# Review on Laser Interaction in Confined Regime: Discussion about the Plasma Source Term for Laser Shock Applications and Simulations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}), the fast absorption and ionization of matter are obtained. A high-pressure plasma ($P>$ GPa) is then created, and a strong shock wave thus propagates inside the material. This mechanism is used in LSP to induce compressive residual stresses (CRS) that improve the fatigue life of the treated part [16,17,18]. Moreover, LSP also helps treated materials fight corrosion, and may be used for underwater applications [12,19,20].

## 2. Fifty Years of Laser/Matter Interaction in Confined Regime: Plasma Generation, Significant Experimental Data, and Summary of Pressure Measurements

#### 2.1. General Considerations about Confined Plasmas

#### 2.2. Laser/Matter Interaction: Plasma Generation and Breakdown in Dielectric Medium

^{2}) is shown. The pressure generated on the target by the plasma (estimated with simulations as presented in Section 4.2) for this intensity is also plotted. The maximum reached pressure is 2.2 GPa, in agreement with the fact that this maximum pressure is a function of the square root of the laser intensity ($P\approx 2.2\sqrt{I}$) [28]. The plasma pressure FWHM (16.3 ns) is approximately 2.2 times the laser pulse duration. In this example, there are no breakdowns and all the laser is interacting with the main plasma on the target surface. However, at higher intensities, this pressure tends to saturate and no longer increases with the intensity: a breakdown plasma appears in the confinement and absorbs the remaining laser energy (which should have been absorbed by the main plasma).

#### 2.2.1. Main Plasma Generation

- At the beginning, when the laser pulse starts irradiating the metal target, only a small fraction of the energy is absorbed by the metal, the other part being reflected. The electrical current generated at the metal surface by the electromagnetic wave tends to heat it by joule heating, on a very small layer called the skin depth and given by$$\delta =\sqrt{\frac{2}{\omega \sigma \mu}}$$Typical values of $\delta $ for aluminum at 1064 nm give a depth skin of 5 nm. This will be the initial size of the plasma, after the metal is vaporized and ionized by joule heating.This first step depends on the initial metal reflectivity, hence it depends on both the used metal and its surface state. However, at the high laser intensities used in the laser shock (>1 GW/cm
^{2}, far from the ablation threshold of metals typically of approximately 0.2 GW/cm^{2}), losses associated with this mechanism have been shown to be negligible [40,41] and independent of the used metal. Furthermore, this step is considered to be almost instantaneous in the whole process. - In a second step, the absorption of the laser energy by the plasma is assumed to be close to 100%. The main mechanism of absorption involved here is the inverse Bremsstrahlung (IB [42], as can be seen in Figure 3). IB is a collisional process involving three particles: a photon, an electron, and an ion. By conservation of the momentum, the photon’s energy is absorbed and the kinetic energy of both the ion and the electron are increased. Furthermore, electrons are accelerated in the electrical field of the laser and they transfer their energy to heavy particles by collision, leading to a global increase in plasma temperature.Thus, this mechanism is highly dependent on the frequency ${\nu}_{ei}$ at which the electron and ion collide.This frequency may be obtained using Lorenz’s model [43,44]:$${\nu}_{ei}\propto \frac{{n}_{e}{Z}_{i}}{{T}^{\frac{3}{2}}}$$Altogether, the optical index of the plasma is given by$$n=\sqrt{{\u03f5}_{p}}$$$${\u03f5}_{p}\approx 1-\frac{{n}_{e}}{{n}_{c}}-i\frac{{n}_{e}{\nu}_{ei}}{{n}_{c}\omega}$$$${n}_{c}=\frac{{m}_{e}{\u03f5}_{0}{\omega}^{2}}{{e}^{2}}$$As the absorption coefficient ${\alpha}_{IB}$ is related to the imaginary part ${n}_{2}$ of the refractive index:$${\alpha}_{IB}=\frac{2{n}_{2}\omega}{c}$$Thus, the absorption by IB is given by$${\alpha}_{IB}\propto \frac{{n}_{e}^{2}{Z}_{i}{\lambda}^{2}}{{T}^{\frac{3}{2}}}$$Finally, if we consider a linear variation of the electronic density from 0 (at the plasma outside surface) to ${n}_{c}$ (in the depth of the plasma) on a total length L, then we have:$${A}_{IB}\propto 1-{\mathrm{e}}^{-\frac{{Z}_{i}L}{{\lambda}^{2}{T}^{\frac{3}{2}}}}$$
- The last step begins at the end of the laser pulse (${t}_{e}\approx 2\tau $, $\tau $ the laser pulse duration (FWHM)). The pressure of the plasma starts to decrease following an adiabatic law of Laplace:$$P\left(t\right)V{\left(t\right)}^{\gamma}=P\left({t}_{e}\right)V{\left({t}_{e}\right)}^{\gamma}$$The pressure maintained in this last part is no longer useful for the shock process. However, the temperature is greatly important as the duration of this slow adiabatic release will mainly drive the thermally affected depth and thus whether the thermal protective coating shall be used.Indeed, a 1D thermal model (temperature ${T}_{1D}$ applied during a duration ${\tau}_{1D}$) gives the following depth that will be affected by the heating:$${z}_{1D}=\sqrt{{\kappa}_{m}{\tau}_{1D}}$$To reduce the thermally affected depth ${z}_{1D}$, one should ensure that the duration ${\tau}_{1D}$ of the plasma’s thermal loading is as short as possible.Furthermore, depending on the used laser spot size, a rarefaction wave propagates from the edges of the plasma towards its center. When it arrives at the center ($t={\tau}_{R}$), a fast release of the pressure occurs which has been previously described by Pirri [46,47]:$$P\left(t\right)\propto {\left(\frac{t}{{\tau}_{R}}\right)}^{-\frac{6}{5}}$$

#### 2.2.2. Breakdown Plasma Generation

- Avalanche ionization (AI)—here, electrons are accelerated by the electrical field of the laser pulse: their kinetic energy (${E}_{k,e}$) increases and becomes sufficient to ionize an atom (${E}_{k,e}>\Delta {E}_{e}$, with $\Delta {E}_{e}$ being the ionization potential). This mechanism evolves exponentially: for one initial electron, and after n avalanche processes, there will be ${2}^{n}$ electrons created.If ${\eta}_{AI}$ is the rate of AI per electron, then the density evolution is given by$${\left[\frac{\mathrm{d}{n}_{e}}{\mathrm{d}t}\right]}_{AI}={\eta}_{AI}{n}_{e}$$The rate of AI, ${\eta}_{AI}$, was calculated by Kennedy [50]. To be effective, this mechanism requires either a minimal number of electrons (seeds) to generate the breakdown or longer laser pulse duration. In the case of water confinement, the initial density has been estimated to be approximately ${10}^{9}$ cm ${}^{-3}$ [48].
- MultiPhotonIonization (MPI)—k (k an integer) photons are simultaneously absorbed by the atom (ionization) if the following condition is obtained:$$\frac{khc}{\lambda}\ge \Delta {E}_{e}$$As this is a quantum process, it can be shown that this mechanism is less effective with a higher value of k (thus a higher value of wavelength $\lambda $).

- The electrons leaving out the focal volume by diffusion. If we use a rate ${\eta}_{d}$ of diffusion per electrons, the variation on the density is given by$${\left[\frac{\mathrm{d}{n}_{e}}{\mathrm{d}t}\right]}_{d}=-{\eta}_{d}{n}_{e}$$
- Losses by recombination: an electron–hole pair is recombined at a rate of ${\eta}_{r}$ per electron and hole. Then, the density variation is given by$${\left[\frac{\mathrm{d}{n}_{e}}{\mathrm{d}t}\right]}_{r}=-{\eta}_{r}{n}_{e}^{2}$$

- By measuring the pressure of the plasma (see the following part concerning the rear-free surface indirect determination of the plasma pressure)—when the breakdown threshold is reached, the pressure stops increasing with the laser intensity and fluctuates around a maximum value associated with the pressure value at the breakdown intensity threshold ${I}_{b}$ [23,25,26,27].

^{2}; ${I}_{b,2\omega}$ = 6 GW/cm

^{2}; ${I}_{b,3\omega}$ = 4 GW/cm

^{2}at 1064 nm, 532 nm and 355 nm, respectively, [26]. More recent experiments performed at 1064 nm and 532 nm, for a pulse duration of 7 ns, gave different results: ${I}_{b,\omega}$ = 8 GW/cm

^{2}and ${I}_{b,2\omega}$ = 10 GW/cm

^{2}[53].

^{2}(when it occurs at the confinement surface as it classically does) to more than 20 GW/cm

^{2}when it occurs at a depth (volume) of water confinement. Hence, the purpose of the WTC is to use a high thickness of confinement to ensure that the laser intensity at the confinement surface is weaker than the known breakdown threshold; thus, the breakdown will only occur in a volume of the confinement, at a higher threshold (at least two times higher). Finally, the maximum pressure of saturation will be increased to 10–12 GPa instead of 6–8 GPa (when a small layer configuration is used).

#### 2.3. Experimental Data and Characterization of Confined Plasmas

^{2}) compared to those used in LSP. The images of the plasma-emitting region generated by 20 ns laser pulses display several spots with bright centers and the resulting spectra are broad and continuous. Indeed, as a consequence of the white light emission from these high-density bright spots, the emission spectra obtained with short laser pulses suffer from broadening and self-absorption. With laser pulses longer than 40 ns (typically 150 ns in their studies), the size of the light emitting region increases and the bright spots disappear. Emission is more homogeneously distributed with the brightest region at the center of the hemispherical ablation plume. In this case, one obtains spectra with clear narrow emission lines at longer delays, because excitation by the later part of a longer pulse expands the plume and reduces its density, therefore reducing the broadening of the emission lines and the self-absorption inside the plasma.

- Fitting of a Planck-like distribution to the continuous background (this usually requires a plasma in thermodynamic equilibrium emitting a type of black body radiation);
- Excitation temperature obtained by comparing the relative intensities of the emission lines of an atomic system;
- Rotational and vibrational temperatures obtained by fitting the rovibrational spectrum of a well-defined electronic transition.

^{2}), there is still a strong need to investigate higher intensities (up to 10 GW/cm

^{2}).

#### 2.4. Development of Diagnostic Systems for Pressure Measurements

**Electromagnetic Gauge**This method uses the displacement of the metal induced by the shock wave when it arrives at the rear surface of the target [89,90,91].As shown in Figure 6, the target is placed in a constant magnetic field B (created by two magnets for example), and two metallic pins are connected to the target and separated by a distance L. Following Faraday’s law of induction, when the shock wave arrives at the rear surface and makes it accelerate, the created displacement ${u}_{f}$ in the magnetic field will induce a voltage ${V}_{U}={u}_{f}BL$, which corresponds to the variation in magnetic flux ($\frac{d{\Phi}_{B}}{dt}=-{V}_{U}$) through the wire loop.As, in this case, the rear surface is free, the measured rear free velocity ${u}_{f}$ is twice the material velocity u (reflection of the shock wave at the metal/air interface). Equation (22) then gives:$${P}_{\mathrm{plasma}}=\frac{\rho ({C}_{0}+S\frac{{V}_{U}}{2BL}){V}_{U}}{2BL}+\frac{2{\sigma}_{{y}_{0}}}{3}+\delta P$$This method has some advantages: it is an easy experimental setup to implement, and signals are directly related to the velocity without complicated treatments to be applied. Furthermore, a good temporal resolution is obtained, which is given by the speed of used oscilloscopes (typical resolution will be 0.3 ns with a 10 GS/s oscilloscope with a bandwidth of 2 GHz).However, there are a lot of possible uncertainties:- −
- The magnetic field may vary during the process, mainly because of the perturbations induced by the plasma;
- −
- The distance between the two electrodes, which is constrained by the size of the laser spot size (a few mm) may also vary and be inaccurately measured. It may also be difficult to ensure that both electrodes are in contact with the target;
- −
- Typical values used for B (0.2 T) and L (1 mm) will give low voltage (40 mV for a maximum rear velocity of 200 m/s) and the signals may thus be noisy.

**PVDF Piezoelectric System**A piezoelectric material, for which PVDF (polyvinylidene fluoride) is often chose, in laser shock experiments [91,92,93], is bonded on the rear-surface of the metal target. When the shock wave arrives at the interface between the metal and the PVDF, a part of the shock wave is transmitted and starts to propagate in the PVDF. As a result, it will undergo elastic deformation, and by a piezoelectric mechanism, a current will be generated. By measuring the electrical current generated between the front and back face of the PVDF, one can deduce the pressure of the shock wave. Thus, by taking into account the mismatch impedance between the metal and the piezoelectric material, the shock wave pressure at the rear surface of the metal can be deduced. Altogether, similarly to the previous measurement of the rear-surface by electromagnetic gauge, the pressure of the plasma is indirectly calculated.When the shock arrives at the interface, moving at a material velocity u, the intensity ${I}_{c}$ is given by: ${I}_{c}\phantom{\rule{3.33333pt}{0ex}}{e}_{0}={P}_{0}\phantom{\rule{3.33333pt}{0ex}}u$ where ${e}_{0}$ is the thickness of the piezoelectric system and ${P}_{0}$ is the ferroelectric polarization.Though this system is quite easy to use and implement, there are a lot of drawbacks:- −
- The material parameters of the piezoelectric medium must be precisely known, which is not often the case (calibration of ${P}_{0}$);
- −
- At high pressure (>5 GPa), the material starts to respond non-linearly, making the pressure measurement inaccurate;
- −
- Some calculations have to be made to estimate the mismatch impedance in order to obtain the pressure at the rear surface of the metal target;
- −
- As with gauges, the electrical signal may be disturbed by the electromagnetic field of the plasma, especially at high laser intensity;

**PDV**Photonic doppler velocimetry (PDV) is based on the concept of heterodyning, but it has only recently been developed as a useful shock-physics diagnostic, thanks to the recent technological advances of the telecommunications industry [94,95]. In its simplest configuration (a light and compact system, with full optical beam transportation through fiber), PDV is a fiber-based Michelson interferometer in which laser light is divided among two paths: the first one containing the target and the other one a stationary reference mirror. The laser Doppler-shifted light coming back from the moving target is combined with the reference light and sent to an optical receiver. The optical interferences which are generated result in a beat frequency $f=\frac{2}{{\lambda}_{0}}|vs.|$, where ${\lambda}_{0}$ is the wavelength of the laser probe. At 1550 nm, which is the typical wavelength of most PDV systems, every km/s of velocity v requires 1.29 GHz of receiver/digitizer bandwidth. The velocity is finally extracted from the recorded signal through a time–frequency analysis, typically performed with short-time Fourier transforms (STFTs) [96]. The main advantages of PDV are its ease of use, its robustness, and its relatively low cost compared to other diagnostics such as VISAR. Moreover, because PDV tracks motion in a frequency-encoded temporal electro-optical signal, it is thus able to record multiple velocities simultaneously. However, like all other diagnostics, PDV also has some drawbacks:- −
- The primary weakness of conventional PDV is its inability to resolve low-velocity transients;
- −
- Conventional PDV measurements are also directionally blind: motion toward and away from the probe lead to the same beat frequency;
- −
- Due to the use of time–frequency analysis to retrieve the velocity, the velocity resolution is inversely proportional to the time resolution. Thus, considering the fact that plasma-induced shock waves involve a short duration (in the nanosecond range), with a required resolution of approximately 1 ns, the resolution on velocities is then limited to approximately 50 m/s.

**VISAR Optical System**The Velocity Interferometer System for Any Reflector (VISAR), developed by Barker in 1972 [101]), is currently the most used system to measure plasma pressure: it also aims to measure the rear-free surface velocity of the target. An accurate optical measurement was performed.As shown in Figure 7, the VISAR is made of two parts (the probe part and the interferometer part):- −
**The probe part**—a single longitudinal mode (${\lambda}_{0}$ with a narrow spectral width) collimated laser beam is focused on the rear surface of the metal target. The scattered reflection of the beam is then collected (using a short focal length) and directed towards the second part to be analyzed. This reflection is wavelength-shifted according to the Doppler–Fizeau effect as the surface is moving under a free velocity ${u}_{f}$ ($\lambda \left(t\right)={\lambda}_{0}(1-\frac{2{u}_{f}\left(t\right)}{c})$).It may be preferable to use an unpolished sample to generate more scattered light compared to the specular reflection that could be deflected out of the interferometer if the sample is becoming excessively deformed (which often occurs for thin samples of ≈100 µm).- −
**The interferometer part**—this second part is a field-compensated Michelson-like interferometer. Calibrated glass enables to delay one arm of the interferometer from the other. This changes the initial path difference (${\delta}_{i}$) in the interferometer, and hence the required velocity to move from one fringe to the next one is also changed: this is the velocity per fringe (VPF) factor. Then, the interference between the signal at a time t (wavelength $\lambda \left(t\right)$) and one at a time $t+\Delta t$ (wavelength $\lambda (t+\Delta t)$) is produced, and the interferogram’s intensity is acquired with a fast PhotoMultiplier (PM). Therefore, there is a translation of the interference fringes as soon as a wavelength-shift occurs. The time resolved measurement of the interference intensity enables one to obtain the corresponding velocity.

The intensity of the interferogram follows the following law: ${I}_{v}\left(t\right)\propto 1+cos(\frac{2\pi {\delta}_{i}c}{{\lambda}_{0}(1-2{u}_{f}\left(t\right))})$. Consequently, a simple code has to be used to analyze the intensity signals obtained on the PM in order to obtain the rear-free surface velocity ${u}_{f}$. Thus, as for EM gauge, the plasma pressure is given by$${P}_{plasma}=\frac{\rho ({C}_{0}+S\frac{{u}_{f}}{2}){u}_{f}}{2}+\frac{2{\sigma}_{{y}_{0}}}{3}+\delta P$$The VISAR system is the most resilient and reliable means of plasma pressure measurement: it is without contact (and therefore not influenced by the process) and it can even be used at very high pressure. However, it is not the easiest system to implement as it must be carefully aligned. Indeed, both the probe laser beam (to be brought under the target) as well as the interferometer system (to analyse the wavelength shift) must be properly adjusted. Furthermore, a wise choice of calibrated glass length has to be made before the measurement beings, and some calculations must be performed to obtain the rear-free surface velocity.Finally, a great temporal resolution of less than 1 ns is achieved, only limited by the PM response-time. For all these reasons, we recommend using this system for pressure measurements as it offers the best compromise between assets and drawbacks.**Method to Measure the Plasma Pressure with VISAR**It is important to measure the plasma pressure as accurately as possible as a function of the used laser parameters. Indeed, this will help perform simulations but also to use specific though not very well known material such as FSW materials which possess specific properties [102]. If one knows precisely, for a given laser configuration, the loading pressure term of the plasma, then it helps to more effectively extract the material parameters from a typical shock-wave measurement, as previously shown.First of all, we recommend using a well-known material, such as pure aluminum with a small thickness, to perform laser-induced shock-wave measurements. This will help prevent the dependency of the results with the material parameters or models.Secondly, we suggest using large laser spot sizes to prevent the interferences of complex phenomena such as edge effects (also called 2D effects) [103,104]. Keeping the shock wave as monodimensional as possible should be ensured whenever possible.Moreover, regarding VISAR optical measurements, one should use an unpolished target in order to avoid a loss of the optical signal.On the other hand, in the case of very thin samples which may become deformed under the shock (thus making the pressure measurement difficult to conduct), we recommend sticking a transparent glass window to the rear surface of the sample, such as a BK7 glass plate for aluminum (as performed in [37]). This window must have a mechanical impedance as close as possible to the target on which it will be stuck. Under these conditions, the shock wave is transmitted between the sample and the window without reflection and the full profile of the shock wave can be measured (as shown in Figure 5). One should note that in this case, the material velocity is measured and is half the value of the free surface velocity.

## 3. Improvements in Experimental Accuracy—Range of Plasma Pressure Data Obtained

#### 3.1. Improvements in Optical Metrology and Better Understanding of Materials Behavior

#### 3.2. Plasma Pressure in Confined Regime—Range of Pressures Obtained through Experiments and Simulations

^{2}). Using the efficiency coefficient $\alpha $ from Fabbro’s model, this corresponds to values ranging from $\alpha $ = 0.15 to $\alpha $ = 0.6.

^{2}); and the upper one with $\alpha $ = 0.6, 532 nm, and 5 ns (breakdown threshold: 10 GW/cm

^{2}).

## 4. A Review on Improvements in Analytical Models and Simulations

#### 4.1. Analytical Models

- The first analytical expression developed to provide the maximum plasma pressure was given by Anderholm within their discovery of the confined regime [21]. The pressure was then given by

^{2}/s) of the confinement and I is the laser intensity (in GW/cm

^{2}).

- This dependency was conceived in 1990 by Fabbro with a more detailed model, described in [28]. The main ideas of this model are as follows:

- The laser pulse is absorbed at the boundary between the confinement and the metal where the plasma is created. This plasma then extends in a 1D geometry (the model is said to be mono-dimensional) with two shock waves generated in both the metal and the confinement (see Figure 10).
- The melting and vaporization of the metal is instantaneous. Then, a plasma absorbs all the laser intensity. This plasma is assumed to behave as an ideal gas and a coefficient $\alpha $ is introduced to determine how much of the internal energy ${E}_{i}$ of the ideal gas is converted into thermal energy ($\alpha {E}_{i}$) or devoted to ionization ($(1-\alpha ){E}_{i}$). As no consideration of the plasma is made in this model, the $\alpha $ coefficient must be experimentally determined (from Equation (30)).
- The absorbed laser energy by the ideal gas is assumed to be converted into two parts: the work of the pressure force (${W}_{P}\propto P\mathrm{d}L$, L the thickness of the plasma), and the increase in the internal energy per volume V. ($\mathrm{d}\left[{E}_{i}V\right]$)The work of the pressure force is used to increase the plasma length by pushing both the confinement and the metal target (see Figure 10). Thus, the pressure will depend on the reduced acoustic impedance Z, given by$$\frac{2}{Z}=\frac{1}{{Z}_{1}}+\frac{1}{{Z}_{2}}$$
- At the end of the laser pulse, the plasma/ideal gas will start to cool down, and the pressure will decrease. This was modeled as a slow adiabatic release which obeys the law of Laplace:$$P{V}^{\gamma}=C$$

- In 2001, Zhang and Yao proposed a similar approach to Fabbro, with 5-layer geometry: the shocked confinement, the unaffected confinement, the plasma, the unaffected metal and the shocked metal [29,30]. A mass equation conservation was added to obtain the density of the plasma, and the mass flow from the metal and the confinement towards the plasma. Moreover, the rarefaction wave phenomenon described by Pirri [46] was also included in this model to take into account the drop in pressure due to the spherical blast wave generated during the laser pulse. Usually, this phenomenon should be considered over longer time (after the laser pulse), but Zhang was developing a model for micro-scale laser processes, thus resulting in an early apparition of this blast wave.
- In 2003, Sollier et al. proposed an extension of Fabbro’s model named ACCIC [31,36]. Similarly to Zhang and Yao [29,30], a mass conservation equation was included so that the whole involved system of equations can become self-closed. The evaporation of the protective coating layer is considered and calculated based on the Hertz–Knudsen theory. The Thomas–Fermi theory for electrons and Cowan’s theory for ions are applied as the equations of state for the confined plasma, and thermal losses to the work piece target and transparent overlay (water) are taken into account. The confined plasma is considered as a gas of neutrals from the target only, without any contribution from the confining overlay, and the laser absorption coefficient comes from experimental measurements. The rarefaction wave phenomena described by Pirri [46] were also included in this model to take into account the drop in pressure and temperature due to the spherical blast wave generated during the laser pulse. The ACCIC model has been used to compute the thermo-mechanical loading used as input conditions in FEM simulations of the LSP treatment [107].
- More recently, it has been experimentally demonstrated that the plasma pressure release was shortened when using small laser spot sizes [37]. Hence, a new model, based on Fabbro’s model, has been proposed, as the previous one was under a mono-dimensional hypothesis which was only valid for large spot sizes. This radius-dependent model (RM) incorporates a plasma-leaking mechanism from the edges, leading to a shortening of the plasma pressure with smaller laser spot sizes. Indeed, this leaking will be proportionally more effective in increasing the plasma volume with small spot sizes, thus resulting in a faster drop of the pressure.

#### 4.2. Numerical Models

- A first numerical code was developed in the late 1970s by Clauer and their team [108,109,110,111], and was named LILA. This code is based on a finite difference method (FDM) resolution of the differential equations governing hydrodynamic phenomena in the plasma. The absorption of the laser light was calculated for both the cold dense metal and the plasma (through IB absorption). For simplification purposes, a unique equation of state (EOS) was used to describe both the metal and the plasma (taking into account thermal motion and thermal ionization).

- In 2003, Colvin et al. developed a model for low-intensity laser drives which was subsequently incorporated in the 2D radiation-hydrodynamics code LASNEX, in order to simulate the confined laser-matter interaction [112]. The elastic–plastic equations of stress wave propagation were treated in a Lagrangian formulation. The equation of state of all the materials was taken as the analytical quotidian equation of state (QEOS), which is reduces to a Gruneisen EOS at low temperatures. Radiation transport was calculated by multigroup diffusion, with opacities calculated from an average atom model. A ray tracing algorithm simulated laser light propagation through the matter, with inverse Bremsstrahlung absorption on free electrons. The authors also added several prescriptions for calculating:
- •
- The correct ionization state and electron densities for metals and insulators as a function of temperature;
- •
- The low-intensity absorption of the laser beam on a solid or liquid metal;
- •
- The photoionization absorption of the low-intensity laser beam in neutral vapor;
- •
- Collisional ionization, three-body recombination, and dielectric breakdown.

- From 2004 to 2009, Ocaña, Morales and their coworkers developed a simulation model named SHOCKLAS consisting of three principal modules, namely HELIOS, LSPSIM, and HARDSHOCK, which dealt with the main aspects of LSP modeling in a coupled way [113,114,115]. HELIOS is a 1-D radiation-magnetohydrodynamics code which is used to simulate the dynamic evolution of laser-created plasmas. HELIOS solves Lagrangian hydrodynamics equations for a single fluid in which electrons and ions are assumed to be co-moving. Energy transport in the plasma can be treated using either a one-temperature (${T}_{i}={T}_{e}$) model (for both electrons and ions) or a two-temperature (${T}_{i}\ne {T}_{e}$) model. Both the electrons and ions are assumed to have Maxwellian distributions defined by their respective temperatures. Material EOS properties are based on either SESAME or PROPACEOS tables, whereas the opacities’ properties are based on tabulated multi-group PROPACEOS data, radiation emission, and absorption terms being coupled to the electron temperature equation. Laser energy deposition is computed using an inverse Bremsstrahlung model, with the restriction that no energy in the beam passes beyond the critical surface.

- In 2005, Wu and Shin from Purdue University brought significant improvement in 1D hydrodynamic numerical codes for confined laser-matter interaction [32,117]. Their code adopts a layer geometry (metal/plasma/confinement) to gain a better understanding of how the laser light is transmitted and reflected during the whole interaction. Altogether, the simulated absorption is more accurate. Furthermore, a model for the breakdown is also provided in order to take into account the saturation of intensity reaching the target when a breakdown plasma occurs.

**$\phantom{\rule{-0.166667em}{0ex}}\u25b3$**is the Laplacian differential operator, $\overrightarrow{E}$ is the electromagnetic field entering the cell, $\omega $ is the angular frequency, and ${n}^{2}$ is the complex refractive index of the cell.

^{2}(Gaussian laser pulse of 7 ns duration at FWHM).

- Most recently, Heya et al. also proposed a 1D code to simulate laser shock interaction [121]. This code, named the integrated simulation code for laser ablation peening (ISLAP), concentrates on the interaction of laser light with the metal target rather than simulating the whole laser shock process. There are three models used in this code: an atomic model, based on a screened hydrogenic model (SHM) and used to obtain the energy levels, population distributions, and ionization states of the metal target; an equation of state (EOS) model to calculate some parameters such as the pressure or the specific heat; and the Cowan model; a laser ablation peening code (LAPCO) to calculate the absorption of the laser light (inverse Bremsstrahlung and resonance absorption) during the process, and also to estimate energy transfers (heating and radiation).

## 5. Applications: Understanding of Laser Shock Generation and Propagation for Aluminum Alloys

^{2}($\Psi $) method (X-ray diffraction) in order to estimate the optimal loading pressure to use. Moreover, fatigue tests were conducted to confirm the previous results and show the effectiveness of LSP treatments on Al alloys. Indeed, a comparison was made to illustrate the enhancement obtained on the fatigue life by the LSP process compared to the one obtained from shot peening.

^{2}) were set. Finally, edge effects ([103]) were also investigated depending on the spatial shape, at the edges, of the laser beam.

## 6. Future Expectations and Improvements

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACCIC | Auto-Consistent Confinement Interaction Code |

AI | Avalanche Ionization |

CCD | Charge-Coupled Device |

CRS | Compressive Residual Stress |

CS | Cross-Section |

DOE | Diffractive Optical Elements |

EBSD | Electron BackScatter Diffraction |

EOS | Equation of State |

FDM | Finite Difference Method |

FEM | Finite Element Method |

FWHM | Full Width at Half-Maximum |

FSW | Friction Stir Welding |

HEL | Hugoniot Elastic Limit |

IB | Inverse Bremsstrahlung |

ICCD | Intensified Charge-Coupled Device |

IP | In Plane |

LAPCO | Laser Ablation Peening Code |

LASAT | Laser Shock Adhesion Test |

LIBS | Laser-Induced Breakdown Spectroscopy |

LSA | Laser Shock Application |

LSD | Laser-Supported Detonation Wave |

LSP | Laser Shock Peening |

LSPwC | Laser Shock Peening without Coating |

LSR | Laser-Supported Radiation wave |

MPI | MultiPhotonIonization |

PDV | Photonic Doppler Velocimetry |

PM | PhotoMultiplier |

PVDF | Polyvinylidene Fluoride |

QEOS | Quotidian Equation of State |

RM | Radius-Dependent Model |

RMS | Root Mean Square |

SHM | Screened Hydrogenic Model |

VISAR | Velocity Interferometer System for Any Reflector |

VPF | Velocity per Fringe |

WTC | Water Tank Configuration |

## Symbols

$\alpha $ | Thermal Efficiency Coefficient (Fabbro’s Model) |

${\alpha}_{IB}$ | Absorption Coefficient by IB |

$\gamma $ | Laplace Adiabatic Coefficient |

$\mathsf{\Gamma}$ | Plasma Coupling Coefficient |

${\mathsf{\Gamma}}_{q}$ | Degenerate Plasma Coupling Coefficient |

$\delta $ | Skin Depth |

${\delta}_{i}$ | Path Difference |

${\delta}_{P}$ | Shock Wave Attenuation |

$\Delta {E}_{e}$ | Ionization Potential |

${\u03f5}_{0}$ | Vacuum Permittivity |

${\u03f5}_{p}$ | Plasma Permittivity |

${\epsilon}_{p0}$ | Reference Plastic Strain |

${\eta}_{AI}$ | AI Rate |

${\eta}_{d}$ | Electron Diffusion Rate |

${\eta}_{r}$ | Electron–Hole Recombination Rate |

${\kappa}_{m}$ | Thermal Diffusivity |

$\lambda $ | Laser Wavelength |

${\lambda}_{DB}$ | De Broglie Length |

$\mu $ | Permeability (Metal) |

${\nu}_{ei}$ | Electron–Ion Collision Frequency |

$\rho $ | Density |

$\sigma $ | Electrical Conduction (Metal) |

${\sigma}_{y0}$ | Elastic Limit |

$\tau $ | Laser Pulse Duration (at FWHM) |

${\mathsf{\Phi}}_{B}$ | Magnetic Flux |

$\omega $ | Angular Frequency |

$\phantom{\rule{-0.166667em}{0ex}}\u25b3$ | Laplacian Differential Operator |

B | Magnetic Field |

c | Speed of Light |

C | Strain Rate Sensitivity |

${C}_{0}$ | Bulk Sound Velocity |

e | Elementary Charge |

E | Laser Energy (per Pulse) |

${E}_{i}$ | Plasma Internal Energy |

${E}_{k}$ | Kinetic Energy |

${E}_{F}$ | Fermi Energy |

${E}_{p}$ | Potential Energy |

$\overrightarrow{{E}_{0}}$ | Electromagnetic Field |

f | Frequency |

h | Planck Constant |

I | Laser Intensity |

${I}_{c}$ | Current Intensity |

K | Strain Hardening Modulus |

${k}_{B}$ | Boltzmann Constant |

${m}_{e}$ | Electron Mass |

n | Optical Index |

${n}_{h}$ | Strain Hardening Parameter |

${n}_{2}$ | Imaginary Part of the Optical Index |

${n}_{c}$ | Plasma Critical Density |

${n}_{e}$ | Electronic Density |

P | Plasma Pressure |

${P}_{0}$ | Ferroelectric Polarization |

S | Hugoniot Constant |

${S}_{l}$ | Laser Surface Spot Size |

t | Time |

${T}_{e}$ | Electronic Temperature |

u | Material Velocity (under Shock) |

${u}_{f}$ | Rear-Free Surface Velocity (under Shock) |

V | Plasma Volume |

${V}_{U}$ | Voltage |

Z | Mechanical Impedance |

${Z}_{i}$ | Ionization State |

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**Figure 2.**Experimental Gaussian laser pulse intensity (in GW/cm

^{2}) and associated plasma pressure (in GPa) profile versus time (in ns), simulated with ESTHER code (Section 4.2).

**Figure 4.**Mechanisms involved in the creation of a plasma by breakdown in the dielectric confinement.

**Figure 5.**Typical experimental (normalized) velocities obtained with and without a BK7 glass plate, with a 0.3 mm Al-foil at 1 GW/cm

^{2}[37].

**Figure 8.**Spatial distribution of the laser intensity in the intermediate field for ${I}_{mean}=4$ GW/cm

^{2}: without DOE (

**left**) and with DOE (

**right**).

**Figure 11.**Full width at half maximum (FWHM) and full width at quarter maximum (FWQM) of the plasma pressure profile, normalized by the pulse duration $\tau $, versus the laser spot size also normalized by $\tau $ [37].

**Figure 12.**Typical laser-induced plasma pressure profile in the confined regime with different models (1D Fabbro’s, RM and Pirri).

**Figure 13.**Temperature of the plasma (aluminum and water part) simulated with ESTHER code at 20 ns (1 GW/cm

^{2}and 7 ns pulse).

**Figure 14.**Density of the plasma (aluminum and water part) simulated with ESTHER code at 20 ns (1 GW/cm

^{2}and 7 ns pulse).

**Table 1.**Properties and parameters (Johnson–Cook model) for Al-2024 and Al-7175 alloys [123].

Alloy | ${\mathit{P}}_{\mathbf{HEL}}$ (MPa) | K (MPa) | ${\mathit{n}}_{\mathit{h}}$ | C | ${\mathit{\epsilon}}_{\mathit{p}0}$ |
---|---|---|---|---|---|

Al-2024 | 1028 | 329 | 0.35 | 0.025 | 0.01 |

Al-7175 | 920 | 0 | 0 | 0.01 | 0.01 |

Alloy | ${\mathit{\sigma}}_{{\mathit{Y}}_{0}}$ (MPa) | K (MPa) | ${\mathit{n}}_{\mathit{h}}$ | C |
---|---|---|---|---|

Al-2024 | 369 | 329 | 0.35 | 0.025 |

Al-2017 (CS/IP) | 260/270 | 700/350 | 0.6/0.3 | 0.035/0.03 |

Al-7075 (CS/IP) | 400/473 | 800/210 | 0.45/0.38 | 0.05/0.033 |

FSW (2017 + 7075) (CS/IP) | 285/340 | 500/200 | 0.35/0.2 | 0.033/0.017 |

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**MDPI and ACS Style**

Rondepierre, A.; Sollier, A.; Videau, L.; Berthe, L.
Review on Laser Interaction in Confined Regime: Discussion about the Plasma Source Term for Laser Shock Applications and Simulations. *Metals* **2021**, *11*, 2032.
https://doi.org/10.3390/met11122032

**AMA Style**

Rondepierre A, Sollier A, Videau L, Berthe L.
Review on Laser Interaction in Confined Regime: Discussion about the Plasma Source Term for Laser Shock Applications and Simulations. *Metals*. 2021; 11(12):2032.
https://doi.org/10.3390/met11122032

**Chicago/Turabian Style**

Rondepierre, Alexandre, Arnaud Sollier, Laurent Videau, and Laurent Berthe.
2021. "Review on Laser Interaction in Confined Regime: Discussion about the Plasma Source Term for Laser Shock Applications and Simulations" *Metals* 11, no. 12: 2032.
https://doi.org/10.3390/met11122032