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We have experimentally investigated the frequency modulation of high-order harmonics in an orthogonally polarized two-color laser field consisting of a mid-infrared 1800nm fundamental pulse and its second harmonic pulse. It is demonstrated that the high harmonic spectra can be fine-tuned as we slightly change the relative delay of the two-color laser pulses. By analyzing the relative frequency shift of each harmonic at different two-color delays, the nonadiabatic spectral shift induced by the rapid variation of the intensity-dependent intrinsic dipole phase can be distinguished from the blueshift induced by the change of the refractive index during self-phase modulation (SPM). Our comprehensive analysis shows that the frequency modulation pattern is a reflection of the average emission time of high-order harmonic generation (HHG), thus offering a simple method to fine-tune the spectra of the harmonics on a sub-cycle time scale.
© 2016 Optical Society of America
1. Introduction
High-order harmonics generated by the interaction of extremely intense laser field with noble and simple polyatomic gases have been extensively studied and utilized to produce an intense coherent XUV or X-ray light source [1–3], to synthesize isolated attosecond pulses (IAPs) or attosecond pulse trains (APTs) by synchronizing harmonics near the cutoff region [4], to probe ultrafast dynamics of tunneling ionization (TI) and rescattering of electron wave packet (EWP) of molecules and atoms with attosecond precision [5–8]. The process, which is frequently referred to as high-order harmonic generation (HHG), has been intuitively clarified by Corkum’s three-step model [9]. This simple model, which includes electron ionization, acceleration and recollision, provides an intuitive understanding of the underlying physics in HHG. However, we need a full account of quantum mechanical theory by solving the time-dependent Schrödinger equation (TDSE) with the strong field approximation (SFA) to precisely describe HHG [10].
In general, the high harmonic spectra will exhibit strikingly complicated features. One of the features that has been widely investigated is the frequency modulation [11–19]. There are two major factors contributing to the frequency modulation. One is the propagation effect [12, 13]. When an intense laser field interacts with a gaseous medium, optical field ionization will occur, and the medium will be ionized, consisting of both neutrals and free electrons. The time-dependent density variation of free electron and neutral will lead to a change of the refractive index of the medium during ionization-the self-phase modulation (SPM), thereby inducing a blueshift both for the fundamental pulse and high order harmonics [14]. The other factor is the nonadiabatic effect of the intensity-dependent intrinsic dipole phase [15–17]. In this effect, the electric field of an ultrashort pulse will vary dramatically during the laser cycle, and the recolliding electrons that give rise to high harmonic emission will gain a phase shift relative to the field. Specifically, the effective amplitude increases during each half laser cycle in the leading edge, while the effective amplitude decreases during each half laser cycle in the trailing edge. Therefore, if the harmonics are majorly generated in the leading edge, the increasing field amplitude will lead to a blueshift in HHG. Conversely, if the harmonics are majorly generated in the trailing edge, the decreasing field amplitude will lead to a redshift in HHG. The net frequency shift of the harmonic depends on the relative contribution of the leading and trailing parts of the laser pulse. For the single color laser field scheme with low to intermediate laser intensities, both the leading and trailing parts contribute equally to HHG, and no net frequency shifts occur.
To date, most groups have utilized extremely intense single color near-infrared 800nm field to study the spectral shift of HHG [12–15]. However, if the multi-color laser field scheme is used for breaking the symmetry of the HHG process, the relative contribution of the leading and trailing parts will be modified, thereby the nonadiabatic spectral blueshift or redshift of high-order harmonics can be achieved. In ref [18], the authors have theoretically studied the nonadiabatic spectral redshift of high-order harmonics in parallelly polarized two-color field, but few studies focus on spectral shift in orthogonally polarized two-color (OTC) laser field, although it has been extensively used to control trajectory selection and intensity modulation [20–22]. According to our previous work [27], we can manipulate electron-ion recollision by MIR-OTC, confine HH emission within few half cycles, and reduce multiple electron trajectories to few or single recollision even by long pulses (~8 cycles in experiment), due to the large phase difference of the harmonic emission that is proportional to the cube of the wavelength, but this cannot be achieved by 800/400nm OTC because of its short driving wavelength. In this sense, the symmetry of HH emission will be broken and the microscopic HHG process can be fine-controlled by a slight change of the two-color delay in the MIR-OTC scheme, and a more pronounced nonadiabatic blue or red shift that is proportional to the cube of the wavelength may occur. Based on the facts above, the frequency modulation of high-order harmonics in an OTC laser field consisting of a MIR 1800nm pulse and its second-order harmonic pulse is demonstrated in detail in this work. Combining the mid-infrared femtosecond pulses with OTC scheme, we have experimentally accomplished fine-tuning of high harmonic spectra simply by changing the relative delay of the two pulses. Our theoretical analyses show the OTC scheme may enable the steering of electron trajectory selection and the relative contribution of the leading and trailing parts of high harmonic emission [20–22].This provides us an easy control of the HHG process and fine tuning of the high order harmonic spectra.
2. Theoretical analyses and experimental results
2.1 Theoretical analyses of the frequency shift in HHG
As we have mentioned above, two effects contribute to the frequency shift: the propagation effect and the nonadiabatic effect of the intensity-dependent intrinsic dipole phase. We will discuss these two effects separately and give a full analysis to quantitatively describe the frequency shift in HHG.
For the propagation effect, previous researchers have discussed the blueshift induced by the SPM [12–14, 18], i.e., the variation of the refractive index of the medium induced by the free electrons. This variation can be expressed,
and the blueshift of the fundamental pulse can be estimated by the following expression [14, 18],Here L is the medium length, is the wavelength of the fundamental pulse, is the gas density, is the ionization probability when the two-color delay is , which can be calculated by the Ammosov-Delone-Krainov (ADK) formula [23], and is the pulse duration. Therefore, for HHG with order q, the total spectral shift is calculated byThe first term on the right side of Eq. (3) is the shift of the fundamental pulse that is directly imposed on the high harmonic spectra, and the second term is the shift of the harmonic that is directly modulated by the free electron density.The other aspect of the propagation effect is caused by neutrals, which is quite similar to free electrons. Although the effect is trivial, we still take it into account, thereby giving an accurate distinguishment between the propagation effect and the nonadiabatic effect. The variation of the refractive index of the medium induced by neutrals is:
Where is the density of neutrals, is the resonance energy between the ground and the first excited state of the medium. Therefore, the spectral shift of the fundamental pulse induced by neutrals is expressed:and the spectral shift of the high harmonic with order q is:the overall shift induced by neutrals is:Therefore, the total spectral shift induced by the propagation effect is calculated by:Next, let’s consider the nonadiabatic spectral modulation induced by the intensity-dependent intrinsic dipole phase. The MIR-OTC field can be expressed:
where, represent the electric amplitude of the fundamental and SH pulse, respectively, is the envelope andis the two-color delay. According to the SFA model [10, 13, 14], the dipole phase of the harmonic with order q generated in our MIR-OTC field can be written as:Here,, denote the ionization time and the recombination time of the harmonic with order q, respectively. is the ionization potential of the medium. This equation suggests the accumulated action of a harmonic in an OTC field can be expressed as a sum of the two actions that are independently acquired along orthogonally polarized two-color laser fields. Like the HHG in a single color field, the dipole phase in our OTC scheme may be roughly written aswhere is a coefficient related to the harmonic order, the selected trajectory [3,20,24,25], and is the laser intensity of the synthesized electric field in Eq. (9). Therefore, a frequency modulation of the harmonic emission is:The observed spectral modulation of this effect in HHG spectra is a time-averaged result of the relative contribution at different recombination moments. Thus, we can introduce an average emission time of each harmonic, which is defined by the weighted average calculation:
where is the weight of the harmonic with order q at different emission times, which can be obtained by the time frequency analysis [11].Then, the nonadiabatic spectral shift calculated by the Eq. (12) is modified as:From the above calculations and discussions, the spectral modulation may provide us an insight into investigating the underlying physics in the HHG process.2.2 Experimental results and discussions
A MIR-OTC experiment has been performed to identify the above-mentioned theoretical analyses, as illustrated in Fig. 1. The CEP-stabilized 50-fs, 1800nm, 1.6mJ mid-infrared driving pulse is generated by a three-stage optical parametric amplifier (OPA) pumped by a commercial Ti: sapphire laser system (45fs, 800nm, 8mJ, 1 KHz). The 1800nm fundamental pulse is guided into the vacuum chamber through a silver mirror (SM). Subsequently, the laser beam passes sequentially through a concave silver mirror (CM, with a focal length of 210mm), a 1.5mm-thick -barium borate (SH-BBO) crystal (type-1 phase matching, used for generating the orthogonally polarized second harmonic pulse with a conversion efficiency up to 59%) and a 0.8-mm-thick calcite crystal (used for controlling the relative time delay between the fundamental and SH pulses). By rotating the calcite crystal with the step of a small angle, the relative delay can be controlled with a time resolution of 90as. Then the collinearly fundamental and SH laser beams are focused into a stainless steel tube with an inner diameter of 1.9 mm. High-order harmonics are generated when the two focused beams go through the steel tube full of the target gas (argon in our experiment). The laser intensities of the fundamental and SH pulses at the focus are estimated to be, and , respectively. A 150-nm-thick aluminum foil is placed after the steel tube to block the residual MIR pulse and low order harmonics. High order harmonics are detected by a variable-space flat-field concave grating spectrograph (1200 lines/mm) equipped with a soft-x-ray CCD camera.
Fig. 1 Schematic diagram of the MIR-OTC experiment. SM: silver mirror; CM: concave mirror; BBO: barium borate crystal; CCD: charge coupled device.
Figure 2(a) shows the experimental harmonic spectra obtained by the MIR-OTC scheme from Ar. In our experiment, a positive delay implies that the 1800nm pulse precedes the SH pulse. It can be clearly seen from Fig. 2(a) that both odd and even harmonics are generated due to the symmetry breaking of the driving field [8]. Besides, the even harmonics are stronger than the odd ones, and the harmonic signals are modulated periodically as the two-color delay changes, which is consistent with previous studies [21, 26]. The most striking feature of the spectra, however, is the spectral shift as a function the time delay. One can clearly see that blueshifts occur throughout the whole HHG spectrum. As the time delay decreases from positive to negative, the centers of each frequency components gradually shift towards the unshifted harmonic positions, i.e., the blueshifts decrease as well.
Fig. 2 Experimental harmonic spectra as a function of the two-color delay ranging from −20fs~20fs (a) and its enlarged spectra 0.4fs~1fs (b) obtained by the MIR-OTC scheme from Ar. The dashed black lines indicate the unshifted harmonic positions on the CCD camera, while the solid black lines indicate the spectral centers of each harmonic at different time delays. Harmonic spectra around H52 (c) and the relative spectral shifts of H52 (d) at delay A, delay B, delay C, delay D and delay E, corresponding to the time delays of 0.145T,0.135T,0.125T,0.115T and 0.105T,respectively. Note that the experimental spectral shifts of H52 in (d) are obtained by taking the spectrum of delay C as a reference.
In order to identify how the two-color delay affects the spectral modulation, we have plotted the harmonic spectra around the time delay of T/8 (T is the optical cycle of 1800nm, the same hereafter) to investigate the fine structures of the spectra, as illustrated in Fig. 2(b) and Fig. 2(c). We choose and compare the spectra of five time delays, i.e., 0.145T, 0.135T, 0.125T, 0.115T and 0.105T, labelled as delay A, B, C, D and E, respectively, for the following considerations. First, at these time delays, the spectra are strong and clean because multiple electron-ion rescatterings are suppressed, and subpeaks between adjacent harmonics are less pronounced [27], thus making it convenient to observe and identify the center of each harmonic. Second, the ionization rate changes little at these five time delays (3.45% according to ADK model), hence the spectral shift induced by the propagation effect remains almost the same according to Eq. (8). Therefore, by comparing the spectra at different time delays, the nonadiabatic spectral shift induced by the intensity-dependent intrinsic dipole phase can be distinguished from the spectral shift induced by the propagation effect. Third, the calibration offset of the X-ray spectrometer can be excluded by comparing the spectra of these time delays. From Fig. 2 (b), we can see that each harmonic exhibits an obvious blueshift. As the time delay changes from delay A to E, the blueshift of each harmonic gradually decreases and the center shifts towards the unshifted position on the CCD. Specifically, we have studied the spectra of H52 at these time delays, as shown in Fig. 2 (c), taking the spectrum of delay C as the reference spectrum. In this case, the experimental relative spectral shifts of delay A to E are shown in Fig. 2(d). We can see that at delay A and delay B, the spectra show blueshifts of 0.09 and 0.06, respectively. At delay D and delay E, however, the spectra show redshifts of 0.04 and 0.06, respectively. The dependence of the spectral shift on the two-color delay can be merely attributed to the nonadiabatic spectral shift because the propagation effect remains almost the same at these time delays. As is discussed above, when the time delay changes, the harmonic emission time may change as well, thus introducing different nonadiabatic spectral shifts.
In order to confirm our proposal, we have performed a time frequency analysis [11] to investigate the harmonic emission during each half cycle based on the following equation:
where is the single atom dipole acceleration calculated by the SFA model [10], is the Morlet wavelet [11]. Parameters in the simulation are chosen to coincide with our experiment. The orthogonally polarized laser pulses are assumed to be Gaussian with a pulse duration of 50fs (FWHM). From Fig. 3, we can clearly see that the harmonic emission is repeated every half cycle and there will be multiple trajectories contributing to HHG because of the long driving pulses (8 cycles). As the time delay changes, the harmonic patterns differ quite significantly. Though the harmonics are generated both on the leading and trailing edges, most of them mainly occur on the trailing edge, thus inducing nonadiabatic redshifts. As the time delay changes from delay A to E, the average emission moment when the relative weight of the harmonic signal is maximum increases. By using Eq. (13) and Eq. (15), we have calculated the average recombination times of H52 when the harmonic emission reaches maximum. The calculated emission moments are 0.322T, 0.333T, 0.342T, 0.351T and 0.366T for delay A, delay B, delay C, delay D and delay E, respectively. The nonadiabatic spectral shift is then calculated by Eq. (14). The order-dependent coefficient is closely related to the dipole phase, [27]. Therefore, is proportional to. Because the moderate intensity is used in our experiment, and harmonics with the orders higher than 50 are near the cut-off region where the long and short trajectories merge together. For 800nm driving laser, the trajectory with the coefficient close to is majorly prominent for a range of laser intensities by the TDSE prediction [28, 29] when the harmonics are near the cut-off. For 1800nm driving laser in our experiment, however, the coefficient is chosen to be in our estimation. Given these considerations, we have calculated the spectral shift of H52, as is shown in Fig. 4 (b).We can conclude the redshifts are0.919,0.952,0.991,1.033 and 1.065 for delay A, delay B, delay C, delay D and delay E, respectively. Similarly, if we take the spectrum at delay C as the reference, the spectra of delay A and delay B will show blueshifts of 0.07 and 0.04, while the spectra of delay D and delay E will show redshifts of 0.04 and 0.072, which is generally consistent with our experimental results that the he spectra of delay A and delay B show blueshifts of 0.09 and 0.06, and the spectra of delay D and delay E show redshifts of 0.04 and 0.06, as is illustrated in Fig. 4 (c).Fig. 3 The time frequency distribution of HHG at five different time delays (a) delay A(0.145T); (b) delay B(0.135T); (c) delay C(0.125T); (d) delay D(0.115T); (e) delay E(0.105T), respectively.
Fig. 4 (a) Laser intensities of the synthesized pulses of the OTC field (solid red line), the fundamental pulse (dashed blue line), and the SH pulse (dashed black line) when the delay is 0.125T. (b) The calculated nonadiabatic spectral shift corresponding to the laser pulse in Fig. 4(a). (c) The relative spectral shifts of H52 obtained by simulation (blue line) and experiment (red line, the same as in Fig. 2 (d)). It is worth noting that both the experimental and the simulated relative spectral shifts are obtained by taking the spectrum at delay C as the reference.
Furthermore, we have also investigated the order dependence of the spectral modulation. Here, we only concentrate on the spectral modulation of even harmonics for the reason that the spectral centers of even harmonics are easy to determine because of their strong and clean spectra. On the contrary, spectral splitting [11] and ultradense harmonic peaks [27] occur for the spectra of odd harmonics, which makes it difficult to determine the spectral shift of each harmonic. According to the time frequency analysis from Fig. 3, we can see that the average emission times of adjacent harmonics will slightly change, i.e., the harmonics are not synchronously emitted, which may result in different nonadiabatic spectral frequency shifts.
Specifically, we have also calculated the average emission times of different harmonics, as illustrated in Fig. 5 (a). We can obviously see that the average emission times increase slightly with the harmonic order, which is consistent with above analysis. Subsequently, the nonadiabatic spectral shifts can be identified by Fig. 4(b) at different recombination times. Therefore, we can simulate the dependence of spectral shifts on the harmonic order, and then compare the calculated results with our experimental results, as shown in Fig. 5(b). It is worth noting that both the experimental and the theoretical spectral shifts are obtained by taking the spectrum at delay C as the reference as well. From Fig. 5(b), we can clearly conclude that the calculated relative spectral shifts agree generally well with experimental results, except for H58 which shows an error of 2~3 times of magnitude. The spectral shifts of harmonics with the order higher than 60 have not been determined because they are too weak to observe.
Fig. 5 (a)Calculated average emission times of different harmonic orders at delay B (blue line), delay C (red line) and delay D (green line). (b)The order dependence of relative spectral shifts obtained by experimental results at delay B (solid blue line) and delay D (solid green line), calculated relative spectral shifts with considering the propagation effect and the nonadiabatic effect at delay B (dashed blue line) and delay D (dashed green line).Noteworthy is the fact that both the experimental and the theoretical relative spectral shifts are obtained by taking the spectrum at delay C as the reference.
3. Conclusions and outlooks
In conclusion, nonadiabatic spectral modulation of high-order harmonic generation has been observed in a MIR-OTC scheme, and coherent XUV sources can be dynamically fine-tuned simply by changing the two color delay. By comparing the relative frequency shift of each harmonic at different two-color delays, the nonadiabatic spectral shift induced by the rapid variation of the intensity-dependent intrinsic dipole phase can be distinguished from the propagation effect. Our SFA results and time frequency analyses demonstrate that slightly changing the two-color delay will change the average emission times of high order harmonics, thereby modifying the nonadiabatic effect. As a consequence, using this simple technique, one can easily control the HHG process and fine tune the high order harmonic spectra. In our experiment, the HHG spectra show the degeneracy of short and long trajectories (near the cutoff), and the emission moments of different harmonics do not change very much, hence the dependence of nonadiabatic spectral shifts on the harmonic order is not so significant. Future experiments can use MIR-OTC pulses with chirped laser fields to preferentially select short or long trajectories on a single atom level [30], which will make the dependence of nonadiabatic spectral shifts on the harmonic order much more pronounced.
Funding
National Natural Science Foundation of China (Grants No. 11127901, No.61221064, No.11134010, No.11227902, No.11222439; and No.11274325); the 973 Project (Grant No. 2011CB808103); Shanghai Commission of Science and Technology (Grant No.12QA1403700).
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