1. Introduction

In recent years, much attention is devoted to understand material properties in the warm dense matter (WDM) regime, a state between condensed matter and plasma [1]. An intense ultra-short laser pulse rapidly heats the material and creates strong gradient plasma, where density and temperature may vary over a scale length much shorter than the wavelength of light [2]. As a consequence, a larger fraction of laser energy can be coupled to the solid before it can hydrodynamically expand [3–5]. Experimental study in WDM state created by ultrashort laser pulses is challenging because material is subjected to very high pressures. As a result, WDM states are extremely short-lived and heated material quickly expands on an ultrafast timescale. The onset time of expansion is about 500 ~ 800 fs for pump intensity < 10¹⁴ W/cm². Before appreciable expansion, sample surface remains sharp and Fresnel law can be used to determine complex dielectric constants from measured reflectivity data. At later times, probe light interacts with both the expansion plasma and the high-density plasma. This increases the difficulty in understanding the measured reflection signals, and we need hydrodynamic simulation to predict density profile of the heated material [6]. One of the ideas to circumvent this problem is to speed-up the measurements. Moreover, at these extreme conditions, some materials are known to undergo phase transitions when parameters, such as density, pressure or temperature is changed. For example, metal-nonmetal transition is observed in mercury upon decreasing density at temperature and pressure close to the critical point [7] whereas molecular fluid oxygen shows an insulator-metal transition when density is increased [8]. Similar effects can be observed as a function of temperature at nearly constant density. To survey these possibilities we need time-resolution in few tens of femtoseconds.

Ultrafast pump-probe technique provides a powerful tool to time-resolve rapid dynamical processes. To achieve high temporal resolution, we need ultrashort pump and probe pulses with durations down to few cycles of the carrier wave, often achieved using non-collinear optical parametric amplifiers [9]. At low pump intensities, these systems can be easily implemented. Temperatures achieved in the samples are less than or close to melting temperature. After excitation, material relaxes to its equilibrium condition, and subsequent relaxation dynamics are tracked by the pump-probe methods. In contrast, investigation in WDM state is difficult due to surface hydrodynamic expansion, material evaporation and other instabilities in the target conditions. All these instabilities increase uncertainty in the experimental data.

Most studies are performed with relatively narrow-band transform limited probe pulse [10, 11]. In these experiments, perturbed reflectivity is recorded at multiple-delay time steps with a time resolution defined by the cross-correlation between temporal intensity profiles of the pump and probe pulse. However, to elucidate the optical behavior of ultrashort pulse laser heated material, it is essential to obtain a broadband dielectric data using either a continuum probe pulse [12] or simultaneous probes at different wavelengths [13]. Dual-angle reflectometry has been successfully used to infer broadband time-resolved dielectric function of the excited semi-conductors [14]. However, it is difficult to use it when the target surface is heated more and evaporation occurred in a short time.

In this paper, we describe our ellipsometric pump-probe experiment with chirped continuum probe pulses. A single reflected continuum pulse yields three of the four Stokes parameters: the s- and p-polarized reflectivity, and an s–p phase difference [15]. In contrast to narrow-band pulse, a chirped pulse probes target reflectivity changes from initially cold to high-temperature condition in a continuous manner owing to the relative temporal delay between continuum spectral components. Thereby, rapid temporal changes in reflection intensity are projected onto different probe wavelengths [16]. For precise chirp correction, we measured chirp by induced phase modulation (IPM) in fused silica. The chirp obtained from liquid bismuth reflectivity measurement is compared and calibrated using chirp measurement due to IPM in fused silica. This double-checking method enabled chirp correction with high precision to detect very early time transient dynamics in ultrashort pulse laser heated liquid bismuth. In particular, we observed a unique behavior in the reflection history of bismuth plasma that appeared well before appreciable surface hydrodynamic expansion.

2. Experiment and results

We use Ti:sapphire amplifier that delivers 25 fs, 700 μJ pulses at 800 nm with variable repetition rate (10 Hz to 1 kHz). This pulse is split by a beam-splitter to make a pump pulse and other part was used to generate a broadband continuum probe pulse by filamentation in a gaseous medium [17, 18], as shown in Fig. 1. Beam with an initial diameter of 12 mm was loosely focused by a concave mirror (R = 3000 mm) into a 115 cm long cell, which contained krypton gas at ~ 1.2 bar. An aperture placed before the concave mirror allowed us to generate stable 12 – 13 cm long single filament inside the gas cell. Typical broadened spectra span from 300 to 950 nm. To observe a uniform spectral response across the detected bandwidth, residual pump light was removed using 2 mm thickness BG-40 (Schott) colored glass filter and an additional thin homemade spectral filter. Broad bandwidth of continuum pulse enables control over the temporal field of view by appropriately chirping the probe pulse. If we optimize the gas molecules, pressure, and focusing optical condition, the spectral broadening in gas cell primarily involves self-phase modulation and broadened pulse maintains a simple time-frequency relation. Therefore, it is easy to introduce a positive chirp in the probe pulse, just by adding a dispersive optical element in its path. However, this not only causes a wavelength dependent delay in origin-time but also affects experimental temporal resolution. Therefore, it is necessary to measure the chirp behavior in order to correlate continuum spectral components with time and to determine optimal temporal resolution with chirped continuum probing.

 

Fig. 1 Schematic of ellipsometric pump-continuum probe system. ND-1, ND-2: Variable ND-filters; HWP, QWP: λ/2 & λ/4 Waveplates; WP-1, WP-2: Wollaston Polarizers; BBS: Al-Coated Beam Splitter (400–700 nm).

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To analyze these effects, we measured transmission change due to IPM in fused silica [19, 20], as shown in Fig. 2(a). 800 nm pump beam was focused inside a 0.58 mm thickness fused silica plate with a spot diameter of 250 μm while the weak continuum probe pulse, chirped to 3.2 ps, was focused to 90 μm spot. Pump intensity was kept below damage threshold of fused silica. Pump and probe polarization was set to p-polarized, and the angle between them was 4°. Typically, 100 laser shots are averaged to give a wavelength-dependent signal at every time step of 6.67 fs. Figure 2(b) displays the pump induced change in optical density, as a function of time delay between the pump and continuum probe. At the start, probe pulse traverses the sample before the pump pulse. Positive maximum of ΔOD occurs when the pump overlaps corresponding continuum spectral component. Central peak shifts to more positive delay times when pump arrival time is changed, i.e. the wavelength dependent delay in origin-time. By tracking positive central peak, we can obtain chirp (relative group delay, t_g(ω)) of the continuum probe pulse, as displayed in Fig. 2(c). Measured chirp shows a quadratic behavior with respect to wavelengths. This is because most optical materials exhibit a quadratic dependence of group velocity in the visible optical range. Thereby, the relative group delay between continuum spectral components also exhibits a quadratic behavior. This measured chirp is used to correct the delay in origin-time at different wavelengths of the continuum.

 

Fig. 2 (a) Experimental setup for induced phase modulation experiment. (b) Changes in the optical density, ΔOD, from the transmission measurement with chirped continuum probe pulse (dynamics at different wavelengths are off-set for clarity). (c) Chirp behavior of the continuum. (d) Dynamics after chirp correction.

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Probe pulse traverses various optical elements (eg. filters, waveplates, lens, etc.); therefore, due to group delay dispersion (GDD) effect, continuum spectral components will propagate inside a dispersive medium with different group velocities. A linearly chirped probe pulse spectrum with Gaussian envelope and bandwidth (Δω) is expressed as;

${\mu }^{\prime }=\alpha /\sqrt{{\alpha }^2+{\beta }^2}$, ${\psi }^{\prime }=\beta /\sqrt{{\alpha }^2+{\beta }^2}$, and ${{\mu }^{\prime }}^2+{{\psi }^{\prime }}^2=1$. Here t_res defines the time resolution of the chirped pulse. For sufficiently broad continuum probe covering entire visible range; α² << β², then time resolution in Eq. (5) is given by;

Equation (6) shows that the time resolution of the chirped continuum probe varies with GDD at instantaneous probe frequency.

IPM dynamics after chirp correction is displayed in Fig. 2(d). Temporal width of the central peak decreases with increasing probe wavelength. This effect is due to the GDD at corresponding continuum spectral components. Φ″(ω) can be calculated from the Sellmeier equation of materials in the probe beam path or can be obtained from the rate of change of measured chirp with respect to angular frequency (∂t_g(ω)/∂ω). The time resolution obtained from Eq. (6) agrees well with the measured temporal width of the central peak, as shown in Fig. 3. Therefore, for different continuum chirp configurations, the time resolution on the order of few tens of femtoseconds can be achieved. This is very helpful in order to investigate early time fast transient dynamics in WDM.

 

Fig. 3 Time-resolution with chirped continuum probing. Temporal width of the central peak at different probe wavelength (dashed line); and time-resolution calculated from the measured GDD using Eq. (6) (solid line).

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Next, we measure polarization dependent reflection intensities from ultrashort pulse laser heated liquid bismuth sample. The experimental setup is as shown in Fig. 1. The sample is positioned inside a high-vacuum chamber. Small pallets of 99.99 % pure bismuth are kept in a 1.5 cm diameter copper crucible and melted. Both the pump and continuum probe pulse are guided inside the vacuum chamber through 2 mm thick fused silica windows. Intense pump beam is incident normally on the sample and tightly focused by Schwarzschild objective to a focal spot diameter of 12 μm, whereas the weak continuum probe arrives at 64° from the target normal and focused to a 30×40 μm elliptical spot. Pump intensity can be varied between 10¹¹ to 10¹⁵ W/cm² using ND-filter. A combination of broadband λ/2 and λ/4 waveplates sets the incident probe beam to linearly polarized at 45° on the sample, so the s-and p-polarized components are equal.

After reflection from the liquid bismuth surface, the probe beam is divided into two identical signals by a broadband beam-splitter. These two signals are guided towards two imaging spectrometers, which are identical to each other except a λ/4 waveplate, with its fast axis oriented at 45°, is inserted in one of the signal’s path. In both imaging spectrometers, each corresponding signal is focused through the narrow slit, dispersed by fused silica prism and further divided into two orthogonal polarized components by Wollaston polarizer before being recorded by 1280×960 pixel monochrome CCD-camera, with a spatial resolution of ~3 μm along the focal spot diameter and spectral resolution of 1 – 4 nm (from 400 to 700 nm).

Since the probe pulse traverses many optical elements from the sample to the imaging spectrometers, therefore, we calibrated each detector using data for cold gold before starting the reflectivity experiment. Temporal delay between pump and continuum probe was adjusted by an optical delay line. At each delay step, two spectral images are acquired, each corresponding to the two detectors, CCD-1 and CCD-2, respectively. Image recorded by CCD-1 gives reflected intensity signals I₁ and I₂ while the image corresponding to CCD-2 gives signals I₃ and I₄. These polarization dependent reflection intensities are analyzed to give the time-dependent material properties, by forming ratios; X = I₂/I₁ = |r_p|²/|r_s|², and Y = (I₃ − I₄)/(I₃ + I₄) = (2|r_s||r_p|sin(δ))/(|r_p|² + |r_s|²) where, r_p and r_s are the amplitudes of the reflected p- and s-polarized probe beam components, δ is the phase difference between s- and p-polarized probe beam components [15, 21]. The advantage of forming the ratios’ X and Y is insensitivity to fluctuations of the probe beam intensity. Additionally, these two ratios are almost orthogonal to each other and give a unique real and imaginary part of dielectric function [6].

An illustration of our detection scheme is as shown in Fig. 4(a), which displays the raw data of plasma reflection history detected for intensity signal I₁ (∝ |r_p|²). Incident pump intensity was 2×10¹³ W/cm², and continuum probe pulse was chirped to 8.65 ps. Each spectral image is recorded at every 100 fs time step and averaged for 60 pump pulse shots. Spectrum in image (1) show continuum probe pulse reflected from the liquid bismuth surface before the pump arrives. In subsequent images (2–8), reflected spectrum clearly shows the plasma interaction region in central 12 μm of the illuminated field. The line outs across the central plasma region, displayed in Fig. 4(b), shows that reflectivity does not change at instantaneous probe wavelength until intense pump and corresponding continuum spectral component overlaps temporally. After pump illumination, plasma is created at surface and probe light is either being strongly absorbed by electrons in the plasma or refracted/scattered by the plasma to cause rapid changes in reflection intensities. Very high intensity during the pump pulse heats a thin layer of liquid bismuth, causing a change in density and temperature and also change ionization state and/or population of excitation levels. All these changes induce modifications to the optical properties of bismuth plasma through real, n(ω), and imaginary, k(ω), part of refractive index.

 

Fig. 4 Time resolved dynamics of bismuth plasma at an incident pump intensity of 2×10¹³ W/cm² and continuum probe beam chirped to 8.65 ps. (a) shows the reflected spectra I₁(∝ |r_p|²), recorded at different time delays between the pump and probe (spectra, 1 – 8, are shown in false color to identify the continuum spectral components). (b) shows the normalized intensity line outs across the central plasma region of the spectra shown in (a). (c) is the normalized dynamics of ratio X without chirp correction. The surface expansion time is indicated by dashed black line in (c).

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Chirp calibration was determined by wavelength at which spectral features in the reflected spectrum appeared at known time delays on absolutely calibrated spectrometers. Figure 4(c) show typical normalized time-resolved dynamics of ratio X. The delay in origin-time from blue to red portion of the continuum is clearly observed. This behavior is similar to one obtained with IPM in fused silica; however, owing to larger positive chirping (8.65 ps) of the continuum probe, it is temporally broader. It is interesting to note that dynamics in Fig. 4(c) shows a delayed early-time response at longer wavelength probes. Furthermore, we analyzed the surface expansion condition by using the Fresnel formulas to construct artificial dielectric function on an assumption of a single sharply defined interface. From these estimated dielectric constants, we reconstruct the expected reflectivity signals for four detector channels. The measured and reconstructed signals start to differ when hydrodynamic expansion becomes significant [6]. Expansion time at all probe wavelengths is plotted in Fig. 4(c), dashed black line. The onset of appreciable hydrodynamic expansion is delayed at shorter wavelength probes. This behavior is in agreement with plasma theory, which shows that the shorter wavelength light penetrates the deeper, high density region of plasma.

Chirp corrected dynamics of ratio X at two different chirp configurations of the continuum probe (2.8 ps and 8.65 ps) are as shown in Figs. 5(a) and 5(b) respectively. The time resolutions, as equated from Eq. (6), are 77 – 52 fs and 133 – 93 fs for 2.8 ps and 8.65 ps probe chirp respectively. The dynamics are recorded at 20 fs time-steps for continuum probe chirped to 2.8 ps while 100 fs for 8.65 ps chirp. At very early times (~ 300 fs), we usually observe a stagnation point where the sudden change of X seems to pause, indicated by dashed lines in Figs. 5(a) and 5(b). The dynamics at different wavelengths are shifted vertically for clarity. Ratio Y also exhibits similar behavior. Interestingly, the onset of this stagnation point appears to change between ~ 260 fs for shorter to ~450 fs for longer wavelength probes. The pause duration at the stagnation point is ~ 150 – 200 fs. Moreover, this sudden change in reflectivity transient at all probe wavelengths appears well before the onset of hydrodynamic expansion. This behavior is also observed for gold [15], where it was indicated that the observed stagnation point may corresponds to the boiling of gold. Similar explanation cannot be made in case of bismuth, which has a low boiling point (1837 K) than gold (3129 K) [22]. The dynamics at stagnation point probably indicates a high temperature phase transition in bismuth plasma.

 

Fig. 5 Chirp corrected time-resolved dynamics of reflectivity ratio X for the continuum probe chirped to 2.8 ps (a) and 8.65 ps (b). Incident pump intensity was 1.8×10¹³ and 2×10¹³ W/cm² respectively. Dynamics at different wavelengths are shifted vertically for clarity. Early time pausing behavior is indicated by dashed black lines and the surface expansion time at corresponding wavelength is shown by dashed red lines in both figures respectively.

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3. Conclusion

We developed ellipsometric pump-probe experiment with chirped continuum probe pulses to detect the optical properties of material heated to extreme density and temperature conditions with sub-100 fs time resolution and a broader wavelength response. The characterization of continuum probe chirp along with its effect on time resolution is discussed experimentally by induced phase modulation in fused silica. For a broad continuum pulse, time resolution varies with GDD at corresponding continuum spectral component. This allowed us to vary the temporal field of view in frequency-time response of bismuth plasma by appropriately chirping the continuum probe beam, and yet preserving high temporal resolution. Remarkably, for different chirps over the continuum probe pulse and scanning delay steps; we are able to time resolve the stagnation behavior, with an identical temporal response, in the transient history of intense ultrashort pulse laser heated liquid bismuth plasma. The time-evolution of stagnation point appeared to be delayed for longer wavelength probes. Especially, this behavior appears well before appreciable surface expansion. We observed these dynamics due to a smooth correlation between time and frequency of our chirped continuum probe pulse.