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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 23724–23735
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First harmonic with wavelength modulation spectroscopy to measure integrated absorbance under low absorption

Peng Zhimin, Ding Yanjun, Jia Junwei, Lan Lijuan, Du Yanjun, and Li Zheng  »View Author Affiliations


Optics Express, Vol. 21, Issue 20, pp. 23724-23735 (2013)
http://dx.doi.org/10.1364/OE.21.023724


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Abstract

Integrated absorbance (IA) can be used to infer gas temperature and concentration directly, in this paper, we proposed a new method that uses the 1st harmonic to measure the IA under low absorption conditions (<10%). Subsequently, a large number of numerical simulations are used to validate the reliability and accuracy of this method, and several absorption lines of CO2 and H2O molecules near 6981 cm–1 are selected to determine the IA and species concentration in experiments. Calculation and experiment results show that the proposed method can accurately measure IA in actual measurements.

© 2013 Optical Society of America

1. Introduction

Tunable diode laser absorption spectroscopy (TDLAS) has been widely employed to measure gas parameters, such as temperature and species concentration in numerous environments, given its properties of non-contact, rapid, and high sensitivity [1

1. A. Farooq, J. B. Jeffries, and R. K. Hanson, “CO2 concentration and temperature sensor for combustion gases using diode-laser absorption near 2.7μm,” Appl. Phys. B 90(3–4), 619–628 (2008). [CrossRef]

5

5. S. Wagner, B. T. Fisher, J. W. Fleming, and V. Ebert, “TDLAS-based in situ measurement of absolute acetylene concentrations in laminar 2D diffusion flames,” Proc. Combust. Inst. 32(1), 839–846 (2009). [CrossRef]

]. In TDLAS, spectral absorbance (SA) or integrated absorbance (IA, integrated value of SA) is a key parameter, and can be used to infer gas parameters conveniently. Since the introduction of TDLAS, direct absorption spectroscopy (DAS) [6

6. X. Liu, J. B. Jeffries, R. K. Hanson, K. M. Hinckley, and M. A. Woodmansee, “Development of a tunable diode laser sensor for measurements of gas turbine exhaust temperature,” Appl. Phys. B 82(3), 469–478 (2006). [CrossRef]

9

9. Y. J. Ding, X. H. Li, Z. M. Peng, and L. Che, “Half-Width Integral Method for Gas Concentration Measuring in Tunable Diode Laser Absorption Spectroscopy,” Spectrosc. Lett. 46(7), 465–471 (2013). [CrossRef]

] has been extensively used to measure gas temperature and concentration because of its simplicity, accuracy in signal analysis process, and ability to make absolute measurements by using the IA. However, drawbacks also exist in DAS, for example, the scanning frequency is usually limited to several kHz or even lower, while the troublesome noise sources, such as laser and 1/f noise, which are the dominating type noises at low frequency. In addition, in high temperature environments, the radiations emitted by hot gas or particle interfere with the laser emission, and will reduce the sensitivity of DAS.

Wavelength modulation spectroscopy (WMS) [10

10. P. Kluczynski, J. Gustafsson, A. M. Lindberg, and O. Axner, “Wavelength modulation absorption spectrometry —an extensive scrutiny of the generation of signals,” Spectrochim. Acta B 56(8), 1277–1354 (2001). [CrossRef]

, 11

11. P. Kluczynski and O. Axner, “Theoretical description based on Fourier analysis of wavelength-modulation spectrometry in terms of analytical and background signals,” Appl. Opt. 38(27), 5803–5815 (1999). [CrossRef] [PubMed]

] uses a high frequency sinusoidal modulation riding on a slowly varying diode laser injection current to produce harmonics. This technique is well known as means to increase signal-to-noise ratio (SNR) because high frequency can reduce laser, 1/f and other noises. However, the traditional 2nd harmonic [12

12. J. A. Silver and D. J. Kane, “Diode laser measurements of concentration and temperature in microgravity combustion,” Meas. Sci. Technol. 10(10), 845–852 (1999). [CrossRef]

14

14. F. Wang, K. F. Cen, N. Li, Q. X. Huang, X. Chao, J. H. Yan, and Y. Chi, “Simultaneous measurement on gas concentration and particle mass concentration by tunable diode laser,” Flow Meas. Instrum. 21(3), 382–387 (2010). [CrossRef]

] detection should calibrate the harmonic signal to a known mixture and condition to recover the absolute gas temperature or concentration. For most practical environments, this is difficult because the gas pressure, scaling factor, and background are unstable. In recent years, scientists have proposed some “calibration-free” methods to measure gas parameters [15

15. J. Henningsen and H. Simonsen, “Quantitative wavelength-modulation spectroscopy without certified gas mixtures,” Appl. Phys. B 70(4), 627–633 (2000). [CrossRef]

20

20. K. Sun, X. Chao, R. Sur, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation diode laser absorption spectroscopy for high-pressure gas sensing,” Appl. Phys. B 110(4), 497–508 (2013). [CrossRef]

]. For example, Hanson et al. [16

16. H. Li, G. B. Rieker, X. Liu, J. B. Jeffries, and R. K. Hanson, “Extension of wavelength-modulation spectroscopy to large modulation depth for diode laser absorption measurements in high-pressure gases,” Appl. Opt. 45(5), 1052–1061 (2006). [CrossRef] [PubMed]

20

20. K. Sun, X. Chao, R. Sur, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation diode laser absorption spectroscopy for high-pressure gas sensing,” Appl. Phys. B 110(4), 497–508 (2013). [CrossRef]

] proposed a useful “calibration-free nf/1f method” for the direct determination of gas temperature or concentration by comparing the experimental signals with their theoretical values. However, uncertainties in gas pressure, spectroscopic constants, laser characteristic parameters, and absorption length will induce discrepancies between the theoretical and experimental values, thus causing measurement errors.

Considering that the data analysis in conventional WMS is more complex than DAS, Stewart et al. [21

21. K. Duffin, A. J. McGettrick, W. Johnstone, G. Stewart, and D. G. Moodie, “Tunable diode laser spectroscopy with wavelength modulation: A calibration-free approach to the recovery of absolute gas absorption line-shapes,” J. Lightwave Technol. 25(10), 3114–3125 (2007). [CrossRef]

25

25. J. R. P. Bain, W. Johnstone, K. Ruxton, G. Stewart, M. Lengden, and K. Duffin, “Recovery of absolute gas absorption line shapes using tuneable diode laser spectroscopy with wavelength modulation—Part 2: Experimental investigation,” J. Lightwave Technol. 29(7), 987–996 (2011). [CrossRef]

] proposed a method to recover the SA using the 1st harmonic residual amplitude modulation (RAM) signal. This method offers all the advantages of DAS (calibration-free, simple of data processing) plus many of the benefits of WMS (smaller noises, higher SNR). However, this method only works well under small modulation indices (m<0.5), and the recovering errors increase sharply as the modulation indices increase. Recent years, to improve the signal’s SNR, they introduced a “shape correction function” [24

24. G. Stewart, W. Johnstone, J. R. P. Bain, K. Ruxton, and K. Duffin, “Recovery of absolute gas absorption line shapes using tunable diode laser spectroscopy with wavelength modulation—part 1: theoretical analysis,” J. Lightwave Technol. 29(6), 811–821 (2011).

, 25

25. J. R. P. Bain, W. Johnstone, K. Ruxton, G. Stewart, M. Lengden, and K. Duffin, “Recovery of absolute gas absorption line shapes using tuneable diode laser spectroscopy with wavelength modulation—Part 2: Experimental investigation,” J. Lightwave Technol. 29(7), 987–996 (2011). [CrossRef]

] to recover the SA under large modulation indices (e.g. m = 2.0). But the problem is that the line shape (determined by gas temperature, pressure, species concentration, as well as the inherent characteristics of the line) is a necessary condition to calculate such correction function. Perhaps we can calculate the “shape correction function” in laboratory conditions when gas parameters are known, this is difficult in practical applications because these parameters are varied or unknown. Based on Stewart’s studies, we proposed a method [26

26. P. Zhimin, D. Yanjun, C. Lu, and Y. Qiansuo, “Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape,” Opt. Express 20(11), 11976–11985 (2012). [CrossRef] [PubMed]

] to recover the SA using the odd harmonics (1st, 3rd, 5th…) rather than only using the 1st harmonic. This method can effectively reduce the recovering errors under large modulation indices. However, the amplitudes of high odd harmonics are weak, making measurements difficult.

In the current paper, taking into account the disadvantages of DAS and Stewart’s method, we propose a new method, which employs the 1st harmonic RAM signal to measure the IA rather than recovering the SA in measurements (according to DAS data analysis, accurate fitting of SA is not a necessary condition to measure gas temperature or concentration, we can also make a precise measurement if the IA is known). This method not only offers all the advantages of the Stewart’s method, and also improves the SNR because larger modulation indices are used. Meanwhile, numerical simulations and physical experiments are used to verify the reliability and accuracy of the proposed method.

2. Theory

The basic principle of DAS has been introduced in many studies [1

1. A. Farooq, J. B. Jeffries, and R. K. Hanson, “CO2 concentration and temperature sensor for combustion gases using diode-laser absorption near 2.7μm,” Appl. Phys. B 90(3–4), 619–628 (2008). [CrossRef]

, 6

6. X. Liu, J. B. Jeffries, R. K. Hanson, K. M. Hinckley, and M. A. Woodmansee, “Development of a tunable diode laser sensor for measurements of gas turbine exhaust temperature,” Appl. Phys. B 82(3), 469–478 (2006). [CrossRef]

], the SA can be inferred directly by using the ratio of the laser transmitted and incident intensities. Subsequently, an appropriate profile (Gaussian, Voigt or Lorentzian; refer to Appendix II) is used to fit the measured SA, and the IA is calculated according to the fitting function. For example, the species concentration can be determined by:
X=+α(v)dvPS(T)L=APS(T)L,
(1)
where X is the species concentration, P (atm) is the total pressure, S(T) (cm−2atm−1) is the line strength, L (cm) is the absorption length, α(v) and A (cm−1) are the SA and IA, respectively.

In Appendix I, we have derived the Y component expression of the 1st harmonic as:
Y=GΔI2[1+(H0H22)]sinψ,
(2)
where G is the electro-optical gain of the detection system, ΔI is the amplitude of intensity modulation, ψ is the phase shift between the intensity and frequency modulation. H0 and H2 are the harmonic coefficients, and their expressions are given in Eq. (15).

When there is no absorption or on both sides far away from the line center, the SA is zero, thus, H0 = H2 = 0, and the background of Y component can be written as:

Yback=GΔI2sinψ.
(3)

In actual measurements, the background signal can be used to normalize the Y component against perturbations to the modulation intensity and phase shift.

Λ=1YYback=H0H22.
(4)

Meanwhile, α(v¯+acosθ)in Eq. (15) can be expanded as a Taylor series around v¯as follows:

α(v¯+acosθ)=α(v¯)+n=1α(n)(v¯)(acosθ)nn!.
(5)

Substituting Eq. (5) into Eq. (15), H0 and H2 can be written as:

{H0=α(v¯)+α(2)(v¯)a24+α(4)(v¯)a464+α(6)(v¯)a62304+...H2=α(2)(v¯)a24+α(4)(v¯)a448+α(6)(v¯)a61536+....
(6)

Substituting Eq. (6) into Eq. (4), the Λ signal can be expressed as:

Λ=α(v¯)+n=11n+1(1n!)2(a2)2nα(2n)(v¯).
(7)

In literature [26

26. P. Zhimin, D. Yanjun, C. Lu, and Y. Qiansuo, “Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape,” Opt. Express 20(11), 11976–11985 (2012). [CrossRef] [PubMed]

], we pointed out that the Λ signal in Eq. (7) can be used to recover the SA under small modulation indices (m<0.5), but the recovering errors will increase sharply as modulation indices increase. In recent research, we found that if we integrate both sides of Eq. (7) into the whole frequency range, we can obtain the following equation:
Λdv¯=α(v¯)dv¯+n=11n+1(1n!)2(a2)2nα(2n)(v¯)dv¯,
(8)
where α(2n)(v¯)dv¯=0 (derivations are given in Appendix II), so we can determine IA by:
A=α(v¯)dv¯=Λdv¯,
(9)
where Λ is the signal obtained by using the date of Y component of the 1st harmonic. When the IA is known, the gas temperature and concentration can be determined like with DAS.

3. Simulation results

As we known, DAS uses the combination of Gaussian and Lorentzian profiles to fit the SA. Inspired by this thought, we have make a great number of simulations, and the calculation results show that the following function with the combination of Gaussian and Lorentzian profiles can be used to fit such Λ signals (usually n≤3), the fitting process will not affect real-time analysis like with DAS.
f(v¯)=p=1napexp[-(v¯v0)2cp2]+q=1naq(v¯v0)2+cq2,
(10)
where ap, aq, cp, and cq are the fitting parameters, and the integrated value of the function can be calculated by the following formula:

A=-Λdv¯=-f(v¯)dv¯=p=1napcpπ+q=1naqcqπ.
(11)

Using the fitting function, we can obtain the fitting curves of the Λ signals as shown in Figs. 1(a)1(c). Figures 1(a1)–(c1) give the fitting residuals between Λ3 signals and their fitting curves with different profiles. According to the fitting functions, the integrated values (denoted by A) of the Λ signals can be calculated, and the results are given in Table 1

Table 1. Integrated Values of the Λ Signals with Different Modulation Indices in Figs. 1(a)1(c)

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.

To investigate the efficacy of this method in dealing with asymmetrical conditions, we applied the proposed method to obtain the Λ signal when the lines overlap one another. Figure 2
Fig. 2 The Λ signals with different modulation indices under overlapping conditions, (a) is the Λ signals and their fitting curves, (b) and (c) are the fitting residuals.
shows the typical simulation results, where the fitting curves of Λ signals are represented by red curves. The integrated values (A1 and A2) of the Λ signals are given in Fig. 2(a), where A0 is the true IA, the relative errors between Ak (k = 1,2) and A0 are −1.26%, 2.54%, respectively.

4. Experiment results

CO2 and H2O molecules have a large number of absorption lines near 6981 cm–1 as shown in Fig. 3(a)
Fig. 3 CO2 and H2O absorption lines near 6981 cm–1, (a) is line strength at 296K, and (b) is the SA profiles (T = 296K P = 100mbar, L = 120.0cm, XCO2 = 20.0%, XH2O = 3.0%).
, where the lines A, B, and C are selected to measure the IA in the following experiments, the spectroscopy constants of the three lines are given in Table 2

Table 2. Spectroscopy Constants for the Selected Lines, Data Are Taken from HITRAN 2008 (296K)

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.

Figure 3(b) shows the SA profiles near 6981cm–1 according to the experiment conditions and spectroscopy constants, where the gas temperature, total pressure, absorption path, CO2 and H2O concentration are 296.0K, 100mbar, 120.0cm, 20.0% and 3.0%, respectively.

Figure 4
Fig. 4 Schematic of the experimental setup used for measuring IA of CO2 and H2O molecules under low absorption.
illustrates the experimental setup. To ensure gas pressure and temperature stability during measurements, the limited vacuum and vacuum leak rate of gas cell are reach 1.0 × 10−4 Pa and 4.0 × 10−5 Pa × m3/s, respectively. The gas cell is immersed in a water thermostatic bath (control accuracy: 0.01°C), the bath temperature can be set in the range of 5°C to 95°C. Prior to each measurement, the gas cell is evacuated by a vacuum pump with an ultimate pressure of 1.0 × 10−4 Pa, and then filled with a CO2-air mixture controlled by two mass flow controllers. A distributed feedback diode laser (NEL NLK1S5EAAA) with a center wavelength of 6981 cm−1 is used as the spectroscopic source, and the laser current and temperature are controlled by a commercial diode laser controller (ITC4001). Light from the fiber-coupled diode laser is passed to a fiber collimator and sent through the gas cell. The optical power exiting from the gas cell is detected using a large-surface Ge photodiode (PDA50B-EC). The detector signals are recorded in a high-speed memory data acquisition card and demodulated by a digital lock-in software (Labview). An external modulation consisting of a 5Hz triangular wave with fast 12,500Hz sinusoidal modulation is fed into the diode laser controller. The modulation depth, scan range, and phase shift of the reference signal are adjusted according to the experimental requirements.

As shown in Fig. 3(b), line A does not have any interference from other lines under low total gas pressure conditions. We can adjust the laser scanning range (0.21 cm–1) and allow it only to scan across the line A. The laser transmitted intensity, as well as the X and Y axes of the 1st harmonic with different modulation depths (a = 0, 0.89 × 10−2, 1.78 × 10−2, 2.67 × 10−2, 3.56 × 10−2, and 4.45 × 10−2 cm–1) are given in Figs. 5(a)
Fig. 5 Laser intensities, X and Y axes of the 1st harmonic with different modulation depths of the line A of CO2 (a = 0, 0.89 × 10−2, 1.78 × 10−2, 2.67 × 10−2, 3.56 × 10−2, and 4.45 × 10−2 cm–1).
5(f), where the total gas pressure, temperature, absorption length, and CO2 partial pressure are 101mbar, 296.5K, 120cm, and 21mbar, respectively. According to the experimental conditions, the theoretical IA, peak absorbance, and modulation indices can be calculated, and the values are approximately 3.647 × 10−3 cm–1, 11.0%, 0, 0.75, 1.50, 2.25, 3.00, and 3.75, respectively. Figure 5(a) is the DAS measurement signals, where the X and Y axes of the 1st harmonic are zero. However, the laser intensities, as well as the X and Y axes evidently change with increasing modulation depth. The background of Y component can be fitted using the data on the both sides far away from line center as shown in graph.

Using the experimental data in Figs. 5(a)5(f), the measurement results of the traditional DAS and proposed method are shown in Fig. 6
Fig. 6 Experimental results of DAS and proposed method (a = 0, 0.89 × 10−2, 1.78 × 10−2, 2.67 × 10−2, 3.56 × 10−2, and 4.45 × 10−2 cm–1), (a) is the simulation results according to the experimental conditions, and (b) is the fitting curves of the experimental data.
, where the average of 4 sequential raw data scans (single scan time is 0.2 second) are used to improve the SNR. In Fig. 6(a), the red dotted lines represent the simulation results with different modulation indices according to the experiment conditions, and the bottom graph gives the residuals between measurements and simulations. The experiment results indicate that the proposed method is effective and feasible. Meanwhile, to obtain the integrated values of the Λ signals, the fitting curves and fitting residuals are shown in Fig. 6(b).

To assess the SNR of the DAS and proposed method, a SNR is defined as follows, where σnoise is the standard deviation of Λ signal when there is no gas absorption [27

27. B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption TDLAS,” Appl. Phys. B 100(2), 367–376 (2010). [CrossRef]

].

SNR=max(Λsignal)3σnoise
(12)

According to experimental data in Fig. 6, the SNR of traditional DAS and Λ signals are approximately 29.7 (DAS), 69.5, 128.2, 108.9, 100.5, and 71.2, respectively. Calculation results show that the proposed method has a higher SNR than DAS, and the SNR of Λ signals increase with the increasing of modulation index, and then decrease.

Using the fitting curves, the integrated values of the measured SA (DAS) and Λ signals (proposed method) are given in Table 3

Table 3. Measurement Results of Integrated Values and CO2 Partial Pressure (theoretical IA is 3.647 × 10−3cm–1)

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, where the measurement error of DAS is about −0.90%. The errors of the proposed method decrease first and then increase, and this may be caused by the SNR of Λ signals, nonlinear frequency modulation, laser scanning range, and fitting errors. So in practical measurements, an extremely small or excessively large modulation index will increase the errors, an optimal modulation index is about 1.5. For example, the error is approximately −0.77% when the modulation index is 1.5, which is slightly better than DAS in the laboratory conditions. However, in harsh environments, the proposed method may have higher accuracy than DAS because high frequency can reduce many types of noises. Substituting the integrated values into Eq. (1), the partial pressures of CO2 are given in Table 3.

To compare with Stewart’s method, Fig. 7
Fig. 7 Experimental results of Stewart’s and proposed methods (a = 0.54 × 10−2, and 2.67 × 10−2 cm–1), (a) is the simulation results according to the experimental conditions, and (b) is the fitting curves of the experimental data.
gives typical measurement results of Stewart’s (m1 = 0.30) and proposed (m2 = 1.48) methods, where the gas temperature, absorption length, and CO2 partial pressure are the same as Fig. 6, the total gas pressure is increased to 202mbar. Experiment results shown that the Λ signal can be used to recover SA under small modulation indices (m<0.5). However, weak modulation will reduce the SNR, an optimal modulation index (e.g. m = 1.5) is desirable in practical measurement to improve the accuracy of gas temperature and concentration, as shown in Fig. 7(b), where δ is the measurement error of IA.

5. Conclusions

In this paper, we presented a method that can be used to measure the IA conveniently by using the 1st harmonic RAM signal. First, we mathematically proved that the IA can be directly determined using the Y component of the 1st harmonic when the absorption is optically thin. Meanwhile, the Gaussian, Voigt, and Lorentzian profiles are used to calculate the integrated values of Λ signals using numerical simulations. In subsequent experiments, the traditional DAS, Stewart’s and proposed method are employed to measure the IA of the transitions of CO2 and H2O molecules near 6981cm–1, and the measurement results are in accord with the expected. Here, it must be pointed out that the background of the 1st harmonic in proposed method will induce measurement errors, especially under very low absorption conditions (e.g. peak absorbance is less than 1.0%). Recently, Stewart’s group has introduced a method to reduce the background and obtained perfect measurement results [28

28. A. L. Chakraborty, K. Ruxton, W. Johnstone, M. Lengden, and K. Duffin, “Elimination of residual amplitude modulation in tunable diode laser wavelength modulation spectroscopy using an optical fiber delay line,” Opt. Express 17(12), 9602–9607 (2009). [CrossRef] [PubMed]

30

30. K. Ruxtona, A. L. Chakraborty, W. Johnstone, M. Lengden, G. Stewart, and K. Duffin, “Tunable diode laser spectroscopy with wavelength modulation: Elimination of residual amplitude modulation in a phasor decomposition approach,” Sens. Actuators B Chem. 150(1), 367–375 (2010). [CrossRef]

].

Appendix I

In WMS, a sinusoidal modulation of the angular frequency ω is riding on a slowly varying diode laser injection current, the instantaneous laser frequency and intensity can be written as:
{v=v¯+acos(ωt+η),I0=I¯+ΔIcos(ωt+η+ψ),
(13)
where a and ΔI are the amplitudes of modulation around v¯ and I¯, which are the slowly varying values of the laser frequency and intensity, η is phase shift of modulation frequency, ψ is the phase shift between the intensity and frequency modulation.

For optically thin samples (peak absorbance is less than 10.0%), the Beer-Lambert law can be expanded by a first-order Taylor series:
ItI0=exp[α(v¯+acosθ)]1α(v¯+acosθ)=1k=0Hkcos(kωt),
(14)
where It and I0 are the laser transmitted and incident intensities, respectively, Hk (k = 0,1,2…) are the harmonic coefficients, and their expressions are given as follows:

{H0=12ππ+πα(v¯+acosθ)dθHk=1ππ+πα(v¯+acosθ)coskθdθ,k=1,2...
(15)

Substituting I0 into Eq. (14), the laser transmitted intensity can be written as:
It=C00+k=1[Ck1cos(kωt)+Ck2sin(kωt)],
(16)
where C00, Ck1 and Ck2 (k = 1,2…) are given as:

{C00=I¯(1H0)ΔI2H1cosψ,C11=I¯H1+ΔI(1H0H22)cosψ,C12=ΔI(1+H0H22)sinψ,Ck1=I¯HkΔI2(Hk1+Hk+1)cosψ,Ck2=ΔI2(Hk1Hk+1)sinψ.
(17)

The reference signals used to detect the X and Y axes of the 1st harmonic can be written as:
{RX=cos(ωt+β),RY=sin(ωt+β).
(18)
where β is the phase shift of the reference signals. Multiplying Eqs. (16) and (18), the outputs of the X and Y components of the 1st harmonic can be written as follows, where G is the electro-optical gain of the detection system.

{X=G2{C11cos(η-β)+C12sin(ηβ)},Y=G2{C11sin(η-β)+C12cos(ηβ)}.
(19)

In practical measurements, we can change the phase shift β so that it is equal to η. And then, the Y component of the 1st harmonic can be written as:

Y=G2C12=GΔI2[1+(H0H22)]sinψ.
(20)

Appendix II

The SA can be described by the Voigt profile, and the function can be written as [31

31. Y. Y. Liu, J. L. Lin, G. M. Huang, Y. Q. Guo, and C. X. Duan, “Simple empirical analytical approximation to the Voigt profile,” J. Opt. Soc. Am. B 18(5), 666–672 (2001). [CrossRef]

]:
α(v¯)=αG(v¯)αL(v¯)=Qln2aπ3/2-exp[ln2(v'v0)2/ΔG2](v¯-v')2+ΔL2dv',
(21)
where αG(v¯) and αL(v¯) are the Gaussian and Lorentzian profiles, respectively, Q is absorption coefficient, v0 is line center, ΔvL and ΔvG are the HWHM of the Lorentzian and Gaussian profiles, Δv is the HWHM of the Voigt profile, and calculated by [32

32. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transf. 17(2), 233–236 (1977). [CrossRef]

].

Δv=0.5346ΔvL+0.2166ΔvL2+ΔvG2.
(22)

Based on the above analysis, the values of +α(2n)(v¯)dv¯in Eq. (8) in Section 2 can be written as following equation, according to the convolution properties.

+α(2n)(v¯)dv=+[αG(v¯)αL(v¯)](2n)dv¯=αG(v¯)(2n1)+αL(v¯)dv¯.
(23)

In Eq. (23), αL(v¯) is an even function and its first derivative is an odd function. Meanwhile, the value of 0+αL(v¯)dv¯ is convergent, so we can obtain:

-+αL(v¯)dv¯=0+α(2n)(v¯)dv¯=0
(24)

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 51206086 and 51176085.

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6.

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S. Hunsmann, K. Wunderle, S. Wagner, U. Rascher, U. Schurr, and V. Ebert, “Absolute, high resolution water transpiration rate measurements on single plant leaves via tunable diode laser absorption spectroscopy (TDLAS) at 1.37μm,” Appl. Phys. B 92(3), 393–401 (2008). [CrossRef]

8.

S. Schilt, “Impact of water vapor on 1.51 μm ammonia absorption features used in trace gas sensing applications,” Appl. Phys. B 100(2), 349–359 (2010). [CrossRef]

9.

Y. J. Ding, X. H. Li, Z. M. Peng, and L. Che, “Half-Width Integral Method for Gas Concentration Measuring in Tunable Diode Laser Absorption Spectroscopy,” Spectrosc. Lett. 46(7), 465–471 (2013). [CrossRef]

10.

P. Kluczynski, J. Gustafsson, A. M. Lindberg, and O. Axner, “Wavelength modulation absorption spectrometry —an extensive scrutiny of the generation of signals,” Spectrochim. Acta B 56(8), 1277–1354 (2001). [CrossRef]

11.

P. Kluczynski and O. Axner, “Theoretical description based on Fourier analysis of wavelength-modulation spectrometry in terms of analytical and background signals,” Appl. Opt. 38(27), 5803–5815 (1999). [CrossRef] [PubMed]

12.

J. A. Silver and D. J. Kane, “Diode laser measurements of concentration and temperature in microgravity combustion,” Meas. Sci. Technol. 10(10), 845–852 (1999). [CrossRef]

13.

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004). [CrossRef]

14.

F. Wang, K. F. Cen, N. Li, Q. X. Huang, X. Chao, J. H. Yan, and Y. Chi, “Simultaneous measurement on gas concentration and particle mass concentration by tunable diode laser,” Flow Meas. Instrum. 21(3), 382–387 (2010). [CrossRef]

15.

J. Henningsen and H. Simonsen, “Quantitative wavelength-modulation spectroscopy without certified gas mixtures,” Appl. Phys. B 70(4), 627–633 (2000). [CrossRef]

16.

H. Li, G. B. Rieker, X. Liu, J. B. Jeffries, and R. K. Hanson, “Extension of wavelength-modulation spectroscopy to large modulation depth for diode laser absorption measurements in high-pressure gases,” Appl. Opt. 45(5), 1052–1061 (2006). [CrossRef] [PubMed]

17.

H. Li, S. D. Wehe, and K. R. McManus, “Real-time equivalence ratio measurements in gas turbine combustors with a near-infrared diode laser sensor,” Proc. Combust. Inst. 33(1), 717–724 (2011). [CrossRef]

18.

A. Farooq, J. B. Jeffries, and R. K. Hanson, “Sensitive detection of temperature behind reflected shock waves using wavelength modulation spectroscopy of CO2 near 2.7μm,” Appl. Phys. B 96(1), 161–173 (2009). [CrossRef]

19.

G. B. Rieker, J. B. Jeffries, and R. K. Hanson, “Calibration-free wavelength-modulation spectroscopy for measurements of gas temperature and concentration in harsh environments,” Appl. Opt. 48(29), 5546–5560 (2009). [CrossRef] [PubMed]

20.

K. Sun, X. Chao, R. Sur, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation diode laser absorption spectroscopy for high-pressure gas sensing,” Appl. Phys. B 110(4), 497–508 (2013). [CrossRef]

21.

K. Duffin, A. J. McGettrick, W. Johnstone, G. Stewart, and D. G. Moodie, “Tunable diode laser spectroscopy with wavelength modulation: A calibration-free approach to the recovery of absolute gas absorption line-shapes,” J. Lightwave Technol. 25(10), 3114–3125 (2007). [CrossRef]

22.

A. J. McGettrick, K. Duffin, W. Johnstone, G. Stewart, and D. G. Moodie, “Tunable diode laser spectroscopy with wavelength modulation: A phasor decomposition method for calibration-free measurements of gas concentration and pressure,” J. Lightwave Technol. 26(4), 432–440 (2008). [CrossRef]

23.

L. Li, N. Arsad, G. Stewart, G. Thursby, B. Culshaw, and Y. D. Wang, “Absorption line profile recovery based on residual amplitude modulation and first harmonic integration methods in photoacoustic gas sensing,” Opt. Commun. 284(1), 312–316 (2011). [CrossRef]

24.

G. Stewart, W. Johnstone, J. R. P. Bain, K. Ruxton, and K. Duffin, “Recovery of absolute gas absorption line shapes using tunable diode laser spectroscopy with wavelength modulation—part 1: theoretical analysis,” J. Lightwave Technol. 29(6), 811–821 (2011).

25.

J. R. P. Bain, W. Johnstone, K. Ruxton, G. Stewart, M. Lengden, and K. Duffin, “Recovery of absolute gas absorption line shapes using tuneable diode laser spectroscopy with wavelength modulation—Part 2: Experimental investigation,” J. Lightwave Technol. 29(7), 987–996 (2011). [CrossRef]

26.

P. Zhimin, D. Yanjun, C. Lu, and Y. Qiansuo, “Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape,” Opt. Express 20(11), 11976–11985 (2012). [CrossRef] [PubMed]

27.

B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption TDLAS,” Appl. Phys. B 100(2), 367–376 (2010). [CrossRef]

28.

A. L. Chakraborty, K. Ruxton, W. Johnstone, M. Lengden, and K. Duffin, “Elimination of residual amplitude modulation in tunable diode laser wavelength modulation spectroscopy using an optical fiber delay line,” Opt. Express 17(12), 9602–9607 (2009). [CrossRef] [PubMed]

29.

A. L. Chakraborty, K. Ruxton, and W. Johnstone, “Influence of the wavelength-dependence of fiber couplers on the background signal in wavelength modulation spectroscopy with RAM-nulling,” Opt. Express 18(1), 267–280 (2010). [CrossRef] [PubMed]

30.

K. Ruxtona, A. L. Chakraborty, W. Johnstone, M. Lengden, G. Stewart, and K. Duffin, “Tunable diode laser spectroscopy with wavelength modulation: Elimination of residual amplitude modulation in a phasor decomposition approach,” Sens. Actuators B Chem. 150(1), 367–375 (2010). [CrossRef]

31.

Y. Y. Liu, J. L. Lin, G. M. Huang, Y. Q. Guo, and C. X. Duan, “Simple empirical analytical approximation to the Voigt profile,” J. Opt. Soc. Am. B 18(5), 666–672 (2001). [CrossRef]

32.

J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transf. 17(2), 233–236 (1977). [CrossRef]

OCIS Codes
(300.1030) Spectroscopy : Absorption
(300.6260) Spectroscopy : Spectroscopy, diode lasers

ToC Category:
Spectroscopy

History
Original Manuscript: September 3, 2013
Revised Manuscript: September 20, 2013
Manuscript Accepted: September 21, 2013
Published: September 27, 2013

Citation
Peng Zhimin, Ding Yanjun, Jia Junwei, Lan Lijuan, Du Yanjun, and Li Zheng, "First harmonic with wavelength modulation spectroscopy to measure integrated absorbance under low absorption," Opt. Express 21, 23724-23735 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23724


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References

  1. A. Farooq, J. B. Jeffries, and R. K. Hanson, “CO2 concentration and temperature sensor for combustion gases using diode-laser absorption near 2.7μm,” Appl. Phys. B90(3–4), 619–628 (2008). [CrossRef]
  2. Y. R. Sun, H. Pan, C. F. Cheng, A. W. Liu, J. T. Zhang, and S. M. Hu, “Application of cavity ring-down spectroscopy to the Boltzmann constant determination,” Opt. Express19(21), 19993–20002 (2011). [CrossRef] [PubMed]
  3. J. Chen, A. Hangauer, R. Strzoda, and M. C. Amann, “VCSEL-based calibration-free carbon monoxide sensor at 2.3 μm with in-line reference cell,” Appl. Phys. B102(2), 381–389 (2011). [CrossRef]
  4. R. Sur, T. J. Boucher, M. W. Renfro, and B. M. Cetegen, “In situ measurements of water vapor partial pressure and temperature dynamics in a PEM fuel cell,” J. Electrochem. Soc.157(1), B45–B53 (2010). [CrossRef]
  5. S. Wagner, B. T. Fisher, J. W. Fleming, and V. Ebert, “TDLAS-based in situ measurement of absolute acetylene concentrations in laminar 2D diffusion flames,” Proc. Combust. Inst.32(1), 839–846 (2009). [CrossRef]
  6. X. Liu, J. B. Jeffries, R. K. Hanson, K. M. Hinckley, and M. A. Woodmansee, “Development of a tunable diode laser sensor for measurements of gas turbine exhaust temperature,” Appl. Phys. B82(3), 469–478 (2006). [CrossRef]
  7. S. Hunsmann, K. Wunderle, S. Wagner, U. Rascher, U. Schurr, and V. Ebert, “Absolute, high resolution water transpiration rate measurements on single plant leaves via tunable diode laser absorption spectroscopy (TDLAS) at 1.37μm,” Appl. Phys. B92(3), 393–401 (2008). [CrossRef]
  8. S. Schilt, “Impact of water vapor on 1.51 μm ammonia absorption features used in trace gas sensing applications,” Appl. Phys. B100(2), 349–359 (2010). [CrossRef]
  9. Y. J. Ding, X. H. Li, Z. M. Peng, and L. Che, “Half-Width Integral Method for Gas Concentration Measuring in Tunable Diode Laser Absorption Spectroscopy,” Spectrosc. Lett.46(7), 465–471 (2013). [CrossRef]
  10. P. Kluczynski, J. Gustafsson, A. M. Lindberg, and O. Axner, “Wavelength modulation absorption spectrometry —an extensive scrutiny of the generation of signals,” Spectrochim. Acta B56(8), 1277–1354 (2001). [CrossRef]
  11. P. Kluczynski and O. Axner, “Theoretical description based on Fourier analysis of wavelength-modulation spectrometry in terms of analytical and background signals,” Appl. Opt.38(27), 5803–5815 (1999). [CrossRef] [PubMed]
  12. J. A. Silver and D. J. Kane, “Diode laser measurements of concentration and temperature in microgravity combustion,” Meas. Sci. Technol.10(10), 845–852 (1999). [CrossRef]
  13. J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B78(3–4), 503–511 (2004). [CrossRef]
  14. F. Wang, K. F. Cen, N. Li, Q. X. Huang, X. Chao, J. H. Yan, and Y. Chi, “Simultaneous measurement on gas concentration and particle mass concentration by tunable diode laser,” Flow Meas. Instrum.21(3), 382–387 (2010). [CrossRef]
  15. J. Henningsen and H. Simonsen, “Quantitative wavelength-modulation spectroscopy without certified gas mixtures,” Appl. Phys. B70(4), 627–633 (2000). [CrossRef]
  16. H. Li, G. B. Rieker, X. Liu, J. B. Jeffries, and R. K. Hanson, “Extension of wavelength-modulation spectroscopy to large modulation depth for diode laser absorption measurements in high-pressure gases,” Appl. Opt.45(5), 1052–1061 (2006). [CrossRef] [PubMed]
  17. H. Li, S. D. Wehe, and K. R. McManus, “Real-time equivalence ratio measurements in gas turbine combustors with a near-infrared diode laser sensor,” Proc. Combust. Inst.33(1), 717–724 (2011). [CrossRef]
  18. A. Farooq, J. B. Jeffries, and R. K. Hanson, “Sensitive detection of temperature behind reflected shock waves using wavelength modulation spectroscopy of CO2 near 2.7μm,” Appl. Phys. B96(1), 161–173 (2009). [CrossRef]
  19. G. B. Rieker, J. B. Jeffries, and R. K. Hanson, “Calibration-free wavelength-modulation spectroscopy for measurements of gas temperature and concentration in harsh environments,” Appl. Opt.48(29), 5546–5560 (2009). [CrossRef] [PubMed]
  20. K. Sun, X. Chao, R. Sur, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation diode laser absorption spectroscopy for high-pressure gas sensing,” Appl. Phys. B110(4), 497–508 (2013). [CrossRef]
  21. K. Duffin, A. J. McGettrick, W. Johnstone, G. Stewart, and D. G. Moodie, “Tunable diode laser spectroscopy with wavelength modulation: A calibration-free approach to the recovery of absolute gas absorption line-shapes,” J. Lightwave Technol.25(10), 3114–3125 (2007). [CrossRef]
  22. A. J. McGettrick, K. Duffin, W. Johnstone, G. Stewart, and D. G. Moodie, “Tunable diode laser spectroscopy with wavelength modulation: A phasor decomposition method for calibration-free measurements of gas concentration and pressure,” J. Lightwave Technol.26(4), 432–440 (2008). [CrossRef]
  23. L. Li, N. Arsad, G. Stewart, G. Thursby, B. Culshaw, and Y. D. Wang, “Absorption line profile recovery based on residual amplitude modulation and first harmonic integration methods in photoacoustic gas sensing,” Opt. Commun.284(1), 312–316 (2011). [CrossRef]
  24. G. Stewart, W. Johnstone, J. R. P. Bain, K. Ruxton, and K. Duffin, “Recovery of absolute gas absorption line shapes using tunable diode laser spectroscopy with wavelength modulation—part 1: theoretical analysis,” J. Lightwave Technol.29(6), 811–821 (2011).
  25. J. R. P. Bain, W. Johnstone, K. Ruxton, G. Stewart, M. Lengden, and K. Duffin, “Recovery of absolute gas absorption line shapes using tuneable diode laser spectroscopy with wavelength modulation—Part 2: Experimental investigation,” J. Lightwave Technol.29(7), 987–996 (2011). [CrossRef]
  26. P. Zhimin, D. Yanjun, C. Lu, and Y. Qiansuo, “Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape,” Opt. Express20(11), 11976–11985 (2012). [CrossRef] [PubMed]
  27. B. Lins, P. Zinn, R. Engelbrecht, and B. Schmauss, “Simulation-based comparison of noise effects in wavelength modulation spectroscopy and direct absorption TDLAS,” Appl. Phys. B100(2), 367–376 (2010). [CrossRef]
  28. A. L. Chakraborty, K. Ruxton, W. Johnstone, M. Lengden, and K. Duffin, “Elimination of residual amplitude modulation in tunable diode laser wavelength modulation spectroscopy using an optical fiber delay line,” Opt. Express17(12), 9602–9607 (2009). [CrossRef] [PubMed]
  29. A. L. Chakraborty, K. Ruxton, and W. Johnstone, “Influence of the wavelength-dependence of fiber couplers on the background signal in wavelength modulation spectroscopy with RAM-nulling,” Opt. Express18(1), 267–280 (2010). [CrossRef] [PubMed]
  30. K. Ruxtona, A. L. Chakraborty, W. Johnstone, M. Lengden, G. Stewart, and K. Duffin, “Tunable diode laser spectroscopy with wavelength modulation: Elimination of residual amplitude modulation in a phasor decomposition approach,” Sens. Actuators B Chem.150(1), 367–375 (2010). [CrossRef]
  31. Y. Y. Liu, J. L. Lin, G. M. Huang, Y. Q. Guo, and C. X. Duan, “Simple empirical analytical approximation to the Voigt profile,” J. Opt. Soc. Am. B18(5), 666–672 (2001). [CrossRef]
  32. J. J. Olivero and R. L. Longbothum, “Empirical fits to the Voigt line width: A brief review,” J. Quant. Spectrosc. Radiat. Transf.17(2), 233–236 (1977). [CrossRef]

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