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A methodology for calibration-free wavelength modulation spectroscopy (CF-WMS) that is based upon an extensive empirical description of the wavelength-modulation frequency response (WMFR) of DFB laser is presented. An assessment of the WMFR of a DFB laser by the use of an etalon confirms that it consists of two parts: a 1st harmonic component with an amplitude that is linear with the sweep and a nonlinear 2nd harmonic component with a constant amplitude. Simulations show that, among the various factors that affect the line shape of a background-subtracted peak-normalized 2f signal, such as concentration, phase shifts between intensity modulation and frequency modulation, and WMFR, only the last factor has a decisive impact. Based on this and to avoid the impractical use of an etalon, a novel method to pre-determine the parameters of the WMFR by fitting to a background-subtracted peak-normalized 2f signal has been developed. The accuracy of the new scheme to determine the WMFR is demonstrated and compared with that of conventional methods in CF-WMS by detection of trace acetylene. The results show that the new method provides a four times smaller fitting error than the conventional methods and retrieves concentration more accurately.
© 2016 Optical Society of America
1. Introduction
In the presence of significant amounts of air pollution, the public pays close attention to the air quality. It has therefore become more and more important to monitor the occurrence of trace gases. In the wake of the development of industry, the detection of process gases, used to regulate the manufacturing process, could lessen the emission of pollutant and improve the productive efficiency [1]. Trace gas detection can also be applied to other fields, e.g. pathological detection [2]. Due to its in situ, non-intrusive and high sensitivity capabilities, tunable diode laser absorption spectroscopy (TDLAS) has, during the last decade, moved from lab towards practical application [3–7].
Mainly because of its simplicity and calibration-free property, direct absorption spectroscopy (DAS) is one of the most versatile detection methodologies of TDLAS. The detection sensitivity, i.e. minimum detectable optical density, of DAS is generally limited to 10−3, primarily because of the excess noise (often so called 1/f noise) of the detection system [8]. In order to improve on the detection sensitivity, one way is to scan the wavelength of laser rapidly, normally in the kHz range. By this, excess noise can be efficiently reduced [7]. An alternative, and more efficient way, is to use a modulation-demodulation technique, such as wavelength modulation spectroscopy (WMS). By modulating and demodulating the wavelength of the light at a frequency at which the background has low amounts of noise and by the use of a narrow bandwidth of demodulation, a WMS system can achieve a lower detection limit than what DAS can demonstrate [9].
In its initial application, the signal of WMS was given only as a certain harmonic components (or a combination of several components) of the modulated absorption signal whereby the signal amplitude is proportional not only to the concentration but also to the power of light, the electronic gain, and the detection phase. To assess a concentration from such a signal implied that the technique needs to be calibrated.
Several efforts have thereafter been pursued to recover information about the concentration from the WMS without calibration. To remove the influences of light power and electronic gain, H. Li et al. suggested, in 2006, a calibration-free WMS measurement based on the 1f normalized 2f signal [10]; As an alternative, in 2011, J. Chen et al. normalized the WMS signals with the zeroth harmonic component to remove the influences of light power and electronic gain [11]; In the same year, G. Stewart et al. developed a methodology that allowed for recovery of the absorption line shape of DAS from the first harmonic component of WMS in the presence of residual amplitude modulation [12]. Compared to other calibration-free methods, Li’s method, referred to as calibration-free wavelength modulation spectroscopy (CF-WMS), is less sensitive to the stray light and suitable for a larger dynamic scope of concentration [3,13]. In this technique, the 1f normalized 2f signal used for concentration assessment was described as
where Xif and Yif are two orthogonal components of the ith harmonic of the signal, each obtained by multiplying the detector signal with the corresponding reference signal, i.e. cos(iωt) and sin(iωt), respectively, followed by integration. Since Rif, which thus is the square root of the sum of Xif2 and Yif2, is proportional to various instrumental factors of the system, e.g. the light power and the electronic gain, the use of the R2f/R1f ratio for gas assessments can provide a useful entity that is independent of the instrumental factors and thereby provides a calibration-free response. This technique is based upon the fact that R1f is nonzero, which indeed is the case when WMS is based upon the modulation of the injection current of a distributed feedback (DFB) diode laser [13]. The concentration is retrieved by fitting a simulated R2f/R1f signal to the measured one, where the former is obtained by digital demodulation of the simulated wavelength-modulated absorption signal that is retrieved, based on the Beer-Lambert law, from the measured non-absorbed laser intensity and the simulated absorbance [13–15]. The simulated absorbance is then proportional to the initial concentration, the gas pressure, the interaction length and the absorption line shape where the latter is determined by the wavelength scan, which includes so called wavelength modulation frequency response(WMFR), defined in section 3.More recently, K. Sun et al. improved this scheme by digital demodulating both of the measured and simulated wavelength modulated absorption signals. Moreover, the simulations are based on the measured non-absorbed laser intensity [13,16]. This removes the influence of the finite responses of analog circuits and takes into consideration all the components of the background intensity, including the nonlinear response of the intensity modulation and various non-absorption losses.
In order to retrieve the concentration from the CF-WMS signal, either the peak value or the line shape of R2f/R1f can be used [10,17]. However, the peak value can be only used under the condition of a pre-given temperature and pressure of the gas sample. Fitting of an entire line shape, on the other hand, can result in a more accurate result since more absorption information is considered. In addition, the fitting can also be used to deduce the gas pressure and resolve the overlapped absorption lines [18–20]. Therefore the line shape fitting is the preferred mode of evaluation.
To perform accurate assessments of trace gas concentration by CF-WMS, it is important that the R2f/R1f is accurately described. For this, it is equally important that the WMFR of the laser is determined accurately [21]. Up to now, in most previous related works, the WMFR has been based upon the assumption that the wavelength response of the laser to the current modulation is the same over the entire scanning range, i.e. the magnitude of wavelength modulation is constant [10,13]. However, it is not always the case. J. Chen suggested that the amplitude of 1st harmonic of WMFR is proportional to the first derivative of a polynomial fit of the wavelength calibration function of sweep when a small current modulation around a bias point, and later Z. Qu applied this idea to CF-WMS [11,22–24]. For example, it was found that for a specific AlGaInAsSb DFB laser, lasing at 3.06 µm, the WMFR (also called FM-index, defined as the amount of tuning in frequency per scanning in current) changed from 0.26 GHz/mA in the beginning of the sweep to 0.815 GHz/mA at the end of the sweep [25]. Since it was shown that the current to wavelength tuning behavior of VCSEL laser can be well described with a second order polynomial, it could be concluded that for her type of laser the WMFR has the linear response to the scanned current [11]. Despite this, it was found that the description of the WMFR laser was still not perfect for a DFB; it could be concluded that the maximum fitted residual of the CF-WMS signal based on DFB laser is still often around 5% [16,22,26].
In order to improve on this situation, so as to increase the measurement accuracy of CF-WMS, this work presents an improved methodology for assessing the WMFR of a DFB laser by use of an high order empirical formula. Although the work is performed on a specific DFB diode laser, the methodology presented is assumed to be generally applicable. It is shown that the parameters of the WMFR function can be rapidly and swiftly assessed by directly fitting the simulated peak-normalized background-subtracted 2f signal with an empirical WMFR expression to the measured one. The accuracy and applicability of the new methodology are verified by a comparison of CF-WMS based on the traditional and new methods for assessment of both the WMFR response and the trace concentration of acetylene in a gas sample.
2. Experimental set up
The experimental setup of CF-WMS is shown in Fig. 1. The light source is a DFB laser (NTT, NLK1L5EAAA) working around 1528 nm, which overlaps with both P5(e) line of ν1 + ν3 band of C2H2 at the wavenumber of 6544.4419 cm−1 and several other relatively weak absorption lines of the same molecule. The line strength of P5(e) line is 9.475 × 10−21 cm−1/molecule·cm−2 and the air-broadened half width coefficient is 0.0903cm−1/atm [27]. A 32 Hz, 1.5 Vpp (peak-to-peak voltage) triangle wave and a 10 kHz, 0.6 Vpp sine wave (corresponding to current modulation amplitude 11mA) produced by two function generators (FG1 and FG2, Tektronix, AFG3102) are sent to the laser controller (ILX Lightwave, LDC-3724C) to scan and modulate the wavelength of DFB laser by injection current, respectively. This modulated laser beam is divided by a splitter into two parts: one is sent into a home-made etalon with a free spectral range (FSR) of 0.022 cm−1 and is recorded directly by the PC for wavelength calibration, while the other part, serving as a probe, passes through the target gas cell. The detected signals by the two photo detectors (PD, Thorlabs, PDA10CS-EC) are acquired by a 10 MHz, 12 bits data acquisition (DAQ) card (NI Corporation, PCI-6115) and sent to the computer. The window of each PDs was tilted to avoid fringes. A LabVIEW programmed double-channel digital lock in amplifier was used to demodulate the measured and the simulated wavelength-modulated absorption signals. The bandwidth of the digital filter with the type of finite impulse response was 10Hz. The concentration of a given analyte in a sample can then be retrieved by fitting the simulated R2f/R1f signal to the measured ratio with the concentration, the phase shifts between the intensity modulation (IM) and the frequency modulation (FM), and the variables of the WMFR model as free parameters. The concentration of target gas is mixed based on the standard gas of 428ppm and pure nitrogen by a home build concentration regulation system based on two mass flow controllers (MKS GM50A). When we perform the measurement, the flow rate to the inlet of the gas cell is 120 sccm (standard cubic centimeter per minute). Since the outlet was open to the air, the gas pressure inside the cell corresponds to the local air pressure which at the time of the experiment was 0.920 atm, assessed by the pressure gauge (MKS, type660).
Fig. 1 Schematic of the experimental setup used for CF-WMS. DFB-distributed feedback diode laser, PD-photodiode, FG-function generation, PC-personal computer.
3.General assessment of the WMFR of a DFB diode laser
The amplitude and the line shape of the CF-WMS signal are strongly influenced by not only the concentration of the analyte but also the WMFR of the laser. An accurate assessment of the concentration therefore requires a correct pre-description of the WMFR. In order to assess and express the WMFR correctly, the wavelength response of the laser was calibrated by use of an etalon when the laser is scanned and modulated. The adjacent transmission peaks of etalon signal are separated by a fixed wavelength interval, equal to the FSR of the etalon. The positions of these peaks, as a marker, therefore can be used to record the relative wavelength variation accurately. The measured and analyzed results are shown in Fig. 2. The squares (black in color) in the upper window of Fig. 2(a) represent the relative wavenumber of laser at the positions of the transmission peaks of the etalon (in cm−1) as a function of time (in ms). The solid curve (red in color) shows a 4th order polynomial fit to the etalon peak positions when the modulation is removed. The WMFR could then be obtained by subtracting these two curves, shown by the square (black in color) in the middle window of Fig. 2(a).
Fig. 2 The measured and fitted WMFR from the DFB laser scrutinized. (a) the laser wavelength vs. time (the injection current is proportional to the controlling voltage); (b) the amplitude of FFT of the fitted WMFR; (c) the amplitude of 1st harmonic vs. time.
A general description of the WMFR can be written as a sum of harmonics of the modulation frequency, each with its own time-dependent amplitude. A first general attempt to model the response of the laser consisted of a sum of three harmonics, each with an amplitude that can be expressed as a 2nd order polynomial in time, i.e. with a form that can be written as
In the first assessments of the WMFR, this expression is fitted to the data with all parameters being free. The resulting fit is shown by the red curve in the middle window of Fig. 2(a) while the residual between the fitted function and the data is shown in the lowest window of Fig. 2(a). The fitting parameters are shown in Table 1. In order to clearly assess the contributions of each of the three harmonics, the amplitude spectrum of the Fourier transform of the fitted function, i.e. Equation (2) based on the parameters given in Table 1, is shown in Fig. 2(b). The figure shows that all three harmonics provide contributions to the WMFR. The existence of the second and third harmonics shows that the laser has a nonlinear wavelength response. The amplitude of the various harmonic components decreases strongly with the order of the harmonic; the amplitude of the second is around 2 order of magnitude smaller than that of the first, while the third in turn is one order of magnitude smaller than that of the second. This shows that the nonlinear responses are fairly weak. A piecewise fitted amplitude of the 1st harmonic based on the dots in the middle panel of Fig. 2(a) as a function of time is shown by black solid markers in Fig. 2(c), while the red line is the plot of the function . This shows that the amplitude of the 1st harmonic basically has a linear response over the scan which is different from existed estimations of WMFR of DFB laser.
Table 1. The fitting result based on Eq. (2).
The data in Table 1 provide a possibility to reduce the number of free parameters and simplify the fit. For example, it can be seen that the part of the 1f component that is linear with time (i.e. b1tmax, in the 4th row, 1st column of data) can contribute with up to 18% to the time-independent response (ai, in the 1st row). The quadratic time-dependent response (i.e. citmax2) contributes only 0.4%. This shows that the latter one can be neglected with respect to the other two without loss of accuracy. Although the amplitude of the 2nd harmonic is around 0.7% of the 1st harmonic (a2 in comparison to a1), which implies that it should be included in a fit, the time-dependence of this harmonic (b2tmax) is small and can be ignored. The contributions of all the other parameters are lower than 0.1% and can be neglected. This implies that it should be possible to describe the variation of laser frequency equally well by the simpler function
where θ2 is the phase difference between the 1st and the 2nd harmonics of wavelength modulation. According to Eq. (3), the FM shows a nonlinear response with respect to the current modulation, which is similar to the IM that normally is given bywhere is the average laser intensity, i1 and i2 are the IM amplitudes of the 1st and 2nd harmonic, and ψ1 and ψ2 are the phase shifts of 1st and 2nd harmonic of intensity modulation relative to the 1st harmonic wavelength modulation, respectively. Based on the measurement, the first IM amplitude, i1 also changes linearly with time, similar to the amplitude of 1st harmonic of WMFR (but with a different sign of slope). Since the expression form of i1 has no extra impact on the latter theoretical analysis, it is consistent with the previous description [10].In semiconductor lasers, there is a strong coupling between the carriers and photons. The modulation of the carrier density in the active layer induces the IM directly, while the FM is caused by changes of the effective refractive index originated from the threshold carrier density [28,29]. Therefore, the changes of the relative phase shifts ψ1 and ψ2-θ2 between the IM and the FM of different harmonics for a special diode laser are all small, respectively. Meanwhile if ψ1 is known, ψ2 can be determined directly from the fitting of laser intensity [10]. Hence, if ψ1 is determined in a series measurements, ψ2 can also be confirmed, and θ2 can only vary in a rather small range.
4. Methodology for swift assessment of the WMFR based on the 2f method
In order to determine the WMFR accurately, due to the narrow profile of the response of the etalon with a relative high finesse and short FSR, the sampling rate of the data acquisition card should be sufficient. Despite this, it is not always trivial to assess the position of the peak features correctly by software, since the symmetric two transmitted modes of etalon will possibly mixed together around each turning point of the sinusoidal wavelength modulation. This makes it impracticable in practice to use an etalon for assessment of the WMFR automatically.
An alternative to determine all these parameters, comprising the concentration, a1, b1, a2, and the phase shifts ψ1, ψ2 and θ2, is to use the CF-WMS signal itself. However, a fit with too many free parameters, would consume a lot of time and could introduce uncertainties. Therefore a simple, rapid, and accurate procedure for the determination of the WMFR needs to be developed.
A way to develop such a procedure is to recognize that the two orthogonal demodulation components of WMS, Xif and Yif in Eq. (1), can be expressed as [30–32]
where A is the instrument factor, including the photoelectric conversion gain and non-absorption loss of light, while Hk is the kth order Fourier coefficient of absorbance, which can be represented as [10,32]where is the Kronecker delta function.According to Eq. (5), in the absence of absorption, the two orthogonal components of 2f background signal can be given as
The absolute magnitude of a background-subtracted 2f signal can therefore be given by
This shows that S2f is dependent on the absorption line shape function and the modulation parameters (i1, i2, ψ1, ψ2, θ2). Of these, i1, i2 and Δψ = (ψ2-ψ1) can be determined from the fitting of the modulated light intensity. As we know, the peak-normalized Voigt line shape function does not change for a given pressure and temperature condition, and suffers negligible variation with concentration in particular under a low concentration condition. Therefore, the background-subtracted peak-normalized 2f signal defined as , is virtually independent of the concentration of the targeted trace gas. Figure 3(a) exemplifies this by a pair of numerical simulations of the peak-normalized background-subtracted 2f signal for two different absorbance value, viz. 0.05% and 1.75% (for a pressure of 0.92 atm). As is shown by the residual in the lower window, the difference in line shape is in no case larger than 0.15%.
Fig. 3 The comparisons of peak-normalized background-subtracted 2f signal at different (a) concentrations, (b) ψ1, (c) ψ2-θ2, and (d) amplitude of WMFR. The lower window in each panel shows the difference of the corresponding two curves.
As mentioned above, as long as the modulation frequency is held constant, the other parameters of the WMFR parametrization, i.e. the relative phase shifts ψ1, ψ2, and θ2, change only marginally during a scan. For example it was found that for the laser used in this work, when the temperature of DFB laser varies from 15° to 35°, the center current from 80 to 120 mA, and the modulation amplitude of current from 5 to 20mA for a modulation frequency of 10 kHz, the phase shift ψ1 ranges from −2.84 to −2.85 rad, ψ2 follows the change of ψ1, and the phase difference, ψ2-θ2, also has a small variation range only from −0.1 to −0.4 rad. The Figs. 3(b) and 3(c) display simulations performed to assess the corresponding influence of the changes of these parameters on the peak-normalized background-subtracted 2f signal. These panels show that line shape differences changes solely 0.1% and 0.3% with the aforementioned changes of the ψ1 and ψ2-θ2, respectively.
On the other hand, Fig. 3(d) shows the two line shapes when the amplitude of WMFR is changed by 5% (for a fixed center frequency). The lower window reveals that such an alteration of the WMFR can have a significant impact (larger than ± 3%) on the 2f signal [the asymmetry of the difference is mainly because of the changes of b1 in Eq. (3)]. This implies that the peak-normalized background-subtracted 2f signal can be used to deduce the parameters of the WMFR, i.e. the a1, b1 and a2, even though there is no knowledge of the concentration or phase shifts. This way of assessing the WMFR function is henceforth denoted the 2f method.
Although the 2f method requires slightly longer time than the traditional fitting method, it needs only to be done in the beginning of a concentration assessment. In addition, with only three free parameters, it provides increased simplicity and accuracy. The WMFR determined for the laser used in the present study by this method is demonstrated in Fig. 4. The dots (black in color) show the measured relative wavelength as a function of time (see the upper scale) while the solid curve (red in color) gives the deduced result based on the new method. The deduced parameters are shown in the figure. The difference in the lower window shows a maximum error of no more than 0.002 cm−1 and demonstrates that there is a good agreement between the deduced result and the measurement. It should be noted that the residual around the wavelength of line center shows better determination of WMFR than other positions. When the expression of WMFR, based on Eq. (3), is obtained, the concentration, ψ1, and θ2 are the left three free parameters for the fitting to CF-WMS signal.
Fig. 4 The comparison of measured frequency response(dots in black) and simulated one based on the 2f method (curve in red). The lower window displays the difference of the two curves.
5. CF-WMS with the new WMFR assessment method
In order to evaluate the validity of the aforementioned expression of WMFR and the new 2f methodology for retrieval of the parameters, an acetylene sample with standard concentration of 428 ± 5ppm (verified by a series of DAS measurement) was assessed by the CF-WMS technique. Figures 5(b)-5(d) compare fits based on three different means to assess the WMFR function to measured data. For comparison, Fig. 5(a) shows the result when the WMFR was represented by the measured wavelength response of the laser.
Fig. 5 The fitting results of CF-WMS signal with different WMFR models. (a) Measured WMFR with 2nd harmonic considered (b) Simulated WMFR with 2f method determined parameters (c) WMFR with a constant amplitude of only 1st harmonic (d) WMFR deduced from the slope of wavelength calibration function of sweep and (e) the modulation amplitude of the 1st harmonic component of the WMFR inferred with different methods, corresponding to those of the panels (a) to (d).
Figure 5(b) shows the fitted results based on the Eq. (3) with the parameters determined by the 2f method. Figure 5(c) shows the result when the WMFR is assumed to be represented by only a single harmonic with a constant amplitude, while panel Fig. 5(d) is based on the WMFR deduced from the slope of the wavelength calibration function of sweep (the derivative of the 4th order polynomial fit of the etalon peak positions when the modulation is removed). The residuals are shown in the lower windows of each panel. The maximum values of the residuals of these four cases were found to be 0.28%, 0.39%, 1.54%, and 1.35%, respectively. The corresponding modulation amplitude and modulation index of the 1st harmonic of the WMFR with different methods are shown in Fig. 5(e). This shows that the 2f method [Fig. 5(b)] can provide a parametrization that mimics the actual 2f-response better than any of the other methods. It should be added, in passing, that the results shown in this work in general show smaller fitting residuals than those of previous works, which typically have residual around 5%. This is attributed to the fact that the previous works were mainly addressing assessments in harsh environment, whereby they suffer from a larger amount of noise [16,22,26].
The evaluated acetylene concentrations are also given in each panel. These show that, as compared with the traditional descriptions of WMFR, our new model, described as Eq. (3) with 2f method determined parameters, can indeed achieve a better fitting for the CF-WMS signal and acquire concentrations with higher accuracy than what the other methods can do. Note that although the fittings, by use of the traditional models, do not seem to be bad, their estimations of concentrations are associated with relatively large errors. As for line shape fitting of wavelength-scanned WMS with an inaccurate WMFR, the fitting program would miscalculate the concentration in order to alleviate the deviation in the wing range.
Finally, samples with a variety of acetylene concentrations (ranging from 11 to 428 ppm) were assessed based on the new WMFR model with its parameters determined by the 2f method. The estimated concentrations are shown in Fig. 6 as a function of the prepared concentration. The square markers (black in color) in the upper window show the measured concentrations while the line (red in color) provides the ideal response, i.e. y = x. The square markers (black in color) in the lower window show the relative error. The dashed line (black in color) shows the zero baseline. The maximum error is 7.5% and the R-square value is excellent, viz. 0.9991. The relative large errors, appearing in the lower concentrations, mainly arise from the mass flow controller which works close to the end of its measurement range.
6. Conclusion
A new high order empirical model for the modulated laser frequency, WMFR, used for CF-WMS, has been developed, based upon combination of measurement and analysis. It was found that a good description of the WMFR could be obtained by using a 1st harmonic component with an amplitude that includes a linear time-dependence and 2nd harmonic component with a fixed amplitude. The parameters are retrieved by fitting the peak-normalized background-subtracted 2f signal with a theoretical simulation. Although our new 2f-method consumes solely a few more seconds at the beginning of measurement as compared with the traditional methods to assess the WMFR parameters, it can achieve a better fitting to the CF-WMS signal and a more accurate concentration assessment. As a demonstration, C2H2 at different concentrations are measured based on our new method and a good agreement is achieved. This simple method can make the application of CF-WMS more reliable.
Acknowledgment
The work was supported by the 973 program (Grant No. 2012CB921603), the National Natural Science Foundation of China (Grant Nos. 61475093, 61127017, 61178009, 61378047, 61275213, and 61205216), the National Key Technology R&D Program (2013BAC14B01), the Shanxi Natural Science Foundation (Grant Nos. 2013021004-1, and 2012021022-1), the Shanxi Scholarship Council of China (2013-011, 2013-01), and the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi. The authors Weiguang Ma and Ove Axner would also like to acknowledge Umeå University’s program “Strong research environments”.
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