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For absorption linewidth inversion with wavelength modulation spectroscopy (WMS), an optimized WMS spectral line fitting method was demonstrated to infer absorption linewidth effectively, and the analytical expressions for relationships between Lorentzian linewidth and the separations of first harmonic peak-to-valley and second harmonic zero-crossing were deduced. The transition of CO2 centered at 4991.25 cm−1 was used to verify the optimized spectral fitting method and the analytical expressions. Results showed that the optimized spectra fitting method was able to infer absorption accurately and compute more than 10 times faster than the commonly used numerical fitting procedure. The second harmonic zero-crossing separation method calculated an even 6 orders faster than the spectra fitting without losing any accuracy for Lorentzian dominated cases. Additionally, linewidth calculated through second harmonic zero-crossing was preferred for much smaller error than the first harmonic peak-to-valley separation method. The presented analytical expressions can also be used in on-line optical sensing applications, electron paramagnetic resonance, and further theoretical characterization of absorption lineshape.
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.
Wavelength modulation spectroscopy is a widely used technique,[22, 23] and the second harmonic signal is also commonly used for the measurement of lineshape parameters.[24] Linewidth of the gas absorption spectrum can be derived by fitting the complete second harmonic signal.
For optically thin conditions (absorbance
Then the second harmonic
To optimize the fitting procedure, Voigt profile
Then the k-th harmonic signals of Voigt profile Hk can be expressed as[27]
Levenberg–Marquardt (LM) algorithm is used as the fitting method. The center wavenumber v0, the integral absorbance A, the Lorentzian linewidth
The optimized WMS spectral fitting procedure was able to extract the absorption linewidth more effectively. However, in some faster measurement occasions, the computing time should be improved further. It can be demonstrated that the absorption linewidth was able to be inferred from the separations of harmonic signals much faster than the fitting method.
It is known that the larger the separations of harmonic signals are, the wider the absorption linewidth will be. The absorption linewidth can be well inferred from the peak-to-valley separation of the first harmonic signal
When collisional broadening was dominated, the absorption lineshape
The modulated Lorentzian lineshape then can be expressed as
Define:
In the cases of k = 1, 2, the first harmonic signal
According to the properties of
(i) Setting
(ii) Setting
![](cpb_27_2_024205/cpb_27_2_024205_ieqn042.gif)
![](cpb_27_2_024205/cpb_27_2_024205_ieqn043.gif)
![](cpb_27_2_024205/cpb_27_2_024205_ieqn044.gif)
![](cpb_27_2_024205/cpb_27_2_024205_ieqn045.gif)
The first harmonic
Both
Let C be equal to the product of
The parameter C is considered to be a guideline that helps to choose a compromise modulation amplitude for the combination of signal amplitudes and sensitivity. When C reaches its maximum, the corresponding β value will be chosen as the optimal value. By using Eq. (
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
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.
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
By approximately calculating the Voigt profile with the weighted combination of Lorentzian and Gaussian functions, the
The ratio of the Lorentzian linewidth to the Doppler linewidth of the target spectrum was calculated to be
The
![]() | Fig. 7. (color online) Absorption linewidth results inferred from the WMS spectral line fitting, ![]() ![]() |
![]() | Table 1.
Results inferred from the WMS spectral line fitting, |
From Table
It has been demonstrated that the WMS fitting procedure was much optimized by replacing Eq. (
In addition, as shown in Table
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
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] |