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.
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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  has been generated. Very recently, Goulielmakis et al. brought through the 100-as-barrier . 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 . 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  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 . Hong et al. generated broadband extreme ultraviolet supercontinuum harmonics by using the combination of a few-cycle laser pulse and a low-frequency field . Zeng et al. generated an ultrabroad extreme ultraviolet supercontinuum spectrum which directly generates an isolated 65as pulse without phase compensation . 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 . 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” . 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 . Strelkov proposed the theory of high-order harmonic generation and attosecond pulse emission by a low-frequency elliptically polarized laser field . 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 . 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 . 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 .
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)23]:
The harmonic spectrum is then obtained by Fourier transforming the time-dependent dipole acceleration a(t):
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 × 1014W/cm2 and their carrier-envelop phases are set as π/2. Currently, the intense, ultrafast laser sources in the midinfrared (1 μm < λ < 5μm) 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) = Ex(t)i + Ey(t)j. The electric fields polarized along x and y directions can be expressed respectively:3], and the parameters β and t0 are used to control the chirp form and are set as 6.25, Td/7.0. ϕ is the carrier-envelope phase of the chirped pulse and chosen as 0.2π. 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 = −0.25T0 (T0 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 , which is shown in Fig. 4. The maximum harmonic order of the quantum path is 815ω0, and the harmonics above 330ω0 are mainly dominated by the peak around t = 0.3T0. 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 = 0.0π, 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 τp = 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.
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.
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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