Abstract

We present a comprehensive theory for heterodyne absorption spectroscopy with phase-modulated light. The general equations presented allow for an arbitrary modulation index and an arbitrary modulation frequency. We use this description for three purposes: First, we review the special cases of so-called frequency modulation and wavelength modulation spectroscopy. Second, we present the additional case of large-index, high-frequency modulation. Third, we present an overview of how the absorption signal depends on the experimental parameters of modulation frequency and modulation index. This overview may be helpful to experimentalists in choosing these parameters, for it provides a systematic understanding of how moving around in parameter space changes certain features of the signal, while leaving other features invariant.

© 1994 Optical Society of America

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  1. M. Cardona, Modulation Spectroscopy, Suppl. 11 of Solid State Physics, F. Seitz, D. Turnbull, H. Ehrenreich, eds. (Academic, New York, 1969), Chaps. 3–4, pp. 89–115.
  2. M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985).
    [CrossRef]
  3. E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (1983).
    [CrossRef]
  4. P. Pokrowsky, W. Zapka, F. Chu, G. C. Bjorklund, “High frequency wavelength modulation spectroscopy with diode lasers,” Opt. Commun. 44, 175–179 (1983).
    [CrossRef]
  5. G. C. Bjorklund, M. D. Levenson, “Sub-Doppler frequency-modulation spectroscopy of I2,” Phys. Rev. A 24, 166–169 (1981).
    [CrossRef]
  6. C. B. Carlisle, D. E. Cooper, H. Preier, “Quantum noise-limited FM spectroscopy with a lead–salt diode laser,” Appl. Opt. 28, 2567–2576 (1989).
    [CrossRef] [PubMed]
  7. W. Lenth, C. Ortiz, G. C. Bjorklund, “Pulsed frequency-modulation spectroscopy as a means for fast absorption measurements,” Opt. Lett. 6, 351–353 (1981).
    [CrossRef] [PubMed]
  8. E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
    [CrossRef] [PubMed]
  9. P. Pokrowsky, W. Herrmann, “Sensitive detection of hydrogen chloride by derivative spectroscopy with a diode laser,” in Laser Spectroscopy for Sensitive Detection, J. A. Gelbwachs, ed., Proc. Soc. Photo-Opt. Intrum.286, 33–38 (1981).
  10. E. D. Hinkley, P. L. Kelley, “Detection of air pollutants with tunable diode lasers,” Science 171, 635–639 (1971).
    [CrossRef] [PubMed]
  11. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980); >G. C. Bjorklund,, “Method and device for detecting a specific spectral feature,” U.S. patent4,297,035 (27October1981).
    [CrossRef] [PubMed]
  12. J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
    [CrossRef]
  13. W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
    [CrossRef]
  14. M. Gehrtz, W. Lenth, A. Y. Young, H. S. Johnston, “High-frequency-modulation spectroscopy with a lead–salt diode laser,” Opt. Lett. 11, 132–134 (1986).
    [CrossRef] [PubMed]
  15. A. Schenzle, R. G. DeVoe, R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
    [CrossRef]
  16. G. C. Bjorklund, M. D. Levenson, W. Length, C. Ortiz, “Frequency modulation (FM) spectroscopy–theory of line-shapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  17. W. Lenth, “Optical heterodyne spectroscopy with frequency and amplitide-modulation semiconductor lasers,” Opt. Lett. 8, 575–577(1983).
    [CrossRef] [PubMed]
  18. D. Hils, J. L. Hall, “Response of a Fabry–Perot cavity to phase modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
    [CrossRef]
  19. J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental techniques,” Appl. Opt. 6, 707–717 (1992).
    [CrossRef]
  20. This Fourier expansion is commonly used in modulation spectroscopy. For background material see, e.g., C. Louis Cuccia, Harmonics, Sidebands, and Transients in Communication Engineering (McGraw-Hill, New York, 1952), Chaps. 15 and 16, pp. 255–280.
  21. Lord Rayleigh, “On approximately simple waves,” Philos. Mag., 5th Ser. 50, 135–139 (1900).
    [CrossRef]
  22. D. E. Coooper, 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]
  23. Here we have used Jo2+2∑n=1∞Jn2=1.
  24. M. Ducloy, “Doppler-free two-photon heterodyne spectroscopy by single-beam frequency modulation,” Opt. Lett. 7, 432–433 (1982).
    [CrossRef] [PubMed]
  25. N. Nayak, G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
    [CrossRef] [PubMed]
  26. E. A. Whittaker, G. C. Bjorklund, “Developments in the theory of frequency modulation spectroscopy,” in Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 232.
  27. 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]
  28. T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28–30 (1982).
    [CrossRef] [PubMed]
  29. H. Lotem, “Extension of the spectral coverage range of frequency modulation spectroscopy by double frequency modulation,” J. Appl. Phys. 54, 6033–6035 (1983).
    [CrossRef]
  30. 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]
  31. P. Werle, F. Slemr, M. Gehrtz, C. Brauchle, “Quantum-limited FM-spectroscopy with a lead–salt diode laser,” Appl. Phys. B 49, 99–108 (1989).
    [CrossRef]
  32. N-Y. Chou, G. W. Sachse, “Single-tone and two-tone AM-FM spectral calculations for tunable diode laser absorption spectroscopy,” Appl. Opt. 26, 3584–3587 (1987).
    [CrossRef] [PubMed]
  33. D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with diode lasers: theory of line shapes and experimental results,” J. Opt. Soc. Am. B. 4, 470–480 (1987).
    [CrossRef]
  34. J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers—comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
    [CrossRef]
  35. G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
    [CrossRef]
  36. R. N. Hager, R. C. Anderson, “Theory of the derivative spectrometer,” J. Opt. Soc. Am. 60, 1444–1449 (1970).
    [CrossRef]
  37. G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), p. 151.
  38. N. M. Blachman, G. A. McAlpine, “The spectrum of a high-index FM waveform: Woodward’s theorem revisited,” IEEE Trans. Commun. Technol. COM-17, 201–208 (1969).
    [CrossRef]
  39. R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
    [CrossRef]
  40. H. Wahlquist, “Modulation broadening of unsaturated Lorentzian lines,” J. Chem. Phys. 35, 1708–1710 (1961).
    [CrossRef]

1992 (1)

J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental techniques,” Appl. Opt. 6, 707–717 (1992).
[CrossRef]

1989 (2)

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

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

1988 (1)

1987 (4)

D. E. Coooper, 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]

N-Y. Chou, G. W. Sachse, “Single-tone and two-tone AM-FM spectral calculations for tunable diode laser absorption spectroscopy,” Appl. Opt. 26, 3584–3587 (1987).
[CrossRef] [PubMed]

D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with diode lasers: theory of line shapes and experimental results,” J. Opt. Soc. Am. B. 4, 470–480 (1987).
[CrossRef]

D. Hils, J. L. Hall, “Response of a Fabry–Perot cavity to phase modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

1986 (1)

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

1985 (3)

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

N. Nayak, G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (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]

1984 (1)

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

1983 (5)

G. C. Bjorklund, M. D. Levenson, W. Length, 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 amplitide-modulation semiconductor lasers,” Opt. Lett. 8, 575–577(1983).
[CrossRef] [PubMed]

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (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]

H. Lotem, “Extension of the spectral coverage range of frequency modulation spectroscopy by double frequency modulation,” J. Appl. Phys. 54, 6033–6035 (1983).
[CrossRef]

1982 (3)

1981 (4)

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, “Sub-Doppler frequency-modulation spectroscopy of I2,” Phys. Rev. A 24, 166–169 (1981).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Pulsed frequency-modulation spectroscopy as a means for fast absorption measurements,” Opt. Lett. 6, 351–353 (1981).
[CrossRef] [PubMed]

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)

E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
[CrossRef] [PubMed]

1971 (1)

E. D. Hinkley, P. L. Kelley, “Detection of air pollutants with tunable diode lasers,” Science 171, 635–639 (1971).
[CrossRef] [PubMed]

1970 (1)

1969 (1)

N. M. Blachman, G. A. McAlpine, “The spectrum of a high-index FM waveform: Woodward’s theorem revisited,” IEEE Trans. Commun. Technol. COM-17, 201–208 (1969).
[CrossRef]

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]

1961 (1)

H. Wahlquist, “Modulation broadening of unsaturated Lorentzian lines,” J. Chem. Phys. 35, 1708–1710 (1961).
[CrossRef]

1900 (1)

Lord Rayleigh, “On approximately simple waves,” Philos. Mag., 5th Ser. 50, 135–139 (1900).
[CrossRef]

Agarwal, G. S.

N. Nayak, G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[CrossRef] [PubMed]

Anderson, R. C.

Arndt, R.

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

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

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]

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (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]

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

T. F. Gallagher, R. Kachru, F. Gounand, G. C. Bjorklund, W. Lenth, “Frequency-modulation spectroscopy with a pulsed dye laser,” Opt. Lett. 7, 28–30 (1982).
[CrossRef] [PubMed]

G. C. Bjorklund, M. D. Levenson, “Sub-Doppler frequency-modulation spectroscopy of I2,” Phys. Rev. A 24, 166–169 (1981).
[CrossRef]

W. Lenth, C. Ortiz, G. C. Bjorklund, “Pulsed frequency-modulation spectroscopy as a means for fast absorption measurements,” Opt. Lett. 6, 351–353 (1981).
[CrossRef] [PubMed]

G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980); >G. C. Bjorklund,, “Method and device for detecting a specific spectral feature,” U.S. patent4,297,035 (27October1981).
[CrossRef] [PubMed]

E. A. Whittaker, G. C. Bjorklund, “Developments in the theory of frequency modulation spectroscopy,” in Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 232.

Blachman, N. M.

N. M. Blachman, G. A. McAlpine, “The spectrum of a high-index FM waveform: Woodward’s theorem revisited,” IEEE Trans. Commun. Technol. COM-17, 201–208 (1969).
[CrossRef]

Brauchle, C.

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

Brewer, R. G.

A. Schenzle, R. G. DeVoe, R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Cardona, M.

M. Cardona, Modulation Spectroscopy, Suppl. 11 of Solid State Physics, F. Seitz, D. Turnbull, H. Ehrenreich, eds. (Academic, New York, 1969), Chaps. 3–4, pp. 89–115.

Carlisle, C. B.

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]

Coooper, D. E.

Cooper, D. E.

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

D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with diode lasers: theory of line shapes and experimental results,” J. Opt. Soc. Am. B. 4, 470–480 (1987).
[CrossRef]

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]

DeVoe, R. G.

A. Schenzle, R. G. DeVoe, R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Ducloy, M.

Gallagher, T. F.

Gehrtz, M.

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

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

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

Gounand, F.

Hager, R. N.

Hall, J. L.

D. Hils, J. L. Hall, “Response of a Fabry–Perot cavity to phase modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Herrmann, W.

P. Pokrowsky, W. Herrmann, “Sensitive detection of hydrogen chloride by derivative spectroscopy with a diode laser,” in Laser Spectroscopy for Sensitive Detection, J. A. Gelbwachs, ed., Proc. Soc. Photo-Opt. Intrum.286, 33–38 (1981).

Hils, D.

D. Hils, J. L. Hall, “Response of a Fabry–Perot cavity to phase modulated light,” Rev. Sci. Instrum. 58, 1406–1412 (1987).
[CrossRef]

Hinkley, E. D.

E. D. Hinkley, P. L. Kelley, “Detection of air pollutants with tunable diode lasers,” Science 171, 635–639 (1971).
[CrossRef] [PubMed]

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Johnston, H. S.

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

Kachru, R.

Kelley, P. L.

E. D. Hinkley, P. L. Kelley, “Detection of air pollutants with tunable diode lasers,” Science 171, 635–639 (1971).
[CrossRef] [PubMed]

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]

Length, W.

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

Lenth, W.

Levenson, M. D.

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

G. C. Bjorklund, M. D. Levenson, “Sub-Doppler frequency-modulation spectroscopy of I2,” Phys. Rev. A 24, 166–169 (1981).
[CrossRef]

Lotem, H.

H. Lotem, “Extension of the spectral coverage range of frequency modulation spectroscopy by double frequency modulation,” J. Appl. Phys. 54, 6033–6035 (1983).
[CrossRef]

Louis Cuccia, C.

This Fourier expansion is commonly used in modulation spectroscopy. For background material see, e.g., C. Louis Cuccia, Harmonics, Sidebands, and Transients in Communication Engineering (McGraw-Hill, New York, 1952), Chaps. 15 and 16, pp. 255–280.

McAlpine, G. A.

N. M. Blachman, G. A. McAlpine, “The spectrum of a high-index FM waveform: Woodward’s theorem revisited,” IEEE Trans. Commun. Technol. COM-17, 201–208 (1969).
[CrossRef]

Moses, E. I.

E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
[CrossRef] [PubMed]

Nayak, N.

N. Nayak, G. S. Agarwal, “Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation,” Phys. Rev. A 31, 3175–3182 (1985).
[CrossRef] [PubMed]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Length, 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, “Pulsed frequency-modulation spectroscopy as a means for fast absorption measurements,” Opt. Lett. 6, 351–353 (1981).
[CrossRef] [PubMed]

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]

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (1983).
[CrossRef]

P. Pokrowsky, W. Herrmann, “Sensitive detection of hydrogen chloride by derivative spectroscopy with a diode laser,” in Laser Spectroscopy for Sensitive Detection, J. A. Gelbwachs, ed., Proc. Soc. Photo-Opt. Intrum.286, 33–38 (1981).

Preier, H.

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

Rayleigh, Lord

Lord Rayleigh, “On approximately simple waves,” Philos. Mag., 5th Ser. 50, 135–139 (1900).
[CrossRef]

Reid, J.

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

Riris, H.

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Roche, K.

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (1983).
[CrossRef]

Sachse, G. W.

Schenzle, A.

A. Schenzle, R. G. DeVoe, R. G. Brewer, “Phase-modulation laser spectroscopy,” Phys. Rev. A 25, 2606–2621 (1982).
[CrossRef]

Silver, J. A.

J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental techniques,” Appl. Opt. 6, 707–717 (1992).
[CrossRef]

Slemr, F.

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

Tang, C. L.

E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
[CrossRef] [PubMed]

Wahlquist, H.

H. Wahlquist, “Modulation broadening of unsaturated Lorentzian lines,” J. Chem. Phys. 35, 1708–1710 (1961).
[CrossRef]

Wang, L-G.

Warren, R. E.

D. E. Coooper, 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]

D. E. Cooper, R. E. Warren, “Two-tone optical heterodyne spectroscopy with diode lasers: theory of line shapes and experimental results,” J. Opt. Soc. Am. B. 4, 470–480 (1987).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), p. 151.

Werle, P.

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

Whittaker, E. A.

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

E. A. Whittaker, P. Pokrowsky, W. Zapka, K. Roche, G. C. Bjorklund, “Improved laser technique for high sensitivity atomic absorption spectroscopy in flames,” J. Quant. Spectrosc. Radiat. Transfer 30, 289–296 (1983).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

Zapka, W.

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[CrossRef]

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[CrossRef]

Appl. Opt. (1)

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[CrossRef]

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[CrossRef]

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[CrossRef]

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H. Lotem, “Extension of the spectral coverage range of frequency modulation spectroscopy by double frequency modulation,” J. Appl. Phys. 54, 6033–6035 (1983).
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[CrossRef]

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[CrossRef]

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[CrossRef]

Opt. Commun. (1)

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

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E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Here we have used Jo2+2∑n=1∞Jn2=1.

This Fourier expansion is commonly used in modulation spectroscopy. For background material see, e.g., C. Louis Cuccia, Harmonics, Sidebands, and Transients in Communication Engineering (McGraw-Hill, New York, 1952), Chaps. 15 and 16, pp. 255–280.

G. N. Watson, A Treatise of the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1966), p. 151.

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Figures (3)

Fig. 1
Fig. 1

Probe spectrum. Moving from top to bottom, more sidebands become significant because M increases. Moving from left to right, the sideband spacing ω m increases.

Fig. 2
Fig. 2

Each subplot shows the FM absorption signal versus laser frequency. Each subplot has the same abscissa and ordinate scales. See Section 6 for further discussion.

Fig. 3
Fig. 3

Enlargements of three subplots from Fig. 2. The ordinate and abscissa scales vary, unlike in Fig. 2, where they were held constant for comparison. Γ = 1 and Ω = 0 throughout. (a) M = 0.1, ω m = 10. (b) M = 10, ω m = 10. (c) M = 0.1, ω m = 0.01.

Equations (27)

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E ( t ) = E o exp [ i ( ω o t + M sin ω m t ) ] = E o exp ( i ω o t ) n = - + J n ( M ) exp ( i n ω m t ) ,
E T ( t ) = E o exp ( i ω o t ) n = - T ( ω n ) J n ( M ) exp ( i n ω m t ) .
E ω m ( t ) 2 = E o 2 ( J o 2 exp ( - 2 δ o ) + n = 1 J n 2 × [ exp ( - 2 δ n ) + exp ( - 2 δ - n ) ] + 2 cos ω m t × n = 0 J n J n + 1 [ exp ( - 2 δ - n ) ( δ - n - 1 - δ - n ) ] + exp ( - 2 δ n ) ( δ n - δ n + 1 ) + exp ( - 2 δ n ) - exp ( - 2 δ - n ) ] + 2 sin ω m t × n = 0 J n J n + 1 [ exp ( - 2 δ - n ) ( ϕ - n - 1 - ϕ - n ) + exp ( - 2 δ n ) ( ϕ n + 1 - ϕ n ) ] ) ,
E ω m ( t ) 2 = E o 2 exp ( - 2 δ o ) [ 1 + 2 cos ω m t n = 0 J n J n + 1 × ( δ - n - 1 - δ n + 1 + δ - n - δ n ) + 2 sin ω m t × n = 0 J n J n + 1 ( ϕ - n - 1 - ϕ - n + ϕ n + 1 - ϕ n ) ] .
E 2 ( t ) = E o 2 exp ( - 2 δ o ) [ 1 + M ( δ - 1 - δ 1 ) cos ω m t + M ( ϕ - 1 + ϕ 1 - 2 ϕ o ) sin ω m t ] .
ω i ( t ) d d t ( ω o t + M sin ω m t ) = ω o + Δ F cos ω m t ,
I T ( ω ) = I o [ 1 - a ( ω o ) - d a d ω | ω o ( ω - ω o ) - 1 2 ! d 2 a d ω 2 | ω o × ( ω - ω o ) 2 - 1 3 ! d 3 a d ω 3 | ω o ( ω - ω o ) 3 ] .
I T ( ω i ) = I o [ 1 - a ( ω o ) - d a d ω | ω o Δ F cos ω m t - 1 2 ! d 2 a d ω 2 | ω o × ( Δ F ) 2 1 2 ( 1 + cos 2 ω m t ) - 1 3 ! d 3 a d ω 3 | ω o ( Δ F ) 3 × 1 4 ( 3 cos ω m t + cos 3 ω m t ) - ] .
I = - d a d ω | ω o Δ F .
δ - n - 1 - δ n + 1 + δ - n - δ n = - 2 ( 2 n + 1 ) Δ δ ,
I = - 4 Δ δ n = 0 ( 2 n + 1 ) J n ( M ) J n + 1 ( M ) .
I ( ω o ) = - 2 Δ δ ( ω o ) M .
I ( ω o ) = - 2 d δ d ω | ω o M ω m .
E T ( t ) = E o exp ( i ω o t ) { [ n k J n exp ( i n ω m t ) + J k T k exp ( i k ω m t ) ] } .
E T ( t ) = E o exp ( i ω o t ) { [ n = - J n exp ( i n ω m t ) + J k ( T k - 1 ) exp ( i k ω m t ) ] } .
E * E = E o 2 [ Σ J n exp ( - i n ω m t ) + J k ( T k - 1 ) * × exp ( - i k ω m t ) ] [ Σ J n exp ( i n ω m t ) + J k ( T k - 1 ) exp ( i k ω m t ) ] .
E ω m 2 = ( T k - 1 ) J k [ J k - 1 exp ( i ω m t ) + J k + 1 exp ( - i ω m t ) ] + J k ( T k - 1 ) * [ J k - 1 exp ( - i ω m t ) + J k + 1 exp ( i ω m t ) ] .
E * E = E o 2 J k [ - δ k 2 cos ω m t ( J k - 1 + J k + 1 ) + ϕ k 2 sin ω m t ( J k - 1 - J k + 1 ) ] .
E * E = E o 2 [ - δ k 4 k M J k 2 ( M ) cos ω m t + ϕ k 4 J k ( M ) J k ( M ) sin ω m t ] .
δ ( ω ) = A Γ 2 ( ω - Ω ) 2 + Γ 2 ,
I = exp ( - 2 δ o ) M ( δ - 1 - δ 1 ) = exp ( - 2 δ o ) M × [ A Γ 2 ( ω - ω m - Ω ) 2 + Γ 2 - A Γ 2 ( ω + ω m - Ω ) 2 + Γ 2 ] .
I max = M δ max = M A .
I k = δ max 4 k M J k 2 ( M ) .
I 1 = δ max 4 M J 1 2 ( M ) .
I = - 2 M ω m d δ d ω
I = 4 M ω m A Γ 2 ( ω - Ω ) [ ( ω - Ω ) 2 + Γ 2 ] 2 .
I max = 3 3 4 A M ω m Γ .

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