1. Introduction

Interest in the physics of pulsed laser ablation (PLA) of solid targets has grown because of its relevance to applications such as laser material processing, including machining, milling, nanostructure growth, and pulsed laser deposition (PLD) methods, as well as analytical methods including laser-induced breakdown spectroscopy (LIBS) and PLA-laser absorption spectroscopy (LAS) [1–11]. The PLA involves a complex amalgamation of physical processes. During ablation, a laser pulse induces rapid phase changes and ionization that form a highly transient plasma environment [5,11]. The plasma expansion is highly dynamic and depends on laser parameters, including the pulse duration, wavelength, energy [6,7], as well as the ambient gas conditions and target characteristics [5,12]. Therefore, it is important to study all aspects of the laser-target interaction, plasma formation, and plasma evolution in order to appropriately understand PLA. The absorbed laser energy, in particular, strongly contributes to plasma formation during PLA [5,13]. The energy deposited in the target is released via thermoelastic expansion, formation, evolution, and relaxation of a plasma, as well as generation and expansion of an acoustic shockwave [5,11].

The shockwave carries a significant amount of information regarding the processes which occur during laser ablation [11,13]. For example, the temporal evolution and geometry of the shockwave are influenced by the energy of the laser pulse and the shape of the generated plasma [13,14]. The plasma shape can be influenced by the beam profile and focusing geometry, pulse duration, and the target geometry. Ultrashort-pulse ablation, for example, usually results in preferential expansion in the target normal direction initiated by Coloumb explosion; whereas, an increased rate of elastic collisions during short-pulse ablation has been shown to cause increased lateral motion of the plume [15]. Further, the nature of the target contributes to plume shape: heavier constituent species, e.g. tungsten, have been shown to cause plume sharpening in low-pressure environments because of larger momentum than lighter constituents [16]. Elongated plasmas formed in liquids (or gases) have been observed to create shockwaves with a cylindrical shape [13]. The information provided by the shockwave about fundamental processes occurring during PLA motivates improving the fidelity of methods for characterizing the shock. Methods commonly used to determine the laser-target coupling efficiency include investigation of the ablation crater [17], mass removal [18], and tracking the expansion rate of the shockwave associated with the LIP [13]. Important factors to consider when studying shockwaves include the desired spatial and temporal resolution as well as accounting for event-to-event variability, which can be caused by fluctuations of the laser pulse energy. Current methods do not correlate measurements to individual laser shots and may not provide reliable information in the cases of shot-to-shot variability, multiple detonation sites [19,20], or when the number of measurements are constrained by limited sample size or measurement time. In this work, we correlate measurements of the energy deposited to the same shockwave instead of repeated realizations. We use shadowgraphic imaging to study the evolution of the shockwave, and we use the Taylor-Sedov blast model to extract the energy deposited in the target which governs the shock expansion rate.

The Sedov-Taylor blast model, initially used to describe blast waves produced by nuclear explosions [1,13], has since found applications in PLD [21,22], interstellar phenomena [2,3], and high-explosive detonations [1,23]. This model can also be used to describe the evolution of a laser-induced shockwave [13,23]. The shockwave radius $R$ varies with time $t$ as

Two conditions constrain the use of Sedov-Taylor model for point explosions. The model is considered valid only (1) after the mass of ablated material equals the mass of the gas enveloped by the shockwave; and (2) before the pressure within the shockwave has come to equilibrium with the ambient gas pressure [1,24]. These two conditions can be associated with the limits to the shockwave radius:

The shockwave radius can be determined by imaging methods like shadowgraphy [27,28], interferometry [29], schlieren photography [30], and holography [31]. High-speed imaging can be coupled with other diagnostic methods, such as spectroscopic measurements, to provide temporal and spatial resolution of the shockwave along with additional information on plume species and concentrations [16,32]. In this work we use shadowgraphy for its relatively simple implementation and data analysis. A common technique used to determine the deposited energy involves measuring the shock dimension at several delays after the initial laser-target interaction. In this approach, the time-dependent radii are fit using the blast model of appropriate geometry [5,25]. Typical implementation of this approach involves scanning the delay of a single, pulsed probe laser with respect to the ablation laser. In this configuration, the shockwave is measured for different ablation laser shots; therefore, reproducing the ablation conditions between shots is vital for accurate measurements. First, shot-to-shot variability in laser parameters such as pulse energy, pulse duration, and beam profile should be sufficiently small. Second, the target surface composition and geometry should not change appreciably between shots. Finally, the local composition, pressure, and temperature of the ambient environment should be constant. In practice, the laser may exhibit significant variability, and the target homogeneity may vary either during translation or because of pitting or drilling from ablation. These factors can contribute to notable changes in ablation conditions between shots.

When an induced plasma cannot be consistently reproduced, or a limitation is set on the number of measurements, a single-shot measurement method is preferred. One such case is when the ablation uses multiple ultrashort laser filaments. Filamentation, which results from nonlinear propagation of intense laser pulses through media, has been combined with analytical spectroscopy methods including LIBS to extend the distance to which compositional measurements can be performed. For laser peak powers that greatly exceed the threshold power for self-focusing, the modulation instability causes stochastic formation of multiple intense filament cores, referred to as multiple filamentation. Previous work [20] shows that the standard scanning approach cannot be used with the blast model to determine the deposited energy in the target in these conditions. Here we demonstrate a modified single-shot method based on shadowgraphy that overcomes this limitation that arises in the application of Sedov-Taylor model to highly variable conditions. We employ a double-pulse probe laser to image the shockwave at two delays and fit the two measured shock dimensions with the blast model. This method reduces the measurement uncertainties for the deposited energy by making two measurements of the same shockwave instead of repeated realizations of the shock wave, where the laser parameters, ambient conditions, and target morphology can vary. We show that the the initial assumption of expansion dimensionality is an important consideration in the application of this method. The proposed method provides several advantages over the existing methods, including the ability to determine the detonation energy for a single ablation laser shot while also providing the resolution to potentially track multiple detonation sites that may arise in the case of nonuniform ablation beam profiles such as those exhibited in multiple filamentation.

2. Methods

The experimental setup is shown in Fig. 1(a). The single-shot, double-pulse (SSDP) method involves using a pair of probe pulses to measure two delays for a single ablation laser shot. The 1064-nm Nd:YAG ablation laser (Surelite, Continuum) with a Gaussian beam profile, pulse energy of 40$\pm$2 mJ, and pulse duration of 10 ns was focused using a 15-cm focal length lens ($f$/25) onto an aluminum target, producing a measured spot size with a diameter of 1.5 mm. The sample was continuously translated so that individual laser shots did not overlap, and an unblemished portion of the sample was always used. This avoided excessive drilling of the sample surface that could influence the shot-to-shot reproducibility of the shock. A flat target is used in order to evaluate the method applied to hemispherical shockwaves. We utilize a double-pulse Nd:YAG laser (1064 nm, 10 ns, <1 mJ, Evergreen, Quantel) system where a single monoblock houses both of the two probe lasers. Both lasers within the monoblock are linearly polarized such that one laser pulse is vertical and the second horizontal. The two cross-polarized probe beams are separated using a thin-film polarizer. The probe pulses are synchronized with respect to the ablation laser using a digital delay generator (DG645, Stanford Research Systems). The interpulse delay $\Delta t$ is defined as the difference between the probe 2 delay ($t_2$) and probe 1 delay ($t_1$). The probes propagate perpendicular to the ablation laser direction and are incident on two CMOS cameras (3.75 µm square pixel dimension, CGE-B013-U, Mightex), with one camera capturing each probe beam. Example shadowgrams of the shockwave measured at different delays are shown in Fig. 1(b). Figure 1(c) overlays the shockwave profiles measured for successive delays in the range of 0.1 µs–1 µs. An image processing algorithm is developed in order to automate measurements of the shock dimensions. The steps for processing a shadowgram are outlined in Fig. 2(a–d). First, the reference beam profile is subtracted from each shadowgram. Second, each reference-subtracted image is binarized using a dynamic threshold. Finally, the shockwave profile is identified, and its radius is measured.

 

Fig. 1. (a) Double-pulse (DP) shadowgraphy setup for imaging the shockwave produced from a single ablation laser shot. T: aluminum target; M: mirror; L$_1$: ablation laser focusing lens, f.l. 15 cm ($f$/25); L$_2$ imaging lens; TFP: thin-film polarizer. Example shadowgrams depicting the shockwaves from the same ablation laser shot at probe delays of (b) 0.5 µs and (c) 2.5 µs. The direction of the ablation laser is depicted by the red arrow, and the locations at which shockwave radius measurements are made is depicted by the white arrows. Shockwave radii are measured using a custom image analysis algorithm. (d) Binarized shock profiles are overlaid onto a single image for delays in the range 0.1 µs–1 µs, showing expansion measured for different ablation laser shots.

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Fig. 2. (a) Example reference beam profile for one of the probes. An example shockwave image at a delay of 2.5 µs (b) before reference-subtraction, (c) after reference subtraction, and (d) after the reference-subtracted image is binarized.

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3. Results and discussion

We compare detonation energies determined using the SSDP approach to those determined using the traditional scanning method, in which the shock dimensions are measured for different ablation laser shots. In the scanning method, measurements are performed in increments of 0.25 µs, from 0.25 µs to 4 µs, in order to reproduce the delays measured with the SSDP technique. The shock radii were measured 100 times for each delay. Figure 3(a) presents the distributions of shock radii with corresponding normal distribution fits for delays of 0.25 µs, 0.5 µs, and 0.75 µs. In Fig. 3(b), the shockwave radii measured at different delays are fit with the Sedov blast model [Eq. (1)]. Each measurement of the shockwave radius represents the average of 100 shots. The standard deviations of the measured radii in Fig. 3(b) range from $\sim$11.2 to $\sim$35.7 µm. Both the expansion dimensionality ($\alpha$) and the deposited energy ($E_d$) are free parameters in the fit. Measurements which do not satisfy the model limits given by Eq. (2) are excluded from the analysis [24,33]. The radius of the shock wave at time zero is assumed to be zero in order to constrain the fit. The detonation energy extracted from the scanning measurements is 28.5 $\pm$ 18.2 mJ for the ablation laser energy of $\sim$40 mJ. The average dimensionality parameter is determined from the fits to be 2.75 $\pm$ 0.11, corresponding to a nearly spherical shock geometry. The Sedov-Taylor blast model is intended for implementation when assumptions of an ideal blast are applied, but applications with LIP have introduced experimental data that deviates from the ideal blast wave such that non integer dimensionalities are observed [1,34]. Jeong et al. [34] investigated this deviation and suggested that though similar, laser induced plasma have inherent differences from an ideal blast that lead to mixed dimensionality. Specifically, they highlighted that the ideal blast wave assumes the mass released to be negligible, but in an LIP, energetic vapor can be present at high power densities and condense to nano and sub-micron particles. Palanco et al. [35] also identified the limitations of the ideal blast model and identified that at longer time scales the LIP shockwave is well approximated as hemispherical ($\alpha =3$), but at early times the front propagates much faster than predicted by the ideal blast wave, and a decrease in $\alpha$ is observed. The deviation is a result of the model not accounting for energy loss or gain following the finite energy deposition and radiative processes such as phase explosion that are specific to LIP. Schmitz et al. [36] provided an extensive summary of laser parameters such as decreased environmental pressure, fluence, and spot size that induce deviations due to their effects on the assumption of an instantaneous release as a point explosion and the assumption of a strong shock. Next, the distributions for the shock radii at each delay from Fig. 3(a) are randomly sampled 500 times in order to generate the distributions of fit parameters which may be determined from this data set, shown in Fig. 4. The expansion dimensionality is well described by a normal distribution with a mean value of 2.75 and a standard deviation of 0.07. On the other hand, the deposited energy is better represented by a right-skewed distribution with a mean energy of 31.3 mJ and a standard deviation of 12.1 mJ. In the experiment, the deposited energy is constrained between 0 mJ and 40 mJ, which may contribute to the observed skew. The distribution of deposited energy is consequently fit with a gamma distribution, shown in red on Fig. 4(b). The gamma distribution [Eq. (4)] is fit with a shape parameter (a) and scale parameter (b). The expected mean ($\mu$) and standard deviation ($\sigma$) of the gamma distribution are determined from the shape and scale parameters by Eqs. (5) and (6). In our analysis, a shape parameter of 6.75 and scale parameter of 4.64 are determined.

 

Fig. 3. (a) Distributions of shockwave radii from 100 measurements are fit with a normal distribution model calculated for delays of 0.25 µs, 0.5 µs, and 0.75 µs.(b) Example fit of the blast model to the time-dependent shockwave radii determined using a scanning approach. The average detonation energy determined from 100 scanning measurements is 28.5$\pm$18.2 mJ.

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Fig. 4. Distributions of 500 measurements for the (a) expansion dimensionality ($\alpha$) and (b) deposited energy ($E_d$). Data are fit with (a) normal and (b) gamma distributions, respectively. Each measurement of the fit parameters is made for a randomly sampled set of shock radii from the distributions in Fig. 3.

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Fig. 5. (a) The expansion dimensionality estimated with the SSDP technique as a function of the interpulse delay with the mean value and one standard deviation from the scanning method highlighted in red and grey, respectively. (b) Dimensionality $\alpha$ estimated with the SSDP method. The values are normalized to the standard deviation of $\alpha$, such that $\langle \alpha \rangle$ and $\sigma _{\alpha }$ are both obtained from the scanning method. The interpulse delay $\Delta t$ is defined as the difference between the probe 2 delay ($t_2$) and probe 1 delay ($t_1$).

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Fig. 6. (a) Energy reconstructed from measurements with various interpulse delays, with the mean value and one standard deviation from the scanning method highlighted in red and grey, respectively. Relative error in the reconstructed deposited energy estimated with the SSDP method when (b) $\alpha$ is unknown, and (c) $\alpha = 2.75$. The values in (b) and (c) are normalized to the standard deviation of the deposited energy estimated with the scanning method. Both (b) and (c) use the same color scale. The interpulse delay $\Delta t$ is defined by the difference between the probe 2 delay ($t_2$) and probe 1 delay ($t_1$).

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Next, we show that the determination of deposited energy using the two-point SSDP method may be improved to better match that determined by the scanning approach with a reasonable assumption or a separate measurement of the dimensionality. In the subsequent analysis, the dimensionality is fixed in the model. First, a reasonable assumption is made of spherically-symmetric expansion with dimensionality $\alpha =3$. As shown in Fig. 6(a), with this assumption the deposited energy is consistently underestimated. When $\alpha$ is assumed to be 3, no interpulse delay results in reconstruction of the deposited energy within one standard deviation of the scanning method. In contrast, when $\alpha$ is a priori known, such as $\alpha$ = 2.75 obtained from the scanning measurement, the reconstructed deposited energy is consistent with the scanning measurement. When $\alpha$ is a free parameter in the two-parameter fit, SSDP yields a large uncertainty in the deposited energy. Figure 6(a) seeks to identify the optimal combination of parameters that minimize the deviation from the true values of deposited energy and dimensionality, when the scanning method reconstruction is taken to be the true value. In the example measurement, when $\alpha$ is unknown, the energy can be determined within one standard deviation of the scanning method at a certain limited range of interpulse delays, but there is no evidence of any generality in the optimal choice of interpulse delay.

To further understand the role of interpulse delay, the deposited energy estimated with the SSDP method is shown normalized to the standard deviation of the scanning method estimate when $\alpha$ is unknown and when $\alpha$ = 2.75 in Fig. 6(b) and (c), respectively.

A color coding is used in Figs. 5 and 6 to indicate the relative deviation of the reconstructed dimensionality and deposited energy from that obtained in the scanning measurement. The a priori knowledge of the dimensionality provides the best agreement in the energy deposited between the proposed SSDP approach and the standard scanning method. A reasonable assumption of symmetry was found to yield consistent but lower energies than the average measured using the scanning approach. Allowing both the dimensionality and deposited energy to vary freely in the fitting procedure yields large discrepancies from the averages mean values obtained from the scanning method. In our experiments with the SSDP method, the scanning measurement is assumed to provide the true value of both the energy deposited and the dimensionality due to its wide acceptance in studies of LIP. When the assumption of a strong shock is not satisfied, the scanning method can present non-integer dimensionalities. In fact, this is often true for LIP applications of the Sedov-Taylor model. Despite this inherent deviation from an ideal blast for LIP, the scanning method is taken as the standard benchmark of measurement in the absence of an alternative measurement. A direct measurement of dimensionality would improve the interpretation of the scanning method results. This would require a more sophisticated measurement, such as two orthogonal shadowgrams or a tomographic reconstruction.

4. Conclusion

The SSDP technique yields two time-delayed shadowgrams of a single shockwave to which the Sedov-Taylor blast model can be fit. For the entire range of interpulse delays used in this study, the deposited energy determined using the SSDP method is in good agreement with the mean obtained from the scanning method if the dimensionality of the shockwave is known. In contrast, when the expansion dimensionality is unknown, a considerable discrepancy in the deposited energy is observed between SSDP and the standard scanning measurement because of the high sensitivity of the model to shock expansion dimensionality. When the blast model is fit to only two radius measurements, the dimensionality and the deposited energy cannot be reliably determined as free parameters in the model. In summary, the SSDP method is determined to be viable when the expansion dimensionality is known or can be determined from previous scanning measurements or by other means [16,29]. In this case, the SSDP approach enables the determination of detonation energy for a single ablation event while also providing the resolution to track potential multiple detonation sites, such as those that may be produced by filamenting femtosecond beams. When integrated as a diagnostic measurement to determine the deposited energy, the SSDP approach could provide insight into the complex ablation mechanisms and improve the understanding of signal and background sources in analytical measurements via techniques such as laser-induced breakdown spectroscopy.