1. Introduction

Attosecond science opens the route to study and control physical processes that occur in less than a fraction of visible light [1, 2]. The unprecedented resolution in time that has sparked long-term interest in generation of attosecond pulses. The generation of attosecond pulse trains through the synthesis of a comb of harmonic with an multi-cycle femtosecond laser pulse has been well established[3, 4, 5, 6], however, straightforward attosecond metrology prefers an isolated attosecond pulse. Besides, shorter attosecond pulse in duration is another important goal we are pursuing for studying attosecond-scale phenomena directly. For example, isolated attosecond pulses shorter than the lifetime of neutral pion π ₀ (82.0 as, plus or minus 2.4 as) are needed for observing this pion directly.

Single attosecond pulses have been achieved by selecting cutoff harmonic from a few-cycle driving pulse[7, 8, 9, 10], but the pulse duration is limited to 250 as due to that the band-width of continuous harmonic is less than 20 eV. Continuing efforts have been paid to broaden the bandwidth of attosecond pulse. It has been studied a weak two-color field can be used for breaking down the symmetry of fundamental field in successive half cycles[11, 12]. By controlling electron dynamics or ionization with two-color field in the few-cycle regime, the bandwidth of supercontinuum can be significantly broadened[13, 14, 15, 16]. As an alternative technique, the polarization gating method with approach 2 optical cycles driving pulses has been proposed, and broadband XUV emission has been demonstrated[17, 18]. However, the requirement for the few-cycle driving pulse is rather stringent: a few-cycle pulse with stabilized carrier-envelope phase is required, which is can only be achieved with the state-of-the-art laser system. Therefore, generation of broadband single attosecond pulse with multi-cycle driving pulse has caused a great deal of concern. It is shown theoretically that the difference between the highest and the second highest photo energies of half-cycle cutoff is significantly enlarged, even a weak SH field or subharmonic is added with suitable phase. Lan et al. has obtained an isolated sub-100-as pulse is produced via a 10-fs fundamental field at 800 nm in combination with a weak subharmonic controlling field[15].

Very recently, Chang et al. have proposed a method to combine a polarization gating with a two-color gating named double optical gating (DOG) to generate isolated attosecond pulse using 10-fs driving pulse at 800 nm[19, 20]. The pulse duration is increased by a factor 2 than conventional polarization gating method. In this paper, we propose to generate polarization gating with laser pulses at 1600 nm, and then modulate ionization ratio between the driving field half-cycles via a SH field. Because HHG efficiency is more sensitive to ellipticity and the larger pondermotive energy when 1600-nm driving pulse is adopted, broader supercontinuum can be generated with longer driving pulses. Calculations show that up to 6 optical cycles (32.4 fs) driving pulses can be employed, and the generating supercontinuum is significantly broadened to 280 eV. The Fourier-transform limit of the bandwidth is 10 as, and 100-as XUV pulses with tunable wavelength can be straightforwardly obtained from the spectrum without chirp compensation.

2. Results and discussions

The modulated polarization gating that used in this work is realized by adding a SH field to the laser pulse with time dependent ellipticity. The polarization gating is created by two counter-rotating, circularly polarized Gaussian pulses with a proper delay between them[21, 22, 23]. The peak amplitude E ₀, frequency ω, pulse duration τ_p, and carrier-envelop phase ϕ_CE are the same for the left and right circularly polarized fundamental pulses propagating in the z direction. T_d is the delay between the two pulses. The electric fields of the combined pulses polarized along x and y directions respectively are

The SH field used for modulating ionization rate is described as

where a _2ω is the ratio of the amplitude between the SH field and the driving pulse. The relative phase delay between the two fields is φ _2ω. τ _2ω is duration of the SH field.

The polarization gating method is based on the strong dependence of the HHG efficiency on the ellipticity. If the ellipticity exceeds a certain threshold value ε, the HHG efficiency drops to about half the value obtained for a linearly polarized field. Thus the time interval δt, wherein the ellipticity is less than the threshold value, is considered as the width of polarization gating[17]

The gate width is determined by the pulse duration, the delay between two pulses and the threshold value of ellipticity. Note that, the gate width should be restricted in driving pulse half cycle in order to generate isolated attosecond pulse with pure polarization gating method. It seems that polarization gate can be applied with any pump pulse duration as long as the delay is large enough. Unfortunately, the intensity of linearly polarized portion of driving pulse versus delay increase follows an exponentially attenuation law.

To utilize longer driving pulse, the threshold value ε must be smaller. The HHG efficiency is more sensitive to ellipticity when longer wavelength driving pulse is adopted, thus the threshold value of ellipticity ε is about 0.1 for 1600-nm and 0.2 for 800-nm laser pulses respectively [24, 25]. This feature can be understood by a competition between the transverse drift and the spread of the electron wave packet. On the one hand the classical model proposed by Corkum [26] has shown that the transverse drift of electron increases with the increasing wavelength of driving pulse, on the other hand the larger spread of the electron wave packet is achieved during the longer motion. When the 1600-nm driving pulses are employed, the larger transverse drift of the electron plays a more important role and it can not be compensated with the spread of the electron wavepacket, leading to stronger ellipticity sensitivity. That’s to say HHG efficiency will drop to about half the value obtained for a linearly polarized field with a smaller ellipticity. We have considered the pure polarization gating at 1600 nm, E ₀=0.075 a.u., corresponding to the intensity 2×10¹⁴ W/cm ², τ_p=3T, and T_d=τ_p, where T is the oscillation period of the field. I_g=4×10¹⁴ W/cm ² is the maximum intensity within the gate. Following, harmonic generation from driving pulse with time dependent ellipticity is calculated with a quantum model[24], and the amplitude of ground state is calculated by using the Ammosov-Delone-Krainov (ADK) nonadiabatic rate[27]. Neon is chosen as the target atoms.

 

Fig. 1. Harmonic spectra generated by pure polarization gating at 1600 nm.

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Figure 1 shows the harmonic spectrum, the supercontinuum is presented from 50th to 450th order. Fig. 2 shows the time-frequency analysis of the harmonic response, the harmonic is efficient only within duration of 0.45 optical cycle. It is estimated from Eq. (4) that the gate width δt is about 0.45 optical cycle. The calculated gate width agrees well with the duration of radiation in Fig. 2. The dramatic result show that longer driving pulses at 1600 nm can be used for generating single attosecond, and give evidence of intensifying the ellipticity dependence of high harmonic signal.

In order to further increase the duration of driving pulses, the modulated polarization gating is employed. The duration of 1600-nm fundamental driving pulse is τ_p=6T, other parameters are chosen as Fig. 1 and Fig. 2. The amplitude ratio of the added SH field, a _2ω, is 45%, corresponding to the intensity I ₁=2×10¹³ W/cm ². The intensity ratio between SH field and driving field I ₁/I_g is 5%. ϕ_CE and φ _2ω are -0.35π and -π respectively. The calculated gate width is about 0.9 optical cycle, thus high harmonics radiate two times in principle over the duration of driving pulse. When the SH field is added, the efficient radiation is reduced from two times to one times due to that the ionization ratio between driving pulse half-cycles is modulated.

 

Fig. 2. Time-Frequency analysis of harmonic response in Fig. 1.

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Fig. 3. Harmonic spectra generated by the modulated polarization gating.

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Fig. 4. Time-frequency analysis of the harmonic response in Fig. 3.

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We plot the spectrum of harmonics in Fig. 3. Two plateaus can been seen obviously. The intensity of harmonics in first plateau is about two order of magnitude higher than that in second plateau. Supercontinuums are presented form 50th order to 420th order and 420th order to 460th order in the two plateaus respectively. In Fig. 4, we present a time-frequency analysis of the harmonic response. Only two arms which are restrict in half an optical cycle with comparable intensity are visible clearly. Other arms are much weaker than these two arms. In order to obtain a better comprehension of the phenomena, we make analysis for HHG of our scheme in detail. Firstly, an electron is ejected in the continuum after tunnelling through the atomic barrier formed by the instantaneous laser field. The adding SH field break down the symmetry of atomical barrier, and then ionization in first half optical cycle of the fundamental field is enhanced and the ionization in the successive optical cycle is depressed, then the electron wave packet with a periodicity of 1 optical cycle is generated. Secondly, the electron is accelerated away from the core and driven back to the parent ion. The path that electron passes through is sensitive to the ellipticity of driving pulse. Large ellipticity will lead to missing the parent ion on its return. We adopt 1600-nm driving pulse to intensify the dependence on ellipticity and the calculated width of polarization gating is about 0.9 optical cycle for 6 optical cycles driving pulse. Finally, the electron recombines with it’s parent ion through the release of an energetic photon. The afore-discussed features determine efficient harmonic radiation occurs within one half optical cycle in our scheme. The first plateau [See Fig. 3] is produced by intense two arms with a relative low peak, and the second plateau is generated by weak arms with a relative high peak. The two-arm structures correspond to the short an long trajectories in the HHG. The phase accumulated in short and long trajectories are different, and the regular modulation in Fig. 3 is created by the interference of these two trajectories.

 

Fig. 5. The temporal profiles of the attosecond pulse by selected the harmonics from 350th order to 400th order. The left and right subpulse correspond to the contributions from the short and long trajectories respectively.

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By selecting the XUV spectrum from 350th to 400th order, it is shown in Fig. 5 that a 100-as pulse is generated and the two subpulses of comparable peak intensities in temporal profile attribute to the short and long trajectories. Note that, the attosecond pulses are extracted form first plateau which is more efficiently generated, this mechanism is in favor of increasing intensity of attosecond pulses. To generate a single attosecond pulse, one of the long or short trajectory must be eliminated. It has been demonstrated that the long and short trajectories have much different phase-match conditions[4], and short trajectory can be selected by focusing the laser pulse before the HHG cell which has been implemented and verified experimently[5, 28]. Furthermore, even considering just a single class of trajectories, the chirp is intrinsic to the XUV generation process, because the harmonic emission time varies quasi-linearly with harmonic frequency. The chirp is positive for short trajectory [See Fig. 4]. Though the 280-ev supercontinuum corresponding to a fourier-transform-limited 10-as pulse,without external phase compensation, the useful bandwidth is limited to a few orders. As have been demonstrated in references [18, 29, 30], the positive chirp of the XUV radiation produced by HHG can be compensated for by the negative group delay dispersion (GDD) materials. The two principles for choosing materials are negative dispersion and high transmittance within the selected bandwidth. Thus, the materials mentioned in references [18, 29, 30] are not suitable for the ultrabroad bandwidth. However, with the rapid development of material science, a proper material may be found in near future. After trajectory selection and chirp compensation, a close to fourier-transform limited pulse can be generated by selecting all the supercontinuum with 280 eV bandwidth.

 

Fig. 6. Temporal profiles of the attosecond pulses generated by harmonics with different central wavelengths. The bandwidth of the selected harmonics is about 40 eV.

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Actually, the ultrabroad bandwidth of supercontinuum can be used to generate single attosecond pulse centered at different wavelengths by selecting different range of the spectrum. As shown in Fig. 6, a series of attosecond pulses about 100 as are generated centered from 21 nm to 4.3 nm. The bandwidth of selected harmonics is about 40 eV. Besides, the time separation between the long and short trajectories gives rise to evident existence of chirp to XUV attosecond pulse generation. The isolated attosecond pulses with tunable wavelength can be used in many fields, such as nanolithography, high resolution tomograph, XUV interferometry and so on.

The contour plots of Fig. 7 show population of ground state P_g, as a function of laser intensity and pulse duration. In Figs. 7(a) and (b), the wavelength of driving pulses is 800 nm and 1600 nm respectively, and I ₁/I_g is set to 5% in both plots. Dudovich et al. have compared the birth of attosecond XUV pulses to electron interferometer[31]. The electron wave function of bound state and free electron overlap, their interference produced an oscillating dipole, leading to attosecond pulse. In other words, it will attain a much lower conversion efficiency when the ground state is depleted. The both plots are divided into two regions, with P_g>0.5 and P_g<0.5. Considering the depletion of ground state, we should choose the parameters only in the range defined by the red shaded area (P_g>0.5).

The contour plots of Fig. 8 show ionization ratio A_ion between driving pulse half-cycles in logarithmic scale, as a function of intensity I_g and the intensity ratio I ₁/I_g. Here, we only calculate the ionization ratio between the two one-half cycles of driving pulse which contribute to the harmonic radiation within the polarization gate. The results are shown in two cases: (a) 4 optical cycles driving pulse at 800 nm and (b) 6 optical cycles driving pulse at 1600 nm. When the depletion of ground state and dependence on ellipticity are negligible, the harmonic efficiency is determined mainly by the ionization rate of the electron. In our mechanism, the distinctness in ionization rate between two successive one-half cycles should be at least two order of magnitude in presence of the SH field. As shown in Fig. 8, driving pulse with high intensity need intense SH field to modulate. The both plots are divided into two regions, with A_ion>2 and A_ion<2.

 

Fig. 7. Dependence of the population of ground state on pulse duration and intensity. Intensity ratio of SH field is set to 5%. Fundamental field wavelength is (a)800 nm and (b) 1600 nm.

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Fig. 8. Dependence of ionization ratio A_ion between two adjacent half optical cycle on I_g and I ₁/I_g, where A_ion is expressed in logarithmic scale. Fundamental laser parameters: (a) 4 optical cycles, 800 nm, (b) 6 optical cycles, 1600 nm.

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Based on the above discussions, the maximum intensity of the driving pulses is limited by two factors: one is the deletion of the ground state (Fig. 7), the other is the ionization ratio between the driving pulse half-cycles (Fig. 8). That’s to say the maximum harmonic order is confined for modulated polarization gating both at 800 nm and 1600 nm. It has been studied the cutoff of HHG locates at 3.17U_p+I_p[26], where U_p is the pondermotive energy and I_p is the ionization potential of the atom. U_p is determined with laser field intensity I_g and wavelength λ, since U_p~I_g and U_p~λ ². Longer driving wavelength will generate higher-order harmonics. For example, we set intensity ratio I ₁/I_g to 5%, the maximum I_g of 4 optical cycles 800-nm and 6 optical cycles 1600-nm fundamental field we can chose are 5.5×10¹⁴ W/cm ² and 4.5×10¹⁴ W/cm ², corresponding to 125-eV and 360-eV cutoff respectively, thus 1600- nm fundamental field benefits of generating broadband supercontinuum with moderate laser intensity.

Experimentally, our scheme can be carried out with a 10 mJ, 30 fs at 1 kHz Ti:sapphire laser system. The laser beam can be split into two beams: one is used as the 800-nm modulating field, and another is used for producing 1600-nm laser pulses via an optical parametric amplifier (OPA). The 1600-nm beam is split and delayed with a quartz plate and then recombined with a quarter waveplate, generating the polarization gating. The 800-nm beam is equipped with a delay stage to control the relative delay between the fundamental field and modulating field.

3. Conclusion

In conclusion, we present a method to generate broadband supercontinuum with modulated polarization gating in multi-cycle regime. Ionization modulated by SH field and intense ellipticity dependence on 1600-nm fundamental field both play an important role in our scheme. Consequently, the 1600-nm fundamental field is beneficial for obtaining broader supercontinuum with longer driving pulses, comparing with the 800-nm fundamental field. The driving pulses we used is up to 6 optical cycles (32.4 fs), which is much easier to generate and manipulate than few-cycle pulses. The present bandwidth of supercontinuum is about 280 eV, corresponding to a fourier transform-limited 10-as pulse, and opens up exciting prospects for the generation of intense, tunable in wavelength, isolated 100-as pulses.