1. Introduction

High-order harmonic generation (HHG) in noble gases, as a very promising approach to generate attosecond pulses [1–4 ], can be well understood by a semi-classical three-step model [5–6 ]. The more precise description of HHG requires some quantum-mechanical theories, including an approximated method, usually referred to the strong-field approximation (SFA) [7–8 ], and the direct integration of time-dependent Schrödinger equation (TDSE) [9–10 ]. Moreover, many schemes have been proposed to extend the cutoff energy and improve the conversion efficiency [11–14 ]. One effective way to enhance the conversion efficiency is based on the multi-color schemes [15–20 ], which cannot only achieve a broadened supercontinuum corresponding to an isolated attosecond pulse [17], but also can achieve even a single harmonic emission used as a monochromatic source [18,19 ]. Recently, it has been theoretically demonstrated that the excitation of atoms to the Rydberg states is another effective way to generate harmonics with both the large cut-off energy and the high conversion efficiency [21–23 ].

In this work, we experimentally demonstrate enhanced HHG from a spatially modulated filament induced by a linearly polarized laser pulse propagating in an argon-filled cell. When shifting the focus position of an elliptically polarized laser pulse over the filament along the on-axis propagation direction, we find an obvious enhancement of harmonic yield by nearly one order of magnitude, which could be interpreted in terms of the double contributions from both the excited states of target atom and the phase-matching effect of harmonic beam. In contrast to the enhancement phenomena, we also observe an obvious suppression of harmonic yield with the intensity reduced by one order of magnitude, which could be attributed to both the ground-state depletion and the plasma effect.

2. Experiment

The experimental schematic is shown in Fig. 1 . A commercial Ti:sapphire femtosecond laser (Coherent, Inc.) is used to produce 8 mJ laser pulses at 800 nm center wavelength with a pulse duration of 45 fs at a repetition rate of 1 kHz. The output pulse is split into two beams, where one beam is used as the driving pulse for the generation of harmonics and the other beam is used as the modulating pulse for the preparation of gas medium. The two beams are respectively focused by two convex lenses with a focal length of 500 mm (the focal-spot diameter is about 90 μm; the depth of focus is10 mm from Z = −5 mm to Z = + 5 mm, where Z_R = 5 mm is the Rayleigh length) to a 30-mm long argon gas cell located in a high-vacuum interaction chamber. The energy of the linearly polarized driving pulse is kept constant at 1 mJ. The peak intensity is estimated to be about 3 × 10¹⁴ W/cm² at the focal spot, while the average intensity is estimated to be about 1.5 × 10¹⁴ W/cm² at the interaction region. The modulating pulse has an elliptical polarization, which is created by a quarter-wave plate that adds a certain ellipticity of about 0.35. It is worth pointing out that the ellipticity is introduced to suppress the harmonic generation [24–26 ]. The standard of introduced ellipticity is to remove the contribution of harmonics from the modulating pulse and therefore ensure that the collected harmonics are primarily generated by the driving pulses. The pulse energy of this modulating pulse is controlled in the range of 0~1.5 mJ by using a half-wave plate and a high extinction film polarizer. For the maximum energy of 1.5 mJ, the peak intensity is estimated to be about 4.5 × 10¹⁴ W/cm² at the focal spot. Moreover, an aperture-iris diaphragm is used to improve the beam profile before the first beam splitter, and a 500-nm thick aluminum foil is used to block the residual driving laser after the gas cell. Finally, the generated high-order harmonics are collected by a homemade flat-field grating spectrometer equipped with a soft x-ray charge-coupled device camera (Princeton Instruments, SX 400). The translation stages, the time-delay line, the energy control and the soft x-ray camera are manipulated by a personal computer.

 

Fig. 1 Experimental schematic.

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3. Results and discussion

Using the above experimental setup, we have experimentally measured the harmonic yield as a function of modulating position from −20 mm to + 20 mm, which is shown by the inset of Fig. 1. Note that the focus position of driving pulse remains unchanged at the center of gas cell to form a femtosecond filament, while the focus position of modulating pulse is active to scan the filament along the on-axis propagation direction. In order to avoid the interference between each other, the time delay of modulating pulse is set to precede about 3 ps with respect to the driving pulse, which sufficiently guarantee the spatial separation between the two pulses. This gas pressure of 30 Torr is optimized for the case of two pulses, and the optimum pressure is higher for the single driving pulse at the same laser conditions. We find that the enhancement of HHG yield is much more obvious at the lower gas pressure, and there is still small enhancement by comparing the HHG yields when they are optimized at their specific pressure. The spatial modulating results are shown in Fig. 2 with the gas pressure of 30 Torr, where Fig. 2(b) is the integrated signal from Fig. 2(a) for both the 25th harmonic and the 21st–31st harmonics. One can find an obvious enhancement of HHG by nearly one order of magnitude at the modulating position of about −4 mm. In contrast to the enhancement, one can also find an obvious suppression of HHG around the modulating position of about + 3 mm.

 

Fig. 2 (a) Experimentally measured harmonic yield as a function of modulating position at the time delay of about + 3 ps when scanning the filament along the on-axis propagation direction, (b) the integrated signal from (a) for both the 25th harmonic and the 21st–31st harmonics.

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It is obvious that the different modulating position corresponds to the different modulating laser intensity and the different phase-matching condition. In order to distinguish these contributions with the different modulating positions, we have further experimentally measured the harmonic yield, not only as a function of time delay between the modulating and driving pulses, which is shown in Figs. 3(a)-3(c) , but also as a function of modulating pulse energy, which is shown in Figs. 3(d)-3(f). The positive time delay corresponds to the situation that the modulating pulse goes through the gas medium before the driving pulse, which represents the preparation process, while the negative time delay is without the preparation. For Fig. 3(a) with the modulating position of −4 mm, the harmonic signal exhibits the fast enhancement at the time delay just after 0 ps, and a stable enhancement plateau with a lifetime of dozens of picosecond for this preparation case is visible. For Figs. 3(b)-3(c) with the modulating positions of 0 mm and + 4 mm, the harmonic signal is immediately reduced just at the time delay after 0 ps, and a stable suppression plateau for these preparation cases is also visible. For Figs. 3(d)-3(f), as a function of modulating pulse energy, the harmonic signal is gradually enhanced at the modulating position of −4 mm, the harmonic signal is first enhanced and then suppressed at the modulating position of 0 mm, while the harmonic signal is gradually suppressed at the modulating position of + 4 mm. Moreover, the combined pulse should be quite elliptical at the temporal overlap between the two pulses, and the suppression of harmonic yield should be expected for all three modulating positions. These suppressions have been indeed observed by using the higher temporal resolution within a small temporal window from −150 fs to + 150 fs, as shown in Fig. 4 .

 

Fig. 3 Experimentally measured harmonic yield as a function of time delay for the different modulating positions of (a) −4 mm, (b) 0 mm and (c) + 4 mm, with the modulating pulse energy of 1.5 mJ. Experimentally measured harmonic yield as a function of modulating pulse energy for the different modulating positions of (d) −4 mm, (e) 0 mm and (f) + 4 mm, with the time delay of about + 3 ps.

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Fig. 4 Experimentally measured harmonic yield as a function of time delay by using the higher resolution within a small temporal window from −150 fs to + 150 fs for the different modulating positions of (a) −4 mm, (b) 0 mm and (c) + 4 mm, with the modulating pulse energy of 1.5 mJ.

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The enhancement plateau with such a long time delay could be the evidence for the existence of a coherent superposition between ground state and excited states, which can improve the conversion efficiency [21–23 ]. We suggest that the modulating pulse at the particular position can optically prepare the excited atoms with the Rydberg states, which could be created by the frustrated tunneling ionization (FTI) [27,28 ]. The suppression plateau could be caused by the ground-state depletion and the plasma effect. It is well known that the plasma effect produced by the high degree of ionization of the medium can also degrade the harmonic efficiency. Besides the effect from the different medium state of target atom, the phase-matching effect including both the microscopic effect and the macroscopic effect should also play a key role in our experiment. The microscopic effect for the single-atom response [18] should be emphasized according to the following simulation in Fig. 5(a) , and the macroscopic effect should also been considered with respect to the gas-pressure-induced phase matching [30], the asymmetric spatial dependence of modulating position, and even the critical limit of free electrons. On the one hand, if considering the self-focusing effect in the interaction region along the direction of laser transmission from Z = + 5 mm to Z = −5 mm, the effective self-focusing length in the case of the modulating position of + 4mm is longer than that in the case of position −4 mm, which leads to that the average modulating laser intensity in the former case is larger than that in the latter case. On the other hand, if considering the direction of laser transmission, the phase-matching conditions are also different between these two cases. For example, the driving laser pulse in the case of position −4 mm may pass sequentially through the neutral atoms, the excited atoms and the ionized atoms, while it would be the reverse order in the case of position + 4 mm. Therefore, the enhancement around the modulating position of −4 mm could be interpreted in terms of the double contributions from both the excited states of target atom and the phase-matching effect of harmonic beam, while the suppression around the modulating positions of 0 mm and + 4mm could be attributed to both the ground-state depletion and the plasma effect.

 

Fig. 5 (a) Calculated yield ratio of HHG as a function of excited population ratio with the TDSE simulation, for 3p + 3d state and 3p + 4s state. (b) Calculated Number of neutrals and ions in the interaction region as a function of modulating laser intensity.

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In order to interpret these results, we perform the TDSE simulation [9,10 ]. The numerical solution of TDSE is based on the expansion of the time-dependent wave function in a series of partial waves, which forms a set of coupled equations between the different angular quantum numbers for the radial wave function. The finite-element discrete variable representation (FE-DVR) method is employed to discretize the radial equations with the advantage of providing block-diagonal sparse matrix representation of kinetic operator and the diagonal matrix representation of the effective potential. The temporal evolution of the wave function is carried out by the Lanczos algorithm. The imaginary time propagation scheme allows us to obtain 3p ground state, and 3d and 4s excited states. The coherent superposition of 3p and 3d (or 4s) states upon varying their population is used as the initial wave function to consider the pre-excitation process of Argon atom. The theoretical HHG yield ratio (S/S₀, S is the harmonic signal from the coherent superposition state consisting of ground state and excited state, and S₀ is the harmonic signal from the ground state) as a function of excited state probability, are shown in Fig. 5(a). Since the lower energy trajectories are more easily captured into the Rydberg states, we theoretically present the two lowest excited states involved in the coherent superposition: 3p + 3d and 3p + 4s. One can see that, in both cases, the HHG yield ratio is gradually improved as the excited population is increased. This is consistent with the experimental observation shown in Fig. 3(d), which indicates that the excited population could be improved by increasing the modulating laser intensity at the particular modulating position of −4 mm. Moreover, the excited state probability can be approximately estimated to be about 0.05~0.1 by comparing the yield ratios between the experimental enhancement shown in Fig. 2(b) and the theoretical enhancement shown in Fig. 5(a). We also theoretically calculate the number of neutrals and ions as a function of modulating laser intensity, as shown in Fig. 5(b). The numerical calculations of the ionization rate use the expression in [29] for the Ammosov-Delone-Krainov (ADK) theory including the Keldysh theory after Coulomb correction. One can see that the neutrals are gradually reduced when the modulating laser intensity is increased, and all the neutrals are almost depleted in the case of the modulating laser intensity increased to about 4~5 × 10¹⁴ W/cm². These are consistent with the experimental observations shown in Figs. 3(e)-3(f), which indicate that the harmonic yield could be suppressed for the ground-state depletion at the modulating positions of 0 mm and + 4 mm. Moreover, the harmonic signals in Figs. 3(e)-3(f), corresponding to Figs. 3(b)-3(c), are not only suppressed but also even split in two, which can be further attributed to the plasma effect.

This work is our firstfruits about the enhanced HHG from spatially prepared filamentation in experiment, and the main aim of this letter is to report the interesting phenomenon and give a qualitative explanation. However, the comprehensive theoretical simulation that involves the phase-matching condition in propagation is still necessary for fully accounting for the underlying physics.

4. Conclusions

In conclusion, an obvious enhancement of harmonic yield by nearly one order of magnitude has been experimentally demonstrated upon scanning an Argon filament along the on-axis propagation direction by using an elliptically polarized laser pulse. In contrast to the enhancement phenomenon, an obvious suppression of harmonic yield has also been observed. Our demonstration of this enhancement phenomenon would indicate that both the excited states of target atom and the phase-matching effect of harmonic beam would play a key role in this experiment.