## Abstract

We propose an efficient method to generate an ultra-broadband supercontinuum by using a three-color field synthesized by a 1600 nm driving pulse and two weak (3200 nm and 400 nm) control pulses. The results show that the acceleration and ionization steps in the HHG process are simultaneously controlled in our scheme, which can not only broaden the bandwidth but also enhance the harmonic yields of the generated spectrum. Thus a high-efficiency supercontinuum with the photon energies ranging from 205 eV to 480 eV is observed. Based on our 3D propagation simulations, we show that the short quantum path of the supercontinuum is well phase-matched and can be selected under suitable propagation conditions. Thereby, the modulation due to quantum path interference is largely reduced and intense isolated sub-100 as pulses with tunable central wavelengths are directly obtained within water window spectral region.

© 2013 Optical Society of America

## 1. Introduction

The development of research on attosecond pulses [1, 2] has opened up a new field of time-resolved studies with high precision, which provides an important tool for investigating and manipulating the ultrafast electron dynamics in atoms and molecules with unprecedented resolutions [3, 4]. Nowadays, high-order harmonic generation (HHG) seems to be the most promising way to produce attosecond pulses. The physical mechanism of HHG can be well understood in terms of the three-step model [5]: ionization, acceleration and recombination. During the recombination, the maximum photon energy obeys the cutoff law, *I _{p}*+3.17

*U*, where

_{p}*U*is the ponderomotive energy and

_{p}*I*is the ionization potential. Such a process occurs every half optical cycle and gives rise to the generation of attosecond pulse trains with a periodicity of half optical cycle [1]. Since straightforward attosecond metrology prefers an isolated attosecond pulse [6, 7], how to generate an isolated attosecond pulse has become one of the most active research directions in attosecond science.

_{p}Up to now, many schemes have been proposed for the production of isolated attosecond pulses. By using a few-cycle laser pulse [8, 9], the emission time of the highest harmonics can be confined within half an optical cycle where the laser field reaches its maximum, which leads to a supercontinuum at the cutoff. Then an isolated attosecond pulse can be filtered out from the supercontinuum. This scheme has been carried out in experiment directly using a two-cycle (5 fs) driving pulse [4]. However, the bandwidth of the supercontinuum is less than 20 eV, thus the duration of the shortest attosecond pulse is only 250 as. To further shorten the pulse duration, Goulielmakis *et al*. [8] employed a sub-4-fs near-single-cycle driving pulse and a 80 as pulse was filtered out in their experiment, which first broke through the 100-as-barrier. Very recently, Zhao *et al*. [10] have produced a 67 as pulse with a 7 fs, 750 nm laser pulse, which is the known shortest attosecond pulse at present. But few-cycle laser pulse is still a challenge for current laser technology, and other approaches to generate a broadband supercontinuum are desired. According to the classical three-step model, the generation of a broadband supercontinuum can be realized by dominating the electron dynamic process in the laser field. It has been theoretically demonstrated that recombination of the electrons can be gated into one half-cycle with the polarization-gating technique [11–13] and then a broadband supercontinuum in the plateau region can be produced. This scheme has been experimentally achieved by Sansone *et al*. [14]. In their experiment, a 130 as isolated attosecond pulse was obtained after compensating the harmonic chirp. Another effective way is to adopt the two-color field. With this method, the electron acceleration process can be effectively controlled. It has been shown theoretically that the difference between the highest and the second highest half-cycle photo energies is significantly enlarged and the broadband supercontinuum with several tens eV and even over 100 eV are produced [15–18]. In addition, the two-color field scheme can also be applied to manage the ionization step for high-efficiency harmonic generation [19–22]. It has been reported by Lan *et al*. [23] that the electron ionization can be restricted within one half cycle with a high ionization rate, which is called “ionization gating”. Then a broadband supercontinuum with high harmonic yields was produced in the plateau region.

In this paper, we present a method to generate isolated broadband attosecond pulses with a three-color laser field. In our scheme, we preform the simulation based on coupled solution of strong field approximation and the Maxwell wave function, including both the single-atom response and macroscopic effects of propagation and phase matching. The single-atom calculations confirm that we successfully achieve the control of both the ionization and acceleration steps in HHG, and an efficient 275-eV supercontinuum with the spectral range from ultraviolet to water window [the spectral range between the *K*-absorption edges of carbon (284 eV) and oxygen (543 eV)] is produced. Macroscopically, with the right phase-matching technique, the short quantum path is selected, which permits us to directly synthesize isolated sub-100 as pulses with tunable central wavelengths within the water window domain.

## 2. Theoretical model

The HHG process in atomic gases can be theoretically described by single-atom response and the copropagation of laser and harmonic beams. In our calculation, we apply Lewenstein model [24] to calculate the harmonic radiation. The time-dependent dipole momentum is described as

*E*(

*t*) is the laser field,

*A*(

*t*) is the corresponding vector potential,

*ε*is a positive regularization constant.

*p*and

_{st}*S*are the stationary momentum and quasi-classical action, respectively. And the

_{st}*g*(

*t*) represents the ground-state amplitude, which is given by ADK tunneling model [25]. Then the harmonic spectrum is obtained by the Fourier transforming the time-dependent dipole acceleration

*a⃗*(

*t*),

*a⃗*(

*t*) = ${\overrightarrow{\ddot{d}}}_{\mathit{nl}}(t)$,

*T*and

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

*q*corresponds to the harmonic order.

The copropagation of the laser and harmonics beams can be simulated by numerically solving Maxwell wave equations for the fields of the laser pulse *E _{l}* and the harmonics

*E*, which are given by

_{h}*P*(

_{nl}*r, z, t*) = [

*n*

_{0}−

*n*(

_{e}*r, z, t*)]

*d*(

_{nl}*r, z, t*) is the nonlinear polarization of the medium.

*n*

_{0}is the gas density and ${n}_{e}(t)={n}_{0}\left[1-\mathit{exp}\left(-{\int}_{-\infty}^{t}w({t}^{\prime})d{t}^{\prime}\right)\right]$ is the free-electron density in the gas, where

*w*(

*t*) is the electron ionization rate. The equations here take into account both temporal plasma induced phase modulation and the spatial plasma lensing effects on the driving field. Equation (3) and (4) can be numerically solved with the Crank-Nicholson method as described in Ref. [26].

## 3. Results and discussion

In our investagation, a 12 fs, 1600 nm pulse is selected as the driving field and the two control fields are a 24 fs, 3200 nm pulse and a 12 fs, 400 nm pulse, respectively. The electric field of the synthesized three-color laser pulse can be expressed as

*E*

_{1},

*E*

_{2}and

*E*

_{3}are the amplitudes of the electric fields of the driving and the control pulses,

*ω*

_{1},

*ω*

_{2}and

*ω*

_{3}are the corresponding frequencies of the three pulses.

*f*

_{1}(

*t*),

*f*

_{2}(

*t*) and

*f*

_{3}(

*t*) are the envelopes for each laser pulse.

*τ*is the time delay of the 3200 nm pulse, which equals three optical cycles of the driving pulse. We choose the intensities of the three fields as:

*I*

_{1}= 3

*×*10

^{14}W

*/*cm

^{2},

*I*

_{2}= 1.5

*×*10

^{13}W

*/*cm

^{2},

*I*

_{3}= 3

*×*10

^{13}W

*/*cm

^{2}, and the corresponding carrier-envelope phases are set as

*ϕ*

_{1}= 1.1

*π*,

*ϕ*

_{2}= 1.0

*π*and

*ϕ*

_{3}= 0, respectively. Note that

*ϕ*

_{2}and

*ϕ*

_{3}, i.e., the carrier-envelope phases of the 3200 nm and 400 nm laser pulses, are not restricted to 1.0

*π*and 0. Our calculations show that (not presented here) the conclusion of this paper still holds for

*ϕ*

_{2}varying from 0.8

*π*to 1.3

*π*and

*ϕ*

_{3}from −0.1

*π*to 0.7

*π*. Here, the target atom is chosen to be helium atom with 24.6 eV of the ionization energy.

To clearly clarify the different roles played by the two control fields in our scheme, we first investigate the harmonics generated in the two-color field schemes, in which only one of the control fields is adopted. According to the three-step model [5], the HHG process can be well described in terms of classical electron trajectories and the ionization rate calculated by the ADK model [25]. Figure 1 depicts the physical picture of HHG with the 1600 nm driving field alone. It’s obvious that there are three main peaks of the classical trajectories (marked as P_{1}, P_{2} and P_{3}) contributing to the harmonic generation and each peak contains two branches, which are called short and long path, respectively. The interference of the short and long trajectories will lead to a coherent structure of the supercontinuum. The bandwidth of the supercontinuum is directly depended on the energy difference between the highest and second highest peaks, and the corresponding ionization rate largely determines the harmonic yields. Filtering the harmonics near the cutoff with a bandpass filter, isolated attosecond pulse can be obtained. Since the maximum kinetic energy of P_{1} is only 35 eV higher than that of P_{2}, the bandwidth of the supercontinuum near the cutoff is inappreciable. While we can successfully control the electron acceleration process by adding a weak 3200 nm control field. The result is displayed in Fig. 2(a). One can clearly see that the maximum kinetic energy of P′_{1} is increased to 400 eV, and that of P′_{2} is only 195 eV. Then a supercontinuum spectrum with a bandwidth of 205 eV can be produced. However, the ionization rate of P′_{1}, from which the supercontinuum originates, is much lower than that of P′_{2}. Therefor, the harmonic yields of the supercontinuum are inefficient, which may result a two-plateau structure in the spectrum. On the other hand, We can also dominate the electron ionization process by using the driving field in combination with a 400 nm control pulse. As shown in Fig. 2(b), the corresponding ionization rates of the two ionization peaks, marked as P″_{1} and P″_{2}, are approximately 1 or 2 magnitudes higher than those in Fig. 1, which leads to the enhancement of harmonic yields in the plateau region. Besides, the electron ionization between P″_{1} and P″_{2} is completely suppressed. Thus we can produce a high-efficiency supercontinuum with a bandwidth of 120 eV, which is corresponding to the energy difference between P″_{1} and P″_{2}.

To confirm the above classical analysis, we calculate the corresponding harmonic spectra as presented in Fig. 3. Figure 3(a) shows the harmonic spectrum with the 1600 nm driving pulse alone. A supercontinuum with bandwidth of only 35 eV (295* ^{th}*–340

*) is obtained near the cutoff region. Such a narrow spectrum is difficult to produce attosecond pulses with short duration. When the 3200 nm pulse is added, the harmonic cutoff is dramatically extended to 437 eV (565*

^{th}*) and a broadband supercontinuum with the bandwidth of 205 eV (300*

^{th}*–565*

^{th}*) is directly produced, as shown in Fig. 3(b) (blue line). Due to the low ionization rates, the harmonic yields of the continuous harmonics are inefficient, leading to an apparent two-plateau structure in the spectrum. Under the control of the 400 nm pulse, the generated harmonic spectrum is shown in Fig. 3(c) (blue line). The harmonics from 167 eV (215*

^{th}*) to 287 eV (370*

^{th}*) become continuous, and the harmonic yields are also several times higher than those of the harmonics with the driving pulse alone (red line). To sum up, by adding only the 3200 nm control field, the oscillating electrons in the field gain much more energy than the well-known value of 3.17*

^{th}*U*, and the acceleration step is successfully controlled for the spectrum extension. What unsatisfactory is the low harmonic yields of the supercontinuum. While only adding the 400 nm control field, the electronic ionization rate is largely improved, and the ionization step is effectively dominated for harmonic yields enhancement. However, the bandwidth of the harmonic is deeply limited by the restricted cutoff energy.

_{p}In our three-color field scheme, both the 3200 nm and 400 nm control fields are employed. We predict that the ionization and acceleration steps can be simultaneously controlled and an efficient broadband supercontinuum can be produced with the scheme. The synthesized electric field and the corresponding classical electron trajectories are presented in Fig. 4. It’s clear that the maximum electron kinetic energy of P_{1} is increased to 440 eV and the corresponding ionization rate is also extremely enhanced, which is nearly 2 magnitudes higher than that in Fig. 1. These results imply that our scheme can not only extend the harmonic cutoff but also enhance the harmonic yields of the harmonic spectrum. The control of ionization and acceleration process in HHG are successfully achieved by adding the two control fields. Additionally, we also note that the left branch of peak P_{1} is bright, while the right one is almost disappeared, which implies that the quantum path selection can be also realized in the scheme.

We further present the harmonic spectrum to confirm the our analysis above. The harmonic spectrum is shown in Fig. 5(a) (blue line). An ultra-broadband supercontinuum through the plateau with the bandwidth of 275 eV (265* ^{th}*–620

*), which is covered by the range from ultraviolet to water window x ray, is successfully generated. Compared to the supercontinuum with the 1600 nm driving pulse alone (red line), the harmonic yields with the synthesized pulse are approximately 1–2 order higher, and the cutoff is extremely extended to 480 eV (620*

^{th}*). Moreover, the modulation in the supercontinuum, which is due to the interference of the long and short quantum paths, is much reduced, implying one quantum path is selected. A deeper insight is obtained by investigating the emission times of the harmonics in terms of the time-frequency analysis method [22]. The result is shown in Fig. 5(b). One can clearly see that the short quantum path (the left branch) of the peak that contributes to the continuous harmonic is much more dominant than the long quantum path (the right branch), resulting a smooth supercontinuum as shown in Fig. 5(a). In the scheme, the short quantum path is selected to dominate the harmonic generation. These results well agree with the classical approaches in Fig. 4.*

^{th}In our scheme, due to the low ionization probability of the gas medium (far below 0.1%), the blue shift of the laser field [27] caused by free electrons can be neglected and thus the gas can be treated as neutral media. Correspondingly, it allows us to adopt the suitable phase-matching conditions to macroscopically enhance the harmonic yields of the short quantum path in the supercontinuum [28]. These conditions can be fully satisfied by adjusting the gas pressure and the position of the laser focus. To demonstrate this issue, we simulate the co-propagation of both laser and harmonic beams in the gas target [26]. To achieve the phase-match of the short path, we adopt a focused laser beam with a beam waist of 35 *μ*m and a 0.5-mm long gas jet with a density of 3.5 × 10^{17}/cm^{3}. The gas jet is placed 1.5 mm after the laser focus. Figure 6 shows the continuous part of macroscopic harmonic spectrum in the three-color pulse (red line). For comparison, the single-atom response is also presented (blue line). One can clearly see that the harmonic yields are further enhanced and the modulation in the supercontinuum is further reduced after propagation, which implies that the harmonics from the short path are well phase-matched and the short path is further selected.

Finally, we investigate the temporal characteristics of the smoothed supercontinuum by applying a square window with a width of 75 eV to the supercontinuum at different orders. And then, the pure isolated 75 as pulses with tunable central wavelengths from 2.9 nm to 4.6 nm are directly obtained as shown in Fig. 7. To further shorten the duration of the attosecond pulse, one can superpose much more harmonics. However, due to the positive harmonic chirp of the short quantum path, it’s difficult to produce a Fourier-transform-limited pulse. By using a material with a negative group delay dispersion, the positive chirp can be availably compensated [10, 29]. With this technique, it is anticipated that isolated pulses of 55 as can be generated corresponding to the bandwidth of 75 eV. It is worth mentioning that such an ultra-broadband supercontinuum enables the generation of attosecond pulse with duration below 24 as (one atomic unit of time) by proper chirp compensation, which is most valuable in probing and controlling the electronic dynamics inside atoms and molecules.

## 4. Conclusion

In conclusion, we propose a three-color field scheme to simultaneously control the ionization and acceleration steps of HHG for the generation of an efficient ultra-broadband supercontinuum. By adding two weak 3200 nm and 400 nm control pulses to a 1600 nm driving pulse, the maximum kinetic energy and the ionization rate of the electrons that attribute to the continuous harmonics are both increased, and a supercontinuum with the spectral range from ultraviolet to water window x ray is generated. Furthermore, the short quantum path of the supercontinuum is well phase-matched and can be selected under the proper propagation conditions, which enables the production of intense isolated sub-100 as pulses with tunable central wavelengths within the water window domain. Experimentally, our scheme can be carried out by using a Ti:sapphire laser system. The two mid-infrared laser pulses can be achieved via the optical parametric amplification (OPA) technology and the 400 nm pulse can be realized through the second harmonic generation. The time delay among the three pulses can be adjusted by a piezoelectric translator stage. And then, we hope our scheme could be applied in practice for detecting and controlling the ultrafast processes in atoms someday.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants No. 60925021, 11234004, 10904045 and the 973 Program of China under Grant No. 2011CB808103.

## References and links

**1. **P. M. Paul, E. S. Toma, P. Breger, G. Mullot, F. Aug, Ph. Balcou, H. G. Muller, and P. Agostini, “Observation of a train of attosecond pulses from high harmonic generation,” Science **292**, 1689–1692 (2001). [CrossRef] [PubMed]

**2. **Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovacev, R. Taieb, B. Carreć, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft X-rays,” Science **302**, 1540–1543 (2003). [CrossRef] [PubMed]

**3. **M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovelv, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature (London) **419**, 803–807 (2002). [CrossRef]

**4. **R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature (London) **427**, 817–821 (2004). [CrossRef]

**5. **P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. **71**, 1994–1997 (1993). [CrossRef] [PubMed]

**6. **I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-harmonic generation of attosecond pulses in the “single-cycle” regime,” Phys. Rev. Lett. **78**, 1251–1254 (1997). [CrossRef]

**7. **M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature (London) **414**, 509–513 (2001). [CrossRef]

**8. **E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science **320**, 1614–1617 (2008). [CrossRef] [PubMed]

**9. **J. J. Carrera, X. Tong, and S. I. Chu, “Creation and control of a single coherent attosecond xuv pulse by few-cycle intense laser pulses,” Phys. Rev. A **74**, 023404 (2006). [CrossRef]

**10. **K. Zhao, Q. Zhang, Michael Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. **37**, 3891–3893 (2012). [PubMed]

**11. **Z. Chang, “Chirp of the single attosecond pulse generated by a polarization gating,” Phys. Rev. A **71**, 023813 (2005). [CrossRef]

**12. **Q. Zhang, P. Lu, P. Lan, W. Hong, and Z. Yang, “Multi-cycle laser-driven broadband supercontinuum with a modulated polarization gating,” Opt. Express **16**, 9795–9803 (2008). [CrossRef] [PubMed]

**13. **W. Hong, P. Wei, Q. Zhang, S. Wang, and P. Lu, “Mid-infrared modulated polarization gating for ultra-broadband supercontinuum generation,” Opt. Express **18**, 11308–11315 (2010). [CrossRef] [PubMed]

**14. **G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulse,” Science **314**, 443–446 (2006). [CrossRef] [PubMed]

**15. **T. Pfeifer, L. Gallmann, M. J. Abel, D. M. Neumark, and S. R. Leone, “Single attosecond pulse generation in the multicycle-driver regime by adding a weak second-harmonic field,” Opt. Lett. **31**, 975–977 (2006). [CrossRef] [PubMed]

**16. **P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A **76**, 011402 (2007). [CrossRef]

**17. **Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. **98**, 203901 (2007). [CrossRef] [PubMed]

**18. **W. Hong, P. Lu, Q. Li, and Q. Zhang, “Broadband water window supercontinuum generation with a tailored mid-IR pulse in neutral media,” Opt. Lett. **34**, 2102–2104 (2009). [CrossRef] [PubMed]

**19. **P. Lan, P. Lu, W. Cao, and X. Wang, “Efficient generation of an isolated single-cycle attosecond pulse,” Phys.Rev. A **76**, 043808 (2007). [CrossRef]

**20. **W. Cao, P. Lu, P. Lan, X. Wang, and Y. Li, “Control of the launch of attosecond pulses,” Phys. Rev. A **75**, 063423 (2007). [CrossRef]

**21. **Y. Zheng, Z. Zeng, X. Li, X. Chen, P. Liu, H. Xiong, H. Lu, S. Zhao, P. Wei, L. Zhang, Z. Wang, J. Liu, Y. Cheng, R. Li, and Z. Xu, “Enhancement and broadening of extreme-ultraviolet supercontinuum in a relative phase controlled two-color laser field,” Opt. Lett. **33**, 234–236 (2008). [CrossRef] [PubMed]

**22. **Q. Zhang, P. Lu, W. Hong, Q. Liao, P. Lan, and X. Wang, “Enhanced high-order harmonic generation via controlling ionization in spatially extended systems,” Phys. Rev. A **79**, 053406 (2009). [CrossRef]

**23. **P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Attosecond ionization gating for isolated attosecond electron wave packet and broadband attosecond xuv pulses,” Phys. Rev. A **76**, 051801 (2007). [CrossRef]

**24. **M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L Huillier, and P. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A **49**, 2117–2132 (1994). [CrossRef] [PubMed]

**25. **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).

**26. **E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimentional model of high-order harmonic generation in the few-optical cycle regime,” Phys. Rev. A **61**, 063801 (2000). [CrossRef]

**27. **E. V. Vanin, A. V. Kim, and A. M. Sergeev, “Excitation of ultrashort bursts of harmonics of the radiation during ionization of a gas by an intense light pulse,” JETP Lett. **58**, 900 (1993).

**28. **Philippe Balcou, Pascal Salieres, Anne L’Huillier, and Maciej Lewenstein, “Generalized phase-matching conditions for high harmonics: The role of field-gradient forces,” Phys. Rev. A **55**, 3204–3210 (1997) [CrossRef]

**29. **Kyung Taec, Kim Chul, Min Kim, Moon-Gu Baik, G. Umesh, and Chang Hee Nam, “Single sub-50-attosecond pulse generation from chirp-compensated harmonic radiation using material dispersion,” Phys. Rev. A **69**, 051805(R) (2004).