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We theoretically investigate the enhancement of a selected spectral range in the high-order harmonic generation driven by the multi-color laser fields. The results show that the central wavelength of the enhanced narrow-bandwidth spectrum can be effectively controlled by adjusting the laser intensity and the time delay between laser pulses of different colors, due to the modified electron trajectory in the shaped laser field. Our approach can offer intense, bandwidth controllable single attosecond pulses for attosecond pump-probe experiments.
©2012 Optical Society of America
With the rapid development of ultrafast laser technology, high-order harmonic generation (HHG) has attracted significant attention in strong field regime due to its important applications in high-contrast X-ray microscopy, molecular tomography, attosecond pulse synthesis and so on [1–17]. In most cases, for applications which require narrow-bandwidth coherent sources (e. g., biological imaging and nanolithography), filters or grating are needed to filter out a specific spectral range in the plateau of the high-order harmonic spectra [18–24], which, however, could introduce significant losses of the precious extreme ultraviolet (XUV) photons. Recently Sansone et al. have experimentally studied polarization gating technique to tune spectra in a narrow range [21]. The numerical simulation based on a single atom model of HHG is in good agreement with the experimental results and the propagation effect involved in HHG experiment will not affect the generation of the narrow-bandwidth XUV emission. In addition, in our previous work we have proposed a purely optical approach to directly obtain the desirable narrow-bandwidth HHG by synthesizing two-color laser pulses [25, 26]. By an orthogonal-polarized two-color laser field, a narrow-bandwidth XUV radiation via HHG was demonstrated and the underlying physics have been revealed by performing classical analysis [25]. Nevertheless, this approach only allows generation of narrow-bandwidth XUV spectrum with limited capacity of bandwidth tunability. On the other hand, we have also proposed a scheme for generation of bandwidth controllable XUV spectrum with enhanced intensity in parallel-polarized two-color laser fields by modified the electron trajectories [26]. However, in this work, a 3-fs, 800-nm laser pulse and a 64-fs, 2400-nm pulse were used, which are too short and unrealistic for real applications [25]. When longer laser pulses were used in the simulation, the enhanced narrow-bandwidth spectrum will disappear. In the current paper, we show that such difficulty can be overcome by constructing a specific light waveform with laser pulses in three different colors, i. g., of three different wavelengths.
In the present work, the synthesized field including three-color laser pulses will interact with argon (Ar) gas, which is numerically calculated by using the Lewenstein model based on the single-active-electron approximation [25, 27–29]. As expressed in the Ref [9], the synthesized laser field consisting of a 5-fs, 800-nm pulse, a 15-fs, 2000-nm pulse and a 25-fs, 2200-nm pulse has the following form:
Here, and (i = 1, 2, 3) denote the angular frequency and the pulse duration of three-color (i.e., 800 nm, 2000nm and 2200nm) laser fields, respectively. We chose the laser intensity and [9]. The parameter is the time delay between the 800-nm laser field and 2200-nm laser field. As described in Refs [25, 30], due to the high-order harmonic signals depending strongly on the quantum-mechanical expectation value of dipole acceleration, the intensity of the nth harmonic can be obtained by calculating the , where is the frequency of the generated XUV photon, and is the dipole matrix element for bound-free transitions [11–14].We first study the narrow-bandwidth XUV radiation appeared in the plateau region of the high-order harmonic spectrum by controlling the waveform of the driving field. As shown in Fig. 1(a) , with a driving field synthesized by 800-nm and 2000-nm pulses, high-order harmonic spectrum has a typical plateau where harmonics at different orders have almost the same intensity. Surprisingly, when a 2200-nm laser field at an optimized delay ( = −0.2fs) is superposed on the two-color field, HHG around 141 eV is enhanced by even one order of magnitude within an extremely narrow spectral range. As indicated in Fig. 1(b), the selected XUV radiation shows a smooth spectral profile and its bandwidth is estimated to be ~3 eV [full width at half maximum (FWHM)], which allows us to generate isolated attosecond pulses with a narrow spectral bandwidth. Due to the limit enhanced spectral bandwidth, this method is not so efficient for the isolated attosecond pulse generation. When the low-order harmonics are filtered, an isolated attosecond can be obtained by performing inverse Fourier transformation in the energy range from 100 eV to 300 eV. Just as shown in Fig. 2(a) , a main attosecond pulse with the 155-as pulse duration accompanied by a satellite attosecond pulse is directly obtained. After phase compensation, an isolated, ~34-as pulse is created, which is close to one atomic unit (i. e., ~24 as), as indicated in Fig. 2(b).
Fig. 1 (a) High-order harmonic spectra driven by two-color field and three-color field. (b) The enhanced harmonic spectrum in a linear scale.
Fig. 2 The temporal profiles of attosecond pulses generated by Fourier transformation of the high-order harmonics in the spectral range from 100 eV to 300 eV (a) without and (b) with the dispersion compensation. All parameters are the same as the red solid line of Fig. 1(a).
More interestingly, the photon energy (i.e., wavelength) of the selected enhanced XUV radiation critically depends on the time delay between the third field at 2200 nm and the other two fields. In Fig. 3 , one can clearly see that when the time delay is adjusted to 0.2 fs from −0.3 fs, the central photon energy of the narrow-bandwidth XUV radiation will gradually shift from ~120 eV to ~250 eV. Therefore, it will be easy to tune the central wavelength of the narrow-bandwidth XUV radiation in an extremely broad spectral range, which shows a significant advantage as compared to the report [26]. With the increase of the time delay, the selected XUV radiation will become weak as well as broad, and finally disappear. In addition, it is also worthy pointing out that this effect of spectral selection can be observed over a large delay scale. That is to say, the requirement of this method on the vibration of time delay or carrier-envelope phase jitter is as strict as Ref [25], which is in favor of experimental performance in the future.
Fig. 3 High-order harmonic spectra generated by the synthesized field for different time delays of the field at 2200 nm.
In order to obtain a clear insight for the generation of narrow-bandwidth, wavelength-tunable XUV radiation with the change of time delay, we perform the corresponding time-frequency analyses for the dipole moments at different time delays. Figures 4(a) -4(f) correspond to the results at = −0.3fs, −0.2fs, −0.1fs, 0fs, 0.1fs and 0.2fs, respectively. The time-frequency analyses clearly show that one short trajectory around the emission time of ~0.5T (T is optical period of 800-nm pulses) has been obviously modified and created additional quantum paths in the folded region of time-frequency curve. That is to say, the electron following these quantum paths will obtain almost the same kinetic energy in the laser field and therefore contribute to the same order harmonic emission when it is captured by the parent ion. Therefore, similar to the previous report [26], the constructive interference between these folded quantum paths leads to the selectively enhanced XUV radiation in a small spectral range. As indicated by red dashed lines in Figs. 4(a)-4(f), with the increase of the time delay, the photon energies in the folded region correspondingly move toward higher energy, which is in good agreement with the results in Fig. 3. Furthermore, with the increase of the time delay, the folded region of quantum paths covers a broader range of photon energy, leading to the increase of the bandwidth of the selected XUV radiation.
Fig. 4 Time-frequency analyses corresponding to high-order harmonic spectra in Fig. 3 at the time delays of (a) −0.3fs, (b) −0.2fs, (c) −0.1fs, (d) 0fs, (e) 0.1fs and (f) 0.2fs.
The above analysis can be further demonstrated by calculating the return kinetic energy and the ionization rate according to the classical three-step model [15] and the Ammosov-Delone-Krainov (ADK) theory [31]. Figure 5 shows that the return kinetic energy (red solid line), the ionization rate (gray filled curve) and the synthesized laser field (blue dashed line) as functions of the birth time of the electron at the time delay of −0.3 fs. In Fig. 5, the return kinetic energy of electron shows a small plateau around 120 eV, leading to the selective enhancement of HHG shown in Fig. 3. Further analysis demonstrates that the electron along all paths between path 1 and path 2 will obtain the same kinetic energy. That is to say, the creation of the folded quantum paths is a dominant mechanism of generating narrow-bandwidth XUV radiation. Moreover, with the driving field with such a specific waveform, the ionization rate of the short quantum path is much higher than that of the long quantum path. As a result, the short quantum path plays an important role on selectively enhanced HHG [32]. Therefore, another reason for this selective enhancement is because the ionization rate is maximum in the folded region of the kinetic energy (see gray filled curve of Fig. 5).
Fig. 5 The synthesized laser field (blue dashed line), return kinetic energy (red solid line) and the ionization rate (gray filled curve) as a function of birth time at the time delay fs.
We would like to point that the central photon energy and bandwidth of the selected enhanced XUV radiation can also be effectively controlled by adjusting the intensity of 2000-nm pulses. As shown in Fig. 6 , at , the photon energies of the selected enhanced harmonic radiation gradually move toward high-energy region with the increase of the laser intensity I2, which can be well understood by three-step model [15]. It also can be seen that the bandwidth has a slight increase with the increase of the laser intensity at 2000 nm. Hence, in our work, both bandwidth and central wavelength of the selected radiation can be more effectively controlled not only by adjusting the time delay but also by changing the laser intensity. Compared with the previous work [26], our method can provide not only more parameters to control the enhanced narrow-bandwidth spectrum but also the higher flexibility for experimental implement.
Fig. 6 High-order harmonic spectra generated by the synthesized field at different laser intensities I2 of 2000-nm pulses when the time delay is fixed at 0 fs.
Finally, we examine the feasibility of experimental demonstration of the generation of tunable, narrow-bandwidth XUV radiation using the above technique. For this purpose, we investigate sensitivity of the high-order harmonic spectra to the fluctuation of laser intensity at three wavelengths as well as to the jitter of the time delay between the three laser fields. Numerical simulation shows that the narrow-bandwidth XUV emission still can be observed when 15% intensity fluctuation and 0.1fs change of the time delay is introduced, allowing for experimental demonstration in the future. Moreover, although our simulation is based on a single atom model of the high-order harmonic generation, we believe that the effect of the spectral confinement still can be observed when the propagation effect is considered. Because the well phase-matched harmonics only can be generated for a fixed harmonic order, the bandwidth of the selected XUV radiation will become narrow during propagation. In principle, our technique also could take advantage of the phase matching if the phase-matching condition is optimized for the wavelength of the narrow-bandwidth XUV emission.
To conclude, we propose a new method for generating an enhanced narrow-bandwidth XUV spectrum in the plateau of the high-order harmonic spectrum. The enhancement of the selected spectral region results from the creation of a folded quantum path which leads to constructive interference for the XUV radiation in the selected waveband. The central wavelength of the enhanced XUV spectrum depends significantly on the time delay between the three-color laser pulses and the laser intensity of 2000-nm pulses. Interestingly the tunable narrowband enhancement of the XUV emission can be exploited for other applications such as XUV spectroscopy with a monochromator or seeding of free electron lasers (FEL). Moreover, it also may provide a potential way to generate an intense isolated attosecond light source. For example, an isolated 34-as laser pulse can be obtained with this scheme, given that the dispersion of the supercontinuum is completely compensated. Thanks to the rapid development of the ultrafast OPA technique, the specialized waveform as mentioned above is obtainable which facilitates experimental demonstration of this technique.
Acknowledgments
C. Zhang and J. Yao attribute equally to this work. 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, 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 and shanghai Postdoctoral Science Foundation funded project (2012M511145 and 12R21416700), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
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