1. Introduction

There has recently been considerable interest in how the generation of an underdense plasma affects the propagation of femtosecond laser pulses in gases. In particular, recent experiments have shown that high peak-power laser pulses (~ GW) generate stabilized light filaments which propagate for distances on the order of 100 m under laboratory conditions [1–5] and kilometers for TW peak powers in the atmosphere. [6] The intuitive explanation of self-focusing (SF), the critical power being around a GW, being balanced by plasma-induced defocusing generated via multi-photon ionization [1] (MPI) yielding stabilized filaments is very attractive. It suggests that a significant portion of the light energy could be “transported” over long distances in the form of “light bullets”.

Brodeur et al. [5] noticed that the length of the filaments observed in their experiments was approximately one Rayleigh range (here we mean the Rayleigh range as defined by the input pulse parameters, not the one based on the filament’s parameters), which is consistent with the moving focus model, and their observation was supported by numerical simulations [7,8]. Despite the intuitive appeal of the moving focus model for visualizing the complex spatio-temporal dynamics of the propagating field, strict application of this simple model cannot explain the experiments recently performed by Lange et al. [9] who presented results of anomalous long-range pulse propagation in a focusing geometry. According to the moving focus model, the “channel” would be confined within the region between the nonlinear (power dependent) and linear foci. In the experiment, however, the pulse propagated beyond the linear focus, a fact they advanced as evidence against the moving focus picture. Lange et al. [9] recently argued in favor of the pulse self-guiding stabilized by plasma defocusing.

Based on our numerics we advanced a more dynamic picture of the channeling phenomena in Ref. [7]. Within this picture pulses form, are absorbed and defocused, and subsequently replenished by new pulses, thereby creating the illusion of one pulse which is self-guided. The new emerging pulses gain their energy from the outer and trailing parts of the pulse which can have enough power to refocus. These results do not resemble the simple self-guiding picture at all for the parameters used in Ref. [7]. Rather, the length of the filaments seems to be roughly linked to a Rayleigh range, in agreement with the moving focus model of Brodeur et al. [5]. The dynamic picture does not change if we use a focusing geometry in our simulations [10]. In this case we also observe the “channel” beyond the linear focus which can be explained within the “replenishment scenario”.

In this article we shall focus on the field evolution during its propagation through air as the input peak power is varied. In particular, we wish to take full advantage of the multi-media capabilities of this journal to present detailed simulations of the spatial dynamic replenishment scenario and its implications for the propagation scale for filaments. Based on the self-waveguiding model, we would expect that the excess energy would tend to prolong the length of the filaments since the trapped filament would have a larger background energy reservoir. However, our simulations show nonmonotonic dependence of the filament’s propagation distance on the input power. Our report also includes the results obtained with or without the inclusion of the delayed Raman-type cubic nonlinearity.

2. Model

We shall start with a brief discussion of our model for pulse propagation in the infrared at a wavelength of 775 nm. Assuming propagation along the z-axis, the equation (in radial symmetry) for the slowly varying electric linearly polarized field envelope ε(r, z, t) in a reference frame moving at the group velocity is:

where the terms on the right-hand-side describe transverse diffraction, group velocity dispersion (GVD), absorption and defocusing due to the electron plasma, MPI, and nonlinear SF. Here ω is the optical frequency, |ε|² the intensity, k = n_bk ₀ = n_bω/c, k″ = ∂ ² k/∂ω)², ρ is the electron density, σ the cross-section for inverse bremsstrahlung, τ is the electron collision time, β ^(K) is the K-photon absorption coefficient, and the nonlinear change in refractive-index for a continuous wave (cw) field is n ₂|ε|². The critical power for self-focusing collapse for cw fields is defined as P_cr = ${\mathrm{\lambda }}_0^2$/2πn_bn ₂ which equals to roughly 1.7 GW for the parameters from Ref. [7]. The normalized response function (characterized by the resonance frequency Ω and the decay Γ)

accounts for delayed nonlinear effects, and f is the fraction of the cw nonlinear optical response which has its origin in the delayed component, and we denoted the Heaviside step function by θ(t). The evolution of the electron density is described by the Drude model

The first term on the right-hand-side of this equation describes growth of the electron plasma by cascade (avalanche) ionization, the second term is the contribution of MPI, and the third term denotes radiative electron recombination. This model was obtained using the following assumptions: The density of plasma is much less than the atomic density, field intensities are below the threshold for the double ionization of the air constituents, but low enough to ensure the multiphoton character of the absorption.

We are interested in solutions of these equations for an initially collimated input Gaussian beam

where P_in is the peak input power, w ₀ the spot size, t_p characterizes the pulse length (full-width-half-maximum of intensity is then ${\tau }_{\mathrm{FWHM}}=\sqrt{2\mathrm{In}2t_p}$). The Rayleigh range of the input beam is given by z ₀ = ${\pi w}_0^2$ n _b/λ₀, and we considered a spot size w ₀ = 0.7 mm giving z ₀ = 2 m and a pulse length τ _FWHM = 200 fs. The numerical results were checked by doubling the space and time resolutions which led to no significant changes in the behavior of the results presented here.

3. Results and Discussion

The evolution of the global maximum over time of the on-axis intensity as a function of propagation distance z is shown in Figs. 1(a)–4(a) for input peak powers ranging from around 4P_cr to 7P_cr . (The dashed lines are the results when the Raman delayed response is identically zero). As a general feature in the behavior of the maximum over time on-axis intensity in Figs. 1(a)–4(a), we can observe that it initially grows explosively due to SF, but is then limited by MPI as well as absorption and defocusing due to the generated electron density. The corresponding global maximum over time of the on-axis density as a function of z is shown in Figs. 1(b)–4(b). After the intensity is limited, it remains fairly constant and leading to a plateau at intensities significantly above 10¹³ W/cm² in all cases (notice that the different cases are differently normalized according to the corresponding input power; the maximum intensity is in all cases approximately 2.7 × 10¹³ W/cm².), and eventually decays.

The initial explosive growth of the field intensity in each case can be understood in terms of the moving focus model as due to the above-critical-power time slices starting to self-focus. Beyond this point the intensity stabilizes due to MPI and exhibits a plateau which later decays. However, there are resurgences of the intensity for larger distances over spatial scales on the order of the length of the first plateau. In addition, we also note the appearance of “notches” in the plots of the maximum over time on-axis intensity. This phenomenon of intensity resurgences is in agreement with the experimental measurements of the filament energy resurgences [5] It was shown in Ref. [7] that the dynamic spatial replenishment scenario whereby an initial pulse is formed, absorbed and defocused by plasma generation, and subsequently replenished by power from the trailing edge of the pulse, manifests itself in these curves as such “notches”. These “notches” are connected to the appearance of the secondary refocused pulse at the trailing edge as it takes over as the global maximum from the decreasing leading peak. We can see a signature of this process in Figs. 5 and 6 where we plot the intensity of the field at two different propagation distances in the case of the input power equal to ~ 4P_cr (compare Fig. 1) and with the delayed nonlinear response included. The original self-focused pulse in Fig. 5 loses its energy while it generates plasma, which defocuses the trailing part of the field. Once the plasma is insufficient to counteract the effect of self-focusing, the trailing edge pulse shown in Fig. 6 is created. A more detailed picture of this dynamics can be obtained from the movies presented later in this report. We would also like to stress that even though the field evolution in space and time is dynamic and complex in our simulations, the fluence distribution, that is the time integrated intensity typically observed in the experiments, stays very well spatially localized [10].

Next, we turn to the question of how the input power affects the pulse propagation. Intuitively, based on the self-waveguiding picture, it would be expected that increased energy of the pulse would lead to a longer propagation distance: There is a larger energy reservoir to balance the losses. However, as we increase the initial power of the pulse in our simulations, the first resurgence of the field (re-self-focusing) happens sooner (Fig. 1–4). Since the intensity of the field roughly maintains the average lengths of the first plateau and the first resurgence, this leads to the overall shortening of the filament. This effective shortening of the filament continues as we increase the input power until the pulse energy allows for another resurgence to occur (see Fig. 3). As we are limited in the maximum input power by numerical constraints we observed only two resurgences in our simulations. We also expect that with the increase of the input power the beam will be more susceptible to small perturbations which can grow faster and eventually lead to the radial symmetry breaking. Such a process leads to the propagation of multiple filaments also observed in the experiments and can be relevant to atmospheric propagation. However, our key observation here is that within the spatial dynamic replenishment scenario, filament propagation is not overly influenced by the peak input power beyond the threshold for filament formation, assuming radial symmetry is preserved, as any extension in the intensity plateau region stabilized by MPI is offset by the initial distance required for SF. This conclusion holds whether the Raman delayed response is included or not; the gross effect of the Raman response is just rescaling the input power, which is defined in terms of the critical power P_cr obtained using the total (cw) nonlinear index.

 

Figure 1. Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 4P_cr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n ₂ without the Raman contribution.)

(b) The corresponding maximum on-axis density as a function of z.

Associated movie (2.2 MB)

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Figure 2. Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 6P_cr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n ₂ without the Raman contribution.)

(b) The corresponding maximum on-axis density as a function of z.

Associated movie (2.4 MB)

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Figure 3. Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 6.5P_cr . The Rayleigh range was in this case ~ 2 m. Dashed line … n ₂ without the Raman contribution.)

(b) The corresponding maximum on-axis density as a function of z.

Associated movie (2.6 MB)

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Figure 4. Global maximum over time of the on-axis intensity as a function of propagation distance z for peak input power of ~ 7P_cr . The Rayleigh range was in this case ~ 2 m. (Dashed line … n ₂ without the Raman contribution.)

(b) The corresponding maximum on-axis density as a function of z.

Associated movie (2.7 MB)

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Figure 5. The intensity of The field for the case shown in Fig. 1 (Raman response included) at the propagation distance of 1 m.

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Figure 6. The intensity of the field for the case shown in Fig. 1 (Raman response included) at the propagation distance of 2.8 m.

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Finally, in order to make the dynamics of the pulse propagation in air more accessible to the reader we have included the whole spatio-temporal evolution of the pulse in the form of a “quicktime” movie. There is one movie associated with each Fig. 1(a)–4(a). Each movie consist of two graphs, one for the intensity of the field versus radial coordinate r and time t, and the other for the density of plasma as a function of r and t. Both graphs are evolving as a function of the propagation distance z. Because the units of the intensity and density in these spatio-temporal graphs are normalized to arbitrary units, we also plotted the corresponding global maximum over time of the on-axis quantities (Figs. 1–4) in the inset. Again, we can observe in these movies the general scenario: The maximum intensity initially grows explosively due to SF, but is then limited by MPI as well as absorption and defocusing due to the electron density. Notice the effect of defocusing on the trailing part of the field during the first “intensity plateau”, and its subsequent refocusing into the trailing peak. Upon further propagation the leading peak decays while the trailing one remains, until it too decays. The notch in the intensity curve is therefore explained as that propagation distance where the increasing trailing peak takes over as the global maximum from the decreasing leading peak. This process can repeat itself if the power in the trailing part of the field is sufficient to refocus again.

Conclusion

In summary, we have presented the results of simulations which attempt to expose the physics underlying long-distance propagation in air. We have shown that the evolution of the field is dynamic, involving the development of a leading-edge pulse that creates plasma, subsequently decays and is replaced by a new pulse. This process can repeat several times as we have shown in the study of the influence of increasing the input peak power in the pulse on its propagation and filaments. Contrary to the naïve expectation of longer propagation of the field in the form of filaments for higher input pulse power, our numerics reveals that the overall length of filaments does not increase monotonically with input power, rather becomes smaller or larger depending on whether or not the re-self-focusing of another pulse occurs.