Abstract

We propose a simple and efficient method to optimize two-color chirped laser pulses by forming a “temporal gate” for the generation of isolated attosecond pulses (IAPs) in soft X rays. We show that the generation process for higher and cutoff harmonics can be effectively limited within the temporal gate, and the harmonic emission interval can be further reduced with the help of phase-matching by only selecting the contribution from short-trajectory electrons. This two-color gating mechanism is verified by increasing the pulse duration, raising the gas pressure, and extending the target cutoff. Compared to the five-color waveform in Phys. Rev. Lett. 102, 063003 (2009) [CrossRef]  , our waveform can be used to generate the IAP in the long-duration laser pulse while the cutoff energy is higher without the reduction of harmonic yields. Our work provides an alternative temporal gating scheme for the generation of IAPs by simultaneously improving the harmonic conversion efficiency, thus making the attosecond soft X rays an intense and highly time-resolved tabletop light source for future applications.

© 2019 Chinese Laser Press

1. INTRODUCTION

The high harmonic generation (HHG) by the interaction of an ultrafast strong laser with a gaseous medium, resulting in light with wavelength covering the extreme ultraviolet (XUV) to X rays [1], has been used as a tabletop light source and has initiated attosecond science [25]. Compared to the bulky, costly, and inconvenient larger light sources, e.g., synchrotron radiation accelerators and free electron lasers, the temporal characteristics of HHG have made it a unique tool for studying time-resolved spectroscopy and for probing electron dynamics. One of the most important applications of HHG is to provide isolated attosecond pulses (IAPs), often with the idea of limiting the efficient generation process only in a fraction of the driving laser period, i.e., the temporal gating.

Since the first generation of IAP in 2001 [6], several different methods have been developed to produce the IAP in the XUV based on the traditional 800-nm Ti:sapphire lasers, including the polarization gating for an isolated 130-as pulse [7,8], the amplitude gating [9] for an 80-as pulse with carrier-envelope phase (CEP)-stabilized few-cycle pulses, and the double optical gating (DOG) for a 67-as IAP [10]. Until recently, the soft X-ray IAPs have emerged with the development of laser technology in the mid-infrared [11,12]. With a 2-μm and multicycle laser, Chen et al. [13] showed the IAP at photon energies up to 180 eV via the transient phase-matching gating [14]. By rotating the wavefront of the driving laser, the continuous harmonic spectra up to the carbon K-edge of 284 eV were presented by Silva et al. [15], indicating the existence of a pulse as short as 400 as. Later on, the same group demonstrated the generation of 0.5-keV supercontinuum spectra to support a 13-as IAP [16]. In 2017, the new record of the shortest pulse with the duration of 53 as in the soft X rays was reported by Li et al. [17] through the mid-infrared polarization gating technique. In the same year, a 43-as pulse was also claimed with a passively CEP-stable mid-infrared laser by Gaumnitz et al. [18]. Very recently, Johnson et al. [19] applied a 1.8-μm laser in the overdriven regime to rapidly modify the beam’s spatiotemporal shape, and high harmonics with photon energies up to 600 eV were generated to support soft X-ray IAPs. However, all the above generation methods have their own limitations. As soft X-ray pulses are highly in demand for attosecond experiments [20], two serious issues need to be resolved. One is how to improve the emission efficiency, since the HHG yields from each atom scale unfavorably with laser wavelength λ, typically as λ4λ6 [2123]. The other is how to develop efficient gating schemes for the IAP, which can be applied in the soft X rays.

In recent years, advances in laser technology have made it possible to generate any optical waveforms by accurately controlling and adjusting the synthesis of two-color or even multicolor laser pulses [2429]. They have been used to control the conversion efficiency and spectral region of HHG by optimizing subcycle laser waveforms (see review in Ref. [30]). For instance, in 2009, Chipperfield et al. [31] presented the so-called “ideal waveform” to dramatically increase the harmonic cutoff energy, which can be roughly accomplished through the synthesis of five-color laser pulses. With only 30% of pulse energy from the longest wavelength component, the five-color waveform enables the harmonic cutoff energy in the XUV with only an 800-nm laser to be efficiently increased into the soft X rays by a factor of 2.5, while the harmonic yields are maintained. In 2014, Jin et al. [32] proposed an optimization scheme to construct the multicolor laser waveform in the single-atom level by considering the macroscopic effects of HHG propagation in the nonlinear medium. They were able to show the enhancement of soft X-ray harmonic yields by 1 to 2 orders of magnitude compared to the single-color field. Therefore, the optimization of multicolor laser pulses has been established as a meaningful way to boost the HHG conversion efficiency in soft X rays. However, it is not straightforward to employ this approach for the generation of IAP [33,34]. So it is very necessary to develop some innovative optimization methods.

Our main goal in this work is to establish a novel scheme to optimize the multicolor laser waveform for the generation of intense soft X-ray IAPs. We suggest that the waveform consisting of two-color chirped laser pulses is optimized to form a “temporal gate” only for selected higher harmonics, which in turn creates an IAP in time. Note that our waveform with only two color components has better performance than the five-color one by Chipperfield et al. [31], i.e., it can produce even longer cutoff without reducing the harmonic yields and generate the IAP in the long pulse. The paper is arranged as follows. In Section 2, we will present the waveform optimization method, HHG propagation equations, and the formulation for the generation of attosecond pulses; in Section 3, we will show the optimized waveforms, and the resulting harmonic spectra and IAPs; the conclusion of this paper is given in Section 4.

2. THEORETICAL METHODS

A. Optimization of the Two-Color Chirped Laser Waveform

In the optimization, we consider the synthesized waveform consisting of two-color chirped laser pulses in the form of

E(t)=E1A1(t)cos(ω1t+α1t2+ϕ1)+E2A2(t)cos(ω2t+α2t2+ϕ2).
Here Ei, ωi, Ai, αi, and ϕi (i=1,2) are the respective amplitude, angular frequency, temporal envelope, linear chirp rate, and phase of each pulse. Due to the addition of chirp, the synthesized waveform does not have periodicity anymore. The optimization is thus executed over the entire temporal pulse instead of one optical cycle of the fundamental pulse, as in the previous studies [32,35,36]. Note that the effects of chirp on the HHG in single- or two-color laser pulses have been substantially discussed [3747]; however, systematically optimizing the chirp for the generation of attosecond pulses has not been achieved.

In order to simplify the optimization procedure, some parameters in Eq. (1) can be fixed as in the following: (i) the first laser’s wavelength is λ1=800nm, considered to be the fundamental, and ϕ1=0; (ii) both colors have the Gaussian envelope with the full width at half-maximum (FWHM) duration of 8 fs (three optical cycles of the fundamental); (iii) the total pulse energy of two colors is fixed, i.e., |E1|2+|E2|2 is a constant. We search parameters {E2,ω2,α1,α2,ϕ2} to maximize the single-atom HHG yields at given cutoff energies.

We use the genetic algorithm (GA) [48] to carry out the optimization, and tens of thousands of iterations are taken to ensure that the converged results are obtained. At each iteration, the quantitative rescattering (QRS) model [49,50] is applied to compute the single-atom HHG. In the QRS, the induced dipole moment of an atomic target is expressed as

D(ω)=W(ω)d(ω),
where d(ω) is the complex photo-recombination transition dipole matrix element, which can be accurately calculated through solving the time-independent Schrödinger equation (TISE) by including the Coulomb potential effect, and W(ω) is the complex electron returning wave packet and can be calculated by the strong field approximation (SFA) [51].

The fitness function in the optimization is written as

F=ωminωmaxω4|D(ω)|2dω.
Here ωmin=ωcutoff10ω0, and ωmax=ωcutoff+10ω0, where ω0 is the angular frequency of the fundamental, and ωcutoff is the preset cutoff frequency (or, equivalently, photon energy). Since the electron ionization occurs in a very short time interval, the enhancement of harmonics in a small specified energy range would automatically enhance a broad range of harmonics. The ionization probability at the end of the laser pulse is set as a constraint, which is less than 2%–5%, to avoid the defocusing of laser by the electron plasma in the gas medium.

B. Macroscopic Propagation in the Nonlinear Medium

To calculate the soft X-ray high harmonics emitted in a gaseous medium, both the single-atom-induced dipole and the macroscopic response are needed. The former is calculated by using the QRS model, and the latter can be accounted for by solving the propagation equations of driving laser and high-harmonic field in the nonlinear medium. Then the near-field harmonics are obtained at the exit face of the gas medium. The details of these propagation equations have been given in Refs. [5256]. Here we only outline the key equations.

The propagation equation of a driving laser in dense and ionized media is affected by the refraction, nonlinear self-focusing, ionization, and plasma defocusing. In the atomic gas medium, its evolution is given by the three-dimensional (3D) Maxwell’s wave equation [56],

2E1(r,z,t)1c22E1(r,z,t)t2=μ0Jabs(r,z,t)t+ω02c2(1ηeff2)E1(r,z,t),
where E1(r,z,t) is the transverse electric field of the driving laser, ηeff is the effective refractive index, and Jabs is the absorption term due to the ionization.

The 3D propagation equation of the high-harmonic field is given by [56]

2Eh(r,z,t)1c22Eh(r,z,t)t2=μ02P(r,z,t)t2,
where P(r,z,t) is the nonlinear polarization caused by the applied optical field E1(r,z,t).

C. Generation of Attosecond Pulses

To trace the harmonic emission in time, we use time-frequency analysis, which is carried out by using the wavelet transform [57,58],

A(t,ω)=Eh(τ)wt,ω(τ)dτ,
with the wavelet kernel wt,ω(τ)=ωW[ω(τt)]. We use the Morlet wavelet as [57]
W(x)=(1/ν)eixex2/2ν2.
ν is a constant, chosen to be 15.

To avoid the complexity of the spatial distribution of high harmonics in the near field, we calculate A(t,ω) at each radial point and then integrate over the radial coordinate as [58,59]

|Anear(t,ω)|2=02πrdr|Eh(r,τ)wt,ω(τ)dτ|2.
For the generation of attosecond pulses, a spectral filter is used to select a range of harmonics (ω1ω2). The total intensity of attosecond pulses in the near field is obtained from [59]
Inear(t)=02πrdr|ω1ω2Eh(r,ω)eiωtdω|2.

3. RESULTS AND DISCUSSION

A. Optimization of the Two-Color Chirped Waveform

In the optimization, Ne is the target atom. In the five-color waveform by Chipperfield et al. [31], 60% and 30% of pulse energy are from 800-nm and longer 1600-nm components, respectively. Note that the longer wavelength is always required for the extension of harmonic cutoff energy. To compare with this waveform, we use the same total pulse energy, i.e., |E1|2+|E2|2=3.0×1014W/cm2, and the fraction of pulse energy from longer wavelength component is limited to less than 30%, while the powerful 800-nm laser is set as the main one. This combination is also reasonable in reality because most wavelength-tunable, CEP-stabilized mid-infrared lasers are pumped by 800-nm Ti:sapphire lasers; thus, the pulse energies of the mid-infrared are usually smaller than those of the pump lasers [6062]. Note that the duration of the two-color waveform is assumed as short as 8 fs to save computation time. We have checked that the optimized waveform does not change much with the increase of pulse duration.

With the goal of optimizing cutoff harmonics around 250 eV, the tentative optimization returns the second wavelength of around 1200 nm and the intensity |E2|2 of 0.9×1014W/cm2 if they are treated as free parameters as well. To further precisely optimize other parameters, we fix these two and search {α1,α2,ϕ2} in Eq. (1). The optimized parameters for the two-color waveform are given in Table 1, and the results are quite stable for different runs.

Tables Icon

Table 1. Optimized Laser Parameters for the Two-Color Waveform at Different Runsa

We first plot the five-color waveform by Chipperfield et al. [31] in Fig. 1(a), and our two-color chirped waveform is shown in Fig. 1(b). Comparing the two waveforms, the most active parts are very similar (as indicated by the green shadow areas), and both have a steep linear slope. We compare the classical excursion distances of return electrons (blue dashed lines), which can produce the cutoff harmonics when they are recombined into atomic ions. For the two-color chirped waveform, the ionized electron has a longer excursion time, and its maximum displacement is larger; thus, it can gain more kinetic energy from the driving laser pulse; additionally, the electric field at the ionization moment is a little bit higher. Note that the waveform in Fig. 1(b) is obtained by performing the optimization for an 8-fs long two-color synthesized pulse.

 figure: Fig. 1.

Fig. 1. Laser waveforms (red lines) of (a) five-color field in Ref. [31] and (b) optimized two-color chirped field. In (a) and (b), the excursion distances of the classical electrons are shown (blue dashed lines), leading to the cutoff harmonics (maximum electron kinetic energies indicated), and the green shadow marks the effective waveform. (c)–(e) The waveforms of two-color chirped fields are extended for longer laser durations of 16, 24, and 32 fs. The green regions show the temporal gate for the generation of continuous high harmonics, leading to IAPs. (o.c. means the optical cycle of an 800-nm laser.)

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We then directly apply the optimized parameters in Table 1 to longer durations in Eq. (1). The resulting waveforms for 16, 24, and 32 fs are plotted in Figs. 1(c)1(e), respectively. We have checked that the ionization level at the end of the laser pulse is less than 5% according to the Ammosov–Delone–Krainov (ADK) model [63,64]. As indicated by the green shadow area, the most effective waveform does not change, and the peak field increases slightly at the longer durations. This clearly shows that a temporal gate is formed in the central part of the laser pulse, which contains the peak field and occurs just once in the whole laser pulse. Only within this time interval are the electrons mainly contributing to the higher and cutoff harmonics ionized, propagated in the field, and recombined to the atomic ions, also indicating the possible generation of IAPs in time. Note that the formation of the temporal gate is caused by the coherence of two chirped laser pulses, which can only be achieved with the proper combination of chirp rates, phases, and amplitudes. On the contrary, the five-color waveform has periodicity, which is not expected to be a good candidate for the generation of IAP by using a longer pulse. We next will show how this temporal gate can enhance the harmonic yield, increase the cutoff energy, and generate the IAP.

B. Temporal Gate for the Generation of Continuous HHG and IAP

In Fig. 2(a), we show the single-atom harmonic spectra of an Ne atom with the two-color optimized waveform [shown in Fig. 1(c)], five-color waveform, and single 800-nm laser, calculated by using the QRS model. The total pulse energy is fixed, and the pulse duration is chosen as 16 fs. Compared to the five-color waveform, our two-color waveform can extend the cutoff energy by about 80 eV without reducing the harmonic yields. Both waveforms are much better than the single-color field in terms of the harmonic cutoff energy. The macroscopic harmonic spectra with three laser fields are shown in Fig. 2(b). Single-atom waveforms are only formed in the center of a gas jet after adjusting macroscopic laser parameters [31,32]. After the propagation in the nonlinear medium, the two-color waveform is still better than the five-color one, which can increase the cutoff energy by about 30 eV, maintain harmonic yields, and produce continuous harmonics. Because of the existence of long wavelength components in the two waveforms, the harmonic yields above 100 eV drop compared to those below 100 eV with the 800-nm laser; however, they are still much stronger compared to the single long-wavelength laser (not shown; see Fig. 2 in Ref. [31]).

 figure: Fig. 2.

Fig. 2. Comparison of (a) single-atom and (b) macroscopic high-harmonic spectra by using single-color laser (800 nm), five-color field (from Ref. [31]), and two-color chirped field [shown in Fig. 1(c)]. Laser duration is fixed at 16 fs for all driving fields. For the macroscopic calculations, a 1-mm-long gas jet is located 0.5 mm after laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 50 μm for all colors. For three cases, the single-atom waveforms are only formed in the center of the gas jet, and the harmonic spectra in (b) are thus normalized according to the individual input pulse energy for easy comparison. Time-frequency pictures of harmonic emission and the temporal profiles of attosecond pulses with (c)–(e) two-color chirped and (f)–(h) five-color fields.

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Moreover, the time-frequency analysis of the HHG, calculated by using Eqs. (6) and (8), can assist us in seeing more information in the harmonic emission. As shown in Fig. 2(c), only one main harmonic burst with both short- and long-trajectory electron emissions occurs with the two-color waveform, compared to the two bursts with the five-color waveform in Fig. 2(f). Due to the efficient phase matching, just short-trajectory electron emissions are left in each burst in Figs. 2(d) and 2(g), further shortening the emission time interval. The faint harmonic signals between 2 and 3 optical cycles originate from the electrons ionized around 0.6 optical cycles, and are difficult to phase-match in the gas medium because of the long electron excursion time. Therefore, Fig. 2(d) obviously shows the temporal gate formed by the two-color chirped waveform in the single-atom level works for generating a single and clean harmonic burst, thus resulting in continuous harmonics with the assistance of macroscopic phase-matching effects.

We then synthesize the continuous harmonics from 130 to 219 eV in Fig. 2(b) by using Eq. (9), and the resulting pulse is shown in Fig. 2(e), which appears to be an IAP with the duration of 192 as. For comparison, the five-color harmonics from 150 to 195 eV in Fig. 2(b) are synthesized to give an attosecond pulse train (APT) in Fig. 2(h). Here we demonstrate the generation of IAP in the soft X rays when the duration of the two-color driving laser is 16 fs. Can the physical mechanism of temporal gate be valid for longer durations?

To answer this question, we use the two-color waveforms with durations of 24 and 32 fs, as shown in Figs. 1(d) and 1(e), to do similar simulations and analysis. The single-atom HHG spectra in Fig. 3(a) show that the cutoff energies do not change, and the macroscopic harmonic spectra above 130 eV are continuous in Fig. 3(b) even though the pulse durations are increased. Time-frequency pictures of harmonic emissions indicate how the single-harmonic bursts are built up in Figs. 3(d) and 3(g). The synthesis of high harmonics in the same spectral region (130–219 eV), as in the case of 16 fs, shows the presence of 240- and 200-as IAP in Figs. 3(e) and 3(h). These results show that the temporal gate indeed takes into effect in the generation of IAP when the pulse duration is increased.

 figure: Fig. 3.

Fig. 3. Harmonic spectra, time-frequency analysis of harmonic emission, and temporal profiles of attosecond pulses, similar to Fig. 2. However, the driving lasers are replaced by the two-color chirped fields with durations of 24 and 32 fs [as plotted in Figs. 1(d) and 1(e)]. (c)–(e) 24 fs and (f)–(h) 32 fs. In (e) and (h), the IAPs with durations of 240 and 200 as are obtained by synthesized high-harmonics in the same photon energy region of 130–219 eV.

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C. Effect of Gas Pressure on the Two-Color Chirped Waveform

In the above calculations, the pressure of the gas medium is low and is kept at 10 Torr. Accordingly, the effects of dispersion, Kerr nonlinearity, and plasma defocusing on the driving laser can almost be ignored. Will the two-color gating mechanism still hold if the pressure is increased and the nonlinear propagation effects become significant? We thus choose the higher pressures of 50 and 100 Torr, and simulate the macroscopic HHG spectra, as shown in Fig. 4(a). With the increase of gas pressure, the harmonic cutoff energy is decreased because of laser defocusing and phase mismatch caused by the excessive free electrons, while the plateau yields are increased due to the increase of the number of emitters. In time, if the photon energy of HHG is larger than 120 eV, only one emission burst comes into being, as shown in the time-frequency pictures in Figs. 4(b) and 4(d). Then the useful spectral regions of high harmonics are selected differently to produce attosecond pulses. In Fig. 4(c), at pressure of 50 Torr, a 200-as IAP is presented by synthesizing the harmonics from 119 to 175 eV, and an IAP with the same duration is also obtained by using the harmonics in the range of 105–155 eV, as shown in Fig. 4(e) at 100 Torr. This illustrates that the temporal gate in the two-color chirped waveform plays a crucial role for the IAP generation when the gas pressure is higher with the severe propagation effects. Note that the effect of gas pressure on the control of the soft X-ray high-harmonic spectrum under the two-color laser scheme has been investigated recently [65].

 figure: Fig. 4.

Fig. 4. (a) Macroscopic harmonic spectra with the two-color chirped field at different gas pressures of 10, 50, and 100 Torr. The pulse duration is 16 fs, and other macroscopic conditions are the same as those in Fig. 2. Time-frequency wavelet analysis of harmonic emission and the generation of IAPs at (b), (c) 50 Torr and (d), (e) 100 Torr. In (c) and (e), 200-as single attosecond pulses are generated by synthesized high-harmonics in slightly different photon-energy regions.

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D. Extension of the Temporal Gate into Higher Photon Energies

To yield an IAP with higher photon energies in the soft X rays, the two-color chirped waveform needs to be modified. We choose the target cutoff energy of 330 eV as an illustration example and run the optimization procedure again. The parameters optimized by the GA are listed in Table 2. The second wavelength is chosen as 1600 nm, and the intensity |E2|2 is 0.9×1014W/cm2, which are determined by the initial optimization. The optimized waveform is plotted in Fig. 5(a), in which a temporal gate with a structure similar to the previous one for the 250-eV cutoff also appears (indicated by the green area).

Tables Icon

Table 2. Optimized Laser Parameters for the Two-Color Waveform at Different Runsa

 figure: Fig. 5.

Fig. 5. (a) Optimized two-color chirped laser pulse (using parameters in Table 2) with laser duration of 8 fs when the target cutoff energy is set as 330 eV; (b) extended waveform for the laser duration of 16 fs. The green areas indicate the temporal gate for the generation of soft X-ray harmonics and IAPs. (o.c. is the optical cycle of the 800-nm laser.) (c) Single-atom harmonic spectra by using optimized waveforms with different preset cutoff energies, showing that the cutoff energy is extended with the decrease of harmonic yields; (d) macroscopic harmonic spectrum with the two-color chirped waveforms (800nm+1600nm) as shown in (b). In the calculation, a 1-mm-long gas jet is located 3.5 mm after the laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 40 μm for two colors. (e), (f) Time-frequency analysis of harmonic emission; and (g) time profile of IAP with high harmonics in the photon-energy range of 169–245 eV.

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To compare with the previous results for 250-eV cutoff, the extended waveform with the pulse duration of 16 fs, as shown in Fig. 5(b), is analyzed. The resulting single-atom HHG spectra in Fig. 5(c) show that the cutoff energy is indeed increased at the cost of the reduction of harmonic yields due to the second laser with a longer wavelength (1600 nm). The macroscopic HHG spectrum in Fig. 5(d) shows the continuous distribution. Hence, an IAP of 305 as can be achieved in Fig. 5(g) by synthesizing high harmonics from 169 to 245 eV. The time-frequency analysis in Figs. 5(e) and 5(f) further shows the function of temporal gate for the generation of HHG and IAP in the single-atom and the macroscopic level, respectively. Note that weak long-trajectory electron emissions after the propagation in Fig. 5(f) can be eliminated by systematically optimizing the macroscopic conditions, which would further reduce the duration of the IAP. This requires a large computation time and should be investigated intensively in separate studies in the future. This example reveals that the photon energy of the IAP generated based on the two-color gating mechanism can be easily switched.

4. CONCLUSIONS

In summary, we suggested a new scheme to optimize the multicolor laser waveform with the inclusion of pulse chirp for the generation of IAP in soft X rays. We took an example of the two-color chirped waveform consisting of a strong 800-nm laser and a weak longer wavelength one, which can efficiently extend the harmonic plateau region. By optimizing the harmonic yields at the cutoff, the temporal gate showed up in the waveform, which greatly limits the generation of higher harmonics in a short time interval. After the interaction of an optimized two-color waveform with a macroscopic medium, continuous harmonics appeared, which results in an IAP in time due to the temporal gating effect clearly seen in the time-frequency analysis of the HHG by, preferably, selecting short-trajectory electron emissions. Such a waveform could be constructed in the lab with the help of supercontinuum generation technology [66]. Compared to the approximate “ideal waveform” composed by the five-color laser by Chipperfield et al. [31], our waveform has fewer wavelength components, generates longer cutoff while keeping the harmonic yields, and provides a temporal gate for the IAP. The temporal gate in the two-color chirped waveform has been carefully tested at some longer pulse durations, some higher gas pressures, and a larger cutoff energy, and it has been demonstrated to work well for the generation of IAP.

Therefore, we can conclude that our two-color gating method has the following advantages: (i) its conversion efficiency for the HHG (and thus for the IAP) is relatively high because of the major pulse energy from 800-nm laser; (ii) it can readily drive the cutoff getting into the spectral region of soft X rays due to the weak longer wavelength component in the waveform; (iii) it can be employed in the long pulse, i.e., the pulse energy of long-duration driving laser can be maintained since there is no need to compress it to a short one, in contrast to most temporal gating methods; and (iv) the photon energy of an IAP can be easily shifted to a higher one either by scaling the waveform with longer fundamental laser or by increasing the second wavelength in the waveform. To further reduce the duration of an IAP in soft X rays, the atto-chirp due to the short-trajectory electron emission could be compensated for by plasma dispersion [67,68] instead of using the metallic filter for XUV pulse [69].

Our study provides a simple and efficient route for generating the IAP in soft X rays, relying on the traditional 800-nm laser. To make the soft X-ray IAP a powerful light source, there are still some challenging problems, for example, shortening its duration, enhancing its intensity, and extending its central photon energy close to kilo-electron-volt. We hope this work can stimulate some further studies on the application and optimization of multicolor laser pulses [7074] towards these directions.

Funding

National Natural Science Foundation of China (11774175, 11704187, 11774174, 11834004).

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2018 (9)

Z. Chang, “Attosecond chirp compensation in water window by plasma dispersion,” Opt. Express 26, 33238–33244 (2018).
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Z. Hong, S. A. Rezvani, Q. Zhang, and P. Lu, “Ultrafast mid-IR laser pulses generation via chirp manipulated optical parametric amplification,” Appl. Sci. 8, 744 (2018).
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2016 (6)

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M. T. Hassan, T. T. Luu, A. Moulet, O. Raskazovskaya, P. Zhokhov, M. Garg, N. Karpowicz, A. M. Zheltikov, V. Pervak, F. Krausz, and E. Goulielmakis, “Optical attosecond pulses and tracking the nonlinear response of bound electrons,” Nature 530, 66–70 (2016).
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2015 (3)

C. Jin, K.-H. Hong, and C. D. Lin, “Optimal generation of high harmonics in the water-window region by synthesizing 800-nm and mid-infrared laser pulses,” Opt. Lett. 40, 3754–3757 (2015).
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E. Balogh, B. Bódi, V. Tosa, E. Goulielmakis, K. Varjú, and P. Dombi, “Genetic optimization of attosecond-pulse generation in light-field synthesizers,” Phys. Rev. A 90, 023855 (2014).
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C.-H. Lu, Y.-J. Tsou, H.-Y. Chen, B.-H. Chen, Y.-C. Cheng, S.-D. Yang, M.-C. Chen, C.-C. Hsu, and A. H. Kung, “Generation of intense supercontinuum in condensed media,” Optica 1, 400–406 (2014).
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K.-H. Hong, C.-J. Lai, J. P. Siqueira, P. Krogen, J. Moses, C.-L. Chang, G. J. Stein, L. E. Zapata, and F. X. Kätner, “Multi-mJ, kHz, 2.1  μm optical parametric chirped-pulse amplifier and high-flux soft X-ray high-harmonic generation,” Opt. Lett. 39, 3145–3148 (2014).
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S. L. Cousin, F. Silva, S. Teichmann, M. Hemmer, B. Buades, and J. Biegert, “High-flux table–top soft X-ray source driven by sub-2-cycle, CEP stable, 1.85-μm 1-kHz pulses for carbon K-edge spectroscopy,” Opt. Lett. 39, 5383–5386 (2014).
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J. Solanpää, J. A. Budagosky, N. I. Shvetsov-Shilovski, A. Castro, A. Rubio, and E. Räsänen, “Optimal control of high-harmonic generation by intense few-cycle pulses,” Phys. Rev. A 90, 053402 (2014).
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C. Jin, G. Wang, A. T. Le, and C. D. Lin, “Route to optimal generation of soft X-ray high harmonics with synthesized two-color laser pulses,” Sci. Rep. 4, 7067 (2014).
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S. Haessler, T. Balčiunas, G. Fan, G. Andriukaitis, A. Pugžlys, A. Baltuška, T. Witting, R. Squibb, A. Zaïr, J. W. G. Tisch, J. P. Marangos, and L. E. Chipperfield, “Optimization of quantum trajectories driven by strong-field waveforms,” Phys. Rev. X 4, 021028 (2014).
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X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90, 023416 (2014).
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Q. Zhang, L. He, P. Lan, and P. Lu, “Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds,” Opt. Express 22, 13213–13233 (2014).
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2013 (2)

E. J. Takahashi, P. Lan, O. D. Mücke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. 4, 2691 (2013).
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K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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C. Jin, A. T. Le, and C. D. Lin, “Medium propagation effects in high-order harmonic generation of Ar and N2,” Phys. Rev. A 83, 023411 (2011).
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H.-S. Chan, Z.-M. Hsieh, W.-H. Liang, A. H. Kung, C.-K. Lee, C.-J. Lai, R.-P. Pan, and L.-H. Peng, “Synthesis and measurement of ultrafast waveforms from five discrete optical harmonics,” Science 331, 1165–1168 (2011).
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C. Jin, A. T. Le, C. A. Trallero-Herrero, and C. D. Lin, “Generation of isolated attosecond pulses in the far field by spatial filtering with an intense few-cycle mid-infrared laser,” Phys. Rev. A 84, 043411 (2011).
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J. Xu, B. Zeng, and Y. Yu, “Extension of harmonic cutoff in a multicycle chirped pulse combined with a chirp-free pulse,” Phys. Rev. A 82, 053822 (2010).
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J. Wu, G.-T. Zhang, C.-L. Xia, and X.-S. Liu, “Control of the high-order harmonics cutoff and attosecond pulse generation through the combination of a chirped fundamental laser and a subharmonic laser field,” Phys. Rev. A 82, 013411 (2010).
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2009 (6)

Y. Niu, Y. Xiang, Y. Qi, and S. Gong, “Single attosecond pulse generation from multicycle nonlinear chirped pulses,” Phys. Rev. A 80, 063818 (2009).
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A.-T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for high-order harmonic generation from molecules,” Phys. Rev. A 80, 013401 (2009).
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P.-C. Li, X.-X. Zhou, G.-L. Wang, and Z.-X. Zhao, “Isolated sub-30-as pulse generation of an He+ ion by an intense few-cycle chirped laser and its high-order harmonic pulses,” Phys. Rev. A 80, 053825 (2009).
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H. Mashiko, S. Gilbertson, C. Li, S. D. Khan, M. M. Shakya, E. Moon, and Z. Chang, “Double optical gating of high-order harmonic generation with carrier-envelope phase stabilized lasers,” Phys. Rev. Lett. 100, 103906 (2008).
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V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351–15364 (2007).
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J. J. Carrera and S.-I. Chu, “Extension of high-order harmonic generation cutoff via coherent control of intense few-cycle chirped laser pulses,” Phys. Rev. A 75, 033807 (2007).
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V. Tosa, H. T. Kim, I. J. Kim, and C. H. Nam, “High-order harmonic generation by chirped and self-guided femtosecond laser pulses. I. Spatial and spectral analysis,” Phys. Rev. A 71, 063807 (2005).
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2001 (3)

M. B. Gaarde, “Time-frequency representations of high order harmonics,” Opt. Express 8, 529–536 (2001).
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D. G. Lee, J.-H. Kim, K.-H. Hong, and C. H. Nam, “Coherent control of high-order harmonics with chirped femtosecond laser pulses,” Phys. Rev. Lett. 87, 243902 (2001).
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2000 (2)

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1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
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1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

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J. Tate, T. Auguste, H. G. Muller, P. Salières, P. Agostini, and L. F. Di Mauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
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Alisauskas, S.

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Figures (5)

Fig. 1.
Fig. 1. Laser waveforms (red lines) of (a) five-color field in Ref. [31] and (b) optimized two-color chirped field. In (a) and (b), the excursion distances of the classical electrons are shown (blue dashed lines), leading to the cutoff harmonics (maximum electron kinetic energies indicated), and the green shadow marks the effective waveform. (c)–(e) The waveforms of two-color chirped fields are extended for longer laser durations of 16, 24, and 32 fs. The green regions show the temporal gate for the generation of continuous high harmonics, leading to IAPs. (o.c. means the optical cycle of an 800-nm laser.)
Fig. 2.
Fig. 2. Comparison of (a) single-atom and (b) macroscopic high-harmonic spectra by using single-color laser (800 nm), five-color field (from Ref. [31]), and two-color chirped field [shown in Fig. 1(c)]. Laser duration is fixed at 16 fs for all driving fields. For the macroscopic calculations, a 1-mm-long gas jet is located 0.5 mm after laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 50 μm for all colors. For three cases, the single-atom waveforms are only formed in the center of the gas jet, and the harmonic spectra in (b) are thus normalized according to the individual input pulse energy for easy comparison. Time-frequency pictures of harmonic emission and the temporal profiles of attosecond pulses with (c)–(e) two-color chirped and (f)–(h) five-color fields.
Fig. 3.
Fig. 3. Harmonic spectra, time-frequency analysis of harmonic emission, and temporal profiles of attosecond pulses, similar to Fig. 2. However, the driving lasers are replaced by the two-color chirped fields with durations of 24 and 32 fs [as plotted in Figs. 1(d) and 1(e)]. (c)–(e) 24 fs and (f)–(h) 32 fs. In (e) and (h), the IAPs with durations of 240 and 200 as are obtained by synthesized high-harmonics in the same photon energy region of 130–219 eV.
Fig. 4.
Fig. 4. (a) Macroscopic harmonic spectra with the two-color chirped field at different gas pressures of 10, 50, and 100 Torr. The pulse duration is 16 fs, and other macroscopic conditions are the same as those in Fig. 2. Time-frequency wavelet analysis of harmonic emission and the generation of IAPs at (b), (c) 50 Torr and (d), (e) 100 Torr. In (c) and (e), 200-as single attosecond pulses are generated by synthesized high-harmonics in slightly different photon-energy regions.
Fig. 5.
Fig. 5. (a) Optimized two-color chirped laser pulse (using parameters in Table 2) with laser duration of 8 fs when the target cutoff energy is set as 330 eV; (b) extended waveform for the laser duration of 16 fs. The green areas indicate the temporal gate for the generation of soft X-ray harmonics and IAPs. (o.c. is the optical cycle of the 800-nm laser.) (c) Single-atom harmonic spectra by using optimized waveforms with different preset cutoff energies, showing that the cutoff energy is extended with the decrease of harmonic yields; (d) macroscopic harmonic spectrum with the two-color chirped waveforms (800nm+1600nm) as shown in (b). In the calculation, a 1-mm-long gas jet is located 3.5 mm after the laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 40 μm for two colors. (e), (f) Time-frequency analysis of harmonic emission; and (g) time profile of IAP with high harmonics in the photon-energy range of 169–245 eV.

Tables (2)

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Table 1. Optimized Laser Parameters for the Two-Color Waveform at Different Runsa

Tables Icon

Table 2. Optimized Laser Parameters for the Two-Color Waveform at Different Runsa

Equations (9)

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E(t)=E1A1(t)cos(ω1t+α1t2+ϕ1)+E2A2(t)cos(ω2t+α2t2+ϕ2).
D(ω)=W(ω)d(ω),
F=ωminωmaxω4|D(ω)|2dω.
2E1(r,z,t)1c22E1(r,z,t)t2=μ0Jabs(r,z,t)t+ω02c2(1ηeff2)E1(r,z,t),
2Eh(r,z,t)1c22Eh(r,z,t)t2=μ02P(r,z,t)t2,
A(t,ω)=Eh(τ)wt,ω(τ)dτ,
W(x)=(1/ν)eixex2/2ν2.
|Anear(t,ω)|2=02πrdr|Eh(r,τ)wt,ω(τ)dτ|2.
Inear(t)=02πrdr|ω1ω2Eh(r,ω)eiωtdω|2.

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