Comparison of laser-produced plasma spatio-temporal electron density evolution measured using interferometry with simulation results

Laser-produced plasmas (LPPs) have been used extensively in various applications, including EUV lithography,¹ higher order harmonic generation,² pulsed laser deposition (PLD),³ acceleration of high-energy particles,⁴ nuclear fusion,⁵ as an analytical tool to probe materials,⁶ and as a laboratory testbed for high-explosion events.⁷ In each application, understanding how the plasma properties (such as density and temperature) evolve over the plasma lifetime is vital for optimizing the plasmas specific to that application. This includes aspects such as conversion efficiency, line emission, nanoparticle generation, molecular formation, particle velocities and distributions, recombination rates, and species concentrations.^1,7–10 The properties of an LPP are greatly influenced by numerous experimental parameters, specifically the laser wavelength, intensity, and pulse duration.^11–13 All of these factors influence the laser–target and laser–plasma coupling, and hence the fundamental properties of the plasma. These parameters determine how energy is deposited into the target material and the resulting plasma characteristics such as density, temperature, and expansion dynamics.¹⁰ Understanding these dependencies is crucial for optimizing LPPs for various applications.

To measure the fundamental plasma properties, a variety of different techniques may be used, including optical emission spectroscopy (OES),¹⁴ absorption spectroscopy,^10,15,16 interferometry,¹⁰ Thomson scattering,^10,17 and various others.^8,18 The adequate method typically depends on the state of the LPP, which can vary over several orders of magnitude in density and temperature.¹⁰ At moderate densities (⁠ $∼$ 1  $×$ 10¹⁵–1  $×$ 10¹⁷ cm⁻³) and temperatures (⁠ $∼$ 0.1–10 eV), the most commonly used method for measuring plasma conditions is OES, due to its experimental simplicity and non-intrusive nature.¹⁰ OES can then be used to determine the evolution of other plasma characteristics such as stoichiometry,¹⁹ plasma chemistry,²⁰ oxidation rate,²¹ and particle formation rate.²² 

However, early time diagnostics (⁠ $≤$ 50 ns) of LPPs using OES can be challenging due to the high-temperature and high-density conditions present in the plasma, which can vary rapidly over time and space. These conditions can lead to significant continuum emission, self-absorption,²³ deviations from local thermal equilibrium (LTE),²⁴ and large gradients in plasma properties,^10,25 which can make experimental measurements difficult or unreliable. Among the plasma diagnostic tools, interferometry and Thomson scattering tools are better suited for measuring early time properties of LPP although several experimental challenges exist.^10,26

Analytical analysis or simulations are also used to estimate the evolution of the early plasma properties. Some of the simulation tools regularly used for the early evolution of LPPs are FLASH or HYDRA codes.^27,28 However, although these codes have been extensively validated at high intensities (>1  $×$ 10¹² W cm⁻²),^29–32 validation at low-moderate laser intensities has been limited. In these regimes, errors resulting from approximations made regarding the laser–target interaction or uncertainties in the equation of state (EOS) models may become more prevalent.^27,29,33

In the present study, FLASH simulations of LPPs generated at a moderate laser intensity (⁠ $∼$ 10 GW cm⁻²) and at times up to 20 ns were validated against experimental measurements of electron density. The role of laser wavelength on the early time evolution of laser density is also evaluated. The LPP was generated by focusing the fundamental and various harmonics radiation from an Nd:YAG laser with 6 ns pulse duration. A Nomarski interferometer was used to measure the spatio-temporal density evolution of the expanding plasma. The evolution of the electron density, temperature, and charge state was then simulated using FLASH code, and the deviations between the experimental and simulated results are discussed.

A schematic of the experimental setup is given in Fig. 1. LPPs were generated by focusing 1064, 532, and 266 nm pulses from an Nd:YAG laser (Continuum Surelite III) at normal incidence onto a 5 mm diameter copper rod. The pulse width of the laser was $≈$ 6 ns full width half maximum (FWHM). The copper target was placed inside a vacuum chamber at air pressures $≤$ 10 mTorr. The laser energy was attenuated to $≈$ 45 mJ using a combination of a half-wave plate and a cube beam polarizer. The laser polarization used for plasma production was linear and vertical. An anti-reflection coated f = 15 cm plano–convex lens was used to focus the laser down to a $∼$ 300  $μ$m diameter spot size, which was determined by measuring the resulting crater size. This resulted in an average laser intensity of $≈$ 10 GW cm⁻² after accounting for transmission losses. Cleaning shots were used to ablate the target before data acquisition to remove any contamination on the target surface. The chamber was positioned on an x–y translator to move the target and avoid drilling.

Nomarski interferometry was performed by propagating a 532 nm, 4 ns FWHM probe pulse from an Nd:YAG laser (Continuum Minilite II) through the plasma and parallel to the target surface. Prior to the plasma, the probe pulse was first passed through a cube polarizer to set the laser polarization to 45°. After propagating through the plasma, the probe pulse was passed through a Wollaston prism to split the pulse into s- and p-polarized beams of equal intensity. A cube polarizer placed directly after the Wollaston prism then recombined the s- and p-polarized beams back into a single polarization, resulting in interference between the two overlapped beams. A lens placed after the cube polarizer was used for focusing the plasma interferogram image onto a CCD camera (ImageSource DZK 33UX250).

To investigate the early time dynamics of the LPP, the electron density of the plasma was measured using Nomarski interferometry. Examples of interferograms captured for 1064 nm excitation wavelength at various times after the plasma onset are shown in the top row of Fig. 2. The interferograms were then converted to phase diagrams using IDEA software,³⁴ with the resulting phase plots shown in the bottom row of Fig. 2. The phase plots were then converted to measurements of the electron density using methods detailed in other studies.¹⁰ Abel inversion was used to obtain spatially resolved maps of the electron density using the Fourier method³⁵ and assuming cylindrical symmetry in the plasma.

Figure 3 shows the log-scale variation in the spatially resolved electron density of the plasma generated using different laser excitation wavelengths (1064, 532, and 266 nm) and measured at 10 and 20 ns after laser ablation. Similar to other studies,²⁵ measurements of the electron density at distances close to the target surface (<100  $μm$⁠) could not be obtained due to the presence of a blackened region in the interferograms (see Fig. 2). The appearance of this blackened region could likely be attributed to significant refraction of the probe laser caused by large gradients in the electron density.^10,25 These regions are indicated by a brown bar at the target surface in Fig. 3. Similarly, electron densities in the outermost regions of the plasma were too low to produce a resolvable change in phase. Based on the camera resolution (2448 × 2048 pixels) and the spacing between adjacent fringes in the interferograms, the minimum measurable change in phase was approximately 0.14 rad. This corresponds to a minimum measurable electron density of about 1  $×1017 cm−3$⁠, assuming a plasma scale length of roughly 1 mm.

The electron densities obtained in Fig. 3 were found to be generally similar at all wavelengths, with densities varying between 2.0 and 3.6  $×$ 10¹⁹ at 10 ns and between 1.2 and 2.5  $×$ 10¹⁹ at 20 ns for different wavelengths at 100  $μ$m away from the target surface. Overall, the magnitudes of the electron densities were slightly larger at shorter wavelengths throughout most regions of the plasma. The plasma dimensions were also noted to be generally similar across all wavelengths, though this may have been caused by the lower limit of detection of the electron density (1  $×1017 cm−3$⁠). These results were found to be consistent with similar studies performed using OES, which also showed limited changes in density as a function of wavelength, albeit at the later stages of plasma evolution.^12,13,36 As noted in these studies, the variation in electron density could likely be attributed to the variation in the optical penetration depth of the laser as a function of wavelength, where^10, $lp∝nc=4π2ε0mee−2λ−2$⁠, $nc$ is the plasma critical density, $ϵ0$ is the vacuum permittivity, $me$ is the electron mass, e is the electric charge constant, and $λ$ is the wavelength. The larger optical penetration depth at shorter wavelengths allowed the laser to penetrate further into the plasma, resulting in a greater degree of laser ablation.^12,37 However, the lower optical penetration depth at longer wavelengths resulted in laser energy being deposited over a shorter length scale, resulting in larger electron temperatures.^36–38 The combination of these two effects resulted in similar values of the electron density at all wavelengths, though with slightly higher values at shorter wavelengths.

Using FLASH code, the experimental results were compared to simulations. FLASH simulations were performed in a 2D cylindrical geometry, with laser ray-tracing utilizing FLASH's 3D in 2D model for laser energy deposition. The laser pulse was simulated using a 10 GW cm⁻², 6 ns square wave with laser wavelengths of 1064, 532, and 266 nm. The laser energy deposition is calculated solely from the inverse bremsstrahlung process. Tabulated equation-of-state (EOS) models were generated for the target (Cu) and 10 mTorr air (nitrogen and oxygen) using PrOpacEOS.³⁹ The initial charge state of the copper was set to 0.02 and an initial temperature of 290 K. Additional information on the physics of the FLASH code can be found elsewhere.²⁷ The resulting spatial evolution of the simulated electron density, temperature, and charge state is shown in Figs. 4–6 for different wavelengths at 10 and 20 ns after laser ablation.

The simulated electron density contours shown in Fig. 4 were found to be in good agreement with the experimental density contours obtained in Fig. 3. Note that the same scale bar was used in both Figs. 3 and 4, for ease of comparison. The simulated electron densities were found to be largest at shorter wavelengths, with peak values varying between 3.6 and 6.3  $×$ 10²⁰ at 10 ns and between 1.4 and 4.7  $×$ 10²⁰ at 20 ns for different wavelengths. The plasma dimensions were also found to be consistent with Fig. 3, though the simulated distributions were slightly more elongated in the forward direction, particularly at 532 nm. However, it should be noted that the full extent of the plasma is not observable in Fig. 4 due to the lower limit of the electron density being set at 1  $×$ 10¹⁷ cm⁻³.

In comparison, Figs. 5 and 6 show the full extent of the plasma, including the lower density regions not shown in Fig. 4. These figures indicate that the length scale of the plasma was larger for longer wavelengths compared to shorter wavelengths, with the axial length varying between 0.7 and 1.1 mm at 10 ns and between 1.5 and 2.1 mm at 20 ns for 266–1064 nm. The larger length scale of the plasma at longer wavelengths could be attributed to higher electron temperatures in the outer regions, as shown in Fig. 5. The higher electron temperature in these regions would have allowed the plasma to reach higher velocities, resulting in a greater degree of expansion. As noted previously, the higher temperatures at longer wavelengths in the outer regions could be attributed to a shorter length scale of laser energy deposition at the forward end of the plasma.

Nonetheless, differences in temperatures were relatively small both spatially and as a function of wavelength, varying between 5 and 7 eV at 10 ns and between 3 and 4.5 eV at 20 ns for all wavelengths. The spatial uniformity in the electron temperature could be attributed to a high electron thermal conductance, with similar uniformity in the electron temperature being observed in other studies.⁴⁰ The similarity in the temperatures at different wavelengths could be attributed to the strong temperature dependence of the radiative cooling rate, where $qrad∼AT4$ (A and T corresponding to a scaling constant and the plasma temperature),²⁷ resulting in rapid equalization in temperatures between wavelengths. The uniformity in electron temperatures resulted in similar uniformity being observed in the charge states, with charge states varying between 2 and 3 at 10 ns and between 1.5 and 2.5 at 20 ns for all wavelengths, as shown in Fig. 6. Note that the high degree of uniformity in the charge states indicated that the differences in electron densities in Fig. 4 were primarily driven by differences in ablation rates, consistent with observations in other studies.¹² 

Additional insight into the plasma expansion process was gained by plotting the variation in the weighted average of the plasma properties as a function of time, as shown in Fig. 7. Weighted averages were calculated using the plasma density, with $n¯=(∫n ρ dV)/(∫ρ dV)$⁠. Generally, all properties were found to peak at the end of the laser pulse (6 ns) and decreased thereafter, with the electron density and temperature varying as $ne∝λ−0.7$ and $Te∝λ0.2$ at 6 ns. These dependencies were at least partially consistent with analytical models of plasma expansion into vacuum,⁴¹ where electron density and temperature are expected to vary as $ne∝λ−0.75$ and $Te∝λ0.5$⁠. The much lower dependence of the simulated electron temperature on laser wavelength could likely be attributed to a significant degree of radiative cooling in the plasma, which is not considered in the analytical model.

Following the end of the laser pulse, the differences in plasma properties at different wavelengths began to rapidly diminish with time. By 10 ns, the average temperatures and charge states were approximately equal for all wavelengths. The rapid equalization of the temperatures at different wavelengths could be attributed to the strong temperature dependence of the radiative cooling rate, where $qrad∼AT4$⁠. As the temperatures equalized, the charge states did as well. By 20 ns, the average electron densities were also approximately equal. For higher density regions, the increased densities led to higher pressures, which in turn caused faster expansion until the densities equalized. However, it should be noted that although the average plasma properties became approximately equal by 20 ns, differences in the spatial distributions of the plasma properties persisted out to longer times, as was shown in Figs. 4–6.

As shown in Figs. 3 and 4, the experimental and simulated electron density contours were found to be highly consistent. A closer comparison of electron density profiles is shown in Figs. 8 and 9. Figure 8 shows the axial comparison of the experimental and simulated density profiles along the plasma centerline at 10 and 20 ns after laser ablation for each wavelength. Figure 9 shows the lateral comparison of the density profiles perpendicular to the target surface at the approximate plasma center-points, corresponding to 0.25 mm at 10 ns and 0.5 mm at 20 ns. Errors in the experimental electron densities were approximated as $±1×1017$ cm⁻³ based on the lower limit of detection in the Nomarski interferometry setup discussed in Sec. II.

As shown in Figs. 8 and 9, slight differences in magnitudes and shapes were observed between the experimental and simulated electron density profiles. At 10 ns, these differences may be attributed to experimental measurement errors caused by large density gradients during the probe pulse.^10,25 By 20 ns, the spatial trends became more consistent, though slight magnitude differences persisted in the axial profiles (Fig. 8). Figure 9 suggests the axial differences were partly due to a dip in the electron density along the plasma centerline. Similar observations in plasma density along the plasma centerline were previously observed in Faraday cup studies of plasma expansion into vacuum.^8,42–44 In these studies, it was found that the ion flux distributions exhibited a clear dip in the ion flux along the plasma centerline. However, similar to the absence of the dip at 266 nm in both the experimental and simulated profiles, the dip was not always present in all studies.^45,46 Moreover, the dip was not always present in all regions of the plasma, generally favoring the lower density regions at the front of the plasma.^8,42 However, the exact cause of these dips was not well understood, and the differences in the dip locations observed in Fig. 9 could be due to a number of reasons. This includes uncertainties in EOS parameters or a lack of fidelity modeling the laser–target interaction near the target surface.^33,47

In addition to the differences in electron densities near the plasma center, Fig. 9 indicates that the experimental profiles were slightly wider compared to the simulated profiles at both 10 and 20 ns. This is consistent with the electron density distributions obtained in Figs. 3 and 4, where the experimental density distributions were found to have a wider, circular shape compared to the longer, more elliptically shaped simulated distributions. The differences in plasma widths and overall shapes of the experimental and simulated profiles could likely be attributed to uncertainties in the opacities used in the PrOpacEOS model. Previous studies have shown that opacities may vary by as much as several tens of percent between different EOS models.³³ These uncertainties can lead to errors in radiation rates, which can play a significant role in determining the plasma width during expansion, with lower radiation rates resulting in larger plasma widths.³¹ Based on the thinner plasma widths obtained in the simulated profiles in Fig. 9, and the more elongated simulated distributions noted in Fig. 4, the radiation rates were likely overestimated in the FLASH simulations. Several smaller sources of error were also likely present, both experimentally and in the simulations. This includes uncertainties in the various laser parameters (e.g., energy, spot size, pulse width), EOS parameters (e.g. heat capacities, thermal conductivities, charge states), the time-integrated nature of the interferometry, among others.

In this study, experimental measurements of electron densities from LPPs generated at moderate laser intensities for times up to 20 ns were used to validate FLASH simulations of LPP expansion into vacuum. Nomarski interferometry was used to measure the early time and spatial evolution of electron density. The study found that differences in peak densities between wavelengths were small, with shorter wavelengths resulting in higher electron densities and longer wavelengths resulting in larger plasma sizes and higher temperatures. The dependence of plasma properties on wavelength was attributed to variations in the optical penetration depth of the laser. Comparisons of simulation results with analytical models showed good agreement in the dependence of electron density on wavelength, with simulations indicating $ne∝λ−0.7$⁠. Experimental results were shown to have good agreement with FLASH simulations, though slight deviations were also present. Comparisons of experimental and simulated profiles indicated that deviations were primarily caused by a reduction in the electron density near the plasma centerline. Similar results were observed in other studies,^8,42–44 though the exact cause was unknown.

This work was partially supported by the Department of Defense (DoD), Defense Threat Reduction Agency (DTRA) under Award No. HDTRA1-20-2-0001. The content of the information does not necessarily reflect the position or the policy of the federal government, and no official endorsement should be inferred. Pacific Northwest National Laboratory is a multi-program national laboratory operated by Battelle for the U.S. Department of Energy under Contract No. DE-AC05-76RL01830.

The authors have no conflicts to disclose.

Mathew P. Polek: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Tirtha R. Joshi: Investigation (equal); Software (equal); Writing – original draft (supporting); Writing – review & editing (equal). Mathieu Bailly-Grandvaux: Writing – original draft (supporting); Writing – review & editing (equal). Rick B. Spielman: Writing – review & editing (equal). Farhat N. Beg: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Sivanandan S. Harilal: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.