Abstract

A variety of frequency-modulation methods for high-sensitivity absorption detection of gas-phase species has evolved in recent years. The distinctions among these methods are mostly semantic. The mathematical derivations for wavelength-modulation spectroscopy and one- and two-tone frequency-modulation spectroscopies are presented; a common terminology is used to permit a comprehensive comparison of predicted detection sensitivities. Applying this formalism, I compare the optimum detection sensitivities of these different methods for a typical laser system, using the same parameters. As long as residual amplitude modulation is minimized by proper adjustment of the detection phase angle, high-frequency wavelength modulation and one- and two-tone frequency-modulation methods all achieve approximately the same sensitivities. The choice among techniques is most strongly driven by the individual laser tuning characteristics, the absorption linewidth, and the detection bandwidth. It is shown that excess laser noise cannot always be excluded from consideration, even at megahertz detection frequencies. Also, detection at harmonics of the modulation or beat frequency may present certain advantages in minimizing residual amplitude-modulation noise.

© 1992 Optical Society of America

Full Article  |  PDF Article

Corrections

Joel A. Silver, "Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods: errata," Appl. Opt. 31, 4927-4927 (1992)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-31-24-4927

References

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  1. D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1186–1190 (1982)
    [CrossRef]
  2. F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
    [CrossRef]
  3. D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short-extended-cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
    [CrossRef] [PubMed]
  4. P. Werle, F. Slemr, M. Gehrtz, C. Bräuchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
    [CrossRef]
  5. C. B. Carlisle, D. E. Cooper, H. Prier, “Quantum noise-limited FM spectroscopy with a lead-salt diode laser,” Appl. Opt. 28, 2567–2576 (1989).
    [CrossRef] [PubMed]
  6. L. G. Wang, D. A. Tate, H. Riris, T. F. Gallagher, “High-sensitivity frequency-modulation spectroscopy with a GaAlAs diode laser,” J. Opt. Soc. Am. B 6, 871–876 (1989).
    [CrossRef]
  7. E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
    [CrossRef] [PubMed]
  8. P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
    [CrossRef]
  9. J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
    [CrossRef]
  10. W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
    [CrossRef]
  11. G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  12. D. E. Cooper, R. E. Warren, “Two-tone heterodyne spectroscopy with diode lasers: theory of line shapes and experimental results,” J. Opt. Soc. Am. B 4, 470–480 (1987).
    [CrossRef]
  13. E. A. Whittaker, C. M. Shum, H. Grebel, H. Lotem, “Reduction of residual amplitude modulation in frequency-modulation spectroscopy by using harmonic frequency modulation,” J. Opt. Soc. Am. B 5, 1253–1256 (1988).
    [CrossRef]
  14. G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1986).
    [CrossRef]
  15. G. Janik, C. Carlisle, T. F. Gallagher, “Frequency modulation spectroscopy with second harmonic detection,” Appl. Opt. 24, 3318–3319 (1985).
    [CrossRef] [PubMed]
  16. D. E. Cooper, T. F. Gallagher, “Double frequency modulation spectroscopy: high modulation frequency with low-bandwidth detectors,” Appl. Opt. 24, 1327–1334 (1985).
    [CrossRef] [PubMed]
  17. D. S. Bomse, J. A. Silver, A. C. Stanton, “Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser,” Appl. Opt. 31, 718–731 (1992).
    [CrossRef] [PubMed]
  18. L.-G. Wang, H. Riris, C. B. Carlisle, T. F. Gallagher, “Comparison of approaches to modulation spectroscopy with GaAlAs semiconductor lasers: application to water vapor,” Appl. Opt. 27, 2071–2077 (1988).
    [CrossRef] [PubMed]
  19. G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
    [CrossRef]
  20. R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
    [CrossRef]
  21. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972).
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    [CrossRef]
  23. M. Gehrtz, W. Lenth, A. T. Young, H. S. Johnston, “High frequency-modulation spectroscopy with a lead-salt diode laser,” Opt. Lett. 11, 132–134 (1986).
    [CrossRef] [PubMed]
  24. W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers.” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
    [CrossRef]
  25. J. A. Silver, A. C. Stanton, “Two-tone optical heterodyne spectroscopy using buried double heterostructure lead-salt diode lasers,” Appl. Opt. 27, 4438–4444 (1988).
    [CrossRef] [PubMed]
  26. D. E. Cooper, J. P Watjen, “Two-tone optical heterodyne spectroscopy with a tunable lead-salt diode laser,” Opt. Lett. 11, 606–608 (1986).
    [CrossRef] [PubMed]
  27. There is some ambiguity over the use of the symbol α in the FM literature. 10–12,24,26,28 in Cooper’s papers, α is used to represent the electric field attenuation coefficient, whereas Lenth and Bjorklund use δ for this purpose and α for the intensity attenuation. Experimental measurements measure attenuation of the laser intensity (which is proportional to the square of the electric field). This is conveniently expressed by Beer’s law, which relates the intensity attenuation to absorbance (using the symbol α) by I/I0 = exp(−α). We therefore use this convention by following the notation of Bjorklund et al.11 For consistency with all earlier notation, the electric field dispersion coefficient is denoted by ϕ.
  28. D. E. Cooper, R. E. Warren, “Frequency modulation spectroscopy with lead-salt diode lasers: a comparison of single-tone and two-tone techniques,” Appl. Opt. 26, 3726–3732 (1987).
    [CrossRef] [PubMed]
  29. R. D. Hudson, Infrared Systems Engineering (Wiley-Interscience, New York, 1969), p. 309.
  30. X. Ouyang, P. L. Varghese, “Reliable and efficient program for fitting Galatry and Voigt profiles to spectral data on multiple lines,” Appl. Opt. 28, 1538–1545 (1989).
    [CrossRef] [PubMed]
  31. M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985).
    [CrossRef]
  32. C. B. Carlisle, D. E. Cooper, “Tunable-diode-laser frequency-modulation spectroscopy using balanced homodyne detection,” Opt. Lett. 14, 1306–1308 (1989).
    [CrossRef] [PubMed]
  33. N. C. Wong, J. A. Hall, “High-resolution measurements of water-vapor overtone absorption in the visible by frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 6, 2300–2308 (1989).
    [CrossRef]
  34. J. Reid, M. El-Sherbiny, B. K. Garside, E. A. Ballik, “Sensitivity limits of a tunable diode laser spectrometer, with application to the detection of N02 at the 100-ppt level,” Appl. Opt. 19, 3349–3354 (1980).
    [CrossRef] [PubMed]
  35. J. A. Silver, A. C. Stanton, “Optical interference fringe reduction in laser absorption experiments,” Appl. Opt. 27, 1914–1916 (1988).
    [CrossRef] [PubMed]
  36. J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain setup conditions in multiple pass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
    [CrossRef] [PubMed]
  37. C. R. Webster, “Brewster-plate spoiler: a novel method for reducing the amplitude of interference fringes that limit tunable-laser absorption sensitivities,” J. Opt. Soc. Am. B 2, 1464–1470 (1985).
    [CrossRef]
  38. N.-Y. Chou, G. W. Sachse, L.-G. Wang, T. F. Gallagher, “Optical fringe reduction technique for FM laser spectroscopy,” Appl. Opt. 28, 4973–4975 (1989).
    [CrossRef] [PubMed]
  39. T. Iguchi, “Modulation waveforms for second-harmonic detection with tunable diode lasers,” J. Opt. Soc. Am. B 3, 419–423 (1986).
    [CrossRef]

1992 (1)

1990 (2)

1989 (7)

1988 (4)

1987 (2)

1986 (5)

1985 (4)

1984 (1)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers.” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

1983 (3)

W. Lenth, “Optical heterodyne spectroscopy with frequency-and amplitude-modulated semiconductor lasers,” Opt. Lett. 11, 575–577 (1983).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

1982 (2)

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1186–1190 (1982)
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
[CrossRef]

1981 (1)

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

1980 (1)

1977 (1)

1965 (1)

R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
[CrossRef]

1963 (1)

G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
[CrossRef]

Arndt, R.

R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
[CrossRef]

Ballik, E. A.

Bjorklund, G. C.

M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
[CrossRef]

Bomse, D. S.

Bräuchle, C.

P. Werle, F. Slemr, M. Gehrtz, C. Bräuchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
[CrossRef]

Bruce, D. M.

Carlisle, C.

Carlisle, C. B.

Cassidy, D. T.

D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short-extended-cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
[CrossRef] [PubMed]

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1186–1190 (1982)
[CrossRef]

Chou, N.-Y.

Chu, F.

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

Cooper, D. E.

El-Sherbiny, M.

Gallagher, T. F.

Garside, B. K.

Gehrtz, M.

Grebel, H.

Hall, J. A.

Harris, G. W.

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Hastie, D. R.

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Hudson, R. D.

R. D. Hudson, Infrared Systems Engineering (Wiley-Interscience, New York, 1969), p. 309.

Iguchi, T.

Janik, G.

Janik, G. R.

Johnston, H. S.

Kebabian, P. L.

Labrie, D.

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

Lenth, W.

M. Gehrtz, W. Lenth, A. T. Young, H. S. Johnston, “High frequency-modulation spectroscopy with a lead-salt diode laser,” Opt. Lett. 11, 132–134 (1986).
[CrossRef] [PubMed]

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers.” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

W. Lenth, “Optical heterodyne spectroscopy with frequency-and amplitude-modulated semiconductor lasers,” Opt. Lett. 11, 575–577 (1983).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
[CrossRef]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Lotem, H.

Mackay, G. I.

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

McManus, J. B.

Moses, E. I.

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
[CrossRef]

Ouyang, X.

Pokrowsky, P.

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

Prier, H.

Reid, J.

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1186–1190 (1982)
[CrossRef]

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

J. Reid, M. El-Sherbiny, B. K. Garside, E. A. Ballik, “Sensitivity limits of a tunable diode laser spectrometer, with application to the detection of N02 at the 100-ppt level,” Appl. Opt. 19, 3349–3354 (1980).
[CrossRef] [PubMed]

Riris, H.

Sachse, G. W.

Schiff, H. I.

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Shum, C. M.

Silver, J. A.

Slemr, F.

P. Werle, F. Slemr, M. Gehrtz, C. Bräuchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
[CrossRef]

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Stanton, A. C.

Tang, C. L.

Tate, D. A.

Varghese, P. L.

Wang, L. G.

Wang, L.-G.

Warren, R. E.

Watjen, J. P

Webster, C. R.

Werle, P.

P. Werle, F. Slemr, M. Gehrtz, C. Bräuchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
[CrossRef]

Whittaker, E. A.

Wilson, G. V. H.

G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
[CrossRef]

Wong, N. C.

Young, A. T.

Zapka, W.

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

Appl. Opt. (14)

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1186–1190 (1982)
[CrossRef]

D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short-extended-cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
[CrossRef] [PubMed]

C. B. Carlisle, D. E. Cooper, H. Prier, “Quantum noise-limited FM spectroscopy with a lead-salt diode laser,” Appl. Opt. 28, 2567–2576 (1989).
[CrossRef] [PubMed]

G. Janik, C. Carlisle, T. F. Gallagher, “Frequency modulation spectroscopy with second harmonic detection,” Appl. Opt. 24, 3318–3319 (1985).
[CrossRef] [PubMed]

D. E. Cooper, T. F. Gallagher, “Double frequency modulation spectroscopy: high modulation frequency with low-bandwidth detectors,” Appl. Opt. 24, 1327–1334 (1985).
[CrossRef] [PubMed]

D. S. Bomse, J. A. Silver, A. C. Stanton, “Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser,” Appl. Opt. 31, 718–731 (1992).
[CrossRef] [PubMed]

L.-G. Wang, H. Riris, C. B. Carlisle, T. F. Gallagher, “Comparison of approaches to modulation spectroscopy with GaAlAs semiconductor lasers: application to water vapor,” Appl. Opt. 27, 2071–2077 (1988).
[CrossRef] [PubMed]

J. A. Silver, A. C. Stanton, “Two-tone optical heterodyne spectroscopy using buried double heterostructure lead-salt diode lasers,” Appl. Opt. 27, 4438–4444 (1988).
[CrossRef] [PubMed]

D. E. Cooper, R. E. Warren, “Frequency modulation spectroscopy with lead-salt diode lasers: a comparison of single-tone and two-tone techniques,” Appl. Opt. 26, 3726–3732 (1987).
[CrossRef] [PubMed]

J. Reid, M. El-Sherbiny, B. K. Garside, E. A. Ballik, “Sensitivity limits of a tunable diode laser spectrometer, with application to the detection of N02 at the 100-ppt level,” Appl. Opt. 19, 3349–3354 (1980).
[CrossRef] [PubMed]

J. A. Silver, A. C. Stanton, “Optical interference fringe reduction in laser absorption experiments,” Appl. Opt. 27, 1914–1916 (1988).
[CrossRef] [PubMed]

J. B. McManus, P. L. Kebabian, “Narrow optical interference fringes for certain setup conditions in multiple pass absorption cells of the Herriott type,” Appl. Opt. 29, 898–900 (1990).
[CrossRef] [PubMed]

X. Ouyang, P. L. Varghese, “Reliable and efficient program for fitting Galatry and Voigt profiles to spectral data on multiple lines,” Appl. Opt. 28, 1538–1545 (1989).
[CrossRef] [PubMed]

N.-Y. Chou, G. W. Sachse, L.-G. Wang, T. F. Gallagher, “Optical fringe reduction technique for FM laser spectroscopy,” Appl. Opt. 28, 4973–4975 (1989).
[CrossRef] [PubMed]

Appl. Phys. B (3)

G. C. Bjorklund, M. D. Levenson, W. Lenth, C. Ortiz, “Frequency modulation (FM) spectroscopy: theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
[CrossRef]

P. Werle, F. Slemr, M. Gehrtz, C. Bräuchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers.” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

J. Appl. Phys. (2)

G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
[CrossRef]

R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
[CrossRef]

J. Geophys. Res. (1)

F. Slemr, G. W. Harris, D. R. Hastie, G. I. Mackay, H. I. Schiff, “Measurement of gas phase hydrogen peroxide in air by tunable diode laser absorption spectroscopy,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

J. Opt. Soc. Am. B (8)

Opt. Commun. (2)

P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Frequency modulation excitation spectroscopy,” Opt. Commun. 41, 369–373 (1982).
[CrossRef]

Opt. Lett. (5)

Other (3)

There is some ambiguity over the use of the symbol α in the FM literature. 10–12,24,26,28 in Cooper’s papers, α is used to represent the electric field attenuation coefficient, whereas Lenth and Bjorklund use δ for this purpose and α for the intensity attenuation. Experimental measurements measure attenuation of the laser intensity (which is proportional to the square of the electric field). This is conveniently expressed by Beer’s law, which relates the intensity attenuation to absorbance (using the symbol α) by I/I0 = exp(−α). We therefore use this convention by following the notation of Bjorklund et al.11 For consistency with all earlier notation, the electric field dispersion coefficient is denoted by ϕ.

R. D. Hudson, Infrared Systems Engineering (Wiley-Interscience, New York, 1969), p. 309.

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (3)

Fig. 1
Fig. 1

Gaussian spectra for one-tone FM detection as a function of modulation frequency. The nominal linewidth (HWHM) is 50 MHz.

Fig. 2
Fig. 2

Signal peak as a percentage of direct absorption signal for different FM methods. Solid and dashed curves denote absorption and dispersion signals, respectively, for one-tone nth harmonic detection. Solid circles represent two-tone FM signals for n = 1 and n = 2. The arrow along the abscissa denotes the absorption half-width.

Fig. 3
Fig. 3

Minimum detectable absorbance versus laser power for different FM techniques. Dotted curves represent signals dominated by RAM-induced noise; dashed curves denote signals limited by excess noise. The detection methods are listed by type (O is one-tone FM, T is two-tone FM, W is low frequency wavelength modulation, HW is high-frequency wavelength modulation) followed by a number corresponding to the detection harmonic. Signals that use dispersion rather than absorption are subscripted by the letter d.

Tables (5)

Tables Icon

Table I Computed Peak Heights and Optimal Modulation Indices for Wavelength-Modulation Spectroscopic Detection

Tables Icon

Table II Designation of Modulation Frequencies and Detection Harmonics Used in Calculations of Minimum Detectable Absorption Calculationsa

Tables Icon

Table III Typical Operating Characteristics for Various Diode-Laser Systems

Tables Icon

Table IV Laser System Parameters Used in Sensitivity Calculations

Tables Icon

Table V Minimum Detectable Absorbance for Different FM Conditions

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

x ω ω 1 / 2 , x m ω m ω 1 / 2 ,
x ( t ) = x 0 + m sin ( ω m t ) ,
S ( t ) = G [ x 0 + m sin ( ω m t ) ] .
I ( x 0 ) = ( 1 ) n n π m m d x G ( x 0 x ) T n ( x / m ) m 2 x 2 , 0 = 1 , n 0 = 2 ,
E ( t ) = E 0 exp [ i ω 0 t + i ϕ ( t ) ] , ϕ ( t ) = β sin ( ω m t ) .
ω ( t ) = ω 0 + βω m cos ( ω m t ) .
m = β x m .
E ( t ) = E 0 [ 1 + M sin ( ω m t + ψ ) ] exp [ i ω 0 t + i β sin ( ω m t ) ] .
I ( t ) = I 0 [ 1 + 2 M sin ( ω m t ) ] ,
M = | I 0 I max | 2 I 0 .
E ( t ) = E 0 exp ( i ω 0 t ) l = + r 1 exp ( i l ω m t ) ,
r l = k = 1 1 a k J l k ( β ) , a 0 = 1 , a ± 1 = ± M 2 i exp ( ± i ψ ) .
I = 2 I 0 [ Re ( Z ) cos ( θ ) Im ( Z ) sin ( θ ) ] , Z = l r l r l n * exp { 1 2 α ( ω 0 + l ω m ) 1 2 α [ ω 0 + ( l n ) ω m ] } exp { i φ ( ω 0 + l ω m ) + i φ [ ω 0 + ( l n ) ω m ] } ,
I = 2 I 0 cos ( θ ) l , m r l r m r l n * r * r m + n * exp { α [ ω 0 + ( l + m ) ω m ] } ,
I RAM ( 1 -tone ) = 2 I 0 l r l r l n * = 2 I 0 j , k = 1 1 a j a k * l J l j ( β ) J l k n ( β ) .
I RAM ( 1 -tone ) = 2 I 0 R ( M ) = { 2 M I 0 sin ( θ + ψ ) n = 1 1 2 M 2 I 0 cos ( θ + 2 ψ + π ) n = 2 0 n > 2 .
I RAM ( 2 -tone ) = 2 I 0 R ( M ) = { 2 M 2 I 0 cos ( θ ) n = 1 1 8 M 4 I 0 cos ( θ ) n = 2 0 n > 2 .
SNR = i s ( t ) 2 ¯ i s n 2 ¯ + i t h 2 ¯ + i RAM 2 ¯ + i e x 2 ¯ ,
SNR = ( e η h ν 0 ) 2 2 P 0 2 | Q ( α , ϕ ) | 2 2 e Δ f [ ( e η h ν 0 ) P 0 ( 1 + M 2 2 ) N + 2 k T eff e R L ] + ( e η h ν 0 ) 2 ( 2 R 2 ( M ) σ P 2 + Δ f f b σ ex 2 ) .

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