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Abstract
Considering the importance of the laser wavelength response and the difficulty in its real-scenario measurement in WMS, a high-accuracy and universal method was developed to characterize the relative wavelength response (RWR) by analyzing the laser current response. A coupling term that depends on both the current scan and the modulation characteristic was introduced to describe the coupling effect between the wavelength scan and modulation. The accuracy of the proposed method was verified with different laser working conditions and scan waveforms. All fitting residuals of the RWR result from the proposed method are smaller than 0.1% of the total scan range and the fitting residual of the ramp scanned WMS is twice smaller than the minimum value from literature. The better calibration-free 2f/1f fitting and more accurate CO2 concentration results also suggest the high accuracy and superiority of the proposed method. Finally, based on the precise prediction of RWR with small scan and modulation indices, the spectral parameters, including line strength and self-collisional broadening coefficient, of CO2 transition at 6976.2026 cm−1 were successfully measured using WMS.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the last several decades, the tunable diode laser absorption spectroscopy (TDLAS) is widely used in various fields, e.g. environmental monitoring [1–3], spectral line parameter measurements [4–7], and diagnostic of combustion [8–10] and plasma [11–13], due to its attractive merits of non-intrusiveness, high selectivity and high accuracy. Specifically, the wavelength modulation spectroscopy (WMS) is preferable in both laboratory and industrial applications, as it offers an improved noise rejection ability and sensitivity [14]. In WMS, the laser wavelength is rapidly modulated with a sinusoidal waveform fm on a superposition of the low-frequency scan fs. By retrieving the nfm harmonic with a lock-in amplifier (LIA), the absorption information is shifted to high frequency and immune to the ubiquitous low-frequency noises. To extract the desired gas properties from the detected harmonic signals, several strategies have been developed, including the in-situ calibration with a known gas mixture [15,16], recovery of absorbance with harmonics [17–21], calibration-free nf/1f fitting (CF-nf/1f fitting) [22,23] and so on.
Among all these techniques, the CF-nf/1f fitting method, where the gas properties are retrieved from the fitting of a simulated nf/1f signal to the detected one processed with the same digital LIA, is widely applied due to its easy operation, no need for a complex analytic model, and wide applicability. In spite of the promising theory of CF- nf/1f method, precise fitting of the nf/1f signals is still challenging since the harmonic signals are influenced by many factors [23]. Particularly, the characterization of the relative wavelength response (RWR) in WMS, which determines the simulated transmitted light intensity, is of great importance to the simulated nf/1f signal. It directly affects the goodness of CF-nf/1f fitting and thus the accuracy of the deduced gas properties [24–27]. In fact, not only the CF-nf/1f fitting, other approaches, except for the calibration approach, also rely heavily on the RWR characterization, as the absorption strength of the target transitions is essentially a function of laser frequency.
However, an accurate characterization of the RWR in WMS is very demanding and not well investigated. As for the real-scenario measurement of RWR (with the coexistence of current scan and modulation) with an etalon, the main difficulties are caused by the tremendous amount of etalon peaks in one scan period that contains hundreds of modulation periods. It requires not only a sufficiently high sampling rate of the data acquisition system to detect but also cumbersome efforts to manually locate and label. This makes the real-scenario measurement of RWR in WMS impracticable in practice. The most commonly used approach is to measure the RWR with fs and fm separately and then add these two RWRs together [22,23,27–29]. In fact, this simplification ignores the strong coupling effect between the wavelength scan and modulation, which is prominently reflected in the varying modulation depth with the bias current over the scan period [30–32]. For example, Kluczynski found that the wavelength modulation index for a DFB laser with a 23 Hz sweep rate varied from 0.26 GHz/mA at the beginning of the sweep to 0.815 GHz/mA at the end [30]. Lytkine also found a second-order-polynomial behavior of wavelength tuning respect to the bias current in VCSELs [31]. The rough estimation of laser RWR in WMS may contribute to the relatively large fitting residual, ∼5%, in the CF-nf/1f fitting. [22–24]
More recently, some efforts have been made to increase the accuracy of RWR characterization in WMS, especially for the ramp (triangular/sawtooth) current scan that provides a relatively simple linear bias current change and evades the troublesome issue of phase shift between the wavelength scan and modulation. For instance, based on the finding of Lytkine [31], Chen Jia [32] proposed to use the first derivative of the RWR with fs to describe the changing modulation coefficient in the ramp-scanned WMS instead of a simple summation. Then Qu [24] successfully utilized this idea in his newly developed CF-WMS method. Meanwhile, Ma et al. also contributed a lot to this topic [25,26]. In addition to the linear time-dependent amplitude of the 1st harmonic of modulation, they introduced a 2nd harmonic component with a constant amplitude to describe the RWR in WMS [26]. By introducing another four parameters in the fitting program, an obvious improvement was achieved in the final 2f/1f fitting residual. To further eliminate the additional fitting parameters, they proposed a method to pre-determined RWR in WMS based on the assumption of 2nd order polynomial description of the RWR with a ramp current scan [25]. Although all these efforts have improved, to some extent, the accuracy of the characterization of RWR in WMS, all these methods rely heavily on the ramp current scan assumption and are undoubtedly only applicable to the ramp scan situation. However, more and more interests are put forward to the sinusoidal-scanned WMS as it is competent in the high-temporally resolved measurement scenarios, including the shock tube and combustion [14,23,33], and is also beneficial to the laser performance, as discussed later. Moreover, under some low-pressure conditions, to avoid the over scan/modulation, the scan and/or modulation amplitude are even too small to be separately measured by an etalon with moderate FSR. Therefore, a universal and practical method to characterize the RWR in WMS is in great demand.
In this paper, by analyzing the relationship between the RWR and the injected current, a more accurate and comprehensive model is proposed to precisely describe the RWR in WMS. A coupling term was introduced to describe the coupling effect between the wavelength scan and modulation. On the basis of this model, a practical three-step method was developed to easily characterize the RWR in WMS with separate current scan and modulation. This method does not only have the merits of high precision, easy operation and wide applicability to different scan waveforms but also makes it possible for the RWR predication with small scan/modulation depths. To verify the accuracy of the proposed method, the RWR under different scan and modulation waveforms and specifications were assessed. Meanwhile, the best-fit CF-2f/1f signal and the further deduced gas properties were compared with different RWR characterization methods. Finally, as an application, the spectral parameters, including line strength and self-collisional broadening coefficient, of CO2 transition at 6976.2026 cm−1, were precisely measured with WMS, based on the accurate RWR prediction using the proposed method.
2. Experimental setup
The schematic of the experimental setup is shown in Fig. 1. To verify the proposed method of the laser characterization in WMS, a typical distributed-feedback (DFB) tunable diode laser (NEL) near 1432 nm was used to probe CO2 absorption transition at 6976.2026 cm−1. A dual-channel function generator (Keysight 33500B) was used to produce the sinusoidal scan waveform fs and modulation waveform fm. After summation, the signal was sent to a commercial laser controller (Thorlabs ITC4001) to tune the injection current of the laser diode. The laser from the fiber pigtail was split into two parts, one of that passed through the gas cell for the gas monitor and the other one was sent into a Fabry-Perot etalon (Thorlabs SA200-12B) for the wavelength characterization. The gas cell has an optical length of L = 52.5 cm. Prior to each experiment, the gas chamber was pumped down to a pressure of 10−2 Pa, and then filled with pure CO2 to target pressures. All the experiments were conducted around room temperature monitored by a mercury thermometer. The Fabry-Perot etalon has two confocal mirrors with a cavity length of 50 mm in air, ensuing a spectral range (FSR) of 0.05 cm−1. Two Ge photodiode detectors (Thorlabs PDA50B2) were used to monitor the transmitted light intensity and etalon signal. By tracking the interference peaks, the RWR can be determined with the proposed method. With the deduced relative wavelength information and the detected transmitted light intensity, the 2f/1f signal, and thus the CO2 spectral parameters can be obtained with a digital lock-in amplifier. It must be mentioned that the wavelength characterization process also can be pre-conducted independently to the gas monitoring, which resolves the trouble of splitting the laser into two parts and also avoids the potential optical etalon. In this case, the injected laser current in both processes should be monitored serving as synchronization of these two independent processes.
3. Theory
In the semiconductor lasers, with a modulation frequency of less than 1 MHz, which is of interest in WMS applications, the total frequency modulation is a vector sum of the effects due to the carrier and thermal effects. Considering the difficulty in the theoretical derivation of the frequency chirp with both effects, which is also out of the scope of this paper, the following empirical equation is used to describe the relationship between the instantaneous frequency (in the units of cm−1) v(t) and the injected current i(t),
where Δtn is the delay of nth order between the laser wavelength response and the tuning current and An is the nth polynomial coefficient. As shown in [26], for the commonly used DFB laser, An decrease sharply with the index n and An (n ≥ 3) are at least three orders of magnitudes smaller than A1. Therefore, only the linear term A1 and the second-order non-linear term A2 are taken into consideration in the following model. It must be mentioned that A2 is non-negligible especially with large scan/modulation depth or with high-frequency scan/modulation.Taking the sinusoidal scanned WMS as an example, the diode laser current is scanned with a low-frequency sinusoidal waveform fs in superposition with an additional high-frequency sinusoidal modulation waveform fm. The feedback current injected to the DFB laser can be monitored directly through the laser controller or with a resistor in series with the DFB laser [25] and can be expressed as follows,
As An (n = 0, 1, 2) is well-known to vary with current tuning frequency, as shown in Fig. 2(a), the wavelength scan vs(t) and modulation vm(t) in WMS are conventionally measured separately and the final RWR is considered as a simple summation of these two responses,
Fig. 2. Measured A1 of DFB laser for different modulation frequencies (a) and ibias (b). The best-fit parameters in (a) are a = 19.2, b=-17.9, c = 0.0048, and vary with the property of the DFB laser.
Therefore, inspired by the model of the instantaneous RWR in WMS in Eq. (5), a simple and practical method for wavelength characterization is proposed. The detailed measurement procedure is shown in Fig. 3(b). Firstly, the wavelength scan response including v0, vs,1, ηs,1, vs,2, ηs,2 can be easily measured with a current tuning frequency of fs and bias of ibias = i0. Then, to quantify the wavelength modulation response and the coupling effect, two pairs of wavelength modulation parameters with different bias currents at fm should be measured. Then, the corresponding va m,1 and vb m,1 in the coupling term in Eq. (5) can be obtained with a linear fit of the above obtained two modulation coefficients Am,1. Here, it is worth noting that although the coupling coefficients can be obtained with any combination among ivalley, i0, and ipeak theoretically, the combination of ivalley and ipeak is suggested for better interpolation accuracy. For other parameters ηm,1, vm,2, ηm,2 in Eq. (5), either pair of results with ibias = ivalley or ibias = ipeak can be used if these parameters will not change significantly with bias current and a small difference in vm,2 and ηm,2 has a negligible effect on the final wavelength results. If it is not the case for some lasers, a similar fitting method for Am,1 with different bias currents can be used to obtain ηm,1. Finally, by substituting all these above-mentioned parameters and the monitored input current scan and modulation curves in the experiments into Eq. (5), the RWR in WMS can be easily calculated.
Fig. 3. (a) Difficulties in the real-scenario measurement (with the coexistence of current scan and modulation) of RWR in WMS; (b) Schematic of the proposed three-step method.
As shown in Fig. 3(b), the above-mentioned three-step wavelength characterizations can be easily implemented, as all the laser current is tuned with a single sinusoid. That is to say, with the proposed method, the complex characterization of RWR in WMS, which is a superposition of two sinusoids with a huge frequency difference in Fig. 3(a), is converted to three simple characterizations of RWR with single sinusoidal tuning, as shown in Fig. 3(b). More importantly, with the proposed method, the RWR in WMS with a small scan and modulation for some low-pressure conditions also can be easily achieved without the requirement of a hyperfine etalon. Under this condition, all the coefficients in Eq. (5) can be measured by enlarging the current scan and modulation amplitudes, as discussed later. In addition, it must be mentioned that the proposed model and simplified method are also applicable to a ramp-scanned WMS by substituting the is(t) in Eqs. (2) and (4) with a proper polynomial. The detailed example and validation will be discussed in Section 4. Last but not least, for better explanation and application of the proposed model and method, a software program RWRinWMS has been written and shared online for the convenience of readers. (https://github.com/claire321/RWRinWMS)
4. Discussions
4.1 Accuracy of the proposed model and method in the wavelength characterization
To validate the accuracy of the proposed model and the simplified measurement method, the RWR with different WMS working conditions, including different laser settings (bias current and temperature) and different combinations of scan and modulation waveforms (the type of scan waveform, frequency, and amplitude) were measured. All the experimental conditions are numbered and listed in Table 1. The measured RWRs together with the best fitting for conditions No. 1, 2, 3A and 4A, are compared in Fig. 4. The conditions and results of No. 3B and 4B will be used and discussed in next section. For each condition in Table 1, two different measurement manners were performed: (1) To validate the model in Eq. (5), the RWR in WMS, i.e. the etalon signal, with a superposition of fs and fm was recorded and then fitted with the model. (2) To validate the purposed simplified three-step method, all the required coefficients in Eq. (5) were measured with a single sinusoidal as shown in Fig. 3(b) and then the RWR was directly calculated and compared with the measured results in procedure (1). It must be mentioned that although the first procedure sounds straightforward theoretically, it is laborious to assess the tremendous etalon peaks and label them correctly, especially at the turning points of the sinusoidal modulation.
Fig. 4. Comparison of the simultaneously measured RWR in WMS with the pre-determined results with the proposed method.
Table 1. Working conditions for the validation of the proposed method.
Figure 4 presents the measured RWR and the corresponding best-fit for different conditions, with the residuals attached. As shown by the blue curves in Figs. 4(a)–4(c), all the best-fit 1σ residuals with the proposed model are smaller than 0.1% of its total scan range, which suggests the high accuracy of the proposed model in describing the RWR in WMS. Meanwhile, the best-fit residual increases slightly with both amplitudes and frequencies of the wavelength scan and modulation. The black curves present the RWR and the residual measured with the proposed three-step method. As can be seen, all the residuals are close to those from the direct measurements, which verifies the accuracy of the proposed simplified method.
As shown in Fig. 4(d), the RWR with a triangular wave scan was also measured using the proposed model and methods. To be comparable with the previous methods [25], the scan and modulation frequencies, and the wavelength range were kept the same with the literature (fs = 20 Hz, fm = 10 kHz, vpp ∼ 1 cm−1). As was mentioned before, a first-order polynomial was used to describe the injected scan current is(t) in Eqs. (2) and (4), and similarly, up to a second-order was included in Eq. (3) to describe the RWR with regard to current. As can be seen, the residuals of the simultaneous model and simplified method are still the same and are only half of the minimal residual from the literature (the minimal σ = 3.1 × 10−3) in a similar condition. This strongly proves the better performance of the proposed method compared to the existing methods. In addition, if compared with Figs. 4(a)–4(c), we can conclude that although the wavelength scan range in Fig. 4(d) is smaller than those with sinusoidal scans, the fitting residual is still larger. Pronounced blending structured residuals are observed in both ends of the ramp, which is caused by the failure in the description of wavelength scan response with a simple high order polynomial. In fact, this also supports the use of sinusoidal waveform in both DAS and WMS, as the swift change in both peak and valley of the ramp/triangle waveform is not friendly to the laser [33].
4.2 Comparison of the calibration-free 2f/1f signal with different wavelength characterization methods
To further reveal the high accuracy of the proposed method and the significance of the wavelength characterization in WMS, pure CO2 gas with a pressure of 40.16 kPa was measured using the calibration-free 2f/1f method. The laser was scanned and modulated under the condition of No. 1 in Table 1. The preset temperature (30.6 °C) and bias current (45 mA) were chosen to ensure a comparable 2f/1f intensity as shown in [25]. Figure 5 compares the measured 2f/1f signals with different wavelength characterization methods and their corresponding best-fit. The Voigt profile was used to describe the absorption line profile under moderate pressure and the Doppler broadening coefficient is fixed at 0.0065 cm−1 based on the experimental temperature (T = 297.5 K).
Fig. 5. Comparison of the 2f/1f signals with different wavelength characterization methods. (a) the conventional method ignoring the coupling term in Eq. (5); (b) the proposed simultaneous measurement model (Eq. (5)) and the simplified three-step method. The fitting residual of the proposed three-step method is shifted off zero for clarity.
Figure 5(a) shows the results where the RWR is determined conventionally ignoring the coupling term in Eq. (5). As can be seen, pronounced structures are observed in the fitting residual, especially near the center lobe and two wings of the absorption lineshape. For comparison, Fig. 5(b) plots the deduced 2f/1f signals and best-fit residuals, where the RWRs are directly measured with the proposed model or the three-step method. As can be seen, the residuals of 2f/1f signal in Fig. 5(b) are almost the same. This is attributed to the comparable accuracy of the proposed model and the simplified methods in the wavelength characterization, which has been validated in Section 4.1. Meanwhile, contrary to the structured residual in Fig. 5(a), only random noise is observed in the residual plots of Fig. 5(b). The 1σ fitting residuals from the purposed methods (∼2.9×10−4) are almost 10 times smaller than that from the conventional method and half of the best results 1σ = 5.8 × 10−4 in [25] with a comparable 2f/1f strength. Furthermore, the best-fit collisional broadenings Δvc, integrated areas IA, and the evaluated CO2 concentrations nCO2 are also compared in Fig. 5. The proposed three-step method inevitably presents a more accurate CO2 concentration 99.7%, agreeing within the uncertainty of the gas specification. Overall, the good agreement of the 2f/1f signals and further the deduced CO2 concentration indicate the high accuracy of the proposed method in the wavelength characterization.
4.3 Performance of predicting the RWR with small scan and/or modulation depth
Besides the ability to determine the RWR simply and accurately, the proposed three-step method is also capable of predicting the RWR with a small scan and/or modulation depth, which is difficult to measure by an etalon with a moderate FSR. For the simpler condition, where only the modulation is too small to be measured, the modulation coefficients and the coupling term can be certainly measured by an expanded current modulation amplitude. However, if the scan range is also small, the coupling term also can be determined by the proposed method, but with an enlarged current scan amplitude for all three steps. Figure 6(a) explains how the coupling term is dealt with in the latter two steps of the proposed method under this condition. As can be seen, owing to the linearity of the laser tuning character (see Eq. (4) and Fig. 2(b)), the modulation coefficients with a small scan depth, as shown by the red line, can be interpolated in the curve with a magnified scan amplitude, as shown in black. Alternatively, more straightforwardly, we can directly substitute the deduced slope and intercept from the extended curve into Eq. (5), and then the desired RWR with a small scan depth can be predicted.
Fig. 6. (a) Schematic showing the prediction of the coupling term with small scan and/or modulation from the proposed method; (b) the predicted RWR of No. 3A in Table 1 (Vpp,s = 34 mA, Vapp,m = 20 mA) from No. 3B (Vpp,s = 68 mA, Vapp,m = 40 mA) with the proposed method.
To verify the accuracy of the above-mentioned prediction, the RWR of case 3A in Table 1 was predicted by a doubled scan and modulation current amplitudes and compared with the benchmark in Fig. 4(c). The detailed specifications of the laser and the enlarged scan and modulation current waveforms are shown as case 3B in Table 1. Figure 6(b) compares the RWR and the best-fit residuals from both approaches. The red dots are the measured etalon peak results with simultaneous scan and modulation the same as those in Fig. 4(c). The black curve shows the predicted RWR from the proposed method with an extended scan and modulation currents. As can be seen from the residual plot, no evident deviation can be distinguished between the measured etalon points and the predicted curve, with a 1σ residual as small as 8.22×10−4. This agreement verified the potential and the accuracy of the proposed method in predicting the RWR with a small scan and/or modulation depth. In addition, this merit of the proposed method lays the foundation for the precious fitting of calibration-free nf/1f under low pressure and will be applied in the next section.
4.4 Measurement of the spectroscopic parameters of CO2 with CF-WMS based on the accurate RWR determination
After validating the accuracy of the proposed method in the wavelength characterization and its promising potential in predicting the RWR with small modulation, the calibration-free 2f/1f method is enabled to precisely measure the spectroscopic parameters under low pressure. In this part, the line strength and collisional broadening coefficient of CO2 transition at 6976.2026 cm−1 was measured with a pressure range of 1 kPa to 18 kPa at T = 297.5 K. The laser parameters (temperature and bias current) and the scan and modulation frequencies are set to the same with case No.1 in Table 1. To avoid excessive scan and modulation indices under low pressure, the laser was scanned with a sinusoidal waveform with Vpp = 10 mA and modulated with Vpp = 4.5 mA, which are 1/4 and 1/5 of the amplitudes in case No.1, respectively. The RWR, which is evidently impossible to be measured by an etalon with a moderate FSR, i.e. 0.05 GHz, was deduced based on the parameters from case No.1 by the proposed method as explained in Sec. 4.3.
Figure 7(a) shows the two examples of the measured and the corresponding best-fit 2f/1f signals under pressures of 4 kPa and 15 kPa. In the fitting process, the Rautian line profile was used with a Doppler broadening fixed at 0.0065 cm−1 according to the experimental temperature and the collisional width, integrated area, and Dicke narrowing to be fitted. The Doppler broadening coefficient is fixed at 0.0065 cm−1 according to the experimental temperature. Better agreements were achieved between the measurements and the best-fit 2f/1f for all conditions compared with those in literature [22,24,26,34,35], which also suggests the accurate description of the wavelength. Furthermore, the best-fit collisional width and integrated area for all pressures are shown in Fig. 7(b), together with the corresponding two-parameter linear fit. The R-square values (R2 = 0.99969 for Δvc and R2 = 0.99993 for A) and the negligibly small intercepts (-1.0 × 10−5 for Δvc and -4.6 × 10−5 for A) reflect the accurateness of the measurements and fitting processes. The line strength inferred from the best-fit slope of the integrated areas is 2.776 (±0.006) × 10−23 cm−1/(mole·cm−2) @ 297.5 K, which corresponds to 2.818 (±0.006) × 10−23 cm−1/(mole·cm−2) @ 296 K. This agrees within uncertainty with the 2.856 × 10−23 cm−2/(mole·cm−2) (2% ∼ 5%) from HITRAN 2016 [36] and the 2.835 × 10−23 cm−2/(mole·cm−2) (<2%) from the latest literature [37]. The CO2 self-broadening coefficient from the slop of collisional width is 1.143 × 10−1 cm−1/atm, which agrees well with that of the best-Rautian-fit 1.13 × 10−1 (<2%) cm−1/atm in [37].
Fig. 7. (a) Examples of the measured and best-fit 2f/1f under low pressure. (b) Measured integrated area, collisional width with different pressures (T = 297.5 K).
5. Conclusions
In this paper, a comprehensive model was proposed to accurately characterize the RWR in WMS. A universal coupling term in the 1st harmonic amplitude was introduced to describe the coupling effect between the current scan and modulation in WMS with any scan waveform (both ramp or sinewave). Based on this model, a novel and practical three-step method was developed to precisely measure the RWR in WMS. In this method, the laser RWR with current scan and modulation are decoupled and measured separately. This method was verified with different laser working conditions and scan waveforms. Tiny residuals, less than 0.1% of the total scan range, were obtained for all conditions and is only half of the minimal residual reported by literature under a similar working condition. Meanwhile, the better fitting of the CF-2f/1f signal and the more accurate deduced gas concentration also suggest the high accuracy and wide applicability of the proposed method. Last but not least, the performance of the proposed method to predict the RWR with small scan and/or modulation depths was also verified. Based on the accurate RWR prediction, the spectral parameters, including line strength and self-collisional broadening coefficient, of CO2 transition at 6976.2026 cm−1, were successfully measured with WMS under low pressure.
Funding
National Natural Science Foundation of China (51676105, 51906120, 11972213); China Postdoctoral Science Foundation (2018M640125); China Postdoctoral Science Foundation (2019T120088); National Basic Research Program of China (973 Program) (2016YFC0201104).
Acknowledgments
The authors would also like to thank Dr. Zhechao Qu at Physikalisch-Technische Bundesanstalt for valuable discussion regarding the laser wavelength response in WMS.
Disclosures
The authors declare no conflict of interest.
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