1 Introduction

The LPP in gaseous media is attracting lots of attention because of its multifold roles in many areas, such as the filament process of an ultrashort laser in air [1–5], the induced spectral blue shifting of a femtosecond pulse in rare gases [6], and phase mismatch during high-order harmonic generations [7–9]. Of these areas the plasma channel created by the laser filament in air has been widely investigated due to its potential applications, such as remote sensing [10, 11], laser-induced lightning [12–14], and others (see [15, 16] and references therein). One critical issue hindering the development of applications is the lifetime of LPP, which requires detailed investigations of dynamics of LPP distributions.

When atoms or molecules are exposed to intense femtosecond laser fields, some of the bounded electrons will be separated from the nuclear cores through a variety of ionization mechanisms. The acceleration and velocity distribution of the LPP in a local thermodynamic equilibrium (LTE) have been widely studied [17–20]. The gaseous medium we used is sufficiently tenuous (about an atmosphere pressure, i.e., \(N_{0} \le 10^{25} \,{\text{m}}^{ - 3}\)), which means a large free path for the charged particles moving about several femtoseconds. This also signifies that the ionized electron cannot reach an adjacent ion. Therefore, the collisions between the electrons can be neglected. There is not enough time for the partially ionized plasma to reach a local thermodynamic equilibrium (non-LTE), which causes the invalidity of the traditional method to compute the electron plasma temperature [21]. A key physical process in this situation is the electron–ion recombination, in which the electron returns to its parent ion with high kinetic energy during the first laser period after ionization [22, 23]. In their models, Corkum et al. [24] calculated the electron’s drift energy in the absence of atomic potential for arbitrarily polarized laser field, and the return kinetic energies have a cutoff at 3.17 U p with the ponderomotive energy U p. As the separated electron returns to the vicinity of the ion, especially under the circumstances of a weak laser field, however, the atomic potential becomes important and should be taken into consideration. A clear picture of the velocity distribution of LPP is important to investigate the plasma density [25] and the energy transfer from driving laser to the electron heating.

The goal of this paper is to develop a method to calculate the velocity and kinetic energy distribution of LPP based on the ionization rate equation for the generation of LPP. The residual energies carried by the LPP are calculated using the velocity distribution, which can be used as the initial conditions for modeling the evolution of LPP after the laser pulses. Our derivation is based on the ionization rate equation for the generation of LPP and the collisionless Newtonian equation for an ionized electron, which means that, besides the case of laser filamentation in air in the multiphoton ionization (MPI) regime (\({\le}10^{14} \,{\text{W/cm}}^{2}\)), this scheme also applies to the tunneling ionization (TI) (\({\le}10^{16} \,{\text{W/cm}}^{2}\)) as long as the ionization rates are provided. Furthermore, the maximal range of motion of an ionized electron should be smaller than the averaged distance among electrons within tens of femtoseconds, which means a gaseous medium is always reasonable for our scheme. The atomic units are adopted throughout this paper except when otherwise specified.

2 Theoretical model

2.1 Time evolution of LPP density

In the MPI and TI regimes, the relativistic effects and magnetic components of the laser field can be dropped. During the laser–matter interaction, the LPP is mainly produced near the leading peaks of the laser pulse, and the evolution of LPP density is determined by the classical rate equation [26] phenomenologically adding an electron–ion recombination term [27]

$$\begin{aligned} \frac{{{\text{d}}N_{\text{e}} \left( t \right)}}{{{\text{d}}t}} & = R_{\text{pro}} (t) - R_{\text{rec}} (t) \\ R_{\text{pro}} (t) & = W\left( E \right)\left( {N_{0} - N_{\text{e}} \left( t \right)} \right) \\ R_{\text{rec}} (t) & = \mathop \int \limits_{ - \infty }^{t} \alpha (t,t_{\text{b}} )R_{\text{pro}} (t_{\text{b}} ){\text{d}}t_{\text{b}} \\ \end{aligned}$$
(1)

where R pro(t) and R rec(t) stand for the electron densities of production and recombination processes per unit time at t, respectively, and N 0 is the initial density of neutral atoms. The introduced recombination coefficient α(t, t b) characterizes the probability of the recombination of an electron born at t b with its parent ion at t. Its values depend on the trajectory of the electron born at t b and the corresponding kinetic energy obtained from the laser field. The probability conservation law requires \(\int_{{t_{\text{b}} }}^{{t_{\text{f}} }} \alpha (t,t_{\text{b}} ){\text{d}}t \le 1\), where t f is the total pulse duration.

The explicit expression of the ionization rate relies on the different interaction regimes characterized by the Keldysh adiabatic parameter \(\gamma_{\text{K}} = \sqrt {I_{\text{p}} /2U_{\text{p}} }\) [28], where I p is the atomic ionization potential, U p = E 2/(4ω L ) is the ponderomotive potential with the applied electric field E, and ω L is its central angular frequency in atomic units. In the multiphoton regime (γ K > 1), the ionization rate reads W(E) = σ K I K, where σ K represents the K-photon ionization cross section, and I ∝ |E|2 is the laser intensity. In the tunneling regime (γ K ≪ 1), the ionization rate coefficient W(E) is calculated by Ammosov–Delone–Krainov (ADK) model [28, 29].

2.2 Velocity distribution

Once released, the dynamics of the released electron born at t b are governed by the Newtonian equation, which reads [24]

$$\frac{{{\text{d}}^{2} }}{{{\text{d}}t^{2} }}{\mathbf{x}}\left( {t,t_{\text{b}} } \right) = - \nabla V\left( {x\left( {t,t_{\text{b}} } \right)} \right) - {\mathbf{E}}\left( t \right)$$
(2)

Here, \({\mathbf{x}}\left( {t,t_{\text{b}} } \right)\) is the position of the electron from its parent ion at t, V(x) is the atomic potential, and \({\mathbf{E}}(t)\) is an incident laser field. Here, the integral time t ranges from the birth time t b to the recombination time τ(t b) or the time when the laser ends, and the recombination time satisfies t = τ(t b) for α(t, t b) = 1.

In calculating the velocity distribution of LPP, the velocity v is an independent variable, while the velocity \({\mathbf{v}}(t,t_{\text{b}} )\) is a practical physical quantity in solving the motion equation of an ionized electron. In order to derive the velocity distribution of LPP, we take v as the reference velocity and discretize it between \([ - {\mathbf{v}}_{{\rm max} } ,{\mathbf{v}}_{{\rm max} } ]\) with a spacing of \(\Delta {\mathbf{v}}\) to measure the velocities of the ionized electron, where \({\mathbf{v}}_{{\rm max} }\) is a sufficiently large value of velocities. Our task is to calculate the density of released electrons whose velocities lie between \(({\mathbf{v}} -\Delta {\mathbf{v}}/2)\sim({\mathbf{v}} +\Delta {\mathbf{v}}/2)\). Taking account of the production of released electrons and the electron–ion recombination, we introduce a parameter β(t, t b) to pick out the electrons born at t b with the matched velocity \({\mathbf{v}}\left( {t,t_{\text{b}} } \right)\),

$$\beta (t,t_{\text{b}} ) = \left\{ {\begin{array}{ll} 1 \hfill & \quad{{\text{if }}\quad{\mathbf{v}} -\Delta {\mathbf{v}} / 2< {\mathbf{v}}\left( {t,t_{\text{b}} } \right) < {\mathbf{v}} +\Delta {\mathbf{v}} / 2} \hfill \\ 0 \hfill & {\quad {\text{otherwise}}} \hfill \\ \end{array} } \right.$$

The value of β(t, t b) depends on the t b-dependent velocity \({\mathbf{v}}\left( {t,t_{\text{b}} } \right) = {\text{d}}{\mathbf{x}}(t,t_{\text{b}} ) / {\text{d}}t\), which is obtained by solving Eq. (2). Therefore, taking the electron–ion recombination into account, the transient distribution can be defined as follows,

$$f(t,v)\Delta {\mathbf{v}} = \mathop \int \limits_{ - \infty }^{t} R_{\text{pro}} (t_{\text{b}} )\beta (t,t_{\text{b}} )\theta [\tau (t_{\text{b}} ) - t]{\text{d}}t_{\text{b}}$$
(3)

where the prefactor R pro(t b) involves the contribution to the velocity distribution from the LPP generated at t b. The recombination time τ(t b) depends on the trajectory of a released electron born at t b, and the recombination probability relates to its kinetic energies obtained from the laser field. Here, the Heaviside step function θ[x], which obeys θ[x] = 1 for x ≥ 0 and θ[x] = 0 otherwise, incorporates the recombination of an electron with its parent ion.

The factor β(t, t b) can be rewritten in terms of the Heaviside step function and reduced to the Dirac function by taking the first order of \(\Delta {\mathbf{v}}\),

$$\begin{aligned} \beta (t,t_{\text{b}} ) & = \theta [{\mathbf{v}} + \Delta {\mathbf{v}} /2 - {\mathbf{v}}(t,t_{\text{b}} )]\theta [{\mathbf{v}}(t,t_{\text{b}} ) - {\mathbf{v}} +\Delta {\mathbf{v}} /2] \\ & \approx \left[ {\theta [{\mathbf{v}} - {\mathbf{v}}(t,t_{\text{b}} )] + \frac{{\Delta {\mathbf{v}}}}{2}\delta [{\mathbf{v}} - {\mathbf{v}}(t,t_{\text{b}} )]} \right] \\ & \quad \times\,\left[{\theta [{\mathbf{v}}(t,t_{\text{b}} ) - {\mathbf{v}}] + \frac{{\Delta {\mathbf{v}}}}{2}\delta [{\mathbf{v}}(t,t_{\text{b}} ) - {\mathbf{v}}]} \right] \\ & = \delta [{\mathbf{v}} - {\mathbf{v}}(t,t_{\text{b}} )]\Delta {\mathbf{v}} + O(\Delta {\mathbf{v}}^{2} ) \\ \end{aligned}$$

where θ[x] + θ[−x] = 1 and ∂θ[x]/∂x = δ[x] are used. Moreover, considering the definitions of θ-function, the values of left and right limits of the functions are different, \(\lim_{{x \to 0^{ - } }} \theta [x] = 0\) and \(\lim_{{x \to 0^{ + } }} \theta [x] = 1\), which means θ[x] · θ[−x] = 0. Therefore, the velocity distribution function is obtained as

$$f(t,v) = \mathop \int \limits_{ - \infty }^{t} R_{\text{pro}} (t_{\text{b}} )\delta [{\mathbf{v}} - {\mathbf{v}}(t,t_{\text{b}} )]\theta [\tau (t_{\text{b}} ) - t]{\text{d}}t_{\text{b}}$$
(4)

By means of the generalized Dirac function \(\delta [T(x)] = \sum\nolimits_{n} {\frac{{\delta [x - x_{n} ]}}{{|T^{\prime } (x_{n} )|}}}\) for an arbitrary continuously differentiable function T(x) of x, where x n is the nth zero point of T(x), we have

$${\beta (t,t_{\text{b}} )} = \sum\limits_{n} {\frac{{\delta (t - t_{n} )}}{{|{\dot{\mathbf{v}}}(t_{n} ,t_{\text{b}} )|}}}\Delta$$
(5)

Here \({\dot{\mathbf{v}}}(t_{n} ,t_{\text{b}} )\) stands for the derivative of \({\mathbf{v}}(t,t_{\text{b}} )\) with respect to time t at t n , and t n is determined by \({\mathbf{v}} - {\mathbf{v}}(t_{n} ,t_{\text{b}} ) = 0\). Therefore, by inserting Eqs. (2) and (5), Eq. (3) can also be rewritten as

$$f(t,v) = \mathop{\sum}\limits_{n} \mathop{\int}\limits_{-\infty}^{t} {\text{d}} t_{\text{b}} \frac{{\theta [t - \tau (t_{\text{b}} )]\delta [t - t_{n} ]R_{\text{pro}} (t_{\text{b}} )}}{{|E(t_{n} ) + \nabla V(x(t_{n} ,t_{\text{b}} ))|}}$$
(6)

The two equivalent expressions (4) and (6) characterize the velocity distribution of released electrons from two different perspectives. In the former, the velocity distribution is obtained by matching of the velocity \({\mathbf{v}}(t,t_{\text{b}} )\) of released electrons born at different instants t b with the reference velocity v. The latter concerns the contribution of released electrons born at t b to the share of a given reference velocity v at t = t n , where t n is calculated by \({\mathbf{v}} - {\mathbf{v}}(t_{n} ,t_{\text{b}} ) = 0\).

During the interaction of the strong laser field with the gaseous medium, a large number of bounded electrons are freed by the laser field, and then, the released electrons can be recaptured by the ions with a certain probability, which causes the nonconservation of the density of electrons. Therefore, the distribution function f(t, v) is normalized to a time-dependent function, \(\int_{ - \infty }^{ + \infty } {f\left( {t,v} \right)} {\text{d}}{\mathbf{v}} = N_{\text{e}} \left( t \right)\). That is to say, the function f(t, v) describes the transient velocity distribution of LPP in the laser pulse duration, which is an extremely non-LTE system.

Analogously, we also calculate the kinetic energy distribution of released electrons,

$$g(t,E_{\text{K}} ) = \mathop \int \limits_{ - \infty }^{t} R_{\text{pro}} (t_{\text{b}} )\delta [E_{K} - E_{K} (t,t_{\text{b}} )]\theta [\tau (t_{\text{b}} ) - t]{\text{d}}t_{\text{b}}$$
(7)

where \(E_{\text{K}} (t,t_{\text{b}} ) = {{\left[ {v(t,t_{\text{b}} )} \right]^{2} }/2}\) is the kinetic energy of the electron born at t b, and \(E_{\text{K}}\) is the reference kinetic energy frame with the maximal value of \(E_{\text{Kmax}} = {{v_{{\rm max} }^{2} } \mathord{\left/ {\vphantom {{v_{{\rm max} }^{2} } 2}} \right. \kern-0pt} 2}\).

3 Numerical simulations

For a linearly polarized laser pulse, we can restrict our work to one-dimensional situation. It can be directly generalized to the higher-dimensional situations.

3.1 Electron–ion recombination

Generally, the electron–ion recombination is described by the quantum transitions between continuum and bound states of the time-dependent atomic system [30, 31]. Obviously, it is a time-consuming task in simulating the femtosecond laser propagating in a medium. In the classical treatments, the cross section for the electron–ion recombination in the plasma is required [32, 33], which is appropriate only for the plasma in the thermodynamic (quasi)-equilibrium states. For the plasma produced by the femtosecond intense laser pulse, it is obviously not the case.

In this paper, we propose an ad hoc ansatz for the electron–ion recombination. As is well known, the lower the kinetic energy of a freed electron is as it comes back to its parent ion, the higher the probability of the electron–ion recombination is. For convenience, we assume that the electron–ion recombination probability obeys the Maxwell distribution \({\mathcal{P}}(v) \propto \exp \left( { - v^{2} /2\sigma } \right)\), where σ is a characteristic energy width depending on the type of the atom. In our simulation, we choose σ = I p as the characteristic energy width. For the velocity, v, of a freed electron arriving at its parent ion, the recombination coefficient α(t, t b) and the recombination time τ(t b) are determined by generating random numbers that satisfy the distribution \({\mathcal{P}}(v)\).

3.2 Velocity partition

The first step to calculate the velocity distribution of LPP is to construct a reference velocity frame. In order to cover all the possible velocities, the range of velocity of released electrons should be determined. For an incident laser pulse with the electric strength E 0 and angular frequency ω L , the maximal velocity is calculated as 2E 0/ω L in the absence of atomic potential. Then, we can evaluate an appropriate maximal value of the reference velocities as v max > 2E 0/ω L . Therefore, the reference box for the velocities is discretized in cells of equal spacing Δv with numbers 0, 1, 2, …, k − 1, k, k + 1, …, N, where N is the total number of cells in the simulation box, as shown in Fig. 1. The minimal value v 0 = −v max and the maximal value v N  = v max are assigned.

Fig. 1
figure 1

(Color Online) Partition of time-dependent velocity \({\mathbf{v}}(t,t_{\text{b}} )\) in the (k + 1)th cell. Here, the partition weighting factor w is determined by the relative distance between \({\mathbf{v}}(t,t_{\text{b}} )\) and the reference velocities v k normalized by the interval Δv

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For a released electron produced at t b, we can obtain its velocity v(t, t b) at time t by numerically solving Eq. (2). The index k of the cell where the velocity v(t, t b) lies in is given by \(k = {{\rm I}{\rm N}{\rm T}}\left[ {\frac{{v(t_{\text{b}} ,t) - v_{0} }}{{\Delta v}}} \right]\), where the operation \({{\rm I}{\rm N}{\rm T}}[x]\) rounds to the value of x to the nearest integers greater than or equal to x.

In order to reduce statistical error in the calculation, we assign the contributions of v(t, t b) to kth and (k + 1)th grids, which is similar to the method adopted in the particle-in-cell (PIC) simulations for the laser–plasma interaction [34]. The partition weighting factor is calculated as w = [v(t b, t) − v k ]/Δv, meaning that p k  = 1 − w and p k+1 = w. Therefore, the grids k and k + 1 share the contribution of the time-dependent velocity v(t b, t) with the weights p k and p k+1, respectively, as shown in Fig. 1. This scheme smooths the velocity distribution of LPP to some extent. It can also be applied to calculate the kinetic energy distributions of the released electrons, and we will not cover them again here.

3.3 Results and discussion

In this paper, our work is mainly restricted to the tunneling regime. For the simulation, we choose a linearly polarized femtosecond laser pulse with a Gaussian profile,

$$E = E_{0} \exp \left( {{{ - t^{2} } \mathord{\left/ {\vphantom {{ - t^{2} } {t_{\text{f}}^{2} }}} \right. \kern-0pt} {t_{\text{f}}^{2} }}} \right)\cos \left( {\omega_{L} t} \right)$$

where E 0 is the electric amplitude, the central angular frequency ω L  = 0.057 corresponding to the wavelength \(\lambda_{0} = 800\,{\text{nm}}\), and the pulse duration \(t_{\text{f}} = 5\,{\text{fs}}\). In solving Eq. (2), a soft-core potential \(V(x) = - 1/\sqrt {x^{2} + a}\) is adopted to avoid the singularity of the pure atomic potential as x → 0, where a is an adjustable parameter used to fit the real atoms. The motion Eq. (2) of a released electron is numerically solved using fourth-order Runge–Kutta method with the initial conditions x 0 and v 0, where x 0 = −I p/E(t b) is the released electron’s birth place, and its initial velocity v 0 is assumed to be zero [35, 36]. For the MPI regime characterized by the Keldysh parameter γ K, the initial position of released electrons should be set to zero because MPI is a kind of vertical transition.

The numerical calculations give the velocity v(τ, t b), which determines the electron recombination probability obeying the Maxwell distribution as discussed above. Here, the time τ is obtained by solving x(τ, t b) = 0, and τ = τ(t b) if the electron and its parent ion recombine. In this paper, we only consider the recombination probability as the freed electron arrives at its parent ion for the first time. For a released electron with a low return velocity, the recombination probability is relatively high, which means that the contribution of the successive excursions of the electron to the electron–ion recombination can be neglected. In the same way, for an electron with a large return velocity, the electron–ion recombination probability is very small, and it is not easy for the electron to reach its parent ion again. That makes our consideration reasonable.

In the simulation, we take the soft-core parameter a = 0.487 for the helium atoms as an example. The electric amplitude is set to E 0 = 0.16 a.u. corresponding to the laser intensity \(I_{0} \approx 9 \times 10^{14} \,{\text{W/cm}}^{2}\), which makes the Keldysh parameter γ K ≈ 0.48. Based on aforementioned parameters and analysis, we present the return times τ(t b) (the black and hollow circles) of released electrons produced at different times t b, as shown in Fig. 2. It is clear that the released electrons are mainly captured at larger probabilities (the green and solid squares) when the laser field changes its polarization direction.

Fig. 2
figure 2

(Color Online) Electron–ion recombination and the corresponding recombination probability obeying the Maxwell distribution with respect to the return velocity of released electrons. The black circles stand for the released times (left vertical axis) versus the recombination times (bottom axis), and the green and solid squares represent the electron–ion probabilities (right vertical axis) relating to the recombination times

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The modified rate Eq. (1) is numerically solved using the fourth-order Runge–Kutta method. In Fig. 3, we depict the evolution of LPP density (the green and solid line) normalized by the density of initial neutral atoms. The blue and dotted line represents the cumulated density of produced electrons during the laser–atom interaction. The cumulated density of recombined electrons is also drawn by the magenta and dashed line. Obviously, the released electrons are always produced around the peaks of the laser pulse. As expected, the electron–ion recombination mainly occurs when the electric field changes its sign, where the electron density decreases quickly.

Fig. 3
figure 3

(Color Online) Densities of released electrons normalized to N 0. The green solid line represents the solution of rate equation with the recombination incorporated, and the magenta and dashed lines stand for the cumulated density of recombined electrons. The cumulated density of produced electrons is denoted by the blue and dotted line

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Using the scheme proposed in Sect. 3.2, we calculate the evolutions of velocity distribution of released electrons, as is shown in Fig. 4 on the logarithmic scale. The electric field of the laser pulse is drawn in the black and dashed line. The parameters presented above give the maximal velocity of an electron oscillating in the laser field, 2E 0/ω L  ≈ 5.6 a.u., which allows to evaluate the maximal reference velocity as v max ≈ 6.0 a.u. in the presence of the atomic potential. To assure the velocity of the electron distributing between most of the grids, we set the number of grids to be N v  = 500. According to the Keldysh criteria for the tunneling ionization, γ K ≪ 1, the released electrons are produced significantly when the electric field satisfies E(t) ≪ E C, where \(E_{\text{C}} = \omega_{L} \sqrt {2I_{p} }\) is a critical field (the thin and black and dashed lines in Fig. 4).

Fig. 4
figure 4

(Color Online) Velocity distribution of released electrons on logarithmic scale. There are three LPP bursts near the leading and its two adjacent peaks of the laser pulse, and the critical electric field (E C) is displayed in thin and black and dashed lines

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We can see that there are mainly three different LPP bursts near three laser peaks where the Keldysh parameter γ K ≪ 1, i.e., the electric strength E(t) ≫ E C, as shown in Fig. 4. For the released electrons produced at t b, the collective motion weighted by the production rate R pro(t b)dt b is applied, and their velocity directions are opposite to the laser polarization. The dynamics of the released electrons are governed by the motion Eq. (2). Because of the very short range of the Coulomb potential, the released electrons are basically accelerated by the laser field after they are produced. With the decrease in the electric strength, i.e., in the last several optical cycles, the velocity of LPP distributes nearly evenly in a larger region ranging from 3.2 to 3.2 a.u., which means the LPP is an extremely non-LTE system. The same analysis and conclusions are also applicable to the kinetic energy distribution of LPP, as shown in Fig. 5 on logarithmic scale. It can be predicted that, after the laser pulse, all these released electrons will collide with each other and their momentum and kinetic energies will exchange. Eventually, the LPP will reach a local thermal equilibrium with the lifetime of several nanoseconds.

Fig. 5
figure 5

(Color Online) Kinetic energy distribution of released electrons on logarithmic scale. The three LPP bursts are also observed near the leading and its two adjacent peaks of the laser pulse, and the critical electric field (E C) is displayed in thin and black and dashed lines

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At this point, we turn to the energy transfer and conversion during the accelerations of LPP in the intense femtosecond laser field. According to the statistical theory of thermal systems, the mean kinetic energies (MKE) of particles are directly related to the macroscopic properties of the system, such as the thermal temperature \(T_{\text{e}} \propto \langle E_{\text{K}} (t)\rangle\), where \(\langle E_{\text{K}} (t)\rangle\) is the time-varying MKE of LPP. By the virtue of the velocity distribution function (4), the evolution of MKE is calculated by

$$\begin{aligned} \langle E_{\text{K}} (t)\rangle & = \mathop \int \limits_{ - \infty }^{ + \infty } \frac{1}{2}v^{2} f(t,v){\text{d}}{\mathbf{v}} \\ & = \mathop \int \limits_{ - \infty }^{t} \frac{1}{2}\left[ {v(t,t_{\text{b}} )} \right]^{2} R_{\text{pro}} (t_{\text{b}} )\theta [\tau (t_{\text{b}} ) - t]{\text{d}}t_{\text{b}} \\ \end{aligned}$$
(8)

where the factor θ[τ(t b) − t] characterizes the decay of the released electrons born at t b through electron–ion recombination. Moreover, the MKE can also be calculated by \(\langle E_{\text{K}} (t)\rangle = \int_{ - \infty }^{ + \infty } {g(t,E_{\text{K}} )E_{\text{K}} {\text{d}}E_{\text{K}} }\) using Eq. (7). The evolution of the MKE of LPP is displayed in Fig. 6, which shows that the MKE always reach its local maximal values as the electric field approaches zero. After being produced near the laser peaks, the released electrons are continually accelerated until the electric field E(t) = 0 for the first time, where they reach the maximum of their kinetic energies. The maximal value of the MKE, which is the third peak of the MKE curve, accounts for the contribution of the released electrons produced at the leading and the next adjacent peaks of the laser pulse because of their collective motions.

Fig. 6
figure 6

(Color Online) Evolution of the MKE of LPP

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As the laser pulse ends, the LPP stores some kinetic energies in the form of residual energies, as shown in Fig. 6. In the follow-up study of the dynamics of LPP, such as the evolution of laser–plasma channel, the residual energies can be used as an initial condition through \(T_{\text{e}} = C\langle E_{\text{K}} (t_{\text{f}} )\rangle\), where C ≈ 3.2 × 106 is a conversion factor between the thermal temperature (in Kelvin) and atomic system of units.

To connect our current research with the LPP filamentation during ultrashort pulse propagation in air, we consider a Gaussian distribution for the axially symmetric laser field as E 0(r) = E max exp (−r 2/r 2f ) in the transverse direction, where r is the transverse variable, \(r_{\text{f}} = 0.9\,{\text{mm}}\) characterizes the waist radius, and \(E_{{\rm max} } = 0.16\,{\text{a}}.{\text{u}}.\) is the maximal strength at the center. Using the ionization–recombination rate Eq. (1), we calculate the generation and recombination of plasma and demonstrate the sensitivity of the density of LPP against variation of the peak electric strength (or laser intensity) in Fig. 7. The similar behaviors as shown in Fig. 3 can also be observed. In Fig. 8a, we present the peak strength E 0(r) and the corresponding Keldysh parameter γ K. The value γ K = 1 that characterizes the TI and MPI mechanisms is displayed in red dotted and dashed line, which gives the critical electric strength \(E_{\text{C}} \approx 0.0765\,{\text{a}}.{\text{u}}.\) at \(r_{\text{C}} \approx 0.773\,{\text{mm}}\). For r > r C, the LPP density and the final MKE are down to close to zero, as shown in Fig. 8b. The non-monotonicity of LPP density and the corresponding final MKE with the increase in radius (r < r C) in Fig. 8b originates from the recombination of ionized electron with its parent ion. The transverse distributions of final MKE and the density of LPP together can be used as the initial conditions of evolution equations of LPP after the laser pulse.

Fig. 7
figure 7

(Color Online) Sensitivity of the density of LPP against variation of the peak electric strength

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Fig. 8
figure 8

(Color Online) Transverse distribution of electric field and LPP. a The electric field and the corresponding Keldysh parameter, and the red dotted and dashed line denotes the Keldysh parameter γ K = 1 and b the final LPP density and the final MKE

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4 Conclusions

In this paper, the velocity distribution is derived by matching the time-varying velocities of LPP with the reference velocity. In order to reduce the errors and smooth the distribution functions, we introduce a velocity partition scheme, in which a value of the time-varying velocity is shared by two adjacent grids with different weighting factors.

In our simulations, the tunneling ionization is the dominant mechanism, and the ionization rate is described by the ADK model. The Newtonian equation is parameterized by the birth time of the released electrons. As expected, the electrons are always recombined with the ions when the electric field changes its sign. The advantage of the rate equation is that we can calculate the cumulative production and the recombination yields of the electrons separately. The simulation results show that the LPP bursts appear around the laser peaks where the Keldysh parameter γ K ≪ 1. And then LPP will be accelerated collectively. As the electric strength decreases, the distribution of velocities of LPP implies that LPP is an extremely non-LTE system. The similar conclusions are made from the calculation of kinetic energy distribution of LPP. From the evolution of the MKE of LPP, we can see that the MKE reach its local maximal values as E(t) ∼ 0, which means that the kinetic energies of the LPP reach its maximum when the electric field changes it sign for the first time.

As a matter of fact, the LPP reserves some kinetic energies as residual energies after the laser pulse. This can be used as the initial condition in the future studies of the dynamics of LPP, for instance, the evolution of laser–plasma channel and LPP filaments. Finally, we calculate the transverse distribution of final MKE of LPP by considering the electric amplitudes with a Gaussian distribution. For the short-pulse propagation in air, only the rate equation of LPP and motion equation of the ionized electron need to be modified. Starting with the initial conditions presented by our research, the LPP undergoes various processes, such as recombination, attachments, and radiations, which all happen in nanoseconds.

By tracing the derivation of the velocity and the kinetic energy distributions, we come to the conclusion that our model is valid for interaction of femtosecond laser pulse with the gaseous medium, where the plasma is produced through optical field ionization, such as tunneling and multiphoton ionization mechanisms. The scheme of calculating the velocity and kinetic energy distributions can also be generalized directly to the case of circularly polarized laser pulse, in which a two-dimensional motion equation of the ionized electron and area-weighted velocity partition are required.