1. Introduction

High-order harmonic generation (HHG) in noble gases, a promising approach for attosecond pulses [1-5], has been well established over the past decades. The underlying physics of the HHG is well described by the semi-classical three-step model [6, 7]. This model describes many properties of the HHG that cannot be explained by the perturbative theory. However, in order to describe the emission process more precisely, a quantum-mechanical theory, known as the strong field approximation (SFA) [8, 9], has been extensively used. In this model, similar to the Feynman’s path integral approach, the harmonic dipole moment can be expressed as the coherent sum over all the different quantum paths contribute to the harmonic emission. These quantum paths are a generalization of classical electron trajectories described by the semi-classical model, which is characterized by the time of tunneling and the time it spends in continuum. For a harmonic in plateau region, there are two electronic trajectories which are most important. One of them spends a longer time (near a full period of laser field) in continuum and is called long trajectory, and the other one (less than a half laser period) is called short trajectory [8, 10]. These trajectories interfere, constructively or destructively, which are depending on their relative dipole phase. In addition, the dipole phase associates to each quantum path of one harmonic order, which is time dependent and gives rise to both spectral broadening and frequency modulation. Moreover, the longer trajectory corresponds to the stronger intensity dependence of dipole phase.

For investigating these quantum paths in harmonic emission, the macroscopic response of harmonics after the propagation can be discriminated at the “downstream” and “upstream” levels [11, 12]. At downstream level, the short or long path contributions in harmonic emission can be selected through a spectral or spatial filtering, because the different laser-intensity dependences of long and short path components lead to a spectral or spatial separation. For example, the interference and the intensity control of the two components are experimentally observed through using a far-field 6-mrad off-axis window (i.e. the spatial filtering) [11]. At upstream level, the driving laser conditions, such as the focusing geometry, the beam profile, the spatial phase and so on, can be optimized for favoring one of quantum-path contributions according to the phase-matching conditions. For example, Liu et al. show that the macroscopic intensities of the two components are separated spectrally by choosing the focal position, and the addition of a diaphragm can further control the short and long path contributions through modifying the beam profile and the spatial phase [12].

In this letter, we present a strategy of both tracing and controlling the quantum-path distributions in harmonic emission at upstream level by applying a diaphragm and choosing the focal position. The macroscopic evolution of quantum-path distributions is experimentally demonstrated through increasing the laser intensity. The red and blue shifts in quantum-path distributions are experimentally traced and clarified in the spatial and spectral domain. We obviously find that the spatial profiles of harmonics are very different and are gradually stretched and split into two components when the driving laser intensity is increased.

2. Experiment

In our experiment, a commercial Ti: sapphire femotosecond laser source (Coherent, Inc) operating at 800 nm wavelength, 45 fs (FWHM) pulse duration and 2 mJ output energy with a repetition rate of 1 kHz is used. High-order harmonics are generated by focusing the laser pulses into a continuous argon gas jet emitted from a nozzle with 0.2 mm inner diameter in the high-vacuum interaction chamber. The focal length is about 400 mm and the laser focus is about 1 mm downstream of the nozzle orifice. The focal position can be moved along the laser propagation direction and the Rayleigh length is about 8 mm (i.e., from Z = −4 mm to Z = + 4 mm. Z< 0 mm means the laser beam is focused after the gas jet while Z> 0 mm means the laser beam is focused before the gas jet). Then, the generated harmonics are detected by a home-made flat-field soft X-ray grating spectrograph and imaged by a soft X-ray CCD (Princeton Instruments, SX400). The spectrograph consists of a gold-coated spherical mirror, a gold-coated cylindrical mirror, a slit, and a Hitachi flat-field grating (1200 grooves/mm). The spherical mirror acts as a spatial integrator in horizontal direction (integral to the slit), the cylindrical mirror acts as an imaging mirror in vertical direction (imaging to the CCD), and the slit acts as a spatial window in vertical direction (the slit width is about 0.2 mm). A diaphragm in front of the lens is used to further control the short and long electron trajectories by modifying the laser beam profile [12]. A 500 nm thick aluminum foil is used in the spectrometer to block the driving pulse. The spectrometer is set to observe the spectral range between 30 eV and 40 eV.

3. Results and discussion

We first optimize the harmonic signal by modifying the laser beam profile, choosing the focal position and adjusting the argon gas pressure. The diaphragm diameter is about 8 mm for suppressing the short-path contributions, and the optimum gas pressure is about 0.3 bars for obtaining the maximum harmonic yield. Then, at different focal positions, we observe the macroscopic evolution of harmonics in the spatial and spectral domain with the increasing laser intensity. The spectral distributions with the increasing laser intensity at two different focal positions (Z = −3 mm is within the Rayleigh range, while Z = −6 mm is outside the Rayleigh range) are shown in Figs. 1(a) and 1(b). They are similar with the experimental results in [11–13]. A clear broadening of harmonic bandwidth is visible, which is consistent with the long-path contributions as observed in [14]. They have confirmed that the short and long path components could be discriminated by increasing the laser intensity: the short-path components are almost independent on the laser intensity while the long-path components are much more sensitive to the laser intensity [14]. In our experiment, we focus the laser beam after the argon gas jet, where the harmonic emissions from both short and long path components are phase-matched. Then, we further suppress the short-path contribution by modifying the beam profile and the spatial phase [12]. As a result, the contributions of short-path components clearly decrease and the long-path components become the dominant one. However, the spectral distributions are different between Figs. 1(a) and 1(b). In Fig. 1(a), the blue shift is the dominant one, just like the experiment results in [11, 12]. While in Fig. 1(b), the red shift is the dominant one, just like the experiment result in [13].

 

Fig. 1 Experimentally measured spectral distributions of harmonics with the increasing laser intensity at two different focal positions of (a) Z = −3 mm and (b) Z = −6 mm. (c)-(d) The corresponding spatial distributions at the laser intensity of 2.5 × 10¹⁴ W/cm².

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Then, we further present the spatial distributions of the red and blue shifts in Figs. 1(c) and 1(d), corresponding to Figs. 1(a) and 1(b) at the laser intensity of 2.5 × 10¹⁴ W/cm². One can see that the spatial distributions of harmonics from the red and blue shifts are very different. They are shown that the harmonic emissions from the blue shift are mainly on the center of spatial profile while that from the red shift are mainly beyond the center of spatial profile. One can also see that the dominant signals are mainly on the center of spatial profile in Fig. 1(c), corresponding to the dominant blue-shift components in Fig. 1(a), while the dominant signals are mainly beyond the center of spatial profile in Fig. 1(d), corresponding to the dominant red-shift components in Fig. 1(b).

Moreover, the detailed evolution of harmonic emission in the spatial and spectral domain with the increasing laser intensity are shown in Fig. 2 and Fig. 3, corresponding to Figs. 1(a) and 1(b). From these figures, one can see that the spatial and spectral evolution of harmonics with the increasing laser intensity can be definitely traced and clarified. On the one hand, as the laser intensity increases, the cutoff region of harmonics extends gradually to higher orders. This is in accord with the famous cutoff law. On the other hand, the harmonic distribution is gradually stretched and separated from the one body in the center of spatial profile to the two separated bodies. The sub body is separated from the parent body and goes away from the center of spatial profile, at the same time the photon energy of the sub body is gradually reduced (i.e., the red shift), then again it goes back to the center of spatial profile, at this time the two bodies (the sub and parent bodies) are already separated in spectral domain. However, the parent body seems always on the center of spatial profile while the photon energy of the parent body is gradually enhanced (i.e., the blue shift).

 

Fig. 2 Experimentally measured harmonic distributions (from 19th to 27th harmonics) with spatial and spectral resolution at different laser intensities and at the focal position of Z = −3 mm. One can see that the parent body in harmonic emission is the dominant one.

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Fig. 3 Experimentally measured harmonic distributions (from 19th to 27th harmonics) with spatial and spectral resolution at different laser intensities and at the focal position of Z = −6 mm. One can see that the sub body separated from the parent body is the dominant one.

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The main difference between Fig. 2 and Fig. 3 is the intensity distribution in harmonic emission in the spatial domain. In Fig. 2, the parent body in harmonic emission on the center of spatial profile is the dominant one, i.e., the blue-shift component is the dominant one. While in Fig. 3, the sub body in harmonic emission (separated from the center of spatial profile, and then back to the center of spatial profile) is the dominant one, i.e., the red-shift component is the dominant one. In other words, the red and blue shifts can be discriminated not only in the spectral domain but also in the spatial domain. Moreover, one can see that not only the short and long path components but also the red and blue shifts in the long-path distribution can be discriminated and controlled by choosing the focal position.

In this letter, the red and blue shifts in macroscopic quantum-path distributions are experimentally traced and clarified in the spatial and spectral domain when the generated harmonic spectra are broadened with the increasing laser intensity. However, the partial evidences have been presented in other references. For example, He et al. theoretically calculated the far-field spatial profile of different harmonic for two different peak intensities of 1 × 10¹⁴ W/cm² and 3.3 × 10¹⁴ W/cm², shown in figure 9 of [15], which presented that the sub body away from the center of spatial profile existed in their lager laser intensity. Jin et al. experimentally demonstrated the spatial distribution of harmonics in the far field using the laser condition with different pulse durations and using the gas jet with different pressures, shown in the Fig. 2 of [16], which presented that the sub body was separated from the parent body and stretched to the red-shift side. These results are consistent with our observations. However, both of them didn’t demonstrate the detailed evolution in the spatial and spectral domain. Moreover, Zaïr et al. calculated the macroscopic spectra of the 15th harmonic in argon as a function of peak laser intensity with a far-field 6-mrad off-axis window, shown in the Fig. 3 of [11], which presented that the dominant component is the red-shift one. However, their theoretical result is different from their experimental result shown in the Fig. 4 of [11], which presented that the dominant component is the blue-shift one with the increasing laser intensity.

 

Fig. 4 Theoretically calculated spectral distributions with the increasing laser intensity for the 21st harmonic emission at the focal positions of (a) Z = −3 mm and (b) Z = + 6 mm. (c)-(d) The spectral distributions without the intense signal on the center (i.e., removes the short-path component signal), corresponding to (a) and (b).

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In order to give some qualitative interpretation, we also generate the HHG theoretically under the experimental conditions based on the SFA model [8-9], including macroscopic propagation effect [17]. When compared with the time-dependent Schrödinger equation (TDSE) model, the SFA model is simple and isn’t time consuming. Theoretically scanning the focal positions from Z = −10 mm to Z = + 10 mm for obtaining the red and blue shifts respectively, the optimum theoretical results of the 21st harmonic emission are shown in Fig. 4, where Figs. 4(a) and 4(c) are at the focal position of Z = −3 mm for the dominant blue-shift signal while Figs. 4(b) and 4(d) are at the focal position of Z = + 6 mm for the dominant red-shift signal. For the blue shift, the optimal experimental result is at the focal position of Z = −3 mm while the theoretical result can be easily obtained at the focal positions from Z = −6 mm to Z = + 4 mm. For the red shift, the optimal experimental result is at the focal position of Z = −6 mm while the theoretical result can be only obtained at the focal positions near Z = + 6 mm. From Figs. 4(a) and 4(b), we can find the intense signal on the center of spectral profile are almost without spectral broadening (belongs to the short-path component), because the intense short-path component in harmonic emission is not suppressed or removed in our calculated model. Therefore, the intense short-path signal is dominant and almost fixed on the center, while the weak long-path signal is moved with the red or blue shift. Figures 4(c) and 4(d) are the spectral distributions with removing the intense signal on the center, which are corresponding to Figs. 4(a) and 4(b). Therefore, from Figs. 4(c) and 4(d), one can see that the blue shift is the dominant one at the focal position of Z = −3 mm while the red shift is the dominant one at the focal position of Z = + 6 mm. These results indicate that the red and blue shifts can be discriminated and controlled by choosing the focal position. Although there are some differences between the quantum-path distributions of harmonics calculated using the SFA model and those calculated using the TDSE model (these differences may lead to large discrepancies in the predictions for the time-frequency characteristics of the harmonics generated in a macroscopic nonlinear medium) [18], and together with some differences between the simulational result and the experimental result (for the position of Z = −6 mm, the experimental result is different from the simulational result), our experimental quantum-path tracing in the spatial and spectral domain with the increasing laser intensity can lead to a more comprehensive understanding of the spatial and spectral distribution and therefore a better control of the HHG. However, a further investigation in theory is still necessary to solve these differences.

4. Conclusions

In conclusion, we experimentally investigate the macroscopic quantum-path distributions in harmonic emission from an argon gas jet, and trace their spatial and spectral evolution with the increasing laser intensity. We find that the spatial profiles of harmonics from the red and blue shifts are very different and are gradually stretched and split into two components when the driving laser intensity is increased. Moreover, the red and blue shifts in quantum-path distributions are experimentally clarified and controlled by choosing the focal position.