Calibration-free wavelength modulated TDLAS under high absorbance conditions

Peng Zhimin; Ding Yanjun; Che Lu; Li Xiaohang; Zheng Kangjie

doi:10.1364/OE.19.023104

1. Introduction

With the development of tunable diode laser-absorption spectroscopy (TDLAS) and with gas concentration measurement precision improvement, the traditional direct absorption method, easily affected by particles concentration, laser intensity fluctuations, and baseline-fitting errors often hamper the accurate determination of line shapes, which satisfy more difficult measurement requirements [1]. During the past ten years, recognizing the power of wavelength modulation spectroscopy (WMS) for highly sensitive measurements, many scientists have applied WMS in many areas to measure gas concentration and have achieved many valuable research results [2–4]. However, one of the key drawbacks of WMS is that the gas concentration cannot be obtained directly from measured harmonic signals in practical measurements. In the early stage of WMS, scientists use a first-order Taylor series to approximate laser transmission when the absorbance is less than 5.0%, derive that the second harmonic peak is a liner with gas concentration, and then determine the gas concentration through calibration experiments [5]. Recently, R. K. Hanson et al. [6,7] developed a method that enables “calibration-free” measurements using WMS, based on residual amplitude modulation (RAM). This method uses the first harmonic (1f) signal to normalize the second harmonic (2f) signal, and the absorption information such as gas concentration can be gained from the 2f/1f signal. More importantly, an expression that can be used to determine the absolute value of the gas concentration has been derived when the absorbance is less than 5.0% [8]. From the above analysis, at present the WMS seem to be limited to optically thin conditions (absorbance is less than 5.0%) [9], but this restriction precludes many interesting cases such as absorbance of more than 5.0%.

To improve the measuring accuracy of TDLAS and widen its scope of application in the industrial field, a second-order Taylor series is employed to approximate laser transmission, and an expression for measuring gas concentration has been derived based on the absorption spectral theory in this paper. Meanwhile, a second-order algorithm has been established, and it is used to determine the NH₃ mole fraction at different absorbance levels in the laboratory.

2. Derivation of second-order algorithm

The basic principle of WMS has been introduced in the literature [10,11]. Some of the basic theory will be presented here in order to explain the derivation process of the second-order algorithm in this study. In WMS, the laser transmission of monochromatic radiation at frequency v (cm⁻¹) through a uniform medium is given by the Beer–Lambert relation and can be expanded in a Fourier cosine series as follows:

τ (v) = \frac{I_{t}}{I_{0}} = \exp [- α (v)] = \exp [- P S (T) L X φ (v)] = \sum_{k = 0}^{\infty} A_{k} \cdot \cos (k ω t),

where τ(v) is the laser transmission; I_t and I₀ are the transmitted and incident laser intensities, respectively; and α(v) is the spectral absorbance. For the isolated transition α(v) = PLS(T)Xφ(v), P (atm) is the total gas pressure, S(T) (cm⁻²atm⁻¹) is the line strength, L (cm) is the optical absorbing path length, X is the mole fraction of the absorbing species, and φ(v) (cm) is the line-shape function for the absorption feature and expressed by the Voigt function [12]. By using the definition ξ = PLS(T), the functions A_k are given as

{\begin{cases} A_{0} = \frac{1}{2 π} \int_{- π}^{π} \exp [- ξ X φ (v)] \cdot d θ, \\ A_{k} = \frac{1}{π} \int_{- π}^{π} \exp [- ξ X φ (v)] \cdot \cos k θ \cdot d θ k = 1, 2, 3.... \end{cases}

The incident laser intensity is also simultaneously modulated and is given by [13]

I_{0} = {\bar{I}}_{0} (1 + i_{1} \cos (ω t + ψ_{1}) + i_{2} \cos (2 ω t + ψ_{2})),

where

{\bar{I}}_{0}

is the average laser intensity, and i₁, i₂, ψ₁ and ψ₂ are the characteristic parameters of the diode-laser. Substituting Eq. (3) into Eq. (1) for harmonic detection, the magnitude of the 1f and 2f signals at the line center of absorption feature can be written as [13]

{\begin{cases} S_{1 f} = \frac{G {\bar{I}}_{0}}{2} {{[i_{1} (A_{0} + \frac{A_{2}}{2}) \cos (ψ_{1})]}^{2} + {[i_{1} (A_{0} - \frac{A_{2}}{2}) \sin (ψ_{1})]}^{2}}^{1 / 2}, \\ S_{2 f} = \frac{G {\bar{I}}_{0}}{2} {{[A_{2} + i_{2} (A_{0} - 1 + \frac{A_{4}}{2}) \cos (ψ_{2})]}^{2} + {[i_{2} (A_{0} - 1 - \frac{A_{4}}{2}) \sin (ψ_{2})]}^{2}}^{1 / 2}, \end{cases}

where the 2f signal is recovered to reduce the RAM background signal, and G is electro-optical gain of the detection system. From Eq. (1), when there is no absorption (A₀ = 1, A_k = 0), the magnitude of the 1f signal can be written as follows, where

R_{1 f}

is the background of the 1f signal:

S_{1 f}^{0} = R_{1 f} = \frac{G {\bar{I}}_{0}}{2} \cdot i_{1} .

2.1 First-order algorithm—2f/1f calibration-free method

In Eq. (1), the laser transmission τ(v) can be expanded by a first-order Taylor series when the absorbance is less than 5.0%.

τ (v) = \exp [- ξ X φ (v)] \approx 1 - ξ X φ (v) = 1 - ξ X \cdot \sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t),

where

φ (v) = \sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t)

; H_k can be expressed as:

{\begin{cases} H_{0} = \frac{1}{2 π} \int_{- π}^{π} φ (v) \cdot d θ, \\ H_{k} = \frac{1}{π} \int_{- π}^{π} φ (v) \cdot \cos k θ \cdot d θ k = 1, 2, 3.... \end{cases}

Comparing Eq. (1) with Eq. (6), we can obtain the following relations:

{\begin{cases} A_{0} = 1 - ξ X H_{0}, \\ A_{k} = - ξ X H_{k} k = 1, 2, 3.... \end{cases}

In most applications, i₂<<1; substituting Eq. (8) into Eq. (4), the 1f, 2f, and 2f/1f signals at the line center can be simplified as

{\begin{cases} S_{1 f} \approx \frac{G {\bar{I}}_{0}}{2} i_{1} = R_{1 f} \\ S_{2 f} \approx - \frac{G {\bar{I}}_{0}}{2} ξ X H_{2} \end{cases} \Rightarrow S = \frac{S_{2 f}}{S_{1 f}} \approx - \frac{P S (T) L X H_{2}}{i_{1}} .

As shown in Eq. (9), the gas concentration (mole fraction) can be determined by the following equation, where the 2f/1f signal (S) is measured through experiments:

X = - \frac{S \cdot i_{1}}{P S (T) L H_{2}} .

2.2 Second-order algorithm

A first-order Taylor series is employed to approximate the laser transmission, which will bring larger approximation errors when the absorbance is more than 5.0%. Compared with the first-order Taylor approximation, the approximation errors will be reduced sharply when the laser transmission is approximated by a second-order Taylor series. For example, the error is only 0.56% when the absorbance reaches to 30.0%, and the laser transmission can be expanded as

τ (v) = \exp [- ξ X φ (v)] \approx 1 - ξ X \cdot \sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t) + \frac{ξ^{2} X^{2} \cdot {[\sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t)]}^{2}}{2},

where

φ (v) = \sum_{k = 0}^{\infty} H_{k} \cdot \cos (k ω t)

; defining that

\sum_{k = 0}^{\infty} T_{k} \cos (k ω t) = {[\sum_{k = 0}^{\infty} H_{k} \cos (k ω t)]}^{2}

, we can obtain the following equations:

{\begin{cases} T_{0} = \frac{1}{2} (H_{0}^{2} + \sum_{n = 0}^{\infty} H_{n}^{2}), \\ T_{k} = \frac{1}{2} \sum_{n = 0}^{k} H_{n} H_{k - n} + \sum_{n = 0}^{\infty} H_{n} H_{n + k} k =1,2,3 .... \end{cases}

Comparing Eq. (1) with Eq. (11), the relationships can be obtained as

{\begin{cases} A_{0} = 1 - ξ X H_{0} + \frac{ξ^{2} X^{2} T_{0}}{2}, \\ A_{k} = - ξ X H_{n} + \frac{ξ^{2} X^{2} T_{n}}{2} k = 1, 2, 3.... \end{cases}

Similarly, substituting Eq. (13) into Eq. (4), the 2f signal can be expressed as

S_{2 f} \approx \frac{G {\bar{I}}_{0}}{2} (- ξ X H_{2} + \frac{ξ^{2} X^{2}}{2} T_{2}),

where T₂ will attain convergence and stability quickly and can be expanded as

T_{2} = \frac{H_{1}^{2}}{2} + 2 H_{0} H_{2} + H_{1} H_{3} + H_{2} H_{4} + H_{3} H_{5} + ... + H_{n} H_{n + 2} n = 1, 2, 3....

To eliminate the errors caused by using S_1f instead of R_1f in the 2f/1f calibration-free method, the second-order algorithm uses the background of the 1f signal (R_1f) to determine the gas concentration, and the 2f/1f signal at the line center can be written as

R = \frac{S_{2 f}}{R_{1 f}} \approx \frac{- P S (T) L X H_{2} + {(P S (T) L)}^{2} X^{2} T_{2} / 2}{i_{1}} .

From Eq. (16), the gas concentration can be determined by the following equation when these parameters (i₁, P, S(T), L, H₂, T₂ and R) are known:

X = \frac{H_{2} + \sqrt{H_{2}^{2} + 2 T_{2} R \cdot i_{1}}}{P S (T) L T_{2}} .

3. Experimental results

So as to validate the measurement accuracy of the second-order algorithm, we take an NH₃-air mixture as the research object. The absorption transition at 6529.184 cm⁻¹ is selected to measure the NH₃ concentration, and the spectroscopic parameters are shown in Table 1 [14].

Table 1. Spectroscopic Parameters for the Selected NH₃ Transition at 6529.184 cm⁻¹

View Table

The schematic of the experimental setup is shown in Fig. 1 . The DFB diode laser (NEL NLK1S5EAAA) produced by NTT Electronics Company is used as the spectroscopic source. The laser current and temperature are controlled by a commercial diode laser controller (ITC4001). Light from the fiber-coupled diode laser emitting near 1531.5 nm (6529.5 cm-1) is passed to a fiber collimator (Thorlabs F280FC-1550) and sent through a gas cell (length is 25.5 cm). The optical power exiting from the cell is detected using a large surface (3 mm diameter) Ge photodiode. The laser is tuned to the line center wavelength 1531.585 nm (6529.184 cm⁻¹) of the selected transition using a free-space near-infrared (NIR) wavelength meter (Bristol 621B). Meanwhile, the diode laser is sinusoidally modulated by 1.0 kHz digital waveforms generated by a signal generator (AFG 3021B), and the modulation depths are adjusted to the optimum value (a = 0.0392 cm⁻¹). The detector signals are recorded by a digital oscilloscope (DPO 4034B) and demodulated by a digital lock-in software.

Fig. 1 Experimental schematic for measuring ammonia’s concentration.

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Prior to each experiment, the cell is evacuated by a vacuum pump to an ultimate pressure of 1.0 Pa to record the background of 1f and 2f signals. Then the cell is filled with the NH₃-air mixture controlled by the two mass flow controllers, and the NH₃ mole fraction can be chosen according to the experimental requirements. Figure 2 shows a typical experimental result. The cell is filled with a mixture gas of 10.0% NH₃, where the total gas pressure and temperature are 0.1 atm and 296K, respectively. Additionally, calculations show that the absorbance and modulation index are about 16.5% and 2.20 in the experiment.

Fig. 2 (a) Incident and transmitted laser intensity versus time; (b) n-f signal amplitudes from the data of curve A and B (P = 0.1atm, T = 296K, L = 25.5cm, X = 10.0%, a = 0.0392cm⁻¹).

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In Fig. 2(a), curves A and B are the incident (no NH₃ absorption) and transmitted (with NH₃ absorption) laser intensities recorded by the digital oscilloscope, and the laser parameter i₁ is about 0.136, which can be determined by curve A. Meanwhile, in Fig. 2(b), a discrete Fourier transform (DFT) is used to gain the n-f signal amplitudes from the data of curves A and B, and the even harmonic amplitudes have been deducted the background signals. The result of the DFT shows that the 1f signal amplitude is very large (R_1f = 459) and the 2f signal amplitude (R_2f = 5.0) is very small under no-NH₃ absorption conditions. However, the 1f signal amplitude decreases gradually (S_1f = 444) and the 2f signal amplitude increases rapidly (S_2f = 194) when the laser is absorbed by NH₃. Here, we can obtain the 2f/1f signals as follows: S = S_2f /S_1f = 0.437, R = S_2f /R_1f = 0.423. Moreover, the values of H₂ and T₂ can be calculated according to the line-shape function and the modulation index (H₂ = −7.56, T₂ = −145.51). Substituting these parameters (S, R, $i_{1}$ , P, S(T), L, H₂ and T₂) into Eq. (11) and Eq. (18), we can infer that the NH₃ mole fractions are about 9.574% and 10.069%, respectively.

In the following experiments, fixed laser parameters, temperature, and pressure remain unchanged (a = 0.0392cm⁻¹, i₁ = 0.136, P = 0.1atm, T = 296 K). The NH₃ mole fraction (from 1.0% to 26.0%) is controlled by the mass flow controllers, and the absorbance changes from 2.0% to 30.0%. Figure 3(a) compares the known NH₃ mole fraction with the measured fraction determined by the first-order and second-order algorithms. Moreover, the absorbance is also plotted in Fig. 3(a) as a function of the NH₃ mole fraction.

Fig. 3 (a) Measured fraction of the first-order and second-order algorithms; (b) measurement errors of NH₃ mole fraction (i₁ = 0.136, P = 0.1atm, T = 296K, L = 25.5cm, a = 0.0392cm⁻¹).

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In Fig. 3(a), the NH₃ mole fractions determined by the first-order and second-order algorithms are in good agreement with the actual values when the absorbance is less than 5.0%. However, whether with the first- or second-order algorithm, the deviations between the measured and actual value increases as absorbance increases, and the former’s deviations are much larger than the latter’s. For example, when the absorbance is 9.41% (X_NH3 = 5.0%), the measured fractions determined by the first-order and second-order algorithms are about 4.878% and 5.021%, respectively. In Fig. 3(b), the deviations of the experimental results from the known NH₃ mole fraction are clearly seen. For example, the relative error does not exceed 2.10%, and even the absorbance is about 30.61% (X_NH3 = 26.0%) when the second-order algorithm is used.

4. Conclusions

A second-order algorithm for measuring gas concentrations is reported in this paper. First, a second-order Taylor series is employed to approximate the laser transmission, and the expression for measuring gas concentrations is derived based on the absorption spectral theory. Second, a second-order algorithm with high precision and wide application is established. Finally, to validate the measurement accuracy of this algorithm, the NH3 mole fractions are determined by the first-order and second-order algorithms, respectively. Compared with the first-order algorithm, the second-order algorithm has greatly improved the gas concentration measurement accuracy and also broadened the scope of application of TDLAS.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 51176085 and China Postdoctoral Science Foundation No 2011M500301.

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Calibration-free wavelength modulated TDLAS under high absorbance conditions

Abstract

1. Introduction

2. Derivation of second-order algorithm

2.1 First-order algorithm—2f/1f calibration-free method

2.2 Second-order algorithm

3. Experimental results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Tables (1)

Equations (17)

Optics Express