We theoretically investigate the X-ray supercontinuum generated by interaction of multi-cycle, chirped polarization gating pulses with the helium gas. It is shown that with this scheme, an isolated sub-50-attosecond pulse can be obtained straightforwardly without any phase compensation. Interestingly, if one selects an extremely broad spectral range near the high-order harmonic cutoff, an isolated and intense sub-24-attosecond pulse can be generated after phase compensation, which could be used to detect and control the electronic dynamics inside the atoms. Furthermore, it is found that the generation of such a broad and smooth X-ray supercontinuum is not so stringent on the selection of the simulated parameters, allowing for the experimental demonstration of this technique in the future.
© 2012 OSA
With the rapid development of ultrafast laser technology in the past two decades, high-order harmonic generation (HHG) has been widely studied in the strong field regime due to its important applications in high-contrast X-ray microscopy [1–3], molecular tomography [4, 5] and short-wavelength coherent light sources [6–8]. Furthermore, HHG also provides an effective way for the generation of the attosecond pulse [9–11]. When a sub-5 fs, the carrier-envelope phase stabilized laser pulse interacts with the gas [12, 13], an isolated attosecond pulse can be achieved, which can be well understood by the three-step model [2, 6, 14]. However, the generation of the few-cycle laser pulse with a high energy is still technically difficult, which only is demonstrated in several laboratories in the world, limiting the widespread use of attosecond technology. Therefore, it is highly desired to generate isolated attosecond pulses with a multi-cycle, high-energy laser field. Generally speaking, the HHG periodically occurs in a multi-cycle laser field and thus corresponds to the attosecond pulse trains. As contrast to the attosecond pulse train, an isolated attosecond pulse, especially the one close to one atomic unit (i. e., ~24 as), is more benefit for measuring and controlling the ultrafast electronic dynamics inside the atoms, such as the inner-shell electronic relaxation and ionization by optical tunneling [4, 15]. Hence, a series of methods have been proposed to generate the isolated attosecond pulses with multi-cycle laser pulses, such as two-color scheme [10, 12, 16–18], the chirped control method [19–22], the polarization gating method [7, 23–26]. Recently, when a chirped control pulse combined with the polarization gating pulses interacts with the gas, an isolated ~70 attosecond pulse is filtered out from the supercontinuum spectrum [27, 28]. In contrast, in our numerical simulation we only use a chirped polarization gating field with the pulse duration of 30 fs, which can be obtained directly from the amplifiers. Numerical results show that, due to the presence of the chirped effects, the symmetry of the electric field is broken, which results in significant extension of cutoff energy of HHG and the X-ray supercontinuum. Therefore, in this scheme, an isolated sub-50 attosecond pulse can be generated after a proper spectral filter, indicating a significant advantage as compared to the previous works [27, 28]. Importantly, an extremely smooth harmonic spectra ranging from 425th to 570th order can synthesize an isolated ~17 attosecond pulse after phase compensation (i. e., assuming a flat spectral phase), which could have potential applications in investigating the ultrafast electronic dynamics with an unprecedented time resolution. Moreover, the generation of the smooth and ultrabroad continuous spectrum is not so stringent to the selection of simulated parameters, allowing for experimental demonstration in the future.
In this work, when the nonlinear chirped effect is considered, the laser field composed of the two counter-rotating polarization gating pulses can be expressed as:20, 29], the chirped pulse with hyperbolic tangent form is difficultly to achieve in experiments, so here the chirp is expressed as a form generally employed in the experiment . In our numerical simulation, we choose the following laser parameters: β = 2.0, t0 = 2σ = 9.68 fs [20, 29], ω0 = 2.36 fs−1 (corresponding to a laser wavelength λ = 800 nm), τ0 = 30 fs and Td = 8 fs. The laser intensity I0 = 1.0 × 1014 W/cm2 corresponds to the electric field of E0 = 2.74 × 108 V/cm. As shown in the previous report , at a low-intensity field below 6.0 × 1014 W/cm2, the distortion of driving pulse is negligible after propagation through the gas medium, and thus the Lewenstein model based on the single-active-electron approximation can be applied to calculate the HHG spectra [32–34] by performing the Fourier transformation of the single-atom dipole momentum [7, 8].
We first investigate the influence of the chirped effects on HHG spectra in the polarization gating technique. As displayed in Fig. 1(a) , in the case without the chirped effect, the electric field waveform of the polarization gating pulses is similar to that of the previous reports [25, 27, 28]. The corresponding HHG spectrum is shown in Fig. 1(b). Clearly, the HHG spectrum shows a deep modulation near the cutoff energy around the 60th harmonic. When the chirped effect is considered, the cutoff energy of HHG spectrum is extended to ~600th order, as shown in Fig. 1(d), which is due to well broken symmetry of the driving field (See Fig. 1(c)). Particularly, the harmonic spectrum above 400th orders is extremely smooth, which will correspond to a well isolated attosecond pulse in the temporal domain.
In order to obtain a clear insight for generation of the dramatically broadened X-ray supercontinuum, we perform time-frequency analyses for dipole momentums in two cases. From Fig. 2(a) , we can clearly see that in this absence of the chirped effect, the harmonic emission occurs every half of an optical cycle of the laser pulse and each emission show up and down arms with nearly equivalent intensities, which is defined as the long trajectory and the short trajectory [12, 16, 19], respectively. Because of the interferences between different quantum trajectories, HHG spectrum is modulated significantly, as indicated in Fig. 1(b). But when the chirped effects are taken into account in the polarization gating technique, there is only a major burst near the emission time of ~1.3T (T is optical period of the laser pulse), as illustrated in Fig. 2(b). In this case, not only the emission from the other half cycles is effectively suppressed, but also a short trajectory is well selected, resulting in an extremely broad and smooth supercontinuum, as shown in Fig. 1(d).
To further understand the underlying physical mechanism behind the dramatically broadened X-ray supercontinuum in the presence of the chirped effects, we depicted the ellipticity of the driving field as a function of time in both cases without and with chirped effects. Figure 3(b) clearly shows that when the chirp effect is introduced, the ellipticity of the driving laser field exhibits a rapid variation. It is well known that HHG strongly depends on the ellipticity of the driving laser field [35, 36]. Generally, when the ellipticity is up to 0.2, the ionized electron cannot revisit its parent ion and thus HHG will be terminated. Therefore, HHG in the case with chirped effects will be confined within a much narrower temporal window as compared to the case without chirped effect (see Fig. 3(a)). As a result, HHG from the other cycles is effectively suppressed. In this way, a single quantum trajectory is selected (see the Fig. 2(b)), leading to a dramatically broadened X-ray supercontinuum.
When the low-order harmonics are filtered, an isolated attosecond pulse can be obtained by performing inverse Fourier transformation for the HHG spectrum. Just as shown in Fig. 4(a) , the supercontinuum in the spectral range from 460th harmonic to 520th harmonic can support an isolated sub-50-attosecond pulse even without any phase compensation, which is shorter than that in the previous reports [27, 28]. More interestingly, if we only select a broader supercontinuum range from 425th harmonic and 570th harmonic to perform inverse Fourier transformation, an isolated ~17-attosecond pulse can be created after phase compensation (see Fig. 4(b)). Such an isolated attosecond pulse with the pulse duration comparable to an atomic unit would have potential applications in detecting and monitoring the ultrafast electronic dynamics inside the atoms.
In order to examine the feasibility of the experimental demonstration of this method, we investigate the influences of the laser intensity, the time delay Td between two counter-rotating chirped polarized pulses and chirp parameter on the HHG spectra and attosecond pulses. As shown in Fig. 5(a) , with the increase or decrease of the laser intensity, the cutoff energy of the HHG changes in the same way. However, we always can observe a smooth and broad supercontinuum in all cases. Correspondingly, by selecting HHG spectrum from the 460th to 520th harmonics, the isolated attosecond pulses can also obtained without any phase compensation though theirs shapes are different. At the laser intensity of 1.1 × I0, the intensity of the HHG spectrum in this spectral range (460th~520th harmonics) is lower as compared to the case at the laser intensity of I0, so that the synthesized attosecond pulse is relatively weak in this case, as shown by dotted line in Fig. 5(b). When the laser intensity is reduced to 0.9 × I0, the attosecond pulse shows a double-peak structure, as indicated by the solid line in Fig. 5(b), which could be the result of interference of the long and short trajectories. Generally, the fluctuation of laser intensity is far less than the current assumption ( ± 10%). Therefore, we believe that an isolated attosecond pulse will remain unchanged if intensity fluctuation of laser pulses is controlled within a reasonable range. In addition, we also examined the influence of the delay Td between two counter-rotating chirped polarized gating pulses on the supercontinuum spectrum as well as the attosecond pulses. As shown in Fig. 6 , both HHG spectrum and isolated attosecond pulses only exhibit a slight change when the delay Td increases to 12 fs from 8 fs.
Lastly, we would like to point out that the supercontinuum can still be effectively broadened when the chirp of pump pulse is reduced to the experimentally available value. As mentioned above, in our simulation, the pulse duration of the Fourier-transform limit is 30 fs in the absence of chirped effect. But when the chirped effect is considered, the spectral bandwidth will increase significantly as shown in the previous report . Therefore, in the case of β = 2.0, the Fourier-transform limit of the pump pulse is shorter than an optical period. Though such a laser pulse cannot be generated, in principle this new scheme is still a simple and effective way to generate sub-100-as isolated attosecond pulses in the multi-cycle regime. Moreover, further simulation shows that a smooth X-ray supercontinuum and a sub-100-as isolated attosecond pulse can still be observed at β = 0.5, as shown in Figs. 7(a) and 7(b), respectively. In this case, the pulse duration of Fourier-transform limit is ~3.6 fs, which is easily achieved by the current pulse compression technique [12, 13].
In conclusion, we theoretically proposed a novel method to generate sub-100-as isolated attosecond pulses in the multi-cycle regime. It is shown that HHG can be effectively confined within an extremely narrow temporal window due to the rapidly varying ellipticity when a chirp is added to the two counter-rotating polarization gating pulses. In this way, the harmonic emission occurring at every half optical cycle is effectively reduced to one time, leading to a well isolated attosecond pulse with the pulse duration comparable to an atomic unit. Such short attosecond pulses could have potential applications in detecting and controlling the ultrafast electronic dynamics with an unprecedented time scale. Lastly, we also demonstrated the experimental feasibility of this method by examining the dependence of both HHG spectra and attosecond pulses on the laser intensity, the delay between two counter-rotating polarization gating pulses and the chirp.
C. Zhang and J. Yao attribute equally to this work. The authors are grateful to Professor Z. Xu, Y. Cheng and Dr. B. Zeng of SIOM for their help. The work is supported by National Basic Research Program of China (Grant No. 2011CB808102), National Natural Science Foundation of China (Grants No. 11134010, No. 60825406, No. 60921004, No. 61008061, No. 11204332, and No. 11104236) and Education Committee Foundation of Jiangsu Province (Grant No. 10KJB140012). C. Zhang gratefully acknowledges the support of K.C.Wong Education Foundation, China Postdoctoral Science Foundation funded project (2012M511145), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
References and links
1. P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. Di Mauro, “Scaling strong-field interactions towards the classical limit,” Nat. Phys. 4(5), 386–389 (2008). [CrossRef]
3. P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys. 3(12), 846–850 (2007). [CrossRef]
4. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]
6. P. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]
7. Y. Li, Q. Zhang, W. Hong, S. Wang, Z. Wang, and P. Lu, “Efficient generation of high beam-quality attosecond pulse with polarization-gating Bessel-Gauss beam from highly-ionized media,” Opt. Express 20(14), 15427–15439 (2012). [CrossRef] [PubMed]
8. C. Zhang, J. Yao, J. Ni, G. Li, Y. Cheng, and Z. Xu, “Control of bandwidth and central wavelength of an enhanced extreme ultraviolet spectrum generated in shaped laser field,” Opt. Express 20(15), 16544–16551 (2012). [CrossRef]
9. Y. Zheng, Z. Zeng, P. Zou, L. Zhang, X. Li, P. Liu, R. Li, and Z. Xu, “Dynamic chirp control and pulse compression for attosecond high-order harmonic emission,” Phys. Rev. Lett. 103(4), 043904 (2009). [CrossRef] [PubMed]
10. F. Calegari, M. Lucchini, K. S. Kim, F. Ferrari, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “Quantum path control in harmonic generation by temporal shaping of few-optical-cycle pulses in ionizing media,” Phys. Rev. A 84(4), 041802 (2011). [CrossRef]
11. A. L. Lytle, X. Zhang, P. Arpin, O. Cohen, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase matching of high-order harmonic generation at high photon energies using counterpropagating pulses,” Opt. Lett. 33(2), 174–176 (2008). [CrossRef] [PubMed]
13. 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(5883), 1614–1617 (2008). [CrossRef] [PubMed]
14. C. Altucci, J. W. G. Tisch, and R. Velotta, “Single attosecond light pulses from multi-cycle laser sources,” J. Mod. Opt. 58(18), 1585–1610 (2011). [CrossRef]
15. W. Hong, P. Lu, P. Lan, Z. Yang, Y. Li, and Q. Liao, “Broadband xuv supercontinuum generation via controlling quantum paths by a low-frequency field,” Phys. Rev. A 77(3), 033410 (2008). [CrossRef]
16. X. Song, Z. Zeng, Y. Fu, B. Cai, R. Li, Y. Cheng, and Z. Xu, “Quantum path control in few-optical-cycle regime,” Phys. Rev. A 76(4), 043830 (2007). [CrossRef]
17. J. Yao, Y. Li, B. Zeng, H. Xiong, H. Xu, Y. Fu, W. Chu, J. Ni, X. Liu, J. Chen, Y. Cheng, and Z. Xu, “Generation of an XUV supercontinuum by optimization of the angle between polarization planes of two linearly polarized pulses in a multicycle two-color laser field,” Phys. Rev. A 82(2), 023826 (2010). [CrossRef]
18. K. Zhao and T. Chu, “A single isolated sub-50 attosecond pulse generation with a two-color laser field by a frequency-chirping technique,” Chem. Phys. Lett. 511(1-3), 166–171 (2011). [CrossRef]
19. M. M. Masoud Mohebbi and S. B. Saeed Batebi, “Generation of 40-as few-cycle pulse through chirp manipulation,” Chin. Opt. Lett. 10(8), 081901–081903 (2012). [CrossRef]
20. P. Zou, Z. Zeng, Y. Zheng, Y. Lu, P. Liu, R. Li, and Z. Xu, “Coherent control of broadband isolated attosecond pulses in a chirped two-color laser field,” Phys. Rev. A 81(3), 033428 (2010). [CrossRef]
21. E. Mansten, J. M. Dahlström, P. Johnsson, M. Swoboda, A. L’Huillier, and J. Mauritsson, “Spectral shaping of attosecond pulses using two-color laser fields,” New J. Phys. 10(8), 083041 (2008). [CrossRef]
22. H. Du and B. Hu, “Propagation effects of isolated attosecond pulse generation with a multicycle chirped and chirped-free two-color field,” Phys. Rev. A 84(2), 023817 (2011). [CrossRef]
24. C. Altucci, R. Velotta, V. Tosa, P. Villoresi, F. Frassetto, L. Poletto, C. Vozzi, F. Calegari, M. Negro, S. De Silvestri, and S. Stagira, “Interplay between group-delay-dispersion-induced polarization gating and ionization to generate isolated attosecond pulses from multicycle lasers,” Opt. Lett. 35(16), 2798–2800 (2010). [CrossRef] [PubMed]
25. Z. Chang, “Chirp of the single attosecond pulse generated by a polarization gating,” Phys. Rev. A 71(2), 023813 (2005). [CrossRef]
26. S. Tang and X. Chen, “Method to generate isolated attosecond pulses with many-cycle laser fields,” Phys. Rev. A 85(6), 063816 (2012). [CrossRef]
27. H. Du and B. Hu, “Broadband supercontinuum generation method combining mid-infrared chirped-pulse modulation and generalized polarization gating,” Opt. Express 18(25), 25958–25966 (2010). [CrossRef] [PubMed]
28. Y. Xiang, J. Miao, Y. Niu, S. Gong, R. Li, and Z. Xu, “Isolated sub-100 attosecond pulse generation driven by a multi-cycle chirped laser pulse and a polarization gating,” J. Phys. B 45(11), 115601 (2012). [CrossRef]
29. P. Li, X. Zhou, G. Wang, and Z. 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(5), 053825 (2009). [CrossRef]
30. Z. Chang, A. Rundquist, H. Wang, I. Christov, H. C. Kapteyn, and M. M. Murnane, “Temporal phase control of soft-x-ray harmonic emission,” Phys. Rev. A 58(1), R30–R33 (1998). [CrossRef]
31. P. Lan, P. Lu, Q. Li, F. Li, W. Hong, and Q. Zhang, “Macroscopic effects for quantum control of broadband isolated attosecond pulse generation with a two-color field,” Phys. Rev. A 79(4), 043413 (2009). [CrossRef]
32. M. Lewenstein, Ph. 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(3), 2117–2132 (1994). [CrossRef] [PubMed]
34. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
35. Z. Chang, “Single attosecond pulse and xuv supercontinuum in the high-order harmonic plateau,” Phys. Rev. A 70(4), 043802 (2004). [CrossRef]
37. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2005).