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It is proposed to use apertures with large acceptance angles to reduce the integration time when studying the emissivity of laser induced plasmas by means of the Abel inversion method. The spatial resolution lost due to contributions of angled lines of sight to the intensity data collected along the plasma plume diameter is restored by a special numerical data processing. The procedure is meant for the laser induced plasma diagnostics and tomography when the integration time needed to achieve a reasonable signal to noise ratio exceeds a characteristic time scale of the plasma state variations which is short especially at early stages of the plasma evolution. It can also be used to improve the spatial resolution in a conventional experimental setup for plasma diagnostics.
The Radon transform is tested as a method for reconstruction of the emissivity distribution of asymmetric laser induced plasmas. Two types of experiments were carried out. First, the plasma asymmetry is introduced via focusing the laser by a cylindrical lens to create plasma plumes elongated along the symmetry axis of the lens. Second, an asymmetric power distribution across the laser beam is created by reflecting the latter from a damaged mirror. Various effects on the quality of the plasma emissivity reconstructed by the Radon tomography method are investigated. The understanding of these effects appears to be essential to design a proper experimental setup to study LIBS plasmas by the Radon tomography method. It is demonstrated that the Radon tomography can successfully be used for experimental studies of asymmetric LIBS plasmas.
Abel inversion applied to a transient laser induced plasma: implications from plasma modeling
(2011)
We test the effects of non-uniformity, non-transparency, and non-stationarity of a laser-induced plasma on the results obtained by the Abel inversion method. The method is commonly used for obtaining spatially resolved emissivity of axially symmetric non-homogeneous radiating objects. Besides the axial symmetry, the plasma is assumed to be optically thin. As the method addresses a certain plasma state, the plasma is required to be stationary during measurements. It is difficult to satisfy the aforementioned conditions for transient laser induced plasmas. As such the plasmas are not stationary; they have steep gradients of thermodynamic parameters that rapidly vary during the plasma evolution. Therefore, any conclusion based on time-integrated measurements and the corresponding data processing should account for these effects. In this work, we use the collision-dominated plasma model to generate time- and spatially resolved synthetic spectra. The spectra are processed by executing the Abel inversion using two numerical algorithms. Thus obtained spatially resolved plasma parameters (emissivity, temperature, and number density) are compared with the exact parameters used to set up the model. In doing so, the accuracy of the Abel inversion method is assessed. Special attention is paid to the dynamic aspect of the expanding plasma and possible errors which result from time-integrated measurements.
An emission tomography of laser-induced plasmas employed in the laser induced breakdown spectroscopy (LIBS) requires signal integration times in a microsecond range during which the LIBS plasma cannot be considered stationary. Consequently, the use of the data for reconstructing the plasma properties under the assumption that the latter does not change significantly during the integration time leads to inaccurate results. To reduce the integration time, it is proposed to measure a plasma absorption in parallel rays using a scanning rectangular aperture whose dimension Δ along the scanning direction is about a characteristic size of plasma plumes (Δ ~ 1 cm) and the other dimension Δp is of the order of a uniformity length of plasma parameters (Δp ~ 10 µm). The aperture is moved step by step along the scanning direction and the total energy of photons coming through the aperture is measured during time T at each position of the aperture. Owing to the large size of the aperture, the integration time T is reduced by a factor ~ Δp/Δ. A numerical data processing is proposed to restore the spatial resolution of the plasma absorption along the scanning direction. It is determined by the scanning step Δs ≤ Δp. Another advantage of the proposed procedure is that inexpensive linear CCD or non-discrete (PMT, photodiode) detectors can be used instead of costly 2-dimensional detectors.
The Radon transform tomography is used for reconstruction of the emissivity distribution in single- (SP) and double-pulse (DP) laser induced plasmas in orthogonal geometry. The orthogonal DP plasma is intrinsically asymmetric and thus suitable for the Radon reconstruction. The DP plasma consists of two plasmas separated by a short time interval of ~ 1 µs. The first plasma is created in air near the surface of a Si wafer and is followed (pre-ablation mode) or preceded (post-ablation mode) by the second plasma induced on this surface. A spectrometer moves in a semi-circular path around the plasma keeping the plasma in the rotation center. The optical detection is arranged so that a thin plasma layer parallel to the target surface is monitored. The axial symmetry of the SP plasma is investigated by comparing data from the Abel inversion taken at different angles and Radon reconstruction. The multi-angle measurements are used to estimate errors of the Abel reconstruction due to asymmetries of the plasma. Time-resolved Radon reconstruction in white light is performed for the DP plasma in both pre- and post-ablation modes. In the former case, the effect of ablated aerosol on the formation of the air plasma is monitored. In the latter case, a formation of an asymmetric compression shock created by the target plasma inside the air plasma is visualized. This observation is supported by computer simulations. An interaction of the two plasmas is studied by spectrally resolved Radon reconstruction revealing a complex distribution of target and ambient species inside the plasma at all studied delay times. Overall, it is demonstrated that Radon-based tomography is an informative tool to study transient asymmetric laser induced plasmas.
A dynamical model of a laser induced plasma with axial symmetry is developed to systematically study the effects of the plasma equation of state, radiation transfer, various transport phenomena (viscosity, thermal conductivity, diffusion), and the ablation surface on the observable quantities such as spectra emitted by LIBS plasmas containing multiple species. Theoretical and numerical foundations of the model are described in detail. It is shown that the plasma spectra simulated with the equation of state based on the energy balance that includes the kinetic (thermal) energy, ionization energy, and energy of electronic excitations in atoms and ions differ significantly from the spectra obtained for plasmas modeled in the ideal gas approximation (where only the kinetic energy is included into the energy balance). Various transport phenomena, such as viscosity, diffusion, and thermal conductivity, are shown to have a little effect on the spectra. Radiation losses are proved to have noticeable effects. The effects of various interactions (adhesion, heat exchange, mass inflows) of the evolving plasma with the ablation surface are also illustrated by numerical simulations for typical LIBS plasmas. The model provides a numerical tool to assess various settings for LIBS plasma experiments as well as to interpret experimental data.
Under the assumption of local thermal equilibrium, a numerical algorithm is proposed to find the equation of state for laser-induced plasmas (LIPs) in which chemical reactions are permitted in addition to ionization processes. The Coulomb interaction in plasma is accounted for by the DebyeHückel method. The algorithm is used to calculate the equation of state for LIPs containing carbon, silicon, nitrogen, and argon. The equilibrium reaction constants are calculated using the latest experimental and ab initio data of spectroscopic constants for the molecules N2 ,C2 ,Si2 ,CN,SiN,SiC and their ions. The algorithm is incorporated into a fluid dynamic numerical model based on the NavierStokes equations describing an expansion of LIP plumes into an ambient gas. The dynamics of LIP plumes obtained by the ablation of SiC, solid silicon, or solid carbon in an ambient gas containing N2 and Ar is simulated to study formation of molecules and molecular ions.
The equation of state for plasmas containing negative atomic and molecular ions (anions) is modeled. The model is based on the assumption that all ionization processes and chemical reactions are at local thermal equilibrium and the Coulomb interaction in the plasma is described by the Debye–Hu¨ckel theory. In particular, the equation of state is obtained for plasmas containing the elements Ca, Cl, C, Si, N, and Ar. The equilibrium reaction constants are calculated using the latest experimental and ab initio data of spectroscopic constants for the molecules CaCl2, CaCl, Cl2, N2, C2, Si2 , CN, SiN, SiC, and their positive and negative ions. The model is applied to laserinduced plasmas (LIPs) by including the equation of state into a fluid dynamic numerical model based on the Navier–Stokes equations describing an expansion of LIP plumes into an ambient gas as a reactive viscous flow with radiative losses. In particular, the formation of anions Cl-, C-, Si-, Cl2, Si2, C2 , CN-, SiC-, and SiN- in LIPs is investigated in detail.
Data processing in the calibration-free laser-induced breakdown spectroscopy (LIBS) is usually based on the solution of the radiative transfer equation along a particular line of sight through a plasma plume. The LIBS data processing is generalized to the case when the spectral data are collected from large portions of the plume. It is shown that by adjusting the optical depth and width of the lines the spectra obtained by collecting light from an entire spherical homogeneous plasma plume can be leastsquare fitted to a spectrum obtained by collecting the radiation just along a plume diameter with a relative error of 10−11 or smaller (for the optical depth not exceeding 0.3) so that a mismatch of geometries of data processing and data collection cannot be detected by fitting. Despite the existence of such a perfect least-square fit, the errors in the line optical depth and width found by a data processing with an inappropriate geometry can be large. It is shown with analytic and numerical examples that the corresponding relative errors in the found elemental number densities and concentrations may be as high as 50% and 20%, respectively. Safe for a few found exceptions, these errors are impossible to eliminate from LIBS data processing unless a proper solution of the radiative transfer equation corresponding to the ray tracing in the spectral data collection is used.