# Single-shot picosecond interferometry for the characterization of laser-driven shock waves

Gagan Saini, Steven E Kooi, Keith A Nelson, Thomas Pezeril

► To cite this version:

Gagan Saini, Steven E Kooi, Keith A Nelson, Thomas Pezeril. Single-shot picosecond interferometry for the characterization of laser-driven shock waves. Optics Express, 2024, 32 (20), pp.35832. 10.1364/oe.535387. hal-04705680

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# Single-shot picosecond interferometry for the characterization of laser-driven shock waves

GAGAN SAINI,<fnref n="1" /> STEVEN E. KOOI,<fnref n="2" /> ID KEITH A. NELSON,<fnref n="1" />,<fnref n="2" /> ID AND THOMAS PEZERIL<fnref n="1" />,<fnref n="3" />,* ID

**Abstract:** In conventional laser-driven shock experiments, an out-of-plane shock wave is launched and is typically detected interferometrically after it propagates through the sample. In such experiments, the target materials are unavoidably optically damaged at each laser shot. This necessitates changing targets after laser exposure, lowering the shot-to-shot reproducibility and data quality. Here we present a Sagnac interferometer combined with an echelon that can split a single femtosecond probe into many beams, very well adapted for single-shot interferometric characterization of laser-induced shock waves. The echelon provides a 10 ps time resolution and a full time window of about 150 ps. The simplicity, stability, and sensitivity of the single-shot Sagnac interferometer technique ease the thorough characterization of picosecond to nanosecond shock waves, specifically for samples available in limited quantities or for samples that are not uniform from one region to the next.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

The responses of materials to shock loading are actively studied in many fields ranging from material and plasma physics, planetary science, and high-pressure chemistry, to new surgical therapies and much more. Shock experiments typically involve high-speed projectile impacts or intense laser pulses focused on an optimized photoacoustic transducer, producing sudden shock compression with various rise and release timescales and strengths of the shock front that depend on the employed technique. The duration of the shock front is often much shorter than the laser pulse that initiated it, a consequence of nonlinear reshaping. As a result, when using nanosecond lasers, which are commonly employed for launching shock waves in materials within the 1-100 GPa pressure range [1–4], resolving the shock front requires sub-nanosecond resolution.

The most commonly used single-shot diagnostic in shock physics is the velocity interferometer system for any reflector (VISAR), which uses fiber-coupled continuous-wave laser light to measure the velocity of a free or windowed surface of a shocked material [5]. The probe light is reflected at normal incidence to the sample surface and recombined with light that has been optically delayed. The modulations in the intensity of the superposed signal provide a measurement of the velocity of the sample surface. More recently, photon Doppler velocimetry (PDV) has been developed, a technique that also probes the sample surface at normal incidence, to extract velocities [6]. These shock diagnostics are coupled to fast electronics for data acquisition that render the measurement limited to the GHz frequency range i.e. to the nanosecond timescale. To enable faster, sub-picosecond time resolution, alternative methods based on femtosecond lasers have been developed [7,8].

In the most basic configuration employing femtosecond lasers, the time profile of the shock breakout event is built up over many repeated single-shot interferometry measurements taken at different nearby sample regions with varied time delays between the shock generating pulse

<footnote n="1">¹Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA</footnote>

<footnote n="2">²Institute for Soldier Nanotechnologies, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA</footnote>

<footnote n="3">³Institut de Physique de Rennes, UMR CNRS 6251, Université Rennes 1, 35042 Rennes, France</footnote>

<footnote>*thomas.pezeril@cnrs.fr</footnote>
and the probe pulse. These measurements require samples available in large quantities and are inherently impaired by shot-to-shot laser instabilities and sample inhomogeneities. More sophisticated single-shot interferometry techniques, based on frequency-chirped femtosecond probes, overcome these limitations, providing a full time window of each single-shot event. It is worth noting that these techniques can sometimes be paired with ultrafast spectroscopic tools to track spectral changes in the sample under shock loading [9,10].

In spectral interferometry [11–18], a femtosecond pulse is frequency-chirped in order to stretch it into a picosecond duration, and is injected into a standard interferometer that interrogates the shocked sample. Given that time and frequency are encoded in the chirped pulse, the time trace is read out on a spectrometer that can resolve each frequency component of the stretched pulse. The total duration of the stretched pulse, that ranges from a couple of picoseconds to several hundreds of picoseconds, dictates the full time window of the experiment. The major technical difficulty in this technique is stretching the pulse to hundreds of picoseconds, a task that requires expertise in tandem gratings, such as those used in amplified Ti:Sapphire systems [19].

In the following, we demonstrate a versatile method based on low-cost optics for spatially stretching an incoming femtosecond pulse up to 150 ps. We use an homemade glass echelon [20–25] to spatially transform a single femtosecond probe into multiple femtosecond beams with sequential time delays in the picosecond range. We demonstrate how to combine these multiple delayed probe pulses in order to interferometrically probe, in a single shot, the shock dynamics of a metallic sample.

## 2. Experimental methodology

A Ti:sapphire femtosecond laser oscillator (Kapteyn-Murnane Laboratories) centered at 800 nm with a bandwidth full width at half maximum (FWHM) of 10 nm is used as the seed pulse for a femtosecond Ti:sapphire amplifier (Titan Quantronix). The femtosecond amplifier utilizes a grating-based pulse stretcher and compressor, with a regenerative amplifier, to produce up to 3 mJ per pulse and a 250 fs pulse duration (FWHM). The laser system is run at a 60 Hz repetition rate to allow the use of a mechanical shutter for selection of a single pulse. A beam splitter is used to split the output laser pulse into a low-energy probe pulse (approximately ten times lower than the sample optical damage threshold at 1 kHz, ensuring that the probe does not cause any unwanted disturbance to the sample) used for optical single-shot interferometric detection, and a high-energy pump pulse, in the mJ range, used to excite the shock wave. The shock generating pump pulse, so-called in the following the "laser shock pulse" is focused to a beam diameter of 100 µm on the back-side of the sample. Note that the laser shock pulse can be directed toward either the back-side or front-side of the sample, at the convenience of the experimentalist or to comply with specific experimental constraints of the sample. The primary advantage of using the back-side configuration for opaque samples, as in our case, is that the probe is not disturbed by the plasma generated by the laser shock pulse.

The probe pulse, after been expanded by a ×2 telescope, goes through a glass echelon composed of 15 steps to form a train of 15 femtosecond pulses that are spatially and temporally separated. The echelon was made of a stack of 1"×1" glass slides of $L = 6$ mm thickness slightly shifted vertically in order to form a stair case with 2 mm step height, see Fig. 1. Each step of the echelon induces an optical path difference $\delta = (n - 1)L$ with $n = 1.5$ the refractive index of the glass. This produces a 10 ps time delay between pulses and a 150 ps time window. The probe pulse-train is then directed into an interferometer.

Sagnac interferometry has been used extensively for monitoring phase changes that are due to acoustic fields on picosecond timescales [26–29]. The particular advantage of this design is its "common-path" geometry. The common-path design obviates the need for active stabilization and renders the interferometer relatively insensitive to thermal and vibrational fluctuations. The Sagnac interferometer, shown in Fig. 2(a), relies on the interference between a reference pulse,
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** (a) Homemade glass echelon composed of 15 glass plates of $L = 6$ mm thickness kept in contact with each other mechanically i.e. not glued together, with no index-matching liquid used, slightly spatially shifted to form a transparent stair case. (b) Upon transmission through the echelon, the femtosecond probe pulse becomes a train of femtosecond pulses with an inter-pulse delay $\delta/c = (n-1)L/c$, with $c$ the speed of light and $n$ the refractive index of the echelon. (c) A high-energy laser pulse, named Laser shock pulse, is used to launch a shock wave in an opaque sample. The shock wave dynamics can be interferometrically resolved by the train of probe pulses once the shock wave reaches the probed region of the sample.</figcaption></figure>

arriving at the sample before the laser shock pulse, and a signal pulse arriving after shock excitation. The probe enters the interferometer through a nonpolarizing beam splitter (NPBS). It is then split into vertical and horizontal polarizations, the signal and the reference beams, respectively, by the conjunction of a $\lambda/2$ waveplate (oriented at $45^\circ$) and the first polarizing beam splitter (PBS1). The beams are recombined at the second polarizing beam splitter PBS2, and passed through a $\lambda/4$ plate (oriented at $45^\circ$). The pulses are focused at normal incidence onto the sample surface using an achromatic convex lens (L2) of 30 cm focal length, giving a probe spot diameter of 30 $\mu$m. As long as the probe spot size is much smaller than the shock front, the measurement can safely be assumed to meet the criterion for point probe detection. It is important to note that since all the probe pulses in the pulse train are collinear, they focus and overlap at the same single spot on the sample surface. After being reflected at the sample, the beams polarizations are flipped by $90^\circ$ by the $\lambda/4$ plate. In turn, PBS2 reverses the paths of the reference and signal beams that are then recombined at PBS1 and partially reflected by NPBS towards the camera path. The camera is a standard scientific CCD camera (Cohu, Inc.) with an exposure time set in the millisecond range, which is several orders of magnitude longer than the duration of the femtosecond probe pulses being captured. In these single-shot experiments, the exposure time is not critical, as the "very slow" camera simply needs to be opened at the arrival of the femtosecond pulses. Since the two probe pulses travel in opposite directions inside the Sagnac interferometer (PBS1-PBS2-M1-M2), these beams cover the same total optical path, making the interferometer relatively insensitive to mechanical vibrations. The probe and the reference beams are made to superpose after passing through a second $\lambda/4$ plate (oriented at $45^\circ$). A calcite beam displacer splits the superposed beams into s- and p- polarizations. An imaging lens L3 is used to obtain two images of the echelon on the camera sensor with crossed
polarizations. The images show the glass echelon steps, each corresponding to specific time delays, covering the full time window from 0 to 150 ps with 10 ps steps, as shown in Fig. 2(b).

<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Optical design of the "common-path" single-shot Sagnac interferometer. L, PBS, NPBS, $\lambda/2$ and $\lambda/4$ stand for convex lens, polarizing beam splitter, non-polarizing beam splitter, half-wave plate and quarter-wave plate respectively. The two reference and signal probe beams are focused at normal incidence onto the front surface of the sample and are counter-propagating inside the Sagnac interferometer path PBS1-PBS2-M1-M2. The optical delay between the reference pulse and the signal pulse at the sample is about 2 ns. It corresponds approximately to the Sagnac optical path length. The signal and reference beams that exit the interferometer cover the same total optical path, are superimposed near collinearly, and interfere at the camera sensor. (b) Crossed-polarized images of the echelon on the camera. Each step corresponds to a specific time delay. Note that the reference and signal beams are intentionally set not perfectly collinear in order to obtain a 1D fringe pattern on the camera. (c) Fringe intensity versus horizontal pixel number, showing the $\pi$-phase shift between s- and p- polarized images.</figcaption></figure>

The $\lambda/4$ plate before the calcite beam displacer is adjusted to achieve phase sensitivities for s- and p- beams that are equal in magnitude and opposite in sign [30]. Mathematically, the intensities of the s- and p- beams on the camera sensor take the conventional form,

$$I_{s,p} = I_0[1 \pm m \cos(\phi + \pi/4)], \quad (1)$$

with $I_0 = (1 + \rho^2)/2$, $m = 2\rho/(1 + \rho^2)$ the interference contrast and $\rho$ the Fresnel coefficient of optical reflection of the sample. In order to extract the phase change induced by the shock wave, we perform the simple operation on the echelon image. For each s- and p-echelon step, we select a narrow horizontal strip to analyze the image intensity versus the horizontal pixel number. An example of such an extraction is shown in Fig. 2(c), illustrating the $\pi$ phase shift between the s- and p-polarizations. The optical phase of the s- and p- interference pattern within the strip was then extracted by fitting the intensity distribution to a sinusoidal function, as described in [31]. Note that the phase could also be extracted using FFT.
In our setup, the timing of the s- and p-probes is arranged such that the s-probe reflects off the sample before the laser shock pulse, making it the reference probe, while the p-probe that reflects after the laser shock pulse serves as the signal probe. The reference phase was then subtracted from the signal phase measured on the shock image. This approach allows for the detection of very small changes in optical phase using the Sagnac interferometer.

### 3. Experiment and data analysis

Femtosecond laser pulses can generate shock waves in metal thin films using frustrated laser ablation. Typically, a femtosecond laser pulse is focused through a transparent substrate onto a metal film, where the laser energy heats the electrons within the skin depth of the metal. The thermal energy of the electrons strongly heats the metal lattice via electron-phonon interactions well beyond the melting point and produces an expanding plasma that launches a shock wave in the metal [32]. Once the laser-excited shock wave reaches the metal free surface, it produces an out-of-plane surface displacement that can be measured interferometrically.

The targets used in the experiments are polycrystalline aluminum films of 200 nm thickness, produced by vapor deposition onto 250 µm thick BK-7 microscope cover slips (Fisher Scientific) and PMMA substrates (GE Plastics). The laser shock pulse energy was set at 130 µJ energy. This laser energy corresponds to an intensity of approximately 6.4 GW/cm², which is significantly higher than the damage threshold for metals but much lower than that for dielectrics [33]. This ensures that the substrate does not affect the transmission of the laser shock pulse and that the shock wave generation process occurs entirely within the metallic sample.

Figure 3 shows the relative phase shift as a function of time delay between the laser shock pulse and the train of probe pulses measured using single-shot Sagnac interferometry. We see no inherent effect of the substrate properties on the shock generation mechanism and the shock dynamics. This is evidenced by identical slopes in Fig. 3 of the best fit linear curve for data points between 0 ps to 110 ps for both glass and PMMA substrates. The pressure achieved in the aluminum film is determined from the corresponding shock and particle velocities derived from the free surface velocity measurements. A measurement of the reflected light from a bare aluminium surface as a shock wave emerges yields a phase shift that is primarily a result of the movement of the free surface. Funk et al. measured phase shifts due to changes in aluminium optical properties at the peak of a pressure wave, but they are orders of magnitude smaller than the phase shifts due to surface motion discussed here [14]. We can therefore safely neglect the purely optical contribution of the shock-induced phase shift.

The one-dimensional surface displacement $\Delta\delta$ is related to the phase shift $\Delta\phi$ by the equation

$$ \Delta\delta = \frac{\lambda}{4\pi}\Delta\phi \qquad (2) $$

where $\lambda$ is the probe wavelength. As shown in Fig. 3, with phase noise around $\Delta\phi \sim 0.05$ radians, the minimum displacement that can be measured in a single shot by our technique is approximately 3 nm, or roughly 1/300th of the wavelength. This demonstrates outstanding sensitivity.

The free surface velocity, $U_{fs}$, is calculated using the change in the phase shift, $\Delta\phi$, and the change in time $\Delta t$ as given by

$$ U_{fs} = \frac{\lambda}{4\pi} \frac{\Delta\phi}{\Delta t} \qquad (3) $$

The slope of phase shift versus time, $\Delta\phi/\Delta t$, as measured by the Sagnac interferometer in Fig. 3, is 0.005 rad/sec. Thus, $U_{fs}$, calculated using Eq. (3), is 0.32 nm/ps (or 320 m/s). For a free surface at low shock pressures, the particle velocity $U_p$ is well approximated as half of the free surface velocity [34,35]. Thus, the particle velocity $U_p \approx U_{fs}/2 = 0.16$ nm/ps in our experiments. The shock speed $U_s$ and pressure $P$ in aluminum can be determined from $U_p$, the aluminum
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Single-shot time-resolved optical phase shifts measured as a laser-induced shock wave reaches the free surface of a 200 nm thick aluminum layer deposited either on a glass or PMMA substrate. The slope of the linear fit is identical for both substrates and is the time-dependent pressure profile. Using the phase shift versus time to extract the free surface velocity, the shock pressure is calculated to be 2.5 GPa.</figcaption></figure>

Hugoniot (experimental $U_s$ versus $U_p$ relation), and the Hugoniot-Rankine equations that account for conservation of mass, energy, and momentum across a shock discontinuity [19]. A more extensive discussion of free surface velocity measurements in thin metal films is available from Gahagan et al. [12]. The shock speed $U_s$ is calculated from the aluminum Hugoniot equation [36]

$$U_s = 1.34U_p + c \quad (4)$$

where $c = 5.38$ nm/ps is the longitudinal sound speed. We obtain $U_s = 5.59$ nm/ps. Then, the shock pressure $P$ is calculated using the relation

$$P = P_0 + \rho_0 U_p U_s \quad (5)$$

where $P_0$ is the initial pressure and $\rho_0$ is the density of aluminum, 2.712 g/cm³. We obtain a shock pressure $P = 2.5$ GPa for both the soft substrate (PMMA) and a hard substrate (glass). As the experimental curve in Fig. 3 becomes linear very quickly, on a timescale below 10 ps, it implies that the pressure remains constant at 2.5 GPa for the rest of the time history. This is consistent with experimental data obtained previously on aluminium films [37].

## 4. Summary and outlook

We have demonstrated the ability of single-shot Sagnac interferometry based on low-cost echelons to provide detailed information about the dynamic properties and pressure profile of laser-generated shocks propagating through metal thin films. We were able to determine that the choice of substrate material does not affect the excited shock pressure. The measured rise time was less than 10 ps and a constant pressure of 2.5 GPa is maintained for a few hundred picoseconds in samples on both PMMA and glass substrates.

We believe that our single-shot interferometric technique could be used not only to probe single-event shock dynamics but also to facilitate data acquisition of repetitive ultrafast phenomena, with shot-to-shot balanced readout, even in cases where far smaller time increments and hundreds of time points are needed, as used in other types of multi-time single-shot measurement [29].
Parallel detection of this sort enables data acquisition on a picosecond timescale without the use of slow mechanical delay lines. For longer time scale single-events, in the ten's of nanoseconds, a tilted cavity device such as in [38], that can provide a train of pulses with inter-delays in the ns range, could be adapted to obtain single-shot interferometry data.

**Funding.** Army Research Office (DAAD-19-02-D-0002, W911NF-18-2-0048); DEVCOM Army Research Laboratory (0601102A); Région Bretagne (SAD CHOCONDE); Direction Générale de l'Armement (ERE 2017 60 0040); Rennes métropole (AES).

**Acknowledgement.** We acknowledge insightful discussions with Dr. David Veysset, Dr. Jet Lem and Dr. Yun Kai. This work was supported by the U.S. Army Research Office through the Institute for Soldier Nanotechnologies, under Contract DAAD-19-02-D-0002 and under Cooperative Agreement Number W911NF-18-2-0048. We acknowledge funding from the DEVCOM Soldier Center and the Assistant Secretary of the Army for Acquisition Logistics and Training, specifically 0601102A Defense Sciences. Thomas Pezeril acknowledges financial support from DGA (Direction Générale de l'Armement) under grant ERE 2017 60 0040 as well as from Région Bretagne under SAD grant Choconde and from Rennes métropole.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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