## Abstract

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.

© 2013 OSA

## 1. Introduction

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 [7]. More importantly than all of that, the recombination of electrons and ions is the root of the HHG according to the three-step model [9] as well as Lewenstein’s quantum model [10].

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) [6]. 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 [11], the corresponding macroscopic polarization of the medium is characterized by the average deviations of the positive and negative charges *P*(*τ*) = *e*∑* _{i}n_{i}*(

*τ*)

*s*(

_{i}*τ*), where

*e*is the electron charge,

*n*(

_{i}*τ*) the number of electrons with the

*i*-th type of displacement

*s*(

_{i}*τ*) relative to their parent ions at time

*τ*.

In the majority of previous investigations in non-relativistic regime (< 10^{18}W/cm^{2}), 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*) =*eN*_{e}(*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*N*_{e}(*τ*) is relating to the projection of the wave function |Ψ(*s*,*τ*)〉 on the continuum state, in other words,*N*_{e}(*τ*) =*N*_{0}[1 − ∑_{ℓ}_{=0}|〈*ϕ*(_{ℓ}*s*)|Ψ(*s*,*τ*)〉|^{2}], where*N*_{0}is the initial density of the neutral atoms,*ϕ*(_{ℓ}*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 [16] 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 [17]. 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 d
*N*_{e}(*τ*)/d*τ*=*w*(*τ*)*N*_{a}(*τ*), where*w*(*t*) is the quasi-static ionization rate depending on the magnitude of laser strength |*E*(*τ*)|,*N*_{a}(*τ*) =*N*_{0}−*N*_{e}(*τ*) the density of remanent neutral atoms. Due to the characteristic of QSE approximation and the positive value of*w*(*τ*),*N*_{e}(*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 [18], 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*E*_{0}of the laser field increases, i.e., ${E}_{0}>{I}_{\text{p}}^{2}/4$ (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

*p*is the pressure of the gaseous medium and

*μ*

_{0}the permeability, and we denote

*P*(

*τ*) =

*P*(

*r*,

*ξ*,

*τ*).

For the sake of brevity, we denote *N*_{e}(*r*, *ξ*, *τ*) by *N*_{e}(*τ*) with no ambiguity. Considering the recombination of electrons with their parent ions, the electron density satisfies

*R*(

*τ*) is the rate of the electronic density. It consists of two parts: the production rate and the recombination rate,

*N*

_{a}(

*τ*) =

*N*

_{0}−

*N*

_{e}(

*τ*) is the density of neutral atoms with

*N*

_{0}the initial density of neutral atoms depending on the applied temperature and pressure. In above equations, the time

*τ*

_{pro}is the releasing time of an electron from its parent ion under a strong femtosecond laser field, and

*α*(

*τ*,

*τ*

_{pro}) characterizes the recombination condition at

*τ*=

*τ*

_{rec}. In the quantum frame, the value of

*α*(

*τ*,

*τ*

_{pro}) ranges from 0 to 1 depending on overlap of the electronic wavepacket with the nuclear-dominated interaction region, which requires some further investigations via the transitions between bound and continuum states. More details about the capture of ionized electrons by ionic cores can be found in [23] 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.*[24] for helium atoms. Meanwhile, we make an ansatz that *α*(*τ*, *τ*_{pro}) = 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 *m*_{e}*ẍ*(*τ*, *τ*_{pro}) = *eE*(*τ*) with two *τ*_{pro}-dependent initial conditions *x*_{0} and *v*_{0} for *τ* < *τ*_{rec}, where *τ*_{rec} is the recombination time of the electron with its parent ion. Generally, the ionization position is *x*_{0} = *I*_{p}/ (*eE* (*τ*_{pro})), and *v*_{0} = 0 is always assumed. The explicit solution of electronic motion equation can be obtained by performing integration over time [*τ*_{pro}, *τ*]

*v*(

*τ*,

*τ*

_{pro}) is the corresponding velocity which is given by

*A*(

*τ*) of the electric field is defined as $A\left(\tau \right)=-{\int}_{-\infty}^{\tau}E\left({\tau}^{\prime}\right)\text{d}{\tau}^{\prime}$. Using Eqs.(3) and (4), we have

Since what really interests us is the recombination parameter *α*(*τ*, *τ*_{pro}) = 1, we investigate the condition *x*(*τ*, *τ*_{pro}) → 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* = −*e*^{2}/(4*πε*_{0}*x*)+*exE*, is given as
${x}_{\text{c}}=\sqrt{e/\left(4\pi {\epsilon}_{0}E\right)}$. When *x*(*τ*, *τ*_{pro}) < *x*_{c}, 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 *τ*_{rec} is determined by

*α*(

*τ*,

*τ*

_{pro}) = 1 for

*τ*=

*τ*

_{rec}, and

*α*(

*τ*,

*τ*

_{pro}) = 0 otherwise. In Fig. 1, we show the releasing time

*τ*

_{pro}and the corresponding return time

*τ*

_{rec}. 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*,

*t*

_{b}) = 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, *τ*_{pro}[*τ*] as the releasing time *τ*_{pro} corresponding to the return time *τ*, and *τ*_{rec}[*τ*] as the return time *τ*_{rec} corresponding to the releasing time *τ*, which is determined by Eq. (5). In this sense, the recombination parameter *α*(*τ*, *τ*_{pro}[*τ*]) = 1, and the recombination rate can be rewritten as *R*_{rec}(*τ*) = ∑_{τpro} *R*_{pro}(*τ*_{pro}[*τ*]).

Using the expression of *R*_{pro}(*τ*) and *R*_{rec}(*τ*), Eq. (2) can be solved explicitly.

*N*

_{pro}(

*τ*) =

*N*

_{1}(

*τ*)+

*N*

_{2}(

*τ*), is made up of two parts,

*N*

_{1}(

*τ*) represents the electron density obtained by traditional rate equation without recombination consideration, and

*N*

_{2}(

*τ*) is the density of electrons which are induced by the laser field for the second (or more) time(s) after they are captured by their parent ions. Then the density of the produced and recombined electrons can be read as

*N*

_{0}for the applied laser intensity, i.e.,

*N*

_{pro}(

*τ*→ +∞)/

*N*

_{0}> 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 × 10

^{15}W/cm

^{2}, 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 〈*N*_{e}(*τ*)*s*(*τ*)〉 as

*x*(

*τ*,

*τ*′) is the electronic displacement relative to its parent ion at time

*τ*, and

*τ*′ is the releasing time of the electron. In order to involve the recombination effects, we introduce the Heaviside function

*θ*(

*τ*) defined as

*θ*(

*τ*) = 1 for

*τ*≥ 0 and

*θ*(

*τ*) = 0 for

*τ*< 0. The derivative of the heaviside function is

*τ*-dependent function

*F*(

*τ*) over (−∞,

*τ*] is

*τ*

_{pro}of electrons that return to the ions at time

*τ*. Moreover, the density of free electron can be re-expressed by

*θ*(

*τ*)-function

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, *J _{P}*(

*τ*) =

*∂P*(

*τ*)/

*∂τ*, that is

*x*(

*τ*,

*τ*) =

*I*

_{p}/(

*eE*(

*τ*)) and

*θ*(

*τ*−

*τ*

_{rec}[

*τ*]) = 1 are used. Notably, the second term on the right-hand side actually vanishes due to the zero return position

*x*(

*τ*,

*τ*

_{pro}[

*τ*]) = 0. However, we should maintain this term for the second derivative of polarization because the return velocity of the electron

*ẋ*(

*τ*,

*τ*

_{pro}[

*τ*])≠ 0, where the single dot refers to the first derivative respect to time

*τ*. The last term means the average current generated by motion of the

*N*

_{e}(

*τ*) electrons, i.e., 〈

*N*

_{e}(

*τ*)

*v*(

*τ*,

*τ*′)〉. Using the electronic motion equation

*m*

_{e}

*ẍ*(

*τ*,

*τ*′) =

*eE*(

*τ*) with

*τ*<

*τ*

_{rec}[

*τ*′], the second derivative of polarization

*P*(

*τ*) is

*v*(

*τ*,

*τ*

_{pro}[

*τ*]) of the electron at time

*τ*is calculated by Eq. (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,

## 3. Discussions

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 *I*_{p} 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 *I*_{p} to excite its bound electrons to the continuum states. Therefore, the analysis above on the laser-atom interaction shows that there are *N*_{pro} electrons released throughout the laser pulse, in which the laser field loses *I*_{p} per ionized electron. Then the energy loss rate *T*_{pro} is proportional to the ionization rate *R*_{pro},

*R*

_{nonRecom}(

*τ*) is the ionization rate without recombination involved, and

*T*

_{ioni}(

*τ*) is the corresponding energy transfer rate from laser field to the ionized atoms. The comparison up to a factor

*I*

_{p}is depicted in 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 by

*θ*-function accounts for decrease of the ionized electrons due to the effects of the electron-ion recombination. Applying the Poynting theorem

*T*(

*τ*) =

*J*(

*τ*) ·

*E*(

*τ*), we have the absorption energy rate of ionized electrons from the laser field

*E*(

*τ*) = −

*∂A*(

*τ*)/

*∂τ*,

*v*(

*τ*,

*τ*′) = (

*e/m*

_{e})[

*A*(

*τ*′)−

*A*(

*τ*)], and the expression of

*N*

_{e}(

*τ*) in terms of

*θ*-function. While the absorption rate of the laser energy without electron-ion recombination can be calculated as [25]

*N*

_{nonRecom}(

*τ*) is the time-dependent density of ionized electrons without recombination involved. As is shown in 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 *T*_{drift}(*τ*) can be built from the kinetic energy obtained by an ionized electron from the laser field, *ρ*(*τ*, *τ*′) = *m*_{e}[*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 *v*_{d} = *v*(+∞, *τ*) = *eA*(*τ*)/*m*_{e} 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 *N*_{e}(+∞) ionized electrons, while the number of free electrons *N*_{e}(*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 *I*_{p} will be also released, which is a kind of atomic processes. During this process, the change of energy conversion can be calculated by

*v*(

*τ*,

*τ*

_{pro}[

*τ*]) is the return velocity of an electron born at

*τ*

_{pro}[

*τ*], and the term enclosed in the curly brace means the energy of an high-energy photon. Ostensibly, the ionization potential

*I*

_{p}comes from the atomic process and is independent of the laser field, but actually the atomic ionization is just caused by the laser field, and the energy conversion of

*I*

_{p}equivalent is carried by an ionized electron. So the emitted energy of electromagnetic normalized to incident single-photon energy

*h*̄

*ω*at

*τ*is

*W*

_{EM}(

*τ*) = {·}/(

*h*̄

*ω*), where the {·} means the portion in the curly brace in Eq. 12,

*h*̄ 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

*W*

_{EM}(

*τ*) is shown in Fig. 6.

## 4. Conclusions

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 *τ*_{pro}, and the electron’s releasing and return times, *τ* = *τ*_{rec}[*τ*_{pro}], 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.

## Acknowledgments

We acknowledge the support from the National Natural Science Foundation of China (Grant No. 10974010).

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