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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 5413–5423
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A modified rate equation for the propagation of a femtosecond laser pulse in field-ionizing medium

Cheng-Xin Yu, Shi-Bing Liu, Xiao-Fang Shu, Hai-Ying Song, and Zhi Yang  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 5413-5423 (2013)
http://dx.doi.org/10.1364/OE.21.005413


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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

1. A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science 280(5368), 1412–1415 (1998) [CrossRef] [PubMed] .

, 2

2. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60, 389–486 (1997) [CrossRef] .

]. However, the presence of electron plasma due to the atomic field-ionization destructs the phase-matching between the fundamental laser and the harmonics [1

1. A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science 280(5368), 1412–1415 (1998) [CrossRef] [PubMed] .

, 3

3. S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A 50(4), 3438–3446 (1994) [CrossRef] [PubMed] .

, 4

4. E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003) [CrossRef] [PubMed] .

], and consequently causes the spectral blueshifting of the laser pulse [3

3. S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A 50(4), 3438–3446 (1994) [CrossRef] [PubMed] .

, 5

5. S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A 46(2), 1084–1090 (1992) [CrossRef] [PubMed] .

, 6

6. M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A 74(5), 053401 (2006) [CrossRef] .

]. Moreover, The free electrons are the main contributor to the laser loss by Joule heating [7

7. F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. 59 (1), 52–55 (1987) [CrossRef] [PubMed] .

]. 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

9. P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993) [CrossRef] [PubMed] .

] as well as Lewenstein’s quantum model [10

10. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994) [CrossRef] [PubMed] .

].

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

6. M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A 74(5), 053401 (2006) [CrossRef] .

]. 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

11. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

], the corresponding macroscopic polarization of the medium is characterized by the average deviations of the positive and negative charges P(τ) = eini(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.

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 = tz/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
2c2Eτξ+2E=n2(p)1c22Eτ2+μ02P(τ)τ2
(1)
where p is the pressure of the gaseous medium and μ0 the permeability, and we denote P(τ) = P(r, ξ, τ).

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 satisfies
dNe(τ)dτ=R(τ)=Rpro(τ)Rrec(τ)
(2)
where R(τ) is the rate of the electronic density. It consists of two parts: the production rate and the recombination rate,
Rpro(τ)=w(τ)Na(τ),Rrec(τ)=τproα(τ,τpro)Rpro(τpro)
where Na(τ) = N0Ne(τ) is the density of neutral atoms with N0 the initial density of neutral atoms depending on the applied temperature and pressure. In above equations, the time τpro is the releasing time of an electron from its parent ion under a strong femtosecond laser field, and α(τ, τpro) characterizes the recombination condition at τ = τrec. In the quantum frame, the value of α(τ, τpro) 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

23. J. Eichler and T. Stohlker, “Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high-Z systems,” Phys. Rep. 439, 1–99 (2007) [CrossRef] .

] 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

24. A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett. 83(4), 706–709 (1999) [CrossRef] .

] 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 me(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]
x(τ,τpro)=x0+τproτv(τ,τpro)dτ
(3)
where v(τ, τpro) is the corresponding velocity which is given by
v(τ,τpro)=emeτproτE(τ)dτ=eme[A(τpro)A(τ)]
(4)
Here the vector potential A(τ) of the electric field is defined as A(τ)=τE(τ)dτ. Using Eqs.(3) and (4), we have
x(τ,τpro)=x0+eme[A(τpro)(ττpro)τproτA(τ)dτ]

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 xc=e/(4πε0E). 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 by
pst(τrec,τpro)eA(τpro)=mx0τrecτpro
(5)
where pst(τ,τpro)=eτproτA(τ)dτ/(ττpro). Obviously, Eq. (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.

Fig. 1 Return time v.s. releasing time of the freed electrons. The subsequent excursions (red pentagrams) of an ionized electron is truncated due to the recombination coefficient α(t, tb) = 1 at the first return time.

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]).

Fig. 2 Density of ionized electrons. The red dashed line represents the recombination-corrected density, and as a comparison, the non-corrected one is shown by the magenta connected asterisks.
Fig. 3 The contours of the laser pulse at (a) the entrance and (b) the exit. The laser pulse is selected with a Gaussian profile, E(r,τ)=E0exp(r2/rf2)exp(τ2/τf2)cos(ωτ), where the angular frequency ω = 2πc/λ with the wavelength λ = 800 nm, the laser intensity is I0 = 3.0×1015W/cm2, and the full width at half maximum (FWHM) τf = 5 fs, rf = 40μm.

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
Ne(τ)s(τ)=τ[1θ(ττrec[τ])]Rpro(τ)x(τ,τ)dτ
where 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
τθ(ττrec[τ])=δ(ττrec[τ])
and the integral of an arbitrary given τ-dependent function F(τ) over (−∞, τ] is
τδ(ττrec[τ])F(τ)dτ=τproF(τpro[τ])
where the right-hand side sums over the releasing time τpro of electrons that return to the ions at time τ. Moreover, the density of free electron can be re-expressed by θ(τ)-function
Ne(τ)=τ[1θ(ττrec[τ]]Rpro(τ)dτ

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 is
P(τ)τ=IpRpro(τ)E(τ)eτproRpro(τpro[τ])x(τ,τpro[τ]]+eτ[1θ(ττrec[τ]]Rpro(τ)x(τ,τ)τdτ
where the releasing position x(τ, τ) = Ip/(eE(τ)) and θ(ττrec[τ]) = 1 are used. Notably, the second term on the right-hand side actually vanishes due to the zero return position x(τ, τpro[τ]) = 0. However, we should maintain this term for the second derivative of polarization because the return velocity of the electron (τ, τpro[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 Ne(τ) electrons, i.e., 〈Ne(τ)v(τ, τ′)〉. Using the electronic motion equation me(τ, τ′) = eE(τ) with τ < τrec[τ′], the second derivative of polarization P(τ) is
2P(τ)τ2=Ipτ[Rpro(τ)E(τ)]+e2meNe(τ)E(τ)2eτproRpro(τpro[τ])v(τ,τpro[τ])
(7)
where the return velocity v(τ, τpro[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,
E(τ)dξ=c2τ2E(τ)dτ12cτ[n2(p)1]2E(τ)τ2dτcμ0Ip2[Rpro(τ)E(τ)]cμ0e22meτNe(τ)E(τ)dτ+cμ0e2meττpro[Rpro(τpro[τ])τpro[τ]τE(τ)dτ]dτ
(8)
where the first term are responsible for the transverse diffraction of the laser pulse, and the second term shows that the laser pulse experiences a dispersion due to the neutrals. The third term plays an important role to the optical field-ionization of the gaseous medium. The penultimate term shows that the ionized electrons gain their kinetic energy from the laser field. The last term involves the recombination processes of electrons with their parent ions. All these terms together dominate the evolution of the laser pulse propagating through the field-ionizing medium. As shown in the contours of the laser pulse in 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.

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 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,
Tpro(τ)=IpRpro(τ),Tioni(τ)=IpRnonRecom(τ)
(9)
where RnonRecom(τ) is the ionization rate without recombination involved, and Tioni(τ) is the corresponding energy transfer rate from laser field to the ionized atoms. The comparison up to a factor Ip 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).

Fig. 4 Comparison of the recombination-corrected production rate with the non-corrected ionization rate of electrons in helium gas. Clearly, the electron production rates depend strongly on the electric field strength. The result of the corrected model almost coincides with that of the non-corrected model during the front edge of laser pulse, while during the back edge of laser pulse the result of the corrected model is significantly larger than that of the non-corrected one.

(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
Jdrift=eτ(1θ(ττrec[τ]))Rpro(τ)ν(τ,τ)dτ
where the introduction of 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
Tdrift(τ)=e22meNe(τ)[A(τ)]2τe2meA(τ)ττ(1θ(ττrec[τ]))Rpro(τ)A(τ)dτ
(10)
where we have used E(τ) = −∂A(τ)/∂τ, v(τ, τ′) = (e/me)[A(τ′)−A(τ)], and the expression of Ne(τ) in terms of θ-function. While the absorption rate of the laser energy without electron-ion recombination can be calculated as [25

25. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999) [CrossRef] .

]
Tdrift(τ)=e22me[A(τ)]2τNnonRecom(τ)
(11)
where NnonRecom(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.

Fig. 5 Comparison of the energy transfer rate to accelerate the ionized electrons between with/without the electron-ion recombination incorporated. Obviously, the electron-ion recombination causes the decrease of the ionized electrons, thus reduces the energy transfer rate.

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
Tdrift(τ)=τ(1θ(ττrec[τ]))ρ˙(τ,τ)Rpro(τ)dτ

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 by
Trecom(τ)=τproRpro(τpro[τ]){12me[v(τ,τpro[τ])]2+Ip}
(12)
where v(τ, τpro[τ]) is the return velocity of an electron born at τpro[τ], and the term enclosed in the curly brace means the energy of an high-energy photon. Ostensibly, the ionization potential Ip 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 Ip equivalent is carried by an ionized electron. So the emitted energy of electromagnetic normalized to incident single-photon energy h̄ω at τ is WEM(τ) = {·}/(h̄ω), where the {·} means the portion in the curly brace in Eq. 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.

Fig. 6 The emission of the high-energy photon. The maximal single-photon energy is about 270 times that of the fundamental laser, which appears at the right side of the main peak of the pulse. Furthermore, there is a periodic harmonic emission with the period of a half cycle of the driving laser field.

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 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.

Acknowledgments

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

References and links

1.

A. Rundquist, C. G. Durfee, Z. H. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft X-rays,” Science 280(5368), 1412–1415 (1998) [CrossRef] [PubMed] .

2.

M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60, 389–486 (1997) [CrossRef] .

3.

S. C. Rae, K. Burnett, and J. Cooper, “Generation and propagation of high-order harmonics in a rapidly ionizing medium,” Phys. Rev. A 50(4), 3438–3446 (1994) [CrossRef] [PubMed] .

4.

E. A. Gibson, A. Paul, N. Wagner, R. Tobey, D. Gaudiosi, S. Backus, I. P. Christov, A. Aquila, E. M. Gullikson, D. T. Attwood, M. M. Murnane, and H. C. Kapteyn, “Coherent soft X-ray generation in the water window with quasi-phase matching,” Science 302(5642), 95–98 (2003) [CrossRef] [PubMed] .

5.

S. C. Rae and K. Burnett, “Detailed simulations of plasma-induced spectral blueshifting,” Phys. Rev. A 46(2), 1084–1090 (1992) [CrossRef] [PubMed] .

6.

M. B. Gaarde, M. Murakami, and R. Kienberger, “Spatial separation of large dynamical blueshift and harmonic generation,” Phys. Rev. A 74(5), 053401 (2006) [CrossRef] .

7.

F. Brunel, “Not-so-resonant, resonant absorption,” Phys. Rev. Lett. 59 (1), 52–55 (1987) [CrossRef] [PubMed] .

8.

I. P. Christov, “Enhanced generation of attosecond pulses in dispersion-controlled hollow-core fiber,” Phys. Rev. A 60(4), 3244–3250 (1999) [CrossRef] .

9.

P. B. Corkum, “Plasma perspective on strong-field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993) [CrossRef] [PubMed] .

10.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994) [CrossRef] [PubMed] .

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22.

N. B. Delone and V. P. Krainov, “Tunneling and barrier-suppression ionization of atoms and ions in a laser radiation field,” Phys. Usp. 41, 469–485 (1998) [CrossRef] .

23.

J. Eichler and T. Stohlker, “Radiative electron capture in relativistic ion-atom collisions and the photoelectric effect in hydrogen-like high-Z systems,” Phys. Rep. 439, 1–99 (2007) [CrossRef] .

24.

A. Scrinzi, M. Geissler, and T. Brabec, “Ionization above the Coulomb barrier,” Phys. Rev. Lett. 83(4), 706–709 (1999) [CrossRef] .

25.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999) [CrossRef] .

OCIS Codes
(320.0320) Ultrafast optics : Ultrafast optics
(020.2649) Atomic and molecular physics : Strong field laser physics

ToC Category:
Ultrafast Optics

History
Original Manuscript: December 13, 2012
Revised Manuscript: February 15, 2013
Manuscript Accepted: February 21, 2013
Published: February 26, 2013

Citation
Cheng-Xin Yu, Shi-Bing Liu, Xiao-Fang Shu, Hai-Ying Song, and Zhi Yang, "A modified rate equation for the propagation of a femtosecond laser pulse in field-ionizing medium," Opt. Express 21, 5413-5423 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5413


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