## Abstract

We propose a method to control the harmonic process by using a two-color field in combination with a 400-nm few-cycle control pulse for the generation of an ultra-broadband supercontinuum with high efficiency. The ionization and acceleration steps in the harmonic process can be simultaneously controlled by using a three-color field synthesized by a 2000-nm driving pulse and two weak 800-nm and 400-nm control pulses. Then an intense supercontinuum covered by the spectral range from 140eV to 445eV is produced. The 3D macroscopic propagation is also employed to select the short quantum path of the supercontinuum, then intense isolated sub-100-as pulses with tunable central wavelengths are directly obtained within water window region. In addition, the generation of isolated attosecond pulses in the far field is also investigated. An isolated 52-as pulse can be generated by using a filter centered on axis to select the harmonics in the far field.

© 2013 OSA

## 1. Introduction

The appearance and development of attosecond pulses have caused a breakthrough in metrology, which provides a powerful tool for resolving and controlling electronic dynamical processes occurring in the sub-femtosecond scale [1–3]. The physical process exploited to produce attosecond pulses is high-order harmonic generation (HHG). To date, it has been experimentally demonstrated that isolated and multiple attosecond pulses can be produced by means of high-order harmonic generation in rare gases [4–10]. In particular, the generation of ever-shorter isolated attosecond pulses has continued to attract much interest. Experimentally, the 100-as barrier has been first brought through with a sub-4-fs near-single-cycle driving pulse by Goulielmakis *et. al.* [6]. Very recently, Zhao *et. al.* [10] produced an isolated 67-as pulse by the double optical gating technique, which is the known shortest attosecond pulse at present. However, it is still a huge challenge to generate the isolated attosecond pulse under one atomic unit of time.

The physical mechanism of HHG can be well understood by the three-step model [11]. In the first step, the electron tunnels through the potential barrier formed by the Coulomb potential and the laser field, then gains kinetic energy moving in the laser field, and finally returns to the ground state by recombining with the parent ion. During the recombination process, a photon with energy equaling to the ionization potential plus the kinetic energy is emitted. According to this model, several schemes have been proposed to generate isolated attosecond pulses, such as using a few-cycle laser pulse [12,13], polarization gating [14–19], ionization gating [20–22], and so on. Another effective way is the two-color or multi-color field scheme [23–36], which can effectively extend the cutoff energy and generate the broadband supercontinuum by controlling the acceleration step. However, many applications are still limited by the low intensity of the isolated attosecond pulses at present. Therefore, new schemes are needed for generating intense isolated attosecond pulses. Very recently, we proposed a method to produce a broadband supercontinuum with high efficiency by enhancing the ionization within half optical cycle with a 400-nm few-cycle laser pulse [37]. In this work, we further introduce this scheme to the two-color field. Microscopically, it is found that the three-color field can simultaneously control the ionization and acceleration steps, leading to the efficient generation of a 305-eV supercontinuum covered by the spectral range from ultraviolet to water window x ray. Macroscopically, the 3D propagation is carried out to select the short quantum path of the supercontinuum, then isolated sub-100-as pulses with tunable central wavelengths are directly obtained. In addition, we also investigate the generation of isolated attosecond pulses in the far field.

## 2. Theoretical methods

The simulation is carried out by taking into account both the single-atom response to the laser pulse and the collective response of the macroscopic gas to the laser and high-harmonic field. The single-atom response is calculated with the Lewenstein model [38]. In this model, the instantaneous dipole moment of an atom is described as (in atom units)

*E*(

*t*) is the electric field of the laser pulse,

*A*(

*t*) is its associated vector potential,and

*ε*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 atom, and

_{p}*d*(

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

*g*(

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

*ω*(

*t″*) is the ionization rate which is calculated by ADK tunneling model [39]:

*Z*is the net resulting charge of the atom, and

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

_{ph}*e*and

*m*are electron charge and mass, respectively.

_{e}The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole acceleration *a⃗*(*t*):

*a⃗*(

*t*) =

*d̈*(

_{nl}*t*),

*T*and

*ω*are the duration and frequency of the driving pulse, respectively.

*q*corresponds to the harmonic order.

The collective response of the macroscopic medium is described by the propagation of the laser and the high harmonic field, which can be written separately [40]

*E*and

*E*are the laser and high harmonic field;

_{h}*ω*is the plasma frequency and is given by ${\omega}_{p}=e\sqrt{{n}_{e}\left(\rho ,z,t\right)/{m}_{e}{\epsilon}_{0}}$ and

_{p}*P*= [

_{nl}*n*

_{0}−

*n*(

_{e}*ρ,z,t*)]

*d*(

_{nl}*ρ,z,t*) is the nonlinear polarization generated by the medium.

*n*

_{0}is the gas density and ${n}_{e}={n}_{0}\left[1-\text{exp}\left(-{\int}_{-\infty}^{t}\omega ({t}^{\prime})\right)d{t}^{\prime}\right]$ is the free-electron density in the gas. This propagation model takes into account both the temporal plasma-induced phase modulation and the spatial plasma lensing effects, but does not consider the linear gas dispersion, the depletion of the fundamental beam during the HHG process and absorption of high harmonics, which is due to the low gas density [40]. Then the induced refractive index

*n*can be approximately described by the refractive index in vacuum (

*n*= 1). These equations can be solved with Crank-Nicholson method. The calculation details can be found in [40].

The far-field harmonic emissions can be obtained from the near-field harmonic emissions at the exit face of a gas medium through a Hankel transformation [41–43]

*J*

_{0}is the zero-order Bessel function,

*z′*is the exit position of a gas medium,

*z*is the far-field position from the laser focus,

_{f}*r*is the transverse coordinate in the far field,

_{f}*r*is the transverse coordinate at the exit face of a gas medium, and the wave vector

*k*is given by

*k*=

*ω/c*.

## 3. Results and discussions

In our scheme, a 2000-nm laser pulse is selected as the driving pulse, and the two control laser pulses are a 800-nm pulse and a 400-nm few-cycle pulse, respectively. The synthesized three-color field has the following form:

*E*

_{1},

*E*

_{2}and

*E*

_{3}are the amplitudes of the driving and control electric fields, respectively.

*ω*

_{1},

*ω*

_{2}and

*ω*

_{3}are the frequencies of the driving and control pulses. ${f}_{1}(t)=\text{exp}\left[-2\mathit{ln}(2){t}^{2}/{\tau}_{1}^{2}\right]$, ${f}_{2}(t)=\text{exp}\left[-2\mathit{ln}(2){t}^{2}/{\tau}_{2}^{2}\right]$ and ${f}_{3}(t)=\text{exp}\left[-2\mathit{ln}(2){t}^{2}/{\tau}_{3}^{2}\right]$ present the profiles of the three laser pulses.

*τ*

_{1}=

*τ*

_{2}= 2

*T*

_{1}and

*τ*

_{3}= 2

*T*

_{3}are the pulse durations of the driving and control pulses(full width at half maximum), where

*T*

_{1}and

*T*

_{3}are the optical periods of the driving and 400-nm control pulses.

*τ*

_{delay}_{2}and

*τ*

_{delay}_{3}are the time delays of the 800-nm and 400-nm control pulses.

*ϕ*

_{1},

*ϕ*

_{2}and

*ϕ*

_{3}are the carrier-envelope phases of the driving and control pulses, and are set as 0, 0 and

*π*, respectively. In calculation, we choose

*E*

_{1}= 0.095

*a.u.*,

*E*

_{2}= 0.021

*a.u.*and

*E*

_{3}= 0.06

*a.u.*, and the time delays of the 800-nm and 400-nm control pulses are set as

*τ*

_{delay}_{2}= −0.45

*T*

_{2}and

*τ*

_{delay}_{3}= 0.45

*T*

_{1}, respectively. Experimentally, this scheme can be carried out by using a Ti: sapphire laser system. The 2000-nm driving pulse can be achieved via the optical parametric amplification (OPA) technology [44,45], and the 400-nm control pulse can be realized through the second harmonic generation. The time delays can be adjusted by a piezoelectric translator stage [46].

In order to clarify the roles played by the two control pulses in our scheme, we firstly investigate the HHG process using the classical three-step model [11]. Since the laser intensity is far below the saturation intensity of the target atom (here the helium atom is chosen), thus the HHG process can be well described in terms of the classical electron trajectories and the ADK ionization rate [39]. Figure 1 presents the classical sketch of the HHG process in the 2000-nm driving pulse. It can be seen from Fig. 1(a) that there are three ionization peaks within the pulse duration. Therefore, as shown in Fig. 1(b), there are three quantum paths (marked as P1, P2 and P3) contributing to the harmonic generation, and their maximum energies are 644*ω*_{1}, 535*ω*_{1} and 326*ω*_{1}, respectively. Consequently, the harmonics above 535*ω*_{1} are mainly emitted by the quantum path P1, and form a supercontinuum with the bandwidth of 67eV. In order to further extend the harmonic cutoff and generate a broadband supercontinuum, we can control the acceleration process by adding a 800-nm control pulse. The results are shown in Fig. 2. One can clearly see that the maximum energy of the quantum path P1 is increased to 702*ω*_{1}, and that of the quantum path P2 is decreased to 470*ω*_{1}. Consequently, the harmonics above 470*ω*_{1} are mainly emitted by the quantum path P1, and form a supercontinuum with the bandwidth of 144eV. We also clearly see from Fig. 2(a) that the corresponding ionization rate of the quantum path P1 is enhanced, which leads to the efficient generation of the supercontinuum. Taking into account the above results, we can conclude that the 800-nm control pulse can simultaneously enhance the yields and extend the cutoff of the generated supercontinuum. To further enhance the yields of the supercontinuum, we introduce a 400-nm few-cycle control pulse to enhance the corresponding ionization rate of the quantum path P1. Figure 3 presents the classical sketch of the HHG process in the three-color field. As shown in Fig. 3(a), there is only one ionization peak within the pulse duration. Therefore, there is only one quantum path P1 contributing to the harmonic generation, as shown in Fig. 3(b). Moreover, the corresponding ionization rate of the quantum path P1 is approximately 2 orders of the magnitude higher than that in Fig. 2(a).

Thus we can conclude that an ultra-broadband supercontinuum with high efficiency can be produced in the three-color field scheme. In this scheme, the 800-nm control pulse is mainly used to extend the cutoff of the quantum path P1, and the 400-nm few-cycle control pulse is mainly used to select the quantum path P1 by enhancing the corresponding ionization rate. Therefore, the parameters of the 800-nm control pulse are chosen to maximize the cutoff of the quantum path P1. It is noted that the amplitude and time delay of the 400-nm few-cycle control pulse must be carefully chosen to successfully select the quantum path P1 and generate a broadband supercontinuum. Our calculations reveal that the main conclusions of this paper can be keep for *E*_{3} > 0.03*a.u.* and *τ _{delay}*

_{3}varying from 0.4

*T*

_{1}and 0.5

*T*

_{1}.

To confirm the above sketch, we calculate the harmonic spectra using the Lewenstein model [38]. Here the neutral species depletion is considered using the ADK model. Figure 4(a) shows the harmonic spectrum in the three-color field (solid blue curve). For comparison, the harmonic spectra in the 2000-nm driving field and in the two-color field are also given, respectively. The harmonic spectrum in the 2000-nm driving field has been shifted down 2 units for clarity. As shown in Fig. 4(a), the overall spectral structure is irregular for the harmonics below 554*ω*_{1}, and only the harmonics near the cutoff are continuous in the 2000-nm driving pulse alone. A supercontinuum with bandwidth of 67eV is produced near the cutoff. The harmonic cutoff can be dramatically extended to 717*ω*_{1} and a broadband supercontinuum with the bandwidth of 144eV can be obtained by adding the 800-nm control pulse. Moreover, the yields of the supercontinuum are stronger than those of the harmonics in the driving pulse alone. When the 400-nm control pulse is added, the harmonics above 226*ω*_{1} is supercontinuous and an ultra-broadband supercontinuum with the bandwidth of 305eV is generated. The modulations on the supercontinuum are due to the interference of the short and the long quantum paths. Moreover, the harmonic intensity is 2 or 3 orders of the magnitude higher than that in the two-color field. Therefore, an intense ultra-broadband supercontinuum exceeding 300eV can be produced with the three-color field scheme. In order to further understand the emission times of the harmonics, figure 4(b) shows the time-frequency distribution in the three-color field. It is clear that there is only one quantum path P1 contributing to the generation of the supercontinuum. And the intensity of the short quantum path is comparable with that of the long quantum path, which leads to the modulations on the supercontinuum. These results are in good agreement with the above classical results in Fig. 3.

To generate an isolated attosecond pulse from the modulated supercontinuum, one quantum path must be eliminated. This can be achieved by adjusting the focus position of the laser pulse relative to the gas jet since the short and long quantum paths have different phase-match conditions [47]. In order to achieve the generation of isolated attosecond pulses, we perform the nonadiabatic three-dimensional (3D) propagation simulations [40] for fundamental and harmonic field in the gas target. We consider a tightly focused Gaussian laser beam with a beam waist of 40*μm* and a 1.5-mm long gas jet with a density of 1.0 × 10^{18}/*cm*^{3}. The gas jet is placed 1mm after the laser focus. Other parameters are the same as in Fig. 3. Figure 5 presents the continuous part of the macroscopic harmonics in the three-color field. For comparison, the single-atom result is also presented (thin red curve). One can clearly see that the interference fringes through the plateau to the cutoff are all removed after propagation, which implies that only one quantum path is further selected. Then a macroscopic supercontinuum with the bandwidth of 305eV can be obtained, which covers the spectral range from ultraviolet to water window x ray.

In the following, we investigate the generation of pure isolated attosecond pulses. The results are shown in Fig. 6. By applying a square window with a width of 50eV to the macroscopic supercontinuum at different orders, isolated sub-90-as pulses are directly obtained. This reveals that the isolated attosecond pulses with tunable central wavelengths from ultraviolet to water window x ray can be produced by this scheme. In addition, it is worth mentioning that the supercontinuum with bandwidth of over 300eV can support the generation of attosecond pulse with duration below 14as with proper chirp compensation.

The characteristics of the macroscopic attosecond pulses also include the spatial properties. Next, we further investigate the temporal profile and the spatiotemporal distribution of the attosecond pulses generated by filtering the 300th–380th and 380th–460th harmonics, which are shown in Fig. 7. As shown in this figure, a pure isolated attosecond pulse is directly obtained for the two cases. It can be judged from the emitted time of the isolated attosecond pulses that the short path of the supercontinuum is well phase-matched after propagation. Moreover, the attosecond pulse is always a single pulse at each radius and has some spatial chirps (The attosecond pulse is generated later off-axis than on-axis.) in the large radius region, as shown in Fig. 7(c) and (d).

Finally, we further discuss the generation of attosecond pulses in the far field. In Fig. 8(a), the spatiotemporal distribution of the attosecond pulse generated by filtering the 300th–380th harmonics in the far field is shown. By using a spatial filter (indicated by a solid white curve, with a radius of 100*μm*) to select harmonics near the axis in the far field, an isolated 80-as pulse can be obtained, as shown in Fig. 8(b). To further shorten the duration of the attosecond pulse, one can superpose much more harmonics. In Fig. 9(a), the spatiotemporal distribution of the attosecond pulse generated by filtering the 270th–420th harmonics in the far field is shown. By using a spatial filter (indicated by a solid white curve, with a radius of 100*μm*) to select harmonics near the axis in the far field, an isolated 52-as pulse can be obtained, as shown in Fig. 9(b).

## 4. Conclusion

In summary, we propose a method to simultaneously control the ionization and acceleration processes of HHG for the generation of an intense ultra-broadband supercontinuum. It is found that the 800-nm control pulse mainly plays a role for controlling the acceleration process of HHG, and the 400-nm control pulse plays a role for controlling the ionization process of HHG in this scheme. Then the maximum kinetic energy and the ionization rate of the electrons contributing to the continuous harmonics are both increased, and an intense supercontinuum covered by the spectral range from ultraviolet to water window x ray is obtained. Moreover, the short quantum path is successfully selected after 3D propagation, which enables the generation of intense isolated sub-100-as pulses with tunable central wavelengths within the water window region. In addition, we also investigate the generation of isolated attosecond pulses in the far field. By using a filter with a radius of 100*μm* centered on axis to select the 270th–420th harmonics in the far field, an isolated 52-as pulse can be obtained.

## Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant No. 91026021, 11075068, 10875054, 11175076 and 10975065), the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2010-k08) and Scholarship Award for Excellent Doctoral Student granted by Ministry of Education.

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