## Abstract

We present a method to control the harmonic process by a mid-infrared modulated generalized polarization gating for the generation of the broadband supercontinuum. Using a mid-IR generalized polarization gating modulated by a weaker mid-IR linearly polarized chirped field, the ionization, acceleration and recombination steps in the HHG 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. Using this method we expect that isolated sub-100 attosecond pulses with tunable wavelength could be obtained straightforwardly.

© 2010 OSA

## 1. Introduction

It is still a dream to image the dynamics of electrons in atoms and molecules
experimentally. This is due to the fact that such motion takes place in an
ultra-short time scale. The pulses much shorter than 150-as are currently
unavailable to probe such fast dynamics. Therefore, much scientific effort has been
put forward for the generation of isolated single attosecond pulses to probe such
fast dynamics. To generate an isolated attosecond pulse, the emission time of the
harmonics should be confined in one-half cycle of the driving pulse. When an
ultra-short few-cycle driving field is used for HHG, the cutoff of the spectrum may
become a continuum which corresponds to one re-collision at the peak of the driving
field. By selecting the cutoff of the spectrum, an isolated single attosecond pulse
of duration about 250 as [1]
has been generated. Very recently, Goulielmakis *et al.* brought
through the 100-as-barrier [2]. In their experiment, a sub-4fs near-single cycle driving pulse
has been employed to generate a 40-eV supercontinuum and a 80-as pulse has been
obtained. Carrera *et al.* extended high-order harmonic generation
cutoff via coherent control of intense few-cycle chirped laser pulses [3]. However, a much broader continuum is
desired in order to further shorten the pulse in the time domain. This requires
producing a continuum not only at the cutoff but also in the plateau of the harmonic
spectrum.

The three-step model [4]
indicates that the processes of HHG are mainly determined by the ionization,
acceleration, recombination steps of the electrons. It has been proposed that the
two-color field can significantly enlarge the difference between the highest and the
second highest half-cycle photon energies and form a broadband supercontinuum, from
which a broadband isolated attosecond pulse can be generated [5–8]. Recently, Takahashi *et al.* experimentally
generated a continuum high-order harmonic spectrum under conditions that gave low
the ionization probability using an infrared two-color multicycle laser field
[9]. Hong *et
al.* generated broadband extreme ultraviolet supercontinuum harmonics by
using the combination of a few-cycle laser pulse and a low-frequency field
[10]. Zeng *et
al.* generated an ultrabroad extreme ultraviolet supercontinuum spectrum
which directly generates an isolated 65as pulse without phase compensation
[11]. Our previous work
proposed a method to generate a broadband supercontinuum using a few-cycle two-color
laser pulse and a high-order pulse, from which an isolated 24-as pulse can be
directly obtained [12]. 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” [13]. It has been shown that HHG depends on the ellipticity of
the pulse sensitively. If the ellipticity of the driving pulse changes from circular
to linear and back to circular, then the harmonics can only emit in the central
portion of the pulse, and an isolated pulse can be generated with multicycle pulses.
Recently, Chang *et al.* generated a broadband supercontinuum in the
plateau by polarization gating which gates the recombination of the electrons into
one half-cycle [14]. Strelkov
proposed the theory of high-order harmonic generation and attosecond pulse emission
by a low-frequency elliptically polarized laser field [15]. Skantzakis *et al.* also
generated high-energy coherent continuum radiation by the interaction of rare gases
with a high-power many-cycle laser field, utilizing the technique [16]. This technique is based on the
strong dependence of the HHG on the ellipticity of the driving pulse. Mashiko
*et al.* recently combined the ionization gating and the
polarization gating techniques to simultaneously gate the ionization gating and the
recombination steps, which is named double optical gating (DOG), and generated
directly an efficient broadband supercontinuum in the plateau [17]. To further reduce the depletion of
the ground-state population by the leading edge of the driving pulse, Feng
*et al.* proposed a generalized double optical gating (GDOG) for
generating isolated attosecond pulses with a relaxed requirement on laser pulse
duration [18,19]. Recently, Yakovlev et al. theoretically
predicted that conversion efficiency can be improved, and isolated attosecond pulses
can be extracted from plateau harmonics by adopting longer laser wavelengths
[20].

In this work, we propose a method to simultaneously control all the steps (ionization, acceleration, and recombination) in the HHG process for the generation of an ultrabroadband supercontinuum. By using a mid-IR generalized polarization gating modulated by a weaker mid-IR linearly polarized chirped field, an ultra-broadband supercontinuum covering the spectral range from ultraviolet to water window x-ray is successfully produced, from which isolated sub-100-as pulses with tunable central wavelengths are obtained straightforwardly.

## 2. Theoretical methods

In our calculation, the Lewenstein model [21,22] is applied to qualitatively give harmonic spectrum in the combination field. In this model, the instantaneous dipole moment of an atom is described as (in atom units)

*E*(

_{f}*t*) is the electric field of the laser pulse,

*A*(

*t*) is its associated vector potential,

*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,

_{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 The

*g*(

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

*ω*(

*t*″) is ionization rate which is calculated by ADK tunneling model [23]:

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

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

_{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 **a**(*t*):

**a**(

*t*) =

*d̈*(

_{nl}*t*),

*T*and

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

*q*corresponds to the harmonic order.

## 3. Results and discussions

In our scheme, the ellipticity-dependent pulse for the generalized polarization
gating can be generated by combining two 2000-nm 20-fs counter-rotating elliptical
polarized pulses with the ellipticity *ε* = 0.5 with
a delay of 20fs. The intensities of these two pulses are 3.4 ×
10^{14}W/cm^{2} and their carrier-envelop phases are set as
*π*/2. Currently, the intense, ultrafast laser sources in
the midinfrared (1 *μm* < *λ*
< 5*μm*) region have been employed experimentally
[24,25]. A 2000-nm 20-fs chirped laser pulse with
10% intensity of the generalized polarization gating pulses is added to
modulate the generalized polarization gating. The laser field for the generalized
polarization gating pulses can be expressed as the combination of two
perpendicularly polarized fields **E**(*t*) =
*E _{x}*(

*t*)

**i**+

*E*(

_{y}*t*)

**j**. The electric fields polarized along x and y directions can be expressed respectively:

*E*

_{0}is the peak amplitudes of the two counter-rotating elliptical polarized pulses.

*T*is the delay between two counter-rotating elliptical polarized pulses.

_{d}*ω*

_{0}and

*τ*are the frequencies and pulse durations of the ellipticity-dependent pulses and the control chirped pulse.

_{p}*δ*(

*t*) = −

*β*tanh[(

*t*–

*t*

_{0})

*/T*] [3], and the parameters

_{d}*β*and

*t*

_{0}are used to control the chirp form and are set as 6.25,

*T*/7.0.

_{d}*ϕ*is the carrier-envelope phase of the chirped pulse and chosen as 0.2

*π. a*is the ratio of the amplitudes between the control and driving pulses. 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 generalized PG pulse via an optical parametric amplifier, and the weaker one is used for generating the chirped control pulse using state-of-the-art pulse compression.

In order to demonstrate our quantum control scheme, we first investigate the HHG
process in terms of the semiclassical three-step model. In our calculation, we
ignored the recombination between the electron and the parent ion, but considered
the electron velocity in y direction. Figures
1(a) and 1(b) show the electric
field and the electron trajectories in the unmodulated generalized polarization
gating, respectively. The ionization and recombination times are shown in unfilled
blue circles and red crosses in Fig. 1(b),
respectively. The electrons can only return to the parent ion where the ellipticity
is under a very small value [14,26]. Namely, the supercontinuum with
the unmodulated generalized polarization gating is attributed to the recombination
control of the electron trajectories. As shown in Fig. 1(b), the cutoff of the supercontinuum locates around the 380th
harmonic. In this gating, the electron dynamics for the supercontinuum are mainly
dominated by the electric field component along the x direction. By properly adding
a control chirped field along x direction in analogy to the conventional two-color
control scheme, the cutoff of the supercontinuum can be extended, and the ionization
can be enhanced, leading to the harmonic yields enhancement. The results are
presented in Fig. 2(a) and Fig. 2(b). From Fig.
2, one can see clearly that the cutoff of the supercontinuum is extended
to the 815th harmonic, which is higher than that in the unmodulated generalized
polarization gating of the 380th harmonic and the electric field value around
*t* = −0.25*T*_{0}
(*T*_{0} is the optical cycle of the driving pulse)
increases significantly compared with the unmodulated generalized polarization
gating, which leads to ionization and harmonic yields enhancement. Taking into
account all of these results, we can conclude that this control scheme can
simultaneously control the ionization, acceleration and recombination steps. In
other words, the control scheme can enhance the yields and extend the cutoff of the
generated supercontinuum simultaneously.

Following, we perform the calculation to confirm above classical approaches by the
Lewenstein model. Here, the neutral species depletion is considered using the ADK
ionization rate. The harmonic spectrum is shown in Fig. 3. As shown in this figure, the spectrum cutoff is around the 815th
harmonic. The spectrum above the 330th harmonics is continuous. The modulations on
the supercontinuum are due to the interference of the short and long quantum paths.
The trajectory with earlier ionization but later recombination times is called the
long trajectory, and the other one with later ionization but earlier emission times
is called the short trajectory. For comparison, the harmonic spectrum in unmodulated
generalized polarization gating only is also given (dotted red line). It is shown
that the modulated generalized polarization gating significantly broadens the
bandwidth of the supercontinuum and enhances the harmonics yields by 1 order. A
deeper insight is obtained by investigating the emission times of the harmonics in
terms of the time-frequency analysis method [27], which is shown in Fig.
4. The maximum harmonic order of the quantum path is
815*ω*_{0}, and the harmonics above
330*ω*_{0} are mainly dominated by the peak
around *t* = 0.3*T*_{0}. The
interference of the short and long quantum paths of the peak leads to the
modulations on the supercontinuum. These results are consistent with those
calculated by the above classical approaches in Fig.
2. Besides, it can be also seen that the intensity of the short path is
higher than that of the long one from Fig.
4(a), which results in the generation of isolated attosecond pulses with
extremely high signal-to-noise ratio.

Furthermore, we investigate isolated attosecond pulse generation by a square window with the width of 50eV. Figure 5 shows the temporal profiles of the generated attosecond pulses by superposing the harmonics with different central frequencies. As shown in Fig. 5, by adding a frequency window with a bandwidth (50eV) of 80 order harmonics to the supercontinuum, isolated sub-100-as pulses with extremely high signal-to-noise ratio are generated directly without any phase compensation. If a broader filtering window would be used, the pulse duration of the generated attosecond pulses will become large. The isolated attosecond pulses with tunable wavelengths can be used in many fields, such as nanolithography, high resolution tomograph and XUV interferometry.

In this work, the chirp parameter *β* and the phase
*ϕ* of the chirped pulse are extremely important in the
process. We further investigate the sensitivity of the harmonic spectrum to the
variation of chirp parameter *β* and the phase
*ϕ* of the chirped pulse, respectively. Figure 6 presents the sensitivity of the harmonic spectrum
to the variation of the phase *ϕ* of the chirped pulse. As
shown in this figure, the harmonic spectrum is sensitive to the the variation of the
phase *ϕ*. For *ϕ* =
0.0*π*, the cutoff location of the harmonic spectrum
decreases obviously. Therefore, the phase *ϕ* should be
precisely controlled. The sensitivity of the harmonic spectrum to the variation of
chirp parameter *β* of the chirped pulse is shown in Fig.7. It is clear that the result is
insensitive to the variation of chirp parameter *β*.

Our quantum control scheme for the generation of the supercontinuum is also suitable
for longer pulse duration. Figure 8 presents
the harmonic spectra with different pulse durations. The harmonic spectra with the
pulse duration *τ _{p}* = 30fs (dotted red
line) and 40fs (solid blue line) have been shifted up 2.0 and 4.0 units for clarity.
One can see clearly that the bandwidth of the supercontinuum significantly increases
with the increasing pulse durations. And the modulations of the supercontinuum
become obvious for long pulse durations. In our scheme, since the ionization
probability is very low, one can carefully adjust the gas pressure or the position
of the laser focus to fully satisfy the phase matching conditions to macroscopically
select the short quantum path.

## 4. Conclusion

In summary, we propose a method to generate the broadband supercontinuum. Using a mid-IR generalized generalized polarization gating, the recombination step in HHG process can be controlled, leading to the generation of a supercontinuum with the bandwidth of 115eV. By adding a weaker mid-IR linearly polarized chirped field, the ionization, acceleration and recombination steps in the HHG process are simultaneously controlled. Using our proposed method, an ultra-broadband supercontinuum from 205 eV to 505 eV should be obtainable. With such a supercontinuum, the generation of isolated sub-100-as pulses would be straightforward. This quantum control scheme for supercontinuum generation can also be used with input pulses of longer duration.

## Acknowledgments

We thank Professor Peixiang Lu and Dr.Weiyi Hong in Huazhong University of Science and Technology for help. This work was supported by Program for New Century excellent Talents in University, National Natural Science Foundation of China under Grant No. 10775062 and 10875054, and by the Fundamental Research Funds for the Central Universities with Grant No. lzujbky-2010-k08.

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