## Abstract

The phase matching of quantum paths in high-order harmonic generation is investigated in a two-color laser field. Our results show that the phase-matching properties of the short and long trajectories can be modulated effectively by adding a weak second harmonic field. This scheme can not only achieve the quantum path selection, but more importantly it can achieve on-axis phase matching for both of the two trajectories simultaneously. A bright interference pattern induced by the simultaneously phase-matched two trajectories can be observed by placing a near-field on-axis filter, which has great potential for experimentally characterizing the full single-atom dipole moment.

© 2011 OSA

## 1. Introduction

The generation of high order harmonic by interaction of an intense laser field with an atomic or molecular target has been widely studied due to its fascinating applications of coherent light production in the extreme ultraviolet range [1,2] and the generation of attosecond pulses [3–5]. The mechanism of high order harmonic generation (HHG) can be well explained by the three-step model [6]. The electron first tunnels through the barrier formed by the Coulomb potential and the laser field, then it oscillates in the laser field, finally, it may return to the ground state by recombining with the parent ion and emit a harmonic photon with energy up to *I _{p}* + 3.17

*U*, where ${U}_{p}={E}_{0}^{2}/\left(4{\omega}^{2}\right)$ is the ponderomotive potential and

_{p}*I*is the ionization potential. During the process of HHG, each harmonic is mainly associated with two domain electron paths (so-called short and long paths) within one and a half optical cycles.

_{p}The quantum path selection can lead to a regular attosecond pulse train or an isolated attosecond pulse [7–11], which has attracted much attention. On the other hand, the interference induced by the contributions of the two paths is also important, due to the potential for experimental characterization of the full atomic dipole moment and demonstration an unprecedented accuracy of quantum path control on an attosecond time scale. However, it is difficult to experimentally observe interference pattern due to the spatial and temporal averaging that smears out the interference signal. Recently, A. Zaïr *et al.* experimentally observed an interference pattern using a multi-cycle laser pulse [12], and the interference pattern is originated from quantum-path interferences [13]. In their experiments [12–15], the analogous phase-matching degrees of the short and long paths are achieved off axis, and then the signal of the interference pattern is very weak to observe. How to simultaneously achieve good phase matching of the short and long trajectories on axis is still difficult.

In this paper, we propose to effectively control the phase-matching properties of the two trajectories in a two-color laser field. It is shown that good phase matching for both the short and long trajectories can be achieved from the on-axis region to the off-axis region. In addition, the bright interference pattern induced by the simultaneously phase-matched two trajectories can be observed by placing a near-field on-axis filter.

## 2. Theoretical model

The theoretical description of HHG takes into account both the single-atom response (SAR) to the laser pulse and the collective response of macroscopic gas to the laser and high harmonic fields. In our simulation, SAR is calculated with Lewenstein model [16,17] and the nonlinear dipole momentum is [in atomic units (a.u.)]

*E*(

*t*) is the electric field,

*A*(

*t*) is the vector potential.

*ɛ*is a positive regularization constant.

*p*and

_{st}*S*are the stationary momentum and quasiclassical action, which are given by

_{st}*I*is the ionization energy of the helium.

_{p}*d*(

*p*) is the dipole matrix element for transitions from the ground state to the continuum state. For hydrogenlike atoms, it can be written as

*g*(

*t*) in the Eq. (1) represents the ground state amplitude:

*ω*(

*t*′) is the ionization rate, which is calculated by Ammosov-Delone-Krainov (ADK) tunnelling model [18]:

*I*is the ionization potential of the hydrogen atom, and

_{ph}*e*and

*m*are electron charge and mass, respectively.

_{e}To simulate the collective response of macroscopic gas, we solve the light propagation for the laser and high harmonic fields in cylindrical coordinate separately [19,20],

*E*and

_{l}*E*are the laser field and high harmonics, respectively.

_{h}*ω*is the plasma frequency and is given by

_{p}The nonlinear polarization of the gas is *P _{nl}* =

*n*

_{0}

*d*.

_{nl}*n*

_{0}and

*n*are the densities of neutral atoms and free electrons. The electron density can be expressed as

_{e}In our simulation, a near-field filtering along one dimension is adopted [12], and then the integration is performed only in the integration window. The near-field harmonics are “projected” into the far field through a Hankel transformation [21].

## 3. Result and discussion

In this work, the two-color field is synthesized by a 30-fs linearly polarized driving pulse with a wavelength of 800 nm and a 30-fs linearly polarized controlling pulse with a wavelength of 400 nm. The intensity of the controlling field is 4% of the driving field throughout this paper. The electric field of the synthesized laser pulse is expressed by

*E*

_{0}and

*E*

_{1}are the amplitudes of the driving and controlling fields.

*f*(

*t*) and

*ω*

_{0}are the envelope and the frequency. A Gaussian envelope shape is adopted and

*ϕ*

_{0}is set as 0.

In order to clearly understand the physical picture of the quantum path selection in the *ω* + 2*ω* field for the single-atom response, we first investigate the HHG process by the semiclassical three-step model and the ADK model [18]. The intensity of the driving pulse is set as 2 × 10^{14} W/cm^{2}. Figures 1(a) and (c) illustrate the electron trajectories and the tunnel ionization rates of the electrons in the two-color field, the electric field of the synthesized two-color pulse is also shown in Fig. 1(c). In the *ω* + 2*ω* field, it is shown that there are two classes of the trajectories characterized by electron travel times in the continuum of about one-half and about an optical cycle, which are called the short and long trajectories. At the time marked by T, the ionization rate corresponding to the short trajectory is nearly equal to that of the long one. Then the contributions of the two trajectories are comparable. Moreover, in the adjacent half optical cycle, the ionization rates corresponding to the two trajectories are both very low (shown in Fig. 1(c)), then the two trajectories with a period of one optical cycle can contribute to the final high-order harmonic yield. For comparison, the electron trajectories and the tunnel ionization rates of the electrons in the driving field alone are also shown in Figs. 1(b) and (d). In the one-color field, the ionization rates corresponding to the two trajectories at the time marked by T1 are much lower than those in the two-color field, which means that the intensities of the short and long trajectories are both enhanced in the two-color field for the single-atom response [5].

Due to different phase matching conditions of the two trajectories, the propagation of the high harmonics through the gas medium should be considered. The laser pulse is assumed as Gaussian beam propagating in the *z*-direction. In the nonadiabatic three-dimensional propagation simulations, we use a 0.5-mm-long neon gas jet with a density of 1.37 × 10^{18}/cm^{3}, which corresponds to a gas pressure of 40 Torr at room temperature. The focuses of the driving and controlling fields are both placed 2 mm before the entrance of the gas media. Figures 2(a) and (b) show the time-frequency distributions of high-order harmonic generation on axis after propagation in the two-color fields. The beam waists of the driving and controlling fields at the focus are both 25 *μ*m in Fig. 2(a). The selection of the quantum paths can be demonstrated by the time-frequency distribution of HHG, which implies the phase matching of the short and long trajectories. As shown in Fig. 2(a), the short trajectories are selected on axis after propagation. This means that the short trajectories are well phase matched on axis while the long ones are phase mismatched. When the beam waist of the driving field at the focus is increased to 35 *μ*m, the short and long trajectories are both well phase matched on axis (shown in Fig. 2(b)), which is beneficial to experimental observation of the quantum path interference. For comparison, the time-frequency distributions of HHG after propagation in the driving fields alone with different waists are shown in Figs. 2(c) and (d). The short trajectories in the driving field with the waist of 25 *μ*m are selected (shown in Fig. 2(c)). When the waist of the one-color field is 35 *μ*m, the long trajectories are more intense than the short ones (shown in Fig. 2(d)), which implies that the on-axis phase matching of the long trajectories is a little better that of the short ones.

To clarify the influence of the laser parameters on the phase matching of the quantum paths, we perform a phase-matching analysis using the graphical method of Balcou *et al.* [22]. Figure 3 shows the phase-mismatching maps for the different quantum path contributions to the 22nd harmonic in the two-color (Figs. 3(a)–(c)) and one-color fields (Figs. 3(d)–(f)). The coordinate of “propagation distance” in Fig. 3 is the distance between the laser focus and gas medium. The phase-mismatching maps of the long trajectory in two-color field and one-color field are shown in Figs. 3(a) and (d), respectively. The waist of the driving field is 25 *μ*m. The phase matching of the long trajectories in the region marked by A in Fig. 3(a) and the region marked by D in Fig. 3(d) is very poor, and then the long trajectories in two-color field and one-color field can weaken, which are consistent with the results in Figs. 2(a) and (c). When the waist of the one-color field is increased to 35 *μ*m, as shown in Figs. 3(e) and (f), the phase matching of the long trajectories in the region marked by E in Fig. 3(e) are better than those in Figs. 3(d) and (f), which results in the strong long trajectories and the weak short trajectories in the driving field alone (shown in Fig. 2(d)). By adding a weak second controlling field to the driving field, the phase matching of the short trajectories becomes better and that of the long ones can be suppressed slightly (the regions marked by C and B shown in Figs. 3(c) and (b)), therefore the well phase matched short and long trajectories with comparable intensities can be achieved on axis in the two-color field. The results are consistent with those in Fig2. (b). In addition, the intensity of the weak second harmonic field is much lower than that of the driving field, small variations of the relative intensity and the relative phase of the two-color field can not significantly affect on the phase matching of the quantum paths. In the single-atom response, the coexistence of the two trajectories is not significantly influenced by the small changes of the relative intensity (2% to 10%) and the relative phase (≤ ±0.2*π*) [5]. Our scheme is also adopted for the other atomic species.

Furthermore, the interference pattern can be observed when the contributions of the two trajectories are comparable. We introduce the parameter *κ* to measure the ratio between the contributions of the long and short trajectories. The contribution of one trajectory is defined by [23,24]

*I*(

_{XUV}*t*) is the intensity of the attosecond pulse, and

*τ*=

_{N}*T*

_{0}/(2

*N*), and N is the number of harmonics superposed to generate the attosecond pulse,

*T*

_{0}is the optical cycle. Figure 4(a) shows that the near-field spatial distribution of the contribution ratio between the two trajectories as a function of the laser intensity. The parameters except the laser intensity are the same as those in Fig. 2(b). When the ratio

*κ*is not too big or small, i.e., the contributions of the short and long trajectories are comparable, a clear interference pattern can be obtained. This interference will be weak when eliminating one quantum trajectory. As shown in Fig. 4(a), the ratio in the region marked by the black dashed rectangle, where the radial distance is smaller than 15

*μ*m and the laser intensity is below about 3.1 × 10

^{14}W/cm

^{2}, is between 0.5 and 2. We apply a near-field spatial filtering along one dimension[12]: a 1

*μ*m wide integration window centered different radial distance (such as 0.5

*μ*m, 1.5

*μ*m, 2.5

*μ*m and so on), and then the near-field interference fringe maps of 22nd harmonic can be obtained (shown in Fig. 4(b)). The interference fringes in the region where the radial distance is smaller than 15

*μ*m can be observed clearly, and this result is consistent with that in Fig. 4(a).

The calculations in Fig. 4 show that the two trajectories are simultaneously phase matched from the on-axis region to the off-axis region. To obtain bright interference pattern, we use a near-field spatial filter with a large window from −5 *μ*m to 5 *μ*m. The dependence of the 22nd and 23rd harmonics on the laser intensity in the far field without and with the near-field filter is shown in Fig. 5. In the first case (Fig. 5(a)), the total near-field harmonics are “projected” into the far field through a Hankel transformation [21], and then the integration is performed over the total spatial profile in far field. The spatial averaging results in blurring the interference pattern. When the near-field filter is performed, only the harmonics in the window from −5 *μ*m to 5 *μ*m in the near field can be “projected” into the far field. A clear interference pattern is observed with a modulation period of about 0.3 × 10^{14} W/cm^{2}, which is close to the expected 2*π*/Δ*α* interference period of the two trajectories. More importantly, the interference signal is the same magnitude as the signal without the near-field spatial filter, and this result makes experimental observation of interference pattern easy. In addition, if we use a filter with a larger window, the far-field interference pattern becomes brighter while the contrast of the corresponding interference could be depressed. In other words, a near-field filter with a small window of 1 *μ*m is adopted, the contrast of the far-field interference pattern can increase while it becomes weaker, which implies that a far-field interference image obtained by using the same filter in Fig. 4(b) is similar to the those in near field.

## 4. Conclusion

In conclusion, we have investigated the phase matching of the quantum paths in high-order harmonic generation in two-color laser fields. Our results show that the phase-matching properties of the short and long trajectories are modulated effectively by adding a weak harmonic field. In our scheme, we can achieve the quantum path selection, which results in a regular attosecond pulse train or an isolated attosecond pulse. On the other hand, the on-axis phase matching for both of the two trajectories can be achieved simultaneously. The calculations show that the two trajectories are well phase matched from the on-axis region to the off-axis region. In this case, the interference pattern induced by the simultaneously phase-matched two trajectories can be observed by placing a near-field on-axis filter with a window from −5 *μ*m to 5 *μ*m, and the interference signal is the same magnitude as the signal without the near-field spatial filter. Our results are beneficial to experimental characterization of the full atomic dipole moment.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants No. 60925021, 10904045, 10734080, and the National Key Basic Research Special Foundation under Grant No. 2011CB808103.

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