## Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape |

Optics Express, Vol. 20, Issue 11, pp. 11976-11985 (2012)

http://dx.doi.org/10.1364/OE.20.011976

Acrobat PDF (4815 KB)

### Abstract

In this paper, a new method for recovering gas absorbance shape using wavelength modulation spectroscopy is proposed. We have mathematically proven that the gas absorbance shape can be directly recovered using the data of *X* and *Y* components of odd harmonics, regardless of the value of the modulation depth. The transitions of NH_{3} near 1531 nm are selected to recover the absorbance shape using numerical simulation and experimental technique. The simulation and experiment results show that our proposed method can simply and accurately recover the gas absorbance shape.

© 2012 OSA

## 1. Introduction

1. 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]

4. N. Goldstein, S. Adler-Golden, J. Lee, and F. Bien, “Measurement of molecular concentrations and line parameters using line-locked second harmonic spectroscopy with an AlGaAs diode laser,” Appl. Opt. **31**(18), 3409–3415 (1992). [CrossRef] [PubMed]

5. A. Farooq, J. B. Jeffries, and R. K. Hanson, “CO_{2} 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. 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]

*f,*and other types of noise in actual measurements [7

7. E. D. Tommasi, G. Casa, and L. Gianfrani, “High precision determinations of NH_{3} concentration by means of diode laser spectrometry at 2.005 μm,” Appl. Phys. B **85**(2–3), 257–263 (2006). [CrossRef]

8. 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]

9. A. Farooq, J. B. Jeffries, and R. K. Hanson, “Sensitive detection of temperature behind reflected shock waves using wavelength modulation spectroscopy of CO_{2} near 2.7μm,” Appl. Phys. B **96**(1), 161–173 (2009). [CrossRef]

10. 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]

14. 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]

*X*and

*Y*components of the first harmonic to recover the gas absorbance shape, and works well under small modulation indices (

*m*<0.2) when the first harmonic profile is close to the true absorbance shape [10

10. 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]

*m*>0.5) are used. As their studies note, the recovering errors increase sharply as the modulation index increases in actual measurements [11

11. W. Johnstone, A. J. McGettrick, K. Duffin, A. Cheung, and G. Stewart, “Tunable diode laser spectroscopy for industrial process applications: System characterization in conventional and new approaches,” IEEE Sens. J. **8**(7), 1079–1088 (2008). [CrossRef]

12. 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 calibrationfree measurements of gas concentration and pressure,” J. Lightwave Technol. **26**(4), 432–440 (2008). [CrossRef]

14. 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]

*X*and

*Y*component expressions of each harmonic based on the absorption and harmonic theories. On this basis, we have mathematically proven that the gas absorbance shape can be recovered directly, when the data of

*X*and

*Y*components of the odd harmonics are processed by our method. Finally, to validate the precision of our method, transitions of NH

_{3}near 1531 nm are selected to recover the gas absorbance shape using numerical simulation and experimental techniques.

## 2. Theory

15. 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]

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

*ω*rides on a slowly varying diode laser injection current, and the instantaneous laser frequency and intensity may be expressed as:where

*a*and Δ

*I*are the amplitudes of modulation around

*ν*

_{1}and

*I*

_{1}, which are the slowly varying values of the laser frequency and intensity. Here,

*a*is named as modulation depth in the following chapter,

*ψ*

_{1}is the phase shift between the intensity and frequency modulation, which is strongly dependent on the modulation frequency. For the isolated transition, the modulation index can be defined as:

*m*=

*a/γ*, where

*γ*is the half width at half-maximum (HWHM) of the transition.

*ν*through a uniform medium is given by the Beer–Lambert relation and can be expanded as follows:where

*τ*(

*ν*) is the laser transmission,

*I*and

_{t}*I*

_{0}are the laser transmitted and incident intensities, respectively,

*α*(

*ν*) is the gas absorbance shape, and the functions

*H*(

_{k}*k*= 0,1,2…) are given as follows:

*I*

_{0}into Eq. (2), the laser transmitted intensity can be written as:where

*C*

_{00},

*C*

_{k}_{1}and

*C*

_{k}_{2}(

*k*= 1,2…) are given as:

*X*and

*Y*components of the

*k*th (

*k*= 1,2...) harmonic can be expressed as:where

*V*is the signal amplitude, and

*β*is the phase shift between the reference and detection signals. Multiplying Eq. (4) and (6), the outputs of the

*X*and

*Y*components of the

*k*th (

*k*= 1,2...) harmonic generated by the lock-in amplifier can be written as follows, where

*G*is the electro optical gain of the detection system.

*τ*(

*ν*) = 1), the

*H*

_{0}= 2 and

*H*= 0 (

_{k}*k*= 1,2…), the outputs of

*X*and

*Y*components of the first harmonic can be written as follows, where

*S*

_{1-}

*is the magnitude of the first harmonic when there is no gas absorption (defining as background signal).*

_{back}*X*and

*Y*components of the odd harmonics are multiplied by sin

*β*and cos

*β*, respectively, and are normalized by the background signal of the first harmonic, we can obtain the following equations:

*τ*(

*ν*

_{1}+

*a*cosθ) can be expanded as a Taylor series around any particular frequency

*ν*

_{1}as follows:

*can be written as follows:*

_{k}*m*=

*a/γ*) cannot be eliminated in Λ

_{1}(the data of

*X*and

*Y*components of the first harmonic are used). Clearly, when the modulation depth is larger, contributions to Λ

_{1}made by the even higher-order terms, especially the second- and fourth-order terms become greater, which is consistent with the conclusion of G. Stewart’s studies [8

8. 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]

12. 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 calibrationfree measurements of gas concentration and pressure,” J. Lightwave Technol. **26**(4), 432–440 (2008). [CrossRef]

_{2}) are used. Similarly, the second- and fourth-order terms can be eliminated in Λ

_{3}, and the second, fourth…up to 2(

*k*-1)-order terms can be eliminated in Λ

*as shown in Eq. (13). Meanwhile, the sum of the even higher-order terms [2*

_{k,}*k*, 2(

*k*+ 1)…order terms] in Λ

*can be near zero when*

_{k}*k*approaches infinity, as per the d'Alembert and Cauchy convergence principles, so we can obtain the following equation:

*Fun*are determined by data of the

_{k}*X*and

*Y*components of the odd harmonics in actual measurements, the phase shift

*ψ*

_{1}between the intensity and frequency modulation can be determined by using the method described in literature [19

19. 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]

## 3. Simulation results

_{3}/air mixture as the research object, and the transitions of NH

_{3}near 1531 nm are selected to recover the gas absorbance shape. The spectroscopic parameters of these transitions are shown in Table 1 [20

20. H. Jia, W. X. Zhao, T. D. Cai, W. D. Chen, W. J. Zhang, and X. M. Gao, “Absorption spectroscopy of ammonia between 6526 and 6538cm^{−1},” J. Quant. Spectrosc. Radiat. Transf. **110**(6–7), 347–357 (2009). [CrossRef]

*a*= 0) represents the calculation absorbance shape calculated by the spectroscopic parameters of the transitions, as shown in Table 1. Gas temperature, pressure, NH

_{3}concentration, absorption path length, and phase shift

*ψ*

_{1}adopted in the simulation are 296 K, 0.1 atm, 3.0%, 25.5 cm, and 45.5°, respectively. Meanwhile, HWHM of each transition is listed in Table 1. Figure 1(a)~1(d) reflect the characteristics of the gas absorbance shape recovered by Λ

*(*

_{k}*k*= 1, 2, 3, 4), which are close to the calculation absorbance shape at decreasing modulation depth. In Fig. 1(a), the absorbance shape recovered by Λ

_{1}only agrees well with the calculation value when the modulation depth is less than 0.008 cm

^{−1}(

*m*≈0.5), and the residual errors increase sharply with increasing modulation depth. Calculations show that the maximum error becomes higher than 25% when the modulation depth reaches 0.016 cm

^{−1}(

*m*≈1.0). This phenomenon can be attributed to the even higher-order terms, especially the second- and fourth-order terms (as shown in Eq. (13), which have made significant contributions to Λ

_{1}. The effects of these terms cannot be neglected at such a large modulation depth. Therefore, the method of G. Stewart et al. cannot be used to recover the gas absorbance shape, especially the modulation index is larger than 0.2.

_{2}is used. The simulation results show that the absorbance shape recovered by Λ

_{2}is in excellently agreement with the calculation value, even if the modulation depth reaches 0.016cm

^{−1}(

*m*≈1.0), as shown in Fig. 1(b). Clearly, the results in Figs. 1(c) and 1(d) show that the modulation depths have a very small influence on the recovery of results when Λ

_{3}and Λ

_{4}are used. For example, the maximum residual errors are no more than 1.75%, even if the modulation depths reach 0.024 cm

^{−1}(

*m*≈1.5) and 0.032 cm

^{−1}(

*m*≈2.0), as shown in Figs. 1(c) and 1(d), respectively. In actual measurements, the modulation index is close to 2.0. Hence, the application of Λ

_{3}or Λ

_{4}to recover the gas absorbance shape is highly accurate in many cases. In addition, simulation results have shown that recovery accuracy can be further improved if the higher odd harmonics (ninth, eleventh, and so on) are used.

_{3}concentration, absorption path length, and phase shift

*ψ*

_{1}adopted in the simulation are 296K, 1.0atm, 3.0%, 25.5cm, and 45.5°, respectively. HWHM of each transition is listed in Table 1. Similar to the characteristics in Fig. 1, Fig. 2 proves that the method established in the present paper can be used to recover a single transition and also can be applied under overlapping transitions.

## 4. Experimental results

^{−1}) distributed feedback diode laser (NEL NLK1S5EAAA) manufactured by NTT Electronics Company is used as the spectroscopic source. The laser current and the temperature are controlled by a commercial diode laser controller (ITC4001). Light from the fiber-coupled diode laser is passed to a fiber collimator (Thorlabs F280FC-1550) and sent through the gas cell (25.5 cm length). The laser wavelength is measured using a free-space NIR wavelength meter (Bristol 621B). The optical power exiting from the cell is detected using a large surface Ge photodiode. An external modulation consisting of a 20 Hz sawtooth ramp with a faster 10 kHz sinusoidal modulation is fed into the diode laser controller. The phase shift

*ψ*

_{1}between the intensity and frequency modulation can be determined using the method described in Ref [15

15. 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]

*β*between the reference and detection signals are adjusted to their optimum values according to the experimental requirements. The detector signals are recorded by a digital oscilloscope (DPO 4034B) and demodulated by a digital lock-in amplifier (SR830). Prior to each experiment, the gas cell is evacuated by a vacuum pump with an ultimate pressure of 1.0 Pa and filled with NH

_{3}-Air mixture controlled by two mass flow controllers.

*P*= 0.1 atm). Hence, in the experiments, we can adjust the laser scanning range and allow it to scan only Line 3. Figure 4 shows a typical experiment result, where the phase shift

*ψ*

_{1}is approximately 45.5° and the modulation depth is 0.03 cm

^{−1}. In the experiment, gas temperature, pressure, NH

_{3}concentration, absorption path length, and phase shift

*β*are 296 K, 0.1 atm, 3.0%, 25.5 cm and 135°, respectively. According to the experimental conditions, the modulation index can be calculated, and the obtained value is approximately 1.95. The data of

*X*and

*Y*components of the first, third, and fifth harmonics are shown in Fig. 4(a). The background of the first harmonic can be calculated using the data on both sides of

*X*

_{1}and

*Y*

_{1}, where no NH

_{3}absorption exists (

*X*

_{1-}

*= 0 mV,*

_{back}*Y*

_{1-}

*= −100 mV, and*

_{back}*S*

_{1-}

*= 100 mV). Subsequently, when the data of*

_{back}*X*and

*Y*components of the first, third, and fifth harmonics are processed using our method, we can recover the absorbance shape, as shown in Fig. 4(b), where the black solid line represents the calculation absorption shape (6529.184 cm

^{−1}), and the residual errors recovered by Λ

_{3}are shown in the bottom graph. In comparison with Fig. 1, the experiment results are in good agreement with the simulation results. The absorbance shapes recovered by Λ

_{1}and Λ

_{2}have become distorted because of the non-negligible effects of the second- and the fourth-order terms when the modulation index is approximately 1.95. However, the residual errors at the bottom figure graph show that the absorbance shape recovered by Λ

_{3}is close to the calculation value in the experiment.

_{3}concentration, absorption path length, and phase shift constants (

*T*= 296K,

*X*= 3.0%,

*L*= 25.5cm,

*ψ*

_{1}= 45.5°), then use Λ

_{4}to recover the absorbance shape under different pressure conditions (0.5 and 1.0 atm). As shown in Fig. 5 , the absorbance shapes recovered by Λ

_{4}are close to the calculation values because the second, fourth, and sixth-order terms have been completely eliminated. Furthermore, if the higher odd harmonics (ninth, eleventh, and so on) are collected in actual measurements, recovery accuracy can be improved.

## 5. Conclusions

_{3}near 1531 nm are selected to recover the absorbance shapes using the numerical simulation and experimental techniques. Results of the simulation and the experiment show that the method established in this paper can work well for recovering gas absorbance shape in actual measurements, regardless of the value of the modulation index.

## Acknowledgments

## References and links

1. | 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 |

2. | 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. |

3. | 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. |

4. | N. Goldstein, S. Adler-Golden, J. Lee, and F. Bien, “Measurement of molecular concentrations and line parameters using line-locked second harmonic spectroscopy with an AlGaAs diode laser,” Appl. Opt. |

5. | A. Farooq, J. B. Jeffries, and R. K. Hanson, “CO |

6. | J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with |

7. | E. D. Tommasi, G. Casa, and L. Gianfrani, “High precision determinations of NH |

8. | 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. |

9. | A. Farooq, J. B. Jeffries, and R. K. Hanson, “Sensitive detection of temperature behind reflected shock waves using wavelength modulation spectroscopy of CO |

10. | 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. |

11. | W. Johnstone, A. J. McGettrick, K. Duffin, A. Cheung, and G. Stewart, “Tunable diode laser spectroscopy for industrial process applications: System characterization in conventional and new approaches,” IEEE Sens. J. |

12. | 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 calibrationfree measurements of gas concentration and pressure,” J. Lightwave Technol. |

13. | 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. |

14. | 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. |

15. | P. Kluczynski and O. Axner, “Theoretical description based on Fourier analysis of wavelength-modulation spectrometry in terms of analytical and background signals,” Appl. Opt. |

16. | P. Zhimin, D. Yanjun, C. Lu, L. Xiaohang, and Z. Kangjie, “Calibration-free wavelength modulated TDLAS under high absorbance conditions,” Opt. Express |

17. | A. N. Dharamsi, “A theory of modulation spectroscopy with applications of higher harmonic detection,” J. Phys. D Appl. Phys. |

18. | J. Henningsen and H. Simonsen, “Quantitative wavelength-modulation spectroscopy without certified gas mixtures,” Appl. Phys. B |

19. | 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. |

20. | H. Jia, W. X. Zhao, T. D. Cai, W. D. Chen, W. J. Zhang, and X. M. Gao, “Absorption spectroscopy of ammonia between 6526 and 6538cm |

**OCIS Codes**

(300.1030) Spectroscopy : Absorption

(300.6260) Spectroscopy : Spectroscopy, diode lasers

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: April 24, 2012

Revised Manuscript: May 6, 2012

Manuscript Accepted: May 7, 2012

Published: May 10, 2012

**Citation**

Peng Zhimin, Ding Yanjun, Che Lu, and Yang Qiansuo, "Odd harmonics with wavelength modulation spectroscopy for recovering gas absorbance shape," Opt. Express **20**, 11976-11985 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11976

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### References

- 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]
- 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]
- 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]
- N. Goldstein, S. Adler-Golden, J. Lee, and F. Bien, “Measurement of molecular concentrations and line parameters using line-locked second harmonic spectroscopy with an AlGaAs diode laser,” Appl. Opt.31(18), 3409–3415 (1992). [CrossRef] [PubMed]
- 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]
- 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]
- E. D. Tommasi, G. Casa, and L. Gianfrani, “High precision determinations of NH3 concentration by means of diode laser spectrometry at 2.005 μm,” Appl. Phys. B85(2–3), 257–263 (2006). [CrossRef]
- 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]
- 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]
- 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]
- W. Johnstone, A. J. McGettrick, K. Duffin, A. Cheung, and G. Stewart, “Tunable diode laser spectroscopy for industrial process applications: System characterization in conventional and new approaches,” IEEE Sens. J.8(7), 1079–1088 (2008). [CrossRef]
- 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 calibrationfree measurements of gas concentration and pressure,” J. Lightwave Technol.26(4), 432–440 (2008). [CrossRef]
- 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).
- 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]
- 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]
- P. Zhimin, D. Yanjun, C. Lu, L. Xiaohang, and Z. Kangjie, “Calibration-free wavelength modulated TDLAS under high absorbance conditions,” Opt. Express19(23), 23104–23110 (2011). [CrossRef] [PubMed]
- A. N. Dharamsi, “A theory of modulation spectroscopy with applications of higher harmonic detection,” J. Phys. D Appl. Phys.29(3), 540–549 (1996). [CrossRef]
- J. Henningsen and H. Simonsen, “Quantitative wavelength-modulation spectroscopy without certified gas mixtures,” Appl. Phys. B70(4), 627–633 (2000). [CrossRef]
- 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]
- H. Jia, W. X. Zhao, T. D. Cai, W. D. Chen, W. J. Zhang, and X. M. Gao, “Absorption spectroscopy of ammonia between 6526 and 6538cm−1,” J. Quant. Spectrosc. Radiat. Transf.110(6–7), 347–357 (2009). [CrossRef]

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