics converges to a large-scale beam self-trapping scenario known since the pioneering work on saturable self-focusing nonlinearities. © 2014 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.5940) Self-action effects. http://dx.doi.org/10.1364/OL.39.004659
Self-focusing is a universal beam dynamics scenario in nonlinear optical physics [1] that has been recognized since the early days of the laser era [2–5]. This effect imposes a fundamental limitation on the maximum peak power in optical systems and materials [1] and gives rise to a broad diversity of physically intriguing and practically significant phenomena, including Kerr-lens mode locking [6] and laser-induced filamentation [7–9]. As the Kerr nonlinearity, which causes self-focusing, inevitably saturates with increasing field intensity, catastrophic beam collapsing, as shown in extensive literature (see, e.g., [5,10,11]), is avoided, opening a phase space for stable beam self-trapping [11] and motivating a quest for stable spatial solitons [12]. In most materials, however, the ultrafast Kerr nonlinearity starts to saturate when yet another fundamental physical phenomenon—field-induced ionization—comes into play. A transverse profile of the electron density induced by an ultrashort pulse is equivalent to a defocusing lens, which can, under certain conditions, balance the Kerr lens, giving rise to a filamentation phenomenon [7], which has been shown to enable long-distance transmission of ultrashort laser pulses [8,9], compression of highintensity field waveforms to few-cycle pulse widths [13], and new schemes of standoff detection [14]. An interplay between ionization and Kerr-effect saturation as mechanisms limiting beam self-focusing leads to a variety of intricate spatiotemporal field evolution scenarios. Since the manifestations of these mechanisms are difficult to decouple from each other, their significance in laser-induced filamentation is a subject of a heated debate and ongoing research [15,16]. In this Letter, we identify a specific filamentation scenario where ionization and the high-order Kerr effect become dominant in spatially separated regions of a laser filament. We demonstrate that a strongly ionized region of a filament can be succeeded by a much longer light channel, where a stable self-guiding spatial mode is sustained by the high-order 0146-9592/14/164659-04$15.00/0
Kerr effect. In the limiting regime of negligibly low ionization, as our analysis shows, this post-filamentation beam dynamics converges to a large-scale beam selftrapping scenario predicted many decades ago in the classical texts on nonlinear optics [5,10,11]. Experiments were performed with an amplified 1030nm output of a solid-state Yb-laser system. The energy of laser pulses, W 0 , was varied from 0.5 to 10 mJ. The laser output with a pulse width τ0 ≈ 200 fs and an FW1∕e2 M beam diameter of 11 mm was focused in atmospheric air with a 250-cm-focal-length lens. A monochromator with a germanium diode were used for spectral measurements. Temporal characterization of laser pulses was performed by means of second-harmonic generation (SHG) frequency-resolved optical gating (FROG) using a 0.1-mm-thick BBO crystal. As the energy of the laser pulse is increased above 1–1.5 mJ in our experimental arrangement, drastic changes in beam divergence and the output beam profile are observed, indicating beam self-trapping. These changes are accompanied by a dramatic spectral broadening, yielding supercontinuum radiation with spectra featuring an extended high-frequency tail and a powerful, strongly modulated low-frequency wing [Fig. 1(a)]. Pulse characterization of the supercontinuum output using SHG FROG reveals a multipeak structure of the supercontinuum field with asymmetric profiles of the temporal and spectral phases [Figs. 1(b)– 1(e)]. Partial chirp compensation with a 6-cm-thick silica block was found to compress this pulse, yielding an intense peak with an FWHM pulse width of 40 fs preceded by an oscillatory prepulse [Figs. 1(f) and 1(g)]. With a more accurate chirp compensation using a pair of SF57 prisms, compression to a pulse width of 25 fs, close to a transform-limited pulse width (21 fs), was achieved. In numerical simulations, we use a model [8,9] based on the field evolution equation that includes the key physical effects, such as dispersion of the medium, beam diffraction, optical nonlinearities because of the cubic © 2014 Optical Society of America
4660
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
U O2 ≈ 12 eV, Kerr nonlinearity coefficients n2 ≈ 5 · 10−19 cm2 ∕W and n4 ≈ −6 · 10−33 cm4 ∕W2 , overall neutral gas density ρ0 ≈ 2.7 · 1019 cm−3 , collision time τc ≈ 350 fs, fraction of Raman nonlinearity f R ≈ 0.5, and a dampedoscillator Raman response function with an oscillation period of 390 fs and a dephasing time of 70 fs. Although we have to take n4 four times higher in its absolute value relative to the n4 from [18] to achieve the best fit with our experimental data, the sign of n4 and its order of magnitude agree well with the result of Loriot et al. [18]. Moreover, at a typical field intensity of I ≈ 20 TW∕cm2 , the overall nonlinear change in the refractive index Δn f R n2 I n4 I 2 in our model deviates from Δn calculated with the model of Loriot et al. [18] by less than 10%. Simulations accurately reproduce experimental results within the entire range of input pulse peak powers studied in experiments (Fig. 1). The key aspects of the spatiotemporal field dynamics are illustrated in Figs. 2(a)–2(f). At the initial stage of field evolution, the focused laser beam undergoes self-focusing, which shifts the focal region toward smaller z relative to the linear focus z0 ≈ 250 cm in Fig. 2(a). As the growing intensity in the selffocusing laser beam induces stronger ionization, the electron density increases [Fig. 2(b)], leading to a stronger defocusing of the laser beam [Fig. 2(a)], eventually giving rise to a formation of a filament in accordance with the
Fig. 1. (a) Supercontinuum spectra generated by 1030-nm, 200-fs Yb-laser pulses behind the self-channeling region at z 4.6 m: (solid line) experiments and (dashed line) simulations. The energy of the input pulse is indicated in the panels. The input spectrum is shown by shading. (b) The experimental SHG FROG trace, (c) the Wigner spectrogram, (d) temporal envelope (solid line) and the phase (dashed line), and (e) the spectrum (solid line) and spectral phase (dashed line) of the laser pulse with τ0 ≈ 200 fs and W 0 ≈ 4 mJ behind the self-channeling region at z 4.6 m. (f) The experimental SHG FROG trace and (g) temporal envelope (solid line) and the phase (dashed line) of the same laser pulse behind the selfchanneling region following a partial chirp compensation with a 6-cm-thick silica block.
and quintic susceptibilities of neutral gas, ionizationinduced nonlinearities, pulse self-steepening, as well as plasma loss, refraction, and dispersion. This equation is solved jointly with the equation for the electron density, where the photoionization rate W calculated using the Popov–Perelomov–Terentyev version of the Keldysh formalism [8,9,17]. A set of parameters used in simulations is chosen in such a way as to model our experiments presented below in this Letter. The input field is assumed to have a Gaussian pulse envelope and a Gaussian beam profile, with an FWHM pulse width of 200 fs, a central wavelength of 1030 nm, the spectrum as shown by shading in Fig. 1(a), and a pulse energy ranging from 0.1 to 10 mJ. Calculations were performed with a standard set of parameters describing the ultrafast nonlinear response of atmospheric air [8,9], modeled as a gas mixture consisting of 80% molecular nitrogen with ionization potential U N2 ≈ 15.6 eV and 20% molecular oxygen with
Fig. 2. (a) Map of the field intensity (on the color-coded dB scale) integrated over the entire pulse with the FWHM beam radius r FWHM (blue curve) and rms beam radius r 0 (rose curve), (b) the on-axis electron density along the beam propagation path, (c) the spatiotemporal map of beam self-trapping, and (d) evolution of the spectrum along the propagation path. (e) Spectrum of the laser pulse behind the self-channeling region at z 4.6 m calculated with the full model (curve 2), with the Raman effect disabled (f R 0) for z > 2.8 m (curve 3), and with self-steepening disabled for z > 2.8 m (curve 1, shading); and (f) the spectrum of the laser pulse behind the self-channeling region at z 4.6 m: (curve 1, shading) experiments, simulations with the full, three-dimensional model (curve 2), and calculations with a one-dimensional model (curve 3).
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
standard filamentation scenario. In our experimental conditions, however, this standard filamentation dynamics is followed by an unusual spatiotemporal field transformation, where one part of the laser beam tends to diverge as it is defocused by the electron density profile, while the second, more intense part of the beam, which is strongly confined to the beam axis, couples into a channel, which can stretch over several meters well beyond the region of strong ionization (Figs. 2 and 3). To visualize this beam dynamics, we examine the FWHM beam R radius, r FWHM R , along with the rms beam radius, r 20 2πr 3 Irdr 2πrIrdr−1 , where Ir is the radial profile of the field intensity. As can be seen from Figs. 2(a) and 3(a)–3(c), while for the linear regime of beam dynamics, observed at the initial stage of field propagation, for z < 2 m, the behavior of r FWHM as a function of z closely follows that of r 0 , behind the ionized region, the behavior of r FWHM z and r 0 z is strikingly different. While the integral average beam radius rapidly increases as a function of z, visualizing a freely diverging fraction of the beam, the FWHM beam radius displays only small variations within the range of z from 2.8 to 4.3 m, indicating that the central, most intense part of the beam is self-trapped. For a beam with W 0 ≈ 4 mJ, the total energy of the self-trapped beam is 0.64 mJ. Figure 2(c) clearly shows that beam self-trapping is not uniform over the pulse, with only the central, most intense part of the pulse effectively coupled into the self-trapped mode. This effect is observed as a drastic change in the on-axis pulse width at z ≈ 2.8 m in the map of Fig. 2(c). Since the pulse
Fig. 3. (a)–(d) Maps of the field intensity (on the color-coded dB scale) integrated over the entire pulse with the FWHM beam radius r FWHM (blue curve) and rms beam radius r 0 (rose curve) calculated using the full model (a), with ionization disabled (b), with the n4 term in Eq. (1) disabled (c), and (d) with a 350-μmradius circular pinhole (white line) placed at z 3 m. (e) Pulse evolution along the propagation path. The initial beam diameter is 11 mm, τ0 ≈ 200 fs, and W 0 ≈ 4 mJ.
4661
width is still quite long even beyond the z ≈ 2.8 m point, dispersion effects play no role in field propagation dynamics, as verified by numerical simulations with dispersion disabled in Eq. (1), excluding temporal soliton effects in this region. Beyond a short region between z 2.25 and 2.52 m, the electron density is very low even on the beam axis [Fig. 2(b)]. Simulations with ionization effects in Eq. (1) disabled for z > 2.8 m show that, within this beam selftrapping region, ionization does not have any influence on the spectra and temporal shapes of laser pulses or on the spatial beam dynamics. Beam self-trapping for z > 2.8 m is thus exclusively because of the saturation of the Kerr nonlinearity. In fact, when ionization effects are switched off in our simulations within the entire beam propagation path, beam self-trapping in the z > 2.8 m region becomes more stable, with ionization-induced variations in r FWHM z almost totally suppressed [see Figs. 3(a) and 3(b)]. On the other hand, when the n4 term is switched off in Eq. (1), no beam self-trapping is possible behind the ionization region [Fig. 3(c)]. Remarkably, the key tendencies of beam dynamics for z > 2.8 m in our conditions are fully consistent with the predictions of the earlier theories of beam self-trapping in media with a saturable self-focusing nonlinearity. In particular, the FWHM radius of the self-trapped beam in our conditions accurately follows the (a∕γP∕P c Karlsson scaling [11], where a 1.2 krFWHM , P is the peak power, P c is the critical power for self-focusing, γ 2 n20 ∕n2sat − n20 , and nsat is the refractive index with saturated Kerr nonlinearity. Approximating the polynomial expansion of the refractive index nI n0 n2 I n4 I 2 used in our model by the Karlsson ansatz [11] n2 I n20 n2 I∕1 γ 2 n−2 0 n2 I, we find γ ≈ 250. For the parameters of the experiment presented in Figs. 2(a) 2(d) and 3(a), the (a∕rP∕P c scaling gives a∕γ ≈ 3 corresponding r FWHM ≈ 105 μm, which agrees very well with the FWHM radius of the self-trapped beam, r FWHM ≈ 100 μm at z 3 m, in our conditions. Alternative scenarios giving rise to beams with suppressed divergence behind the ionization region, including post-ionization long-range self-channeling of ultrashort laser pulses in air [19,20], involve a quasilinear dynamics of beams with intensities below ionization threshold, Bessel-like beams [21], and post-filamentation beam channeling sustained by the photon bath [22,23]. Numerical simulations have been performed to check whether any of these effects is involved in postfilamentation beam self-trapping in our experiments. As shown in Fig. 3(d), a 350-μm-radius circular pinhole blocking the photon bath placed in the beam at z 3 m (white line) does not prevent post-filamentation beam self-trapping, as shown by the black line in Fig. 3(d). On the other hand, a circular obstacle with a radius of 150 μm blocking the central part of the beam at z 3 m immediately stops beam self-trapping. The same effect is achieved by artificially switching off optical nonlinearities at z 3 m, showing that postfilamentation beam self-trapping in our experiments is indeed due to the saturable self-focusing nonlinearity. Moreover, unlike modeling in [22], which, as the authors admit, fails to reproduce the long-wavelength part of the
4662
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
spectrum, motivating further in-depth studies, our simulations provide an accurate fit for the experimental spectra [Fig. 1(a)]. Since our simulations suggest that most of the spectral broadening occurs in the beam self-trapping region [Fig. 2(d)], where the beam dynamics can be to a good approximation considered as one-dimensional [5,10,11], the question arises as to how accurately the supercontinuum spectra and pulse shapes observed in our experiments can be described in terms of a simple one-dimensional model of field evolution. We address this question by modeling field evolution in the beamself-trapping region, i.e., from z 2.8 to 4.6 m, using a 1D equation, ∂A∕∂z iω0 n2 T~ −1 c δη1 − f R I Z f R Rη − η0 Iη0 ; r; zdη0 A;
(1)
where A ≡ Aη; r; z is the field envelope, I ≡ Iη; r; z jAη; r; zj2 is the field intensity, r is the transverse coordinate, η is the retarded time, ω0 is the central frequency, T~ 1 iω−1 0 ∂∕∂η, and Rθ is the Raman response function. This 1D model [dash-dotted line in Fig. 2(f)] enables a reasonably accurate description of the long-wavelength wing of supercontinuum spectra, providing a good agreement with both three-dimensional simulations (dashed line) and experimental results (solid line). This confirms that the red-shifted part of the supercontinuum spectra is generated in our experiments in an effectively onedimensional, guided-wave propagation mode, supported by a saturable Kerr nonlinearity, exactly as expected from the earlier studies of beam self-trapping in media with a saturable self-focusing nonlinearity [5,10,11]. This is consistent with 3D simulations, showing that the redshifted spectral components are mainly produced in the z > 2.8 m region [Fig. 2(d)]. Simulations performed with the Raman term artificially switched off (i.e., with f R 0) indicate that the enhancement of the red part of the supercontinuum spectra is because of the Raman effect [Fig. 2(e)]. Dispersion effects do not play any noticeable role in the z > 2.8 m region, leading to virtually no changes in the pulse envelope in the self-trapped beam [Fig. 3(e)]. The red-shifted spectral features arise in this region as a part of a standard self-phase modulation scenario in a medium with an inertial third-order nonlinearity [24]. In the high-frequency part of the spectrum, as can also be seen from Fig. 2(f), the 1D model tends to overestimate the spectral intensity and the extent of spectral broadening, showing that the spatial and temporal effects leading to the blue shifting (such as self-steepening and space-time focusing) are strongly coupled, requiring a full 3D analysis.
This research was supported in part by the Russian Foundation for Basic Research (projects nos. 13-0201465, 13-02-92115, 14-02-00784) and the Welch Foundation (Grant No. A-1801). The work on ultrashort pulse characterization was supported by the Russian Science Foundation (project no. 14-12-00772). Reserach by A. A. V. is supported in part by the Dynasty Foundation. References 1. Y. R. Shen, The Principles of Nonlinear Optics (WileyInterscience, 1984). 2. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). 3. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965). 4. G. A. Askar’yan, Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1657 (1961)]. 5. W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256 (1968). 6. D. E. Spence, P. N. Kean, and W. Sibbett, Opt. Lett. 16, 42 (1991). 7. A. Brown, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, Opt. Lett. 20, 73 (1995). 8. A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47 (2007). 9. L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007). 10. A. H. Piekara, J. S. Moore, and M. S. Field, Phys. Rev. A 9, 1403 (1974). 11. M. Karlsson, Phys. Rev. A 46, 2726 (1992). 12. Y. Silberberg, Opt. Lett. 15, 1282 (1990). 13. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, Appl. Phys. B 79, 673 (2004). 14. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, Science 301, 61 (2003). 15. P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, Phys. Rev. Lett. 104, 103903 (2010). 16. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, Phys. Rev. Lett. 106, 153902 (2011). 17. L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965) [Zh. Eksp. Teor. Fiz. 47, 1945 (1965)]. 18. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, Opt. Express 17, 13429 (2009). 19. G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, Appl. Phys. B 79, 379 (2004). 20. G. Méchain, C. D’Amico, Y.-B. André, S. Tzortzakis, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, E. Salmon, and R. Sauerbrey, Opt. Commun. 247, 171 (2005). 21. A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. Di Trapani, Phys. Rev. Lett. 92, 253903 (2004). 22. Y. Chen, F. Théberge, C. Marceau, H. Xu, N. Aközbek, O. Kosareva, and S. L. Chin, Appl. Phys. B 91, 219 (2008). 23. J.-F. Daigle, T.-J. Wang, S. Hosseini, S. Yuan, G. Roy, and S. L. Chin, Appl. Opt. 50, 6234 (2011). 24. T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
4659
Post-filament self-trapping of ultrashort laser pulses A. V. Mitrofanov,1,2,3 A. A. Voronin,1,2 D. A. Sidorov-Biryukov,1,2 G. Andriukaitis,4 T. Flöry,4 A. Pugžlys,4 A. B. Fedotov,1,2 J. M. Mikhailova,1,6 V. Ya. Panchenko,3 A. Baltuška,4 and A. M. Zheltikov1,2,5,* 1 2 3
Russian Quantum Center, ul. Novaya 100, Skolkovo, Moscow Region 1430125, Russia
Physics Department, International Laser Center, M.V. Lomonosov Moscow State University, Moscow 119992, Russia
Institute of Laser and Information Technologies, Russian Academy of Sciences, Shatura, Moscow Region 140700, Russia 4 Photonics Institute, Vienna University of Technology, Gusshausstrasse 27-387, 1040 Vienna, Austria 5 6
Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544-5263, USA *Corresponding author: [email protected] Received May 20, 2014; revised June 18, 2014; accepted June 23, 2014; posted June 24, 2014 (Doc. ID 212277); published August 5, 2014 Laser filamentation is understood to be self-channeling of intense ultrashort laser pulses achieved when the selffocusing because of the Kerr nonlinearity is balanced by ionization-induced defocusing. Here, we show that, right behind the ionized region of a laser filament, ultrashort laser pulses can couple into a much longer light channel, where a stable self-guiding spatial mode is sustained by the saturable self-focusing nonlinearity. In the limiting regime of negligibly low ionization, this post-filamentation beam dynamics converges to a large-scale beam self-trapping scenario known since the pioneering work on saturable self-focusing nonlinearities. © 2014 Optical Society of America OCIS codes: (190.7110) Ultrafast nonlinear optics; (190.5940) Self-action effects. http://dx.doi.org/10.1364/OL.39.004659
Self-focusing is a universal beam dynamics scenario in nonlinear optical physics [1] that has been recognized since the early days of the laser era [2–5]. This effect imposes a fundamental limitation on the maximum peak power in optical systems and materials [1] and gives rise to a broad diversity of physically intriguing and practically significant phenomena, including Kerr-lens mode locking [6] and laser-induced filamentation [7–9]. As the Kerr nonlinearity, which causes self-focusing, inevitably saturates with increasing field intensity, catastrophic beam collapsing, as shown in extensive literature (see, e.g., [5,10,11]), is avoided, opening a phase space for stable beam self-trapping [11] and motivating a quest for stable spatial solitons [12]. In most materials, however, the ultrafast Kerr nonlinearity starts to saturate when yet another fundamental physical phenomenon—field-induced ionization—comes into play. A transverse profile of the electron density induced by an ultrashort pulse is equivalent to a defocusing lens, which can, under certain conditions, balance the Kerr lens, giving rise to a filamentation phenomenon [7], which has been shown to enable long-distance transmission of ultrashort laser pulses [8,9], compression of highintensity field waveforms to few-cycle pulse widths [13], and new schemes of standoff detection [14]. An interplay between ionization and Kerr-effect saturation as mechanisms limiting beam self-focusing leads to a variety of intricate spatiotemporal field evolution scenarios. Since the manifestations of these mechanisms are difficult to decouple from each other, their significance in laser-induced filamentation is a subject of a heated debate and ongoing research [15,16]. In this Letter, we identify a specific filamentation scenario where ionization and the high-order Kerr effect become dominant in spatially separated regions of a laser filament. We demonstrate that a strongly ionized region of a filament can be succeeded by a much longer light channel, where a stable self-guiding spatial mode is sustained by the high-order 0146-9592/14/164659-04$15.00/0
Kerr effect. In the limiting regime of negligibly low ionization, as our analysis shows, this post-filamentation beam dynamics converges to a large-scale beam selftrapping scenario predicted many decades ago in the classical texts on nonlinear optics [5,10,11]. Experiments were performed with an amplified 1030nm output of a solid-state Yb-laser system. The energy of laser pulses, W 0 , was varied from 0.5 to 10 mJ. The laser output with a pulse width τ0 ≈ 200 fs and an FW1∕e2 M beam diameter of 11 mm was focused in atmospheric air with a 250-cm-focal-length lens. A monochromator with a germanium diode were used for spectral measurements. Temporal characterization of laser pulses was performed by means of second-harmonic generation (SHG) frequency-resolved optical gating (FROG) using a 0.1-mm-thick BBO crystal. As the energy of the laser pulse is increased above 1–1.5 mJ in our experimental arrangement, drastic changes in beam divergence and the output beam profile are observed, indicating beam self-trapping. These changes are accompanied by a dramatic spectral broadening, yielding supercontinuum radiation with spectra featuring an extended high-frequency tail and a powerful, strongly modulated low-frequency wing [Fig. 1(a)]. Pulse characterization of the supercontinuum output using SHG FROG reveals a multipeak structure of the supercontinuum field with asymmetric profiles of the temporal and spectral phases [Figs. 1(b)– 1(e)]. Partial chirp compensation with a 6-cm-thick silica block was found to compress this pulse, yielding an intense peak with an FWHM pulse width of 40 fs preceded by an oscillatory prepulse [Figs. 1(f) and 1(g)]. With a more accurate chirp compensation using a pair of SF57 prisms, compression to a pulse width of 25 fs, close to a transform-limited pulse width (21 fs), was achieved. In numerical simulations, we use a model [8,9] based on the field evolution equation that includes the key physical effects, such as dispersion of the medium, beam diffraction, optical nonlinearities because of the cubic © 2014 Optical Society of America
4660
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
U O2 ≈ 12 eV, Kerr nonlinearity coefficients n2 ≈ 5 · 10−19 cm2 ∕W and n4 ≈ −6 · 10−33 cm4 ∕W2 , overall neutral gas density ρ0 ≈ 2.7 · 1019 cm−3 , collision time τc ≈ 350 fs, fraction of Raman nonlinearity f R ≈ 0.5, and a dampedoscillator Raman response function with an oscillation period of 390 fs and a dephasing time of 70 fs. Although we have to take n4 four times higher in its absolute value relative to the n4 from [18] to achieve the best fit with our experimental data, the sign of n4 and its order of magnitude agree well with the result of Loriot et al. [18]. Moreover, at a typical field intensity of I ≈ 20 TW∕cm2 , the overall nonlinear change in the refractive index Δn f R n2 I n4 I 2 in our model deviates from Δn calculated with the model of Loriot et al. [18] by less than 10%. Simulations accurately reproduce experimental results within the entire range of input pulse peak powers studied in experiments (Fig. 1). The key aspects of the spatiotemporal field dynamics are illustrated in Figs. 2(a)–2(f). At the initial stage of field evolution, the focused laser beam undergoes self-focusing, which shifts the focal region toward smaller z relative to the linear focus z0 ≈ 250 cm in Fig. 2(a). As the growing intensity in the selffocusing laser beam induces stronger ionization, the electron density increases [Fig. 2(b)], leading to a stronger defocusing of the laser beam [Fig. 2(a)], eventually giving rise to a formation of a filament in accordance with the
Fig. 1. (a) Supercontinuum spectra generated by 1030-nm, 200-fs Yb-laser pulses behind the self-channeling region at z 4.6 m: (solid line) experiments and (dashed line) simulations. The energy of the input pulse is indicated in the panels. The input spectrum is shown by shading. (b) The experimental SHG FROG trace, (c) the Wigner spectrogram, (d) temporal envelope (solid line) and the phase (dashed line), and (e) the spectrum (solid line) and spectral phase (dashed line) of the laser pulse with τ0 ≈ 200 fs and W 0 ≈ 4 mJ behind the self-channeling region at z 4.6 m. (f) The experimental SHG FROG trace and (g) temporal envelope (solid line) and the phase (dashed line) of the same laser pulse behind the selfchanneling region following a partial chirp compensation with a 6-cm-thick silica block.
and quintic susceptibilities of neutral gas, ionizationinduced nonlinearities, pulse self-steepening, as well as plasma loss, refraction, and dispersion. This equation is solved jointly with the equation for the electron density, where the photoionization rate W calculated using the Popov–Perelomov–Terentyev version of the Keldysh formalism [8,9,17]. A set of parameters used in simulations is chosen in such a way as to model our experiments presented below in this Letter. The input field is assumed to have a Gaussian pulse envelope and a Gaussian beam profile, with an FWHM pulse width of 200 fs, a central wavelength of 1030 nm, the spectrum as shown by shading in Fig. 1(a), and a pulse energy ranging from 0.1 to 10 mJ. Calculations were performed with a standard set of parameters describing the ultrafast nonlinear response of atmospheric air [8,9], modeled as a gas mixture consisting of 80% molecular nitrogen with ionization potential U N2 ≈ 15.6 eV and 20% molecular oxygen with
Fig. 2. (a) Map of the field intensity (on the color-coded dB scale) integrated over the entire pulse with the FWHM beam radius r FWHM (blue curve) and rms beam radius r 0 (rose curve), (b) the on-axis electron density along the beam propagation path, (c) the spatiotemporal map of beam self-trapping, and (d) evolution of the spectrum along the propagation path. (e) Spectrum of the laser pulse behind the self-channeling region at z 4.6 m calculated with the full model (curve 2), with the Raman effect disabled (f R 0) for z > 2.8 m (curve 3), and with self-steepening disabled for z > 2.8 m (curve 1, shading); and (f) the spectrum of the laser pulse behind the self-channeling region at z 4.6 m: (curve 1, shading) experiments, simulations with the full, three-dimensional model (curve 2), and calculations with a one-dimensional model (curve 3).
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
standard filamentation scenario. In our experimental conditions, however, this standard filamentation dynamics is followed by an unusual spatiotemporal field transformation, where one part of the laser beam tends to diverge as it is defocused by the electron density profile, while the second, more intense part of the beam, which is strongly confined to the beam axis, couples into a channel, which can stretch over several meters well beyond the region of strong ionization (Figs. 2 and 3). To visualize this beam dynamics, we examine the FWHM beam R radius, r FWHM R , along with the rms beam radius, r 20 2πr 3 Irdr 2πrIrdr−1 , where Ir is the radial profile of the field intensity. As can be seen from Figs. 2(a) and 3(a)–3(c), while for the linear regime of beam dynamics, observed at the initial stage of field propagation, for z < 2 m, the behavior of r FWHM as a function of z closely follows that of r 0 , behind the ionized region, the behavior of r FWHM z and r 0 z is strikingly different. While the integral average beam radius rapidly increases as a function of z, visualizing a freely diverging fraction of the beam, the FWHM beam radius displays only small variations within the range of z from 2.8 to 4.3 m, indicating that the central, most intense part of the beam is self-trapped. For a beam with W 0 ≈ 4 mJ, the total energy of the self-trapped beam is 0.64 mJ. Figure 2(c) clearly shows that beam self-trapping is not uniform over the pulse, with only the central, most intense part of the pulse effectively coupled into the self-trapped mode. This effect is observed as a drastic change in the on-axis pulse width at z ≈ 2.8 m in the map of Fig. 2(c). Since the pulse
Fig. 3. (a)–(d) Maps of the field intensity (on the color-coded dB scale) integrated over the entire pulse with the FWHM beam radius r FWHM (blue curve) and rms beam radius r 0 (rose curve) calculated using the full model (a), with ionization disabled (b), with the n4 term in Eq. (1) disabled (c), and (d) with a 350-μmradius circular pinhole (white line) placed at z 3 m. (e) Pulse evolution along the propagation path. The initial beam diameter is 11 mm, τ0 ≈ 200 fs, and W 0 ≈ 4 mJ.
4661
width is still quite long even beyond the z ≈ 2.8 m point, dispersion effects play no role in field propagation dynamics, as verified by numerical simulations with dispersion disabled in Eq. (1), excluding temporal soliton effects in this region. Beyond a short region between z 2.25 and 2.52 m, the electron density is very low even on the beam axis [Fig. 2(b)]. Simulations with ionization effects in Eq. (1) disabled for z > 2.8 m show that, within this beam selftrapping region, ionization does not have any influence on the spectra and temporal shapes of laser pulses or on the spatial beam dynamics. Beam self-trapping for z > 2.8 m is thus exclusively because of the saturation of the Kerr nonlinearity. In fact, when ionization effects are switched off in our simulations within the entire beam propagation path, beam self-trapping in the z > 2.8 m region becomes more stable, with ionization-induced variations in r FWHM z almost totally suppressed [see Figs. 3(a) and 3(b)]. On the other hand, when the n4 term is switched off in Eq. (1), no beam self-trapping is possible behind the ionization region [Fig. 3(c)]. Remarkably, the key tendencies of beam dynamics for z > 2.8 m in our conditions are fully consistent with the predictions of the earlier theories of beam self-trapping in media with a saturable self-focusing nonlinearity. In particular, the FWHM radius of the self-trapped beam in our conditions accurately follows the (a∕γP∕P c Karlsson scaling [11], where a 1.2 krFWHM , P is the peak power, P c is the critical power for self-focusing, γ 2 n20 ∕n2sat − n20 , and nsat is the refractive index with saturated Kerr nonlinearity. Approximating the polynomial expansion of the refractive index nI n0 n2 I n4 I 2 used in our model by the Karlsson ansatz [11] n2 I n20 n2 I∕1 γ 2 n−2 0 n2 I, we find γ ≈ 250. For the parameters of the experiment presented in Figs. 2(a) 2(d) and 3(a), the (a∕rP∕P c scaling gives a∕γ ≈ 3 corresponding r FWHM ≈ 105 μm, which agrees very well with the FWHM radius of the self-trapped beam, r FWHM ≈ 100 μm at z 3 m, in our conditions. Alternative scenarios giving rise to beams with suppressed divergence behind the ionization region, including post-ionization long-range self-channeling of ultrashort laser pulses in air [19,20], involve a quasilinear dynamics of beams with intensities below ionization threshold, Bessel-like beams [21], and post-filamentation beam channeling sustained by the photon bath [22,23]. Numerical simulations have been performed to check whether any of these effects is involved in postfilamentation beam self-trapping in our experiments. As shown in Fig. 3(d), a 350-μm-radius circular pinhole blocking the photon bath placed in the beam at z 3 m (white line) does not prevent post-filamentation beam self-trapping, as shown by the black line in Fig. 3(d). On the other hand, a circular obstacle with a radius of 150 μm blocking the central part of the beam at z 3 m immediately stops beam self-trapping. The same effect is achieved by artificially switching off optical nonlinearities at z 3 m, showing that postfilamentation beam self-trapping in our experiments is indeed due to the saturable self-focusing nonlinearity. Moreover, unlike modeling in [22], which, as the authors admit, fails to reproduce the long-wavelength part of the
4662
OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
spectrum, motivating further in-depth studies, our simulations provide an accurate fit for the experimental spectra [Fig. 1(a)]. Since our simulations suggest that most of the spectral broadening occurs in the beam self-trapping region [Fig. 2(d)], where the beam dynamics can be to a good approximation considered as one-dimensional [5,10,11], the question arises as to how accurately the supercontinuum spectra and pulse shapes observed in our experiments can be described in terms of a simple one-dimensional model of field evolution. We address this question by modeling field evolution in the beamself-trapping region, i.e., from z 2.8 to 4.6 m, using a 1D equation, ∂A∕∂z iω0 n2 T~ −1 c δη1 − f R I Z f R Rη − η0 Iη0 ; r; zdη0 A;
(1)
where A ≡ Aη; r; z is the field envelope, I ≡ Iη; r; z jAη; r; zj2 is the field intensity, r is the transverse coordinate, η is the retarded time, ω0 is the central frequency, T~ 1 iω−1 0 ∂∕∂η, and Rθ is the Raman response function. This 1D model [dash-dotted line in Fig. 2(f)] enables a reasonably accurate description of the long-wavelength wing of supercontinuum spectra, providing a good agreement with both three-dimensional simulations (dashed line) and experimental results (solid line). This confirms that the red-shifted part of the supercontinuum spectra is generated in our experiments in an effectively onedimensional, guided-wave propagation mode, supported by a saturable Kerr nonlinearity, exactly as expected from the earlier studies of beam self-trapping in media with a saturable self-focusing nonlinearity [5,10,11]. This is consistent with 3D simulations, showing that the redshifted spectral components are mainly produced in the z > 2.8 m region [Fig. 2(d)]. Simulations performed with the Raman term artificially switched off (i.e., with f R 0) indicate that the enhancement of the red part of the supercontinuum spectra is because of the Raman effect [Fig. 2(e)]. Dispersion effects do not play any noticeable role in the z > 2.8 m region, leading to virtually no changes in the pulse envelope in the self-trapped beam [Fig. 3(e)]. The red-shifted spectral features arise in this region as a part of a standard self-phase modulation scenario in a medium with an inertial third-order nonlinearity [24]. In the high-frequency part of the spectrum, as can also be seen from Fig. 2(f), the 1D model tends to overestimate the spectral intensity and the extent of spectral broadening, showing that the spatial and temporal effects leading to the blue shifting (such as self-steepening and space-time focusing) are strongly coupled, requiring a full 3D analysis.
This research was supported in part by the Russian Foundation for Basic Research (projects nos. 13-0201465, 13-02-92115, 14-02-00784) and the Welch Foundation (Grant No. A-1801). The work on ultrashort pulse characterization was supported by the Russian Science Foundation (project no. 14-12-00772). Reserach by A. A. V. is supported in part by the Dynasty Foundation. References 1. Y. R. Shen, The Principles of Nonlinear Optics (WileyInterscience, 1984). 2. R. Y. Chiao, E. Garmire, and C. H. Townes, Phys. Rev. Lett. 13, 479 (1964). 3. P. L. Kelley, Phys. Rev. Lett. 15, 1005 (1965). 4. G. A. Askar’yan, Sov. Phys. JETP 15, 1088 (1962) [Zh. Eksp. Teor. Fiz. 42, 1657 (1961)]. 5. W. G. Wagner, H. A. Haus, and J. H. Marburger, Phys. Rev. 175, 256 (1968). 6. D. E. Spence, P. N. Kean, and W. Sibbett, Opt. Lett. 16, 42 (1991). 7. A. Brown, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, Opt. Lett. 20, 73 (1995). 8. A. Couairon and A. Mysyrowicz, Phys. Rep. 441, 47 (2007). 9. L. Berge, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, Rep. Prog. Phys. 70, 1633 (2007). 10. A. H. Piekara, J. S. Moore, and M. S. Field, Phys. Rev. A 9, 1403 (1974). 11. M. Karlsson, Phys. Rev. A 46, 2726 (1992). 12. Y. Silberberg, Opt. Lett. 15, 1282 (1990). 13. C. P. Hauri, W. Kornelis, F. W. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, Appl. Phys. B 79, 673 (2004). 14. J. Kasparian, M. Rodriguez, G. Méjean, J. Yu, E. Salmon, H. Wille, R. Bourayou, S. Frey, Y.-B. André, A. Mysyrowicz, R. Sauerbrey, J.-P. Wolf, and L. Wöste, Science 301, 61 (2003). 15. P. Bejot, J. Kasparian, S. Henin, V. Loriot, T. Vieillard, E. Hertz, O. Faucher, B. Lavorel, and J.-P. Wolf, Phys. Rev. Lett. 104, 103903 (2010). 16. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, Phys. Rev. Lett. 106, 153902 (2011). 17. L. V. Keldysh, Sov. Phys. JETP 20, 1307 (1965) [Zh. Eksp. Teor. Fiz. 47, 1945 (1965)]. 18. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, Opt. Express 17, 13429 (2009). 19. G. Méchain, A. Couairon, Y.-B. André, C. D’Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, and R. Sauerbrey, Appl. Phys. B 79, 379 (2004). 20. G. Méchain, C. D’Amico, Y.-B. André, S. Tzortzakis, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, E. Salmon, and R. Sauerbrey, Opt. Commun. 247, 171 (2005). 21. A. Dubietis, E. Gaižauskas, G. Tamošauskas, and P. Di Trapani, Phys. Rev. Lett. 92, 253903 (2004). 22. Y. Chen, F. Théberge, C. Marceau, H. Xu, N. Aközbek, O. Kosareva, and S. L. Chin, Appl. Phys. B 91, 219 (2008). 23. J.-F. Daigle, T.-J. Wang, S. Hosseini, S. Yuan, G. Roy, and S. L. Chin, Appl. Opt. 50, 6234 (2011). 24. T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
