Approximate Voigt function formula for laser-induced breakdown spectroscopy fitting

Siying Chen; Yiwen Jia; He Chen; Pan Guo; Qixiang Xu; Lifu Wang; Yinchao Zhang

doi:10.1364/AO.416677

Applied Optics
Vol. 60,
Issue 14,
pp. 4120-4126
(2021)
•https://doi.org/10.1364/AO.416677

Approximate Voigt function formula for laser-induced breakdown spectroscopy fitting

Siying Chen, Yiwen Jia, He Chen, Pan Guo, Qixiang Xu, Lifu Wang, and Yinchao Zhang

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Siying Chen, Yiwen Jia, He Chen, Pan Guo, Qixiang Xu, Lifu Wang, and Yinchao Zhang, "Approximate Voigt function formula for laser-induced breakdown spectroscopy fitting," Appl. Opt. 60, 4120-4126 (2021)

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Abstract

Accurate and rapid spectrum fitting is very important for quantitatively analyzing laser-induced breakdown spectroscopy (LIBS). The Voigt function is often used to fit LIBS spectral lines. We propose a new approximate Voigt function formula. Based on the classic Lorentz–Gauss linear combination formula, a summation term was added that contained a specific convolution operation to improve the Voigt function’s calculation and fitting accuracy. This formula can be used for the approximate calculation of the Voigt function with an overall accuracy of 0.31% and a full width at half-maximum internal accuracy of 0.25% when the ratio of Lorentzian linewidth to Gaussian linewidth is 1:1. The formula was then applied to LIBS data processing to fit four element spectral lines of calcium (Ca-393.37, 396.85, and 422.67 nm) and potassium (K-766.49 nm). The fitting results showed that this new approximate formula could fit at least seven data points, and compared with the complex plane partition method and the classic linear combination formula, the new formula had better fitting speed and accuracy.

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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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	Maximum Residual
	$L : G = 1 : 1$	$L : G = 2 : 1$	$L : G = 1 : 2$	$L : G = 10 : 1$
Linear combination formula	0.0077	0.0085	0.0029	0.0009
New approximate formula	0.0031	0.0073	0.0017	0.0006
Complex plane partition method	0.0016	0.0063	0.0016	0.0006

Linewidth Ratio	Grid Point Spacing	Maximum Residual	Central Residual
$L : G = 1 : 1$	1	0.0013	$4.16 \times 1 0^{- 5}$
	0.5	0.0038	0.0038
	0.2	0.0031	0.0025
	0.1	0.0031	0.0025
	0.01	0.0031	0.0025
	0.001	0.0031	0.0025
$L : G = 2 : 1$	1	0.0019	$8.93 \times 1 0^{- 5}$
	0.5	0.0045	0.0039
	0.2	0.0066	0.0037
	0.1	0.0073	0.0037
	0.01	0.0080	0.0037
	0.001	0.0080	0.0037
$L : G = 1 : 2$	1	0.0025	0.0011
	0.5	0.0014	0.0012
	0.2	0.0016	0.0012
	0.1	0.0017	0.0012
	0.01	0.0018	0.0012
	0.001	0.0018	0.0012
$L : G = 10 : 1$	1	$3.36 \times 1 0^{- 9}$	$1.43 \times 1 0^{- 10}$
	0.5	$7.83 \times 1 0^{- 4}$	$5.00 \times 1 0^{- 5}$
	0.2	$5.18 \times 1 0^{- 4}$	$1.24 \times 1 0^{- 4}$
	0.1	$5.74 \times 1 0^{- 4}$	$9.65 \times 1 0^{- 5}$
	0.01	$5.96 \times 1 0^{- 4}$	$8.60 \times 1 0^{- 5}$
	0.001	$5.97 \times 1 0^{- 4}$	$8.56 \times 1 0^{- 5}$

Wavelength	Parameters	Complex Plane Partition	Linear Combination	New Formula
393.37 nm	SSR	$2.0026 \times 1 0^{3}$	$5.0689 \times 1 0^{3}$	$1.7497 \times 1 0^{3}$
	$R^{2}$	0.9998	0.9996	0.9999
	Time (s)	0.0934	0.0274	0.0029
396.85 nm	SSR	$1.0659 \times 1 0^{4}$	$9.3314 \times 1 0^{3}$	$8.4352 \times 1 0^{3}$
	$R^{2}$	0.9976	0.9979	0.9981
	Time (s)	0.0832	0.0028	0.0066
422.67 nm	SSR	$1.2683 \times 1 0^{4}$	$9.4749 \times 1 0^{3}$	$9.4636 \times 1 0^{3}$
	$R^{2}$	0.9988	0.9991	0.9991
	Time (s)	0.0570	0.0018	0.0027
766.49 nm	SSR	$9.4021 \times 1 0^{3}$	$1.6052 \times 1 0^{4}$	$9.2133 \times 1 0^{3}$
	$R^{2}$	0.9995	0.9992	0.9995
	Time (s)	0.0302	0.0112	0.0036

	Maximum Residual
	$L : G = 1 : 1$	$L : G = 2 : 1$	$L : G = 1 : 2$	$L : G = 10 : 1$
Linear combination formula	0.0077	0.0085	0.0029	0.0009
New approximate formula	0.0031	0.0073	0.0017	0.0006
Complex plane partition method	0.0016	0.0063	0.0016	0.0006

Linewidth Ratio	Grid Point Spacing	Maximum Residual	Central Residual
$L : G = 1 : 1$	1	0.0013	$4.16 \times 1 0^{- 5}$
	0.5	0.0038	0.0038
	0.2	0.0031	0.0025
	0.1	0.0031	0.0025
	0.01	0.0031	0.0025
	0.001	0.0031	0.0025
$L : G = 2 : 1$	1	0.0019	$8.93 \times 1 0^{- 5}$
	0.5	0.0045	0.0039
	0.2	0.0066	0.0037
	0.1	0.0073	0.0037
	0.01	0.0080	0.0037
	0.001	0.0080	0.0037
$L : G = 1 : 2$	1	0.0025	0.0011
	0.5	0.0014	0.0012
	0.2	0.0016	0.0012
	0.1	0.0017	0.0012
	0.01	0.0018	0.0012
	0.001	0.0018	0.0012
$L : G = 10 : 1$	1	$3.36 \times 1 0^{- 9}$	$1.43 \times 1 0^{- 10}$
	0.5	$7.83 \times 1 0^{- 4}$	$5.00 \times 1 0^{- 5}$
	0.2	$5.18 \times 1 0^{- 4}$	$1.24 \times 1 0^{- 4}$
	0.1	$5.74 \times 1 0^{- 4}$	$9.65 \times 1 0^{- 5}$
	0.01	$5.96 \times 1 0^{- 4}$	$8.60 \times 1 0^{- 5}$
	0.001	$5.97 \times 1 0^{- 4}$	$8.56 \times 1 0^{- 5}$

Abstract

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Equations (9)

Applied Optics