Filamentation of high-power femtosecond laser pulses in gases [1] allows one to achieve the intensity of ~100 TW/cm2 for Ti/Sa pulses in the light channel with the diameter of ~100 μm and the length of several meters [2]. Nonlinear processes occur efficiently under such conditions. As a result, the coherent radiation of the supercontinuum from terahertz (THz) to ultraviolet (UV) spectral range is emitted [3–7]. New spectral components in high- and low-frequency range propagate as a conical emission in case of filamentation in the medium with normal dispersion [8]. Conical emission in the visible spectral range was observed experimentally about 20 years ago [1, 9, 10]. The infrared (IR) radiation outgoing from the filament in air was found to be localized on its axis [1, 9–12]. With the decrease in the frequency down to 20 THz (increase in the wavelength up to 15 μm), the radiation diverges from the filament at an angle up to ~50 mrad [13].

Near-infrared spectral range is the only part of the air filament continuum, which propagates forwardly with its maximum located on the beam axis. The experiments [14, 15] show that these newly born IR components of the spectrum are separated from the fundamental radiation and shift to the long wavelength range with propagation distance. Figure 1 represents the spectra of the self-shifted IR component at different propagation distances measured in Refs. [14, 15]. Moreover, IR component turned to be localized on the filament axis and have the stable duration of approximately 30 fs, which remains almost unchanged along the filament [15]. Later experiments [16, 17] demonstrate that four-wave mixing between IR component and fundamental radiation at 800 nm leads to the generation of the radiation in the visible range. The theoretical explanation of this phenomenon has been done using the slowly varying envelope approximation approach [14]; however, the explicit agreement between the experiment and the simulations has not been attained. Other attempts were made recently to semiquantitatively interpret the observed bullet based on the anomalous dispersion in the laser-produced plasma [15–17].

Fig. 1
figure 1

Spectra of the isolated IR component obtained experimentally under filamentation of 50 fs, 2.5 mJ, 4 mm FWHM diameter loosely focused pulse [14] (a) and 55 fs, 4.5 mJ, 1.3 mm FWHM diameter collimated pulse [15] (b) at several distances

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In this paper, we for the first time to our knowledge systematically reproduce the robust forwardly directed bullet-type radiation from 800-nm femtosecond filament. Our 3D+ time axially symmetric simulations with consideration of the fast oscillating carrier wave show the stages of the bullet formation such as the bullet appearance at 860 nm soon after the beginning of the filament, its forward propagation and gradual energy growth. The simulation results are in good agreement with the experimental data [14, 15].

For the simulation of filamentation phenomenon, we use forward Maxwell equation [18]. We assume that the light field E(r, τ, z) is linearly polarized and axially symmetric (r is the transverse coordinate, τ is the time in moving reference frame, and z is the propagation coordinate). Let us represent the field E(r, τ, z) in the Fourier domain:

$$E(r,\tau ,z) = \int_{-\infty}^{\infty} {\hat{E}}(r,\omega ,z)e^{i\omega \tau } {\text{d}}\omega,$$
(1)

where ω is a frequency. Then the equation for the evolution of \(\hat{E}(r,\omega ,z)\) spectral components is

$$\frac{{\partial \hat{E}(r,\omega ,z)}}{\partial z} = - i\left( {k(\omega ) + \frac{{\Delta _{ \bot } }}{2k(\omega )}} \right)\hat{E}(r,\omega ,z) - \frac{2\pi }{cn(\omega )}\left( {\hat{J}(r,\omega ,z) + i\omega \hat{P}(r,\omega ,z)} \right) ,$$
(2)

where Δ⊥ = r −1 ∂/∂r (r∂/∂r) is the transverse Laplacian, k(ω) = ωn(ω)/c is a wave number, P is a nonlinear polarization, J is an electric current, n(ω) is the refractive index, and c is the speed of light. Spectral components of the polarization \(\hat{P}(r,\omega ,z)\) and the current \(\hat{J}(r,\omega ,z)\) are determined similarly to the light field in Eq. (1).

Nonlinear polarization P can be expressed as [19]

$$P(\tau) = \chi_{xxxx}^{(3)} \left\{ {(1 - g)E^{2} (\tau) + g\int_{ - \infty }^{\tau } {E^{2} (t^{\prime})H(\tau - t^{\prime})\;{\text{d}}t^{\prime}} } \right\}E(\tau) ,$$
(3)

where χ (3) xxxx is the third-order nonlinear susceptibility and g is a fraction of the inertial nonlinearity (we assume g = 1/2). Inertial response function can be written as

$$H(\tau ) =\Theta (\tau )\;\Omega ^{2} \exp \left( { - \frac{{\Gamma \tau }}{2}} \right)\frac{{{{\text{sin}}\varLambda }\tau }}{\Lambda },$$
(4)

where Λ2 = Ω2 – Γ2/4, Ω = 20.6 THz, Г = 26 THz, and Θ(τ) is a step function.

The current J = J free + J abs includes free electron current J free and the absorption current J abs, which simulates energy losses due to multiphoton or tunnel ionization. J free can be obtained from the equation:

$$\frac{{\partial J_{\text{free}} (\tau )}}{\partial \tau } = \frac{{e^{2} }}{m}N_{e} (\tau )E(\tau ) - \nu_{c} J_{\text{free}} (\tau ) ,$$
(5)

where e and m are electron charge and mass, respectively, and ν c  ≈ 5 THz is an electron-neutral collision frequency [20]. Ionization-induced free electron concentration is determined by the expression

$$N_{e} (\tau ) = N_{0} \left( {1 - \exp \left[ { - \int_{ - \infty }^{\tau } {R\left( {E(t^{\prime})} \right){\text{d}}t^{\prime}} } \right]} \right),$$
(6)

where N 0 = 2.7 × 1019 cm−3 is the atmospheric air molecular density and R is the light-field-dependent ionization probability.

Absorption component of the current is

$$J_{\text{abs}} (\tau ) = \frac{{W_{I} }}{E(\tau )}\frac{{\partial N_{e} (\tau )}}{\partial \tau },$$
(7)

where W I is the ionization potential.

Initial conditions at z = 0 are chosen as both time and space gauss-shaped function:

$$E(r,\tau ,z = 0) = E_{0} \;\text{exp}\;\left( { - \frac{{r^{2} }}{{2a_{0}^{2} }} - \frac{{\tau^{2} }}{{2\tau_{0}^{2} }}} \right)\,\text{cos}\omega_{0} \tau ,$$
(8)

where E 0 is an electric field amplitude, 2τ 0 is a duration by e −1 of maximum, 2a 0 is a diameter by e −1 of maximum, and ω 0 is a central frequency. According to the experiments, we chose the pulse energy of 3 mJ, a diameter of 1 mm and duration of 54 fs. Carrier frequency ω 0 corresponds to the wavelength of 800 nm. As compared to our earlier work [14], where the light bullet was not pronounced, we employed the correct consideration of the diffraction operator due to the usage of the forward Maxwell model, we used excellent resolution combined with a large frequency range considered (0.125–8000 THz with the grid step of 0.125 THz), and we concentrated on the diligent study of the frequency–angular spectrum of the pulse in the filament. These three factors have made the reproduction of the experimental results on the infrared light bullet formation possible.

The dependence of the light field intensity on propagation distance z is presented in Fig. 2. The start of the filament is at the distance z = 45 cm. The conical emission develops from the very beginning of the filament. The conversion efficiency to both the infrared and the visible parts of the emission stabilizes by the distance of z = 98 cm (Fig. 3). The angle of the conical emission in the mid-infrared range is ~10 mrad in good agreement with the experimental study [13]. However, there exists the spectral range (310–350 THz or 860–960 nm marked with white dashed lines in Fig. 3), where the radiation propagates forwardly and stays localized on the axis. This radiation is expected to correspond to IR light bullet observed in the experiments.

Fig. 2
figure 2

Dependence of the peak intensity of 800-nm laser radiation (solid line) and energy of IR component (red dots) on the propagation distance z

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Fig. 3
figure 3

Angular–frequency spectrum of the radiation at the distance z = 98 cm (logarithmic color scale). White dashed lines indicate the frequency range where the light bullet exists

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Let us concentrate on this wavelength range specific part of the 860–960 nm supercontinuum properly in both spectral and temporal domains. One can see in Fig. 4 the unimodal structure that gradually separates from the 800-nm fundamental radiation. The first sign of this separation shows up at the propagation distance z = 61 cm (Fig. 4a). Further on the separation becomes well pronounced by z = 82 cm and results in the formation of the minimum between the fundamental radiation and the IR component (Fig. 4b). This separation is directly associated with the overall shift of the bullet intensity maximum toward the long-wave part of spectrum (compare Fig. 4b, which corresponds to z = 82 cm, and Fig. 4c, which corresponds to z = 98 cm). Since the angular divergence of the self-shifted IR component is less than 1 mrad, one can claim that it propagates along the filament axis with high accuracy in agreement with the experimental observation [21].

Fig. 4
figure 4

Angular–frequency spectra of radiation at several distances (logarithmic color scale)

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Figure 5a–c represents temporal distribution of the on-axis electric field at the frequencies below 353 THz (wavelengths above 850 nm). One can see the soliton-like structure in the leading front of the pulse: Its envelope is robust and remains the same for at least half a meter of propagation distance after its formation (Fig. 5a–c, the blue lines). Light bullet occupies of about 10 periods of the light field oscillations (Fig. 5a–c), i.e., about 30 fs as was derived from spectral phase interferometry for direct electric field reconstruction (SPIDER) measurements [15]. The overall shift of the bullet toward the long-wavelength side of the spectrum constitutes of the order of 50 nm in both the simulation and the experiment (Figs. 1, 5d–f). The energy of the light bullet increases slightly with the propagation distance z and by the end of the filament reaches of about 5 % of the initial pulse energy (Fig. 2, red dots). It corresponds to 12 GW in case of the duration of 30 fs, i.e., about one critical power for self-focusing of the radiation with the central wavelength of 900 nm. The conversion efficiency into the IR bullet in the experiment [14] was about 9 % of the initial energy of 3.8 mJ in the 50 fs pulse focused by 3-m lens at the distance of 3.7 m from the laser output (see Fig. 1 in Ref. [14]). This is in adequate agreement with the results of our simulation. The authors of Ref. [15] reported the larger quantity, namely 30–50 % of the initial 800 nm pulse energy, transmitted to the IR bullet. We note that this percentage was calculated as a fraction of the energy of the radiation transmitted through the on-axis aperture with the diameter 300, 500 or 1000 μm, while the initial beam size at the beginning of filamentation was 1300 μm. Naturally, the overall initial beam size as well as the conical emission has not passed through the on-axis aperture. Thus, when comparing with our simulation results, this high conversion efficiency number of 30–50 % should be decreased by more than a factor of 2. Here we imply that the area of the ring surrounding the on-axis aperture is larger than the aperture area itself. A factor of 2–3 decrease of 30–50 % conversion efficiency is in reasonable agreement with our simulation results.

Fig. 5
figure 5

On-axis light field (a–c) and spectral (d–f) distributions of the radiation with the frequencies <353 THz (the wavelengths >850 nm). The blue line represents the light field envelope

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The IR bullet exists in the front of the pulse (τ < 0, see Fig. 5), where Kerr effect and diffraction rule its evolution. Hence, the appearance of the minimum in the spectrum at the wavelength of about 850 nm (see Fig. 4b, c) and subsequent separation of the IR bullet from the fundamental radiation of 800 nm is the result of the fine interaction between diffraction, Kerr self-focusing and self-phase modulation. Self-phase modulation leads to the local spectral maxima formed at certain frequencies in the course of propagation [22]. In contrast to the slowly varying envelope approach [see, e.g., 10], the corrected diffraction operator cΔ⊥/2ω used in Forward Maxwell Eq. (2) [18] acts selectively on different frequencies. The joint effect of self-phase modulation due to Kerr nonlinearity and diffraction results in the separation of a bullet in frequency–angular domain as shown in Fig. 4. Physical explanation of this phenomenon is the energy transfer in the nonlinear Kerr medium from 800 nm to the frequencies seeded by self-phase modulation and propagating mainly in the forward direction due to their formation in the leading front of the pulse, which is unperturbed by the plasma.

In conclusion, we have identified the propagation regime in air in the course of which the confined near-infrared robust light bullet is formed during filamentation. We for the first time have found numerically based on the solution of the forward Maxwell equations with the carrier wave that the bullet consists of about 10 cycles of the near-infrared wavelength corresponding to the bullet’s carrier frequency. This temporal distribution is in agreement with the experimentally registered duration of the bullet. Moreover, the carrier frequency of the simulated bullet becomes smaller with the propagation. This shift corresponds to several tens of nanometers, which was observed in Refs. [14, 15]. Finally, the bullet accumulates the energy during the filament. The peak power of the bullet is slightly above the critical power for self-focusing for its carrier frequency. Thus, this confined light structure can persist to propagate forwardly for a long distance after the end of the plasma column. This phenomenon was experimentally found and called “postfilament propagation” [21], and our simulations explain the physics of this result.