1. Introduction

Attosecond science can describe and trace ultrafast electron dynamics in an atom or a molecule [1]. High harmonic generation (HHG) is a most effective way to generate an isolated attosecond pulse; it can also be used as a source of soft X-ray [2]. A HHG spectrum has a general characteristic: It decreases rapidly for first few harmonics, then exhibits a broad plateau, and finally ends up with a sharp cutoff. A harmonic generation process can be well described by the semi-classical three-step model [3, 4], which says that an electron first tunnels out of a Coulomb potential barrier suppressed by the field and moves away from the parent ion; then the freed-electron is pulled back and accelerated when the laser field direction is reversed; finally it recombines with the parent ion and emits a harmonic photon. A large number of schemes have been proposed to produce an isolated attosecond pulse, such as a polarization gating technique [5], two-color field control [6], spatially inhomogeneous fields [7], and so on.

Recently, a method was put forward to generate an isolated attosecond pulse by using a Terahertz (THz) field, which is an interesting topic of research due to its potential applications [8]. Yuan et al [9] present a scheme for producing a single circularly polarized attosecond pulse by a few-cycle elliptically polarized laser field combined with a THz field. An isolated circularly polarized attosecond pulse with the duration of about 114 as (1 as =10⁻¹⁸ s) is obtained, and the transverse kinetic energy as a functions of the ellipticity and terahertz field strength is also considered. A comparison is made [10, 11] between a HHG in a linearly polarized laser field combined with a THz field and with a static electric field. The propagation effect of a 800 nm laser pulse combined with a strong THz field in neon gas is also studied [12].

It should be mentioned that the harmonic plateau in HHG spectrum can be extended when two perpendicularly polarized two-color laser fields are combined with a static electric field [13]. However, such a strong static field is hard to achieve in experiment. The generation of a THz field with a peak intensity reaching tens of kV/cm and even exceeding MV/cm has already been achieved [14]. Thus a HHG process can be controlled by a THz field instead of a static electric field.

When an atom or molecule is irradiated by a circularly polarized laser pulse, the ionized electron can hardly return to the vicinity of the parent ion and emit energetic photons. Attosecond pulses generated in this way are very weak due to low efficiency of HHG. The HHG by using circularly polarized laser pulses and magnetic fields have been investigated before [15, 16], which illustrates recollision model of the ionized electron with its parent ion and the result show that the magnetic fields can be used to enhance the efficiency of HHG. In this paper, we investigate the HHG and isolated attosecond pulse generation of ${\mathrm{H}}_2^{+}$ in a circularly polarized laser pulse combined with a THz field. The harmonic intensity can be greatly enhanced and a continuum spectrum can be formed when a THz field is added. The emission time of harmonics in terms of the time-frequency analysis, the temporal evolution of the probability density of electron wave packet, and the semi-classical three-step model and the classical electron trajectory theory are also investigated to illustrate the physical mechanism of HHG. By superposing the harmonics in the range of 216–249 eV, an isolated attosecond pulse with a duration of about 69 as can be achieved.

2. Theoretical method

In our previous work [17], using a one-dimensional model of ${\mathrm{H}}_2^{+}$, we studied the HHG by numerically solving the non-Born-Oppenheimer time-dependent Schrödinger equation. Our results show that the nuclear motion leads to an overall weaker harmonic signal as compared to fixed-nuclear case, but the harmonic spectrum becomes smoother with fewer modulations. In this work, we consider fixed-nuclei ${\mathrm{H}}_2^{+}$ exposed to a circularly polarized laser pulse combined with a THz field. The HHG and attosecond pulse generations are explored based on the single active electron approximation by numerically solving a 2D time-dependent Schrödinger equation using the second-order splitting-operator fast Fourier transform algorithm [18]. The initial wave function is constructed by the imaginary time-propagation method. The two-dimensional time-dependent Schrödinger equation (in atomic units) is

The three-step model, which makes the ionized electron to have a non-zero initial transverse velocity, can be used in the circularly polarized laser field [20], and the transverse displacement caused by the external field is compensated by an initial transverse velocity [21, 22]. In order to apply the three-step model to the analysis of the HHG in the circularly polarized laser pulse, we consider the effect that the electron has non-zero initial velocity.

According to the three-step model, the Coulomb field of the ${\mathrm{H}}_2^{+}$ is neglected and the initial positions x₀ and y₀ are zero [21]. The trajectory can be written as $x(t)={\displaystyle \int {}_{t_0}^t\left[{\displaystyle \int {}_{t_0}^{t^{\prime }}{E}_x(t^{″})}d{t}^{″}+v_{x0}\right]}d{t}^{\prime }$ and $y(t)={\displaystyle \int {}_{t_0}^t\left[{\displaystyle \int {}_{t_0}^{t^{\prime }}{E}_y(t^{″})}d{t}^{″}+v_{y0}\right]}d{t}^{\prime }$, where v_x0 = v_‖ cos α+v_⊥ sin α and v_y0 = v_‖ sin α + v⊥ cos α are the initial velocity components, v_‖ and v_⊥ are the initial parallel velocity and transverse velocity, respectively, and α is the angle between the ionizing field vector and the coordinate axis at the ionization time t₀. The initial velocity parallel to the ionizing field can be set as v_‖ = 0, because the electron leaves the molecule by tunneling. A proper initial velocity can be determined by solving x(t) = 0 and y(t) = 0.

3. Results and discussion

We first study the harmonic generation of ${\mathrm{H}}_2^{+}$ by a 6 cycle circularly polarized laser pulse of λ = 800 nm with the peak intensity I_x = I_y = 5 × 10¹⁴W/cm² (E_x0 = E_y0=0.1194 a.u.). The relative phase is φ = −0.1π. Figure 1 shows that the cutoff energy of the harmonic spectrum in this circularly polarized laser field (the solid black line) is about 92eV, which follow the rule of I_p+2U_p (the maximum electron energy in circularly polarized laser field is 2U_p [23]), but the plateau structure are not pronounced, which illustrate that the electron hardly returns to its parent ion, leading to a low intensity of the HHG. When we add a THz field with the peak intensity E_THz = 0.06 a.u. in the y direction, however, the harmonic intensity can be greatly enhanced by about 4–6 orders of magnitude in contrast with the purely circularly polarized laser pulse. The cutoff of the harmonic spectrum is extended to 369 eV. A continuum spectrum with a spectral width of about 225 eV from 144 eV to 369 eV can be achieved.

The emission time of harmonics in terms of the time-frequency analysis method is investigated to further understand the physical mechanism of HHG. Figures 2(a) and 2(b) present the time-frequency distributions of HHG corresponding to the case of the circularly polarized laser pulse and to the case of circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u., respectively. Because the added THz field is along the y direction, the intensity of the y component will be enhanced. Thus, we focus on the y component and explore the corresponding physical mechanism of HHG. Figure 2(a) shows that the emission of photon exhibits an evident periodic property, which will lead to a train of pulses. Besides, the intensity of HHG is very low. Figure 2(b) shows that there are three photon energy peaks, which are labeled as P₁, P₂, and P₃. It has been shown that there are two trajectories for each peak (a trajectory with earlier ionization time but later emission time is called the long trajectory, and a trajectory with later ionization time but earlier emission time is called the short trajectory). The intensities of the harmonics in P₁ and P₂ are comparatively strong, whereas the intensity of the harmonics in P₃ is weaker in contrast with that in P₁ and P₂; thus the contribution of P₃ can be neglected. In addition, for the energy above 144 eV, the peak P₁ almost makes no contribution to HHG; thus only the short trajectory of P₂ is selected, and its corresponding emission time is about 3.4 o.c. (o.c. denotes optical cycle). The harmonics between 144 eV and 369 eV is in the continuum part of the spectrum, which is consistent with the HHG in Fig. 1. In addition, in comparison with the case of circularly polarized laser pulse, the intensity of HHG in the circularly polarized laser pulse combined with the THz field is greatly enhanced.

 

Fig. 1 High-order harmonic spectra of ${\mathrm{H}}_2^{+}$ in a circularly polarized laser field (solid black line) and a circularly polarized laser field combined with a THz field with the peak intensity E_THz = 0.06 a.u. (dotted red line).

Download Full Size | PDF

 

Fig. 2 Time-frequency distributions of HHG (the y-component) corresponding to (a) the solid black line in Fig. 1 (the circularly polarized laser pulse) and (b) the dotted red line in Fig. 1 (the circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u.).

Download Full Size | PDF

We now look at the temporal evolution of the probability density of electron wave packet to further understand the motion of the electron and the physical insight into HHG. The electron wave packet for the case of the circularly polarized laser pulse is shown in Fig. 3(a₁)–(a₁₀). We can see from each figure that the electron spreads out clockwise, whereas as the time evolves, the electron density jet rotates anti-clockwise and moves backward. As the time evolves, most of ionized electrons circularly move away and hardly recombine with the parent ion in the circularly polarized laser pulse. When the THz field with the peak intensity E_THz = 0.06 a.u. is added, the probability density of electron wave packet changes dramatically: As the time goes by, more electrons ionize at t = 2.3 o.c. After t = 3.2 o.c. [Fig. 3(b₅)], large number of electrons are gradually pulled back and recombined with the parent ion; then after t = 4.5 o.c. [Fig. 4(b₁₀)], many electrons are recombined with the parent ion again. A clear picture of the electron’s two-time recombination is presented, which corresponds to P₂ and P₁ of the time-frequency distribution of HHG in Fig. 2(b). The increase of ionization rate and a large number of electrons recombined with the parent ion both helps to increase HHG efficiency. The key point is that the main mechanism of HHG is the electron recombined with the parent ion. Thus, the THz field can provide a proper transverse momentum, which drives a large number of electrons gradually pulled back and recombined with the parent ion. This is the main reason for the enhancement of the HHG. Based on above, the increase of the ionization probability can also enhance the harmonic efficiency, which is the second reason for the enhancement of the HHG.

 

Fig. 3 Comparison of the temporal evolutions of the probability density of electron wave packet for the circularly polarized case [(a₁)–(a₁₀)] and the circularly polarized laser pulse combined with the THz field case [(b₁)–(b₁₀)] with the peak intensity E_THz = 0.06 a.u..

Download Full Size | PDF

 

Fig. 4 Electron trajectories for the case in (a) the circularly polarized laser pulse, (b) the circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u..

Download Full Size | PDF

We also use the classical three-step model, where the electron has non-zero initial transverse velocity, to further explain the mechanism of HHG irradiated by the circularly polarized laser pulse [20, 21]. Figure 4 shows the electron trajectories for the case of the circularly polarized laser pulse and the circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u., respectively. The classical electron trajectories are obtained when the electrons are ionized from 2.4 o.c. to 3.0 o.c. with non-zero initial transverse velocity, because of the transverse velocity distribution of the electron wave packet at the exit of the tunnel in the HHG process. These electrons are recombined with the parent ion around 3.2 o.c., which can be seen in Fig. 3(b₅). As shown in Fig. 4(a) for the case of the circularly polarized laser pulse, the electron can return to the parent ion with proper transverse velocity. Since the distance traveled by the electron is very short, it cannot be accelerated and it will gain less energy. On the contrary, if the electron moves longer distance, the electron can be effectively accelerated and it will gain more energy. Figure 4(b) shows that the distance of the electron motion in the circularly polarized laser pulse combined with the THz field is much farther than in the circularly polarized laser pulse. Thus, the cutoff of harmonic can be extended.

Thus a small quantity of electrons can return to the parent ion with suitable transverse velocity [see Fig. 3(a₁)–(a₁₀)]. The travelling distances of these recombined electrons are so short that little energy can be obtained [Fig. 4(a)]. When the THz field is added, large number of electrons are pulled back and recombined with their parent ion [see Fig. 3(b₁)–(b₁₀)]. These recombined electrons move farther distance and gain more energy [Fig. 4(b)].

Figure 5 shows the ionization probability calculated by $P(t)=1-\exp \left[-{\displaystyle {\int }_{-\infty }^{t}W(t^{\prime })d{t}^{\prime }}\right]$ where the ionization rate W(t′) is determined using the ADK model [24], and the energy as a function of the ionization time and emission time, respectively. We first discuss the case of the circularly polarized laser pulse. Fig. 5(a) shows the emission of photon occurring at each cycle and the interference of different quantum trajectories that results in modulation in the harmonic spectrum, which is consistent with the solid black line in Fig. 1. When we add the THz field in the circularly polarized laser pulse as shown in Fig. 5(b), the ionization probability is enhanced by about one order of magnitude compared to that in the pure circularly polarized case. Besides, there are three main peaks contributing to the HHG, marked as A, B, and C, respectively. The ionization probability of electrons before 2.15 o.c. is very low [see the red dotted line in Fig. 5(b)], which causes even lower harmonic efficiency. Thus the harmonics emission around 2.61 o.c. can be ignored, which is in agreement with Fig. 2(b), i.e., the intensity is weaker in peak P₃ than that in peaks P₁ and P₂. We can see that the electrons ionized from 2.36 o.c. to 3.15 o.c. are returned at about 3.45 o.c. with the energy 369 eV and the electrons ionized from 3.39 o.c. to 4.16 o.c. are returned at 4.48 o.c. with the energy 144 eV. The peak C makes no contribution to HHG for the energy above 144 eV, and the contribution of the harmonics above 144 eV only comes from the short trajectory of B. A supercontinuum between the 144 eV and 369 eV with the bandwidth of 225 eV can be obtained. In summary, by adding the THz field the periodicity of electron’s recombination in the circularly polarized laser pulse field is destroyed and the quantum paths contributing to the harmonics can be controlled.

 

Fig. 5 Ionization probability and the dependence of the energy on the ionization (blue circles) and emission times (green triangles) for the case in (a) the circularly polarized laser pulse, (b) the circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u..

Download Full Size | PDF

Figure 6 shows the temporal profile of an attosecond pulse by superposing several orders of the harmonics (the vertical coordinate is normalized for comparison). For the case of the circularly polarized laser pulse combined with the THz field, the plateau structure is a supercontinuum spectrum with less modulation. Thus, by superposing any range of the harmonic spectrum, an isolated attosecond pulse can be generated. By superposing the harmonics in the range of 216–249 eV, an isolated attosecond pulse with the duration of about 69 as is generated as shown in Fig. 6.

 

Fig. 6 Temporal evolution of the attosecond pulse generated by the circularly polarized laser pulse combined with the THz field with the peak intensity E_THz = 0.06 a.u. in the range of 216–249 eV.

Download Full Size | PDF

4. Conclusions

In conclusion, we have theoretically investigated the HHG and the attosecond pulse generation of ${\mathrm{H}}_2^{+}$ using a circularly polarized laser pulse combined with a THz pulse. When a THz field is added, a supercontinuous harmonic plateau of 225 eV can be formed, which can be explained by the temporal evolution of the probability density of electron wave packet that presents a clear picture of the electron’s two-time recombination. The corresponding time-frequency characteristic of high-order harmonics shows that only the short trajectory has contribution to HHG. We have used the semi-classical three-step model, in which the electron has non-zero initial velocity, to interpret the physical insight of HHG and have shown that the periodicity of electron’s recombination is destroyed and the quantum paths contributing to the harmonics can be controlled by adding a THz field. Finally, an isolated attosecond pulse of about 69 as has been generated by inverse Fourier transformation with a spectral width of 33 eV, ie, from 216 eV to 249 eV.