First harmonic with wavelength modulation spectroscopy to measure integrated absorbance under low absorption

Peng Zhimin; Ding Yanjun; Jia Junwei; Lan Lijuan; Du Yanjun; Li Zheng

doi:10.1364/OE.21.023724

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 [1–5]. 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) [6–9] 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, 11] 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 [12–14] 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 [15–20]. For example, Hanson et al. [16–20] 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. [21–25] 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, 25] 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] 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, 6], 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 = \frac{\int_{- \infty}^{+ \infty} α (v) d v}{P S (T) L} = \frac{A}{P S (T) L},

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 = \frac{G Δ I}{2} \cdot [- 1 + (H_{0} - \frac{H_{2}}{2})] \sin ψ,

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:

Y_{back} = - \frac{G Δ I}{2} \sin ψ .

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

Λ = 1 - \frac{Y}{Y_{b a c k}} = H_{0} - \frac{H_{2}}{2} .

Meanwhile, $α (\bar{v} + a \cos θ)$ in Eq. (15) can be expanded as a Taylor series around $\bar{v}$ as follows:

α (\bar{v} + a \cos θ) = α (\bar{v}) + \sum_{n = 1}^{\infty} \frac{α^{(n)} (\bar{v}) {(a \cos θ)}^{n}}{n!} .

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

{\begin{cases} H_{0} = α (\bar{v}) + \frac{α^{(2)} (\bar{v}) a^{2}}{4} + \frac{α^{(4)} (\bar{v}) a^{4}}{64} + \frac{α^{(6)} (\bar{v}) a^{6}}{2304} + ... \\ H_{2} = \frac{α^{(2)} (\bar{v}) a^{2}}{4} + \frac{α^{(4)} (\bar{v}) a^{4}}{48} + \frac{α^{(6)} (\bar{v}) a^{6}}{1536} + .... \end{cases}

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

Λ = α (\bar{v}) + \sum_{n = 1}^{\infty} \frac{1}{n + 1} {(\frac{1}{n!})}^{2} {(\frac{a}{2})}^{2 n} α^{(2 n)} (\bar{v}) .

In literature [26], 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:

\int_{- \infty}^{\infty} Λ \cdot d \bar{v} = \int_{- \infty}^{\infty} α (\bar{v}) \cdot d \bar{v} + \sum_{n = 1}^{\infty} \frac{1}{n + 1} {(\frac{1}{n!})}^{2} {(\frac{a}{2})}^{2 n} \int_{- \infty}^{\infty} α^{(2 n)} (\bar{v}) \cdot d \bar{v},

where

\int_{- \infty}^{\infty} α^{(2 n)} (\bar{v}) \cdot d \bar{v} =0

(derivations are given in Appendix II), so we can determine IA by:

A = \int_{- \infty}^{\infty} α (\bar{v}) \cdot d \bar{v} = \int_{- \infty}^{\infty} Λ \cdot d \bar{v},

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

To validate the reliability and precision of the proposed method, the Gaussian, Voigt and Lorentzian profiles (refer to Appendix II) are taken as research objects. The simulation results are given in Fig. 1, where the abscissa is represented by ( $\bar{v}$ –v₀)/Δv, v₀ is the line center, Δv is the half width at half-maximum (HWHM) of SA. In Figs. 1(a)–1(c), black curves represent the true SA, and the integrated values of SA are 0.2131, 0.2706, and 0.3137Δv, respectively. The Λ signals with different modulation indices are recovered using the 1st RAM harmonics and marked as hollow circles. Calculation results show that we can employ the Λ₁ signals (m = 0.5) to recover the SA, and the recovering errors at line center are approximately −5.0%. However, the errors will increase as the modulation indices increase, this conclusion consistent with our and Stewart’s previous studies [21–26].

Fig. 1 Λ signals and their fitting curves with different modulation indices, where Δv_L and Δv_G represent the HWHM of the Lorentzian and Gaussian profiles, respectively, (a), (b) and (c) are the simulation results of Gaussian, Voigt and Lorentzian profile, respectively, (a1), (b1) and (c1) are the fitting residuals between Λ₃ signals and their fitting curves with different profiles.

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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 (\bar{v}) = \sum_{p = 1}^{n} a_{p} \cdot \exp [- \frac{{(\bar{v} - v_{0})}^{2}}{c_{p}^{2}}] + \sum_{q = 1}^{n} \frac{a_{q}}{{(\bar{v} - v_{0})}^{2} + c_{q}^{2}},

where a_p, a_q, c_p, and c_q are the fitting parameters, and the integrated value of the function can be calculated by the following formula:

A = \int_{- \infty}^{\infty} Λ \cdot d \bar{v} = \int_{- \infty}^{\infty} f (\bar{v}) \cdot d \bar{v} = \sum_{p = 1}^{n} a_{p} c_{p} \cdot \sqrt{π} + \sum_{q = 1}^{n} \frac{a_{q}}{c_{q}} \cdot π .

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 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 A_k (k = 1,2) and A₀ are −1.26%, 2.54%, respectively.

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.

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4. Experiment results

CO₂ and H₂O molecules have a large number of absorption lines near 6981 cm^–1 as shown in Fig. 3(a), 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.

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, X_CO2 = 20.0%, X_H2O = 3.0%).

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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 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.

Fig. 4 Schematic of the experimental setup used for measuring IA of CO₂ and H₂O molecules under low absorption.

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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)–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.

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).

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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, 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).

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.

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

SNR = \frac{\max (Λ s i g n a l)}{3 σ_{n o i s e}}

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, 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.

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

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To compare with Stewart’s method, Fig. 7 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.

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.

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To investigate the efficacy of the proposed method in dealing with lower absorption or asymmetrical conditions, Fig. 8(a) shows the measurement results of line A of CO₂ under lower absorption, where the total gas pressure, temperature, and absorption length are the same as those in Fig. 6, but the CO₂ partial pressure is only 7.1 mbar. According to the above parameters, the peak absorbance, theoretical IA, and modulation index are 3.83%, 1.233 × 10⁻³cm^–1, and 1.53, respectively. The integrated value of Λ signal is 1.220 × 10⁻³cm^–1, and the measurement error is approximately −1.05%. Likewise, Fig. 8(b) shows the measurement results for lines B and C of water vapor (saturated water vapor in laboratory air is about 2.86% at 296.5K), where the total gas pressure is 275mbar. The temperature and absorption length are the same as those in previous experiments. Using the fitting curve, the water vapor concentration can be calculated and the value is about 2.97%.

Fig. 8 Λ signals and their fitting curves under lower absorption and asymmetrical conditions, (a) is for line A of CO₂, and (b) is for lines B and C of H₂O.

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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 [28–30].

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:

{\begin{cases} v = \bar{v} + a \cos (ω t + η), \\ I_{0} = \bar{I} + Δ I \cos (ω t + η + ψ), \end{cases}

where a and ΔI are the amplitudes of modulation around

\bar{v}

and

\bar{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:

\frac{I_{t}}{I_{0}} = \exp [- α (\bar{v} + a \cos θ)] \approx 1 - α (\bar{v} + a \cos θ) = 1 - \sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t),

where I_t and I₀ are the laser transmitted and incident intensities, respectively, H_k (k = 0,1,2…) are the harmonic coefficients, and their expressions are given as follows:

{\begin{cases} H_{0} = \frac{1}{2 π} \int_{- π}^{+ π} α (\bar{v} + a \cos θ) \cdot d θ \\ H_{k} = \frac{1}{π} \int_{- π}^{+ π} α (\bar{v} + a \cos θ) \cos k θ \cdot d θ, k = 1, 2... \end{cases}

Substituting I₀ into Eq. (14), the laser transmitted intensity can be written as:

I_{t} = C_{00} + \sum_{k = 1}^{\infty} [C_{k 1} \cdot \cos (k ω t) + C_{k 2} \cdot \sin (k ω t)],

where C₀₀, C_k₁ and C_k₂ (k = 1,2…) are given as:

{\begin{cases} C_{00} = \bar{I} (1 - H_{0}) - \frac{Δ I}{2} H_{1} \cos ψ, \\ C_{11} = - \bar{I} H_{1} + Δ I (1 - H_{0} - \frac{H_{2}}{2}) \cos ψ, C_{12} = Δ I (- 1+ H_{0} - \frac{H_{2}}{2}) \sin ψ, \\ C_{k 1} = - \bar{I} H_{k} - \frac{Δ I}{2} (H_{k - 1} + H_{k + 1}) \cos ψ, C_{k 2} = \frac{Δ I}{2} (H_{k - 1} - H_{k + 1}) \sin ψ . \end{cases}

The reference signals used to detect the X and Y axes of the 1st harmonic can be written as:

{\begin{cases} R_{X} = \cos (ω t + β), \\ R_{Y} = \sin (ω t + β) . \end{cases}

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.

{\begin{cases} X = \frac{G}{2} \cdot {C_{11} \cdot \cos (η - β) + C_{12} \cdot \sin (η - β)}, \\ Y = \frac{G}{2} \cdot {- C_{11} \cdot \sin (η - β) + C_{12} \cdot \cos (η - β)} . \end{cases}

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 = \frac{G}{2} \cdot C_{12} = \frac{G Δ I}{2} \cdot [- 1 + (H_{0} - \frac{H_{2}}{2})] \sin ψ .

Appendix II

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

α (\bar{v}) = α_{G} (\bar{v}) * α_{L} (\bar{v}) = Q \cdot \frac{\sqrt{\ln 2 a}}{π^{3 / 2}} \int_{- \infty}^{\infty} \frac{\exp [- \ln 2 \cdot {(v^{'} - v_{0})}^{2} / Δ_{G}^{2}]}{{(\bar{v} - v^{'})}^{2} + Δ_{L}^{2}} d v^{'},

where

α_{G} (\bar{v})

and

α_{L} (\bar{v})

are the Gaussian and Lorentzian profiles, respectively, Q is absorption coefficient, v₀ is line center, Δv_L and Δv_G are the HWHM of the Lorentzian and Gaussian profiles, Δv is the HWHM of the Voigt profile, and calculated by [32].

Δ v = 0.5346 \cdot Δ v_{L} + \sqrt{0.2166 Δ v_{L}^{2} + Δ v_{G}^{2}} .

Based on the above analysis, the values of $\int_{- \infty}^{+ \infty} α^{(2 n)} (\bar{v}) d \bar{v}$ in Eq. (8) in Section 2 can be written as following equation, according to the convolution properties.

\int_{- \infty}^{+ \infty} α^{(2 n)} (\bar{v}) d v = \int_{- \infty}^{+ \infty} {[α_{G} (\bar{v}) * α_{L} (\bar{v})]}^{(2 n)} d \bar{v} = α_{G} {(\bar{v})}^{(2 n - 1)} * \int_{- \infty}^{+ \infty} α_{L}^{'} (\bar{v}) d \bar{v} .

In Eq. (23), $α_{L} (\bar{v})$ is an even function and its first derivative is an odd function. Meanwhile, the value of $\int_{0}^{+ \infty} α_{L}^{'} (\bar{v}) d \bar{v}$ is convergent, so we can obtain:

\int_{- \infty}^{+ \infty} α_{L}^{'} (\bar{v}) d \bar{v} = 0 \Rightarrow \int_{- \infty}^{+ \infty} α^{(2 n)} (\bar{v}) d \bar{v} = 0

Acknowledgments

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

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	Gaussian profile,		Voigt profile		Lorentzian profile
	A (Δv)	Error (%)	A (Δv)	Error (%)	A (Δv)	Error (%)
True IA	0.2131		0.2706		0.3137
Λ₁	0.2127	–0.19	0.2725	0.70	0.3140	0.10
Λ₂	0.2118	–0.61	0.2726	0.74	0.3149	0.38
Λ₃	0.2107	–1.13	0.2729	0.85	0.3161	0.77
Λ₄	0.2101	–1.41	0.2723	0.63	0.3164	0.86
Λ₅	0.2091	–1.88	0.2722	0.59	0.3157	0.64

	Lines	v₀ (cm^–1)	S₀ (cm^–1/molecule × cm^–2)	γ_self (cm^–1)	γ_air (cm^–1)	E″(cm^–1)
CO₂	A	6982.0679	5.931E–23	0.103	0.077	81.940
H₂O	B	6980.7554	1.009E–22	0.265	0.052	888.599
H₂O	C	6980.5414	4.034E–22	0.450	0.089	508.812

	DAS	Proposed Method
	DAS	Λ₁	Λ₂	Λ₃	Λ₄	Λ₅
A ( × 10⁻³cm^–1)	3.614	3.581	3.619	3.583	3.521	3.458
Error (%)	–0.90	–1.81	–0.77	–1.75	–3.45	–5.18
X_CO2 (mbar)	20.80	20.62	20.84	20.63	20.28	19.91

	Gaussian profile,		Voigt profile		Lorentzian profile
	A (Δv)	Error (%)	A (Δv)	Error (%)	A (Δv)	Error (%)
True IA	0.2131		0.2706		0.3137
Λ₁	0.2127	–0.19	0.2725	0.70	0.3140	0.10
Λ₂	0.2118	–0.61	0.2726	0.74	0.3149	0.38
Λ₃	0.2107	–1.13	0.2729	0.85	0.3161	0.77
Λ₄	0.2101	–1.41	0.2723	0.63	0.3164	0.86
Λ₅	0.2091	–1.88	0.2722	0.59	0.3157	0.64

	Lines	v₀ (cm^–1)	S₀ (cm^–1/molecule × cm^–2)	γ_self (cm^–1)	γ_air (cm^–1)	E″(cm^–1)
CO₂	A	6982.0679	5.931E–23	0.103	0.077	81.940
H₂O	B	6980.7554	1.009E–22	0.265	0.052	888.599
H₂O	C	6980.5414	4.034E–22	0.450	0.089	508.812

First harmonic with wavelength modulation spectroscopy to measure integrated absorbance under low absorption

Abstract

1. Introduction

2. Theory

3. Simulation results

4. Experiment results

5. Conclusions

Appendix I

Appendix II

Acknowledgments

References and links

Cited By

Figures (8)

Tables (3)

Equations (24)

Optics Express