Abstract

The capability of retrieving spectral information from line shapes recorded by two-tone frequency-modulation spectroscopy (TTFMS) is investigated. A TTFMS theory accounting for dispersion and nonlinear distortion of diode laser frequency modulation response is presented. The adequacy of the theory for a detailed modeling of line shapes recorded with high resolution is examined. An extensive error analysis of line parameters (i.e., width, intensity, and line center) retrieved by a nonlinear least-squares fitting procedure is made. Plots of residual errors with characteristic signatures that are due to incorrectly assigned modulation parameters and choice of line profile are presented. In least-squares fits to experimental oxygen data with a Voigt profile influence from collisional (Dicke) narrowing is clearly exhibited, and when we used a collisionally narrowed line profile deviations of the model were reduced to less than 0.2%. We demonstrate that accurate quantitative measurements by TTFMS over a wide range of concentrations, temperatures, and pressures are possible.

© 1996 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  61. M. Imai, K. Kawakita, “Measurement of direct frequency modulation characteristics of laser diodes by Michelson interferometry,” Appl. Opt. 29, 348–353 (1990).
    [CrossRef] [PubMed]

1995

D. C. Benner, C. P. Rinsland, V. M. Devi, M. A. H. Smith, D. Atkins, “A multispectrum nonlinear least squares fitting technique,” J. Quant. Spectrosc. Radiat. Transfer 53, 705–721 (1995).
[CrossRef]

1994

1993

L. C. Philippe, R. K. Hanson, “Laser diode wavelength-modulation spectroscopy for simultaneous measurement of temperature, pressure, and velocity in shock-heated oxygen flows,” Appl. Opt. 32, 6090–6103 (1993).
[CrossRef] [PubMed]

C. L. Brummel, L. A. Philips, “Error analysis in rotationally resolved spectra: least-squares and Monte Carlo methods,” J. Mol. Spectrosc. 159, 287–299 (1993).
[CrossRef]

M. P. Arroyo, R. K. Hanson, “Absorption measurements of water-vapor concentration, temperature, and line-shape parameters using tunable InGaAsP diode laser,” Appl. Opt. 32, 6104–6116 (1993).
[CrossRef] [PubMed]

V. G. Avetisov, A. I. Nadezhdinskii, A. N. Khusnutdinov, P. M. Omarova, M. V. Zyrianov, “Diode laser spectroscopy of water vapor in 1.8 μm: line profile measurements,” J. Mol. Spectrosc. 160, 326–334 (1993).
[CrossRef]

A. S. Pine, V. N. Markov, G. Buffa, O. Tarrini, “N2, O2, H2, Ar, and He broadening in the ν1 band of NH3,” J. Quant. Spectrosc. Radiat. Transfer 50, 337–348 (1993).
[CrossRef]

F. S. Pavone, M. Inguscio, “Frequency and wavelength modulation spectroscopies: comparison of experimental methods using an AlGaAs diode laser,” Appl. Phys. B 56, 118–122 (1993).
[CrossRef]

1992

J. A. Silver, “Frequency modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
[CrossRef] [PubMed]

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

T. Giesen, R. Schieder, G. Winnewisser, K. M. T. Yamada, “Precise measurements of pressure broadening and shift for several H2O lines in the ν2 band by argon, nitrogen, oxygen, and air,” J. Mol. Spectrosc. 153, 406–418 (1992).
[CrossRef]

D. S. Baer, R. K. Hanson, “Tunable diode laser absorption diagnostics for atmospheric pressure plasmas,” J. Quant. Spectrosc. Radiat. Transfer 47, 455–475 (1992).
[CrossRef]

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[CrossRef]

V. Dana, J.-Y. Mandin, C. Camy-Peyret, J.-M. Flaud, J. P. Chevillard, R. L. Hawkins, J.-L. Delfau, “Measurements of collisional linewidths in the ν2 band of H2O from Fourier-transformed flame spectra,” Appl. Opt. 31, 1928–1936 (1992).
[CrossRef] [PubMed]

A. S. Pine, “Self-, N2, O2, H2, and He broadening in the ν3 band Q branch of CH4,” J. Chem. Phys. 97, 773–785 (1992).
[CrossRef]

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

S. Adler-Golden, J. Lee, N. Goldstein, “Diode laser measurements of temperature dependent line parameters for water vapor near 820 nm,” J. Quant. Spectrosc. Radiat. Transfer 48, 527–535 (1992).
[CrossRef]

1991

G. Morthier, F. Libbrecht, K. David, P. Vankwikelberge, R. G. Baets, “Theoretical investigation of the second-order harmonic distortion in the AM response of 1.55 μm F-P and DFB lasers,” IEEE J. Quantum Electron. 27, 1990–2002 (1991).
[CrossRef]

H. Riris, C. B. Carlisle, D. E. Cooper, L. Wang, T. F. Gallagher, R. H. Tipping, “Measurement of the strengths of 1–0 and 3–0 transitions of HI using frequency modulation spectroscopy,” J. Mol. Spectrosc. 146, 381–388 (1991).
[CrossRef]

T. J. Johnson, F. G. Wienhold, J. P. Burrows, G. W. Harris, “Frequency modulation spectroscopy at 1.3 μm using InGaAsP lasers: a prototype field instrument for atmospheric chemistry research,” Appl. Opt. 30, 407–413 (1991).
[CrossRef] [PubMed]

1990

C. B. Carlisle, D. E. Cooper, “Tunable diode laser frequency modulation spectroscopy through an optical fiber: high sensitivity detection of water vapor,” Appl. Phys. Lett. 56, 805–807 (1990).
[CrossRef]

M.-S. Lin, S.-Y. J. Wang, N. K. Dutta, “Measurements and modeling of the harmonic distortion in InGaAsP distributed feedback lasers,” IEEE J. Quantum Electron. 26, 998–1004 (1990).
[CrossRef]

M. Imai, K. Kawakita, “Measurement of direct frequency modulation characteristics of laser diodes by Michelson interferometry,” Appl. Opt. 29, 348–353 (1990).
[CrossRef] [PubMed]

1989

N. C. Wong, J. L. Hall, “High-resolution measurement of water-vapor overtone absorption in the visible by frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 6, 2300–2308 (1989).
[CrossRef]

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]

P. Werle, F. Slemr, M. Gehrtz, C. Brauchle, “Quantum-limited FM-spectroscopy with a lead-salt diode laser,” Appl. Phys. B 33, 99–108 (1989).
[CrossRef]

C. B. Carlisle, D. E. Cooper, “Tunable-diode-laser frequency-modulation spectroscopy using balanced homodyne detection,” Opt. Lett. 14, 1306–1308 (1989).
[CrossRef] [PubMed]

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

M. Carlotti, “Global-fit approach to the analysis of limb-scanning atmospheric measurements,” Appl. Opt. 27, 3250–3254 (1988).
[CrossRef] [PubMed]

A. C. Stanton, J. A. Silver, “Measurements in the HCl 3–0 band using a near-IR InGaAsP diode laser,” Appl. Opt. 27, 5009–5015 (1988).
[CrossRef] [PubMed]

C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
[CrossRef]

L. 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]

1987

M. Osinski, J. Buus, “Linewidth broadening factor in semiconductor lasers: an overview,” IEEE J. Quantum Electron. QE-23, 9–29 (1987).
[CrossRef]

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]

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]

K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A band,” J. Mol. Spectrosc. 121, 1–19 (1987).
[CrossRef]

1986

1985

1984

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

K. Y. Lau, A. Yariv, “Intermodulation distortion in a directly modulated semiconductor injection laser,” Appl. Phys. Lett. 45, 1034–1036 (1984).
[CrossRef]

P. L. Varghese, R. K. Hanson, “Collisional narrowing effects on spectral line shapes measured at high resolution,” Appl. Opt. 23, 2376–2385 (1984).
[CrossRef] [PubMed]

1982

J. Humlicek, “Optimized computation of the Voigt and complex probability functions,” J. Quant. Spectrosc. Radiat. Transfer 27, 437–444 (1982); F. Schreier, “The Voigt and complex error function: a comparison of computational methods,” J. Quant. Spectrosc. Radiat. Transfer 48, 743–762 (1992).
[CrossRef]

S. Kobayashi, Y. Yamamoto, M. Ito, T. Kimura, “Direct frequency modulation in GaAlAs semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 582–595 (1982).
[CrossRef]

1981

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

1977

1974

F. Herbert, “Spectrum line profiles: a generalized Voigt function including collisional narrowing,” J. Quant. Spectrosc. Radiat. Transfer 14, 943–951 (1974).
[CrossRef]

1968

E. E. Whiting, “An empirical approximation to the Voigt profile,” J. Quant. Spectrosc. Radiat. Transfer 8, 1379–1384 (1968); J. J. Olivero, R. L. Longbothum, “Empirical fits to the Voigt line width: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 17, 233–236 (1977).
[CrossRef]

1967

S. G. Rautian, I. I. Sobelman, “Effect of collisions on the Doppler broadening of spectral lines,” Sov. Phys. Usp. 9, 701–716 (1967).
[CrossRef]

1961

L. Galatry, “Simultaneous effect of Doppler and foreign gas broadening on spectral shapes,” Phys. Rev. 122, 1218–1223 (1961).
[CrossRef]

1953

R. H. Dicke, “The effect of collisions upon the Doppler width of spectral lines,” Phys. Rev. 89, 472–473 (1953).
[CrossRef]

Adler-Golden, S.

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

S. Adler-Golden, J. Lee, N. Goldstein, “Diode laser measurements of temperature dependent line parameters for water vapor near 820 nm,” J. Quant. Spectrosc. Radiat. Transfer 48, 527–535 (1992).
[CrossRef]

Agresta, D. L.

Arroyo, M. P.

Atkins, D.

D. C. Benner, C. P. Rinsland, V. M. Devi, M. A. H. Smith, D. Atkins, “A multispectrum nonlinear least squares fitting technique,” J. Quant. Spectrosc. Radiat. Transfer 53, 705–721 (1995).
[CrossRef]

Avetisov, V. G.

P. Kauranen, V. G. Avetisov, “Determination of absorption line parameters using two-tone frequency-modulation spectroscopy with diode lasers,” Opt. Commun. 106, 213–217 (1994).
[CrossRef]

V. G. Avetisov, A. I. Nadezhdinskii, A. N. Khusnutdinov, P. M. Omarova, M. V. Zyrianov, “Diode laser spectroscopy of water vapor in 1.8 μm: line profile measurements,” J. Mol. Spectrosc. 160, 326–334 (1993).
[CrossRef]

V. G. Avetisov, P. Kauranen, “High-resolution measurements using two-tone frequency-modulation spectroscopy with diode lasers,” submitted to Appl. Opt.

Baer, D. S.

D. S. Baer, R. K. Hanson, “Tunable diode laser absorption diagnostics for atmospheric pressure plasmas,” J. Quant. Spectrosc. Radiat. Transfer 47, 455–475 (1992).
[CrossRef]

Baets, R. G.

G. Morthier, F. Libbrecht, K. David, P. Vankwikelberge, R. G. Baets, “Theoretical investigation of the second-order harmonic distortion in the AM response of 1.55 μm F-P and DFB lasers,” IEEE J. Quantum Electron. 27, 1990–2002 (1991).
[CrossRef]

Benner, D. C.

D. C. Benner, C. P. Rinsland, V. M. Devi, M. A. H. Smith, D. Atkins, “A multispectrum nonlinear least squares fitting technique,” J. Quant. Spectrosc. Radiat. Transfer 53, 705–721 (1995).
[CrossRef]

L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
[CrossRef]

Bien, F.

Bjorklund, G. C.

Bomse, D. S.

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H. Riris, C. B. Carlisle, L. W. Carr, D. E. Cooper, R. U. Martinelli, R. J. Menna, “Design of an open path near-infrared diode laser sensor: application to oxygen, water, and carbon dioxide vapor detection,” Appl. Opt. 33, 7059–7066 (1994).
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C. B. Carlisle, D. E. Cooper, “Tunable diode laser frequency modulation spectroscopy through an optical fiber: high sensitivity detection of water vapor,” Appl. Phys. Lett. 56, 805–807 (1990).
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L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
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L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
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C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
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Jönsson, B.

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

G. Morthier, F. Libbrecht, K. David, P. Vankwikelberge, R. G. Baets, “Theoretical investigation of the second-order harmonic distortion in the AM response of 1.55 μm F-P and DFB lasers,” IEEE J. Quantum Electron. 27, 1990–2002 (1991).
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V. G. Avetisov, A. I. Nadezhdinskii, A. N. Khusnutdinov, P. M. Omarova, M. V. Zyrianov, “Diode laser spectroscopy of water vapor in 1.8 μm: line profile measurements,” J. Mol. Spectrosc. 160, 326–334 (1993).
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V. G. Avetisov, A. I. Nadezhdinskii, A. N. Khusnutdinov, P. M. Omarova, M. V. Zyrianov, “Diode laser spectroscopy of water vapor in 1.8 μm: line profile measurements,” J. Mol. Spectrosc. 160, 326–334 (1993).
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L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
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Philips, L. A.

C. L. Brummel, L. A. Philips, “Error analysis in rotationally resolved spectra: least-squares and Monte Carlo methods,” J. Mol. Spectrosc. 159, 287–299 (1993).
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A. S. Pine, V. N. Markov, G. Buffa, O. Tarrini, “N2, O2, H2, Ar, and He broadening in the ν1 band of NH3,” J. Quant. Spectrosc. Radiat. Transfer 50, 337–348 (1993).
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C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
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S. G. Rautian, I. I. Sobelman, “Effect of collisions on the Doppler broadening of spectral lines,” Sov. Phys. Usp. 9, 701–716 (1967).
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J. Reid, D. Labrie, “Second-harmonic detection with tunable diode lasers: comparison of experiment and theory,” Appl. Phys. B 26, 203–210 (1981).
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D. C. Benner, C. P. Rinsland, V. M. Devi, M. A. H. Smith, D. Atkins, “A multispectrum nonlinear least squares fitting technique,” J. Quant. Spectrosc. Radiat. Transfer 53, 705–721 (1995).
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L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
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H. Riris, C. B. Carlisle, L. W. Carr, D. E. Cooper, R. U. Martinelli, R. J. Menna, “Design of an open path near-infrared diode laser sensor: application to oxygen, water, and carbon dioxide vapor detection,” Appl. Opt. 33, 7059–7066 (1994).
[CrossRef] [PubMed]

H. Riris, C. B. Carlisle, D. E. Cooper, L. Wang, T. F. Gallagher, R. H. Tipping, “Measurement of the strengths of 1–0 and 3–0 transitions of HI using frequency modulation spectroscopy,” J. Mol. Spectrosc. 146, 381–388 (1991).
[CrossRef]

C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
[CrossRef]

L. 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).
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T. Giesen, R. Schieder, G. Winnewisser, K. M. T. Yamada, “Precise measurements of pressure broadening and shift for several H2O lines in the ν2 band by argon, nitrogen, oxygen, and air,” J. Mol. Spectrosc. 153, 406–418 (1992).
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Silver, J. A.

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D. C. Benner, C. P. Rinsland, V. M. Devi, M. A. H. Smith, D. Atkins, “A multispectrum nonlinear least squares fitting technique,” J. Quant. Spectrosc. Radiat. Transfer 53, 705–721 (1995).
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Supplee, J. M.

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[CrossRef]

C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
[CrossRef]

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L. S. Rothman, R. R. Gamache, R. H. Tipping, C. P. Rinsland, M. A. H. Smith, D. C. Benner, V. M. Devi, J. M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S. T. Massie, L. R. Brown, R. A. Toth, “The HITRAN molecular database: editions of 1991 and 1992,” J. Quant. Spectrosc. Radiat. Transfer 48, 469–507 (1992).
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H. Riris, C. B. Carlisle, D. E. Cooper, L. Wang, T. F. Gallagher, R. H. Tipping, “Measurement of the strengths of 1–0 and 3–0 transitions of HI using frequency modulation spectroscopy,” J. Mol. Spectrosc. 146, 381–388 (1991).
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[CrossRef]

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G. Morthier, F. Libbrecht, K. David, P. Vankwikelberge, R. G. Baets, “Theoretical investigation of the second-order harmonic distortion in the AM response of 1.55 μm F-P and DFB lasers,” IEEE J. Quantum Electron. 27, 1990–2002 (1991).
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[CrossRef]

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C. B. Carlisle, H. Riris, L.-G. Wang, G. R. Janik, T. F. Gallagher, A. L. Pineiro, R. H. Tipping, “Measurement of high overtone intensities of HBr by two-tone frequency-modulation spectroscopy,” J. Mol. Spectrosc. 130, 395–406 (1988).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (20)

Fig. 1
Fig. 1

Spectral distribution of the laser field, two-tone frequency modulated at ν1 and ν2, with β = 1.0 and M = 0.

Fig. 2
Fig. 2

Ratio of the first five sideband group amplitudes to the carrier group amplitude versus the FM index β with M = 0.

Fig. 3
Fig. 3

(a) Theoretical nondistorted TTFMS line shape calculated for β = 1.0 and M = 0 and with the intermediate frequency Ω chosen larger than the absorption linewidth for illustration. (b) Differences between distorted line shapes and the nondistorted line shape in (a) showing the effect of second-harmonic and second-order intermodulation distortion at frequencies 2ν1 (and 2ν2) and ν1 ± ν2, respectively, calculated for ζ, ζ± = 0.02 and ϑ, ϑ± = π.

Fig. 4
Fig. 4

Experimental TTFMS spectrum of a Fabry–Perot transmission resonance recorded with νm = 570 MHz and β ≈ 1.5. The second-order intermodulation distortion at frequency ν1ν2 is clearly visible.

Fig. 5
Fig. 5

Comparison of TTFMS line shapes calculated for νm/Δ = 0.3, 1.0, 4.0 with y = 1.0, β = 1.0, and M = 0, and the corresponding direct absorption line shape. The TTFMS line shapes are normalized to the peak amplitude of the direct absorption line shape.

Fig. 6
Fig. 6

(a) Comparison of TTFMS line shapes calculated for M = 0 (dashed curve) and M = 0.03 (solid curve) with y = 1, β = 0.7, Ψ = π/2, and ν ¯ m = 1 . 0 . (b) Difference (solid curve) between the two line shapes in (a) and the residual (dashed curve) obtained from a least-squares fit using M = 0 to the asymmetric line shape with M = 0.03. The scale is normalized to the TTFMS line-shape (M = 0.03) peak-to-peak amplitude.

Fig. 7
Fig. 7

Induced relative broadening error δyM versus the relative AM index error δM. The data were obtained from least-squares fits to synthetic line shapes calculated for different Ψ with y =1.0, β = 1.0, M = 0.05, and ν ¯ m = 1 . 0 . The solid curves are the induced errors that were calculated by the expressions presented in the text.

Fig. 8
Fig. 8

Difference (solid curve) between line shapes that were calculated for M = 0 and 0.03, respectively, with Ψ = π, y = 1.0, β = 1.0, M = 0.03, and ν ¯ m = 1 . 0 , and the residual (dashed curve) obtained from a least-squares fit using Ψ = π/2 to the line shape calculated with Ψ = π and M = 0.03. The scale is normalized to the TTFMS line-shape (Ψ = π) peak-to-peak amplitude.

Fig. 9
Fig. 9

Induced relative broadening error δyΨ versus the AM–FM phase difference Ψt. The error δyΨ was obtained from least-squares fits using differently fixed Ψ to synthetic line shapes calculated for different Ψt with y = 1.0, β = 1.0, M = 0.05, and ν ¯ m = 1 . 0 . The solid curves represent the induced errors that were calculated by the expressions presented in the text.

Fig. 10
Fig. 10

(a) Induced relative broadening parameter error δyβ, (b) induced relative integrated line intensity error δSβ, (c) amplitude of the residuals Re as functions of νm/Δ. The data were obtained from least-squares fits where δβ = −0.1 to synthetic line shapes calculated for β = 0.7, 1.0, 1.3 with y = 1.0 (Δ = 1.47σ) and M = 0.

Fig. 11
Fig. 11

Residuals that were obtained from least-squares fits using incorrectly assigned FM indices to synthetic line shapes calculated for νm/Δ = 0.7, 1.0, 2.0 with y = 1.0, β = 1.0, and M = 0. The scales are normalized to the corresponding line-shape peak-to-peak amplitude (Re = 10−3). Above each plot is the corresponding FM index error δβ and the induced errors δyβ and δSβ.

Fig. 12
Fig. 12

Coefficients (a) Qβy and (b) QβS versus νm/Δ that were used in the expressions presented in the text to calculate FM index-induced errors in the retrieved broadening and integrated line intensity parameters. The coefficient QβS was plotted for Gaussian (y = 0), Lorentzian (y =∞), and Voigt (y = 1.0) profiles.

Fig. 13
Fig. 13

(a) Induced broadening error δyβ, (b) induced integrated intensity error δSβ, (c) FM index error δβ as functions of νm/Δ calculated for a magnitude of the residuals of 10−3 from the data presented in Fig. 10.

Fig. 14
Fig. 14

Line-shape extent Xe, defined as the standardized halfwidth at 0.01 of the line-shape peak-to-peak value, as a function of νm/Δ calculated for β = 0.7, 1.0, 1.3 with y = 1.0 and M = 0. The corresponding extent of the direct absorption line shape is indicated by the horizontal dashed line.

Fig. 15
Fig. 15

TTFMS line-shape peak-to-peak amplitude versus νm/Δ calculated for β = 0.7, 1.0, 1.3 with y = 1.0 and M = 0.

Fig. 16
Fig. 16

Comparison of experimental TTFMS spectra of a Fabry–Perot transmission resonance recorded with phase-sensitive detection at θ = 180° and 88° and corresponding theoretical line shapes. A dispersion line shape calculated for θ = 90° is also illustrated. The diode laser was frequency modulated at 570 ± 5.4 MHz with β = 0.7 and M = 0.01.

Fig. 17
Fig. 17

Comparison of theoretical (a) absorption and (b) dispersion line shapes that were calculated using a Voigt profile with y = 1, β = 1.0, M = 0, ν ¯ m = 1 , and Ω = 0.02σ.

Fig. 18
Fig. 18

Ratio of the dispersion (IΩϕ)pp and absorption (IΩα)pp intensity components normalized to Ω/Δ versus νm/Δ. The data were obtained from synthetic TTFMS line shapes that were calculated using Gaussian (y = 0), Lorentzian (y = ∞), and Voigt (y = 1.0) profiles.

Fig. 19
Fig. 19

(a) TTFMS line shape calculated with a Galatry profile (G) for y = 1, z = 1, β = 0.7, M = 0, and ν ¯ m = 1 , and the residuals that were obtained from a least-squares fit using Voigt (V) and Rautian–Sobelman (R) profiles. (b) Corresponding direct absorption line shape and residuals.

Fig. 20
Fig. 20

Comparison of the results from least-squares fits to experimental data. (a) TTFMS recording of the R15Q16 oxygen transition at a total pressure of 200 Torr. (b) Residuals from a fit using Voigt (V) and Rautian–Sobelman (R) profiles. (c) Residual obtained when second-harmonic distortion is included in the TTFMS theory. The curves are offset for clarity.

Tables (1)

Tables Icon

Table 1 Results from Least-Squares Fits to Synthetic Data with Random Noise

Equations (48)

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E 1 ( t ) = E 0 [ 1 + M sin ( 2 π ν 1 t + Ψ ) ] [ 1 + M sin ( 2 π ν 2 t + Ψ ) ] × exp { i 2 π ν c t + i β [ sin ( 2 π ν 1 t ) + sin ( 2 π ν 2 t ) ] } ,
E 1 ( t ) = E 0 exp ( i 2 π ν c t ) n , m = + r n r m exp [ i 2 π ( n ν 1 + m ν 2 ) t ] ,
r n ( β , M , Ψ ) J n ( β ) + M 2 i [ J n 1 ( β ) exp ( i Ψ ) J n + 1 ( β ) exp ( i Ψ ) ] .
E ( t ) = E 0 exp [ i 2 π ν t 1 2 α ( ν ) i ϕ ( ν ) ] ,
E 2 ( t ) = E 0 exp ( i 2 π ν c t ) n , m = + r n r m exp [ i 2 π ( n ν 1 + m v 2 ) t ] exp ( 1 2 α n , m i ϕ n , m ) ,
I ( t ) = c ε 0 2 E 0 2 n , m , n , m r n r m r n * r m * exp { i 2 π [ ( n n ) ν 1 + ( m m ) ν 2 ] t } × exp [ 1 2 ( α n , m + α n , m ) i ( ϕ n , m ϕ n , m ) ] .
I Ω ( t ) = c ε 0 2 E 0 2 exp ( 2 π i Ω t ) n , m , r n r m r n + 1 * r m 1 * × exp [ 1 2 ( α n , m + α n + 1 , m 1 ) i ( ϕ n , m ϕ n + 1 , m 1 ) ] + c . c .
exp [ 1 2 ( α n , m + α n + 1 , m 1 ) i ( ϕ n , m ϕ n + 1 , m 1 ) ] = exp [ 1 2 ( α n , m + α n + 1 , m 1 ) ] [ 1 i ( ϕ n , m ϕ n + 1 , m 1 ) 1 2 ( ϕ n , m ϕ n + 1 , m 1 ) 2 + ] exp [ 1 2 ( α n , m + α n + 1 , m 1 ) ] i ( ϕ n , m ϕ n + 1 , m 1 )
I Ω ( t ) = ( I Ω α + Ω ϕ ) cos ( 2 π Ω t ) + ( I Ω ϕ + Ω α ) sin ( 2 π Ω t ) ,
I Ω α c ε 0 E 0 2 n , m Re ( r n r m r n + 1 * r m 1 * ) × exp [ 1 2 ( α n , m + α n + 1 , m 1 ) ] ,
I Ω ϕ c ε 0 E 0 2 n , m Re ( r n r m r n + 1 * r m 1 * ) ( ϕ n + 1 , m 1 ϕ n , m ) ,
Ω α c ε 0 E 0 2 n , m Im ( r n r m r n + 1 * r m 1 * ) × exp [ 1 2 ( α n , m + α n + 1 , m 1 ) ] ,
Ω ϕ c ε 0 E 0 2 n , m Im ( r n r m r n + 1 * r m 1 * ) ( ϕ n , m ϕ n + 1 , m 1 ) .
| ϕ n , m ϕ n + 1 , m 1 | | ϕ 0 , 0 ϕ 1 , 1 | 1 2 α ( ν 0 ) Ω Δ ( Ω Δ < 0 . 2 ) ,
α ( ν 0 ) Ω Δ < 4 × 10 3 ,
( I Ω ϕ ) pp / ( I Ω α ) pp A Ω Δ , 0 . 5 A < 1 . 5 ,
( Ω ϕ ) pp / ( I Ω α ) pp β ( Ω Δ ) 2 [ 1 + ( Δ ν m ) 2 ] M β | cos Ψ | ,
( Ω α ) pp / ( Ω ϕ ) pp 10 .
I Ω ( θ ) I Ω α cos θ + I Ω ϕ sin θ,
I Ω α c ε 0 E 0 2 n , m Re ( r n r m r n + 1 * r m 1 * ) × exp { α [ ν c + ( n + m ) ν m ] } .
Ω Δ < 0 . 02 1 + ( Δ / ν m ) 2 .
I Ω α = c ε 0 E 0 2 k R k exp [ α ( ν c + k ν m ) ] ,
R k ( β , M , Ψ ) n Re ( r n r k n r n + 1 * r k n 1 * )
φ ( t ) = β sin ( 2 π ν 1 t ) + β sin ( 2 π ν 2 t ) + ζ sin ( 4 π ν 1 t + ϑ ) + ζ sin ( 4 π ν 2 t + ϑ ) + ζ + sin [ 2 π ( ν 1 + ν 2 ) t + ϑ + ] + ζ sin [ 2 π ( ν 1 ν 2 ) t + ϑ ] ,
E ( t ) = E 0 exp ( i 2 π ν c t ) n r n exp ( i 2 π n ν 1 t ) m r m × exp ( i 2 π m ν 2 t ) p = 1 1 J p ( ζ ) exp [ i p ( 4 π ν 1 t + ϑ ) ] × p = 1 1 J p ( ζ ) exp [ i q ( 4 π ν 2 t + ϑ ) ] r = 1 1 J r ( ζ + ) × exp { i r [ 2 π ( ν 1 + ν 2 ) t + ϑ + ] } s = 1 1 J s ( ζ ) × exp { i s [ 2 π ( ν 1 ν 2 ) t + ϑ ] } .
E ( t ) = E 0 exp ( i 2 π ν c t ) n , m [ ( r n + δ n ) ( r m + δ m ) + δ n , m + + δ n , m ] exp [ i 2 π ( n ν 1 + m ν 2 ) t ] ,
δ n ( β , ζ , ϑ ) J n 2 ( β ) J 1 ( ζ ) exp ( i ϑ ) + J n + 2 ( β ) J 1 ( ζ ) exp ( i ϑ ) ,
δ n , m ± ( β , ζ ± , ϑ ± ) J n 1 ( β ) J m 1 ( β ) J 1 ( ζ ± ) exp ( i ϑ ± ) + J n + 1 ( β ) J m ± 1 ( β ) J 1 ( ζ ± ) exp ( i ϑ ± )
E ( t ) = E 0 ( t ) n , m ( r n r m + A n , m ) exp [ i 2 π ( n ν 1 + m ν 2 ) t ] ,
A n , m J m ( β ) δ n + J n ( β ) δ m + δ n , m + + δ n , m .
r n r m r n + 1 * r m 1 * r n r m r n + 1 * r m 1 * + J n + 1 ( β ) J m 1 ( β ) A n , m + J n ( β ) J m ( β ) A n + 1 , m 1 * .
r n r m r n + 1 * r m 1 * r n r m r n + 1 * r m 1 * + [ J n + 1 ( β ) J m 1 ( β ) + J n 1 ( β ) J m + 1 ( β ) ] A n , m
α ( ν ν 0 ) S K ( x , y , z ) σ π ,
V ( x , y ) y π exp ( ζ 2 ) y 2 + ( x ζ ) 2 d ζ = Re [ w ( x , y ) ] ,
R ( x , y , z ) Re [ w ( x , y + z ) 1 π z w ( x , y + z ) ] ,
G ( x , y , z ) 1 π Re ( 0 d τ exp { i x τ y τ + 1 2 z 2 × [ 1 z τ exp ( z τ ) ] } ) ;
α n , m S K ( x + n ν ¯ 1 + m ν ¯ 2 , y , z ) σ π ,
J n + 1 ( β ) = 2 n β J n ( β ) J n 1 ( β ) ,
r n = J n ( β ) ( 1 + n M β ) .
J n ( β ) = 1 2 [ J n 1 ( β ) J n + 1 ( β ) ] .
Δ / σ = y / 2 + ( y 2 / 4 + ln 2 ) 1 / 2 ,
δ y M 3 . 8 Δ ν m ( 1 + 1 y ) ( M β ) 2 Q M ( δ M , Ψ ) , δ S M 1 . 5 Δ ν m δ y M ( 0 . 3 < y < 3 , 0 . 25 < ν m / Δ < 4 , β < 1 . 3 , M < 0 . 1 ) ,
Q M ( δ M , Ψ ) [ 1 + cos 2 ( 2 Ψ ) + ( 1 + sin Ψ ) δ M ] δ M ,
( d ν 0 ) M σ 0 . 6 ( 1 + y ) Δ ν m M β sin Ψ δ M ( 0 . 3 < y < 3 , 0 . 25 < ν m / Δ < 4 , β < 1 . 3 , M < 0 . 1 , 0 Ψ π ) .
δ y Ψ 2 . 1 Δ ν m ( 1 + 1 y ) ( M β ) 2 Q Ψ ( Ψ , Ψ t ) , δ S Ψ 1 . 5 Δ ν m δ y Ψ ( 0 . 3 < y < 3 , 0 . 25 < ν m / Δ < 4 , β < 1 . 3 , M < 0 . 1 ) ,
Q Ψ ( Ψ , Ψ t ) ( sin Ψ t sin Ψ ) 2 1 ( π / 8 Ψ 7 π / 8 ) .
δ y β Q β y ( 1 + 1 y ) β 2 . 5 δβ,
δ S β ( 2 . 2 + Q β S β 1 . 5 ) δβ,

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