1. Introduction

High harmonic generation (HHG) in atoms is a promising scheme to produce extreme ultraviolet (XUV) emission and an attractive test bed for ultrafast coherent control of electron dynamics [1]. The underlying mechanism has been intensively and widely studied. In general, HHG can be described by semi-classical three-step model [2, 3], strong-field approximation (SFA) [4] and Feynman’s path-integral approach in quantum-mechanical theory frame [5].

The quantum path is an effective tool to interpret the underlying physics of the interaction between the intense laser field and atoms. In the theory of path-integral, the dipole moment is described by infinite-dimensional functional integral. However, SFA and saddle point analysis simplifies this integral into the sum over a few orbits [5, 6]. These orbits represent that an electron is released at a certain moment and recombines into the ionic core at the later moment.

Most of investigations including both in the theory and in the experiment show that the two shortest orbits referred to as short and long quantum path have dominant contribution in HHG [7]. The contribution from much longer electron trajectory can be neglected due to quantum wave packet diffusion. The long-short quantum-path interference has also been observed experimentally by changing the driving laser intensity [8]. The interference fringe can be reconstructed by only considering the two shortest quantum paths.

Recently, a surprising phenomenon that the harmonic yield with rapid oscillation with the period of 6 nm~20 nm depending on wavelength region has been theoretically investigated [9–12], which is attributed to the interference of five quantum paths. The result shows that high-order electron returning trajectories can unexpectedly contribute to harmonic yield in the HHG process, which is also proved by numerical solution of the time-dependent Schrodinger equation [13]. However, these theoretical investigations are only previously performed in the interaction of “isolated” atom with driving laser field. In order to supply experimental guidance and demonstrate the phenomenon can be observed in the experiment, macroscopic propagation effect is necessary to be investigated.

In this paper, we propose an alternative scheme to control the quantum-path interference by using a chirped driving laser pulse. By scanning chirp parameter, the effective central wavelength of driving pulse can be precisely adjusted. The high-order path interference with the contributions from the four shortest quantum paths is identified in the single atom response. With a far-field spatial filtering, we demonstrate that the high-order interference can also be observed when high harmonic propagation in a macroscopic medium is considered. Consequently the experimental identification of the role of high-order electron returning trajectories is feasible.

2. Theory and results

2.1 single-atom response

The HHG of single-atom dipole response can be calculated by using Lewenstein model [4], which is valid in the regime of SFA. According to this model, the single-atom dipole is expressed as a product of three probability amplitudes [14, 15]

where the three probability amplitudes (A_ion, A_pr, A_rec) correspond to tunneling ionization from binding state, acceleration through laser field and recombination with ion core, respectively.

Up to numerical constants the three amplitudes are (atomic units are used throughout unless otherwise indicated)

where I_p is ionization potential, n(t_i) is the probability of neutral atom ionized at the moment t_i, w_i(t_i) is the ionization rate calculated by the ADK model [16], and A_f(t) is the vector potential of the driving electric field E_f(t). The connection between n(t_i) and $n\left(t_i\right)=1-\exp \left(-{\int }_{\infty }^{t_i}{w}_t\left(\tau \right)d\tau \right)$. The expression of A_pr contains two parts that one is the exponential term representing atomic phase accumulating along a given path and the other is pre-exponential term representing electron wave function diffusion. A_rec is relevant to the atomic dipole matrix element for the bound-free transition.

For a given recombination moment t in the expression of p(t), the sum over all possible electron trajectories P(t_i) is large, leading to the quantum-path interference. These trajectories P(t_i) with different ionization moments t_i are obtained by solving the stationary point equation ${\int }_{t_i}^t{A}_f\left(t\text{'}\right)dt\text{'}=\left(t-t_i\right)A_f\left(t_i\right)$. The equation represents that a classical electron may be freed at several possible moments t_i with zero initial velocity and recombines with the parent ionic core at the same later moment t. For a given recombination moment t, these possible ionization moments can be arranged in accordance with their magnitude (t ⁽¹⁾ i>t ⁽²⁾ _i>t ⁽³⁾ _i>t ⁽⁴⁾ _i>…). The maximum moment t ⁽¹⁾ _i corresponding to the trajectory P(t ⁽¹⁾ _i) represents the shortest excursion time. Obviously, P(t ⁽¹⁾ _i) and P(t ⁽²⁾ _i) refers to frequently discussed long and short quantum paths. P(t ⁽³⁾ _i), P(t ⁽⁴⁾ _i) and etc refers to high-order electron returning trajectories.

The contribution from only one trajectory P(t ^(j) _i) to the dipole moment can be expressed as P ^(j)(t)=e-^iπ/4 A_ion(t ^(j) _i)A_pr(t ^(j) i,t)A_rec(t ^(j) i,t)+c.c. The total dipole moment is therefore expressed as $p\left(t\right)=\sum _{{t_i}^{\left(j\right)}}{p}^{\left(j\right)}\left(t\right)$. We use this model to calculate single-atom dipole response and to analyze the contributions from different quantum paths independently.

Let us firstly investigate the effect of the chirp of driving laser pulses in the single-atom response. For a linearly polarized chirped laser field, its time-varying carrier envelope phase (CEP) can be expressed as $\phi \left(t\right)=-\beta \tanh \left(\genfrac{}{}{0.1ex}{}{t-t_0}{\tau }\right)$[17]. The chirp is governed by three parameters β, τ, t ₀, where β controls the frequency sweeping range, τ controls the steepness of ω(t) and t ₀ controls the centre of frequency sweeping.

For illustrating the quantum-path interference, the harmonic spectrum is calculated for argon atoms subjected to the 30 fs (FWHM) linearly polarized Gaussian laser field with a CEP of φ(t), a central wavelength of 800 nm, a peak intensity of 1.6×10¹⁴ W/cm ², and a focus waist of 55µm.

 

Fig. 1. Single atom harmonic spectra at the focus as a function of chirp parameter β. Note that β<0 corresponds to positively chirped pulse and β>0 corresponds to negatively chirped pulse.

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Figure 1 shows single-atom harmonic spectrum at the focus as a function of chirp parameter β(τ=t ₀=60 fs). According to SFA, harmonic phase is approximately proportional to laser intensity. Rapidly varying intensity of ultra-short laser pulse leads to time-dependent harmonic phase, which causes high-order harmonics are negatively chirped through a phase modulation process when the laser intensity is lower than the saturation intensity [18]. Since harmonics are emitted per half-cycle of fundamental laser pulse, this harmonic chirp can be coherently controlled by introducing the time-dependent phase, which shapes temporal structure of driving laser pulse, or in other words, makes the pulse be chirped. In this case, the harmonic chirp could be compensated by using a positively chirped (PC) fundamental laser pulse and be enhanced by using a negatively chirped (NC) fundamental pulse [19, 20]. As a result, the dramatic sharpening of the harmonic branches in Fig. 1 can be nicely explained, i.e., the structure of harmonics is well defined and discrete for PC fundamental pulse (β<0), but broad and continuous for NC fundamental pulse (β>0).

The laser intensity we adopt in the simulation is lower than saturation intensity. Therefore, the high-order harmonics are generated mostly at the peak of driving laser pulse. Since the driving laser is chirped as β parameter varies, the effective driving frequency at which harmonics emit also varies correspondingly, so that harmonic central frequencies are shifted with different β, which leads to the appearance of sloping harmonic branches in Fig. 1. Interference fringe appears along these sloping branches. The overlapping of adjacent harmonic orders leads to adjacent harmonic interference as well, which is more pronounced at β>0 than that interference at β<0. The harmonic yield is modulated with β, which is mainly due to the four shortest quantum paths interference (see below). On both sides of each central harmonic frequency, considering central frequency shifting, some additional fine spectral structure is also visible. It’s even more pronounced for the lower harmonic orders. The phenomena can be attributed to the contribution of more quantum paths and stronger phase modulation effect.

 

Fig. 2. Integrated harmonic yield (20~50 eV) of a single atom as a function of chirp parameter β by taking into account the contributions from all possible quantum paths (Royal-bold-solid curve), 4 shortest quantum paths (Olive-thin-solid curve), 3 shortest quantum paths (Red-bold-dashed curve), 2 shortest quantum paths (Violet-thin-dashed curve) and only 1 quantum path (Black-dotted curve), respectively.

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In order to illustrate how many quantum paths contribute considerably to the interference patterns, we calculate the argon harmonic yield integrated in the spectral region from 20 eV to 50 eV [9] and the results are shown in Fig. 2. The Royal-bold-solid curve in Fig. 2, including the contribution of all possible quantum paths to harmonic yield, reveals fluctuant behavior of the integrated harmonic yield with a period of δβ ≈11. When harmonic yield is calculated including up to four shortest quantum paths, the modulation period and approximate modulation depth can be reconstructed (see Olive-thin-solid curve in Fig. 2), indicating that the chirp dependent interference is originated from four significantly contributing quantum paths. The interference shows that multiple electron scattering is important for HHG.

However, a more exactly full convergence requires more than four quantum paths, leading to the fine structure shown in Fig.1.

The multi-quantum-path-interference (MQPI) effect is insensitive to the duration, the peak intensity and central wavelength of driving laser field. However, the laser intensity should be low enough to ensure that plasma effect is weak, which can reduce defocusing and self-phasing modulation effect on fundamental laser field. The chirp parameters can affect the figure such as Fig. 1 (harmonic spectrum as a function ofβ), i.e. the harmonic spectrum is more inclined for the smaller chirp parameter τ. However, the oscillation of integrated harmonic yield (20~50 eV) can also be observed for various τ and t ₀, since the effective frequency, at the pulse peak where harmonics are mostly emitted, is also varied by scanning β.

In our simulation, the pulse duration is fixed as chirp parameter β varies, which implies the bandwidth of fundamental laser is slightly modified. However, since harmonic yield is integrated in an energy region which is much larger than this modification, from a theoretical point of view, the analysis is proper when we only discuss the chirp effect.

The range of parameter β acquired for the observation of at least one oscillation depicted in Fig. 2 calls for δβ≈11, corresponding to which the effective wavelength at the peak of driving pulse is varied by δλ_eff≈28 nm. It’s represented that the effective driving wavelength can be precisely adjusted by scanning β. For β=11, the instantaneous wavelength within the total duration of the driving pulse is varied by δλ≈70nm, which represents the chip is not very intense for ultra-short driving pulse, such that it’s not very difficult to implement experimentally.

2.2 macroscopic response

In order to demonstrate the contribution from high-order electron return trajectories may be observed in the experiment, let us turn to consider the HHG in a macroscopic medium.

It has been demonstrated that the macroscopic harmonic spectrum is quite different from that of single-atom, because the phase-matching effect can select electron trajectory in the process of harmonic propagation. In general, the fundamental driving laser is Gaussian both in space and in time. For a macroscopic argon gas medium, different argon atoms perceive different laser intensity. In order to avoid the intensity interference [8], we need to eliminate this intensity difference by some means. Non-uniformity along the on-axis direction can be reduced by making the Rayleigh length of the driving laser pulse be longer than the gas target length. On the other hand, the transverse non-uniformity can induce spatial chirp and ultimately lead to beam divergence of harmonic emission at the exit of gas target. We can overcome this transverse non-uniformity by using spatial filtering in the far field.

Meanwhile, in order to observe the chirp related interference in the macroscopic response, more quantum paths should be involved to contribute to the macroscopic harmonic emission. However, the phase-matching condition is different for different quantum path, and the contribution of different quantum paths to the macroscopic harmonic yield is different [21]. The minimum phase variation over the medium length corresponds to an optimum phase-matching condition. When the laser beam is focused after the gas target, the phase matching is efficiently achieved on axis for the shorter path and off axis for the longer path. In this situation, there is an optimum off axis angle at which the quantum-path interference is the most pronounced for the macroscopic response [8].

Propagation equation is numerically calculated by going to a moving coordinate frame and performing the slowly-evolving wave approximation [14, 22], and the single-atom response as the radiation source is calculated by the SFA. We consider that a 1 mm long, 10 Torr argon gas target centered at 3 mm before the laser focus. According to the parameters of the driving laser pulse mentioned above, defocusing and self-phasing modulation effect for the fundamental laser pulse can be neglected.

 

Fig. 3. Macroscopic harmonic spectra as a function of chirp parameter β. (a) near-field result without a spatial filtering. (b) far-field result with an off-axis spatial filtering from 3 to 6 mrad.

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Figure 3(a) shows the macroscopic harmonic spectrum as a function of β in the near-field region without spatial filtering, whereas Fig. 3(b) shows the result obtained by off-axis filtering including only the contribution integrated from 3 mrad to 6 mrad off axis in the far field. Along each harmonic central frequency, considering central frequency shifting, there is no intensity modulation without spatial filtering, but there is obvious intensity modulation similar to the single-atom response when off-axis spatial filtering is performed. That’s because the spatial average effect due to laser intensity gradient smoothes the interference fringe when the spatial filtering is not performed. However, some fine fringes beside central frequency are also visible in Fig. 3, which originates from the overlap of adjacent harmonics and can be attributed to the long path contribution from different half-cycles of the fundamental laser pulse [23, 24]. This is even more pronounced for the lower harmonic orders due to the stronger phase modulation effect. By comparing Fig. 3(a) and Fig. 3(b), we can conclude that the off-axis spatial filtering plays an important role in observing the chirp dependent quantum path interference in a macroscopic medium.

 

Fig. 4. Macroscopic integrated harmonic yield (20~50 eV) as a function of chirp parameter β. (a) near-field result without a spatial filtering. (b) far-field result with an off-axis spatial filtering from 3 to 6 mrad.

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In order to reveal this chirp dependent interference further, we calculate the macroscopic harmonic yield integrated in the spectral region from 20 eV to 50 eV as a function of β, and the results are shown in Fig. 4. In Fig. 4(a) the near field harmonic yield as a function of β is monotone without obvious modulation, indicating the quantum path interference is smoothed due to the spatial averaging. However, as shown in Fig. 4(b) the harmonic yield obtained with 3~6 mrad off-axis spatial filtering is significantly modulated, which exhibits the oscillation with the same period (δβ≈11) as that of the single-atom response shown as the Olive-thin-solid curve in Fig. 2. A comparison between Fig. 4(b) and Fig. 2 indicates that the interference among the four shortest quantum paths can be identified in the macroscopic harmonic emission by off-axis spatial filtering.

3. Summary

In conclusion, we have demonstrated the chirp dependence of multi-quantum-path interference (MQPI) in both the HHG for the single-atom and that for a macroscopic medium. For the single-atom response, the interference is identified from four significantly contributing quantum paths. For the macroscopic response, the quantum-path interference is also observed by using off-axis spatial filtering from 3mrad to 6mrad. This kind of chirp interference shows that high-order electron trajectories of multiple scattering with ionic core play an unexpectedly significant role during HHG. Three-dimensional propagation simulation further shows that the multi-quantum-path contribution can be observed experimentally.