1. Introduction

The recent progress in the tabletop laser system has led to the generation of high-intensity (I>10¹⁴ W/cm²) optical pulses with duration down to a few periods of the carrier frequency [1,2]. This opened new perspectives in light-matter interaction because the external field may increase significantly before the atom is fully ionized. In particular, generation of harmonic orders higher than 300 is observed which is a predecessor of the newly-emerging field of extreme nonlinear optics [3,4].

When matter is exposed to a strong fast-increasing field the transient ionization becomes the dominating mechanism which determines the atomic response. This affects both the single-atom interactions and the propagation of laser pulses through gas medium where macroscopic effects like dispersion take place. Because of the very short pulse duration the ionization increases rapidly along the leading front of the pulse so that the plasma dispersion that is due to the increasing concentration of free electrons leads to significant changes in the pulse profile [5]. However, the leading front of the pulse still propagates in a low-ionised gas so that it experiences the dispersion due to the neutrals. In addition, the laser pulse experiences loss due to the ionization, which is also a field-dependent effect. Therefore, the correct description of light-matter interaction in the single-cycle time scale requires a method which involves a self-consistent solution of the wave equation and the time-dependent Schroedinger equation. This method appears as an alternative of the well known quasistatic approximation where the ionization is calculated by using tunneling ionisation rate formulas (see e.g. [6,7]). An important flaw in the quasistatic approximation is that it ignores the contribution to the ionisation rate that is due to the return of the electron wave packet to the core. In this way, this is a fully adiabatic approximation which holds only for low frequencies and long pulses [8].

Although the method proposed here requires larger computer resources as compared to the methods based on the quasistatic theory, it is more accurate in that it takes into account the nonadiabatic effects that arise in the ionization of the atom by sub-10 fs laser pulses [9]. The predictions of our method are compared with the results of the quasistatic theory for parameters close to the experiment.

2. Results of the model

Here we use a reduced form of the scalar wave equation written in a local frame of reference moving with the speed of light along the direction of propagation z:

where the first term in the right-hand side of Eq.(1) describes the dispersion that is due to the transient plasma, the second term accounts for the ohmic power dissipation that is due to the ionization [10], and the third term describes the dispersion by the neutrals; n(p) is the index of refraction as a function of the pressure of the neutral atoms (e.g. [11]): p=p₀ [1-P(E)], where p₀ is the initial pressure of a gas of atoms with concentration N, and with an ionization potential I_p . In Eq.(1) P(E) is the ionization probability calculated by solving in a self-consistent manner the Schroedinger equation for a hydrogen-like atom in a field E(t), and by projection of the wave function on the first three bound states of the unperturbed atom:

where α is a smoothing parameter which allows us to tune the binding energy of the ground state to match I_p [9]. In principle, by using Eqs.(1)–(2) we can take ab initio into account some important effects that accompany the light-matter interaction such as the dispersion of the neutral gas and the self-phase modulation that is due to the Kerr nonlinearity [12]. However, inasmuch as the concrete set of levels of the model atom may differ from that of the gas under consideration, the contribution of the field-induced dipole moment is neglected, and the (non-resonant) dispersion is introduced empirically into Eq.1. Also, in most cases of practical interest the self-phase modulation can be neglected because the ionisation increases rapidly at the leading front of the pulse which reduces the density of the neutrals.

The most striking difference between the quasistatic approximation of light-atom interaction and the (ab initio) solution of the Schroedinger equation (see e.g. [13]) is that the former relies on the concept of ionization rate. It is important to stress that the rate is a positively defined quantity while the time derivative of the ionization probability as calculated by the Schroedinger equation is not. This effect can be neglected for pulses much longer than the period of the carrier frequency because of the time averaging over many periods that takes place in this case. However, in a single-cycle time scale (very short pulse) the return of the wave packet to the core at each half-cycle of the increasing laser field modulates strongly the ionization yield, which next modulates the laser pulse through the medium polarization. In order to show this effect we calculated by using Eq.(2) the atomic ionisation by a laser pulse with duration 5 fs (λ=800 nm) and peak intensity 2.5 10¹⁴ W/cm². Fig.1 shows the ionization probability and the rate as a function of time, for α=0 in Eq.2.

 

Fig. 1. Time dependence of the ionization (a) and the ionization “rate” (b) for a hydrogen atom. Solid line-numerical results; dashed line-quasistatic approximation.

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It is seen from Fig.1 that unlike the quasistatic approximation, the ionization obtained from the exact solution of the Schroedinger equation is an oscillating curve, so that the ionization “rate” may acquire negative values (Fig.1(b)). These negative “rates” result from the return of the electron wavepacket to the core at each half-period of the carrier frequency of the laser pulse. However, Eq.1indicates that the negative values of the time derivative of the ionization probability may lead to a negative loss experienced by the laser pulse. With other words, the atomic polarization restores periodically energy back to the pulse. This effect is ignored by the quasistatic theory where the ionization rate is a positively defined quantity. Inasmuch as the changes in the pulse shape are most easily seen from the intensity (envelope) of the field, we retrieve the envelope of the pulse after the propagation. Although the intuitive procedure for envelope retrieval includes an averaging of the field over several periods of the carrier frequency, here we use another very efficient formal procedure based on Fourier analysis. First, we take the Fourier transform of the field and then cut the negative frequency part of the resulting spectrum. After applying an inverse Fourier transform, the modulus of the complex field corresponds to the envelope of the initial oscillating field.

Figure 2 shows the output pulse after propagation through a gas with pressure 25 Torr over a distance of 0.5 cm. Figure 2(a) shows the case when only the loss term is taken into account in the right-hand side of Eq.(1). It is seen that the pulse envelope at the output exhibits oscillations (Fig.2(b)), an effect that is not clearly seen in the wave picture (Fig.2(a)). In fact, these oscillations increase each time when the field approaches zero, which enhances the loss term in Eq.(1). Moreover, it can be seen that the oscillations of the envelope are enhanced by the non-adiabatic calculation (solid line in Fig.2(b)). Besides that, Fig.2 shows that the exact calculation predicts a delay of the field with respect to the quasistatic case, which stems from the delayed non-adiabatic response of the atom under the influence of a strong ultrashort pulse [8]. When the plasma and the neutrals dispersion are taken into consideration, the difference between the predictions of the two models becomes even more pronounced (Fig.2(e)). The difference in the shapes of the propagating pulses predicted by the quasistatic theory and by the Schroedinger equation is most clearly seen in Fig.2(c),(f) where additional time averaging is done (similarly to the response of a slow detector). This difference in the pulse shapes is not trivial and may lead to significantly different results, e.g. when phase matching conditions are important (e.g. in high harmonic generation).

 

Fig. 2. Time dependence of the field (a,d), the envelope (b,e), and the time averaged envelope (c,f) of 5 fs pulse at the output of the gas medium. (a-c)-with the loss term only; (d-f)-with all terms. Solid line-numerical results; dashed line-quasistatic approximation.

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Fig. 3. Time dependence of the field (a), the envelope (b), and the time averaged envelope (c) of 10 fs pulse at the output of the gas medium, with all terms in Eq.1. Solid line-numerical results; dashed line-quasistatic approximation.

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It is instructive to observe the difference between the quasistatic regime and the exact solution for longer pulses. Figure 3 represents the case of 10 fs pulse with intensity 2.2 10¹⁴ W/cm² after propagation through a gas with density 25 Torr and length 0.5 cm. The comparison between the averaged pulse envelopes in Fig.3(c) and in Fig.2(f) shows that the result of the exact solution approaches the predictions of the quasistatic model when the pulse duration is increased. These results indicate that the quasistatic approximation does not provide sufficient accuracy in the single-cycle regime of light-atom interaction and therefore ab initio solution of the Schroedinger equation should be used. An example of such a regime that is of large practical interest is the generation of attosecond pulses by phase matched high harmonic generation in gas medium [9].

Another evidence for the breakdown of the quasistatic approximation is provided by the dependence of the atomic ionization on the absolute phase of a laser field [14]. Figure 4(a) shows the maximum field for a 5 fs laser pulse with time dependence E(t)=E_o(t)cos(ωt+φ), where the envelope E_o(t) has a maximum when φ=0, as a function of the absolute phase of the field φ. It is seen that the maximum field is highest for cosine carrier and lowest for sine. The phase dependence of the calculated ionization for peak intensity 1.75 10¹⁴ W/cm² (which is higher than the barrier suppression intensity 1.3 10¹⁴ W/cm² of the model atom) is shown in Fig.4(b). It is clearly seen that the ionization calculated by the Schroedinger equation has a well defined minimum similar to the minimum of the laser field in Fig.4(a), whereas the quasistatic result shows no dependence of the ionization on the absolute phase of the pulse. We found that this result is valid in both tunneling and barrier-suppression regimes of ionization although the quasistatic formula is not strictly valid for the latter. In general, it can be shown that the phase dependence of the ionization in the barrier suppression regime is close to the phase dependence of the amount of photons when the field exceeds the “appearance value” E_app=${\mathrm{I}}_{\mathrm{p}}^2$/4.

 

Fig. 4. Dependence of the maximum field amplitude (a) and of the final ionization (b) on the absolute phase of the laser pulse. Solid line-numerical results; dashed line-quasistatic approximation.

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3. Conclusions

In conclusion, we describe theoretically the propagation of sub-10 fs optical pulses in gas medium beyond the wide-spread quasistatic approximation. It is shown that the non-adiabatic effects that arise due to the return of the electron wave-packet to the core modify the atomic ionization and lead to significant difference in the output pulse profiles. Therefore, the self-consistent solution of the wave equation and the Schroedinger equation provides more accurate description of the propagation effects especially when phase matching conditions are important, e.g. in high-harmonic and attosecond pulse generation.