1. Introduction

The interaction of intense femtosecond laser pulses with solid targets leads to conditions relevant to extremely high-energy-densities [1,2]. There are numerous applications of matter at extreme conditions - generation of ultrafast x-rays [3,4], mega-gauss magnetic fields [5,6], or acceleration of electrons to relativistic energy for fast ignition scheme of inertial confinement of laser fusion [7,8]. High intensity ultrashort lasers can achieve these conditions because the ionized solid does not get enough time to expand within the femtosecond pulse duration of the exciting laser pulse, leading to coupling of laser energy close to the solid density. At relativistic intensities (> 10¹⁸ W/cm²), the energy transfer from the laser to the solid density plasma electrons can become quite efficient and results in large flux of hot electrons channeling through the solid [9]. Efforts to further increase the energy coupling are underway, through modification of the target surface [10] and optimization of the laser parameters like pulse energy, duration, and intensity contrast [11–13]. In particular, the femtosecond peak – picosecond pedestal contrast (i.e., picosecond intensity contrast) has significant influence on the efficiency of energy coupling [14]. However, comparative experimental studies in this regard have been limited, especially at ultra-high contrast (≥ 10⁹) for high power lasers. Such studies are essential for experiments with laser intensities ≥ 10¹⁸ W/cm² at target, since the pedestal may significantly alter the efficiency of laser to plasma coupling [15]. Direct time resolved measurements of the electron transport inside the solid target demonstrating the effect of intense, ultra-high contrast lasers, are essential for a comprehensive understanding of the interactions and processes involved. Coupling the time resolved transport study with the measurement of various emissions like coherent transition radiation (CTR), optical transition radiation (OTR) or Cherenkov radiation would further enhance the understanding of the physics involved in the interactions and energy coupling mechanisms like ponderomotive acceleration [16], resonance absorption [17], Brunel heating [18] and anharmonic resonance excitation [19].

The propagation of ultra-short pulse induced energetic electrons through a dielectric is a very interesting problem from both application and physics point of view [20,21]. The phenomenon is entirely different from the laser induced solid-filamentation experiments performed at low laser intensities [22], where low energy (~few eV) electrons are generated inside the dielectric [23]. The motion of a flux of charged particles inside a dielectric is space charge limited. For fast electron energies of few 100 keV to few MeV, the theoretical space charge limited propagation distance is of the order of few µm [21]. However, fast electrons have been shown to propagate through dielectrics for considerable distances, going up to few 100 µm in experiments [9,20,24]. This is possible due to the ionization by the strong space charge field itself (breakdown field ~5 × 10¹⁰ V/m) and the collisional ionization processes, which furnish sufficient return current for beam propagation [21,25,26]. The depth of penetration is primarily influenced by the collision cross-section of the dielectric atoms, the energy distribution of the fast electrons, and the flux of the electron beam, and has been studied via modeling and simulation [21,27].

In this work, we report the effect of laser picosecond intensity contrast on the transport of fast electrons in a dielectric medium [15]. The fast electrons are generated by the interaction of relativistic intensity (> 10¹⁸ W/cm²), 25 – 40 fs, 800 nm laser pulses with an appropriately prepared 10 mm BK7 glass target. The transport dynamics of the electrons inside the target as well as plasma expansion in vacuum are studied using shadowgraphy with 400 nm probe pulses. The critical density corresponding to 400 nm is ~7 × 10²¹ cm⁻³, thereby enabling the visualization of the dynamics of high density electrons (at or above this value, perhaps extending to solid densities). It may be noted that earlier Gremillet et. al. [24], and Sarkisov et. al. [9] had used probes at longer wavelengths (1057 and 529 nm respectively) to study the time-resolved dynamics of fast electrons, setting a lower threshold of densities accessed by shadowgraphy to ~1 × 10²¹ cm⁻³ and ~4 × 10²¹ cm⁻³ respectively. They therefore integrated the measurements over a larger number of electrons occupying a larger volume of target space behind the critical layer of the pump laser, and perhaps accessing electrons over a larger energy range. This makes it difficult to look exclusively at electrons at higher densities. A unique feature of this study is that we employ two different high power femtosecond laser systems under otherwise identical conditions. One of these has a picosecond contrast of ~10⁶ and another, ~10⁹. It is observed that in the case of higher contrast laser, long (~1 mm) relativistic fast electron streaks are obtained at early times (~few picoseconds) inside the target, while the plasma expansion in vacuum becomes observable much later (~tens of picoseconds). For the lower contrast laser, observable plasma expansion in vacuum starts even before the arrival of the main femtosecond peak due to the ionization caused by the large pre-pulse, while the electrons travel in the form of an ionization cloud inside the target at much lower speeds after the arrival of the main pulse. The study of electron transport inside the target and in vacuum shed light on the efficiency of the energy coupling mechanisms between the laser and the plasma.

The paper is arranged as follows: Section 2 describes the experimental setup in details; results are presented in Section 3 followed by a summary of the study and detailed discussion in Section 4. Conclusions from the study are presented in Section 5.

2. Experimental setup

The experiments are carried out at the Ultrashort-Pulse High Intensity Laser Laboratory (UPHILL) at the Tata Institute of Fundamental Research (TIFR), Mumbai using 100 TW high contrast and a 20 TW low contrast laser systems, both having pulse width ~30 fs. Figure 1(a) shows the schematic of the experimental setup. An 800 nm, p-polarized laser pulse is split into a pump and a probe beam using a 5% beam-splitter near the entrance to the experimental vacuum chamber. The pump beam is focused by a 75 mm diameter, 45°-off-axis parabola of effective focal length 175 mm (i.e., f/2.3) on the surface of a BK7 glass target of thickness 10 mm.

 

Fig. 1 (a) Schematic of the experimental setup; BS: beam splitter, M₀₁ – M₁₁: mirrors, L₀₁ – L₀₃: focusing lenses, TDS: time delay stage, OAP: off-axis parabola, BGF: blue-green filter, BBO: beta-Barium Borate (second harmonic generator) crystal, T: target (BK7), CCD: charged coupled device, NDF: neutral density filter, VC: vacuum chamber. (b) Magnified view of the pump-probe schematic on the target. (c) Laser spot-size imaging and measurement for the high contrast laser. (d) Intensity contrast measurement using third order auto-correlator for both the laser systems (high contrast – 100 TW, low contrast – 20 TW).

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The probe beam is guided through a time delay stage to a second harmonic generating beta-barium borate (BBO) crystal (type-I), to generate a 400 nm pulse. This is passed through a Schott-BG39 blue-green filter that transmits only the 400 nm beam, which is then gently focused by a lens (focal length ~40 cm) along the vacuum-glass interface of the target perpendicular to the target normal. Figure 1(b) shows a magnified view of the shadowgraphy scheme at the target-vacuum interface. At the location where the pump beam strikes the target, the probe beam samples a circular area of diameter ~2 mm, with 80% of the beam passing through the target, and 20% through vacuum, to record the laser-matter interaction on both sides of the interface [Fig. 1(b)]. The edges of the target are highly polished to minimize distortions in imaging, making the interface clearly discernible. The 400 nm transmitted probe is guided to an imaging lens (focal length ~15 cm) and taken out of the vacuum chamber via a quartz window to a CCD camera giving the shadowgraphs. The shadowgraphy system has an optical resolution of 4 µm at a magnification of 15X.

In addition, an achromatic imaging lens (focal length ~20 cm) is used to collect the optical emissions from the rear-side of the target, along the normal [Fig. 1(a)]. The collected emission is guided to a time gated Intensified-CCD (ICCD) camera for recording the images. The rear-side emission imaging system has an optical resolution of 5 µm, at a magnification of 4X. Both the CCD (for shadowgraphy) and ICCD (for rear-side emissions) have two BG-39 filters (~10⁻⁸ transmission at 800 nm per filter) before their entrance aperture to eliminate unwanted signal from stray 800 nm scatter or leakage. Additionally, a number of neutral density filters are used to control the intensity of light entering the cameras, and various interference filters are employed for selecting different spectral windows.

Figure 1(c) shows the image of the focused laser spot taken for the high contrast laser pulse. The focused laser spot profile for the low contrast laser system is very similar to Fig. 1(c). From the image, an average FWHM of ~11 μm is obtained. Figure 1(d) shows the plots of the third order autocorrelation signal using a third order autocorrelator (SEQUOIA) for both laser systems, from which the laser peak to pedestal intensity contrast can be determined. For the high contrast (100 TW) system, a picosecond contrast of ~5 × 10⁹ at −25 ps isobtained, while the low contrast (20 TW) system gives a contrast of 5 × 10⁶ at −25 ps with pre-pulse spikes at −11 ps and −20 ps.

Experiments are conducted at a peak intensity of I_o ~4.7 × 10¹⁸ W/cm² (total incident energy on target ~134 mJ) for both laser systems for the contrast comparison study. From the contrast data in Fig. 1(d), it can be inferred that for the higher contrast laser, the intensity becomes > 10¹⁴ W/cm² at about −500 fs where multi-photon ionization processes dominate, and reaches ~10¹⁶ W/cm² around −200 fs, when over-the-barrier ionization occurs. This creates plasma density of the order of > 10²¹ cm⁻³ over a short length scale. In comparison, for the low contrast laser the pedestal reaches ~10¹⁴ W/cm² at ~-2.5 ps (i.e., at much earlier time compared to the high contrast). There is also a possibility of pre-pulse spikes causing ionization at ~11 ps before the arrival of the main pulse [Fig. 1(d)].

3. Experimental results

3.1 Fast electron transport

It is well known that, under our experimental conditions the laser accelerated electrons in the target consist of two distinct species [9,24], a “very fast” population of density n_e ~10¹⁸ cm⁻³, with average energy > 1 MeV, and a “fast” population with average energy < 1 MeV and n_e > 10²¹ cm⁻³ [28]. The plasma channels created by the highest density fast electron population propagating in the bulk dielectric are optically dense compared to 400 nm wavelength probe beam (400 nm critical electron density ~7 × 10²¹ cm⁻³; 800 nm critical density ~2 × 10²¹ cm⁻³). Therefore, the propagation of the fast electrons can be observed via the shadowgraphy of these overdense plasma channels. The energy characteristics of the fast electron jet is strongly dependent upon the contrast of the laser since the length scale of the pre-plasma influences the collective processes involved in laser-plasma coupling. This in turn influences the length of the overdense plasma channel created. The observations with the two different contrasts used in this experiment are presented in the following sub-sections.

3.1.1 High contrast results

The results obtained with the high-contrast laser are discussed first. Figure 2(a) shows a shadowgraph in absence of the 800 nm pump interaction with the target. The vacuum-target interface appears as a dark band with Llyod’s mirror like interference pattern visible on the vacuum side, created due to a part of the gradually converging probe beam reflecting off the interface [Fig. 1(b)]. This shadowgraph furnishes the reference of the interaction zone in presence of only the probe beam. Such “probe only” shadowgraphs are taken at fresh positions of the target each time before the pump pulse is shot to ensure that shadows are due to the fast electron jets, and not artifacts.

 

Fig. 2 Shadowgraphs obtained with the high contrast laser for a BK7 target. (a) Target – vacuum interface in the absence of any pump pulse (probe only). (b) Typical fast electron jet streak in the target at a time delay of 5.0 ps after main pump interaction. Corresponding linear intensity profiles are plotted on the shadowgraph to demonstrate the difference between absence and presence of electron jet streaks. (c) An enlarged image of the section (~1 mm length) marked by the dashed rectangular boundary in part (b), showing the gradual divergence of the electron jet.

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Figure 2(b) shows a typical shadowgraph of a fast electron jet streak induced plasma channel at probe time delay t_d = 5.0 ps using the high contrast laser. It may be noted that the zero of the probe delay time is taken to be the shadowgraph just before the fast electron jet makes an appearance. The estimated maximum error in the high contrast shadowgraph experiment is ~ ± 0.4 ps in time, which is of the order of overdense pre-plasma scale length as described before [Fig. 1(d)]. The pump laser falls on the target edge from the right (vacuum) side of the edge. From the target edge, a dark streak emanates towards the left depicting the ionized plasma channel created by the collimated fast electron jet, which penetrates deep into the glass target. An enlarged view of the streak is shown in Fig. 2(c), where it is observed that the vertical width of the channel increases deeper inside the target from 13 µm (~laser focus diameter) to 37 µm, indicating lateral spread of the electron jet, which may be attributed primarily to beam instabilities due to oppositely directed flow of the fast electron jet and corresponding return current, with secondary contributions from factors like Coulombic repulsion, elastic scattering, and magnetic field induced defocusing [9,24,25,29]. The divergence of the fast electron jet induced channel is estimated to be ~2° [Fig. 2(b)]. The pump laser convergence angle is quite tight at ~16°, which ensures a large divergence inside the target for the leading edge of the pulse below 10¹⁴ W/cm². This reduces the probability of laser induced ionization inside the target to values much below the critical density of the 400 nm probe pulse. A normalized linear intensity profile of the central axial region corresponding to the jet is shown in the bottom part of Fig. 2(b). Similar normalized linear intensity profile is also shown in Fig. 2(a) at the same region for comparison. The minimum of the intensity profile inside the dark band of the interface can be taken as the target edge [dashed magenta line in Figs. 2(a) and 2(b)] and designated as the origin. The extent of the fast electron streak inside the target is determined by comparing with the probe-only shadowgraph. For the shadowgraph in Fig. 2(b), the streak length is estimated to be ~1.2 mm.

Figures 3(a)–3(f) shows the early time evolution of the electron jet as it propagates inside the target. It is observed that the plasma channel due to the fast electron jet extends up to 1.2 mm in t_d = 5.0 ps. Careful viewing of the shadowgraphs and comparison with the probe-only shadowgraph reveal that the jet is indeed originating right at the interface. The time dependent length of the plasma channel plotted in Fig. 3(g), demonstrates a monotonic increase. To minimize the effect of systematic errors that may arise during the determination of streak appearance time and streak length on the estimation of the fast electron speed, a straight line fitting is performed on the data. A slope of ~199 µm/ps is obtained from the straight line fit, which is the average speed of fast electron propagation. This speed of ~0.66c (c is the speed of light in vacuum) corresponds to a kinetic energy of ~174 keV (total energy ~0.7 MeV). The Ponderomotive hot electron temperature scaling is given by the expression $T_{hot}\cong 0.511[\sqrt{\left(1+I_{18}/1.37\right)}-1]$, where I₁₈ is the absorbed laser intensity in the units of 10¹⁸ W/cm². To estimate the percentage of laser energy absorption in the target, a separate experiment is performed, where the high contrast laser pulse is focused on a 100 µm thick optically polished fused silica glass in the same setup. Reflected (specular) and transmitted laser energies are measured using identical large-aperture (~10 cm) pyroelectric detectors (Ophir: PE100BB-DIF-SH), from which the laser energy absorbed in the target can be estimated. We obtain ~52% absorption for incident I_o ~4.7 × 10¹⁸ W/cm² at 45° (Adak et. al., to be published), which gives the calculated hot electron temperature T_hot ~341 keV. The contribution from the harmonics generated near the critical surface to the reflected laser energy has been found to be negligible.

 

Fig. 3 (a)–(f) Shadowgraphs demonstrating the temporal evolution of fast electron jet streak measured using high contrast laser pulses at early times. (g) Fast electron channel length vs time delay plot for estimation of the corresponding electron jet speed. (h)–(m) Shadowgraphs demonstrating the temporal evolution of fast electron jet streaks at later times, along with the plasma expansion into the vacuum. (n) Plot of vacuum side plasma expansion radius as a function of time delay for high contrast pulses.

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Figures 3(h)–3(m) present the electron dynamics with high contrast laser pulses at later times (t_d > 20 ps). Multiple, shorter streaks extending up to ~0.8 mm are observed instead of a single collimated long streak, accompanied by expansion of plasma on the vacuum side . A lateral expansion of the plasma from ~13 µm [c.f., Fig. 2(c)] at early times to ~200 µm [c.f., Fig. 3(m)] later on is also observed at the laser – glass interface. The occurrence of multiplebeamlets may be attributed to instabilities associated with the laterally and axially expanding plasma at the interface at longer times. Due to the short pulse width and high contrast of the laser, the plasma expansion into the vacuum is much delayed. The expansion velocity can be estimated from the slope of the linear fit to the data of time delay vs expansion radius obtained by analyzing the shadowgraphs of Figs. 3(h)–3(m), and is shown in Fig. 3(n). The estimated expansion velocity is ≅ 0.38 µm/ps = 3.8 × 10⁷ cm/s << c.

3.1.2 Low contrast data and comparison with high contrast data

Figures 4(a)–4(e) shows the shadowgraphs corresponding to the experiments with the low contrast laser. The effect of ~3 orders of magnitude lower contrast is evident from the plots. Plasma expansion into vacuum is observed even at earlier times (~4 ps). There are no observable long fast electron streaks (single or multiple) similar to Figs. 3(a)–3(f) in these shadowgraphs. The penetration of electrons into the target occurs over a much longer time and is visible as short jets at ~21 ps, after which a semi-isotropic cloud of electrons is observed. The observations are very similar to those made by Sarkisov et. al. [9], where semi-isotropic electron cloud was seen to propagate inside the medium. It may be noted that Sarkisov et. al. had a contrast of 10⁶ at ~1 ns in their experiment, and used a 529 nm probe with a critical density ~50% lower than the 400 nm probe used in the current experiment. The penetration depth of the electrons in the target is estimated inside the target by averaging over the whole cloud [Figs. 4(a)–4(e)].

 

Fig. 4 (a)–(e) Low contrast laser pulse shadowgraphs demonstrating the temporal evolution of fast electron cloud inside the target, along with the plasma expansion into the vacuum. (f) Temporal variations of, (left-axis) average channel length of electron cloud penetration in to the target, and (right-axis, same scale) Radius of plasma expansion in the vacuum side, for the low contrast laser.

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The temporal evolution of the penetration of the electron cloud in the target and expansion of the plasma in vacuum are shown in Fig. 4(f). The speed of electron penetration inside target and rate of plasma expansion out to the vacuum are estimated from the slopes of the linear fits to the monotonically increasing data [Fig. 4(f)]. The average electron penetration speed is ~5 × 10⁸ cm/s, ~40 times less than the fast electron jet speed of ~0.66c obtained for the high contrast experiment [Fig. 3(g)]. It may be noted that this estimated speed is for high density electrons. There could of course be lower density electron bunches moving at relativistic speeds, which indicates a lower laser energy coupling in the low contrast case compared to the high contrast laser. The lower energy coupling can be attributed to the large laser pedestal and pre-pulse spikes, making the pre-plasma scale length long and dense enough to reflect much of the laser peak power when it arrives at the interaction region. The vacuum plasma expansion speed is ~4.4 × 10⁷ cm/s [Fig. 4(f)].

3.2 Rear-side optical emission

As an additional piece of evidence for the relativistic nature of the electron jets obtained for the high contrast laser, we studied the rear-side optical emission. The estimated electron jet speed of 0.66c in the high contrast laser experiment is superluminal for wavelengths smaller than 800 nm in BK7 glass. Therefore, these electrons would emit Cherenkov radiation in the blue-green spectral region during their propagation through the medium, which would be a part of the emissions collected from the rear-side of the target [30,31]. The rear-side emission is collected by the ICCD camera at different time delays with a minimum gate width of 4 ns (below which the camera is unable to record the images) having an uncertainty of ± 0.2 ns. The images are analyzed to investigate the evolution of the rear-side emission, and give the information regarding Cherenkov emission which occurs for a brief period at early times, typically within the initial 0.5 ns. The images are taken within 1 ns of the rear-side emission arriving at the camera, with time delay (t_d) interval of 0.25 ns. Since the ICCD gate width is > t_d, the recorded images are successively subtracted to obtain the differential Cherenkov emission profiles at differential times (Δt); i.e., the image taken at t_d ~0 is subtracted from t_d ~0.25 ns to obtain the first differential image, then t_d ~0.25 ns is subtracted from t_d ~0.50 ns to obtain the second, and so on. The Δt corresponding to the first image is designated as zero, followed by Δt = 0.25, 0.50, and 0.75 ns for the subsequent images. A typical single shot Cherenkov image for Δt = 0.5 ns is shown in Fig. 5(a). From the intensity information a meanFWHM ~198 µm can be estimated, which can be taken as the diameter of the Cherenkov cone projected at the target rear.

 

Fig. 5 (a) Cherenkov emission image (background subtracted) after a differential time interval of 0.5 ns for the high contrast laser. (b) Differential peak intensity of Cherenkov emission obtained with three different interference filters (400, 520, 650 nm) of bandwidth ~10 nm, indicating the spectrum of the Cherenkov emission.

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Figure 5(b), shows the differential peak intensity at Δt = 0.50 ns obtained by placing different interference filters with central frequencies 400, 520, and 600 nm (all of bandwidth 10 nm), in front of the ICCD, one at a time. The intensity has been corrected for the individual transparencies of the filters and the spectral response of the ICCD. It is observed that the intensity is maximum for the 520 nm filter (blue-green region). The observation of Cherenkov radiation with its spectral maxima falling in the blue-green region substantiates the observation of fast electrons in the shadowgraphy experiments.

4. Discussion

We consider high intensity (> 10¹⁸ W/cm²), femtosecond (~30 fs) laser induced fast electron transport in a transparent dielectric for two laser systems having three orders of magnitude different femtosecond to picosecond contrast. As pointed out earlier, the use of a 400 nm probe beam shadows only high density electrons (> 7 × 10²¹ cm⁻³) generated during the laser-plasma interaction. It has been observed that at high picosecond contrast (~10⁹), a large fraction (~52%) of the peak laser intensity is coupled to the plasma electrons generated at the vacuum-glass interface enabling long collimated transport of high density fast electrons inside the dielectric medium at relativistic speeds (~0.66c). The plasma expansion towards vacuum is observable at much longer timescales (~30 ps) for the high contrast laser.

In comparison, the laser system with picosecond contrast of ~10⁶ has a large pre-plasma scale length, which limits the coupling of laser energy to the solid density plasma electrons. Thus a much lower number of fast electrons are injected into the dielectric, with bulk of the electrons expanding as a cloud inside the medium with an order of magnitude lower speed than the fast electrons obtained with the high contrast laser. The expansion speeds of the plasma towards vacuum are similar for both the contrasts implying that the increase in the electron energy for the high contrast case, affects both the sheath field and the outward plasma pressure almost equally for a high density electron bunch.

It is of interest to look at the effect of larger intensity (~10¹⁹ W/cm²) on the dynamics of the highest density electrons for the laser with high picosecond intensity contrast. Figures 6(a)–6(f) shows the temporal dynamics of the fast electrons at early times for the high contrast laser at an intensity I_o ~1.7 × 10¹⁹ W/cm² (incident energy on target ~495 mJ), nearly an order of magnitude higher than the intensity used for the experiments presented above (Fig. 3). Faint multiple streaks are observed accompanying a broad jet at early times up to 1.2 ps, with typical width ~19 µm [Figs. 6(b), 6(c)]. The jet broadens to a width of about 40 µm by 2.1 ps, and penetrates the medium as an almost parallel beam up to ~0.5 mm. In the subsequent shadowgraphs at 2.9 and 7.1 ps, it is observed that multiple jets with widths ranging from 20 – 60 µm, travel together giving the appearance of a broad beam of width ~100 µm [Figs. 6(e), 6(f)]. Several of these jets merge and separate, with the most energetic of them penetrating up to ~0.9 mm at 7.1 ps [Fig. 6(f)]. The appearance of multiple jets at early times may be attributed to various instabilities that come into play at high beam electron densities [21,25].

 

Fig. 6 (a)–(f) Shadowgraphs demonstrating the temporal evolution of fast electron jet streak measured using high contrast laser pulses with a larger intensity (~10¹⁹ W/cm²), at early times. (g) Fast electron channel length vs time delay plot for estimation of the corresponding electron jet speed. (h)–(m) Shadowgraphs demonstrating the temporal evolution of fast electron streaks at later times, along with the plasma expansion into the vacuum. (n) Plot of vacuum side plasma expansion radius as a function of time delay for high contrast pulses.

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The average length of the plasma channel due to fast electron jet penetration is plotted as a function of the time delay (t_d) in Fig. 6(g). The penetration length increases almost uniformly up to 2.9 ps, after which it slows down considerably, most likely due to resistive or ionization instabilities. Two separate line fits are performed as shown in Fig. 6(g). From the slopes it is observed that the jets travel relativistically up to 2.9 ps with a velocity of about 210 µm/ps ~0.70c, corresponding to a kinetic energy ~207 keV (total energy ~0.72 MeV), and then going down to about 76 µm/ps by 7.1 ps, which is no more relativistic. From absorption measurements described in Section 3.1.1, ~61% coupling is obtained for I_o ~1.7 × 10¹⁹ W/cm² at 45° (Adak et. al., to be published), which gives the calculated hot electron temperature as, T_hot ~1 MeV.

Figures 6(h)–6(m) shows the evolution of the fast electrons, accompanied by plasma expansion in vacuum at later times (≥ 32 ps). The electron jets penetrating the medium are spread over a width > 200 µm. The length of the individual filaments with densities above the 400 nm critical density are however are limited to < 0.9 mm. On the vacuum side, plasma expansion occurs at an average speed of about 3.6 × 10⁷ cm/s, as shown in the expansion radius vs time delay plot of Fig. 6(n), which is again similar to the numbers obtained for the 10¹⁸ W/cm² experiments with high and low contrast lasers.

It is interesting to recall the observations of Gremillet et. al. [24] who reported collimated relativistic jets of width < 20 µm, penetrating up to 1 mm inside the medium, accompanied by an almost hemispherical electron cloud of diameter ~1 mm travelling at ~0.5c, for a laser intensity of ~10¹⁹ W/cm². They used a 1057 nm probe with a critical density 7 times lower than what has been used in this work.

From the above estimation of coupled laser intensity and electron kinetic energy, several basic parameters related to fast (hot) electron generation and transport can be calculated as follows. The hot electron density n_eh can be estimated from the balance of energy fluxes viz., $f{I}_o=n_{eh}{v}_h{T}_{hot}$, where f is the absorption fraction, v_h is the hot electron velocity, and T_hot is the hot electron temperature ~kinetic energy of the fast electrons. The parameters are calculated for the experimentally obtained results from the high contrast laser at the two intensities, and are tabulated below in Table 1.

Table 1. Estimation of Basic Parameters Associated with Fast Electron Transport for High Contrast Laser.

View Table

5. Conclusion

In conclusion, we show that picosecond intensity contrast plays a major role in the electron transport process. It is observed that the ultra-high contrast (≥ 10⁹) significantly enhances the absorbed intensity of the laser by the plasma, which is not usually expected in plain targets. This large laser to plasma-electron coupling leads to the creation of intense, energetic flux of electrons, which opens up a plethora of research opportunities in the field of relativistic transport of charged particles through a dielectric medium. Generation of transition radiations, induction of mega-gauss magnetic fields deep inside materials [6,10,32], and fast ignition [7,29], are some of the applications where, such fast electrons are expected to play a major role.