Applied Physics B

, Volume 110, Issue 1, pp 123–130

3D Raman bullet formed under filamentation of femtosecond laser pulses in air and nitrogen

Authors

  • Daria Uryupina
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
  • Nikolay Panov
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
  • Maria Kurilova
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
  • Anna Mazhorova
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
    • École Polytechnique de MontréalGénie Physique
  • Roman Volkov
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
  • Stepan Gorgutsa
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
    • Institut National de la Recherche Scientifique-EMT
  • Olga Kosareva
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
    • International Laser Center and Faculty of PhysicsLomonosov Moscow State University
Article

DOI: 10.1007/s00340-012-5261-9

Cite this article as:
Uryupina, D., Panov, N., Kurilova, M. et al. Appl. Phys. B (2013) 110: 123. doi:10.1007/s00340-012-5261-9

Abstract

Complex experimental study of spectral, spatial and temporal behaviors of the IR shifted component observed under filamentation of the collimated femtosecond laser beam (80 GW, 50 fs, 805 nm) in molecular gases showed that this component behaves like a Raman soliton. Namely, it is confined in all domains: (a) it propagates within the filament core, (b) it has a stable duration of 30 fs along the filament, and (c) its spectrum shifts as a whole from 820 to 870 nm on the distance of 2 m from the filament start. A simple model explaining the origin of anomalous group velocity dispersion in the plasma channel of a filament is suggested.

1 Introduction

Filaments created during propagation of an ultrashort laser pulse in gases and solids have gained increasing attention in the last years. A lot of application areas are widely discussed nowadays ranging from atmospheric optics to photonic micro-devices formation and generation of a few-cycle optical and THz pulses [15]. The filamentation phenomenon is intrinsically non-linear process arising due to the interplay between Kerr self-focusing and plasma induced de-focusing of a laser beam. That means that the filament itself has high non-linear susceptibility. Since light intensity inside a filament core goes beyond 10 TW/cm2 and the filament length is much longer than the Rayleigh length, different non-linear optical processes gain high efficiency. This was approved in numerous experiments on white light generation [6], four-wave mixing [7, 8], coherent Raman scattering [9], third harmonic production [10], THz emission [11, 12], probe pulse polarization rotation [13, 14], and other phenomena.

Most of these experiments dealt with changes in a radiation spectrum (including new spectral components generation and/or their angular distribution). Recently it was shown that spectral broadening might be accompanied by pulse self-compression in noble gases, and a few cycle pulses were obtained from the filament created by the 800 nm, 30–50 fs, 3–5 mJ laser pulse [1518]. The central wavelength of the compressed pulse was nearly the same as of the initial pulse creating the filament.

The other behavior was observed in the air and some other molecular gases: red shifted spectral component grows up under filamentation of 800 nm femtosecond laser pulses [9, 19, 20]. In [19] this phenomenon was explained by the self-phase modulation. In [9] it was shown that this component propagates after the filament termination as a slightly divergent smooth beam in the same direction as the initial laser beam. The spectrum of IR-shifted component moves also with the distance from the filament start toward longer wavelengths, while its spectral shape remains nearly unchanged [9]. In [9] the new red shifted component was explained by the rotational Raman process, but numerical simulations failed to reproduce the experimentally observed data. Such a behavior looks almost like as in the case of self-shifted Raman soliton in usual [21] and photonic [22] fibers with broadband anomalous GVD, but the medium inside a filament has normal group velocity dispersion (GVD) near 800 nm. Hence, the mechanisms of formation of this component remained unclear and investigations of its temporal and spatial properties could shed a light on its origin.

Our study presents a thorough experimental characterization of the Raman-shifted component in spectral, spatial and temporal domains. We explored pre-collimated beam geometry to trace the Raman shifted component behavior at long distances. For the first time we showed that this component has a soliton-like structure being confined in 3D inside a filament core. We also suggest a simple model of refraction in a filament pointing at the origin of the anomalous GVD in its plasma channel.

2 Experimental arrangement and techniques

The scheme of our experimental setup is shown in Fig. 1. A single filament was created by the laser pulse delivered by a Ti:Sapphire laser system with the central wavelength 805 nm, pulse duration 55 fs, spectral width of 23 nm, energy per pulse up to 10 mJ, and repetition rate of 10 Hz. The beam splitter formed two beams with pulse energies of 4.5 and 3.5 mJ. The first beam creates a single filament, while the second beam serves as a reference for the SPIDER temporal profile measurements. Peak power of the first beam (almost 80 GW) exceeded well the critical self-focusing threshold (Pcr ~5 GW for femtosecond filamentation in air, argon or nitrogen [23]), but was still below than the onset of multi-filamentation. This was also due to arbitrary poor M2 = 1.7 parameter of our radiation [24].
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Fig. 1

Experimental set up (see description in the text)

A telescope consisting of the plane-convex lens (focal length 4.52 m) and the convex mirror (focal length 1 m) reduced the 4.5 mJ beam diameter to 1.3 mm FWHM to generate arbitrary short (2–3 m in length) single filament. The collimated laser beam entered the evacuated tube, which had a variable length (from 2 to 4 m) and was filled with air, nitrogen or argon. The tube had 0.6 mm thick input and output quartz glass windows. The tube was equipped with an exchangeable movable aperture with the diameter d of 300–1,000 μm positioned at any desired place along the filament. The moveable aperture stopped the filament at the desired position by cutting off certain energy from the background reservoir [25] and simultaneously selected the central part of the filament [18]. The beam became divergent after the aperture thus preventing damage to the output window by the radiation.

Spectral phase and duration of the output pulse were measured using the SPIDER technique [18, 26, 27]. Energy passed through the aperture was assessed by placing the wedge in the beam immediately after the output window of the tube and measuring the attenuated energy of the reflected pulse by the pyroelectric detector. The energy measurement setup was calibrated at peak powers of the initial beam well below the self-focusing threshold and without aperture inside the tube. The same wedge was used to measure a spectrum of the output radiation by the Solar S150-II spectrometer.

3 Experimental results

Figure 2a presents typical single-shot spectra of radiation passed through apertures with different diameter d. An aperture was positioned at the distance L ~140 cm from the point at which the filament appeared (hereafter all the distances are counted out from this position). Measurements were made in air at a pressure of 1 atm. The spectrum obtained with the widest aperture (d = 1 mm, black line in the Fig. 2a) consists of the two distinct components. The first component is the fundamental one broadened up to ~40 nm (mostly to the longer wavelength side). The next component is red shifted to 870 nm and has the spectral FWHM of 15–20 nm. Note that the very different spectra (with huge wing on the short wavelength side) were observed under filamentation of loosely focused [9] or collimated [18] laser beam in argon. This also supports that the IR shifted component appears due to rotational Raman process, which is possible for the molecular gases, but does not exist for the noble ones.
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Fig. 2

Spectra of radiation under filamentation in the air (pressure 1 atm) obtained at the distance L = 140 cm using apertures with different diameter d (a) and at different distances L using aperture with d = 500 μm (b)

The aperture with d = 500 μm (blue line in the Fig. 2a) cuts off the fundamental component leaving the red-shifted one nearly unchanged. The spectrum preserves if the aperture with d = 300 μm was placed into the beam (see Fig. 2a, red line). Hence, it follows from our measurements that the observed Raman component propagates inside the thin filament core, while the fundamental radiation occupies the outer part of the filament. In particular, this explains results published in [9], where authors observed that Raman shifted component forms the high spatial quality low divergent beam propagating in the same direction as the fundamental one.

The same changes in the spectrum with aperture size were observed at distances L ranging from 10 to 200 cm (the latter value was close to the point where the filament stops due to energy leakage), i.e., if the aperture size was below 500 μm the spectrum generally contains only the red-shifted component. This component undergoes the monotonic spectral shift with an increase in the distance L (see Fig. 2b). Our findings coincide with data obtained with the loosely focused laser beam [9], but spatial filtering using aperture allows us to view the Raman component more clearly. It is interesting to note that this component existed already at the very beginning of the filament and was not observable if the spectrometer is placed immediately after the convex mirror. Hence, the Raman component is formed during the filament formation due to phase self-modulation and rotational Raman processes and experienced monotonic spectral red shift for the longer distances. This was confirmed recently in the numerical experiment [28].

Even more complicated behavior was observed if nitrogen was used instead of air (Fig. 3, d = 500 μm). Data obtained at L = 70–80 cm shows that much more energy preserved at the fundamental frequency in the latter case (compare red curves in Figs. 2b, 3a). This can be due to the lower non-linearity of the nitrogen gas as compared with the air because of the molecular oxygen impact [19]. The second red-shifted component appears in the nitrogen at arbitrary long distances when the first component shifted from its original spectral position quite far (see the blue curve for L = 110 cm in Fig. 3a). Spectral amplitude of the first component gradually decreases along the filament, while the spectral amplitude of the second component increases. Moreover, the third component arises at L = 180 cm (black curve in Fig. 3a). At the same time, the amount of energy at the fundamental frequency decreases with L.
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Fig. 3

Spectra of radiation under filamentation in nitrogen obtained at different distances L at pressure 1 atm (a) and 0.5 atm (b) using aperture with d = 500 μm

The case of the nitrogen gas appearance of numerous Raman shifted components can be explained by the re-focusing phenomenon [15]: radiation in the reservoir undergoes multiple re-focusing events along the filament propagation. This leads to the non-monotonic behavior of such quantities as plasma channel diameter, peak intensity of radiation, etc. Hence, the Raman shifted component forms at each refocusing event provided intensity near the fundamental frequency is high enough. This is true for the case of the nitrogen gas but not with air. The first re-focusing event appears at L ~100 cm at the experimental conditions quite close to our [18], while the distance to the second one from the first focusing point is much smaller. Hence, the origin of multiple spectral maxima in nitrogen indeed can be due to consecutive re-focusing events.

Figure 4 presents the amount W of energy passed through the aperture (d = 500 μm, L = 140 cm) in dependence on the nitrogen gas pressure. This energy was ~1 mJ at pressures below 0.5 atm (the transmittivity t = 0.22 ± 0.05 corresponds to the transmittivity t ~0.2 of the same aperture for the collimated beam without self-focusing). The transmittivity increases steadily with pressure and equals ~0.65 ± 0.15 (W ~2.5–3 mJ) at the pressure of 1 atm. One could conclude that the red-shifted Raman component takes nearly half of the energy of the initial lase pulse, taking into account data in Fig. 2a.
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Fig. 4

The dependence of the energy W passed through the aperture with d = 500 μm of the nitrogen pressure P. Initial pulse energy 4.5 mJ, L = 140 cm

In Fig. 3b we show spectra obtained at the nitrogen gas pressure of 0.5 atm corresponding to the filamentation “threshold” (see Fig. 4). These spectra contain the red-shifted component, but it does not experience further spectral shifting with the distance L. This again points at the fact that initially the Raman component appears during the filament formation, while spectral self-shifting comes into play during propagation of this component in the filament created by the fundamental radiation (see also the green curve corresponding to the L = 10 cm in Fig. 2b).

We also measured temporal shape of the Raman shifted component using the SPIDER technique. The SPIDER set up provides for additional spatial filtering and eliminates radiation coming not from the filament core [18]. This allows us to use the 700 μm aperture for measuring temporal structure at different points along the filament (larger aperture makes SPIDER measurements easier lowering simultaneously problems associated with the beam pointing instability). The SPIDER signal was very stable in our experimental conditions. Typical single-shot data obtained at the L = 140 cm is presented in Fig. 5a. One can see that the Raman component is confined in time having a duration slightly less than the duration of the initial laser pulse. Moreover, this pulse has no chirp, and its spectral and temporal phases are nearly flat.
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Fig. 5

The typical temporal envelope of the Raman component retrieved from the SPIDER data (a) and the time–frequency diagram retrieved from the SPIDER data using the PG FROG representation (b). Nitrogen pressure 1 atm, d = 700 μm, L = 140 cm

The high quality of the pulse is well demonstrated by Fig. 5b, which represents the time–frequency diagram of the pulse calculated from the SPIDER data using the PG FROG algorithm [29]. We revealed that the temporal shape of the Raman component is nearly the same for any distance L up to the end of the filament (see Fig. 6).
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Fig. 6

The dependence of the FWHM duration τ of the Raman component of the distance L. Nitrogen pressure 1 atm, d = 700 μm

4 Discussion and estimates

Let us now consider how the spectrally and temporally confined, IR-shifted light bullet could appear during the filamentation. The commonly used equation governing changes in spatial, temporal and spectral properties of radiation inside a femtosecond filament is [30, 31]:
$$ 2ik_{0} \frac{\partial E}{\partial z} = \hat{T}^{ - 1} \Updelta_{ \bot } E - k_{0} \left( {k^{\prime\prime}\frac{{\partial^{2} E}}{{\partial \tau^{2} }} + \hat{D}E} \right) + 2k_{0}^{2} \left( {\hat{T}\Updelta n_{k} + \hat{T}^{ - 1} \Updelta n_{p} } \right)E - ik_{0} \alpha E, $$
(1)
where E is an envelope of the electric field, z is a propagation coordinate, k0 is a wave number at the carrying frequency ω0 corresponding to the central wavelength λ0, τ is a time in a coordinate system propagating with a group velocity, \( \hat{T} = 1 - {i \mathord{\left/ {\vphantom {i {\omega_{0} }}} \right. \kern-0pt} {\omega_{0} }}{\partial \mathord{\left/ {\vphantom {\partial {\partial \tau }}} \right. \kern-0pt} {\partial \tau }} \), \( k^{\prime\prime} \) is a second-order dispersion coefficient at ω0, \( \hat{D} \) is a high-order dispersion operator, Δnk and Δnp are Kerr and plasma nonlinearities, α is a multiphoton absorption coefficient. Air and nitrogen are molecular gases, therefore Δnk contains both instantaneous electronic and inertial Raman responses:
$$ \Updelta n_{k} = \frac{{n_{2} }}{4}\left( {\left| {E(x,y,z,\tau )} \right|^{2} + \int\limits_{ - \infty }^{\tau } {\Uptheta (\tau - \tau^{\prime})\Upomega^{2} \exp \left( { - \frac{{\Upgamma (\tau - \tau^{\prime})}}{2}} \right)\frac{{\sin \left( {\Uplambda (\tau - \tau^{\prime})} \right)}}{\Uplambda }\left| {E(x,y,z,\tau^{\prime})} \right|^{2} d\tau^{\prime}} } \right), $$
(2)
where n2 is the Kerr coefficient of the gas at ω0, Λ2 = Ω2 – Γ2/4 (for the air at normal conditions Ω = 20.6 THz, Г = 26 THz) and Θ(τ) is the Heaviside step function.

The code, solving Eq. (1), was extensively used in our previous studies, including filament formation, pulse self-compression, polarization rotation, etc. [14, 18, 24, 31]. Unfortunately, the current state of this code is not appropriate to shed light onto the problem under consideration. This mainly is due to the fact that group velocity dispersion \( k^{\prime\prime} \) and other dispersion coefficients (in the operator \( \hat{D} \)) are fixed in the code, i.e., not affected by fast changes to the filament index of refraction due to plasma generation.

Formation of a soliton in a Kerr medium, including formation of a Raman soliton-like structures [22], is possible only if this medium has anomalous GVD [21]. At the same time, the air and nitrogen gas have the normal dispersion in the optical range [32], hence soliton-like structures cannot be formed in these media.

Changes in the GVD could come from various nonlinear processes in a filament. The total index of refraction of the filament can be written in the following form:
$$ n(\omega ) = 1 + A + B\omega^{2} + n_{2} (\omega )I(\omega ) - \frac{{2\pi e^{2} N_{e} }}{{m_{e} \omega^{2} }} + n_{\rm{HOKE}} (\omega ), $$
(3)
where me and e are electron mass and charge respectively, Ne is the concentration of a free electrons, ω is the radiation frequency, I is the laser pulse spectral intensity, and A = 2.73 × 10−4, B = 5.79 × 10−37 c2 for the air under normal conditions [32]. First three terms in (3) describe the index of refraction of the air, the fourth one comes from the Kerr impact, the next term is the filament plasma response within the Drude model, while the last one is due to the high-order Kerr effect (HOKE, [33]). The most reasonable physical mechanism of the HOKE under filamentation is the impact of highly excited atoms due to the multi-photon resonant absorption in the Rydberg states [34]. In [33] the term nHOKE was written as the expansion of powers of the intensity I in temporal domain. However, wave functions of the Rydberg states are very broad in space and one needs nonperturbative methods of their analysis instead. The susceptibility per one electron for the Rydberg states has nearly the same absolute value and frequency dependence as the susceptibility of a free electron [35]. Hence one can rewrite (3) as
$$ n(\omega ) \approx 1 + A + B\omega^{2} + n_{2} (\omega )I(\omega ) - \frac{{2\pi e^{2} N_{e} }}{{m_{e} \omega^{2} }}\left( {1 + \alpha (\omega )\frac{{N_{R} }}{{N_{e} }}} \right) $$
(4)
where NR stands for the concentration of atoms in a Rydberg state, and \( \alpha (\omega ) \approx 1 \) is a slowly varying function of the frequency ω [35].
The GVD \( k^{\prime\prime}(\omega ) \) (\( k(\omega ) = {{\omega n(\omega )} \mathord{\left/ {\vphantom {{\omega n(\omega )} c}} \right. \kern-0pt} c} \)) inside a filament is:
$$ k^{\prime\prime}(\omega ) = \frac{{\partial^{2} k}}{{\partial \omega^{2} }} \approx 6B\omega - \frac{{4\pi e^{2} N_{e} }}{{m_{e} \omega^{3} }}\left( {1 + \alpha (\omega )\frac{{N_{\text{R}} }}{{N_{e} }}} \right) $$
(5)

Here we took into account that \( n_{2} (\omega ) \) dependency is slow [36], while spectral power density of the pulse envelope amounts at frequencies 10–100 times lower than the frequency ω0, and one can neglect normal Kerr impact to the GVD. This can be applied for a few cycle pulses and/or in the case of the leading front sharpening due to a non-linear dispersion [37]. The ratio NR/Ne varies in time and strongly depends on the laser field strength, so the issue of HOKE-to-plasma impact on filamentation is widely discussed nowadays [33, 3840]. In the recent numerical study [41] it was shown (by direct solving the 3D Shrodinger equation for the Xe atom with an ionization potential of 12.13 eV in a strong laser field) that for intensities above 20 TW/cm2 the ionization probability surpasses few times the excitation one. Thus, we can put \( \frac{{N_{\text{R}} }}{{N_{e} }} \ll 1 \) in (5), and neglect the HOKE impact below. Note, that the HOKE-induced anomalous GVD can be essential at the pulse front and/or at the plasma filament periphery.

Since \( k^{\prime\prime}(\omega \to 0) \to - \infty \) and \( k^{\prime\prime}(\omega \to + \infty ) \to + \infty \) the equation \( k^{\prime\prime}(\omega ) = 0 \) has a solution corresponding to the non-zero electron concentration \( N_{e}^{\text{R}} \) at which the GVD inside the filament equals zero at a given frequency ω:
$$ N_{e}^{\text{R}} (\omega ) = \frac{{3Bm_{e} \omega^{4} }}{{2\pi e^{2} }}. $$
(6)
If \( k^{\prime\prime}(\omega ) = 0 \) is satisfied at the frequency nonperturbative ω0, the GVD is anomalous for all ω < ω0. Consequently, the relation (6) determines the key condition in the Raman soliton formation inside the filament. Figure 7 presents the dependence of \( k^{\prime\prime}(\omega_{0} ) \) inside the filament on the concentration Ne. The shaded area corresponds to the area of anomalous GVD. Estimates show that \( N_{e}^{\text{R}} \) = 3.26 × 1016 cm−3 at λ0 = 800 nm (air under normal conditions).
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Fig. 7

The dependence of the second-order dispersion coefficient inside the filament \( k^{\prime\prime} \) of the free electrons concentration Ne (at λ = 800 nm). The shaded area corresponds to the anomalous GVD where Raman soliton is possible

Experimental determination of the electron concentration inside the plasma channel of the filament is intensively investigated but a difficult task, because diameter of the plasma channel is small, below 100 μm, and the electron concentration rapidly changes at the leading front of the laser pulse. Besides this concentration is still low for the easy implementation of commonly used plasma physics methods. Different indirect methods were used for this goal, including optical interferometry [42, 43] and in-line holography [44], THz probing [43], plasma fluorescence [45], and secondary electrical discharges [46]. Experimental conditions (focal distance and numerical aperture, gas composition, contamination and humidity, etc.) and methods used (in particular, spatial and temporal averaging of data) have the prominent impact on the inferred properties of the plasma channel. In the most cases, the experiments provide with averaged over time and space values for the electron concentration ranging from 1015 to a few times 1017 cm−3. Electron concentrations above 3 × 1016 cm−3 were reported even if the filament is launched by the loosely focused laser beam [44, 45].

Consequently, the above-mentioned conditions for the formation of anomalous GVD inside the plasma channel of the filament can be satisfied at the back front of the laser pulse, and at Ne = 6 × 1016 cm−3 the GVD \( k^{\prime\prime} \) =  –22 fs2/m (instead of \( k^{\prime\prime} \) = 16 fs2/m for the air under normal conditions). To confirm that a Raman soliton could be formed inside the filament we made numerical simulation using the 1D variant of the Eq. (1) with \( k^{\prime\prime} \) = –22 fs2/m and other parameters taken for the air under normal conditions. Data presented in the Fig. 8 clearly supports this idea (see also experimental data in the Fig. 2b). Additional approval of the condition (6) comes from the data plotted in Fig. 3 for the nitrogen gas at pressures of 1 and 0.5 atm. Indeed, the condition (6) fails at the lower pressure since the filament does not form and electron concentration is relatively low. This explains why we saw the red-shifted component in this case, but it did not experience further spectral shifting.
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Fig. 8

Simulated spectra of radiation in the filament core obtained at different distances L (air at a pressure of 1 atm with \( k^{\prime\prime} \) = –22 fs2/m)

5 Conclusions

Thus, our study confirmed the importance of more complex description of the filamentation process, especially if collimated beams are used to produce long filaments in an open air. It also opens up new possibilities for efficient generation of collimated ultrashort pulses in IR and visible bands.

The experimental study of the spectral, spatial and temporal behavior of the IR shifted component observed under filamentation of the collimated femtosecond laser beam in molecular gases showed that this component behaves like a 3D light bullet:
  1. (1)

    it propagates within the filament plasma core,

     
  2. (2)

    it has a stable duration of 30 fs along the filament, and

     
  3. (3)

    its spectrum shifts as a whole from 820 to 870 nm along the filament.

     

This “soliton-like” light bullet can gain up to 30–50 % of the energy of the fundamental pulse. It is formed during the filament formation but its spectral shift starts if the pressure is above the certain threshold (0.7–0.8 atm for the nitrogen gas). We observed this 3D light bullet in the air and nitrogen gas at different pressures, but nothing was observed with argon gas. This supports the idea that the bullet is formed due to the rotational Raman process. The type of molecular gas used to launch the filament is also important: depending on its non-linearity, ionization threshold, etc. few shifting spectral components could be formed (in nitrogen gas three spectral components were observed separated by 20–30 nm).

Properties of the 3D light bullet are almost like as of the well-known self-shifted Raman soliton in optical fibers. It is impossible to describe such a soliton-like structure formation within the framework of the simplified commonly used model of the filamentation process with fixed GVD coefficient. The key equation of this approach describes a filament based on the fundamental generalization of the slowly varying amplitude method [47] and is built for the field envelope. It does not describe changes in the GVD due to plasma and Kerr impacts. The frequency dependence in plasma current may be taken into account in the forward Maxwell equation [48] or unidirectional pulse propagation equation [49]. A thorough simulation of the 3D Raman light bullet formation is underway in our group.

Acknowledgments

The authors wish to thank prof. L. Berge for fruitful discussions. This work was partially supported by the Russian Foundation for Basic Research (Grants #12-02-01368-a, #11-02-12061-ofi-m-2011, #12-02-31341-mol-a, #12-02-33029-mol-a-ved), the Council of the President of the Russian Federation for Support of Young Scientists (No. 5996.2012.2) and Leading Scientific Schools (No. 6897.2012.2), Ministry of Education and Science of Russian Federation (Grant #8393). N.P. also acknowledges support from the Dynasty Foundation.

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© Springer-Verlag Berlin Heidelberg 2012