A calibration-free method for laser-induced breakdown spectroscopy based on a high-repetition-rate fiber laser and non-gated detectors

Abstract

The combination of laser-induced breakdown spectroscopy based on fiber lasers (FL-LIBS) and non-gated detectors takes advantages of cost-effective, fast response, and stable long-term operation, has great potential for rapid detection and online diagnosis. However, conventional calibration-free (CF) method struggle to achieve satisfactory analytical accuracy in FL-LIBS measurements with such detectors. This is because the plasma properties certainly change during the acquisition time which is much longer than plasma lifetime. Therefore, this work develops a novel calibration-free method to address the limitations in FL-LIBS using non-gated detectors. The methodology comprises two principal components: spectra simulation via time-integration method and determination of unknown parameters utilizing particle swarm optimization (PSO); hence, the new calibration-free method is designated as PSO-SSCF. Overall, it exhibits superior accuracy on the quantitative analysis of standard TC4 titanium alloys. Compared to conventional CF method, reductions in average relative errors (AREs) range from 1.539% to 7.631% for aluminum, 22.631% to 29.173% for vanadium, and −1.071% to 0.714% for titanium. PSO-SSCF even outperforms the time-integrated calibration-free (TICF) method with an iCCD gated detector. Moreover, PSO-SSCF shows good repeatability with relative standard deviation (RSD) less than 5%, and achieves sub-second computation time via GPU acceleration. In a summary, this work provides a feasible calibration-free method for FL-LIBS, facilitating the application of LIBS in scientific and industrial fields.

Graphical abstract

Keywords

laser-induced breakdown spectroscopy / fiber laser / calibration-free method / particle swarm optimization

1 Introduction

Laser-induced breakdown spectroscopy (LIBS) has emerged as a potential atomic spectroscopy technique taking advantages of quickness, in-situ, non-contact, and online diagnosability. In traditional LIBS systems, solid-state lasers with low repetition rate (≤ 100 Hz, such as Nd:YAG) are utilized as the excitation sources [1, 2]. However, these lasers experience energy decline after longtime operation, particularly in workplaces with contaminants [3] or in complex industrial conditions [4]. Unstable output energy reduces the accuracy of LIBS analysis. Therefore, conventional LIBS systems are still faced with challenges in 8 hours (even 7 × 24 hours) industrial fields.

Compared to common solid-state lasers in LIBS, fiber lasers exhibit high repetition rate, good flexibility, excellent thermal management, and stable output energy. Furthermore, some of them cost less than one-tenth the price of solid-state lasers. In recent years, fiber lasers have been equipped in more than 60% of laser processing systems in manufacturing. Since M. Baudelet reported the implementation of a fiber laser in a LIBS system [5], an increasing number of scholars have been drawn to LIBS based on fiber lasers (FL-LIBS) [6−9], and have devoted efforts to device upgrade, spectra preprocessing, spectra classification, and calibration curves. These studies continue to promote the application of FL-LIBS.

Although FL-LIBS possesses several advantages, low output energy (generally ≤ 1 mJ) results in short plasma lifetime, decreasing the signal-to-background ratio (SBR) of spectra. In order to acquire high-quality spectra, an effective approach is to accumulate a large amount of plasma in one collection. Intensified CCD/CMOS (iCCD/iCMOS) working in trigger mode can capture several time-resolved plasma signals through repeatedly opening a high-speed shutter during exposure time [10−12]. However, high cost, large size, and strict environment requirements render these gated detectors more suitable for laboratories rather than industrial settings. Therefore, non-gated detectors like CCD/CMOS gradually become the preferred devices for LIBS researchers [13−15]. CCD/CMOS are more maneuverable and cost-effective, but they are difficult to realize tens of kHz exposure rate in trigger mode. Besides, the exposure time of non-gated detectors, which is much longer than the fiber-laser-induced plasma lifetime, makes trigger mode meaningless. Consequently, non-gated detectors usually work in continuous mode to accumulate plasma within exposure time. However, plasma properties like electron temperature and density continuously change throughout the observation period. The fixed values calculated from conventional calibration-free (CF) method [16] are unable to describe these changing plasma properties. This problem decreases the analytical accuracy of conventional CF. Hou et al. [17] proposed a time-integration calibration-free (TICF) method for traditional LIBS using non-gated detectors. Their method rewrites the intensity expression into a time-integration formula, utilizes power functions to describe the temporal evolution of electron temperature and density, and improves the accuracy of calibration-free analysis. However, gated detectors are still necessary for determining the parameters of power functions in TICF. Therefore, a feasible calibration-free method for FL-LIBS only using non-gated detectors is still unclear.

Referring to the TICF method, we propose a novel calibration-free method based on spectra simulation and particle swarm optimization (PSO) [18], abbreviated as PSO-SSCF. Unknown parameters involved in the simulation process were determined by PSO. Experiments were conducted using only a compact spectrometer equipped with CMOS detectors. The analytical performance of PSO-SSCF was validated against conventional CF through the elemental determination of titanium alloys. This work aims to improve the calibration-free accuracy of FL-LIBS, and to promote the application of LIBS in long-term online monitoring.

2 Experiment

The FL-LIBS system used in experiments is shown in Fig. 1. A 1064 nm fiber laser (RFL-P30Q, Wuhan Raycus Fiber Laser Technologies Co., Ltd., China, less than 800 USD) was applied, with a power of 12.51 W, a repetition rate of 30 kHz, and a pulse duration of 120 ns. A Czerny−Turner spectrometer (AvaSpec-ULS2048CL-EVO, Avantes, Netherlands) equipped with CMOS detectors was used for spectral acquisition. The wavelength range of the spectra was from 240 to 415 nm with a resolution of about 0.1 nm. The laser was scanned by placing the sample on a motorized X−Y stage (ZP110-50, Beijing Lyseiki, China). Standard TC4 titanium alloys (China Shipbuilding Industry Group Co., Ltd., with certified concentrations listed in Table 1) were used as the samples. The optical collector was oriented at approximately 45 degrees above the stage. PSO-SSCF was implemented using PyTorch 2.1.1 on a GPU (NVIDIA GeForce RTX 3060 Ti GDDR6X).

While the system was operating, a series of Gaussian beams passed through a convex lens (f = 200 mm) to ablate the sample. The laser spot at the sample surface was about 72.25 μm in diameter and 10.17 J/cm² (3.05 × 10⁵ kW/cm²) in energy density. The motorized X−Y stage was set to move at 7.5 mm/s. The CMOS detectors operated in continuous mode with an exposure time of 7 ms, capturing about 210 plasma events (corresponding to 30 000 shots/s × 7 ms) per spectrum. Due to the high-repetition-rate output of the fiber laser, hundreds of spectra could be acquired in a few seconds. The average of 10 spectra was taken as one measurement. Each sample was measured 50 times.

The analytical lines are listed in Table 2. Because the photo-response of CMOS detectors in this work was weak before 300 nm, the analytical lines were selected within the range of 300−415 nm. Ti I and Ti II exhibited numerous lines, whereas only four V I lines, one V II line, two Al I lines, and no Al II lines were observed.

3 Methodology

Spectra simulation and parameter assumption are the cores of the new method. The unknown parameters in spectra simulation are assumed to be different values randomly generated by PSO. Each value will be evaluated based on the similarity between simulated spectra and observed spectra.

3.1 Spectra simulation

Because plasma intensity evolves during spectral acquisition, if observed spectra are considered as the accumulation over time, simulated spectra can be expressed in the time-integration form:

(1)I^{\text{sim}}={\displaystyle \int i^{\text{sim}}\text{d}t,}

where i^{\text{sim}} is the stimulated intensity at a moment. The expression of i^{\text{sim}} can be theoretically deduced. Based on spontaneous emission theory, Boltzmann distribution and Saha ionization equation [19], the spectral intensity of optically thin lines at a moment is defined as

(2){\iota }_0=F_N\chi {\displaystyle \frac{\text{hc}{A}_{ki}{g}_k}{\text{4π}\lambda }}\exp \left(-{\displaystyle \frac{E_k}{k_{\text{B}}T}}\right),

(3)\chi =\{\begin{array}{l}{\displaystyle \frac{R}{U_s^l\left(T\right)\cdot \left(1+R\right)}},\text{for atomic lines};\\ {\displaystyle \frac{1}{U_s^{ll}\left(T\right)\cdot \left(1+R\right)}},\text{for ionic lines},\end{array}

(4)R={\displaystyle \frac{U_s^l\left(T\right){\hslash }^3{N}_{\text{e}}}{2U_s^{ll}\left(T\right)\cdot {\left(2\text{π}{m}_{\text{e}}{k}_{\text{B}}T\right)}^{1.5}}}\exp \left({\displaystyle \frac{E_{ion,s}}{k_{\text{B}}T}}\right),

where \hslash is the Planck constant; c is the light velocity in vacuum; \text{π} is the ratio of circumference to diameter; k_{\text{B}} is the Boltzmann constant; m_{\text{e}} is the electron mass; \lambda is the central wavelength of spontaneous emission; subscripts i and k indicate the lower and the upper levels; A_{ki} is the spontaneous radiation transition probability; g_k is the degeneracy of upper level k; E_k is the energy of upper level k; T is the electron temperature; U_s^l\left(T\right) and U_s^{ll}\left(T\right) are the partition functions of element s at temperature T; superscripts l and ll indicate the atomic and ionic lines; E_{ion,s} is the first ionization energy of element s; N_{\text{e}} is the electron density; F_N is defined as the element number density.

In fact, spectral lines affected by the self-absorption effect hardly exhibit optically thin intensity. In order to reflect the real situation, the blackbody radiation referenced (BRR) method [20] is added to the spectral simulation. A blackbody L at wavelength \lambda and temperature T is defined by Eq. (5). By incorporating the self-absorption effect simulation, spectral intensity at a moment is modified from Eq. (2) to Eq. (6),

(5)L={\displaystyle \frac{2\text{π}h{c}^2}{{\lambda }^5}}\cdot {\left[\exp \left({\displaystyle \frac{hc}{\lambda k_{\text{B}}T}}\right)-1\right]}^{-1},

(6)\iota =L\cdot \left[1-\exp \left(-{\displaystyle \frac{{\iota }_0}{L}}\right)\right].

All the parameters contained in Eq. (6) are obtained from NIST database (www.nist.gov/pml/atomic-spectra-database) except N_{\text{e}}, T, and F_N. The electron density N_{\text{e}} is estimated by Saha−Boltzmann ionization equation [21]:

(7)N_{\text{e}}={\displaystyle \frac{2\cdot {\left(2\text{π}{m}_{\text{e}}{k}_{\text{B}}T\right)}^{1.5}}{h^3}}{\displaystyle \frac{I^l{A}_{ki}^{ll}{g}_k^{ll}}{I^{ll}{A}_{ki}^l{g}_k^l}}\exp \left(-{\displaystyle \frac{E_{ion,s}+E_k^{ll}-E_k^l}{k_{\text{B}}T}}\right),

where I^l, A_{ki}^l, g_k^l and E_k^l as the spectral parameters belong to atomic lines; I^{ll}, A_{ki}^{ll}, g_k^{ll} and E_k^{ll} belongs to ionic lines. Because the trend of temperature decrease is initially severe and then gradual [17, 22−24], the electron temperature T is approximately profiled as a power function in this method:

(8)T\left(t\right)=T_0{t}^{\alpha },

where T_0 is the initial temperature; \alpha indicates the decrease trend. By substituting Eq. (8) into Eq. (2) and Eq. (6), two spectral intensity expressions are rewritten as the functions of time. The certain expression of simulated spectra thereby is modified as

(9)I^{\text{sim}}={\displaystyle \int \iota \left(t\right)\text{d}t.}

3.2 Unknown parameters determination

Thus far, three parameters T_0, \alpha, and F_N are unknown in Eq. (9). These unknown parameters are treated as an undetermined vector \left[T_0,\alpha ,F_N\right], whose values will be determined through assumptions. Because it is impractical to assume F_N in the whole real number field, F_N is narrowed into a range, where self-absorption coefficient as a function of F_N drops from 0.99 to 0.01. The self-absorption coefficient is defined as the ratio of the spectral intensity given by Eq. (6) to that given by Eq. (2) [25], where T is set to 10 000 K for normalization and simplification. In addition, observed spectra require correction before comparison with simulated spectra:

(10)I^{\text{obs}}=F_{\text{cor}}{I}_{{}^0}^{\text{obs}},

where I_{\text{0}}^{\text{obs}} represents the observed spectrum after preprocessing; F_{\text{cor}} is defined as the correction factor adjusting the intensity into an appropriate magnitude order. F_{\text{cor}} depends on optical collection efficiency and spectral preprocessing, and thus a specific experimental system has a fixed F_{\text{cor}}. Nevertheless, F_{\text{cor}} is unnecessary to be precise. If reference samples are available, F_{\text{cor}} can be inferred from the preliminary experiments; if not, F_{\text{cor}} can be estimated as

(11)F_{\text{cor}}={\displaystyle \frac{t_p{L}_{{\lambda }_{max},T=10000\text{K}}}{I_{max}^{\text{obs}}}},

where I_{max}^{\text{obs}} is the maximum intensity in an observed spectrum; {\lambda }_{max} is the wavelength of I_{max}^{\text{obs}}; L_{{\lambda }_m,T=10000\text{K}} is the blackbody radiation intensity at {\lambda }_{max} at 10 000 K; t_p is the magnitude order of plasma lifetime (generally on the order of 10⁻⁶ s).

PSO generates multiple undetermined vectors \left[T_0,\alpha ,F_N\right] and searches for the optimal one among them. The evaluation approach involves assessing the similarity between the simulated spectrum corresponding to each undetermined vector and the observed spectrum. Fitness z is defined to characterize the similarity:

(12)z={\displaystyle \frac{1}{u}}{\displaystyle \sum _u{\left[\ln \left({\displaystyle \frac{I^{\text{sim}}}{I^{\text{obs}}}}\right)\right]}^2,}

where u is the total number of atomic and ionic lines for an element. The iterative process can be described as the candidate solutions moving toward the optimal position at prescribed velocities within a three-dimensional space. The PSO parameters include population size p, maximum iteration m, inertia weight w_i, cognitive weight w_c, and social weight w_s. Population size decides the number of generated vectors, which explores the solution space more effectively in a large size but requires more computation time. Maximum iteration defines the epoch in which the process should terminate. Inertia weight controls the influence of the previous velocity on the current velocity. Cognitive weight influences how much a vector is attracted to its previous best position. Social weight influences how much a vector is attracted to the previous global best position. In order to avoid premature convergence and improve robustness, w_i is set to twice the value of w_c and w_s.

3.3 Implementation process

Figure 2 shows the process of the PSO-SSCF method. The value of F_{\text{cor}} is estimated by Eq. (11). The range of F_N is deduced from the reasonable self-absorption coefficients. Then, the element (generally a matrix element) exhibiting abundant lines is selected for spectral simulation. PSO starts to explore the optimal vector in the solution space. To shorten the computation time, PSO executes parallel computation [26]. When the epoch arrives at the maximum iteration, F_N of the element selected before is determined from the optimal vector as well as the electron temperature curve. Subsequently, based on the electron temperature curve, the F_N of other elements are solved respectively corresponding to the highest similarity between simulated spectra and observed spectra. Because the problem can be described as finding the minimum point of Eq. (12), where F_N is the only one variable, Nelder−Mead algorithm is recommended to use at this step. Finally, the elemental concentration is determined according to all the F_N values.

4 Result and discussion

Since the detectors working in continuous mode are unable to avoid bremsstrahlung, the background appears at the bottom of the spectra. The peak-clipping algorithm is introduced to remove the background [27], which terminates when the difference in the area of the fitting background between two generations is less than 0.1%.

The spectra with background removed finishing background removed were regarded as the observed spectra in the PSO-SSCF method. Taking TC4-2 titanium alloy as an example, the observed spectrum is shown in Fig. 3. The maximum intensity was located at 375.94 nm. Therefore, the value of F_{\text{cor}} was estimated to be 10¹⁰ based on the blackbody radiation intensity at 375.94 nm at 10 000 K.

Figure 4 shows the self-absorption coefficients of Ti I 399.86 nm, Ti II 324.86 nm, V I 411.52 nm, V II 355.68 nm, and Al I 394.40 nm. These curves roughly determined the range of F_Nvalues within 10²⁸ to 10³² in this work.

The matrix element titanium was selected for simulation. The undetermined vectors [T₀, α, F_N] were processed by PSO with the configuration: p=50, m=20, w_i=0.6, w_c=0.2, and w_s=0.2. When PSO finished the iterations, the optimal vector was obtained from the global best solution. The optimal vector determined the electron temperature, the electron density, and the F_N of titanium. Figure 5 shows the electron temperature and electron density curves of six samples.

Existing studies demonstrate that the electron density of fiber-laser-induced plasma exceeds that of conventional laser-induced plasma by over an order of magnitude. It indicates that the plasma lifetime in FL-LIBS was much shorter than that in conventional LIBS (10⁻⁶−10⁻⁵ s). Consequently, plasma in the experiments readily met the local thermodynamic equilibrium (LTE) theory required by PSO-SSCF. The electron density results presented in Fig. 5(b), which comply with the McWhirter criterion defined as Eq. (13), further confirm the fulfillment of LTE theory [28],

(13)N_{\text{e}}\left(T\right)>1.6\times {10}^{\text{12}}\sqrt{T\left(t\right)}{\left(\text{Δ}{E}_{ki}^s\right)}^3.

The analytical performance of PSO-SSCF was compared with conventional CF and TICF. All approaches processed a common set of spectral data (50 measurements per sample) obtained from one experiment. Results shown in Fig. 6 indicate that self-absorption effect evidently reduced the accuracy of analysis. With the BRR method, conventional CF attained average relative errors (AREs) of results ranging from 3.086% to 15.022% for aluminum, 24.552% to 43.177% for vanadium, and 0.176% to 0.917% for titanium. In comparison, PSO-SSCF demonstrated lower AREs: 1.547% to 7.391% for aluminum, 1.661% to 14.837% for vanadium, and 0.203% to 1.247% for titanium. Although conventional CF was a little more accurate for titanium in TC4-1, the overall performance of PSO-SSCF was obviously better. Furthermore, PSO-SSCF was even superior to TICF, although TICF needs an iCCD to measure the decay of the plasma emission. The low pulse energy (< 1 mJ) in FL-LIBS resulted in short plasma lifetime (< 1 μs), low SNR of single plasma, and large error in time-resolved temperature measurement in TICF. Therefore, the results demonstrate PSO-SSCF stands out as the best scheme in FL-LIBS.

Moreover, the repeatability of PSO-SSCF was assessed by the relative standard deviation (RSD). Figure 7 illustrates a comparison of the RSD of results between conventional CF and PSO-SSCF. PSO-SSCF showed RSD of results less than 5%, which was similar to that of conventional CF. It should be noted that the reproducibility of results is primarily affected by spectral fluctuation for conventional CF, while for PSO-SSCF, whether PSO exhibits good convergence also affects its reproducibility.

The best fitness at each epoch across 100 independent runs is presented in Fig. 8(a), which demonstrates good convergence for PSO, confirming the repeatability of PSO-SSCF. A large population size ensures better convergence but significantly increases computational time. The size was determined to be 50 after comprehensive consideration and experiments with different population sizes. Figure 8(b) in particular demonstrates that PSO calculations cost less than a second with GPU acceleration when population size was 50. As a result, the GPU reduces the computation time of PSO-SSCF to sub-second, achieving a 60 times speedup over a CPU-only implementation.

5 Conclusion

Addressing the limitations in FL-LIBS, the PSO-SSCF method allowing calibration-free analysis on spectra acquired by non-gated detectors is proposed in this work. This method utilizes PSO to generate multiple candidate values for electron temperature and element number density. It then evaluates the similarity between the observed spectra and the simulated spectra determined by these candidate parameters. Finally, the parameters corresponding to the simulated spectrum exhibiting the highest similarity are determined as the solution. Versus conventional CF, PSO-SSCF reduces AREs by 1.539%−7.631% for aluminum, 22.631%−29.173% for vanadium, and 1.071%−0.714% for titanium. Furthermore, PSO-SSCF was even superior to TICF, although TICF needs an iCCD to measure the decay of the plasma emission. It also shows good repeatability (RSD < 5%) and fast computation time (< 1 s), which was enabled by GPU acceleration.

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