1 Introduction

Laser-produced breakdown plasma in gas medium is involved and plays an important role in many technologies and applications, such as laser-induced breakdown spectroscopy (LIBS), laser-induced spark initiation, and laser propulsion [1–3]. For process improvement and optimization, it is crucial to understand laser-induced gas breakdown plasma, and therefore, lots of relevant investigations have been carried out (e.g., [4–7]).

However, despite the previous research work, studies on nanosecond (ns) laser pulse interactions with gas breakdown plasma confined in a microhole have been limited. Jang et al. [8], Dunbar et al. [9] have studied the shock wave due to laser-induced gas breakdown plasma confined in a tube for the applications in laser cleaning. Sasoh and colleagues [10–13] have studied laser-induced gas breakdown plasma confined in a tube for the applications in laser propulsion. However, in these studies [8–13], the tube diameters are often in the millimeter range, and detailed results on some of the important plasma properties, such as plasma temperature, are often not given.

Gamaly and colleagues [14, 15] have performed experimental and theoretical studies on laser-material interactions inside the bulk of transparent solids, and the investigations show that the interactions can generate super-hot plasma, and extremely high temperatures and pressures inside the transparent solid. Badziak et al. [16, 17] have studied the laser-induced cavity pressure acceleration (LICPA) process, where the acceleration can be realized through the pressure of plasma generated by laser ablation and confined in a cavity. It has been found in Refs. [16, 17] that LICPA can have much higher energetic efficiency of acceleration than the conventional ablative acceleration. However, the plasma in Refs. [14–17] is often generated by laser interaction with a solid target, instead of laser-induced gas breakdown plasma.

In this paper, ns laser pulse interactions with gas breakdown plasma in a microhole will be studied. Due to the microscale size of the hole and the very short ns timescale, experimental measurements are very difficult. Hence, a physics-based model will be used as the investigation tool, where two-dimensional (2D) axisymmetric gas dynamic equations are solved numerically. The effects of microhole sidewall confinement on plasma properties (such as temperature, pressure, and plasma-induced thrust) will be studied. As shown later, under the studied conditions it has been found that the microhole confinement effect can yield higher plasma temperature, pressure, and thrust (over the same surface area) than that without the microhole sidewall confinement, which implies potential advantages for related applications such as LIBS [1] and laser propulsion [2].

2 Model

Figure 1 shows the schematic diagram for the model in this work. In Fig. 1a it is assumed that the microhole has a diameter of 100 μm, whose bottom is located at z = 0. A nanosecond (ns)-pulsed and 1064-nm laser beam is assumed to propagate in the –z direction. It is assumed that the hole bottom is made of material that is transparent to the incoming laser beam. The laser beam (right after it passes through the hole bottom) has a pulse energy of 0.02 mJ and a full pulse duration of 6 ns with a triangular temporal pulse shape (i.e., the laser pulse starts at t = 0 and ends at t = 6 ns). In this study, it is assumed that the laser beam intensity follows a Gaussian profile at any cross section of z = constant, and the beam has a constant diameter of 30 μm that does not change with z. At t = 0, it is assumed that the microhole is filled with argon, whose temperature and pressure is 450 K and 0.15 MPa, respectively, except a preexisting, spherical argon breakdown plasma region (which has a spherical shape, a diameter of 30 μm, and a center at r = 0 and z = −50 μm). In the breakdown plasma region, the argon temperature and pressure at t = 0 are assumed to be 40,000 K and 37.3 MPa, respectively. The laser pulse starts at t = 0, after which the laser interactions with the preexisting plasma will begin. Figure 1b shows the schematic diagram for laser interactions with breakdown plasma over a flat surface at z = 0, where all the conditions are the same as in Fig. 1a, except that there is no microhole sidewall confinement.

Fig. 1
figure 1

Schematic of the model setup for ns laser pulse interactions with initial preexisting a gas breakdown plasma confined in a microhole and b gas breakdown plasma over a flat surface without the hole sidewall confinement (as shown in the figure, at t = 0, the initial plasma region is surrounded by the ambient non-ionized argon region)

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The evolution of argon gas inside the microhole (including the plasma region) is governed by the 2D axisymmetric gas dynamic equations [18–20]:

$$ \frac{\partial }{\partial t}\left[ \begin{array}{l} \, \rho \hfill \\ \rho u_{\text{r}} \hfill \\ \rho u_{\text{z}} \hfill \\ E_{\text{t}} \hfill \\ \end{array} \right] + \frac{1}{r}\frac{\partial }{\partial r}\left[ \begin{array}{l} \, r\rho u_{\text{r}} \hfill \\ \, r\rho u_{\text{r}}^{2} \hfill \\ \, r\rho u_{\text{r}} u_{\text{z}} \hfill \\ r(E_{\text{t}} + P)u_{\text{r}} \hfill \\ \end{array} \right] + \frac{\partial }{\partial z}\left[ \begin{array}{l} \, \rho u_{\text{z}} \hfill \\ \, \rho u_{\text{r}} u_{\text{z}} \hfill \\ \, \rho u_{\text{z}}^{2} \hfill \\ \, (E_{\text{t}} + P)u_{\text{z}} \hfill \\ \end{array} \right] = \left[ \begin{array}{l} \, 0 \hfill \\ \, - \partial P/\partial r \hfill \\ \, - \partial P/\partial z \hfill \\ \frac{1}{r}\frac{\partial }{\partial r}(rk_{\text{e}} \frac{\partial T}{\partial r}) + \frac{\partial }{\partial z}(k_{\text{e}} \frac{\partial T}{\partial z}) + \alpha_{\text{IB}} I \hfill \\ \end{array} \right] $$
(1)

where \( \rho \) is the density of argon, \( u_{\text{r}} \) and \( u_{\text{z}} \) represent the velocities in r and z direction, respectively, \( E_{\text{t}} \) denotes the total energy, including both the kinetic energy and the internal energy, T is temperature, \( P \) is the pressure, \( k_{\text{e}} \) represents thermal conductivity due to free electrons (if they exist), \( \alpha_{\text{IB}} \) is the inverse bremsstrahlung optical absorption coefficient due to free electrons, and \( I \) denotes the laser beam intensity. When solving the equations, the no-slip and adiabatic boundary conditions are applied at the solid surfaces of the hole bottom and the hole sidewall.

In solving Eq. (1), the argon gas equation of state (EOS) (which relates the gas internal energy and pressure with its density and temperature) is needed for the entire temperature range involved in the simulation. At sufficiently low temperatures, the argon ionization degree and hence the free electron number density are negligible, and hence, the ideal gas EOS can be applied. For the high-temperature region, obvious argon ionization may occur, which will affect its EOS, and in this case, the argon EOS is developed based on the free electron number density (due to the argon ionization) that can be obtained through the Saha equation [21] with the argon ionization potentials taken from Ref. [22].

In this study, because the 1064-nm laser photon energy is much smaller than the argon ionization potential, the bound-free absorption of laser photons is expected to be very small [21]. Therefore, it is assumed that the absorption of the laser beam energy is mainly due to the inverse bremsstrahlung effect, and the corresponding optical absorption coefficient, \( \alpha_{\text{IB}} \), is calculated by [23–25]:

$$ \alpha_{\text{IB}} = \alpha_{\text{ei}} + \alpha_{\text{ea}} $$
(2)
$$ \alpha_{\text{ei}} = \frac{{3.7 \times 10^{8} }}{{\nu^{3} \sqrt T }}Z^{2} \left[ {1 - \exp \left( { - \frac{h\nu }{{k_{\text{B}} T}}} \right) } \right]n_{\text{e}} n_{\text{i}} $$
(3)
$$ \alpha_{\text{ea}} = \frac{{e^{2} n_{\text{e}} }}{{\pi mc\nu^{2} }}n_{\text{a}} \sigma_{\text{c}} \sqrt {\frac{{8k_{\text{B}} T}}{\pi m}} $$
(4)

where \( \alpha_{\text{ei}} \) and \( \alpha_{\text{ea}} \) are the electron–ion and electron–atom inverse bremsstrahlung absorption coefficients, respectively, \( \nu \) is the photon frequency, \( Z \) denotes the ionic charge, \( h \) is the Planck constant, \( k_{\text{B}} \) is Boltzmann constant, \( n_{\text{e}} \) and \( n_{\text{i}} \) represent the electron and ion number densities, respectively, \( m \) is the electron mass, \( c \) is the speed of light, e is electron charge, \( n_{\text{a}} \) is atom number density, and \( \sigma_{\text{c}} \) represents the electron–atom collision cross section.

The plasma electron thermal conductivity, \( k_{\text{e}} \), is calculated using the following expression [25]:

$$ k_{\text{e}} = \delta_{T} \cdot 20 \cdot \left( {\frac{2}{\pi }} \right)^{3/2} \frac{{\left( {k_{\text{B}} T} \right)^{5/2} k_{\text{B}} }}{{\sqrt m \cdot e^{4} \cdot Z \cdot \ln \Uplambda }} $$
(5)

where \( \Uplambda = \frac{12\pi }{{Ze^{3} }}\sqrt {\frac{{\varepsilon_{0}^{3} k_{\text{B}}^{3} T^{3} }}{{n_{\text{e}} }}} \) [23] (\( \varepsilon_{0} \) is the dielectric constant of free space) and \( \delta_{\text{T}} \) represents a constant related to Z [25].

When Eq. (5) fails due to unreasonable values of \( \ln \Uplambda \), the plasma electron thermal conductivity can be calculated using [26]:

$$ k_{\text{e}} = \frac{5}{2} \cdot \left( {\frac{2}{\pi }} \right)^{1/2} \left( {\frac{52}{15} + \frac{16}{15}\frac{\sqrt 2 }{Z}} \right)^{ - 1} \frac{{\left( {k_{\text{B}} T} \right)^{5/2} k_{\text{B}} }}{{\sqrt m \cdot e^{4} \cdot Z \cdot B}} $$
(6)

where B is given by: \( B = \frac{1}{2}\frac{\eta - 1 - \ln \eta }{{\left( {1 - \eta } \right)^{2} }} \) and \( \eta = \frac{1}{{\Uplambda^{2} }} \).

Equation (1) is solved numerically using the finite difference method, where the essentially non-oscillatory (ENO) scheme given in Ref. [27] has been employed as the numerical technique to solve the equation. To clearly reveal the confinement effect of the microhole sidewall on the breakdown plasma, numerical simulations will also be performed for the plasma evolution over a flat surface without the hole sidewall confinement as shown in Fig. 1b. The laser parameters, the preexisting plasma conditions, and the ambient non-ionized argon parameters at t = 0 are the same as introduced earlier. Similarly, the no-slip and adiabatic boundary conditions are also applied at the flat surface at z = 0.

3 Results and discussions

Figure 2a, b shows the model-predicted spatial distributions of pressure at different times for laser–plasma interactions with and without the hole sidewall confinement, respectively. At t = 2 ns, the pressure distributions with and without hole sidewall confinement are almost the same, because the pressure wave front has not reached the hole sidewall yet. At t = 18 ns, in Fig. 2a the high-pressure wave front inside the hole has already reached and been reflected by the hole sidewall. Obviously, the sidewall inhibits the further plasma expansion and the pressure wave front propagation in the r direction. As a result of the pressure wave front reflection by the sidewall, relatively large high-pressure red regions are formed in the plot at t = 18 ns in Fig. 2a. For the situation without hole sidewall confinement (plasma over a flat surface at z = 0), the pressure wave front will continue propagating in r direction without being reflected, and the high-pressure red regions are smaller in the plot at t = 18 ns in Fig. 2b. The pressure distributions in Fig. 2a, b are very different at t = 18 ns, and the difference becomes even more significant at t = 30 ns, when the maximum pressure inside the microhole is around 5.2 MPa in Fig. 2a, while that without the hole sidewall confinement is only around 3.2 MPa in Fig. 2b. At t = 30 ns, the high-pressure region inside the microhole (in Fig. 2a) is much larger than that without the hole sidewall confinement (in Fig. 2b).

Fig. 2
figure 2

The model-predicted spatial distributions of pressures at different times (for a, the domain size for each plot is 100 μm × 400 μm, and for b the domain size is 400 μm × 400 μm for each plot; at t = 0, the initial plasma (ionized argon) region has a temperature of 40,000 K and a pressure of 37.3 MPa, and the ambient argon has a temperature of 450 K and a pressure of 0.15 MPa; hole diameter is 100 μm for Fig. 2a)

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Figure 3a, b shows more clearly the pressure wave front reflection process with and without microhole sidewall confinement, respectively. At t = 7 ns, in Fig. 3a the pressure wave front has just started touching the solid surfaces of the hole bottom and the sidewall. Then the high-pressure wave fronts are reflected from both the hole bottom and the sidewall (see the three small red regions in the plot at t = 9 ns in Fig. 3a). The pressure wave reflected from the microhole sidewall will strongly affect the magnitudes and distributions of pressures as shown in Fig. 2 (and also temperatures shown later in Fig. 5) inside the microhole. In Fig. 3b, because there is no hole sidewall, the pressure wave front is only reflected from the flat solid surface at z = 0, and there is no pressure wave reflection from the hole sidewall. Hence, only one small high-pressure red region can be observed near the flat solid surface at z = 0 in the plot at t = 9 ns in Fig. 3b.

Fig. 3
figure 3

The model-predicted spatial distributions of pressures showing the reflection of high-pressure wave front by: a microhole bottom and sidewall and b flat solid surface at z = 0 without hole sidewall confinement (the domain size for each plot in Fig. 3a, b is the same as that in Fig. 2a, b, respectively; the argon initial conditions are the same as in Fig. 2; the hole diameter is 100 μm for Fig. 3a)

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Figure 4a shows the temporal variations of model-predicted thrust on the solid surface at z = 0 (in the region of r < r 0, where r 0 is the hole radius). The force exerted by argon on the solid surface at t = 0 is used as the zero reference when the results are plotted. Before t = ~11 ns, the thrust variations for plasma both with and without the microhole sidewall confinement are almost the same. After t = ~11 ns, the thrust due to plasma without hole sidewall confinement drops very quickly, while the thrust due to plasma inside the microhole first increases to an even higher value due to the hole confinement effect, and then the thrust decreases, but still maintains at an obviously higher value than that without the hole sidewall confinement. Therefore, for the time range shown in Fig. 4a, the impulse over the same solid surface area (the region of r < r 0 at z = 0) is much higher for plasma confined in the microhole, which implies a potential advantage for this kind of plasma in laser propulsion or other related applications. Figure 4b shows the ratio of the thrust with hole sidewall confinement over that without the confinement (the ratio is defined as 1.0 when both thrusts are 0). It can be seen that at the beginning, the thrusts with and without the hole sidewall confinement are almost the same, and hence, the ratio is around 1.0. However, after t = ~11 ns, the ratio increases above 1.0 and reaches a peak value larger than 10 at t = ~22 ns. This large ratio implies a significant enhancement effect on the thrust due to the hole sidewall confinement.

Fig. 4
figure 4

The model-predicted a thrust exerted on the solid surface at z = 0 (within r < r 0 region, where r 0 is the hole radius) with and without hole sidewall confinement and b the ratio of the thrust with hole sidewall confinement, Thrust_h(t), over that without hole sidewall confinement, Thrust_f(t) (the argon initial conditions are the same as in Fig. 2; the hole diameter (2r0) is 100 μm)

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Figure 5a, b shows the model-predicted distributions of temperatures at different times with and without the hole sidewall confinement, respectively. At t = 4 ns, the temperature distributions in Fig. 5a, b are almost the same, because the plasma front and the associated pressure wave front have not reached the microhole sidewall, and hence, the sidewall confinement effect has not been “felt.” It should be noted that due to the absorption of the energy from the incoming laser beam, the peak temperature of the plasma at t = 4 ns is ~42,000 K, which is a little higher than its initial temperature of 40,000 K. By t = 18 ns, in Fig. 5a the sidewall confinement effect has already been “felt” due to its restriction on plasma expansion in r direction and its reflection of pressure wave. Therefore, at t = 18 ns the temperature distribution in Fig. 5a is very different from that in Fig. 5b. At t = 18 and 30 ns, the peak temperature in Fig. 5a is around ~35,000 and ~32,000 K, respectively, which is much higher than that in Fig. 5b (less than ~30,000 K). The results in Fig. 5 suggest that the microhole sidewall confinement effect may yield higher plasma peak temperatures. This may indicate potentially more intense optical emission of the plasma, which implies potential advantages for this kind of plasma in the LIBS application that utilizes plasma optical emission for elemental analysis [1] (e.g., the detection limit of LIBS may be potentially improved). Certainly, research work in the future is still needed to further verify this.

Fig. 5
figure 5

The model-predicted spatial distributions of temperatures at different times (for a, the domain size is 100 μm × 400 μm for each plot, and for b the domain size is 400 μm × 400 μm for each plot; the argon initial conditions are the same as in Fig. 2; the hole diameter is 100 μm for Fig. 5a)

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For the results in Figs. 2, 3, 4, and 5, as mentioned in “Model” section of this paper, the initial (at t = 0) plasma (ionized argon) region has a temperature of 40,000 K and a pressure of 37.3 MPa, and the ambient argon has a temperature of 450 K and a pressure of 0.15 MPa, and the hole diameter is 100 μm (for the case with the hole sidewall confinement). The microhole sidewall confinement effects may be different when the initial plasma region parameters, the initial ambient argon parameters, and/or the hole diameters are different. This has been shown in Figs. 6, 7, and 8.

Fig. 6
figure 6

The model-predicted thrust ratio (as defined in Fig. 4) for different plasma initial conditions (the hole diameter and the initial ambient argon region parameters are also given in the figure)

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Fig. 7
figure 7

The model-predicted thrust ratio (as defined in Fig. 4) under different initial conditions of the ambient non-ionized argon region

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Fig. 8
figure 8

The model-predicted thrust ratio (as defined in Fig. 4) for different hole diameters (the initial conditions of the plasma and the ambient argon region are also given in the figure)

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Figure 6 shows the thrust ratio for different plasma initial conditions while the other conditions are the same. As defined in Fig. 4, the thrust ratio is the ratio of the thrust with hole sidewall confinement over that without hole sidewall confinement, where the thrust is the force exerted on the surface at z = 0 within the r < r 0 region (r 0 is the hole radius). Figure 6 shows that when the initial plasma region has higher temperature and pressure, the ratio starts increasing above 1.0 earlier, but reaches a lower peak value. That means, under the studied conditions in Fig. 6, with higher initial plasma temperature and pressure, the hole sidewall starts affecting the thrust earlier, but the maximum relative enhancement on the thrust will be reduced.

Figure 7 shows the thrust ratio under different initial conditions of the ambient non-ionized argon region, while the other conditions are the same. It can be seen from the figure that when the initial ambient argon has higher temperature and pressure, the thrust ratio has a larger peak value, which implies that the hole sidewall confinement has more significant relative enhancement on the thrust.

Figure 8 shows the thrust ratio for different hole diameters, while the other conditions are the same. It can be seen that when the hole diameter is smaller, the ratio starts increasing above 1.0 earlier and also reaches a higher peak value. That means, under the studied conditions in Fig. 8, with a smaller hole diameter, the hole sidewall starts affecting the thrust earlier, and the maximum relative enhancement on the thrust will also be increased.

4 Conclusions

Nanosecond laser pulse interactions with preexisting argon breakdown plasma in a microhole have been studied using a physics-based model based on 2D axisymmetric gas dynamic equations. It has been found that under the studied conditions the spatial confinement effect of the microhole sidewall can obviously enhance the plasma temperature, pressure, and thrust (over the same solid surface area), which suggests potential advantages of the microhole spatial confinement in LIBS, laser propulsion, and other related practical applications. It has also been found that under the studied conditions in Figs. 6, 7, and 8, the maximum relative enhancement for the thrust will be larger when the initial plasma has lower temperature and pressure, the initial ambient argon has higher temperature and pressure, or the hole diameter is smaller.

It is good work in the future to further verify and study how to utilize the potential advantages of the microhole spatial confinement on gas breakdown plasma in LIBS and laser propulsion applications. It is also good work in the future to verify and improve the model through experimental observations, which are certainly very challenging due to the very small size of the microhole and also the very short ns timescale involved.