## Abstract

We propose a method to control the harmonic process by a mid-IR modulated polarization gating for the effective generation of an ultra-broadband supercontinuum in the neutral rare-gas media. Using a mid-IR polarization gating pulse modulated by a weaker 800-nm linearly polarized pulse, the ionization, acceleration and recombination steps in the harmonic process are simultaneously controlled, leading to the efficient generation of an ultra-broadband supercontinuum covered by the spectral range from ultraviolet to water window x ray under the low ionization rate. The right phase-matching technique is employed to macroscopically select the short quantum path of the supercontinuum, then isolated sub-100-as pulses with tunable wavelengths are directly obtained. This supercontinuum also supports the pulse duration far below one atomic unit of time.

©2010 Optical Society of America

## 1. Introduction

The appearance and development of isolated attosecond pulse pave the way for the study and control of the ultrafast electron dynamics with unprecedented resolutions [1]. Nowadays, isolated attosecond pulses based on the high-order harmonic generation (HHG) have been successfully produced in experiment, and many efforts have been paid to broaden the bandwidth of the supercontinuum and shorten the pulse duration for the potential applications with much higher time resolutions. The 100-as-barrier has recently been first brought through by Goulielmakis *et al*. [2]. In their experiment, a sub-4-fs near-single-cycle driving pulse has been employed to generate a 40-eV supercontinuum and a 80-as pulse has been filtered out. It is still extremely challenging to generate a broadband supercontinuum that can support the attosecond pulse duration under one atomic unit of time, since the duration of the driving pulse can hardly be further compressed.

The three-step model [3] indicates that the time-frequency characteristics of HHG are mainly determined by the ionization, acceleration and recombination of the electrons in the intense laser field. This classical picture suggests that many “knobs” can be introduced to control the electron trajectories for the broadband supercontinuum generation. It has been proposed that the recombination of the electrons can be “gate” into one half-cycle by using the few-cycle polarization gating(PG) technique, then a broadband supercontinuum in the plateau region is produced [4]. This technique is based on the strong dependence of the HHG on the ellipticity of the driving pulse. Another effective way is to adopt a wave-form-controlled two-color field to significantly enlarge the difference between the highest and the second highest half-cycle photon energies, i.e., “gate” the ionization step, and then forms a broadband supercontinuum with several tens eV near the cutoff [5–7]. However, the yields of the supercontinuum are much lower than that in the plateau, due to the intrinsic dynamic properties of the corresponding electron wave packet [8]. Lan *et al*. also found that the two-color field can restrict the ionization peak within one half cycle and also enhance its amplitude, which is dubbed “ionization gating”, then the high-efficiency harmonics in the plateau merge to a broadband supercontinuum [6]. Mashiko *et al*. recently combine the ionization gating and the polarization gating techniques to simultaneously gate the ionization and recombination steps, which is named double optical gating(DOG), and generate directly an efficient broadband supercontinuum in the plateau [9].

Due to the *λ*
^{2} scaling of the ponderomotive energy, a mid-IR driving pulse can significantly extend the harmonic cutoff, providing the possibility to generate much broader supercontinuum with much higher photon energies [10, 11]. Additionally, the ionization rate in the mid-IR driving pulse is quite low, which allows one to adopt the right phase-matching technique [11] to macroscopically enhance harmonic yields from one quantum path. Our previous work has also shown that the two-color controlling scheme in the mid-IR regime is much more effective than that in the near-IR regime, and a 155-eV supercontinuum in the *water window* spectral region [the spectral range between the *K*-absorption edges of carbon (284 eV) and oxygen (543 eV)] has been produced from the neutral (nonionized) media [12]. The DOG technique has also been applied in the mid-IR region to produce broadband supercontinuum with higher efficiency but lower harmonic cutoff than that in the PG pulse [13]. In this paper, we propose a method for the first time, to the best of our knowledge, to simultaneously control all the steps (ionization, acceleration and recombination) in the HHG process for the efficient generation of an ultra broadband supercontinuum. By using a mid-IR PG few-cycle pulse modulated by a weaker near-IR linearly polarized field, a high-efficiency ultra-broadband supercontinuum covered by the spectral range from ultraviolet to water window x ray is successfully produced in the neutral gas.

## 2. Theoretical model

In our calculation, the Lewenstein model [14] is applied to qualitatively gives harmonic spectrum in the two-color field. In this model, the instantaneous dipole moment of an atom is described as(in atomic units)

$$\phantom{\rule{3.2em}{0ex}}\times {d}^{*}\left[{p}_{\mathrm{st}}\left(t\prime ,t\right)-A\left(t\right)\right]d\left[{p}_{\mathrm{st}}\left(t\prime ,t\right)-A\left(t\prime \right)\right]$$

$$\phantom{\rule{3.2em}{0ex}}\times \mathrm{exp}\left[-i{S}_{\mathrm{st}}\left(t\prime ,t\right)\right]{E}_{f}\left(t\prime \right)g\left(t\prime \right)+c.c..$$

In this equation, *E _{f}*(

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

*A*(

*t*) is its associated vector potential,

*ε*is a positive regularization constant,

*w*(

*t*

^{′}) is the ionization rate, which is calculated by the ADK theory [15].

*p*and

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

_{st}where *I _{p}* is the ionization energy of the atom.

*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 the equation(1) represents the ground state amplitude:

*w*(*t*
^{′}) is the ionization rate, which is calculated by Ammosov-Delone-Krainov (ADK) tunnelling model [15]:

where

where Z is the net resulting charge of the atom, *I _{ph}* is the ionization potential of the hydrogen atom, and

*e*and

*m*are electron charge and mass respectively.

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

where *a*⃗(*t*) = *d*̈_{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 media also plays an important role in the harmonic generation. The propagation of the laser and the high harmonic field can be described separately by [16]

where *E _{f}* and

*E*are laser and harmonic fields, ${\omega}_{p}=e\sqrt{4\pi {n}_{e}(\rho ,z,t)/{m}_{e}}$ is the plasma frequency, and

_{h}*P*(

_{nl}*ρ*,

*z*,

*t*) = [

*n*

_{0}-

*n*(

_{e}*ρ*,

*z*,

*t*)]

*d*(

_{nl}*ρ*,

*z*,

*t*) is the nonlinear polarization generated by the media.

*n*

_{0}is the gas density and

*n*(

_{e}*t*) =

*n*

_{0}[1 -

*exp*(- ∫

^{t}

_{-∞}

*w*(

*t*

^{′})

*dt*

^{′})] is the free-electron density in the gas. The equations here take into account both temporal plasma induced phase modulation and the spatial plasma lensing effects on the driving field. They does not consider the linear gas dispersion and the depletion of the fundamental beam during the HHG process, which is due to the low gas density (75 torr at room temperature in our scheme) [16]. Then the induced refractive index n can be approximately described by the refractive index in vacuum (n=1). Equation (9) and (10) can be numerically solved using the Crank-Nicholson method. The calculation details can be found in [16].

## 3. Result and discussion

In our scheme, the mid-IR PG pulse is generated by combining two 2000-nm 20-fs counter-rotating circularly polarized pulse with a delay of 20 fs. The intensities of these two pulses are both 3 × 10^{14}
*W*/*cm*
^{2}, and their carrier-envelop phases are both set to *π*/2. An 800-nm 25-fs pulse with 10% intensity of the PG pulse is added to control the HHG process. The polarization direction of this control pulse is parallel to that of the linearly-polarized portion of the PG pulse. Its relative phase delay to the PG pulse is chosen to be zero. The electric fields of the combined pulse polarized along x and y directions can be expressed respectively:

The control field is described as

Where *ω*
_{0} and *ω*
_{1}, *τ _{p}* and

*τ*

_{2ω}are the frequencies and pulse duration of the mid-infrared driving pulse and the control pulse, respectively.

*a*

_{ω1}is the ratio of the amplitude between the control and driving pulses.

In order to clearly demonstrate the physical picture of this quantum control scheme, we first investigate the HHG process using the classical three-step model. Since the laser intensity is far below the saturation intensity of the target atom (here the helium atom is chosen), then the HHG process can be well depicted in terms of the classical electron trajectories and the ADK ionization rate [15]. Firstly, we show the classical picture of the harmonic process in a PG pulse alone, which is depicted in Fig. 1(a) and 1(b). Figure 1(a) are the electron trajectories in the PG pulse. The ionization and recombination times are show in blue dots and red circles, respectively. Figure 1(b) are the ionization rate and the electric field components parallel (green solid line) and perpendicular(blue dashed line) to the linearly-polarized direction of the PG pulse. The supercontinuum with the PG pulse is attributed to the recombination control of the electron trajectories: the electrons can only return to the parent ion where the ellipticity is under a very small value [4, 13]. In this recombination gating, the electron dynamics for the supercontinuum are mainly dominated by electric field component along the linearly polarized direction. In analogy to the conventional two-color control scheme [6], a weaker control field can be introduced to control the acceleration step for spectrum extension or the ionization step for harmonic yields enhancement. The ionization control in a PG pulse, i.e., double optical gating(DOG), has also been investigated by Chang et al. [4] and Zhang et al. [13]. The DOG can produce more efficient supercontinuum than the conventional PG but with shorter harmonic cutoff. The acceleration control in a PG pulse has never been reported. Here we would show that these two motivations can be both achieved in such a modulated gating, implying that three steps for HHG are simultaneously controlled. The sketch is shown in Fig. 1(c) and 1(d). With such modulated PG pulse, the acceleration of the electrons is controlled and the harmonic cutoff is extended to 450 eV, which is higher than that in the unmodulated PG pulse of 380 eV shown in Fig. 1(a). We also clearly see from Fig. 1(d) that the corresponding ionization rate for the supercontinuum is enhanced. Taking into account the above results, we can conclude that this control scheme can simultaneously the yields and the cutoff of the generated supercontinuum. we further calculate the harmonic spectrum using Lewenstein model [14] to confirm the classical sketch above. Here the neutral species depletion is considered using the ADK ionization rate. The harmonic spectrum is shown in Fig. 2(a) (blue thick curve). The inset is the detail structure in the lower orders region. A perfect ultra-broadband supercontinuum through the plateau with the cutoff of 455 eV, which is covered by the range from ultraviolet to water window x ray, is successfully generated. Compared to the supercontinuum with a unmodulated PG pulse (grey dashed curve), the harmonic yields with the modulated PG pulse are 2–3 order higher, and the cutoff is also extended by 70 eV. A deeper insight is obtained by investigating the emission times of the harmonics in terms of the time-frequency analysis method, which is shown in Fig. 2(b). One can clearly see that two bright branches corresponding to the short and long quantum paths. Their interference leads to the modulations in the supercontinuum.

The mid-IR pulse can produce harmonics in the nearly nonionized media. In our scheme, the calculated ionization probability is below 0.1%, then such neutral media enable one to adopt the right phase-matching technique to macroscopically enhance the harmonic yields of a single quantum path in the supercontinuum. In this technique, the phase-matching condition for either quantum path (short or long) can be fully satisfied by adjusting the gas pressure and the position of the laser focus. To demonstrate this issue, we perform the nonadiabatic three-dimensional (3D) propagation simulations [4] for both fundamental and harmonic fields in the gas target. In order to realized the phase-matching condition of a single path of the supercontinuum through the plateau, we consider a tightly focused laser beam with a beam waist of 30 *μm* and a 0.3-mm long gas jet with a density of 2.6×10^{18}/*cm*
^{3}. The gas jet is placed 1 mm after the laser focus. Figure 3 shows spectrum of the macroscopic harmonics in the modulated PG pulse (red curve). For comparison, the single-atom result is also presented (blue line). One can clearly see that the interference fringes through the plateau are all removed after propagation, which implies that the continuous harmonics from a single path are perfectly phase-matched.

This perfectly smoothed ultra-broadband supercontinuum enables the generation of pure isolated sub-100-as pulse with tunable central wavelengths from ultraviolet to water window x ray. The result is shown in Fig. 4. By applying a square window with a width of 50 eV to the super-continuum at different central frequencies, isolated 80-as pulses with extremely high signal-to-noise ratio are directly obtained. This supercontinuum with the bandwidth of over 400 eV can support the attosecond pulse duration below 10 as with proper chirp compensation.

## 4. Conclusion

In conclusion, we propose a scheme to simultaneously control the ionization, acceleration and recombination of HHG in mid-IR regime for effective generation of a ultra-broadband super-continuum. Using a mid-IR PG pulse modulated by a weaker 800-nm linearly polarized pulse, the maximum kinetic energy and the ionization rate of electrons ionized within the polarization gating, which attribute to the supercontinuum generation, are both increased. This dramatic control of the HHG process enables the effective generation of a supercontinuum covered by the spectral range from ultraviolet to water window x ray is observed. Additionally, the low ionization rate in this scheme enables one to adopt the right phase-matching technique to macroscopically select a single quantum path and produce isolated sub-100-as pulse with tunable central wavelengths. Experimentally, this control scheme can be carried out with a Ti: sapphire laser system. The laser beam is split into a stronger beam and a much weaker one. The stronger beam is used to produce the 2000-nm mid-infrared PG pulse via an optical parametric amplifier, and the weaker one is used as the control pulse.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants No. 60925021, 10904045 and the National Basic Research Program of China under Grant No. 2006CB806006.

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