1. Introduction

Ultrashort femtosecond lasers have been extensively used for impulsive excitation of solids. The excited medium, in a highly nonequilibrium state [1, 2], experiences various ultrafast phase transitions like lattice disordering [3, 4], nonthermal melting [5] and ablation [6]. The electrons get preferentially heated by the oscillating light field to a temperature many orders of magnitude higher than that of the cold lattice. After the laser irradiance, the medium remains in a two-temperature nonequilibrium phase [7] during the electron-lattice coupling process, lasting up to a few picoseconds [1]. Evolution of such elusive transient phases can be monitored by time-resolved techniques, employing ultrashort electron bunches [3, 8] or optical pulses [1, 5, 6] as probes. For example, ultrafast electron diffraction has been used to capture the dynamics of lattice structures in solids [3]. A recent study [9] explores the ultrafast electron diffuse scattering technique to reveal the short and long range lattice disorder. The time-resolved diffuse scattering intensity measurement [9] is very sensitive towards short-range order of lattice and is effective in capturing solid-to-liquid transition and any local lattice disordering. Moreover, ultrashort electron bunches can also be used in a shadowgraphy mode to capture transient distribution of fast electrons and the associated electric field [10]. On the other hand, the optical pulses have shown their efficacy in unraveling the transient electrical and thermal properties of solids [11, 12]

In general, a time-delayed optical pulse is incident on the excited target to elucidate the temporal evolution of reflectivity or transmissivity of the hot-dense plasma [13, 14]. Sub-picosecond laser-driven plasmas have a very steep density gradient [15], leading to an inhomogeneous density profile. The penetration of the probe pulse in a highly inhomogeneous plasma is limited only up to its critical density layer (n_c), given as, $n_c=\left(1.1\times {10}^{21}/{\lambda }_{\mu m}^2\right)$ cm⁻³, where λ_μm is the wavelength of the probe in μm [15]. This wavelength-dependent cut-off feature is helpful in the determination of local plasma properties. Therefore, for an extensive mapping of such inhomogeneous plasma, it is imperative to use a probe laser consisting of multiple wavelengths. Femtosecond pulses have been used for a long time to study transient non-equilibrium conditions of matter caused by rapid excitation [1, 2, 5, 19, 20]. In recent times, moderate intensities (10¹² W cm⁻²) have produced solids at ∼ eV temperature scale. In our experiment however, the target is very rapidly (sub-picosecond time scale) heated up to 100‘s eV by intense lasers (I > 10¹⁶ W cm⁻²) and highly inhomogeneous plasma with steep spatial variation is created. A localized probing of such inhomogeneous plasma is possible only by simultaneously launching coherent, harmonic probes, each having few nm spectral widths, as opposed to a broadband supercontinuum probe, typically employed in transient spectroscopy techniques.

In this manuscript, we present time-resolved reflectivity and transmissivity of a transient hot-dense plasma by simultaneously launching fundamental (800 nm), second harmonic (400 nm) and third harmonic (266 nm) probes. As our choice of wavelengths spans the NIR-VIS-UV regime, we can access a broad range of local plasma densities (10²¹ − 10²²) cm⁻³ in a single laser shot. From the sub-picosecond time-resolved reflectivity of the three harmonic probes (ω, 2ω, and 3ω), we have estimated the local electron-ion collision frequency at n_c, 4n_c and 9n_c respectively. Moreover, the simultaneous reflectivity and transmission measurements, performed over 100’s of picosecond time-scale, reveal the long-term dynamics of plasma absorption and the role of surface ablation.

2. Experimental setup

The experiment (setup shown in Fig. 1) was performed using a Ti:sapphire 20 TW chirped-pulse amplification laser system at the Tata Institute of Fundamental Research, Mumbai. A p-polarized 30 fs pulse, focused by an f/4 off-axis parabola to a spot size of 20 μm, was used to excite the 100 μm thick fused silica. The peak intensity of the pump laser was ∼ 2×10¹⁷ W cm⁻². The nanosecond pre-pulse contrast was 5×10⁻⁶, while contrast at 5 picosecond was 10⁻⁵. A probe line was derived from the main pump beam by a 5 % beam-splitter. Two β -BaB₂O₄ (BBO) crystals were used to generate second and third harmonic probe pulses. The first BBO (2mm thick, 29.2°-cut, type-I) was used to generate 400 nm light, after which the second BBO (200 μm thick, 44.3°-cut, type-I) was placed to generate 266 nm probe. The BBO crystals were kept just after the lens that focused the probe beams on the target with near normal (∼ 5°) incidence. This ensured that all the three probes were focused on the same plane. In order to avoid any possible dispersion of the harmonic probes while propagating through the transparent quartz windows of the vacuum experimental chamber, the entire harmonic generation set up was placed inside the chamber. The focal spot of all the three probes were found to be nearly 60 μm. All the three harmonics in the reflected and transmitted probe were separated by a grating and fed individually into different photodetectors to extract time-resolved reflectivity and transmissivity of the plasma.

 

Fig. 1 Schematic of the experimental setup. A time-delayed multicolor probe pulse samples the plasma excited by 30 fs pump pulse. The reflected and transmitted probe signal is recorded in a photodiode (PD) after being dispersed by a grating. The inset shows spectra of all three harmonics used as a probe. The individual harmonics are normalized to their peak amplitude.

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3. Multi-color-reflectivity measurement

Figure 2 shows the evolution of the reflectivity captured by a three-color probe. At negative time-delay, reflected probe signals are flat for all the three wavelengths, indicating low reflectivity of the glass target. After t=0, there is a rapid rise in the reflectivity (rise time < 1 ps). The simultaneous rise of the reflectivity trends of all three harmonics indicates that the free electron density quickly reaches the level of 10²² cm⁻³. After reaching the peak, the reflectivity begins to fall down slowly over a period of nearly 10 ps with a decay rate that is different for different wavelengths. An exponential fit gives decay rates of (1.6 ± 0.1) ps, (2.6 ± 0.3) ps and (4.0 ± 0.3) ps for a probe of 266 nm, 400 nm and 800 nm wavelength respectively.

 

Fig. 2 Time-resolved reflectivity for the three harmonic probes for a fused silica target. The solid lines are the exponential decay curves fitted to the decaying part of reflectivity signal.

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The wavelength dependence of the reflectivity decay rates can be explained using a model based on collisional absorption of the probe light in the expanding plasma. For light incident normally on a plasma of an exponential density profile (n_e = n_cexp(−z/L)), the collisional absorption rate is proportional to exp(−8ν_ei(n_c)L/3c) [15], where ν_ei(n_c) is the electron-ion collision frequency estimated at the critical surface, c is the speed of light and L is the plasma scale length. The choice of density profile, being either linear or exponential, does not affect our estimations. At any time t, L is given as c_st, where c_s is the expansion velocity of the critical surface. Given this, the collisional absorption rate can be written as exp(−t/τ), where τ = 3c/(8c_sν_ei(n_c)), defines the rate at which probe is getting absorbed. Both the parameters c_s and ν_ei(n_c) has a temperature dependence, given as c_s ∝ T^1/2, and ν_ei(n_c) ∝ T^−3/2. However, for time scales > 1 ps, variation of temperature in the (10²¹ − 10²² cm⁻³) density range is negligible [16, 17]. Therefore, the wavelength-dependence of the reflectivity decay rate can only come from local plasma density dependence of the ν_ei(n_c). As ν_ei(n_c) ∝ λ⁻², the decay life time (τ) will follow a wavelength dependence given as τ ∝ λ². This implies that longer wavelengths launched inside the plasma will decay with longer decay time. This trend is clearly seen in the experimental results, as shown in Fig. 2.

4. Estimation of electron-ion collision frequency

Evaluation of ν_ei(n_c) at different critical density layers require estimation of τ (given in Fig. 2) and c_s. To determine c_s, we resorted to a pump-probe shadowgraphy [18] measurement, which captures the extent of plasma expansion towards vacuum. A time-delayed second harmonic probe was used to grab a snapshot of the interaction region by a high-resolution imaging line, shown in Fig. 3. During initial time (t < 5 ps) two jets of fast electrons [18] can be seen moving into the bulk target with a speed estimated to be nearly c/3 (Fig. 3(b), encircled by white dotted line). On longer time scales (t > 50 ps), a much slower expansion of the plasma, associated with the bulk hydrodynamic motion of ions can be seen on the front side of the target (Fig. 3(e) – 3(h)). Figure 3(i) shows the plasma expansion speed estimated from a series of shadowgrams. The plasma expansion speed (c_s) can be taken to be ∼ 10⁷ cm/s for early time (t ≤ 50 ps), also confirmed independently by Doppler spectrometry measurement (shows expansion speed of ∼ 5×10⁶ cm/sec after delays of first few picosecond, Fig. 3(j)). By taking the decay rate τ of 1.6 ps, 2.6 ps and 4.0 ps for 266 nm, 400 nm and 800 nm respectively, the collisional frequency, ν_ei(n_c) at different densities (9n_c, 4n_c and n_c) comes to be 7×10¹⁴, 4×10¹⁴, and 2.8×10¹⁴ s⁻¹, respectively. The results are summarized in Table 1. The estimated collision frequencies show an increasing trend with the plasma density.

 

Fig. 3 Transverse shadowgraphic snapshots captured by a time-delayed second harmonic probe pulse. The white arrow in figure (b) indicates the direction of the pump pulse exciting the front surface of 10 mm thick glass target at intensity of 2×10¹⁷ W cm⁻². (a) – (d) shadowgrams at early time-delays (t < 20 ps) and (e) – (h) shadowgrams at long time delays (t > 50 ps). (i) plasma expansion speed, estimated from the plasma expansion towards vacuum. (j) plasma expansion speed for early time delays estimated from Doppler spectrometry.

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Table 1. Wavelength dependent reflectivity decay rate and estimated collision frequencies. λ : probe wavelength, n_c: critical density of probe, τ: reflectivity decay rate, ν_ei(n_c): electron-ion collision frequency estimated at n_c.

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Furthermore, in order to decipher the transient absorption associated with the hot-dense plasma, we have carried out simultaneous reflectivity and transmissivity measurements, shown in Fig. 4(a). For t < 0 ps, the transmitted signal shows high level of transmission (∼ 92 %) from the cold fused silica, while the reflected signal is about ∼ 8 %. Formation of a dense plasma mirror, driven by the explosive ionization of cold glass by the pump pulse, leads to a sudden dip in the transmission (falls to ∼ 25 %), while the reflectivity rises rapidly to ∼ 22 %. A few picoseconds later, the reflectivity drops to a low level (∼ 2 %) due to absorption of light in the expanding plasma plume, as shown in Fig. 3. On a longer time scale of 100’s of picosecond, both reflectivity and transmissivity show a slow recovery trend. However, due to saturation in the reflectivity (at ∼ 4 %) and in the transmissivity (at ∼ 50 %), the signals fail to recover to the initial values of the normal silica. In order to probe the losses in the signal, we have quantified the loss (L) as L = (1 − (R + T)), shown in Fig. 4(a).

 

Fig. 4 (a) simultaneous time-resolved measurement of reflectivity (green triangle) and transmission (orange square) of second harmonic probe from a fused silica target. The violet circle indicates signal loss (L) = 1− (R + T), where R and T are the measured second harmonic reflectivity and transmissivity, respectively. (b) time-resolved transmission of 800 nm, 400 nm and 266 nm probes for fused silica target.

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During initial time (t < 50 ps) the signal (L) increases rapidly to a high level (∼ 70 %), whereas it decays slowly on longer time scale of 100’s of ps, finally saturating at the level of ∼ 50 %. The loss in the reflectivity and transmission signal could be ascribed to the absorption occurring in the plasma and scattering losses happening due to ablation of the surface. During early time scales (t < 50 ps), after which a significant plasma ablation can be seen (Fig. 3(e) – 3(h)), light scattering occurs, but absorption plays a major role in the loss of signal. However, on a much longer time scale (t > 200 ps), the loss due to scattering from the ablated surface contributes the most. In the intermediate zone (t = 50–200 ps), both processes could be contributing significantly towards the signal loss.

5. Multi-color-transmissivity measurement

Finally, to investigate the wavelength-dependence on the transmission of light through a dense plasma, we present results of time-resolved transmission of 800 nm, 400 nm and 266 nm probes, shown in the Fig. 4(b). Similar to the reflectivity measurements, the recovery rate of the transmitted signal differs for different wavelengths. By fitting (a − b × exp(−t/τ)) functional form, the rate τ for 800, 400 and 266 nm are (176 ± 41) ps, (244 ± 138) ps and (277 ± 276) ps respectively. The transmitted signal can be modeled as T = T₀(1 − e^σn_el), where σ, n_e and l are inverse-bremsstrahlung absorption cross-section, electron-density and length of the plasma slab, respectively. Assuming a linear temporal decay of the excited electron density (n_e = n₀(1 − αt)), the transmitted recovery rate (τ) can be given as τ = 1/(σlαn₀). The wavelength dependence of the τ primarily comes from the inverse-bremsstrahlung absorption cross-section. As σ ∝ λ³, the τ ∝ λ⁻³ relation implies that the recovery rate will be longer for the shorter wavelength. The transmitted measurements in the Fig. 4(b) clearly highlights this trend.

6. Conclusion

In conclusion, we have presented a time resolved pump-probe reflectivity and transmissivity of hot overdense plasmas using three harmonic probes, spanning the NIR-VIS-UV spectral band. The ultrafast transient reflectivity and transmissivity measurements indicate the nature of explosive ionization created by the femtosecond pump pulse. The wavelength dependent decay rate of the reflectivity helps in the determination of local electron-ion collision frequency at three density regions, covering a range of (10²¹ − 10²²) cm⁻³. Simultaneous reflectivity and transmissivity measurements reveal the interplay of plasma absorption and ablation in the signal loss. The technique of multicolor probing method could be useful in the extensive mapping of highly inhomogeneous plasma in a single laser shot.