Spectral linewidth is an important parameter in electron paramagnetic resonance (EPR),^[1] nuclear magnetic resonance (NMR)^[2] as well as molecular absorption spectroscopy.^[3] The accurate measurement of absorption linewidth is conducive to the analysis of spectral properties, for example, determination of the intermolecular collisional coefficient depends on the accurate measurement of the linewidth.^[4] In wavelength modulation spectroscopy (WMS), fast and accurate linewidth determination can be used to correct the measurement errors of gas concentration and (or) temperature due to linewidth broadening in real time.^{[5, 6]} Therefore, it is essential to rapidly and accurately measure the absorption linewidth.

Measurements of absorption linewidth have received a great deal of attention.^[7–10] In optical spectroscopy, the absorption linewidth can be obtained directly by spectrum measurement, but the accuracy of this method is limited by the low resolution of the spectrometer and small absorbance of the target spectrum. Practically, the absorption linewidth can be derived by nonlinear least squares fitting of the direct absorption spectrum (DAS) signal, which is also used to measure the intermolecular collisional coefficient under laboratory conditions.^[11–13] However, the measurement accuracy is often restricted by baseline fluctuation and lower absorbance. Wavelength modulation spectroscopy has been developed to meet lower absorbance.^[14–17] The absolute gas absorption lineshape can be recovered from wavelength modulation by using the phasor decomposition method (PDM), however, it is more costly as it requires more sophisticated RF lock-in equipment.^{[18, 19]} Absorption linewidth can also be obtained by nonlinear least squares fitting of the complete harmonic signals,^{[20, 21]} which is also applicable for overlap lines cases. However, the traditional numerical process of simulating the harmonic signals is quite time consuming. Researchers have also proposed to retrieve the Lorentzian linewidth from the separation of the first or second harmonic signals,^[8–10] which can infer the absorption linewidth more rapidly. However, at present, these methods are all numerical computations without any analytical solution.

In this paper, an optimized WMS spectral fitting method was demonstrated to calculate the absorption linewidth effectively, and the analytical expressions for relationships between Lorentzian linewidth and the separation of first harmonic peak-to-valley, and second harmonic zero-crossing were derived to infer the absorption linewidth much more rapidly. The transition of CO₂ centered at 4991.25 cm⁻¹ was used to verify the presented spectral fitting method and the analytical expressions.

The absorption spectrum of CO₂ centered at 4991.25 cm⁻¹ was selected as the target spectrum to verify the WMS spectral line fitting method and analytical expressions demonstrated in section 2.

The experimental setup is constructed based on tunable laser absorption spectroscopy with an NIR-infrared laser, a laser controller, an amplified detector, a homemade digital lock-in amplifier (DLIA), an embedded processor, an F–P etalon and a white cell, as shown in Fig. 4. The laser, from Nanoplus Germany, is a butterfly packaged distributed feedback laser (DFB) with central wavelength of 2004 nm and current tuning rate of 0.045 cm⁻¹/mA. The laser’s injection current and temperature are controlled by the laser controller (ILX Lightwave, LDC-3908) with current noise of rms and temperature stability of ±0.01 °C. The F–P etalon is used to monitor the laser wavelength. The DLIA is used to demodulate the first and second harmonics.^[33]

Fig. 4.

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	Fig. 4. (color online) Schematic diagram of the overall experimental setup.

The laser frequency of the scanning range was linearized by using the F–P etalon (FSR = 1.5 GHz). The laser injection current scanned from 75 mA to 105 mA with a 10-Hz repetitive ramp ride with a 2-kHz sinusoid. The laser working temperature is 33.9 °C. The filter parameters of the DLIA were set to 10 ms and 18 dB/oct, respectively. The ambient air is filled to the white cell used as target gas. Before experiment, the optical configuration was carefully adjusted, including the laser incidence and the field mirror inside the white cell, to ensure excellent experiment conditions. The path-length of the white cell is set to 10 meters, which is directly measured by using an optical frequency domain reflectometer with relative uncertainty of 10⁻⁶.^[34]

First, the absorption linewidth of the target spectrum was derived by fitting the measured signal with LM algorithm. Set three initial parameters: v₀, A, and as free parameters, was set as a constant (0.0046 cm⁻¹) corresponding to T = 296 K. The best-fitted parameter was derived after 126 nonlinear iterations according to the procedure as shown in Fig. 1. The fitting results are shown in Fig. 5. Then the absorption linewidth (Voigt) was calculated to be 0.0766 cm⁻¹.

Fig. 5.

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	Fig. 5. (color online) The measured and simulated signals, modulation depth a = 0.084 cm−1, the best fitted parameters , and Voigt linewidth was calculated to be 0.0766 cm−1. The unit a.u. is short for arbitrary units.

By approximately calculating the Voigt profile with the weighted combination of Lorentzian and Gaussian functions, the signal can be simulated quickly and accurately, thus greatly decreasing the calculation time of the fitting algorithm. In fact, for a processor configured as Intel(R) Core(TM) i5-4590 CPU@3.30 GHz, the fitting procedure needed about 1.7 s of calculation time compared with the 22 s needed in the case of the traditional numerical fitting method reported in SubSection 2.1. The absorption linewidth can also be inferred even much faster through peak-to-valley or zero-crossing separation.

The ratio of the Lorentzian linewidth to the Doppler linewidth of the target spectrum was calculated to be . So the target spectrum can be well described as a Lorentzian lineshape. On the basis of the analysis in Subsection 2.2, considering both the measurement sensitivity and SNR, the modulation depths were estimated and set to be 0.051 cm⁻¹ and 0.084 cm⁻¹ (corresponding to the sinusoidal current amplitudes of 1.65 mA and 2.72 mA) for applying peak-to-valley and zero-crossing separation to infer absorption linewidth, respectively. The measured and signals are shown in Figs. 6(a) and 6(b) respectively. As shown in Fig. 6, the peak-to-valley separation of the signal was located at , absorption linewidth was calculated by using Eq. (12) to be ; the zero-crossing separation of the signal was located at , absorption linewidth then was calculated by using Eq. (13) to be .

Fig. 6.

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	Fig. 6. (color online) (a) signal of the target CO2 absorption, the modulation depth a = 0.051 cm−1, peak to valley separation , the inferred absorption linewidth was 0.0802 cm−1; (b) signal of the target CO2 absorption, the modulation depth a = 0.084 cm−1, zero-crossing separation , the inferred absorption linewidth was 0.0755 cm−1.

The and signals were continuously collected and the corresponding absorption linewidth were inferred from three different methods: WMS spectral line fitting, peak-to-valley separation, zero-crossing separation. The results are shown in Fig. 7 and Table 1. All of the above results were compared with the result from the HITRAN database which was regarded as the true value.

Fig. 7.

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	Fig. 7. (color online) Absorption linewidth results inferred from the WMS spectral line fitting, peak-to-valley separation and zero-crossing separation respectively with 1-s time interval.

Table 1.

Table 1.

Table 1. Results inferred from the WMS spectral line fitting, peak-to-valley separation and zero-crossing separation respectively. . Methods Absorption Relatively Uncertainty/ linewidth/cm−1 error/% 3σ/%a Hitran database[35] 0.0738 – – WMS spectral fitting 0.0761 3.1 1.5 peak-to-valley 0.0801 8.5 2.0 zero-crossing 0.0754 2.2 1.7 a σ is the measured standard deviation.

Table 1. Results inferred from the WMS spectral line fitting, peak-to-valley separation and zero-crossing separation respectively. .

From Table 1, we can see that the absorption linewidths obtained by the presented zero-crossing separation and WMS spectral line fitting were very close to the figures simulated by the HITRAN database. We can also see from Table 1 that the WMS fitting method shows the smallest measurement uncertainty, which possesses a better immunity of noise, since the fitting procedures have taken both the whole Lorentzian and Doppler lineshapes into consideration. In addition, the WMS spectra fitting method was also suitable for spectral overlapping cases.

It has been demonstrated that the WMS fitting procedure was much optimized by replacing Eq. (1) with Eq. (4). However, it still needs 1.7 s to complete one single linewidth measurement, which can hardly meet many high time resolved requirements. Compared with the WMS spectral fitting method, for the same processor, the computing time with the peak-to-valley or zero-crossing separation method was less than which is 6 orders faster and can be negligible in almost all applications.

In addition, as shown in Table 1, the relative errors of the measured absorption linewidth by the presented peak-to-valley or zero-crossing separation method are 8.5% and 2.2% respectively. It is not surprising that the zero-crossing separation method is more accurate than the peak-to-valley separation method. The peak-to-valley separation method relies upon the positions of peak and valley of the signal whose slope is parallel to the x axis, however, the zero-crossing positions of the signal are very steep, which leads to a higher signal to noise ratio and more accurate measurement. Because of the geographical differences, the air composition deviation exists between our laboratory and the air USA model in the HITRAN database, resulting in some deviation between the measured results and the figure from the HITRAN database. Besides, the error of the modulation depth a, the electromagnetic noise, and optical fringes of the experimental system could also attribute to the measurement error. Moreover, it should be pointed out that the analytical expressions for relationships between Lorentzian linewidth and the separations of first and second harmonics established in this paper are only suitable for spectrum that can be well described as Lorentzian lineshape, the presented WMS spectral fitting method is still recommended if both Lorentzian and Doppler linewidth are comparable.

In this paper, three methods for inferring absorption linewidth were presented: WMS spectral line fitting, peak-to-valley separation of first harmonic, and zero-crossing separation of second harmonic. The spectral fitting procedure was optimized by calculating the Voigt profile with the weighted combination of Lorentzian and Gaussian functions, which largely decreased the computing time. In order to meet the high time resolution required in applications, the analytical expressions for relationships between Lorentzian linewidth and the separations of 1st harmonic peak-to-valley and 2nd harmonic zero-crossing were deduced, which computing speed is 6 orders faster than the spectral fitting method. To the best of our knowledge, the zero-crossing method performed better than the peak-to-valley method. When both Lorentzian and Doppler linewidth are non-negligible, the optimized spectral fitting is still recommended as the preference.