1. Introduction

High harmonic generation (HHG) from gas medium is able to generate highly coherent light beam in the extreme ultraviolet (XUV) and soft X-ray range. Such coherent XUV beam is important for attosecond pulse generation, ultrafast electronics monitoring, magnetic dynamics detection, free electron laser seed, and far-field imaging with nanometer resolution [1–5], to name a few. HHG is an extreme nonlinear process and its output photon flux is determined by multiple parameters in experiment. The relationships between the HHG photon flux and individual experimental parameter including the driving laser’s wavelength [6], pulse energy [7], pulse duration [8,9], focusing geometry [10], gas pressure [11–13], and gas medium length [12] have been studied separately. However, there are strong couplings between these relationships and one of them may be significantly modified by the variation of another experimental parameter. There is still no clear guidance how to make the trade-off between two or multiple relationships in practical design. In principle, HHG spectrum is reproduced through iteratively solving the time-dependent Schrödinger equation, which accounts for the microscale single atom response, and Maxwell equation, which accounts for the macroscale propagation of driving laser and XUV radiation [14,15]. Repeating these calculations for all possible experimental conditions will reveal the global optimal design of the system. However, the huge workload and the lengthy time required to execute such calculation [16] make it very hard to be implemented in practice. In real experiments, simplified models [11,12] and rule of thumb [17] are used as the general guidance. In these models, the radial and temporal variation on phase-matching conditions are usually ignored to reduce the computational workload. However, these simplified numerical predictions are suffered by the large discrimination with the experimental results. E. Rogers et al. [18] propose a spatio-temporal model of HHG and show the potential impact of laser intensity variation along radial direction on the HHG flux. However, the accuracy of such model and its impact on guiding HHG optimization is not yet fully verified in experiments.

In this paper, we experimentally study HHG from Ar and propose a practical method to quickly investigate its dependence on gas pressure. Firstly, a spatio-temporal model is proposed to calculate the high order harmonic flux. Besides the ionization rate and phase-matching condition considered in [18], the radial and temporal variation of quantum diffusion effect is also included in our model. Secondly, HHG radiation from Ar are experimentally measured at different gas pressures from 30 mbar to 200 mbar. Compared with 1D on-axis HHG model, our model shows much better accuracy on predicting the HHG flux. It successfully reproduces the variation of HHG flux with respect to the gas pressure on most observable 11 odd harmonics (23^rd to 43^rd). It proves that radial and temporal variations of experimental parameters are able to significantly modify the observable HHG and should be included in the experimental design. Thirdly, we also find that most (∼82%) observable HHG comes from a highly condensed multi-layer capsule-like spatial structure, which only occupies a tiny portion (∼4.17%) of the whole interaction region. Therefore, the computational workload of the HHG flux optimization can be dramatically reduced through only calculating HHG within above spatial volume without compromising the accuracy. Lastly, the proposed model also shows that time-varied spatial chirp exists among different attosecond pulses within an attosecond train generated by a multi-cycle driving laser pulse.

2. Theoretical method

In HHG experiment, high power ultrafast laser pulses are focused into gas medium. The intense laser field ionize the gas atom and accelerate the freed electrons. Within every half cycle of the laser field oscillation, some electrons may return and recombine with their parent atom. The accumulated kinetic energy of the electron is released to emit radiation in XUV region. After propagation through the residual gas medium and XUV optics, the XUV radiation intensity in certain harmonic frequency ω_q detected by spectrometer is estimated by [13]

Equation (1) gives an institutive description of the semi-classical model of HHG in [14]. On the right side of Eq. (1), the first item, $\rho ({z,\; r,t} )$ proportional to $P \cdot [{1 - \eta ({z,\; r,t} )} ]$, accounts for the number of gas atoms able to participate the HHG process, where $\; \eta ({z,\; r,t} )$ is the instant ionization ratio [15]. The second term accounts for the possibility of a gas atom ionized at time ${t_i}$ and able to emit XUV radiation later. It is determined by two major factors: the ionization rate $w({z,r,t} )$ and excursion time $\tau ({z,r,t} )$. According to [19], $A({z,r,t} )\propto \sqrt {w({z,r,t} )/{\tau ^3}({z,r,t} )} $. The excursion time of electron is calculated for individual electron trajectory using the method in [20]. The third term accounts for the XUV radiation absorbed by the residual neutral gas before it exits the gas medium. The last term on the right side of Eq. (1) accounts for the accumulated phase when the XUV radiation propagates to the end of the gas medium.

3. Results and discussions

To test the proposed model, we perform HHG experiments with a Ti: Sapphire laser system that delivers 1.2 mJ, 30 fs pulses at a repetition rate of 1 kHz. The beam is focused with a 500 mm focal length lens to a 60 µm diameter spot inside a vacuum chamber. The HHG spectrum is measured by an X-ray spectrometer after pass through an Al filter with thickness of 500 nm, which is used to block the driving laser. The details of our experimental setup can be found in [23]. The backing pressure of Ar is continuously increased from 30 mbar to 200 mbar and the HHG spectra are recorded. Total 11 harmonics from 23^rd to 43^rd harmonic orders from Ar are clearly observed, as shown in Fig. 1.

 

Fig. 1. Experimental HHG spectra from Ar gas pressure varied from 30 mbar to 200 mbar.

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Three theoretical models, including a one-dimensional spatio-temporal model (M1) in [13] and another spatio-temporal model M2 in [18], which does not include the quantum diffusion effect, and our proposed model M3 are used in numerical simulations with above experimental parameters as input. First of all, the spatio-temporal distribution of the laser electric field is calculated for an ideal Gaussian pulse on a Gaussian beam. The spatial and temporal location of the local electric field intensity able to generate targeted harmonic ${\omega _q}$ is identified. Secondly, the related ionization time, excursion time, ionization rate, ionization ratio and the corresponding wave-vector mismatch $\Delta k({z,r,{t_i}} )$ are calculated using ADK model in [15] and Eq. (2). Finally, the harmonic intensity is calculated use Eq. (3). Above calculation is repeated for the 11 harmonic orders observed in our experiments at various backing gas pressures from 30 mbar to 200 mbar.

Figure 2 shows the comparison between simulation results and experimental results. For 35^th to 43^rd harmonics, model M2 and our model predict that the HHG flux will monochromatically increase with the gas pressure in the range from 30 mbar to 200 mbar. In contrary, the 1D on-axis model M1 predicts that the XUV flux of these harmonics will decrease with respect to the gas pressure. In these harmonics, the on-axis coherent length ${L_{coh}}$ is smaller than the absorption length ${L_{abs}}$ due to the high ionization ratio at the center of the laser beam. The on-axis absorption loss dominants the on-axis HHG generation and the total observable HHG decrease with the pressure. However, at the radial position far away from the center of the beam, the phase-matching condition is still valid over a long off-axis coherent length due to the relatively lower ionization ratio in this area. The off-axis HHG dominates the off-axis gas absorption loss. The boost of off-axis HHG compensates the loss of on-axis HHG and the overall HHG flux increases with the gas pressure as predicated in model M2 and our model M3 as well as shown in the experiments.

 

Fig. 2. Flux of individual harmonics from 23^rd to 43^rd order generated from Ar with varied backing pressure form 30 mbar to 200 mbar predicted by numerical simulation using 1D on-axis model M1 in [13], model M2 in [18], proposed model M3 in this paper and observed in experiments using a 1.2 mJ, 30 fs, 800 nm laser pulse.

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For 27^th to 33^rd harmonics, moderate agreement between experiments and simulation results is obtained. The model M2 and our model M3 predict that the optimal gas pressure is about 150 mbar to 170 mbar for 25^th to 29^th harmonic. Also after the optimal gas pressure, the total flux of HHG will maintain over a broad gas pressure range from 150 mbar to beyond of 200 mbar. Such predictions are in good agreement with the experiments, where the optimal gas pressures for 25^th to 29^th harmonics wandering from 110 mbar to 150 mbar and the total HHG flux maintains almost constant over a broad gas pressure range, as shown in Fig. 2. The HHG flux detected at pressures beyond 150 mbar in experiments are relative lower than prediction. We believe it is due to the saturation of X-ray CCD, which is also can be found in Fig. 1. Therefore, the real experimental HHG flux is expected to be slightly higher and more consist with the theoretical prediction. In contrary, the one-dimensional on-axis model will slightly shift the optimal gas pressure to lower pressure range. More important, the bandwidth of the gas pressure is underestimated using one-dimensional on-axis model. The HHG flux drops quickly with further increasing of the gas pressure beyond 150 mbar, which is not match with experimental observation.

For 23^rd and 25^th harmonics, model M2 and our model M3 show some deviation with the experimental results especially in the pressures region below 100 mbar. On one hand, in these low pressure region, the gas medium length, e.g. ${L_{abs}} \approx 6.3\; mm@30\; mbar$, is compatible to the Rayleigh length of driving laser. Equation (3) in [13] is not valid in this region and there is not enough space for impactful HHG flux to build up. On the other hand, due to the low gas density, the positive dispersion induced by neutral gas atom is not enough to compensate the negative dispersion caused by the free electrons, the HHG flux is contributed by radiation with large mismatch ${ {|\Delta \varphi } |_{3{L_{abs}}}}|> \pi $ as shown in following discussion. The destructive interference between different HHG radiation cause very weak HHG flux able to be observed by the XUV detector.

Compared with model M2, our model M3 includes the quantum diffusion effect, which shows minor effect on the observable HHG flux. Especially when the system is operated nearby the phase-matching condition, the macroscale phase-matching dominates the HHG process and the quantum diffusion effect can be ignored as shown 29^th to 43^rd harmonics in Fig. 2. However, in the system operated in a situation relatively far away from its phase-matching condition, e.g. very low gas pressure or short gas medium length, the quantum diffusion effect may need to be considered in certain applications as shown in 23^rd to 27^th harmonics in Fig. 2.

In Fig. 3(a), the HHG flux is shown as a function of the time and the radial position in logarithmic scale. Figure 3(b) shows the radial distribution of 27^th harmonics generated within the half time cycle [-13.2 fs, -13.0 fs]. The phase-matching HHG generated in the radial range from 30.96 µm to 32.76 µm (marked as green rectangle) dominates the total HHG flux at 27^th harmonic. The quick drop of HHG flux beyond 33 µm is due to the fact that the peak intensity of laser pulse in this region is too weak to generate 27^th harmonic. The oscillation and decay of HHG flux around and below 31 µm is due to the strong destructive interference between HHG generated at different position along the propagation direction. Figure 3(c) shows the total and phase-matching HHG flux of 27^th harmonic as a function of gas pressure. In the low gas pressure region below 110 mbar, the flux of HHG is dominated by the mismatched HHG radiation. The strong destructive interference between these mismatched HHG results in low total flux of HHG at the end of the gas medium. For the high pressure region, the total flux is dominated by the phase-matching HHG. At the optimal gas pressure, ∼82% total HHG flux is contributed by the phase-matching HHG from the very dense spatial volume, which only occupy ∼4.17% the volume of the whole interaction region in the gas medium. This result indicates a possible way to dramatically decrease the workload of HHG computation. As an example, Fig. 4 shows 27^th HHG flux as a function of the radial position and the ionization time for three different pressure completed within few minutes. In order to get a better comparison, the 27^th harmonic flux generated in the first two half cycles of driving laser is enlarged and shown in the insert, where the radial ranges of 27^th harmonic are labeled. In Fig. 4(a) and (b), the radial distribution of 27^th harmonic at the end of gas medium is continue. However, when pressure is 0.17 bar, the 27^th harmonic distribution along radial direction is discrete at the exit of gas medium, as shown in the insert of Fig. 4(c). It shows the spatial distribution of the HHG generated in the gas medium is able to be modulated through control the backing pressure, which potentially benefits the analysis of the spatio-spectral structure in HHG reported recently [24].

 

Fig. 3. (a) Distribution of 27^th order HHG flux (log scale) in terms of radial position (r) and ionization time (t_i) for p = 150 mbar. (b) The radial distribution of 27^th order HHG flux within the half time cycle of [-13.2 -13.0] fs (marked as purple rectangle in (a)). The green box shows the contribution of phase-matching HHG ${ {|\Delta \varphi } |_{3{L_{abs}}}}|\le \pi $. The sharp drop of HHG flux is around 33 µm, beyond which the laser intensity is too weak to generate targeted harmonic. (c) Calculated 27^th HHG intensity with respect to the backing pressure. The red line and the blue line show the total HHG flux and phase-matching HHG intensity respectively. The percentage of phase-matching HHG to the total HHG at 115 mbar, 120 mbar, 150 mbar and 200 mbar are labeled.

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Fig. 4. Flux distribution of phase-matching ($|{\Delta \varphi {|_{3{L_{abs}}}}} |\le \pi $) 27^th harmonic with respect to the radial position (r) and ionization time (t_i) for different gas pressures. The radial ranges of 27^th harmonic generated in the first two half cycles of driving laser are labeled in the insert.

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Figure 5(a) is an illustration of the spatio-temporal structure of HHG in four-dimensional space. The gas medium is marked by the blue line cube and locate in the center of Gaussian beam. The transient phase-matching of q^th-order HHG is satisfied on a multi-layer capsule structure as shown in Fig. 5(b). Each layers indicate the transient phase-matching within certain half-cycle of the driving laser. As a result, several ring-shape radiations of q^th harmonics spread along time axis with an interval of half laser oscillation cycle as shown in Fig. 5(c). Each of these ring radiation contributes to the construction of the attosecond pulse generated within such half-cycle. As a result, time-varied spatial chirp may exist among different attosecond pulses generated by a multiple cycle driving laser pulse. As shown in Fig. 4, the spatial distribution of these attosecond pulses vary with gas pressure. It provides a potential means to modify the attosecond pulse chirp through properly select the gas medium length, gas pressure and spatial phase plate or filters.

 

Fig. 5. (a) Illustration of multi-layer capsule structure of 27^th HHG under perfect phase-matching. (b) Enlarged HHG structure in (a). (c) Dynamics of HHG generation. Each layer represents a HHG pulse (marked with different colors) is generated within different half time cycles of the driving laser pulse.

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There are several limitations in our study. Firstly, our model is not valid for highly ionized gas medium, where the driving laser may not able to be assumed as an ideal Gaussian beam any more. Secondly, in Eq. (3) the phase mismatch is assumed to be independent with respect to the axial position. Such assumption is valid only in loosely focusing geometry. For tightly focused beam, a more sophisticated calculation with the integration along the z direction in Eq. (3) is needed. Thirdly, the temporal profile and CEP phase of the driving laser pulse are assumed to be constant. The nonlinear dispersion, self-phase modulation and self-compression are not considered in our calculation. The variation on driving laser pulse temporal profile or CEP phase may cause modification not only on the phase-matching condition but also the electron trajectory. Therefore, the calculation of the phase-matching condition and excursion time in this study are not valid. Fourthly, in our calculation, the gas medium length is assumed to be longer than the absorption length of HHG. Therefore, it is not applicable for the HHG experiments using very short medium length (<1 mm) or ultrashort pulse duration. In those experiments, the short medium length and narrow pulse time window limit the phase mismatch and the on-axis element dominates the total observable HHG flux. Lastly, only the phase-velocity-based phase-matching condition is considered in our model. Therefore, it is only able to predict the flux at discrete harmonic order but not able to calculate the full spectrum of the HHG output.

4. Summary

In summary, we investigate the spatio-temporal dependence of HHG flux in gas medium with both theoretical analysis and experiments. A theoretical model, including both spatial and temporal variation of the ionization rate, quantum diffusion, and phase-matching condition, is proposed to predict the observable HHG flux under different experimental condition. To test the accuracy of the model, a series HHG spectra from Ar with different gas pressures from 30 mbar to 200 mbar are measured in experiments. Compared with 1D on-axis HHG model, our model is able to reproduce the variation of the HHG photon flux with respect to the gas pressure in much more accurate means. It proves that the radial variation on experimental factors such as laser intensity and phase can significantly modify the phase-matching condition and result in non-ignorable impact on the total HHG flux. We also find that when phase-matched HHG radiation dominates the process, most observable HHG flux (∼82%) comes from a condensed multi-layer capsule-like spatial volume (∼4.17%) within the total laser gas interaction region. Therefore, the computational workload can be dramatically reduced through only calculating within that condensed spatial volume. Our model also shows that the time-varied spatial chirp may exist among the attosecond pulses generated by an idea multi-cycle Gaussian pulse on Gaussian beams.