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Abstract
A novel method with high accuracy and easy implementation was proposed based on the sinewave-scanned direct absorption spectroscopy (DAS) in this paper. A fitting routine in the time domain was developed to simultaneously deduce the baseline and more importantly, absorbance through the explicit baseline expression offered by the sinewave scan. This method effectively solves the difficulties of baseline determination and provides more accurate wavelength calibration compared with conventional DAS. The accuracy and performance with narrow scans and high frequency were experimentally verified using CO transition at 4300.699 cm−1, from which the inferred line strength agrees well with HITRAN 2016. Meanwhile, a more accurate N2 collisional broadening was provided and the speed-dependent collisional broadening coefficient of this transition was reported for the first time.
© 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 for temperature and species concentration measurement in both laboratory and industry due to its non-intrusiveness and high accuracy [1–3]. It has been developed into two major approaches: direct absorption spectroscopy (DAS) and wavelength modulation spectroscopy (WMS). In particular, WMS, being introduced later, has drawn many attentions owing to its effective noise rejection and improved robustness through harmonic detections. However, DAS still has impregnable position in many situations for its wide applicability, easy implementation and relatively simpler data interpretation. More importantly, the ability of directly measuring absorbance profiles makes DAS uniquely advantageous especially in the application of spectroscopic parameters measurement [4–7], although much attempt has been made to recover the absorbance line shape using the harmonic signals in WMS [8–12].
Accurate determination of the incident light intensity (baseline) I0 is crucial for achieving absorbance profile in DAS. Conventionally, it is accomplished by a polynomial fitting of the non-absorbed portions of the transmitted light in time domain [5,6]. Then the absorbance profile can be obtained based on the Beer-Lambert law with the help of an interference etalon. As a result, this approach clearly has the following problems: (1) Triangular or sawtooth waveforms that are intended to ensure a relatively simple baseline form have large bandwidth requirement on the detection system due to their inherent high frequency components. Thus, the modulation frequency is typically limited in the range from 100 Hz to 10 kHz [13], which degrades the time resolution of this method in some applications. (2) The real non-absorbed portions are difficult to extract due to line overlapping, line broadening and limited scan range of typical diode lasers. Specifically, the absorption is still around 1% of its peak value even at the position 10 times of the HWHM away from the line center if the collisional broadening is dominant. Therefore, the accuracy of the baseline fitting is very limited, which will inevitably affect the determination of final target, absorbance profile [14]. As mentioned in [15], a 1% error in the baseline fit for a 1% absorption feature (for example) yields a 100% peak measurement error in DAS.
Some efforts have been made to solve the troublesome baseline issue. In some conditions, the incident light can be experimentally measured if the optical path can be purged [5–7]. While, this is certainly unrealistic when the light pass is not confined by a chamber, such as combustion diagnostic and practical field sensing. Meanwhile, this time-division strategy is hard to get rid of the influence of laser intensity fluctuations and the light pass may be shifted at different working conditions. Introducing a reference path that monitors laser intensity variation provides another solution [16–18]. This is, however, at the expense of having a more complicated system, and potential errors from additional optical etalons. In addition, for the mid-infrared wavelength region, the light splitting is not convenient due to the immature and expensive optical fiber.
In this paper, we propose a novel and easy-operating method based on the sine wave modulation in DAS. A fitting routine is developed based on the explicit baseline expression and accurate wavelength calibration ensured by the sine wave modulation. With this method, the incident light intensity and the more important absorbance profile can be synchronously obtained by fitting the transmitted light in time domain. Thus, the influence of the baseline uncertainty on the absorbance measurement can be effectively eliminated. In addition, the temporal resolution of DAS can be effectively improved. As a verification of the proposed method, the CO absorption transition (2←0) R (11) at 4300.699 cm−1 was investigated in a static cell, with conventional DAS (C-DAS) running in parallel for comparison. Then, its performance with high modulation frequency f = 20 kHz and small scan index (defined by the ratio of wavelength scan range and full-width at half-maximum of the absorption) was evaluated. Finally, this method was applied to measure the spectroscopic parameters of CO R (11) transition. A more accurate N2-collisional coefficient and a first-time reported speed-dependent broadening coefficient for this transition were presented.
2. Method
In contrast to C-DAS where the injected laser is modulated with triangular or sawtooth wave, here, the current is modulated sinusoidally. In consideration of the non-linearity effect, the instantaneous modulated wavelength can be described as follows,
where is the center wavelength. a1, a2 are the linear and nonlinear frequency modulation (FM) amplitudes, and η1, η2 are the linear and nonlinear FM phase difference. By monitoring the FM with an optical etalon, all these parameters can be readily obtained through a least-square fit of the etalon signal using Eq. (1).Analogous to the wavelength modulation, the instantaneously modulated incident intensity can be written as,
where is the center intensity. i1, i2 are the linear and nonlinear intensity modulation (IM) amplitudes, and θ1, θ2 are the linear and nonlinear IM phase difference between the IM and the FM. For a test absorber at temperature T, pressure P and having a mole fraction of χ, the absorbance can be described as,where A is the integrated absorbance area and S(T) is the linestrength of a particular transition at temperature T. φ(v) is the lineshape function which is widely described by Voigt, Rautian [19], Galatry [20] or quadratic-speed-dependent Voigt profile (qSDVP) [21]. Substituting the instantaneous frequency into the Beer-Lambert law, the instantaneous transmitted light intensity would be,Based on the analysis above, it is clear that the transmitted light intensity is a function of f (t, , A, ΔvD, Δvc) where ΔvD and Δvc are the Gaussian and collisional broadening parameters in the lineshape function. If the speed-dependence is considered, it would also depend on the speed-dependent broadening and shift coefficient, as discussed later. Therefore, all these lineshape parameters and baseline can be directly obtained by a least-square fitting of the measure transmitted light signal with t as an independent parameter. The reason for conducting fitting in time domain is that the baseline can be well described by an explicit expression offered by the sinewave modulation. Here, we use MATLAB to perform the fitting process, in particular the function FIT which minimizes the mean-square error between the experimental data and the theoretical one through a nonlinear Levenberg-Marquardt algorithm.
3. Experimental setup
The experimental setup used to verify our proposed method is shown in Fig. 1. A distributed-feedback (DFB) tunable diode laser (NORCADA Canada) was used to interrogate CO absorption transition (2←0) R (11) at 4300.699 cm−1. The current and temperature of the laser was controlled with a commercial controller (Thorlabs ITC4001). A fiber-coupled Fabry-Perot etalon (Thorlabs SA200-18C) with a free spectral range (FSR) of 0.05 cm−1 was used to characterize the relative wavelength by tracking the interference peaks. The test laser beam was sent through the absorbing gas cell with a pass length of L = 52.5 cm. The transmitted light was collimated onto a HgCdTe detector (Vigo PVI-2TE-3) and recorded by an oscilloscope. To achieve a higher signal-to-noise ratio, the detected signal was averaged over 128 scans for each measurement. Prior to each experiment, the gas chamber was pumped down to a pressure of 10−2 Pa, and then filled with 1.05% CO/N2 mixture to target pressures. All the experiments were conducted near room temperature (296 K) monitored by a type-B thermocouple (Omega).
For better comparison of the proposed strategy and C-DAS, the laser was separately modulated with triangular wave and sine wave of the same frequency f = 100 Hz. It must be mentioned that the current modulation amplitude of both triangular and sinusoidal wave was set to 24 mA (0.0498 cm−1/mA @100 mA) to ensure a good non-absorbed part for the baseline fitting in C-DAS. The comparison was made under atmospheric pressure to attest the potential of the proposed method in real applications. To further demonstrate its high accuracy and adaptability, experiments were conducted in various cases of low pressures, high modulation frequency and small scan index.
4. Discussions
4.1 Comparison with triangular-wave-scanned C-DAS method
Figure 2(a) shows the detected transmitted signal and etalon curves in the case of triangular wave scan at the pressure of 100.8 kPa. Each etalon fringe peak is marked with a circle, and the wavelength interval between two adjacent peaks equals the FSR (0.05 cm−1). By fitting theses etalon fringe peaks in terms of time and relative wavelength, the instantaneous wavelength can be determined. As shown by the blue curve in Fig. 2(a), non-linearity and asymmetry in both rising and falling edges are pronounced. Severe bending is observed right after the turning point of current as marked by the red circles in the wavelength curve. That is to say, the wavelength changes slowly with large current in the falling edge of light intensity (black curve V2→V1), while the rising edge (green curve V2→V3) presents the opposite trend. The residual of wavelength fitting is shown in the bottom panel of Fig. 2(a). Even though a polynomial as high as 4th order was used to fit the rising and falling edges of wavelength curve, the fitting residual is still quite large, especially in the two bending regions. This, to some extents, accounts for the limited accuracy of the absorbance profile measurement with triangular modulation in C-DAS.
Fig. 2 Results from C-DAS with a triangular modulation amplitude of 24 mA and P = 100.8 kPa. (a) detected It, etalon fringe peaks and the polynomial fit of the modulation wavelength with the fitting residual attached in the bottom panel; (b) measured absorbance, best-fit qSDVP profiles and corresponding residuals.
Figure 2(a) also illustrates a typical handling of the incident light intensity I0 in C-DAS. To achieve the best performance of C-DAS and thus an objective comparison with the proposed methods, the non-absorbed portions, as shown by the red segment in Fig. 2(a), were automatically optimized by minimizing the final fitting residual through a MATLAB program. It must be mentioned that the final fitted results are very sensitive to the determination of non-absorbed parts. A 3rd order polynomial was used to fit the baseline I0 for both edges, with which the absorbance can be finally determined based on the Beer-Lambert law. The measured absorbance and best-fit qSDVP lineshape from C-DAS are shown in Fig. 2(b). The standard deviations of the fitting residuals are 8.27 × 10−4 and 8.88 × 10−4 for the falling and rising edges, respectively. The detailed best-fit parameters for both edges are listed in Table 1. The collisional width Δvc and integrated area A for both edges have 2% discrepancy, and the deduced CO concentrations χ are around 3% lower than its nominal values.
Table 1. Comparison of the fitting parameters from C-DAS and the proposed method
Figure 3 illustrates the results from the proposed method with a sine wave modulation under the same experimental condition. Figure 3(a) exhibits the detected transmitted light signal and etalon peaks. Unlike the wavelength conversion with triangular-wave modulation where the rising and falling edges have to be handled separately, here the etalon fringe peaks in one complete sinusoidal period can be well fitted by Eq. (1). By fitting the detected It as a function of time t with the calculated value from Eq. (4), the incident light intensity I0 and absorbance can be inferred simultaneously, as shown by the black and red dashed lines. The phase difference θ1 between IM and FM is found to be 1.087π at f = 100 Hz. In fact, this shifting behavior is much easier to characterize compared with the bending feature under triangular modulation, which leads to a much smaller fitting residual in the bottom panel of Fig. 3(a) compared with that in Fig. 2(a). This thus, suggests one advantage of the proposed methods. The measured absorbance calculated from the detected It and fitted I0 and its best-fit qSDVP profile are shown in Fig. 3(b). The tiny fit residual suggests the high accuracy of the fitting process. The best-fit parameters for both falling and rising edge are also listed in Table 1. The measured CO concentration 1.05%, averaged by both edges, agreed within uncertainty with the gas specification. In addition, the fitting residuals for both edges with 1σ = 1.40 × 10−4 and 1.55 × 10−4 are only one fifth of those from C-DAS, which also proves the better performance of the proposed method. Moreover, it must be mentioned that although the rising and falling edges were separately analyzed for comparison with C-DAS, they can actually be processed together owing to the good continuity of light intensity and wavelength with sinewave modulation. This full use of information in both edges contributes a lot to measurements with small modulation index, as discussed later. In addition, from the perspective of date processing, the proposed method has distinct advantages over C-DAS, as the manual determination of non-absorbed part is highly sensitive to the operator and the automatic optimization procedure is fairly time consuming in the C-DAS method.
Fig. 3 Results from the proposed method with a sine wave modulation and P = 100.8 kPa. (a) measured It, etalon curve and the best-fit I0, It and residual for the wavelength fitting; (b) best-fit absorbance and residual.
4.2 Performance with high-frequency modulation and small scan index
As a well-known fact, the temporal resolution of a diagnostic method is of great importance, especially for monitoring some unsteady objects or field applications. The temporal resolution of C-DAS governed by the modulation frequency is usually deficient because of the large bandwidth requirement of triangular or sawtooth on the detection system. In contrast, the proposed should have high enough temporal resolution, since the sinewave modulation is used instead of triangular one. Therefore, the performance of the proposed method with modulation with f = 20 kHz was then verified under the same condition. The laser scan index was kept around 10 by increasing the current modulation amplitude from 24 mA to 55 mA. The phase difference θ1 between IM and FM increases from 1.087π for f = 100 Hz to 1.128π for f = 20 kHz, which agrees well with findings in [22]. The best-fit absorbance and residual are shown in Fig. 4(a). It is clear that even with a modulation frequency of 20 kHz, the best-fit residual is still quite small and the inferred Δvc and A have 1.2% and 0.8% error, respectively, compared with those from f = 100 Hz.
Fig. 4 (a) Best-fit absorbance with a modulation frequency of 20 kHz from the proposed methods at pressure of 100.8 kPa. (b) Relationship between tuning coefficient and modulation frequency.
Except for the restrictions on the modulation frequency, C-DAS also has rigorous requirements on wavelength modulation range to ensure a non-absorbed part for baseline fit. On one hand, the tuning coefficient of laser drops dramatically with increasing modulation frequency. As demonstrated in Fig. 4(b), the tuning range for 20 kHz is less than half of that for 100 Hz with the same 24 mA modulation amplitude. That is to say, a larger current modulation amplitude have to be applied to achieve an enough scan index. However, this is usually unpractical due to the limited current operation scale of diode lasers. Moreover, for some high pressure conditions, where the collisional broadening is much larger, the scan index cannot be guaranteed by simply increasing current amplitude. Consequently, it is of great necessary to check the performance of the proposed method with small scan index. Figure 5(a) shows the results from the proposed method with a scan index of 3.8. By fitting the detected It(t), the incident light I0(t) can be fully restored as shown by the upper blue dashed curve. Evidently, the wavelength scan is too small to ensure desirable non-absorbed parts, which are indispensable for determining I0 in C-DAS. The best-fit absorbance and fitting residual are also attached in Fig. 5(a). It is clear that the absorbance is non-zero even at the two ends, reflecting the narrowness of the scan and the absence of non-absorbed parts. The best-fit collisional broadening and integrated area are Δvc = 0.0578 cm−1 and A = 0.0357 cm−1, with relative differences of only 0.3% and −0.6% compared to the results with scan index of 10.2. Meanwhile, the best-fit collisional broadening Δvc and integrated area A with different scan indexes from 3.2 to 10.2 are plotted in Fig. 5(b). The relative difference with respect to results with scan index of 10.2 are also attached right below. It can be seen that the relative difference of Δvc and A are still quite small, being less than 0.3% and 1.0%, respectively. This is undoubtedly fairly challenging for C-DAS approach.
Fig. 5 (a) Detected light intensity and best-fit results with a modulation index of 3.8; (b) Comparison of best-fit collisional broadening Δvc and integrated area A with different modulation indexes. RE denotes the relative error compared with results from f = 100 Hz.
4.3 Typical application: measurement of spectroscopic parameters including speed-dependent collisional broadening
After checking the performance of the proposed method with different modulation frequencies and scan indexes at atmospheric pressure, it was then applied to measure the spectroscopic parameters of CO R (11) transition, including speed-dependent collisional broadening at reduced pressures. The measured absorbance and the corresponding best-fit Voigt and qSDVP profiles under the pressures of 16.51 kPa and 6.46 kPa are shown in Fig. 6. In both fitting processes, the Doppler broadening ΔvD is fixed at 0.005 cm−1 based on the experimental temperature. The residuals from Voigt profile fitting present a prominent gull-wing structure, as Voigt is incapable of accounting collision-induced velocity changes (i.e., Dicke narrowing) and the speed-dependence, both of which play an important role in the investigated pressure [23]. Obviously, the gun shape residual is effectively eliminated in the qSDVP profile after introducing speed-dependent collisional-broadening and -shifting coefficients, as shown in Figs. 6(b) and (c).
Fig. 6 (a) Measured absorbance and best fit qSDVP profiles for 16.51 kPa and 6.46. kPa; (b) and (c) are the residuals of best-fit Voigt and qSDVP profile for 16.51 kPa and 6.46 kPa, respectively.
To further infer the CO-N2 collisional broadening coefficient and speed-dependent collisional broadening coefficient for the transition at 4300.699 cm−1, the absorbance of 1.05% CO-N2 mixture were measured at 11 pressure conditions ranging from 3.4 to 16.61 kPa at T = 296 K. This pressure range was chosen to ensure that Doppler contribution and collisional width are comparable, and thereby guarantee a more accurate speed-dependent collisional broadening coefficient measurement. The modulation amplitude was set to be constant for all pressures, as the proposed method has no restriction on the scan index. Figure 7(a) shows the detected transmitted light intensities It and best-fit incident light I0 and It for several pressures. Figure 7(b) exhibits the measured absorbance and best-fit qSDVP profiles, with the fitting residuals attached right below. As can be seen, all maximum residuals are smaller than 0.15% of the peak absorbance with standard deviations in the range of 1.1~1.4 × 10−4.
Fig. 7 (a) Measured transmitted light intensities and best-fit I0 and It for different pressures; (b) Measured absorbance and best-fit qSDVP profiles; (c) collisional width and integrated area versus pressure determined from qSDVP fit and the corresponding two-parameter linear fit; (d) measured γ2, CO-N2 × P and its two-parameter linear fit used to infer γ2, CO-N2.
Furthermore, the best-fit collisional width Δvc and integrated area A for all pressures are plotted in Fig. 7(c), together with corresponding two-parameter linear fit. Although the collisional-broadening coefficient and line strength can be inferred from its slope alone, two-parameter linear fit was performed to prevent any potential offset in the pressure measurement from affecting the determination of target coefficient. The R-square values (R2 = 0.99995 for Δvc and R2 = 0.99999 for A) are close to 1 and the intercepts (7.5 × 10−5 for Δvc and −1.2 × 10−5 for A) are negligibly small, suggesting the accurateness of the measurements and fitting processes. The inferred linestrength S (296 K) = 0.0650 (0.6%) cm−2/atm, agrees well with that from HITRAN 2016, S (296 K) = 0.0651 cm−2/atm [24]. It must be mentioned that the relative standard uncertainty marked in the bracket was estimated including the following uncertainty components: pressure (0.5%), gas temperature (0.25%), integrated area A (0.1%), slope of the A vs P (0.1%), path length (0.7%), mole fraction of gas mixture (0.1%) and FSR of Fabry-Perot etalon (0.2%). The N2-broadening coefficient γCO-N2 (296 K) inferred from the slope of collisional widths is 0.0577 (0.7%) cm−1/atm with the uncertainty calculated in the similar fashion with the line strength including uncertainty components of Δvc (0.3%) and slop of the Δvc vs P (0.2%). The measured value is 2.2% larger than that from HITRAN 2016 γCO-N2 (296 K) = 0.05644 cm−1/atm. This may be caused by the utilization of Voigt profile in paper [25], which ignores the collisional and speed dependent narrowing effects and finally results in an underestimation of the collisional width. The measured γ2, CO-N2 × P and two-parameter linear fit used to deduce the speed-dependent collisional broadening coefficient γ2,CO-N2 are shown in Fig. 7(d). The negligibly small intercepts and goodness of fit imply the proper choice of line shape and the fitting processes. The speed-dependent collisional broadening coefficient for this transition derived from the slope is γ2, CO-N2 = 0.0060 (3%) cm−1/atm, which is, to our knowledge, reported for the first time.
5. Conclusions
In view of the limitations of conventional DAS, a high-accuracy and easy-operating DAS method was proposed based on the sinewave modulation. This method thoroughly considers the non-linear characteristics of FM and IM, making it possible to determine the baseline I0 and more importantly the absorption line profile simultaneously by fitting the detected transmitted light intensity in time domain. Therefore, the accuracy of absorbance measurement is immune to the uncertainty in the baseline determination. This method was applied experimentally to measure the CO R (11) transition at 4300.699 cm−1 as a preliminary verification, with conventional C-DAS running in parallel for comparison. As a result, the proposed method presents one fifth smaller fitting residual and a more accurate CO concentration compared with C-DAS, indicating its advantageous over C-DAS in terms of accuracy and simplicity. Meanwhile, the good performance of the proposed method with high-frequency modulation and small scan index grants DAS high temporal resolution in some field applications. The deviation of Δvc and A remains below 0.3% and 1.0%, respectively even with a scan index as small as 3.2. Finally, this method was applied to measure the line parameters of CO R(11) transition. A more accurate N2-broadening coefficient γCO-N2 was obtained and the speed-dependent broadening coefficient γ2, CO-N2 = 0.0060 (3%) cm−1/atm was reported for the first time to our knowledge.
Funding
National Natural Science Foundation of China (NSFC) (51676105); National Key R&D Program of China (2016YFC0201104)
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