Modeling an inhomogeneous optically thick laser induced plasma: a simplified theoretical approach

Abstract

A simplified theoretical approach is developed for an optically thick inhomogeneous laser induced plasma. The model describes the time evolution of the plasma continuum and specific atomic emission after the laser pulse has terminated and interaction with a target material has ended. Local thermodynamic equilibrium is assumed allowing the application of the collision-dominated plasma model and standard statistical distributions. Calculations are performed for a two-component Si/N system. The model input parameters are the number of plasma species (or plasma pressure) and the ratio of atomic constituents. Functions are introduced which describe the evolution of temperature and size of the plasma. All model inputs are experimentally measurable. The model outputs are spatial and temporal distributions of atom, ion and electron number densities, evolution of an atomic line profile and optical thickness and the resulting absolute intensity of plasma emission in the vicinity of a strong non-resonance atomic transition. Practical applications of the model include prediction of temperature, electron density and the dominating broadening mechanism. The model can also be used to choose the optimal line for quantitative analysis.

Introduction

In recent years, many workers have modeled laser breakdown on a solid surface or in the gas phase. For example, a simple model of laser–metal interaction was proposed by Lunney and Jordan [1]. A single absorption cross-section was assigned to both atoms and ions to account for bound–bound and bound–free transitions in the plasma. These transitions, along with inverse bremsstrahlung, were suggested to play an important role in the dynamics of laser ablation. The model was used to calculate plasma absorption, average ion energy and ablation depth for an iron target irradiated with a excimer laser at 248 nm. The calculated values showed good agreement with experimentally measured parameters. Hermann et al. [2] obtained information about the plasma ignition process by comparison of UV-ablation on carbon and metal targets. Callies and colleagues [3] investigated the UV-ablation mechanism. For excimer lasers, the effect of inverse bremsstrahlung was shown to play a minor role, while a strong shielding was presented. This was explained by the Mie absorption of laser light on condensed clusters. Yalçin et al. [4] studied the influence of ambient conditions on the laser-air spark and found evidence in support of a laser-supported radiation wave model.

The effect of varying laser pulse width, from fs to μs and laser power was also carefully investigated in terms of formation of seed electrons which resulted in an electron avalanche following dielectric breakdown [5], [6]. However, after the laser pulse terminated and the supply of energy into the plasma ended, the plasma expanded, gradually relaxed and decayed like any other type of high temperature, high electron density plasma obtained by any other means (e.g. a pinch discharge or a plasma in a shock wave tube). At this stage, the plasma ‘forgets’ about its laser origin and has many similarities with these conventional plasmas which have been well characterized and widely described in the literature (see, for example, [7], [8]).

A more realistic approach to modeling analytical plasmas has to take into account the fact that such plasmas are optically thick. Therefore, lines will be self-absorbed. In addition, temperature inhomogeneities exist along the line of sight of observation leading to self-reversed lines. These problems have been considered in old [9], [10][11], [11](a), [11](b)[12], [13], [14], [15], [16] as well as recent literature [17], [18], [19]. To give a few examples, Tondello et al. [13] observed spatially resolved, time integrated spectra from Be (IV) in the soft X-ray region. The plasma was initiated on a beryllium target by a 10 J, 10 ns ruby laser in vacuum. Line profiles with asymmetrical shift reversal were calculated under the assumption of local thermodynamic equilibrium (LTE) and uniform temperature distribution across the plasma. General agreement was reached between the computed and experimental profiles. Malvezzi et al. [14] proposed a model which took into account Stark and Doppler broadening, Doppler shift and optical opacity. The plasma was induced on a polyethylene target in vacuum and analyzed in the XUV spectral range (2–30 nm). The observed profiles of the Lyman C (VI) lines were reproduced numerically and their build-up within the plasma was explained. Tallents [15] developed a method of determining laser-produced plasma conditions from the shape of a spectral line for an optically thick plasma. Model parameters, such as the plasma emissivity, absorption coefficient and ion velocity were adjusted until a close fit between the computed and experimental profiles was obtained. The experimental data from Malvezzi et al. [14] were used to verify the model. Hermann et al. [17] performed time- and space-resolved diagnostics of the early stage plasma (≤200 ns) induced by an excimer laser on a Ti target in low pressure nitrogen. It was shown that self-absorption has an influence on spectral line profiles even for transitions between highly excited levels. A model of a non-uniform plasma divided into two uniform zones of different densities and temperatures was applied to describe self-absorbed and self-reversed line profiles. The authors also concluded that measurements of electron temperature by the Boltzmann plot method may predict temperature values that are too high.

In the present paper, we are interested in analogies with other plasmas and the applications of well known theoretical tools for characterization of a laser induced plasma (LIP). The model will remain semiempirical, and most of the information will be taken from experiments. Some simplifying assumptions (e.g. local thermodynamic equilibrium, LTE) will be introduced allowing a standard thermodynamic description. Equilibrium and non-equilibrium aspects in plasma spectroscopy have been thoroughly discussed, for example, in references [20], [21], [22]. In our approach, we will consider an optically thick inhomogeneous plasma. The model will describe plasma emission in the vicinity of a strong self-reversed non-resonance line. The Si (I) 288.16 nm line was chosen which has a lower level of only 6269 cm⁻¹ (0.78 eV) above the ground state. According to our preliminary spectral observations, this line is strongly self-reversed in the LIP in air. Of course, the fact itself that the line is self-reversed implies that we are dealing with a heterogeneous, thick plasma where temperature gradients exist. For many years, these plasmas have been of great interest in astrophysics where information about celestial objects could only be extracted from spectroscopic information. The appropriate spectroscopic methods and models were, thus, developed to obtain important plasma parameters like temperature, temperature distribution, electron density. These same spectroscopic methods and models can also adequately describe dense laboratory plasmas like the LIP.