We theoretically demonstrate the generation of extreme ultraviolet supercontinua in an orthogonally polarized two-color few-cycle laser field. We show that the ionized electrons can be driven back to their parent ion after traveling along curved trajectories in a plane perpendicular to the beam propagation direction, giving rise to a train of attosecond pulses at different polarization angles. A single isolated attosecond pulse can be obtained by blocking the low-order high harmonics, which contribute to the formation of the satellite pulses.
©2008 Optical Society of America
Waveform-controlled few-cycle light pulse has been playing a key role in attosecond science and technology, e.g., single isolated attosecond pulse generation and attosecond metrology [1–2]. Although currently the major approach to controlling the waveform of a few-cycle laser pulse is to control its carrier-envelope-phase (CEP), it has been realized that further advancing the waveform control technique by shaping the optical electric field and manipulating its polarization state within an optical cycle would enable the waveform synthesis, which, ultimately, would allow attosecond electron wavepacket control in all forms of matters . Nevertheless, at present, it is still a great technical challenge to create an arbitrary optical waveform, which requires coherent superposition of light waves spanning several octaves ; whereas for some specific applications, desirable optical waveform can still be obtained by combining a few-cycle wave with a multi-cycle wave at different wavelengths. The use of the two-color laser field has shown great potential in generation of single isolated attosecond pulse [4–6], selection of quantum path , and control of emission time of attosecond pulses . Furthermore, it has been known for a long time that laser polarization is another important parameter for steering the electron motion. By rapidly sweeping the laser polarization state within one optical cycle, the shortest attosecond pulse (~130as) reported to date has been demonstrated using polarization gating technique . In addition to the polarization gating technique using two temporally delayed counter-circularly-polarized few-cycle laser pulses, other approaches to rapidly changing the polarization state of an optical field utilize orthogonally polarized two-color laser field [10–12], two-color coplanar field mixing [13–14], combination of coplanar circular and static fields , combination of low-frequency linearly polarized field and high-frequency elliptically polarized field , and so on, by which either single attosecond pulses or attosecond pulse trains can be generated. Particularly, it has been shown recently that by superimposing two orthogonally polarized few-cycle laser pulses, one fundamental and the other its second harmonic, one would be able to either generate an intense single isolated attosecond pulse, or perform a single-shot molecular orbital tomography . However, in this approach, the single isolated attosecond pulse is obtained by polarization-selective reflection from a metallic multilayer mirror . In general, such an extreme ultraviolet (XUV) polarizing mirror is strongly wavelength-dependent, thus it can only be used for a limited bandwidth . In addition, the use of the metallic multilayer mirror could cause extra attenuation for the generated attosecond pulse. In this paper, we aim at constructing an orthogonally polarized two-color few-cycle optical field, in which a single isolated attosecond pulse can simply be obtained by blocking the low-order high harmonics using a piece of suitable thin metal foil. Moreover, we present the polarization characteristics of the generated attosecond pulses.
2. Classical trajectories of electrons in light field
To begin with, we consider an orthogonal two-color laser field consisting of a 9fs/800nm pulse linearly polarized in X direction and a 9fs/1300nm pulse in Y direction. The two pulses are equally intense and their peak intensities are chosen as 6×1014W/cm2. The model atom in the simulation is helium (He). The method of the simulation for HHG has detailed description in Ref. , which is based on the single-active atom approximation . The expression of the synthesized field can be written as
Here E is the peak amplitude of the electric fields of the two laser pulses at different wavelengths; ω 1 and ω 2 are the frequencies of the 800nm and the 1300nm pulses, respectively; and τ=9 fs is the temporal duration (FWHM) of the two pulses. The electric fields of the two pulses are shown in Fig. 1(a), and the evolution of the electric field in three-dimensional (3D) space is shown in Fig. 1(b).
According to the well-known three-step model , in order for HHG to occur, an electron initially in a bound state must first be tunnel ionized, then the electron travels freely in the oscillating light field until being driven back to its parent ion, and finally it recombines with the parent ion through the release of an energetic photon. Therefore, in order to enable the HHG in the orthogonal two-color few-cycle laser field, there must be at least one electron trajectory that could allow the electron to return to its parent ion with sufficient kinetic energy. In Figs. 2(a)–2(c), we present three classical electron trajectories in the X-Y plane to show that the electrons ionized at three times t 1, t 2, and t 3, which were indicated in Fig. 1(a), do have chances to revisit their parent ion after certain amounts of travel times. It is noteworthy that in the orthogonal two-color field, the returning electron usually will not come back to its exact initial position, as can be seen in Figs. 2(a)–2(c). As an example, the closest distance between the returning electron and its parent ion in Fig. 2(c) is approximately 60 atomic units (~3nm). However, in these cases, because of the spreading of the electron wavepacket, the returning electron still can have a probability to recombine with its parent ion. On the other hand, as we can also find in Fig. 2(d), the electron tunnel ionized at the time t 4 will never come back to its parent ion. For comparison, we also present the classical trajectory of the electron ionized in the single-color 9fs/1300nm laser field at the time t 4, as shown in Fig. 2(e). In this case, the electron can be driven back to the parent ion in the linearly polarized 1300nm laser field to contribute to HHG. Therefore, we conclude that the superimposition of the orthogonally polarized 9fs/800nm pulse upon the 9fs/1300nm pulse can destroy the undesirable electron trajectories by taking the advantage of the rapid variation of the polarization state within an optical cycle, facilitating generation of single isolated attosecond pulses.
3. Quantum trajectory analyses
To demonstrate the HHG in the orthogonally polarized two-color laser field, we calculate both X- and Y- components of the harmonic spectrum, as shown in Fig. 3(a). In order to give a clear view, we purposefully shift the Y-component of the HHG spectrum to lower location. The X-component of the HHG spectrum [the upper curve in Fig. 3(a)] shows a smooth XUV supercontinuum covering almost the entire plateau region. In contrast, in the lower curve in Fig. 3(a), significant spectral modulation appears in the low energy range, indicating the existence of multiple electron recolliding events capable of contributing to the generation of low-order high harmonics polarized in the Y direction. The spectral modulation originates from the interference between the low-order harmonics emitted at different times. Looking back at Figs. 2(a) and 2(c), we can see that the electrons tunnel ionized at the times t 1 and t 3 are moving in a direction nearly parallel to the Y direction when they are approaching their parent ion upon return. Therefore, for the electron tunnel ionized near these two times, they will produce additional undesirable attosecond pulses mainly polarized in the Y direction.
Figure 3(c) compares the HHG efficiency in the two-color field with that in the single-color linearly polarized light fields. It can be seen that the HHG efficiency in the orthogonal two-color field is lower than that in the single 800nm light field, but is still higher than that in the single-color 1300nm light field. It should be noted that in Fig. 3(c), the intensity spectrum of harmonics generated in the two-color field is obtained by summing up the intensities of both the X- and Y-polarized harmonics.
A deeper insight can be obtained by performing time frequency analyses for the X- and Y- components of the high-order harmonic spectra, as shown in Figs. 4(a) and 4(b), respectively. The X-axis and Y-axis of the time-frequency diagram show the photon energy and the time at which the photon is emitted, respectively. It can be seen in Fig. 4(a) that only one electron trajectory exists, which is contributed by the electron ionized at the time t 2; whereas in Fig. 4(b), two additional trajectories contributed by the electrons ionized at the times t 1 and t 3 also appear. This difference is caused by the non-symmetric electron motion in the X and Y directions, supported by the electron trajectories shown in Fig. 2. In addition, a similar time-frequency structure is observed in Fig. 4(a) and 4(b) for the electron trajectories contributed by the electron ionized near the time t 2, indicating that this electron recollides with its parent ion at an angle near 45° with respect to either the X or the Y axis. All these results are consistent with the electron trajectories shown in Fig. 2.
4. Single attosecond pulse generation
In principle, it is possible to select the XUV supercontinuum polarized in either the X or Y direction using a piece of metallic multilayer mirror , given that the wavelength regime of the XUV supercontinuum is suitably designed by properly choosing a gas atom and the driving laser intensities. However, in this theoretical simulation, we take a simpler and more straightforward approach to obtain the single isolated attosecond pulse by only blocking the low-order harmonics. Compared in Figs. 5(a), 5(b), and 5(c) are the temporal profiles of the attosecond pulses generated by performing inverse Fourier transformations of the XUV supercontinua in different spectral regions. In order to reveal the polarization characteristics of the generated attosecond pulses, we present the electric fields of the attosecond pulses in 3D space, with their corresponding temporal envelopes (here the temporal envelope is defined as the summation of the intensities of the X- and Y- components) presented in Figs. 5(d)–5(f). First, the selected spectral range is from 25eV to 325eV, resulting in an attosecond pulse train as shown in Figs. 5(a) and 5(d). In this case, a main attosecond pulse is accompanied by several satellite attosecond pulses, whereas the polarization states of the attosecond pulses in the pulse train are different. Next, we block the low-order harmonics and select a spectral range from 130eV to 325eV, and the corresponding attosecond pulse is shown in Figs. 5(b) and 5(e). As we expect, a single isolated attosecond pulse with a pulse duration of ~147as is produced. Theoretically, the XUV supercontinuum spanning from 130eV to 325eV can support a single isolated attosecond pulse with a pulse duration of ~20as, provided that dispersion compensation over such a broad range is accomplished. However, such dispersion compensation is difficult to achieve in experiment. Therefore, we take a simple approach in order to further shorten the attosecond pulse duration. As can be found in Fig. 3(b), the phase profiles over the X- and the Y-components of the XUV supercontinua both appear relatively flat in the spectral region of ~200eV-260eV. The selection of this spectral region for synthesizing the single isolated attosecond pulse leads to a further reduction of the pulse duration to ~115as, as indicated by Figs. 5(c) and 5(f).
Comparing Figs. 5(a), 5(b) and 5(c) with Fig. 2(b), we find that the polarization direction of the main pulse is almost parallel to the moving direction of the recolliding electron ionized at the time t 2 indicated in Fig. 1(a). It also can be seen in Figs. 5(b) and 5(c) that the two satellite pulses before and after the main pulse are mainly polarized in the Y axis direction, which are almost parallel to the moving directions of the recolliding electrons ionized at t 1 and t 3 in Fig. 1(a), as indicated by Figs. 2(a) and 2(c). It is noteworthy that the polarization direction of high-order harmonic emission is usually not the same as the direction of the electron motion when there is a strong external electric field existing at the time of the electron recolliding with the parent ion [20–22]. However, in our simulation of attosecond pulse generation in the orthogonal two-color field, we found that the electrons will mainly be ionized at the peak of the light field where the tunnel ionization rate is the highest, and then revisit their parent ions when the light field approaches zero. Similar physical picture has already been presented in the well-known three-step classical model of HHG . Due to this reason, we can safely ignore the influence of the external field on the polarization state of the attosecond pulse, which implies that the polarization direction of attosecond pulse will be mainly determined by the direction of electron motion. In addition, we have systematically investigated how the single attosecond XUV pulses generated in the orthogonal two-color field will be influenced by the relative phase between the driving pulses. It is found that the best phase delay for obtaining the broadest XUV supercontinuum is zero, as we have chosen in our simulation.
To summarize, we theoretically investigate the HHG in orthogonally polarized two-color few-cycle laser field, by which single isolated attosecond pulses can be obtained. The 9fs/800nm pulse used in our simulation can easily be generated by hollow fiber compression technique ; whereas the 9fs/1300nm pulse used in our simulation has a temporal duration about two optical cycles, which could be obtained from an optical parametric amplifier (OPA) . It is inevitable that there are slight fluctuations of the laser parameters in the experiments. We find in our simulation that the HHG spectrum in the orthogonal two-color field is relatively insensitive to either the fluctuation of the laser intensity or of the pulse duration, even if these two parameters are increased or reduced by 10%. However, the HHG spectrum is much more sensitive to the phase delay between the two pulses. Our simulation shows that the phase delay between the two pulses should be controlled within ±0.05π of the 800nm wave, which is corresponding to an optical delay of ±20nm. Such accuracy is attainable using a piezoelectric translator . Therefore, the scheme proposed here appears feasible for an experimental demonstration in the near future. Furthermore, since the orthogonal two-color field can effectively control the electron wavepacket in the 2D space, we envisage that this unique property can trigger a series of interesting phenomena in molecular HHG process, where the HHG efficiency can be critically dependent on the angle between the laser polarization direction and the molecular axis . In another word, the molecular HHG can be sensitive to the angle between the moving direction of the recolliding electron and the molecular axis, providing an additional way to eliminate undesirable satellite attosecond pulses. Thus, the rapid variation of the moving direction of the recolliding electron in the orthogonal two-color laser field can be employed for further confining the temporal window of the high-order harmonic emission in a molecular medium, facilitating generation of single isolated attosecond pulses.
Finally, we would like to point out that in comparison with the other schemes we proposed previously for generating single attosecond XUV pulses that use only linearly polarized pulses [6–7], the scheme presented here shows several important advantages. First, the total ionization ratio of He atom in the orthogonal two-color field is calculated to be less than 6%, which is much lower than that in the previous cases and will significantly reduce the influence of the propagation effect in generation of single attosecond pulse; second, the temporal duration of the 800nm pulse is 9fs, which is more than three optical cycles and consequently, should be much easier to obtain compared to the 6fs few-cycle pulse used in our former proposals [6–7]; and third, as we have mentioned above, the combination of the rapid polarization variation in the orthogonal two-color driving field and the polarization sensitivity in molecular HHG offers a unique opportunity for simplification of the single attosecond pulse generation process and for further shortening of the attosecond pulse duration. The generation of attosecond pulses from molecules in polarized light fields is to be explored in the future work.
The authors would like to acknowledge Dr. Zhinan Zeng for providing initial code for HHG simulation. This work was supported by the National Basic Research Program of China (Grant No. 2006CB806000), Shanghai Commission of Science and Technology (Grant No. 07JC14055), and National Natural Science Foundation of China (Grant No. 10523003).
References and links
1. A. Baltuška, Udem Th., M. Uiberacker, M. Hentschel, E. Goulielmakis, Gohle Ch., R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003). [CrossRef] [PubMed]
2. R. Kienberger and F. Krausz, “Attosecond metrology comes of age,” Phys. Scrip. T110, 32–38 (2004). [CrossRef]
3. P. B. Corkum and F. Krausz, “Attosecond science,” Nature Phys. 3, 381–387 (2007). [CrossRef]
4. T. Pfeifer, L. Gallmann, M. J. Abel, P. M. Nagel, D. M. Neumark, and S. R. Leone, “Heterodyne mixing of laser fields for temporal gating of high-order harmonic generation,” Phys. Rev. Lett. 97, 163901 (2006). [CrossRef] [PubMed]
5. Y. Oishi, M. Kaku, A Suda, F. Kannari, and K. Midorikawa, “Generation of extreme ultraviolet continuum radiation driven by a sub-10-fs two-color field,” Opt. Express 14, 7230–7237 (2006). [CrossRef] [PubMed]
7. 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, 043830 (2007). [CrossRef]
8. 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]
9. 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 pulses,” Science 314, 443–446 (2006). [CrossRef] [PubMed]
10. M. Kitzler, X. Xie, A. Scrinzi, and A. Baltuska, “Optical attosecond mapping by polarization selective detection,” Phys. Rev. A 76, 011801(R) (2007). [CrossRef]
11. C. M. Kim, I.J. Kim, and C. H. Nam, “Generation of a strong attosecond pulse train with an orthogonally polarized two-color laser field,” Phys. Rev. A 72, 033817 (2005). [CrossRef]
12. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. Jaeyun Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94, 243901 (2005). [CrossRef]
13. D. B. Milošević, W. Becker, and R. Kopold, “Generation of circularly polarized high-order harmonics by two-color coplanar field mixing,” Phys. Rev. A 61, 063403 (2000). [CrossRef]
14. D. B. Milošević and W. Becker, “Attosecond pulse trains with unusual nonlinear polarization,” Phys. Rev. A62, 011403 (R) (2000). [CrossRef]
15. S. Odžak and D. B. Milošević, “Attosecond pulse generation by a coplanar circular and static field combination,” Phys. Lett. A 355, 368–372 (2006). [CrossRef]
16. V. D. Taranukhin, “Attosecond pulse generation by a two-color field,” J. Opt. Soc. Am. B 21, 419–424 (2004). [CrossRef]
17. D. Schulze, M. Dörr, G. Sommerer, J. Ludwig, P. V. Nickles, T. Schlegel, W. Sandner, M. Drescher, U. Kleineberg, and U. Heinzmann, “Polarization of the 61st harmonic from 1053-nm laser radiation in neon,” Phys. Rev. A 57, 3003–3007 (1998). [CrossRef]
18. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000). [CrossRef]
20. Antoine Ph., B. Carré, Anne L’Huillier, and M. Lewenstein, “Polarization of high-order harmonics,” Phys. Rev. A 55, 1314–1324 (1997). [CrossRef]
21. B Borca, A. V. Flegel, M. V. Frolov, N. L. Manakov, Dejan B. Milošević, and A. F. Starace, “Static-electricfield-induced polarization effects in harmonic generation”, Phys. Rev. Lett. 85, 732–735 (2000). [CrossRef] [PubMed]
22. S. Bivona, G. Bonanno, R. Burlon, and C. Leone, “Polarization and angular distribution of the radiation emitted in laser-assisted recombination,” Phys. Rev. A 76, 031402(R) (2007). [CrossRef]
23. A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett. 86, 111116 (2005). [CrossRef]
24. C. Vozzi, F. Calegari, E. Benedetti, S. Gasilov, G. Sansone, G. Cerullo, M. Nisoli, S. De Silvestri, and S. Stagira, “Millijoule-level phase-stabilized few-optical-cycle infrared parametric source,” Opt. Lett. 32, 2957–2959 (2007). [CrossRef] [PubMed]
25. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft Xrays,” Science 302, 1540–1543 (2003). [CrossRef] [PubMed]
26. M. Lein “Molecular imaging using recolliding electrons,” J. Phys. B: At. Mol. Opt. Phys. 40, R135–R173 (2007). [CrossRef]