1. Introduction

Attosecond pulses are important tools for studying and controlling the motion of electrons inside atoms. Until now, high-order harmonic generation (HHG) is the only method that has produced single attosecond pulses in experiment. The high-order harmonic spectrum has a very characteristic and universal shape: it falls off for the first few harmonics, then exhibits a plateau where all the harmonics have the same strength, and ends up with a sharp cut-off [1]. One can understand the high-order nonlinear response of an atom by considering the following three steps [2,3]: ionization, acceleration and recombination.

The successful generation of isolated attosecond pulses is an important step towards attosecond physics since single attosecond pulses are more promising for applications in time-resolved spectroscopy. Obtaining single attosecond pulses remains an important problem. There are currently two ways that could be used to solve this problem. The first is to use a short enough (∼7 fs) linearly polarized fundamental pulse such that the emission of the cut-off harmonics (i.e., the highest harmonics which can only be emitted at the peak intensity) is naturally confined to a fraction of the laser oscillation period at the peak of the pulse. The HHG pumped by an ultrashort (7fs) laser pulse has led to an isolated 650 attosecond pulse [4]. The second way is to generate harmonics with a pulse whose polarization changes with time [5–10]. HHG efficiency is the highest for linearly polarized pump pulse and rapidly decreases with increasing pump pulse ellipticity [11,12]. Temporal modulation of pump pulse ellipticity confines the xuv emission to a temporal gate where the polarization is very close to linear, as shown in recent experiments [13–15]. Different methods have been proposed to generate ellipticity time-gate [10,13,16]. Recently, such gate was obtained by superposing two few-cycle counter-circularly polarized pulses [17]. In the scheme, single attosecond pulses and xuv supercontinuum are produced in the high-order harmonic plateau. Besides the methods mentioned above, the optimization scheme of the exciting laser phase through a genetic algorithm [18] was applied to synthesis single attosecond pulses. In Ref. [19], single attosecond pulse is obtained using appropriately chirped exciting laser pulse of longer duration (8-16fs).

In this work, a technique of using a two-color pump pulse with time-dependent ellipticity to produce single attosecond pulses via HHG is proposed, different from the previous work where the gating via the control over ellipticity is realized with one-color field. The two-color field is created by the fundamental field and its second harmonic. The fundamental field is left-circularly polarized and the second harmonic is right-circularly polarized. The advantage of this technique is possible to generate strong single attosecond pulses using long (20 fs) pump pulse because it is much easier to obtain a high-energy long pulse than to obtain a high-energy few-cycle laser field.

2. Theory and simulation results

The peak field amplitude E ₀, pulse duration τ_p and carrier-envelope phase (CEP) φ are the same for the two pulses. The carrier frequencies of the two fields are ω and 2ω respectively. The delay between them is T_d, which is an integral number of optical periods. The electric fields of the two counter-circularly polarized pulses propagating in the z direction are

where x̂ and ŷ are unit vectors in the x and y directions respectively. The electric field of the synthesis pulse is

And the time-dependent ellipticity of the synthesis pulse in the x direction is showed by Eq. (4).

 

Fig. 1. The ellipticity (in the x direction) of the synthesis pulses. Solid line: two-color synthesis field and dashed line: one-color synthesis field.

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In Fig. 1, the blue solid line shows the calculated ellipticity (in the x direction) of the two-color synthesis pulse using Eq. (3) when T_d=τ_p=20 fs. The red dashed line depicts the ellipticity (in the x direction) of one-color synthesis pulse under otherwise identical conditions. The blue solid curve oscillates with time and the envelope of it is in agreement with the red dashed curve. The ellipticity of one-color synthesis pulse is dependent on the pulse duration and the time delay between them. The width of the ellipticity time-gate is broadened with the decreasing of the time delay. And in the contrary, the width is narrowed with the increasing of the time delay. However, in the case of the two-color synthesis field, the width of the ellipticity time-gate is hardly affected by the time delay between the two pulses. But the change of the time delay gives effects on the modulation depth of the ellipticity curve. The ellipticity of two-color synthesis field is influenced by two different frequencies which resulted in the oscillation of the ellipticity curve. We got a much shorter ellipticity time-gate than in the case of one-color synthesis field. The peak intensity of the synthesis field is also controlled by the time delay in both cases of the two-color and one-color synthesis fields. The moderate time delay, which results in a moderate intensity of synthesis field, is selected in this work.

The fundamental field (left-circularly polarized) and its second harmonic (right-circularly polarized) are assumed to have the same Gaussian pulse shape with a pulse duration τ_p of 20 fs. The center wavelength of the fundamental field is 800 nm. φ =0 is selected. The peak intensity of the fundamental field and its second harmonic are both 4×10¹⁴W/cm². The peak intensity of the linear portion is 5.66×10¹⁴W/cm². In this work, “few-cycle pulse” refers to 7 fs linearly polarized field with intensity of 8×10¹⁴W/cm², unless specified otherwise, which is showed by Eq. (5).

Neon atom (I_p=21.56eV) is considered in our calculation.

Our numerical simulation of the HHG process is based on the current general quantum mechanical model by Lewenstein et al. [17,20], which is practical in quantitative analysis of HHG process. We calculated the ionization rate of electron by Ammosov-Delone-Krainov tunneling ionization formula [22]. All the simulation results: the harmonic spectra and the temporal profiles of attosecond pulses are obtained with this model.

 

Fig. 2. High-order harmonic spectra with (a) two-color field and (b) few-cycle field. In (a), blue solid line and red dashed line depict the harmonic emission in the x and y directions respectively, while their relative intensities are labeled arbitrarily.

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The harmonic spectrum and the attosecond pulse profile are calculated. Figure 2(a) shows the high harmonic spectra in the x (blue solid line) and y directions (red dashed line) obtained with two-color synthesis field. Two spectra are plotted in one figure for comparison, while their relative intensities are labeled arbitrarily. The harmonic spectra produced by two-color synthesis field (Fig.2 (a)) give even order harmonics as well as odd order harmonics. The harmonic spectrum (Fig. 2(b)) produced by the few-cycle field contains odd order harmonics only. The second harmonic of the synthesis field resulted in the appearance of even order harmonics. The harmonic spectrum produced by two-color synthesis field is much stronger than that produced by few-cycle field of comparable intensity. These conclusions are agree with previous results of two-color pump pulse [23,23].

 

Fig. 3. Temporal profiles of attosecond pulse with (a) two-color field and (b) few-cycle field. Physical conditions are the same as in Fig. 2. The high-order harmonic pulses centered at two different frequencies: 51 ħω and 111 ħω (both in the cut-off). The inset of fig.3 (a) shows the harmonic emission in the x (blue solid line) and y (red dashed line) directions with two-color pump pulse.

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A Gaussian spectral window is multiplied by the harmonic spectrum, and an inverse Fourier transform is performed to obtain the attosecond pulses in the time domain. In our calculations, a Gaussian window with the full width at half maximum (FWHM) of 6 eV is used. Figure 3(a) shows the single attosecond pulse centered at 51 ħω in the x direction in the case of two-color field and the FWHM of the attosecond pulse is about 470 attosecond. For comparison, we give the single attosecond pulse profile (Fig. 3(b)) centered at 111 ħω produced by linearly polarized few-cycle pump pulse of comparable intensity. Obviously, the intensity of the attosecond pulses in Fig. 3(a) is much stronger than that in Fig. 3(b). In the inset of Fig. 3(a), the blue solid line and the red dashed line depicted the harmonic emissions in the x and y directions respectively with two-color synthesis pump pulse.

The two-color synthesis field is created by two counter-circularly polarized pulses with the duration of 20 fs and two pulses are separated by 20 fs. Although the ellipticity (blue solid line in Fig. 1) is similar to the curve in the case of one-color synthesis field (red dashed line in Fig. 1), there is a fast modulation which is different from the case of one-color field. With this modulation, a much shorter time-gate can be produced which supports the generation of the single attosecond pulse even with 20fs long pump pulse. Because the harmonic emissions, especially in the cut-off region, have a strong dependence on the ellipticity of the pump pulse, the harmonic spectrum with the highest order can only be emitted by the peak intensity. So, although there are many short ellipticity time-gates, only the time-gate around the peak intensity can produce the high-order harmonics of the cut-off region. So the single attosecond pulse can be obtained by the inverse Fourier transform of the spectrum filtered from the cut-off region by a Gaussian window. In our work, from Fig. 2(a) we also can see, although the spectrum is strongly modulated, the spectrum in the cut-off region is not separated to be one harmonics alone. There always have several harmonics in each part. So we think that the spectrum in the cut-off region (Fig. 2(a)) contains some continuum spectra close to each other. To select part of harmonic spectrum centered properly, a nearly continuum spectrum can be filtered out by a Gaussian window. This also claims the function of the time-gate to generate a single attosecond pulse.

 

Fig. 4. Attosecond pulses in the x direction produced by two-color synthesis pump pulse with different CEP. (a) φ =0; (b) φ =π/4; (c) φ =π//2: (d) φ =3π//4; (e) φ =π.

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In this scheme, the CEP also plays an important role. For the CEP (φ =0) we selected, single attosecond pulse is produced in the x direction. If the CEP is changed with π/2 value, more than one attosecond pulses are produced in the x direction while a single attosecond pulse is produced in the y direction. The modulation cycle of the CEP is π. In Fig. 4, attosecond pulses centered at 51 ħω in the x direction produced by two-color synthesis pump pulse with different CEP are given. The single attosecond pulse can only be produced at the CEP=0 and π. So, similar to the previous work, phase stabilized pulses are always needed.

Using a three-dimensional code including the three dimensional wave equation for the harmonic field, the propagation effect, i.e., absorption, dephasing and defocusing can be considered. And the long-path contribution will be suppressed by the propagation effect. According to the work by Chang et. al. [17], the propagation effect may be benefit to the generation of single attosecond pulses.

3. Summary

In conclusion, a new technique for the generation of single attosecond pulses is proposed. Two-color field used as pump pulse is the synthesis of 20 fs fundamental field (left-circularly polarized) and its second harmonic (right-circularly polarized) separated by 20 fs. For the introduction of second harmonic, a much shorter ellipticity time-gate is obtained. And single attosecond pulse is produced in the cut-off. The single attosecond pulse produced by two-color field in the cut-off is about two orders magnitude stronger that produced by few-cycle field of comparable intensity.