Attosecond pulse generation by applying a weak static electric field to a few-cycle pulse

The high-order harmonic generation process under the combination of a few-cycle pulse and a static electric field was investigated in this work. A linear harmonic cutoff extension was observed with its dependence on the relative strength ratio of the static electric field with respect to a single-color, 2.5 optical cycle (oc), 800 nm, 1.4×1015 W cm−2 few-cycle pulse as the fundamental driving field. Exploiting the relative strength ratio tuning from 0 to 0.1, a linear continuum width extending on the XUV spectrum up to 191 eV, which supports the creation of an 18 attosecond isolated attosecond pulse, was generated. Moreover, classical trajectory calculation and time–frequency analyses for explaining the mechanism are also presented.

3 field and a parallel static electric field exhibiting a multiplateau structure with much longer plateaus than in the case of the presence of a linearly polarized laser field alone.
In this paper, we report on HHG from a combination of a single-color 2.5 optical cycle (oc), 800 nm few-cycle pulse as the fundamental driving field and a static electric field. The main theoretical results showed significant linear harmonic continuum extension by increasing the relative strength ratio of the static electric field with respect to the peak intensity of a fewcycle pulse. The concept of harmonic continuum is defined as the energy difference between the largest and the second-largest harmonic cutoffs (HCOs) on HHG spectra. Then, Fourier transformation is carried out to simulate the creation of IAP by synthesizing harmonics from continuum on HHG spectra with intensity I (t) = | N A q exp[−iw q t + iϕ(w q )]| 2 , where A q is the phase width and ϕ q is the phase of the qth harmonic. According to this equation, the duration of the synthesized IAP is allowed to be shortened to T 0 /2N under the assumption that all the harmonics have the same phase; in other words, continuum with a broad spectral width and a flat profile leads to the creation of intense and short IAP. Thus, we superpose a static electric field onto a few-cycle pulse in order to obtain a continuum satisfying the above quality supporting the prerequisite for the creation of intense and short IAP.
This paper is organized as follows. In section 2, the field amplitude E 2 as well as the ionization probability of the helium gas target from the combination of a single-color, 800 nm few-cycle pulse as the driving field and a static electric field is given. The results indicate the expectation of broad continuum on HHG spectra by superposing a static electric field on a few-cycle pulse. In section 3, through a quasi-classical picture of HHG, an expression for the continuum bandwidth on HHG spectra as a function of the driving field intensity, wavelength and relative strength ratio is given. In section 4, by performing our time-dependent wave packet code, the dependence of HHG and IAP creation on the relative strength ratio of a static electric field is shown. In section 5, we carry out classical trajectory calculation and time-frequency analyses to shed light on the static electric field modulation scheme. Section 6 presents the conclusion. In addition, the time unit in this paper is the optical cycle of an 800 nm driving field, while the harmonic unit is the integral time of the fundamental few-cycle pulse frequency.

Effect of the static electric field on the driving pulse field
In our simulation, the laser parametric source is a 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse, which is usually generated by an amplified Ti:sapphire laser system in the laboratory. The pulse duration 2.5 oc is equal to 6.667 fs, which corresponds to two and a half periods of the 800 nm laser field. A weak static electric field is employed to modulate the driving field with its maximum intensity of 1.4 × 10 14 W cm −2 . Although achieving a static electric field with such a high intensity is difficult with current technology, a low-frequency field can be a good alternative in practice.
The expression for the combination field can be expressed as where E 1 is the peak strength of the few-cycle pulse, τ is the full-width at half-maximum (FWHM) of the few-cycle pulse, w is the frequency of an 800 nm few-cycle pulse and k is the relative strength ratio of a static electric field with respect to the peak strength of the few-cycle pulse. We choose the few-cycle pulse shape of f (t) = cos 2 [π/2 × t/τ ]; thus the few-cycle pulse in our simulation has finite duration and satisfies the physical prerequisite τ 0 E(t ) dt = 0 [44][45][46]. When k = 0, the corresponding case is for a few-cycle pulse alone. In our simulation, the gas target used is a helium atom; a soft-core potential is used with the formula V (x) = −1/ √ a + x 2 . In order to match the experimental value of 24.6 eV for the ionization potential I P of helium, the parameter a = 0.484 is chosen. Figure 1(a) shows a comparison between E 2 of a 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 fewcycle pulse alone and a combination of a few-cycle pulse and a weak static electric field with k = 0.05 and k = 0.1. The HHG process has the period of half the optical cycle of the driving pulse; the difference between the electric field strengths of a neighboring half-cycle induces the energy difference HCO generated by each half-cycle. For this reason, we compare the intensity ratio E 2 0 /E 2 1 , which is the peak intensity of the combination field versus that of the side peak. The intensity ratio E 2 0 /E 2 1 of the above three cases is 1.223, 1.508 and 1.846, respectively, whereas the insets of figure 1(a) show an intensity ratio of 1.776 from the 4 fs, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse alone. A noticeable enhancement of the intensity ratio E 2 0 /E 2 1 from 1.223 to 1.846 was observed by increasing the relative strength ratio k from 0 to 0.10. It is worth noting that when the relative strength ratio increases up to 0.1, the intensity ratio of the driving field even surpass the case of a 4 fs, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse alone, which is difficult to be achieve in current laboratory conditions.
In order to investigate the behavior of the ionization probability dependence on the relative strength ratio k, we employed the Ammosov-Delone-Krainov (ADK) [47] model to compare ionization probability in helium under different driving fields. Figure 1(b) shows the ionization probability generated by the 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse alone (green solid line), superposed by a static electric field with relative strength ratios k = 0.05 (blue dashed dotted line) and k = 0.1 (red dashed line). We observed a decrease in ionization probability before t = 0 oc, and a significant increase in ionization probability at the time with a light delay after t = 0 oc when a static electric field is applied. In that case, harmonic emission in the subcycle of the driving field with the largest magnitude is afforded by a large number of freed electrons. Thus, we may expect a larger HCO and a continuum on HHG spectra with higher intensity.  , and a few-cycle pulse superposed by a static electric field with relative strength ratio k = 0.05 (blue dashed dotted line) and k = 0.1 (red dashed line). We employed the quasi-classical picture to depict the harmonic generation under the combination of a few-cycle pulse and a static electric field. Clearly, there are five larger peaks labeled from A to E in all the driving fields in figure 2(a). According to the TSM, firstly, the helium gas target is ionized at peak A; secondly, freed electrons will be accelerated by peak B; finally, due to the change of direction of the linearly polarized fewcycle pulse in each half-cycle, electrons will be pulled back to recombine between B and C. In the above case, harmonic photons with an energy of E k plus I p are emitted. Since the HHG process has a period of half the optical cycle of the driving field, each driving field in figure 2(a) has three independent processes: A to B to C, B to C to D and C to D to E. We labeled the three processes as P1, P2 and P3, respectively. Next, we introduce two concepts in our work.
First, HCO is defined as to the maximum frequency of the emitted harmonic photons in each sub-cycle harmonic emission. Then the harmonic continuum bandwidth can be defined as the energy difference between the two largest HCOs.
A gas target atom was ionized at t = t i ; after being accelerated, it was pulled back to recombine with parent ions at t = t f ; the velocity obtained during this process is given by where E(t) is the temporal profile of the driving field. During each sub-cycle harmonic emission, HCO is contributed by those recombined electrons that have maximum kinetic energy; this is approximated by where U p is the ponderomotive potential that is expressed as U p = E 2 4w 2 . In this paper, HHG is under the combination of a few-cycle pulse and a static electric field E = E(t) + E s , so that equation (2) HHG from a few-cycle pulse alone is given by The time spent during each sub-cycle harmonic emission from P1, P2 to P3 is approximately be the same, so that Firstly, we consider the HCO, which is determined by the ionization potential of the gas target and the kinetic energy. The energy of harmonic photons in P1 is given by Putting equations (4) into equation (5), we obtain the HCO from P1 under the combination of a few-cycle pulse and a static electric field: where n 1 is the HCO order from P1 under the combination field and n 1 is the HCO order from P1 under a few-cycle pulse alone. The HCO from the corresponding P2 is given by where n 2 is the HCO order from P2 under the combination field and n 2 is the HCO order from P2 under a few-cycle pulse alone. It is worth noting that the direction of HCO shift from P1 and P2 is inverted because a fewcycle pulse field changes its direction every half-optical cycle, whereas a static electric field does not. The entire extension of harmonic continuum on HHG spectra is obtained by subtracting n 1 from n 2 : According to equation (4b), the velocities V 2 and V 1 obtained in a few-cycle driving field are dependent on the time of acceleration t and the strength of the electric field; we take an approximation V ∝ T × E, while due to the half-circle period of HHG, t is proportional to the period of the few-cycle pulse t ∝ T ∝ λ. Finally, we generate the harmonic continuum bandwidth by taking the above linear approximations: where k = E s /E 0 is the relative strength ratio of the static electric field with respect to the few-cycle pulse field. By taking the approximation, we obtain the following: the continuum bandwidth on HHG spectra is directly proportional to the intensity of the few-cycle pulse, square of the wavelength of the few-cycle pulse and the relative strength ratio k of the static electric field. In order to illustrate this relation, we calculate the continuum bandwidth as a function of the relative strength ratio k, the few-cycle pulse's intensity and the few-cycle pulse's wavelength by taking classical trajectory calculation. In this simulation, a single-color 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse is employed as the fundamental field and helium with an ionization of 24.6 eV is used. By varying the relative strength ratio from 0 to 0.1 in each 0.004, we observed a linear increase in continuum bandwidth with increasing relative strength ratio, which is comparable with the theoretical prediction in equation (9). Next, fixing the relative strength ratio k, a linear increase in the continuum with increasing intensity of the few-cycle pulse is shown in figure 2(c). Finally, in figure 2(d), the results of our classical trajectory calculation show that the continuum bandwidth is a second-order function of the fewcycle pulse wavelength. Thus, the continuum bandwidth on the HHG spectrum estimated from equations (9) is clearly discernible from figures 3(b)-(d) through classical trajectory calculation.

Numerical results
We investigated the HHG dependence on the relative strength ratio of the static electric field in extending the continuum bandwidth. Firstly, the largest and the second-largest HCO were calculated from a combination of a 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse and a static electric field with a relative strength ratio varying from 0 to 0.1 in steps of 0.004. Figure 4(a) shows the linear dependence of both HCOs on the relative strength ratio k of the static electric field. A noticeable linear continuum extension is clearly discernible. Next, we analyze the HHG spectra as a function of relative strength k by quantum calculation through our time-dependent wave packet code [48][49][50][51][52][53][54][55][56][57][58][59]. In figure 3(b), the intensity of HHG spectra is plotted as a function of harmonic order and the relative strength ratio k. The relative strength ratio k is shifted from 0 to 0.1 in steps of 0.004. Each color bar in figure 3(b) represents an HHG spectrum, and both the largest and second-largest HCOs are clearly visible in each bar. We observed the largest HCO shifted linearly to higher order, whereas the second-largest one moved linearly to lower order with an increase in the relative strength ratio. The consistent results of both classical and quantum calculations proved equation (9). We will now show the HHG spectra under a few-cycle pulse alone (green solid line), a combination of a few-cycle pulse and a static electric field with k = 0.05 (blue dashed line) and k = 0.1 (red solid line). As shown in figure 4(a), by increasing k from 0 to 0.05 to 0.1, the second-largest HCO decreases from 167 to 147 to 128, while the largest HCO increases from 189 to 219 to 252, leading to extension of the continuum bandwidth from 34 to 110 to 191 eV. By taking Fourier transformation of the harmonics from the continuum on HHG spectra, IAP with duration from 81.3 attoseconds (as) to 25.6 as to 18.1 as were created. At the same time, a noticeable intensity increase of the created IAP was observed; as can be seen from figure 4(b), the intensity of the created IAP under a combination field with k = 0.1 is 30 orders greater than the corresponding case under a few-cycle pulse alone.

Classical trajectory calculation and time-frequency analyses
In order to better clarify the static electric field's effect on HHG, we report classical trajectory calculation and time-frequency analysis in figure 5. In figure 5(a), each point represents the harmonics photon energy and the corresponding recombination time. The harmonic cutoff photons from a neighboring sub-cycle shows an energy difference of 40.7 eV under a 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 few-cycle pulse alone, 72.6 eV under the combination field of the same few-cycle pulse and a static electric field with k = 0.02. The three-dimensional time-frequency analysis from the above two cases is shown in figures 5(b) and (c), respectively. Time-frequency analysis is performed by taking the Gabor transformation of the dipole acceleration [60,61] code. The intensity of harmonic photons is plotted as a function of time and harmonic order in figures 5(b) and (c). Five emission peaks are clearly observed; we label the largest three from A1 to A3, from the top to the bottom. While the energy difference between the largest HCO A2 and the second-largest HCO A2 was broadened by adding a static electric field with the relative strength ratio k = 0.02, this induces the extension of continuum bandwidth on the HHG spectra.

Conclusion
In summary, we reported HHG from a combination of a single-color 2.5 oc, 800 nm, 1.4 × 10 15 W cm −2 pulse and a static electric field. We obtained the expression for continuum bandwidth, which is a function of the relative strength ratio k and the few-cycle pulse's intensity and wavelength through the classical perspective. We demonstrated the relation by performing classical calculation and the time-dependent wave packet code. Our numerical results show a significant linear extension of continuum bandwidth on HHG spectra by superposing the static electric field onto the few-cycle pulse. After Fourier transforming the harmonics from the continuum on HHG spectra, a 18 as IAP with improved intensity was created from a combination of a few-cycle pulse and a static electric field with k = 0.1. Finally, classical trajectory calculation and time-frequency analysis were used to shed light on the physical scheme.