A recombination rate of electron-ion in the strong-field atomic process is phenomenologically introduced into the ionization rate equation, and therefrom an ionization and recombination rates equation (IRRE) is obtained. By using the extended IRRE, the propagation equation of an intense femtosecond laser pulse in the gaseous medium is re-derived. Some new physical behaviors and characteristics caused by the introduced recombination rate are discussed in detail.
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Of all the highly nonlinear phenomena in the strong femtosecond laser-atom interaction, higher-order harmonic generation (HHG) emerges as the most promising spatially and temporally coherent sources for extreme ultraviolet (EUV) and soft X-ray [1, 2]. However, the presence of electron plasma due to the atomic field-ionization destructs the phase-matching between the fundamental laser and the harmonics [1, 3, 4], and consequently causes the spectral blueshifting of the laser pulse [3, 5, 6]. Moreover, The free electrons are the main contributor to the laser loss by Joule heating . More importantly than all of that, the recombination of electrons and ions is the root of the HHG according to the three-step model  as well as Lewenstein’s quantum model .
The complete description of the strong laser-matter interaction involves both the response of a single atom (via time-dependent Schrödinger equation, TDSE) and the collective effects of gaseous medium to the laser pulse (via Maxwell equations) . Microscopically, the exposure of an atom to an intense femtosecond laser pulse causes a great suppression of the inner Coulomb barrier, and the bound electron can tunnel through the barrier and escape away from its parent ion. According to the classical electrodynamics , the corresponding macroscopic polarization of the medium is characterized by the average deviations of the positive and negative charges P(τ) = e∑ini(τ)si(τ), where e is the electron charge, ni(τ) the number of electrons with the i-th type of displacement si(τ) relative to their parent ions at time τ.
In the majority of previous investigations in non-relativistic regime (< 1018W/cm2), there are mainly two approaches for the analysis of the atomic microscopic processes:
- The preferred one certainly is the TDSE formalism. In this formalism, the response of a single atom to the laser pulse can be realized by the transitions among the internal quantum states. The advantages of the TDSE are that it can describe precisely the dynamics of the laser-atom interaction in the form of electronic wave packets [6, 12, 13, 14, 15]. The polarization of the medium is usually described by P(τ) = eNe(τ)〈Ψ(s, τ)|ŝ|Ψ(s, τ)〉, where |Ψ(s, τ)〉 is the single-electron wave function determined by TDSE, and the position operator ŝ = s in the coordinate representation. The electron density Ne(τ) is relating to the projection of the wave function |Ψ(s, τ)〉 on the continuum state, in other words, Ne(τ) = N0 [1 − ∑ℓ=0 |〈ϕℓ(s)|Ψ(s, τ)〉|2], where N0 is the initial density of the neutral atoms, ϕℓ(s) the ℓ-th bound state wave function. The problem with this treatment of the electron density and the atomic polarization is that the generated and recombined electrons are not dealt with separately, and all the atoms inside the laser pulse are considered as a simple collection of many single-atomic responses to the laser field under the identical conditions. Furthermore, the numerical solution of the three-dimensional TDSE requires larger computer resources and is greatly time-consuming, especially for the three-dimensional propagation of the laser pulse.
- An important alternative is the semiclassical approach with a striking concept of ionization rate. For example, in the tunneling ionization (TI) regime, the ionization rate is given by the Ammosov-Delone-Krainov (ADK) theory  under the quasi-static electric field (QSE) approximation, which can be traced back to Laudau’s work on the ionization of hydrogen atom under a weakly static-electric field . The advantage of ADK theory is that it has a simple analytic form and is very low time-consuming. Notably, the central idea of the ADK model is the ionization rate depending on the integral of a probability current through a closed surface that surrounds an atom. In a traditional way on this model, the evolution of ionized electronic density is calculated by the rate equation dNe(τ)/dτ = w(τ)Na(τ), where w(t) is the quasi-static ionization rate depending on the magnitude of laser strength |E(τ)|, Na(τ) = N0 − Ne(τ) the density of remanent neutral atoms. Due to the characteristic of QSE approximation and the positive value of w(τ), Ne(τ) counts the total ionized electrons tunneling the Coulomb barrier, and it increases monotonously along the laser pulse. However, the truth is obviously not the case. The rapid change of the laser’s direction can drive the free electrons back to their parents ions, and there are big chances for the electrons to recombine with the ions, which means that the recombination of an electron with its parent ion is an indispensable process. In , Chris-tov had discussed the differences and relations between the QSE approximation of light-atom interaction and the (ab initio) solution of the Schrödinger equation. As the amplitude E0 of the laser field increases, i.e., (in atomic units), the ADK theory increasingly overestimates the ionization rate and tends to lose its validity. The over-barrier ionization (OBI) (or barrier-suppression ionization, BSI) becomes the dominant ionization mechanism. Therefore, some authors [19, 20, 21, 22] had generalized the ADK ionization rate w(τ) to the OBI regime by means of Airy function for a linearly and circularly polarized strong laser field.
Motivated by the greatly time-consuming calculations of TDSE and the absence of electron-ion recombination effects in the semiclassical model, we introduce the recombination rate to the QSE-based ionization rate equation, which avoids the the computationally expensive TDSE calculations. Furthermore, we rederive the propagation equation that describes the evolution of an intense femtosecond laser pulse in a gaseous medium accompanying with the revised ionization rate equation, and investigate the energy conversion processes. The derivation of the propagation equation is based on a co-moving frame with the light pulse traveling at light speed c, τ = t − z/c, ξ = z, and the international system of units (SI unit) is adopted unless otherwise specified.
2. Theoretical model
Since the propagation effects of an intense femtosecond laser in a gaseous medium plays a pivotal role in enhancing the conversion efficiency from driving laser into HHG, it is necessary to investigate the evolution of the laser-induced plasma density along with the laser pulse. Using the slowly-varying-envelope (SVE) approximation, the three-dimensional propagation of strong femtosecond laser E = E(r, ξ, τ) through a gaseous medium can be described by
For the sake of brevity, we denote Ne(r, ξ, τ) by Ne(τ) with no ambiguity. Considering the recombination of electrons with their parent ions, the electron density satisfies23] and references therein.
In order to highlight the effects of the modified ionization rate equation and avoid the time-consuming calculation of Airy function in OBI regime, we use the numerical results for the values of ionization rate given by Scrinzi et al. for helium atoms. Meanwhile, we make an ansatz that α(τ, τpro) = 1 at the first return time of an ionized electron. Although it will overestimate the density of recombined electrons, it does not change the mathematics expressions of the ionization rate equation and the below propagation equation.
The dynamics of an electron in a strong laser field is described by meẍ(τ, τpro) = eE(τ) with two τpro-dependent initial conditions x0 and v0 for τ < τrec, where τrec is the recombination time of the electron with its parent ion. Generally, the ionization position is x0 = Ip/ (eE (τpro)), and v0 = 0 is always assumed. The explicit solution of electronic motion equation can be obtained by performing integration over time [τpro, τ]Eqs.(3) and (4), we have
Since what really interests us is the recombination parameter α(τ, τpro) = 1, we investigate the condition x(τ, τpro) → 0, which determines the recombination of an electron with its parent ion. At this moment, Coulomb effects can not be neglected. The extreme of distorted Coulomb potential by external strong field, V = −e2/(4πε0x)+exE, is given as . When x(τ, τpro) < xc, Coulomb potential will be in dominant status, so that the free electron can be captured by its parent ion. In this paper, we neglect the two or more collisions between an electron and its parent ion before they are recombined. The return time τrec is determined byEq. (5) has a similar form with the saddle-point equation except a nonzero initial position. Then our ansatz on the recombination parameter can be expressed as α(τ, τpro) = 1 for τ = τrec, and α(τ, τpro) = 0 otherwise. In Fig. 1, we show the releasing time τpro and the corresponding return time τrec. Because of the non-zero initial position and the non-planar electric field, some of the electrons will take several optical cycles to return to their parent ions. The ansatz that α(t, tb) = 1 at the first return time of an ionized electron leads to the truncation of the subsequent excursions which are marked out with red pentagrams.
For the convenience of discussions below, we denote the following two time-pairs, τpro[τ] as the releasing time τpro corresponding to the return time τ, and τrec[τ] as the return time τrec corresponding to the releasing time τ, which is determined by Eq. (5). In this sense, the recombination parameter α(τ, τpro[τ]) = 1, and the recombination rate can be rewritten as Rrec(τ) = ∑τpro Rpro(τpro[τ]).
Using the expression of Rpro(τ) and Rrec(τ), Eq. (2) can be solved explicitly.Fig. 2. Notably, the total produced electron density is larger than the initial atomic density N0 for the applied laser intensity, i.e., Npro(τ → +∞)/N0 > 1, which just corroborates the above arguments. That is to say, an important result from the introduction of the electron-ion recombination is that some atoms have been ionized for two or more times during the laser pulse. It means that more laser energies will be absorbed by the atoms to release their electrons, and less laser energies are used to accelerate the ionized electrons during the laser-atom interaction, which will be discussed in the next section. To highlight this result in the case of single ionization, we have chosen a sufficiently and non-relativistically strong laser field with intensity of 3 × 1015W/cm2, which indicates the OBI dominant regime for helium atoms. (Some other laser parameters are presented in the caption of Fig. 3.)
Because the electrons born at different times have different displacement relative to their parent ions, we write the total deviations per volume defined by an averaged form 〈Ne(τ)s(τ)〉 as
Based on the above efforts and preparations, we turn to the analysis of the atomic polarization. The induced current density is given by the first derivative of the atomic polarization respective to time, JP(τ) = ∂P(τ)/∂τ, that isEq. (4). Substituting Eq. (7) into Eq. (1) and integrating the two sides over time (−∞, τ], one can obtain the wave equation of the intense laser pulse propagating through gaseous field-ionizing medium, Fig. 3, there is an evident blue-shifting near the center of laser pulse at the exit because of the stronger electric strength which leads to a larger density of electronic plasmas. The ionized electrons transport the laser energy from the front of the pulse to the back due to their accelerations and decelerations, as well as recombination with their parent ions. Furthermore, the transverse diffraction of the laser is also displayed.
To verify the validity of the scheme proposed above, we investigate the microscopic energy conversion processes for the incident laser pulse at the entrance. Comparing to the semiclassical methods without electron-ion recombination incorporated, the recombination effects have changed the density of neutral atoms and ionized electrons that interact with the laser field. As is known to all, there are mainly three kinds of energy conversions in the optical field-ionizing medium, in which the electron-electron collisions in the usual laser-plasma interactions are always neglected due to their very short accelerating time. The primary process is the atomic field-ionization, then the ionized electrons gain their kinetic energy from the laser field, and finally they release their kinetic energy plus the Coulomb potential energy Ip in the form of high-order harmonics if they can return to their own parent ions. Contrast with the total energy conversion, the rate of energy transfer can better reveal the details of the temporal conversion process, which directly affects the laser profile.
(i) The ionization loss
When an atom is irradiated by an laser field, it absorbs an ionization potential energy Ip to excite its bound electrons to the continuum states. Therefore, the analysis above on the laser-atom interaction shows that there are Npro electrons released throughout the laser pulse, in which the laser field loses Ip per ionized electron. Then the energy loss rate Tpro is proportional to the ionization rate Rpro,Fig. 4. During the front edge of laser pulse, the two curves almost coincide, indicating that there is almost no electron-ion recombination. While the electron-ion recombination results in the increase of the neutral density during the right side of pulse peak, and then more laser energies are needed to excite the atoms, which is just the multiple ionization of helium atoms. This kind of laser loss process can be embodied by the third term on the right-hand side in Eq. (8).
(ii) The Joule heating process of ionized electrons
As an electron that is born at τ′ is moving in the laser field, its velocity is denoted by v(τ, τ′). Then the electric currents in the laser pulse can be evaluated by25] Fig. 5, the introduction of the electron-ion recombination results in the decline of the ionized electrons, so do the laser energies which are needed to accelerate the electrons. Moreover, the negative value of the energy transfer rate means that the electrons give their kinetics back to the laser field.
An equivalent expression for Tdrift(τ) can be built from the kinetic energy obtained by an ionized electron from the laser field, ρ(τ, τ′) = me[v(τ, τ′)]2/2, so the rate of the total kinetic energy changes at time τ can be calculated as
The final drift velocity of electrons born at τ′ is given by vd = v(+∞, τ) = eA(τ)/me at the end of the laser pulse. Therefore, the first term in the laser line in Eq. (10) refers to the total kinetic energy gained by the Ne(+∞) ionized electrons, while the number of free electrons Ne(τ) will be decreased due to the electron-ion recombination.
(iii) The electron-ion recombination process
The third kind of energy conversion of laser field is the electron-ion recombination process, which is mainly responsible for the emission of high-order harmonics. An ionized electron gains a kinetic energy resulting from a drift velocity until it returns to the ion by the change of laser’s direction. Besides, when an electron recombines with its parent ion, the binding energy Ip will be also released, which is a kind of atomic processes. During this process, the change of energy conversion can be calculated byEq. 12, h̄ is the reduced Planck constant and ω the angular frequency of applied laser pulse. For the incident laser pulse at the entrance, the time-dependent radiant energy WEM(τ) is shown in Fig. 6.
We extended the contents of the existing ionization rate equation to involve the electron-ion recombining process based on the semiclassical model. The results show that for a given return time τ of the ionized electrons the recombination rate is directly relating to the production rate at ionization time τpro, and the electron’s releasing and return times, τ = τrec[τpro], are determined by the saddle-point-like equation with nonzero ionization position. Furthermore, this field-bounded electron-ion recombination retards the ionization saturation of medium and some atoms have to experience the multiple ionizations, which results in more energy losing of the incident laser in the atomic ionization process.
For the propagation of an intense femtosecond laser pulse in the gaseous medium, combining the extended IRRE, we redefine the non-linear polarization of the medium by using the mean displacements of the ionized electrons, and consequently, a brand new evolution equation of an intense laser pulse propagating in the gaseous medium is presented, which includes the atomic ionization, electronic acceleration and electron-ion recombination processes. From the new evolution equation of light propagation the laser energy transfer and conversion are analyzed numerically. Since the atomic field-ionization are described by time-dependent Schrödinger equation, and the dynamics of ionized electrons are governed by Newtonian equation, the scheme proposed in this paper is valid for ionizing gaseous media and for non-relativistic laser fields.
We acknowledge the support from the National Natural Science Foundation of China (Grant No. 10974010).
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