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Abstract
A highly sensitive, calibration-free method, wavelength modulation - direct absorption spectroscopy (WM-DAS), is proposed by extracting the characteristic frequencies of modulated laser intensity. This method combines the advantages of measuring absolute absorbance profile from calibration-free DAS with the enhanced noise rejection and high sensitivity of WMS. The preliminary validation experiment of H2O transitions shows that the WM-DAS method improves the accuracy of the absorbance profile measurement with a best-fit standard residual of 4.9 × 10−5. This new method might inspire new approaches to high-precision measurement of spectral line parameters and gas information with weak absorptions or industrial applications.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Tunable diode laser absorption spectroscopy (TDLAS) has been widely investigated in both laboratory and industry because of its non-intrusiveness and high accuracy [1–3]. The absorbance profile, which is of key importance in TDLAS, can be used to determine gas temperature, concentration and many line parameters, including collision broadening, Dicke narrowing factor and line strength. Direct absorption spectroscopy (DAS) [4–6] and wavelength modulation spectroscopy (WMS) [7, 8] have been developed into the two most common methods in TDLAS.
Over the past decades, DAS has been extensively used to measure gas parameters because of its straightforward understanding and implementation, and ability to measure absolute absorbance profile. However, troublesome noise, such as “dark noise”, “light noise” and “proportional noise” restrict the measurement accuracy of DAS [4]. For example, the scanning frequency is usually limited to several kHz, while most troublesome noise sources, such as vibration, laser and 1/f noise, also dominate at low frequency. Fortunately, some researchers have focused on improving its sensitivity and stability by optimizing the measurement system and have achieved many valuable results [4–6]. Meanwhile, plenty of improvements still can be achieved by further optimizing TDLAS measurement methods, which also motivates this paper.
The WMS employs a sinusoidal modulation signal superimposed on a slow-varying diode laser injection current to produce harmonic signals using a lock-in amplifier. This technique has high sensitivity and signal-to-noise ratio (SNR) through harmonic detection, and many measurement results have shown that WMS is more sensitive by one to two orders of magnitude than DAS. In traditional WMS, the 2nd harmonic [9] is widely used because its peak value is proportional to the gas concentration if the absorption is optically thin. However, one of the key drawbacks of this method is that we should calibrate the 2nd harmonic signal to a known mixture and condition to recover the absolute concentration. In order to avoid the complex calibration, Hanson et al. [10] proposed a calibration-free method which employs the first harmonic to normalize higher harmonics (denoted as nf/1f), and the gas temperature and concentration can be directly determined by comparing the experimental harmonic signals with theoretical ones. This method shows good results in the studied high pressure and temperature environment. However, gas temperature, pressure, species concentration, laser characteristic parameters, spectroscopic constants, and absorption length are the necessary parameters for calculating such harmonic signals. Uncertainties in these parameters will induce discrepancies between simulation and measurement results, and thus lead to measurement uncertainties.
Considering the importance of absorbance profile and advantages of the WMS, recently, some efforts have been made to recover the absorbance profile using harmonic signals [11–13]. Stewart et al. [11, 12] proposed a phasor decomposition method to recover the absorbance profile using the first harmonic on the basis of residual amplitude modulation. This method works well with small modulation indices (m<0.2) when the first harmonic shape is close to the absorbance profile. However, the recovering errors increase sharply as the modulation index increases in actual measurements. To reduce the recovering errors, in our recent research [13], we proposed a method that employs additional higher odd harmonics (3rd, 5th…) to enhance the recovering accuracy with large modulation indices. Results of the simulation show that this method works well for recovering absorbance profile, regardless of the value of the modulation index. However, the higher odd harmonics always come with low SNR and are difficult to detect in actual measurements.
Inspired by the above researches, in this paper, we propose a novel calibration-free wavelength modulation-direct absorption spectroscopy (WM-DAS) method to accurately recover the absolute absorbance profile based on FFT analysis. This new method combines the advantages of DAS (calibration-free, simple) with those of WMS (enhanced noise rejection and high sensitivity by harmonic detections). The collision broadening and Dicke narrowing parameters of the H2O transition at 7185.597 cm−1 were investigated using the proposed method with weak absorption (peak absorbance around 2.5%), and the results were in excellent agreement with literature [14] conducted with strong absorptions. Finally, the ambient temperature and water concentration in air were precisely measured using this method with the H2O line pair (7185.597 and 7185.344 cm−1).
2. Method
The basic principle of the traditional WMS has been introduced widely in the literatures [7, 15]. A sinusoidal modulation of an angular frequency ω is superimposed on a slow-varying diode laser injection current to modulate laser wavelength. In the purposed WM-DAS, the low frequency wavelength scan is removed, and only high frequency sinusoidal modulation is used as shown in Fig. 1(a), where the red and black curves represent the transmitted and reference (or incident) laser intensities, respectively. The instantaneous laser intensity can be expanded into a Fourier series as follows:
where Xk and Yk are the kth Fourier coefficients, and ω is the angular frequency of the sinusoidal modulation. In traditional WMS, the nonlinearity of the output frequency with injection current tuning is always ignored when the modulation depth is small. However, this nonlinearity phenomenon is non-negligible with large modulation. In order to improve the calibration accuracy of laser wavelength, we consider a more comprehensive instantaneous laser wavelength as follows:where ai and φi are the modulation depth and initial phase of the ith-order frequency, respectively. η is the initial phase of the base frequency and φ1 = 0, (cm−1) is the laser center wavelength. The blue curve in Fig. 1(a) shows an example laser wavelength of a distributed feedback diode (DFB) laser near 7185.5 cm−1 which is calibrated with a modulation frequency of f = 100 Hz. It is clear that the duration of the V1V3 period is longer than that of the V1V2. This inequality becomes more significant as the modulation frequency and depth increase, which proves that the nonlinearity of the laser wavelength cannot be neglected, especially with high modulation frequency and large modulation depth in WM-DAS.
Fig. 1 (a) Instantaneous light intensity and wavelength (RI is reference intensity, TI is transmitted intensity). (b) Fourier coefficients Xk and Yk of the TI with 100 cycles of sine waves, f1 = 1979 Hz and f2 = 3958 Hz are the noise’s FFT coefficients.
Equation (1) describes the relationship between light intensity and time. In the proposed method, the relationship between light intensity and wavelength must be established to recover the absorbance profile. Here, by defining:
The phase of ωt can be expressed as following:
where V1→V2 and V1→V3 denotes the left and right periods of the laser wavelength, respectively, as shown in Fig. 1(a):Substituting Eq. (4) into Eq. (1) yields the following expressions for the light intensity:
In Eq. (2), the second order expression is used to describe the laser wavelength nonlinearity effect in this paper, and the laser wavelength can be written as follows:
Substituting Eq. (3) into Eq. (6), the laser wavelength can be expressed as follows:
Based on the above analysis, the absorbance profile can be recovered through the following process. First, Fast Fourier Transform (FFT) is used to obtain the Fourier coefficients Xk and Yk of light intensity. And then, substituting these Fourier coefficients into Eq. (5) yields the relationships between light intensity and the variable x. Meanwhile, an etalon or wavelength meter is used to calibrate the laser wavelength versus time as shown in Fig. 1(a), where parameters such as a1, a2 and φ2 can be inferred by fitting the laser wavelength according to Eq. (6). Substituting these parameters into Eq. (7), we can determine the relationships between the laser wavelength and the variable x. Finally, the relationships between light intensity and wavelength can be reconstructed, and the absorbance profile is recovered by [16]:
where α(ν) is the absorbance profile, It(v) and I0(v) are the reconstructed transmitted and incident laser intensities (versus wavelength), respectively.To verify the validity and accuracy of the proposed WM-DAS method, the H2O absorption transition at 7185.597 cm−1 was investigated through both simulation and experiment. The corresponding spectral line parameters, including collision coefficient, Dicke narrowing coefficient and line strength, have been reported in detail in literature [14]. The temperature, pressure, relative humidity, and optical length used in the simulation are set to be similar to the following experimental condition: 292.5 K, 0.2 atm, (23.6 ± 0.1) % and 53 cm. The H2O volume concentration in the atmospheric condition can be then calculated (0.5227 ± 0.0022) %. The blue curve in Fig. 1(a) shows the instantaneous laser wavelength, which can be simulated by Eq. (6). The corresponding parameters are: = 7185.535 cm−1, a1 = 0.2509 cm−1, a2 = 0.02181 cm−1, η = 1.125π, φ2 = 1.041π and f = 100 Hz. Substituting these parameters into Eq. (7) yields the relationship between the laser wavelength and the variable x. The nonlinearity phenomenon of laser wavelength with laser tuning and the calibration experiments will be discussed in detail in a future paper.
In numerous applications, particularly in harsh environments, the laser intensity fluctuations, particle concentration, and troublesome noises hamper the accuracy of determining absorbance profile. To investigate the efficacy of the proposed method in dealing with these difficulties, we added a variety of random frequency noises into the transmitted and incident signal, as indicated by the red and black curves in Fig. 1(a). Traditional DAS is incapable in dealing with these kinds of noise signals with an inherent frequency. All the spectral line parameters used in the simulation were adopted from the HITRAN 2016 database [17], and the line profile is assumed to be Voigt given in Eq. (9). Figure 1(b) shows the Fourier coefficients Xk and Yk of the transmitted light with 100 cycles of sine waves. Substituting nf (n = 1, 2 …) coefficients into Eq. (5) and combining Eq. (7) for wavelength transformation yields the relationship between transmitted light intensity and wavelength, as shown by the red curve in Fig. 2(a). To compare the different trends of the recovered light intensity (versus wavelength) and the detected light intensity (versus time), the time-dependent light intensity without noise during the V1V2 time period is also given in Fig. 2(a). In our proposed method, only the Fourier coefficients of an integer multiple of the modulation frequency (nf) were used in the recovery process, and the influences at other frequencies (f1 = 1979 Hz, f2 = 3958 Hz) can be easily eliminated in data processing. Similarly, the incident light was recovered as indicated by the red dashed curve in Fig. 2(a). The red dashed curve in Fig. 2(b) illustrates the absorbance profile calculated from the recovered transmitted and incident light signal according to Eq. (8). The residual of the restructured absorbance with the theoretical one is smaller than 10−12.
Fig. 2 (a) Comparison of the time-dependent (without noise) and the reconstructed wavelength-dependent light intensity. (b) Comparison of the recovered absorbance profile and the theoretical one.
3. Experiment and discussions
The transition of H2O at 7185.597 cm−1 was used in our experiment to validate the proposed method. A detailed schematic of the experiment is shown in Fig. 3. A distributed feedback diode laser with a center wavelength of 7185 cm−1 is driven by a commercial diode laser controller (Thorlabs ITC4001) and modulated with a sine signal of f = 100Hz produced by a function generator (Keysight 33220A). The reason that relatively low frequencies, says 100 Hz, are selected, is to show how the low-frequency noises, such as electronic, 1/f and vibration noises can be eliminated in this method. In actual measurements, an optimized modulation frequency according to the specific measurement system, including the vibration, the circuit noise and so on, is certainly suggested. The laser is split into two beams by the optical fiber and received by a pair of similar Ge photodiode detectors (Thorlabs PDA50B-EC). The detector signals are recorded by a digital oscilloscope (DPO 4034B). One beam passes though the L = 53cm long absorbing gas cell, while the other propagates through ambient air serving as the reference. In addition to monitoring the fluctuations of the laser intensity, this reference path also eliminates influence from the water vapor outside the gas cell by carefully matching the two optical lengths L1 + L2. The laser wavelength is calibrated using an etalon (Thorlabs SA210-12B) with a free-spectral range (FSR) of 0.05 cm−1. Prior to each experiment, the gas chamber is pumped down to 10−2 Pa, and then filled with ambient air to 0.2 atm. The ambient temperature, pressure and humidity are the same as those used in the previous simulation part.
Figure 4(a) shows the recovered transmitted and reference light intensities (versus wavelength) from the experimental data. Analogous to the simulation, the red and blue curves represent the V1V2 and V1V3 periods, respectively. Although these two curves have the same starting and ending points, the trends in the two curves vary distinctly. This is caused by the nonlinearity of laser wavelength and the phase difference between the wavelength and the intensity modulation, which both aggravate with increasing modulation frequency. The black curve in Fig. 4(a), which is close to linear, is the average intensity of the V1V2 and V1V3 periods. The absorbance profile can be recovered on the basis of any of these three intensities according to Eq. (8). The scatter in Fig. 4(b) illustrates the measured absorbance profile deduced from the average light intensity in Fig. 4(a). In contrast with other literatures [14, 18, 19], the absolute absorbance in this experiment is relatively small with a peak absorbance of 2.56%.
Fig. 4 (a) The reconstructed transmitted and reference light intensity of the H2O transition at 7185.597 cm−1 with 100 cycles of sine waves. (b) Measured absorbance profile and best-fit Voigt and Rautian profiles.
The line parameters such as integrated absorbance, collisional broadening and the Dicke narrowing coefficients are obtained by fitting a theoretical line shape to the measured absorbance profile. The Voigt profile (VP) and Rautian profile (RP) are used in this work, and the line profiles are defined as follows [16]:
withwhereHere, γC (cm−1) is the collision half width at half maximum (HWHM), γD (cm−1) is the Doppler HWHM, v0 (cm−1) is the line center wavelength, β (cm−1atm−1) is the Dicke narrowing coefficient, p2 (atm) is the partial pressures of the perturber molecules. During the fitting process, a Levenberg–Marquardt nonlinear fitting procedure was used to simultaneously optimize the quadratic polynomial baseline and the line profile to the experimental data.The best-fit Voigt and Rautian profiles and their residuals are given in Fig. 4(b). The Voigt profile with a fixed Gaussian FWHM of γD = 0.01037 cm−1, presents pronounced gull-wing signature residual, which is similar to the findings in literatures [14, 18] and may be related to collision narrowing. The maximum residual is 5.2 × 10−4 (around 2% of the peak absorbance), and the standard residual is 1.5 × 10−4. The best-fit Rautian profile with a fixed Gaussian FWHM effectively removes the w-shaped residual and reduces the maximum residual to 1 × 10−4, and standard residual to 4.9 × 10−5, which is several times smaller than that in literatures [18, 19]. Meanwhile, traditional DAS is also used to measure the absorbance profile at the same conditions. The laser was tuned with a 100 Hz triangle wave, and the average of 100 sequential scans is used to improve the SNR. Measurement results show that the standard residuals of VP and RP are about 2.7 × 10−4 and 9.3 × 10−5, respectively.
Table 1 compares the experimentally derived best-fit parameters with those from the HITRAN 2016 database and those reported in the literature [14]. The collisional broadening and Dicke narrowing coefficients using a Rautian profile and the proposed method agree very well with those corresponding values reported in the literature. It is worth noting that the temperature in the literature is 296 K (not exactly 292.5 K) and the collision partner is pure N2 (rather than air), which may account for the small deviation. For the transition at 7185.597 cm−1, collisional broadening apparently differs from the values from HITRAN 2016 database, and this suggests the importance of high-precision measurement of spectral parameters, especially at high temperature.
Table 1. Comparison of the measured line parameters of H2O with those from literature and HITRAN 2016
Furthermore, the two transitions of H2O at 7189.344 cm−1 and 7189.541 cm−1, which have even weaker absorptions, were also measured using the proposed method under the same condition. The recovered absorbance profile and the best-fit Rautian profile are shown in Fig. 5(a). The detailed best-fit parameters are listed in Table 1 and labeled as lines 2 and 3. To our best knowledge, the measurements of Dicke narrowing parameters together with the collision coefficients for such weak absorptions with traditional DAS have not yet been reported. It can be seen that γH2O-Air for line 2 agrees well with that from the HITRAN 2016. Although deviations exist for γH2O-Air of line 3, the value is still acceptable considering the extremely weak absorption (peak absorbance 0.27%) and the uncertainty in the values reported in the database. It is also clear from Table 1 that the Dicke narrowing coefficient β for lines 1 and 3 are almost the same, while it is much larger than that of line 2. This is mainly due to the higher quantum state of lines 1 and 3, which indicates a larger rotational spacing of the transition compared to the thermal energy, kT [20]. To assure the stability and repeatability of the proposed method, a time-series measurement of the above-mentioned H2O transitions over 60 minutes is conducted and the recovered absorbance profiles are shown in Fig. 5(b). This clearly proves that the proposed method has good reproducibility and accuracy [21].
Fig. 5 (a) Recovered absorbance profile and its best-fit Rautian profile for H2O transitions at 7189.344 cm−1 and 7189.541 cm−1. (b) A time-series measurement over 60 min to assure stability and repeatability.
The ambient temperature and the H2O concentration were deduced based on the integrated absorbance of lines 1 and 2. For two-line thermometry, the gas temperature was determined from the ratio of the experimentally determined integrated absorbances of two transitions with different temperature dependence, and can be calculated using the following expression [22]:
where A and E” represent the integrated absorbance and lower-state energy, whose values are given in Table 2. S(T0) is line strength at some reference temperature, h (J·s) is Planck’s constant, c (cm/s) is the speed of light, k (J/K) is Boltzmann’s constant, hc/k has a numerical value of 1.4388 (cm·K).Table 2. Line parameters of H2O from measurement and HITRAN 2016 database (T0 = 296K)
Substituting these parameters into Eq. (12), the resolved gas temperature is about 293.3 K, which is in very good agreement with the results measured by a thermometer (292.5 K). Meanwhile, the H2O volume concentration in the atmospheric condition can be calculated by:
where S(Tmea) is the line strength at measurement temperature (293.3K). Line 1 and 2’s strengths are about 0.0191 and 0.0156 cm−1 atm−1, respectively. Substituting the pressure (P = 0.2atm), optical length (L = 53cm), line strengths and integrated absorbance into Eq. (13), the water vapor concentrations are about 0.5157% (Line 1) and 0.5165% (Line 2), respectively. These results are also in very good agreement with the results measured by a thermometer and a hygrometer (0.5227%).4. Conclusion
In conclusion, the proposed WM-DAS method can be used to precisely recover the crucial absorbance profile via extraction of the characteristic frequencies of the modulated transmitted light. Much more information, including gas temperature, concentration, pressure and spectral line parameters (collisional broadening, the Dicke narrowing coefficient, and line strength) can be subsequently determined from the recovered absorbance profile. The proposed technique combines the benefits of measuring the absolute absorbance profile of calibration-free DAS with enhanced noise rejection and high sensitivity of WMS. With an optimized modulation frequency, the proposed method is capable to effectively eliminate noises resulting from vibrations, power, particles, and laser fluctuations. The preliminary validation experiment of H2O transitions shows that the WM-DAS method improves the accuracy of the absorbance profile measurement with a standard fitting residual of 4.9 × 10−5. If combined with optimized modulation frequency and laser intensity stabilization techniques, the fitting standard residual is expected to be as low as 10−6. Since WM-DAS exhibits the advantages from both WMS and DAS, it has huge potential in the following areas:
- (1) High-precision measurement of spectral line parameters for many weak absorption lines.
- (2) High-accuracy measurement of gas temperature and concentration under weak absorption conditions in NIR, such as tomography of small size flames and trace gas monitoring.
- (3) In situ gas sensing in harsh and complicated industrial fields, for example, the escaped ammonia monitoring in power plants.
Funding
National Natural Science Foundation of China (NSFC) (51676105); National Key R&D Program of China (2016YFC0201104)
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