Formation and evolution of multiple-core structures in laser-sustained plasmas

We report on the formation of multiple-core structures in laser-sustained plasmas (LSPs) through the utilization of an improved two-dimensional laser-thermal-hydrodynamically coupled fluid model. Our findings reveal that steady-state LSPs exhibit various temperature structures, including single, double, and triple cores, as the input laser power increases incrementally. The spatiotemporal distributions of these multiple-core structures are demonstrated, and the presence of an LSP core is identified when the rate of enthalpy change becomes positive. This behavior is predominantly influenced by factors such as laser power absorption, radiation, and diffusion along the laser path. The obtained results provide explicit insights into the temperature structures within LSPs, thereby potentially contributing to the advancement of LSP-based light sources for optical wafer defect inspection in the semiconductor industry.

Laser-sustained plasma (LSP) is a distinct form of plasma that absorbs laser energy and remains stable over time [1].Unlike other plasma sources generated by electrical gas discharge, LSP can exist without an external electrical field and far from electrodes as a point source, exhibiting unique features, such as high plasma temperature, high ionization, and high brightness, which holds potential for a wide range of applications [2][3][4].Since Raizer [5] proposed the concept of continuous optical discharge, more commonly referred as LSP, which was later experimentally demonstrated by Generalov et al [6].Most of the previous studies on LSP with its application in laser propulsion mainly paid attention to the conversion of internal energy to kinetic energy.Nowadays, novel applications of LSP emerge continuously such as utilized as light source for optical inspection in the semiconductor industry.Although some types of LSP equipment have been developed for commercial wafer inspection [7][8][9], the fundamental mechanism of LSP is rather complicated, which can be largely affected by laser input, working gas properties, thermal-hydrodynamics, and other parameters.Most of the LSP experiments and the diagnostics are limited to macroscopic properties, and thus spatiotemporally resolved results are rather lacking.Rafatov established a numerical model to investigate the effects of laser beam focusing [10], gas flow [11], and gravitational field [12] on LSP properties, which enables the optimization of LSP performance.More recently, the minimum sustaining power of LSP has been experimentally measured for different working gases [13] or under various laser conditions [14].Lu et al conducted experimental measurements of argon-LSP using optical emission spectroscopy to estimate the electron temperature and density [15].Experimental results on the effects of laser power and F-number on the size, radiation, and sustaining power of xenon LSP, as well as the presence of instabilities, have been reported by Hu et al [16].Shi et al experimentally discovered that a ring mode laser, compared to a Gaussian mode, can generate LSP with a smaller volume and higher brightness [17].However, the physical mechanisms underlying LSP are not fully understood due to its inherent complexity and explicit understandings of the LSP towards novel parameter regime are still awaiting for the design and optimization of LSP devices.
High pressure and high temperature conditions are usually required for LSP operation, which is beneficial for achieving a point light source for optical inspection.However, under specific conditions, LSP can undergo a transition from a single-core to a multiple-core structure in terms of spatial temperature distribution.Although this transition results in increased LSP brightness, it introduces severe challenges to the performance of focusing systems due to the deviation from a point light source.The existence of a multiple-core LSP has been numerically observed by Jeng and Keefer [18].Zimakov et al [19] experimentally demonstrated the presence of a dual-core LSP (generated using a fiber Yb laser with λ = 1.07 µm in Gaussian mode) by measuring the radiation intensity.The formation mechanism of the multiple-core LSP was attributed to the refraction of laser propagation within the plasma, which has not yet been explicitly confirmed by other studies.Therefore, the exploration of the mechanism underlying multiple-core LSP is still in its early stages, which is essential for developing high-quality point light sources for wafer defect inspections.
In this Letter, we employ a laser-thermal-hydrodynamically coupled LSP model to investigate the mechanisms involved in the formation of multiple-core LSP structures.Our simulation is based on the steady-state model proposed in [20], where further improvements have been made to incorporate the time-dependent terms.This refinement enables us to accurately capture the temporal evolution of LSP within the computational domain.Given the high pressure and temperature conditions exhibited by LSP, our model adopts the assumption of local thermodynamic equilibrium.Consequently, the plasma state is described by three macroscopic physical quantities: pressure p, temperature T, and velocity u.The fundamental continuity equations of the mass, momentum, and energy are applied to the LSP model, which, respectively, are expressed as ∂ ∂t where ρ is the mass density, τ is the viscosity stress tensor, C p is the specific heat capacity at constant pressure, k T is the thermal conductivity, P abs is the absorbed laser power in the LSP, and P rad is the radiative power of the LSP.In this model, ρ is related to p and T by the perfect gas law p = ρRT/M mol where M mol is the molar mass and R is the perfect gas constant.The viscosity stress τ is given as where µ v is the dynamic viscosity coefficient and Ī is the identity tensor.The interaction between laser and plasma is described by the Lambert-Beer's law [20] dI ds = −αI, ( where I = I(r, z) is the laser intensity, α is the absorption coefficient, and s is the unit vector along the laser propagation.The absorbed laser power P abs is calculated by P abs = αI.
The schematic diagram of the LSP setup is depicted in figure 1(a).A CO 2 laser (wavelength λ = 10.6 µm) is focused into the plasma using a lens.The plasma is initiated through the application of an external high-voltage (HV) pulse between the electrodes.As illustrated in figure 1(b), the LSP absorbs energy from the laser via the inverse bremsstrahlung process, i.e., free-free (ff) absorption, and experiences energy losses through radiation and thermal diffusion.The radiation emitted by the plasma comprises free-free (ff) emission, free-bound (fb) emission, and bound-bound (bb) emission.The computational domain employed in the LSP model is presented in figure 1(c) and is based on a two-dimensional (2D) cylindrical coordinate system.The origin of coordinate O = (r, z) = (0, 0) corresponds to the position of laser focus.The domain exhibits a symmetrical axis aligned with the center of the laser path and features a side wall maintained at a temperature of 300 K.The cylindrical chamber has a diameter of 70 mm and a length of 200 mm.Here the chamber size is much larger than the LSP size, and its effect on the LSP characteristics could be less important compared with laser parameters.The laser is introduced from the left side of the domain and exits through the right side.The dark-red region is the laser zone whereas the plasma zone located to the left of the laser focus is much smaller.The F-number is set to f/7, determined based on the laser beam radius (indicated by the dashed line), which is defined as the position where the intensity drops to 1/e 2 of its axial value.The working gas used in the simulations is xenon, which has a relatively strong ultraviolet emission spectral intensity of interest for application.The inlet velocity is set to zero, and the pressure at the outlet is fixed at 5 atm (1 atm = 101 325 Pa) for all the cases studied.The chamber pressure (though slightly higher) approximates to 5 atm due to the weak pressure gradient, and thus the spatial pressure variation is negligible.Figure 2 demonstrates the presence of the multiple-core structures observed in the LSP as the laser power increases.The core of the LSP is defined as the local maximum temperature within the plasma.From the spatial temperature distribution of the steady-state LSP, it can be observed that for laser powers of 300 W, 400 W, and 800 W, the LSP exhibits single-, dual-, and triple-core structures, respectively.These cores are located near the symmetric axis, as the intensity of the Gaussian beam laser is highest at the center of each transverse plane.For the case of a laser power of 300 W, a single-core LSP is formed near the focal point, with an offset of approximately 37 mm against the laser propagation direction.The absorbed power by LSP core is determined by the laser power difference before and after the core along propagation axis, where the laser power is calculated by P(z) = 2π ´R 0 I(r, z)r dr.The plasma temperature of the core, denoted as T core , is approximately 10 4 K, and it absorbs around 42% of the laser power.When the laser power is set to 400 W, a dual-core LSP is generated, with the first (left) core shifted towards the left.The first and second cores absorb approximately 25% and 24% of the laser power, respectively.These cores exhibit similar core temperatures and dimensions.For the case of a laser power of 800 W, a triple-core LSP is observed, with the isothermal contours of the plasma also exhibiting similarities.The first, second, and third cores absorb approximately 36%, 32%, and 11% of the laser power, respectively.
To sustain each core of the LSP, there exists a minimum laser power requirement (could be different for each core), which is often referred to as the threshold laser power.If the remaining power of the laser passing

(a)
The spatial-temporal evolution of the temperature T, starting from the initial state and evolving towards the steady state.(b) The variation of laser power along the propagation path at t = 0 ms, t = 0.14 ms, t = 0.48 ms, and t = 15 ms.(c) The rate of enthalpy change, dH/dt, in the vicinity of the laser focus when the second core is formed at t = 0.14 ms and the third core is formed at t = 0.48 ms.
through the upstream core is sufficiently large, a new LSP-core can be generated, which is the fundamental principle for the formation of multiple-core structures in LSP.The threshold laser power is influenced by factors such as the laser radius (position) and the profile of the temperature distribution.Therefore, the presence of the second or third LSP-core is not solely determined by the remaining power of the laser but also by the temperature distribution, which is in turn influenced by the temporal evolution of the LSP.
In the simulation, an arc plasma is initiated using a high-pulse, and the initial temperature distribution T (r, z) is assumed to follow a 2D Gaussian distribution with a temperature maximum at 20 000 K. The full width at half maximum along the r-axis and z-axis is set to 2 mm and 40 mm, respectively.Although the effect of the ignition source is not studied, there are no constraints on the initial plasma, and the LSP dynamics can be consistently evaluated for given initial temperature distributions.Although an experimental demonstration of a dual-core LSP has been achieved by Zimakov et al [21], a comprehensive theoretical explanation and numerical validation for such phenomena have not been provided in prior studies.In this work, we present the temporal formation of multiple-core LSP structures based on the simulation results, which are not caused by the refraction of laser propagation within the plasma.Notably, the observation of a triple-core LSP, which has not been reported previously, could be attributed to the challenges associated with generating a strong ignition.
It is obvious that the evolution after ignition can affect the finial steady state of LSP, which can be captured by the established LSP model.The temperature evolution can be explained by the rate of enthalpy change in the LSP, which is given by dH dt = P abs − P diff − P rad , (6) where according to equation ( 3 is the heat diffusion power density. Figure 3(a) depicts the time-dependent evolution of LSP temperature along the symmetrical axis, showcasing the initial formation of a single-core structure, followed by a dual-core structure at t = 0.14 ms, and eventually a triple-core structure at t = 0.48 ms.At t = 0 ms, the first core is generated near z = −20 mm.At this location, the laser power experiences a sharp decline to zero (as shown in figure 3(b)), indicating almost complete absorption by this core at t = 0 ms.Consequently, the laser is unable to propagate through the first core and heat the plasma in the downstream (rightward) region.As a result, the temperature at the focal point temporally decreases.As the LSP core moves upstream (leftward), the laser radius at the LSP core location gradually increases due to the focusing effect, while the size of the moving LSP core remains relatively constant.Consequently, the LSP core becomes unable to fully obstruct the laser propagation, allowing a portion of the laser power to reach the focal point.This fraction of laser power can become even larger as the LSP evolves.At t = 0.14 ms, the laser power absorbed by the plasma at the focal point surpasses the loss power (radiation and thermal diffusion).As a result, the rate of enthalpy change at the focal point becomes positive (as shown in figure 3(c) lefthand), initiating an increase in the LSP temperature at the focal point.This process can be understood as the ignition of plasma at the focal point by the laser, leading to the formation of the second LSP core.Similarly, at t = 0.48 ms, the rate of enthalpy change at the focal point becomes positive (as shown in figure 3(c) righthand), signifying the laser-induced ignition of plasma at the focal point and the subsequent formation of the triple-core LSP.However, once the third core is formed and moves upstream, although the laser can still penetrate through it, the LSP temperature at the focal point noticeably decreases (see figure 3(a)).This cooling effect results in a significant decrease in the absorption coefficient, making the absorbed laser power at the focal point insufficient to generate an additional LSP core.
We analyze the temperature distribution and propagation of the multiple-core LSP at different time points, considering the energy components.Figure 4(a) presents the temperature distribution at 1 ms, 5 ms, and 20 ms.At t = 1 ms, the triple-core LSP structure has formed and begins propagating upstream.The propagation speed is time-dependent, which is estimated to be 2.5 ± 0.5 m s −1 during 1-5 ms, and 1 ± 0.2 m s −1 during 5-20 ms.Over time, the speed of the triple-core structure propagation gradually decreases, and the LSP reaches a steady state at t = 20 ms.The decomposed power densities of P abs , P diff , P rad , and dH/dt along the symmetrical axis are illustrated in figures 4(b)-(e).In terms of P abs , the laser power is primarily absorbed at the fronts of each LSP core, with the first core having the highest magnitude of absorbed power density.It is important to note that the total absorbed laser power corresponding to the LSP-cores decreases as the laser propagates from left to right.Specifically, the first, second, and third cores absorb 36%, 32%, and 11% of the laser power, respectively.The first LSP core is sustained by central heating from the laser, while the downstream LSP-cores (e.g. the third LSP core in figure 2(c)) may be sustained by annular heating, where the highest absorbed power density shifts away from the symmetrical axis.The P rad term is positively correlated with the local temperature within a certain range (e.g. 10 000−20 000 K), resulting in stronger radiation at the fronts of each LSP-core.Due to the larger temperature gradient, the diffusion component P diff is more pronounced at the front of the first core, gradually decreasing as the LSP evolves towards a steady state.During the propagation of the LSP-cores, there is a reversal of dH/dt from positive to negative at the fronts of each LSP-core.This indicates that the LSP-cores move with a heating region at the front followed by a cooling region (e.g.z = −40 mm, t = 1 ms).As a result, the LSP-cores initially propagate faster, slow down over time, and eventually reach a steady state where dH/dt = 0.
In summary, we have successfully demonstrated the formation of multiple-core structures in xenon LSP using an improved 2D time-dependent laser-thermal-hydrodynamically coupled fluid model.By increasing the laser power, we observed the presence of single, double, or triple core structures in the spatial temperature distribution of LSP.Through the analysis of the underlying physical mechanisms, we found that the formation of a multi-core LSP is influenced by both the remaining laser power and the temperature distribution.The temporal evolution of LSP revealed that when the rate of enthalpy change becomes positive, a new LSP-core can be sustained at the focal point as the laser travels through the existing LSP-core.For the triple-core LSP, we observed that the first core is generated in front of the focal point, while the subsequent cores are generated at the focal point itself.All the cores move upstream until a steady state is reached, indicated by a zero rate of enthalpy change.Base on the findings of this work, one can understand the formation of a new LSP core when the enthalpy becomes positive at the focal point.The analysis of energy components provided insights into the role of each component in the evolution of LSP, which confirmed that the enthalpy change plays a crucial role in the formation of multiple-core LSP.Although an increase in laser power can lead to the multi-core LSP generation, the threshold of laser power causing the generation of a new LSP core needs to be further investigated.In future work, the threshold of laser power and the refraction of laser in LSP, as well as the initial arc plasma distribution (e.g.temperature and dimensions), will be considered and comprehensive parametric studies under various operating conditions will be conducted.Further exploration will strongly support the development of optimized LSP point light sources for semiconductor industrial applications.

Figure 1 .
Figure 1.(a) Schematic illustration of the LSP generation system.An arc plasma is ignited by a high-voltage pulse and sustained by a laser focused on it.(b) Fundamental physical processes in LSP.Energy absorption in LSP occurs through ff absorption (inverse bremsstrahlung), while energy loss transpires through radiation, including free-free (ff), free-bound (fb), and bound-bound (bb) emissions, as well as thermal diffusion mechanisms.(c) Computational domain depicting a cylindrical geometry with a symmetric axis and side wall.

Figure 2 .
Figure2.The 2D spatial distribution of the temperature, denoted as T, in the steady-state LSP configuration is depicted for different laser power levels: (a) 300 W, (b) 400 W, (c) 800 W. The LSP chamber is filled with xenon gas at a pressure of 5 atm.The Gaussian laser beam propagates along the z-axis from left to right.The focal point of the laser is located at z = 0, where the laser has a radius of 0.12 mm.

Figure 4 .
Figure 4. (a) The spatial distributions of temperature T along the symmetry axis at t = 1 ms, 5 ms, and 20 ms, respectively.Decomponents of (b) the absorbed laser power density P abs in LSP, (c) the radiative power density P rad , (d) the heat diffusion power density P diff , and (e) the rate of the enthalpy change dH/dt at the steady state t = 20 ms.