1. Introduction

Electron-ion recombination in plasma has been studied for decades, most often using microwave methods [1, 2], in which gaseous atoms or molecules were ionized by quasi-steady microwave pulses, and the subsequent electron density decay during the so-called afterglow period was measured. Further experiments have been carried out by ionizing molecular gas with electron beams [3, 4] or by X-ray [5], with pulse durations in the time scale of μs or ns. However, in those experiments complex positive ions were produced before the afterglow period, bringing great difficulties to quantitative analysis. Recently plasma decay after high-voltage nanosecond discharge was studied [6]. Initial positive ions created in this way are less complex, but only in the condition of low gas density. Femtosecond laser developed in the past 20 years can produce dense plasma in such a short time that there is no time for interactions between atoms or positive ions, making it possible to observe the adiabatic kinetics. Such attempts have been done in air by observing interference stripe of fs laser filament [7, 8], but the relationship between plasma kinetics and stripe shape is still obscure.

High harmonic generation (HHG) is an ultrafast nonlinear process, which can happen when intense femtosecond laser fields interact with atoms or molecules. It is commonly described by a semiclassical model [9]: the electron freed by tunnel ionization is accelerated and driven back to the parent ion, resulting emitting of XUV photons. The harmonics represent the response of a neutral atom or molecule, and emit within only one or two optical cycles, therefore HHG can be used as an ultrafast probe and an effective optical method to trace the kinetics in plasma decay.

In our experiment the electron density was measured from HHG using two-color laser fields. The intense, frequency-doubled 400 nm beam and the weak, fundamental 800 nm beam, with an adjustable time delay, play the role of ionization and probe, respectively. The intensities of the harmonics vary with the time delay adjusted in ~40 ps, which reveals the electron density evolution at each probe time. Although HHG is highly nonlinear and the single atom response is not exactly known, we can take the IR harmonics without pre-ionization as reference. Considering phase mismatch and reabsorption as the main modification of the harmonics, information of plasma decay rate and time-evolution of electron density can be extracted by fitting the harmonic spectral intensities.

2. Experiment and results

The experimental setup is shown in Fig. 1 . Fundamental pulses of 50 fs, 800 nm at a repetition rate of 10 Hz were generated from a Ti: Sapphire amplifier laser system and split by a beam splitter (BS1). In one arm, the orthogonally polarized frequency-doubled pulses were generated by inserting a type I BBO crystal with thickness of 1mm. The 800 nm IR beam and the 400 nm (blue) beam were recombined by a dichroic beam splitter and then focused into the gas jet by a fused silica lens (f = 300 mm). As the blue beam could have a different focus position due to larger divergence angle after BBO and different reflectivity in focal lens, a lens group was inserted in the IR arm to make the two beams focus together. Total pulse delay was controlled by a delay line, while phase stability was not needed.

 

Fig. 1 Experimental setup for the pump-probe HHG device. BBO: 1mm, type I. Lens group adjusting the focus position: f 1 = −1350 mm, f 2 = 3000 mm, f 3 = 5000 mm. The distance of f1 - f2 and f2 - f3 were 480 mm and 65 mm, respectively.

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The supersonic argon gas (purity 99.95%) was injected into the chamber by a pulsed nozzle (diameter 500 µm, backing pressure 5 bar). The density in the interaction region was about 10¹⁷ - 10¹⁸ cm⁻³ and the gas temperature was about 100 K. Energy of each IR and blue pulse was 450 µJ and 420 µJ, respectively. The nozzle was adjusted near the center of the focus position. Under these conditions, the high-harmonic spectra were recorded (Fig. 2 ) by a soft X-ray grating and X-ray CCD when the IR pulse propagated before, simultaneously with, and after the blue pulse, respectively.

 

Fig. 2 High-harmonic spectra under different time delays between two pulses. a: the IR pulse propagates before the blue pulse. The blue beam harmonics (BH) show no suppression. b: Strong even orders indicate that a good temporal and spatial overlap of the two beams is obtained. c-f: The leading blue pulse generates dense plasma and brings to the IR harmonics strong suppression, and as the delay time adds the suppression decreases because of plasma decay.

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As shown in Fig. 2(a), when the IR pulse propagates before the blue pulse, the blue beam harmonics (BH) show no suppression. By adjusting the time delay, the IR and the blue pulse meet temporally together and the strong even-order harmonics [10, 11] can be seen in Fig. 2(b), which indicates a good temporal and spatial overlap of the two driving laser fields. As the delay time increases, the leading blue pulse brings to the harmonics of the IR pulse strong suppression, which gradually decreases.

3. Discussion

From the well known cut-off law [9], the calculated peak intensity of the blue pulse is ~5.2 × 10¹⁴ W/cm², much stronger than that of the IR pulse which is ~1.54 × 10¹⁴ W/cm², because of better focusing of the shorter wavelength pulse. The electron density ionized by an IR or blue pulse is calculated by a non-adiabatic ionization model [12], which includes quasi-static tunneling as well as the wavelength-dependent multi-photon contribution to the ionization probability, since the Keldysh parameters of the blue light field (~1.0) and the IR light field (~0.92) are both ~1. We use a temporally Gaussian-typed pulse envelop for the IR and blue pulses:$f(t)=exp[-2ln(2)t^2/{\tau }^2]$, where $\tau$is the full-width half-maximum (FWHM) pulse duration of 50 fs in the calculation. As can be seen in Fig. 3 , when a blue pulse propagates ahead, the neutral atoms are mostly ionized. The following IR pulse, if arrive to the interaction position after a very short time interval such as only a few picoseconds (Fig. 2(c)), can only interact with the left ions, which have a much lower ionization rate than the neutral atoms and negligible HHG efficiency. During longer time interval, the ionized electrons recombine with ions and form neutral atoms, thus harmonics can be generated under the driving of the IR pulse (Fig. 2(d-f)). On the contrary, the atoms are weakly ionized (~5%) after a leading IR pulse and little suppression on BH can be found.

 

Fig. 3 Calculated electron densities ionized by an IR pulse and a blue pulse. The atoms are entirely ionized by the blue pulse but partially ionized (~5%) by the IR pulse.

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The recombination process of electrons with positive ions in noble gas has been investigated for decades, and the concept of dissociative recombination [13] is widely adopted to explain the rapid recombination, where initial molecular positive ions Ar₂⁺ are necessary and regarded as effective capturers of the free electrons. However, in our experiment, the cold atoms (~100 K) are almost totally ionized in such a short time (50 fs) that those atoms and ions have no time to combine, so the existence probability of Ar₂⁺ is very low and the positive ions are mostly Ar⁺. We express the time-evolution of electron density analytically as [7]

However, the modification of high harmonics from phase mismatch [14, 15] and neutral atom re-absorption [16] can be important. Harmonics with orders below 25 can be strongly reabsorbed by neutral Ar atoms, depending on the atom density and the ionization cross-section $\sigma q$ [16] of the qth order. Phase mismatch occurs when the coherent radiation from each target atom adds destructively, due to the differing phase velocities of the harmonics and the fundamental pulses, so the harmonic yield will be suppressed in the case of strong ionization. For example, it can be seen from Fig. 2(a) that although the intensity of the blue light is much stronger than the IR light and yields cut-off order up to 13, the intensities of the blue beam harmonics are not much stronger than the IR harmonics. Phase mismatch plays an important role: When the peak of pulse comes to the interaction area, a large part of the atoms are already ionized (Fig. 3). The generated electrons can reduce the HHG yield, due to phase mismatch, and other non-linear effects. The atom depletion can also reduce the HHG yield of the blue beam. The electron-induced phase mismatch can be expressed approximately as $\Delta k_q(t)=n_e(t)q{r}_e{\lambda }_0$ [14], where$r_e$, ${\lambda }_0$represent classical electron radius and input wavelength, respectively. Neutral gas dispersion is small compared with electron-induced phase mismatch in the case of low atom density and large fraction of ionization [17]. Other transverse effects such as self-defocusing may play an important role in the first few picoseconds, however it can be seen from Fig. 2(c) that the cutoff order at 5.7 ps is lower but can still reach up to H25, indicating that the suppression of the IR pulse intensity is not so strong. Moreover, the interaction length (L ~1.5 mm) is short, thus these transverse effects can be omitted in most delay time. Therefore we can use the model [14]

With the discussion above, the initial atom (electron) density $n_0$ and the decay coefficient $\beta$are obtained from fitting of [E(t)/E₀]² for typical orders of H19-23, as can be seen in the insets of Fig. 4 , where $n_0$ = 7 × 10²³ m⁻³ and $\beta$ = 2.2 × 10⁻¹² m³/s. Therefore time-evolution of electron density is obtained by Eq. (1) and shown in Fig. 4. The decay rate in gaseous plasma is related to specific experiment conditions, such as atom or molecule density [4], electron temperature [2], or other factors like gas mixing [3, 5]. As to gaseous Ar, the decay coefficient we get conforms to former results [2, 13] in magnitude (10⁻¹²-10⁻¹³ m³/s). But as we have suggested, in our experiment, neutral atoms are ionized by the fs laser pulses, so there are no initial chemical reactions involved, which offers a direct measurement to probe the mechanism of electron-ion recombination. Moreover, as shown in Fig. 4, there is a rapid variation of the electron density in less than 20 ps, so an ultrafast and direct probe is necessary. The experiments with probing methods of observing fluorescence [18] or electrical signals [19] may not have a good performance in high temporal resolution, while HHG can be used as an effective probing approach.

 

Fig. 4 The calculated time-evolution of electron density. Insets: fitting of [E(t)/E₀]² for typical orders of H19-23.

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4. Conclusion

In summary, using a two-color pump-probe method, dynamics of electron density is measured by HHG process. Information of plasma decay rate and time-evolution of electron density can be extracted from the decay kinetics, considering atom depletion, harmonics re-absorption and phase mismatch, under the condition of an intense fs pulse interacted with gas media. This method can also be extended to other atoms or molecules. These information can be useful in further study of HHG phase matching, ion-electron interaction, and other plasma influencing processes in the ps and fs time scale.