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Abstract
We developed a post-processing optical imaging model based on two-dimensional axisymmetric radiation hydrodynamics. Simulation and program benchmarks were performed using laser-produced Al plasma optical images obtained via transient imaging. The emission profiles of a laser-produced Al plasma plume in air at atmospheric pressure were reproduced, and the influence of plasma state parameters on radiation characteristics were clarified. In this model, the radiation transport equation is solved on the real optical path, which is mainly used to study the radiation of luminescent particles during plasma expansion. The model outputs consist of the electron temperature, particle density, charge distribution, absorption coefficient, and corresponding spatio-temporal evolution of the optical radiation profile. The model helps with understanding element detection and quantitative analysis of laser-induced breakdown spectroscopy.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser-produced plasmas (LPPs) have attracted wide attention as ideal laboratory plasma sources [1], ion sources [2,3], and light sources [4,5]. They have been widely used in laser-induced breakdown spectroscopy [6], pulsed laser deposition of thin films [7], and inertial confinement fusion [8]. An accurate and deeper understanding of their radiative and dynamical behaviors is helpful for further applications in these fields. At present, spatio-temporal resolution spectroscopy [9,10], interference methods [11,12], and transient imaging [13–15] are used to understand the radiation characteristics and dynamic evolution of LPPs. Owing to the intuitiveness of transient imaging, it has been widely used to obtain the size [15,16], shape, and species distribution of plasma [17–19], covering a wide range of ablation environments, time scales, and laser parameters [18–20].
Time evolution analysis of an LPP transient image is very helpful in understanding plasma expansion. Combined with theoretical simulation, such analysis helps with understanding the evolution of plasma parameters and the underlying mechanism [17]. However, few theoretical simulations of plasma optical imaging have been reported. In general, the contour images of the plasma temperature output of the radiation hydrodynamics model are compared with 2D images of the plasma plumes obtained by an ICCD camera [21], but this comparison is based on limited understanding of plasma. This is because plasma optical imaging results are determined by multiple parameters, not just temperature [22,23]. The FESTR [23] and SPECT3D [24] codes can output the image and backlit image of plasma, which are calculated using parallel rays. However, the radiation collected by the detectors does not only come from parallel rays, and the imaging system is not considered. Therefore, the simulation results obtained by these codes cannot be directly compared with the experimental results.
We have developed a two-dimensional (2D) axisymmetric radiative hydrodynamics model for LPPs [25]. The physical model was used to describe the following phenomena: (1) heating of the target by laser energy deposition on the target surface and heat conduction inside the target, (2) evaporation and condensation on the target surface, (3) vapor breakdown and plasma formation, (4) laser radiation absorption by the plasma and LPP heating and ionization, and (5) dynamic evolution of plasma. The laser energy deposition on the target surface and the heat conduction inside the target are described by a heat conduction equation and the boundary conditions with the laser source term [26,27]. The laser-plasma interaction and dynamic evolution of the plasma are described by gas dynamic equations [27]. The Knudsen layer (KL) boundary condition is used for the target-vapor interface [28,29]. The internal energy and pressure of plasma are expressed by the equations of state (EOSs) based on a real-gas approximation [30]. The model is effective when the surface temperature of the target does not reach the critical temperature during laser ablation [31]. The parameters of the Al target and corresponding boundary conditions for simulation were obtained from early work [25]. The model outputs the temperature, pressure, density, and velocity distributions of the target and plasma.
At present, moderate-intensity ns-laser irradiation of a solid target is simulated using the radiation hydrodynamics model in ambient argon, air, and helium of different pressures [25]. Therefore, the goal of this paper is to develop a new model for reconstructing the optical image of an LPP plume based on the above radiation hydrodynamics model. The model constructs a series of real optical paths based on the imaging system and considers the contribution of emissivity and opacity to the radiation profile. The aim is to reconstruct the radiation profile of Al LPPs through a series of transient image measurements of plasmas and the results of radiation hydrodynamics simulation. Furthermore, we investigate the radiation and dynamic evolution of Al plasma and the benchmarks of the radiation hydrodynamics model.
2. Experiment
A planar aluminum target (99.999%) was irradiated using a Nd:YAG laser (Beamtech Optronics Dawa-200) emitting radiation at a wavelength of 1064 nm for 8 ns at full width at half maximum (FWHM). The laser pulse, with an energy of approximately 10 mJ, was focused perpendicularly onto the target surface by a lens with 100-mm focal length. The target holder was positioned on a 3D translation stage to avoid forming deep craters on the target surface. An ICCD camera (iStar, sCMOS-18U-E3, Andor), oriented parallel to the target surface, was used to obtain 2D images of the plasma plume. The sensor matrix on the ICCD camera consisted of 2160 × 2560 pixels (1 pixel = 6 µm × 6 µm). Emission profiles of the plumes were obtained in the wavelength range of 180 to 850 nm aided by an imaging system consisting of a lens (f = 50 mm). The imaging ratio was adjusted by changing the position of the lens and detector matrix relative to the plasma. A digital-pulse-delay generator (DG645, Stanford Research Systems) synchronized the entire experimental setup and registered temporally resolved measurements.
The experiment was carried out in air at atmospheric pressure. Figure 1 shows the transient images of Al LPPs with a delay time of 200-900 ns, where the gate width of the camera is 20 ns. The plasma plume profile increases while the intensity decreases. The changes in strength and contour are relatively large, and after 600 ns, the changes in both are relatively small owing to the air confinement of the plasma. In addition, the images appear grainy/blurred, which may be caused by weak plasma radiation and the imaging system does not eliminate the aberration.
3. Optical imaging model
To obtain the emission profile of the plasma plume, a post-processing optical imaging model is established in Cartesian coordinates based on the local thermodynamic equilibrium (LTE) assumption. Assuming that z = 0 mm is the target surface, the symmetry axis of the plasma coincides with the z axis. The plasma temperature and particle number density output by the radiative hydrodynamic model are used as input parameters of this model. Figure 2 shows the schematic of the model. The gray cylindrical calculation area encapsulates the plasma. A plano-convex lens perpendicular to the y axis is used as the imaging system. The emission profile of the plasma is imaged on the sensor of the ICCD camera. Using the paraxial thin-lens imaging approximation, we assume that the imaging ratio is 1:1, the focal length of the lens is f = 50 mm, and the distance between the center of the lens and the symmetry axis of the plasma is 100 mm. The emission profile of the plasma cross section will be imaged on the sensor with equal size.
The modeling process is divided into four steps. (1) The sensor surface is composed of several pixels, and each pixel collects photons as an independent unit. Each pixel receives the radiation from the corresponding area $\epsilon $ on the cross section of the plasma, where the size and position of $\epsilon $ are determined by the object–image relationship and the pixel size, as shown in Fig. 3(a). (2) Let the geometric center of the vector surface element $d\boldsymbol{S}$ on $\epsilon $ be ${\boldsymbol N^{\prime}}$, and the geometric center of the vector surface element $d{\boldsymbol \sigma }$ on the lens surface $\mathrm{{\cal L}}$ be ${\boldsymbol P}$. The ray-tracing path belongs to straight line $N^{\prime}P$. If the line $N^{\prime}P$ goes through the cylinder and intersects with its surface at points ${\boldsymbol N}$ and ${\boldsymbol M}$, then the line segment NM is the ray-tracing path, as shown in Fig. 3(b). (3) The radiation transport equation is solved on the straight-line segment NM along the NM direction. Therefore, its analytical solution can be expressed as
Calculating the effective absorption coefficient needs the charge distribution, electron density, and energy level population of the plasma. If the temperature and particle number density are known, then the charge distribution and electron density of the plasma can be obtained by solving the Saha equations. The energy level population satisfies the Boltzmann distribution [33].
Figure 4 shows the flow chart of the plasma optical imaging simulation. The simulation is divided into three steps. (1) The experimental parameters, which consist of the laser power density, laser walength, target type, melting point, ambient gas type, and pressure, are input along with the corresponding boundary conditions into the 2D axisymmetric radiation hydrodynamics program, and the state parameters such as plasma temperature and particle number density are output on the 2D uniform grid. The plasma population information at each grid point is calculated using the Saha equations and Boltzmann distribution, and the effective absorption coefficient is calculated. On this basis, a three-dimensional cylindrical computational domain is obtained by rotating the 2D data grid 360° around the z axis. (2) The size and position of each $\epsilon $ in the plasma cross section are determined according to the size and position of each pixel on the sensor matrix and the imaging relation. (3) The number of photons collected for each pixel is calculated.
4. Results and analysis
In general, the temperature of the plasma output from the radiation hydrodynamics model is directly compared with the emission profile of the plasma plume obtained from the ICCD camera to obtain better consistency [21,25]. However, the comparison is superficial and perceptual, and cannot reveal the internal relationships between various physical quantities. Figure 5(a) shows the temperature of the air-laser-produced Al plasma simulated using the radiation hydrodynamics model. The corresponding laser power density is 109 W/cm2, the laser wavelength is 1064 nm, the pressure is 105 Pa, and the delay time is 200 ns. Figures 5(b) and (c) show the experimental and simulated optical images of the plasma plume under the same conditions, respectively, with an imaging scale of 1:1. The optical imaging results are similar to the temperature output by the radiation hydrodynamics program, but there are differences in detail. First, in the region z > 0 mm, the temperature profile is significantly larger than the experimental and theoretical optical imaging profiles, because the contribution of optical imaging mainly comes from the bound–free and free–free plasma radiation. As the temperature of the plasma edge region decreases, the bound–free and free–free radiations of the plasma decrease significantly. Second, the high-temperature region of the plasma is at approximately z = 0.3 mm, and the strong-radiation region of the plasma plume reflected by the optical image is at approximately z = 0.1 mm. The significant difference is mainly caused by the misalignment of the symmetry axis of the imaging system with the high-temperature region of the plasma plume and the radiation transport in the plasma. When the symmetry axis of the imaging system is far from the target surface and misaligned with the high-temperature region of the plasma, the plasma is ‘squinted’ by the imaging system. In addition, the large temperature density gradient in the laser plasma leads to significant opacity. When the radiation passes through the plasma with significant opacity along the ‘squint’ path, the difference mentioned above is formed. Finally, we find that the optical image also has the radiation profile of the plasma plume in the region z < 0 mm, which is caused by the imaging angle and target reflection.
Fig. 5. (a) Theoretically simulated temperatures, (b) experimental and (c) theoretical images of laser-produced Al plasmas with a delay time of 200 ns.
To test the model output, a laser-produced Al plasma in ambient air with a pressure of 105 Pa was used to check the experimental and theoretical agreement for the time evolution of the optical imaging contours and intensities. Figure 6 shows the comparison between the 2D images of the plumes obtained by the ICCD camera and the model output in ambient air at 105 Pa. The delay times are 200, 400, and 600 ns. We find that the simulation results reproduce the plasma profile well. The region z < 0 mm on the image is mainly reflected from the target surface to further illustrate the consistency between the experiment and theory. The intensities of the plasma plume image are also compared on the symmetry axis in Fig. 7. Good agreement has been obtained at the three delay times. The absolute quantum effect of the ICCD camera cannot be determined, with the result that the dimensions of the two intensities cannot be unified, but this does not detract from the consistency of their intensity evolution with delay time. The results indicate that the plasma parameters provided by the radiation hydrodynamics model are suitable, and the optical imaging model is also successful. In addition, there are differences between the simulated and experimental images in details, which may be caused by the LTE assumption and the neglect of aberration in the model.
Fig. 6. Comparison between the 2D images of the plasma plumes obtained by fast photography (left column) and output by the model (right column) in ambient air with a pressure of 105 Pa. The delay times are 200, 400, and 600 ns.
Fig. 7. Experimental (points) and theoretical (solid lines) optical imaging intensities on the symmetry axis.
The effective absorption coefficient is a very important parameter for characterizing the radiation of any plasma because it reflects the ability of the plasma to emit and absorb radiation. The coefficients ${\kappa _{bf}}$ and ${\kappa _{ff}}$ are calculated from a semi-empirical formula [34] that has been verified by I. B. Gornushkin et. al. [34,35]. The ${\kappa _{bb}}$ calculation contains 123 strong lines consisting of 29 Al lines, 48 N lines, and 46 O lines. The atomic data are from the NIST database [36]. The plasma temperature and density determine the ion charge and population, which in turn determine the effective absorption coefficient. The relationship between them is illustrated by the relevant parameters on the plasma symmetry axis with a delay time of 200 ns. The mass density and temperature are given in Fig. 8(a), the particle number density and electron number density are shown in Fig. 8(b), and the effective absorption coefficient at 744 nm on the image symmetry axis are presented in Fig. 8(c). The plasma temperature reaches a maximum of approximately 25000 K in the core region (z = 0.3 mm), whereas the mass density is minimal at the same position. This leads to a minimum value of ${\kappa _\nu }$ there. In plasma, the number of Al, N, and O atoms generally dominates, whereas high temperatures in the core region cause a large number of atoms to be ionized. In the plasma edge region (z = 0.5 mm), the plasma interacts with the air, and a large number of N and O atoms are excited and ionized, resulting in a maximum value of ${\kappa _\nu }$ there. ${\kappa _{ff}}$ and ${\kappa _{bf}}$ are generally larger than ${\kappa _{bb}}$. Because the first two absorption coefficients are functions of electron density, ${\kappa _\nu }$ rapidly decreases. Finally, near the target surface, the values of ${\kappa _{ff}}$ and ${\kappa _{bf}}$ are approximately 10 times that of ${\kappa _{bb}}$, but in the plasma edge region, the values of the three absorption coefficients are equivalent.
Fig. 8. Comparison of plasma parameters on the symmetry axis: (a) mass density (blue) and temperature (red), (b) particle number density (lines) and electron density (points), and (c) effective absorption coefficient (lines) at 744 nm.
5. Conclusion
In summary, we have established a post-processing optical imaging model based on 2D axisymmetric radiation hydrodynamics using the real imaging optical path and radiation transport equation. This model reproduces the emission profile and time evolution of an Al plasma plume in air at atmospheric pressure. The effective plasma absorption coefficient was obtained, the influence of the plasma state parameters on the radiation characteristics was clarified, and the radiation hydrodynamics model was indirectly benchmarked. In future, it will improve agreement between simulation and experiment in image details that we will try to eliminate aberrations during the experiment in order to obtain better imaging images. The model is expected to support the study of laser-generated plasma radiation and dynamic evolution of luminescent particles.
Funding
National Natural Science Foundation of China (11874051, 11904293, 12064040); The Science and Technology Project of Gansu Province (20JR5RA530); The Funds for Innovative Fundamental Research Group Project of Gansu Province (20JR5RA541); The Key Cultivation Program of Northwest Normal University (NWNU-LKZD2021-02).
Acknowledgments
Mark R. Kurban from Liwen Bianji (Edanz) edited a draft of this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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