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Abstract
Tunable attosecond pulses are necessary for various attosecond resolved spectroscopic applications, which can potentially be obtained through the tuning of high harmonic generation. Here we show theoretically, using the time-dependent Schrödinger equation and strong field approximation, a continuously tunable spectral shift of high-order harmonics by exploiting the interaction of two delayed identical infrared (IR) pulses within the single-atom response. The tuning spans more than twice the driving frequency (∼2ω) range, for several near-cutoff harmonics, with respect to only one control parameter: the change in delay between the two IR pulses. We show that two distinct mechanisms contribute to the spectral shift of the harmonic spectra. The dominant part of the spectral shift of the harmonics is due to the modulation of the central frequency of the composite IR-IR pulse with respect to delay. The second contribution comes from the non-adiabatic phase-shift of the recolliding electron wavepacket due to the change in amplitude of the subcycle electric field within the double pulse envelope. For optical few-cycle pulses this scheme can produce tunable attosecond pulse trains (APT), and in the single-cycle regime the same can be used for tuning isolated attosecond pulses (IAP). We quantify the dependence of tuning range and tuning rate on the laser pulse duration. We envision that the proposed scheme can be easily implemented with compact in-line setups for generating frequency tunable APT/IAP.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Finely tunable ultrashort XUV pulses are necessary for various applications, such as in ultrafast photoelectron spectroscopy [1,2], time- and angle-resolved photoemission spectroscopy [3,4], high resolution transient absorption spectroscopy [5] etc. High intensity tunable XUV and Xray pulses of attosecond temporal width can be obtained readily from large scale synchrotron sources [6], however for a table-top experimental set up one may rely on high harmonic generation (HHG) based XUV sources [7].
HHG based sources are tuned by changing the fundamental frequency of the femtosecond laser system, or by selecting the desired portion of the spectrum by using suitable filter [8]. These techniques may suffer from the constraint of limited tunability or loss of coherence. One efficient way of obtaining a tunable table-top XUV and soft-Xray source could be through varying the spectral shift of harmonics in the HHG process.
Previous studies have shown spectral shifts in HHG due to different mechanisms. For molecular systems spectral red-shift has been demonstrated which originated from the reduction in ionization energy between recombination time and ionization time [9], a non-adiabatic red-shift mechanism has been theoretically demonstrated in [10] where enhanced excitation of localized long lifetime excited states shifts the harmonic generation in the falling edge of the short laser pulses. For solids red-shift in the higher order plateau has been demonstrated theoretically which comes from the indirect step-by-step excitation of higher conduction bands [11].
For atomic systems spectral tuning of high-order harmonics has been achieved experimentally and theoretically through various mechanisms such as, the control of laser energy and chirp [12], using two-color laser pulses [13–17], nonadiabatic blueshift of harmonics due to the generation in the rising edge of the pulse [18], due to ionization of the medium [19], and using the carrier-envelope phase (CEP) [20]. Attosecond twin-pulse control and generation of fractional higher order harmonics with energy tuning have also been demonstrated by generalized kinetic heterodyne mixing [21–23]. Most of these methods are complex to implement and offer limited tuning range and only discrete tunability.
As HHG in atoms is found to be the most promising table-top candidate in terms of intensity and coherence, it is desirable to achieve the maximum possible tuning of harmonics through the temporal shaping of the driving pulse. Schemes of spectral tuning of harmonics by using two delayed IR pulses have been proposed and experimentally verified recently [24–30]. Tuning of cutoff order occurs due to delay-controlled variation in the amplitude of the composite IR-IR pulses [24–26,28]. Furthermore, tuning of spectral shift and width of harmonics have been shown to arise from a similar tuning of the composite pulses [27,29,30]. However, the full potential of IR-IR pulse for generating widely tunable harmonics and its capability to generate tunable isolated attosecond pulses have not yet been explored.
Here we demonstrate the tuning of harmonic spectra over $2\omega$ in the cutoff harmonics theoretically, using two delayed replica IR pulses interacting with an atomic system. This can be utilized to readily produce a tunable XUV source in a very simple manner as demonstrated in [28]. Employing the time-dependent Schrödinger equation and the strong field approximation we compute the HHG spectra with respect to the delay between the two IR pulses which depicts the prominent spectral shift. Two distinct mechanisms contribute to the spectral shift of the harmonics: the frequency modulation of the driving IR-IR pulse with respect to delay and the non-adiabatic response of the electron to the shape of the delayed IR-IR laser pulse envelope. For optical few-cycle pulses this scheme is found to give tuning of attosecond pulse trains (APT), and in the single-cycle regime the same can be used for tuning isolated attosecond pulses (IAP). We also quantify the dependence of tuning range and tuning rate on the duration of the IR pulse. The paper in organized as follows: after briefly introducing our model in section 2 we present the results and the discussion in section 3 which is followed by a concluding section.
2. Theoretical model
We solve the time-dependent Schrödinger equation (TDSE) in reduced dimensionality (1d)
As the combined delayed IR-IR pulse is linearly polarized the 1d TDSE is expected to reveal the mechanism in the studied parameter regime. In addition, we have also verified our results with the calculation employing the strong field approximation due to Lewenstein for the harmonic spectra and the spectra-map. To observe the harmonic production with respect to time we have used the short time Fourier transformation, with a Gabor window. Propagation effects (plasma generation, self focusing, absorption and dispersion) are not considered here. The presented single atom response calculations could be compared with experiments with thin gas jet and loose focusing geometry [34,35].
The electric field of the combined delayed IR-IR pulse is defined as
where, the temporal envelope of each IR pulse is $f(t)=\cos ^2\left (\frac {t}{T_{pulse}}\right )$ and their amplitudes are $E_{01}$ and $E_{02}$. The CEP of both IR pulses is $\phi _{ce}=\pi /2$ (sin pulse) throughout this work.3. Results and discussions
Figure 1 shows the spectral intensity of the harmonics vs the delay ($\tau$) between the two IR replica pulses (hereafter this will be called as spectra-map) for 5 optical cycles pulse (the number of complete optical oscillation is 5 in each replica IR pulse) with $E_{01}=E_{02}=0.06$ au. Figure 1(a) shows the spectra-map near the cutoff regime computed using the TDSE with the Gaussian potential and with the soft-core Coulomb potential as shown in Fig. 1(e). To independently verify the observed systematic features in the spectra-map we have performed a calculation using strong-field approximation (SFA) [36,37] as presented in Fig. 1(b). The overall agreement between the two methods is very good in this regime of harmonic order and other part of the spectra-map (not shown here) except for the fact that the SFA slightly overestimates the cut-off and hence shows a larger width of the “harmonic band”. Note that, for further analysis we take recourse of TDSE simulations for both soft-core and Gaussian potentials. In Fig. 2 we have plotted the harmonic spectra for different $\tau$ which falls in the range of delay as shown in the spectra-map (Fig. 1(a)) in the harmonic order region: 51-71 of the fundamental angular frequency ($\omega =0.057$ au).
Fig. 1. The harmonic spectra-map for a delay range $\tau = 0.8 - 1.2 T_{oc}$. (a) HHG for the Gaussian potential, and (b) SFA calculation. (c) Peak positions of the harmonic bands (solid line) vs delay, obtained from (a). The dashed lines represent delay-dependent spectral position of $N^{th}$ harmonic with respect to the central frequency of the driver. (d) The difference between the calculated maxima positions and that of the $N^{th}$ multiple of the central frequency of IR-IR driver. (e) and (f) are same as (a) and (c) for soft-core Coulomb potential.
Fig. 2. The harmonic spectra (a) for different delays $\tau =$ 0.9 $T_{oc}$, 1.0 $T_{oc}$ and 1.1 $T_{oc}$ for a range N=51–71 orders, (b) $\tau =$ 0.0, 0.1 $T_{oc}$, 0.2 $T_{oc}$ and 0.3 $T_{oc}$ for a range N=61–81 harmonic order, (c) $\tau =$ 1.7 $T_{oc}$, 1.8 $T_{oc}$, 1.9 $T_{oc}$, 2.0 $T_{oc}$ and 2.1 $T_{oc}$ over N=31–45 harmonic order.
For each harmonic peak shown in Fig. 1(a) we have obtained the position of the maxima ($\sim$ centroid) and plotted it with respect to the delay as shown in Fig. 1(c) (in solid lines). One can see from Fig. 1(a)-(c): (1) clear harmonic comb structures separated by two harmonic orders, and (2) a widely tunable spectral shift of more than two harmonic orders for a change in delay of $0.2~T_{oc}$ as we vary the delay $\tau =0.9~T_{oc}$ to $\tau = 1.1~T_{oc}$. Thus the harmonic comb is spanning the complete spectral range ($2\omega$) in this cut-off region. Both the central frequency and bandwidths of the IR-IR driver pulse manifest in a rather complicated manner in the HHG spectra. In this article we will mostly focus on understanding the origin of the wide tunability of the spectral shift. The shift of the harmonics with respect to the delay can be understood in terms of two important contributing factors: (a) adiabatic contribution, arising from a variation in the central frequency of the composite IR-IR pulse, and (b) non-adiabatic contribution, related to the phase shift of the recolliding electron wavepacket due to the subcycle change in amplitude of the electric field. In the following we will discuss these two contributions in detail.
3.1 Adiabatic contribution
To quantify the variation in the central frequency of the composite driving pulse, the spectrum of the IR-IR pulse for a few sample delays are shown in Fig. 3(c). In Fig. 3(a) we show the spectrum of IR-IR pulses for a delay range of $\pm 0.2~T_{oc}$ around the delay position $\tau =1T_{oc}$ in color map. The peak frequency [frequency at which the intensity is maximum] and central frequency [defined as $\omega _{IR-IR}(\tau )=\int \omega |E_{IR-IR}^{\tau }(\omega )|^2d\omega /\int |E_{IR-IR}^{\tau }(\omega )|^2d\omega$] (see cross and the dashed lines in Fig. 3(a),(d),(e) on top of the color maps) are plotted on the color map. Note that, the central and peak frequency are nearly identical in the chosen parameter range, we have used the peak frequency for further analysis in this paper. For $\tau =1T_{oc}$ the central frequency is the same as that of the single IR pulse ($\omega =0.057~$au or $\lambda =800~$nm). The continuous shift of the central frequency of the IR-IR pulse versus the the IR-IR delay near $\tau =1T_{oc}$ can be clearly seen here. For $\tau < T_{oc}$ the central frequency shifts towards the blue side (higher than $\omega =0.057~au$) and for $\tau > T_{oc}$ the central frequency shifts towards red side (lower than $\omega =0.057~au$). Intuitively, the time delay between the two IR pulses corresponds to a phase-shift between their spectral components in the Fourier-domain. The interference effect in the spectral domain modifies the resulting spectrum of the composite pulse leading to a shift in the central frequency [27]. This frequency shift of the IR-IR composite driving pulse with respect to the delay clearly explains the main contribution to the shift of the harmonic comb near-cutoff. In Fig. 1(c) we have plotted the peak of the harmonic bands as marked in Fig. 1(a) along with the harmonic frequency position of the driver IR-IR pulse with dashed lines. Quantitatively, the shift in frequency of the driver from $\tau =0.9T_{oc}$ to $\tau =1.1T_{oc}$ is $\delta \omega =\sim 0.0025~au$ which translates to a much higher shift in the harmonic orders. For example, for $n=58$ order of the fundamental IR ($\omega =0.057 ~au$) the shift is $n*\delta \omega \sim 2.5$, as observed in Fig. 1(c). For the soft-core Coulomb potential the maxima positions of the harmonic bands along with the $n$ times the central frequency of the driver IR-IR pulse are shown in Fig. 1(f) in solid and dashed lines, respectively. The results for the Gaussian and soft-Coulomb potentials are qualitatively same.
Fig. 3. (a) Frequency spectra vs delay of the driving IR-IR pulse near the first maxima (i.e., around $\tau =1 T_{oc}$). (b) Electric-field profile of the IR-IR pulse for different delays $0.8T_{oc}, 0.9T_{oc}, 1.0T_{oc}, 1.1T_{oc}$ and $1.2T_{oc}$, and (c) corresponding spectra. (d),(e) Same as (a) near $\tau =0$ and around $\tau =2 T_{oc}$. Central (peak) frequency of the delayed pulses are marked as dashed line (cross) on the color maps.
A similar mechanism can be observed for the near zero delay position i.e., for $\tau =0$ and for $\tau =2T_{oc}$ as shown in Fig. 2(b),(c), respectively. In Fig. 3(d),(e) we have presented the spectrum of IR-IR pulses for a delay range of $0.2~oc$ near the central delay position $\tau =0$ and for $\tau =2T_{oc}$, respectively, along with the peak and central frequencies. Clearly for $\tau =0$ the delay dependent shift of the driving IR-IR pulse is smaller, so as the shift of the harmonics (see Fig. 2(b)). On the other hand the central frequency shift of the driving IR-IR pulse is higher with respect to the delay near $\tau =2T_{oc}$ as shown in Fig. 3(e) which is reflected as pronounced shifts of the generated harmonics (see Fig. 2(c)).
Thus we see that the modulation of the central frequency of the driving IR-IR pulse partially explains the total frequency tuning of the generated high harmonics. In Fig. 1(d) we have plotted the difference of the peak/central frequencies of the generated harmonic comb with respect to the harmonics of the driver along the harmonic bands (for example, as marked by 1, 2 along the representative dashed lines in Fig. 1(a),(b),(e)). It is clearly evident from this graph that there is a second important contributing factor to the frequency tuning of the generated harmonics. Specifically (see in Fig. 4(d)-(f)) (i) there is a red-shift of the harmonics throughout the generating range near $1T_{oc}$, (ii) the red shift increases with increasing the offset ($\Delta \tau =|\tau -T_{oc}|$) delay from $\tau =1T_{oc}$.
Fig. 4. (a)–(c) The peak positions of the harmonic bands in the adiabatic driving limit (see text for detail). The corresponding plots for the original driving are shown in (d)–(f).
3.2 Non-adiabatic contribution
To understand the origin of the non-adiabatic shift of the harmonic spectra, we have computed the harmonic spectra with respect to different delay in the adiabatic limit. To this end we have designed the driving pulse with a trapezoidal envelope of 12oc length with 1oc rising time from zero to maximum amplitude, 10oc constant amplitude and 1oc falling time back to zero. We take the frequency of the driver as $\omega _{driver}(\tau )=\omega _{IR-IR}^{center}(\tau )$ and the intensity of the driver as $I_{driver}(\tau )=I_{IR-IR}^{center}(\tau )$. Therefore, now we drive the atomic system with the modified frequency, i.e., the central frequency of the composite IR-IR driver at given $\tau$ value, along with the reduced contribution of the non-adiabatic effect due to the change in intensity over the pulse length. We have extracted the central (peak) position of the harmonic bands (as explained earlier) over the range of delays and plotted along with the $n^{th}$ order of the driving central frequency ($\omega _{IR-IR}^{center}(\tau )$) in Fig. 4. For a comparison we have plotted the same for the harmonics generated with the original IR-IR driving. Clearly the shift of the harmonic band peaks (centers) other than that due to the shift of the driving pulse’s frequency is reduced near zero for $0$, $1T_{oc}$ and $2T_{oc}$ ((a)-(c) of Fig. 4) in the adiabatic limit. This explains that the red (blue in some cases) shifts are due to the non-adiabatic dynamics of HHG in the short pulse limit.
Further, the temporal origin of the red-shift can be identified from the time-frequency analysis of the harmonic generation. The time-frequency response of of the near-cutoff harmonics generated at different delays around $1T_{oc}$ are presented in Fig. 5. The pulse centers vary from $t= 2.9-3.1 ~T_{oc}$ in this delay range from $0.8-1.2 ~T_{oc}$ (see the vertical dashed line at $3.0T_{oc}$) From the time-frequency response (TFR) it is clear that the harmonics are dominantly generated in the “falling edge” of the IR-IR pulse. In this scenario, the major part of the contributing return electron trajectories experience a drop in the electric field within its return (with respect to the conventional flat pulse) time, $\sim T/2$. This leads to an additional phase shift of the return electron amplitude which in turn leads to the shift of the harmonic peaks towards the red side [10,18].
Fig. 5. Time-frequency responses of the harmonic generation for different delays are shown in color map (log scale). Color scales are given on the right-side of each map. Corresponding driving electric fields are shown in dashed lines in each plot (electric field scales are within $\pm 0.15$ au in y-axis).
3.3 Tuning of IAP with a near single-cycle driver
We further show that the proposed scheme of double IR pulses can produce tunable isolated attosecond pulses in the regime of single-cycle carriers. To this end we have presented the spectra-map in Fig. 6(a), where each IR pulse is of 1.5 optical cycles long and the carrier-envelope phase is kept same as before (i.e., $\pi /2$).
Fig. 6. (a) Spectra-map for IR-IR driving obtained through TDSE calculation with Gaussian potential. Each IR pulse is 1.5 optical cycles long with other parameters kept same as in Eq. (1). In the inset of (a) IR-IR driving pulses for different delays are presented. (b) The XUV pulse profiles for the delay range $0-0.35$ optical cycles are presented as color map (log scale). Color scales are given on the right side of each map.
The spectral tunability of the cutoff regime of the harmonic spectra-map is clearly visible (denoted as “abcdef” for guiding the eye) which ranges from a continuum of $\sim 15$ orders near zero-delay $\tau =0$ to $\sim 5$ orders near $\tau =0.35T_{oc}$ spanning over 20 harmonic orders. We have computed the corresponding temporal profiles of the pulses generated in this spectral range. In Fig. 6(b) the color map of these temporal profiles in the delay range is shown. The temporal width of these IAP changes from $\sim 0.005T_{oc} (13~as)$ to $\sim 0.02T_{oc} (53~as)$ in this delay range.
3.4 Variation of tunability with respect to pulse length
A quantification of the tuning range with respect to pulse length is very important. To this end we have computed the variation of the tuning range vs delay for several pulse durations from 40 cycle to 3 optical cycle. We have shown the spectral peak positions of the harmonic band corresponding to $39^{th}$ harmonic order near $1T_{oc}$ delay between the two IR pulses using both soft-core Coulomb and Gaussian potentials in Fig. 7(a). In order to make a comparison of the tuning range of near-cutoff harmonics, we have kept the cut-off harmonic nearly identical, for which the pulse amplitude for the 3 cycle pulse was increased appropriately. To quantify, we have computed the tuning range (defined as the change in harmonic peak position for a delay range, $\Delta \tau =0.3T_{oc}$ about delay position $1T_{oc}$, marked as $\Delta \Omega$ in Fig. 7(a) for a five-cycle pulse) and the tuning rate (defined as $\frac {1}{\Omega _{mean}}\Delta \Omega /\Delta \tau$, where $\Omega _{mean}$ is the mean position of the maxima of the harmonic band in the range as shown in Fig. 7(a)) as shown in Fig. 7(b),(c), respectively. Comparison of tuning range and tuning rate of a representative 39 order harmonics shows that tuning range increases as one decreases the pulse length. It is also clear from the plots that the tuning range and tuning rate are nearly the same for both Gaussian and soft-core Coulomb potentials, although the detailed structure of HHG may vary for the two potentials [38]. The range of tuning varies from $\sim 0.04$ harmonic order for long pulses ($\sim 40$ optical cycles long) to $\sim 7$ harmonic order near 3-cycle pulse with the corresponding tuning rate from $\sim 0.003-\sim 0.6$ per optical cycles, respectively.
4. Conclusions
We have theoretically shown that the time-delayed identical IR pulses leads to a wide spectral tuning of atomic HHG in excess of $\sim 2\omega$ frequency range, for several near-cutoff harmonics. We found that the observed spectral shift results from two contributions: (a) from the shift in the central frequency of the driving IR-IR pulse with varying delay, and (b) from to the non-adiabatic response of the recolliding electron wavepacket to the change in the laser amplitude within an optical cycle. Both, the adiabatic and nonadiabatic component, depend on the delay. For single-cycle pulses, this double-pulse scheme could be used as a source for tunable IAP whereas for the few-cycle regime this results in tunable APT. Finally we have quantified the tuning range and tuning rate with respect to pulse duration. Such tunability of harmonics would be highly useful in application of attosecond pulses and pulse trains to probe atomic and molecular dynamics. Of course, one has to remember that these results have been obtained within a single atom picture. The macroscopic propagation and phase matching can alter the experimental result. It would be interesting to explore how the IR-IR pulse-shaping scheme controls the generation of lower-order harmonics for various driving frequencies. We envision that the work will find applications in various time-resolved spectroscopic studies including tunable seeding of XFEL sources, tunable RABBITT and will motivate experiments to generate widely tunable HHG.
Fig. 7. (a) Peak position of the 39 harmonic order band vs IR-IR time delay near $1T_{oc}$ delay. Filled circles and corresponding fits in dashed line (soft-core Coulomb) and open squares and fit (Gaussian potential). Pulse durations from 40 to 3 optical cycles are marked on the plot. (b), (c) Tuning range and tuning rate for several pulse durations for Gaussian and soft-core Coulomb potentials, as extracted from the data in (a). Shaded regions show sub-5 optical cycle regime.
Funding
Science and Engineering Research Board.
Acknowledgments
AM would like to acknowledge the IISER Mohali post doctoral fellowship and SERB post doctoral fellowship. KPS acknowledges Max-Planck Society for funding through Partners group and DST India for SERB/CRG grant.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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