Abstract

A theoretical description of the wavelength-modulation (WM) spectrometry technique is given. The formalism is based on Fourier analysis and can therefore correctly handle arbitrary large frequency-modulation amplitudes. It can also deal with associated intensity modulations as well as wavelength-dependent transmission effects. It elucidates clearly how various Fourier components of these entities combine with those of the line-shape function to yield separately the final analytical and background nf WM signals. Explicit expressions are given for the 2f and the 4f signals. It is shown, among other things, that the 4f technique in general gives rise to smaller background signals (and therefore larger signal-to-background ratios) than does the 2f technique when the background is dominated by etalon effects from short cavities and that a finite intensity modulation necessarily leads to an out-of-phase nf WM signal. The formalism is also able to elucidate clearly that a linear intensity modulation is not sufficient to cause any 2f background residual–amplitude–modulation signals (as was the general consensus until recently in the literature) but that 2f background signals instead can exist only in systems with either wavelength-dependent transmission or a laser with nonlinear intensity modulation.

© 1999 Optical Society of America

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  1. J. C. Camparo, C. M. Klimack, “Laser spectroscopy on a ‘shoestring’,” Am. J. Phys. 51, 1077–1081 (1983).
    [CrossRef]
  2. J. C. Camparo, “The diode laser in atomic physics,” Contemp. Phys. 26, 443–477 (1985).
    [CrossRef]
  3. T. Imasaka, “Analytical molecular spectroscopy with diode lasers,” Spectrochim. Acta Rev. 15, 329–348 (1993).
  4. J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).
  5. A. W. Mantz, “A review of the applicability of tunable diode-laser spectroscopy at high sensitivity,” Microchem. J. 50, 351–364 (1994).
    [CrossRef]
  6. P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997).
    [CrossRef] [PubMed]
  7. 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]
  8. K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).
  9. P. Werle, “Spectroscopic trace gas analysis using semiconductor diode lasers,” Spectrochim. Acta Part A 52, 805–822 (1996).
    [CrossRef]
  10. J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
    [CrossRef] [PubMed]
  11. D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1185–1190 (1982).
    [CrossRef] [PubMed]
  12. E. I. Moses, C. L. Tang, “High-sensitivity laser wavelength-modulation spectroscopy,” Opt. Lett. 1, 115–117 (1977).
    [CrossRef] [PubMed]
  13. 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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
    [CrossRef]
  14. D. M. Bruce, D. T. Cassidy, “Detection of oxygen using short external cavity GaAs semiconductor diode lasers,” Appl. Opt. 29, 1327–1332 (1990).
    [CrossRef] [PubMed]
  15. C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
    [CrossRef]
  16. A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
    [CrossRef]
  17. 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]
  18. J. A. Silver, A. C. Stanton, “Optical interference fringe reduction in laser absorption experiments,” Appl. Opt. 27, 1914–1916 (1988).
    [CrossRef] [PubMed]
  19. V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
    [CrossRef]
  20. P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 216–221 (1990).
    [CrossRef]
  21. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorption and dispersion line shapes,” Opt. Lett. 5, 15–17 (1980).
    [CrossRef]
  22. J. L. Hall, L. Hollberg, T. Baer, H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
    [CrossRef]
  23. W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
    [CrossRef]
  24. 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]
  25. M. Gehrtz, G. C. Bjorklund, E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1526 (1985).
    [CrossRef]
  26. 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]
  27. 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]
  28. J. M. Supplee, E. A. Whittaker, W. Lenth, “Theoretical description of frequency-modulation and wavelength-modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
    [CrossRef] [PubMed]
  29. G. R. Janik, C. B. Carlisle, T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1986).
    [CrossRef]
  30. O. E. Myers, E. J. Putzer, “Measurement broadening in magnetic resonance,” J. Appl. Phys. 30, 1987–1991 (1959).
    [CrossRef]
  31. A. M. Russel, D. A. Torchia, “Harmonic analysis in systems using phase sensitive detectors,” Rev. Sci. Instrum. 33, 442–444 (1962).
    [CrossRef]
  32. G. V. H. Wilson, “Modulation broadening of NMR and ESR line shapes,” J. Appl. Phys. 34, 3276–3285 (1963).
    [CrossRef]
  33. R. Arndt, “Analytical line shapes for Lorentzian signals broadened by modulation,” J. Appl. Phys. 36, 2522–2524 (1965).
    [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. M. L. Olsen, D. L. Grieble, P. R. Griffiths, “Second derivative tunable diode laser spectrometry for line profile determination I. Theory,” Appl. Spectrosc. 34, 50–56 (1980).
    [CrossRef]
  36. D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
    [CrossRef]
  37. J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
    [CrossRef]
  38. J. Gustafsson, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb by 2f-wavelength modulation diode laser absorption spectrometry—experimental verification of simulations,” Spectrochim. Acta Part B 53, 1895–1905 (1998).
    [CrossRef]
  39. J. Gustafsson, O. Axner, “Theoretical investigation of the temperature dependence of the 2f-wavelength modulated diode laser absorption signal,” Spectrochim. Acta Part B 53, 1827–1846 (1998).
    [CrossRef]
  40. J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.
  41. 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]
  42. X. Zhu, D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B 14, 1945–1950 (1997).
    [CrossRef]
  43. 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]
  44. 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]
  45. 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]
  46. H. C. Sun, E. A. Whittaker, “Novel etalon fringe rejection technique for laser absorption spectroscopy,” Appl. Opt. 31, 4998–5002 (1992).
    [CrossRef] [PubMed]
  47. 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]
  48. An alternative definition of the 2f and the 4f WM-signal fractions, which can be considered more convenient under some experimental conditions, is to relate the actually measured nth signal component to the zeroth signal component rather than to the unmodulated signal, as was done in Eqs. (18)–(21), i.e., as SAS,neven(ν̅d, ν̅a) = ΓAS,neven(ν̅d, ν̅a)SAS,0even(ν̅d, ν̅a). This will, however, give the same expression for the nf WM-signal fractions as do expressions (48) and (49) under the conditions considered [i.e., when T0e ≫ Tkk>0e and IL,0e > IL,1e ≫ IL,2e].
  49. Our choice of expression for the modulated laser intensity [Eq. (6)] however, suggests that the linear intensity modulation, given by IL,0(νc)κ1νa cos(ϕ1), should depend on the laser intensity at the center frequency. This is, however, not the case in reality but rather an artifact from our method of writing the intensity modulation. To correct for this artifact when calculating the nf WM spectra, we use the following expression for the wavelength dependence of the intensity-modulation coefficient: κ1(νd) = κ1(0)/[1 + κ1(0)νd].

1998

J. Gustafsson, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb by 2f-wavelength modulation diode laser absorption spectrometry—experimental verification of simulations,” Spectrochim. Acta Part B 53, 1895–1905 (1998).
[CrossRef]

J. Gustafsson, O. Axner, “Theoretical investigation of the temperature dependence of the 2f-wavelength modulated diode laser absorption signal,” Spectrochim. Acta Part B 53, 1827–1846 (1998).
[CrossRef]

1997

D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
[CrossRef]

J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
[CrossRef]

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

X. Zhu, D. T. Cassidy, “Modulation spectroscopy with a semiconductor diode laser by injection-current modulation,” J. Opt. Soc. Am. B 14, 1945–1950 (1997).
[CrossRef]

P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997).
[CrossRef] [PubMed]

1996

P. Werle, “Spectroscopic trace gas analysis using semiconductor diode lasers,” Spectrochim. Acta Part A 52, 805–822 (1996).
[CrossRef]

1995

A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
[CrossRef]

1994

A. W. Mantz, “A review of the applicability of tunable diode-laser spectroscopy at high sensitivity,” Microchem. J. 50, 351–364 (1994).
[CrossRef]

J. M. Supplee, E. A. Whittaker, W. Lenth, “Theoretical description of frequency-modulation and wavelength-modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
[CrossRef] [PubMed]

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]

K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

T. Imasaka, “Analytical molecular spectroscopy with diode lasers,” Spectrochim. Acta Rev. 15, 329–348 (1993).

J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).

1992

1990

1989

1988

1987

1986

1985

1984

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

1983

J. C. Camparo, C. M. Klimack, “Laser spectroscopy on a ‘shoestring’,” Am. J. Phys. 51, 1077–1081 (1983).
[CrossRef]

1982

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]

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

1980

1977

1965

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

1963

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

1962

A. M. Russel, D. A. Torchia, “Harmonic analysis in systems using phase sensitive detectors,” Rev. Sci. Instrum. 33, 442–444 (1962).
[CrossRef]

1959

O. E. Myers, E. J. Putzer, “Measurement broadening in magnetic resonance,” J. Appl. Phys. 30, 1987–1991 (1959).
[CrossRef]

Arndt, R.

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

Axner, O.

J. Gustafsson, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb by 2f-wavelength modulation diode laser absorption spectrometry—experimental verification of simulations,” Spectrochim. Acta Part B 53, 1895–1905 (1998).
[CrossRef]

J. Gustafsson, O. Axner, “Theoretical investigation of the temperature dependence of the 2f-wavelength modulated diode laser absorption signal,” Spectrochim. Acta Part B 53, 1827–1846 (1998).
[CrossRef]

J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
[CrossRef]

D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
[CrossRef]

J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.

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.

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.

Camparo, J. C.

J. C. Camparo, “The diode laser in atomic physics,” Contemp. Phys. 26, 443–477 (1985).
[CrossRef]

J. C. Camparo, C. M. Klimack, “Laser spectroscopy on a ‘shoestring’,” Am. J. Phys. 51, 1077–1081 (1983).
[CrossRef]

Carlisle, C. B.

Cassidy, D. T.

Chekalin, N.

J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.

Chou, N.-Y.

Cooper, D. E.

Franzke, J.

J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).

Gallagher, T. F.

Gehrtz, M.

Grieble, D. L.

Griffiths, P. R.

Groll, H.

K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

Gustafsson, J.

J. Gustafsson, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb by 2f-wavelength modulation diode laser absorption spectrometry—experimental verification of simulations,” Spectrochim. Acta Part B 53, 1895–1905 (1998).
[CrossRef]

J. Gustafsson, O. Axner, “Theoretical investigation of the temperature dependence of the 2f-wavelength modulated diode laser absorption signal,” Spectrochim. Acta Part B 53, 1827–1846 (1998).
[CrossRef]

J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
[CrossRef]

J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.

Hall, J. L.

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

Hanson, R. K.

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 spectrometry,” 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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Hobbs, P. C. D.

P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997).
[CrossRef] [PubMed]

P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 216–221 (1990).
[CrossRef]

Hollberg, L.

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

Imasaka, T.

T. Imasaka, “Analytical molecular spectroscopy with diode lasers,” Spectrochim. Acta Rev. 15, 329–348 (1993).

Janik, G. R.

Johnston, H. S.

Klimack, C. M.

J. C. Camparo, C. M. Klimack, “Laser spectroscopy on a ‘shoestring’,” Am. J. Phys. 51, 1077–1081 (1983).
[CrossRef]

Kuritsyn, Y.

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

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.

Liger, V.

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

Ljung, P.

D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
[CrossRef]

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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Mantz, A. W.

A. W. Mantz, “A review of the applicability of tunable diode-laser spectroscopy at high sensitivity,” Microchem. J. 50, 351–364 (1994).
[CrossRef]

Moses, E. I.

Myers, O. E.

O. E. Myers, E. J. Putzer, “Measurement broadening in magnetic resonance,” J. Appl. Phys. 30, 1987–1991 (1959).
[CrossRef]

Niemax, K.

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
[CrossRef]

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).

K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).

Olsen, M. L.

Philippe, L. C.

Preier, H.

Putzer, E. J.

O. E. Myers, E. J. Putzer, “Measurement broadening in magnetic resonance,” J. Appl. Phys. 30, 1987–1991 (1959).
[CrossRef]

Reid, J.

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1185–1190 (1982).
[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]

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]

Rojas, D.

D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
[CrossRef]

J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
[CrossRef]

J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.

Russel, A. M.

A. M. Russel, D. A. Torchia, “Harmonic analysis in systems using phase sensitive detectors,” Rev. Sci. Instrum. 33, 442–444 (1962).
[CrossRef]

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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Schnell, A.

J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).

Schnürer-Patschan, C.

A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
[CrossRef]

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).

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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

Stanton, A. C.

Sun, H. C.

Supplee, J. M.

Tang, C. L.

Tate, D. A.

Torchia, D. A.

A. M. Russel, D. A. Torchia, “Harmonic analysis in systems using phase sensitive detectors,” Rev. Sci. Instrum. 33, 442–444 (1962).
[CrossRef]

Wang, L.-G.

Warren, R. E.

Webster, C. R.

Werle, P.

P. Werle, “Spectroscopic trace gas analysis using semiconductor diode lasers,” Spectrochim. Acta Part A 52, 805–822 (1996).
[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]

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]

Young, A. T.

Zhu, X.

Zybin, A.

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
[CrossRef]

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

Am. J. Phys.

J. C. Camparo, C. M. Klimack, “Laser spectroscopy on a ‘shoestring’,” Am. J. Phys. 51, 1077–1081 (1983).
[CrossRef]

Appl. Opt.

D. T. Cassidy, J. Reid, “Atmospheric pressure monitoring of trace gases using tunable diode lasers,” Appl. Opt. 21, 1185–1190 (1982).
[CrossRef] [PubMed]

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, “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]

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

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]

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]

J. M. Supplee, E. A. Whittaker, W. Lenth, “Theoretical description of frequency-modulation and wavelength-modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
[CrossRef] [PubMed]

P. C. D. Hobbs, “Ultrasensitive laser measurements without tears,” Appl. Opt. 36, 903–920 (1997).
[CrossRef] [PubMed]

H. C. Sun, E. A. Whittaker, “Novel etalon fringe rejection technique for laser absorption spectroscopy,” Appl. Opt. 31, 4998–5002 (1992).
[CrossRef] [PubMed]

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

Appl. Phys. B

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]

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

Appl. Phys. Lett.

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

Appl. Spectrosc.

Contemp. Phys.

J. C. Camparo, “The diode laser in atomic physics,” Contemp. Phys. 26, 443–477 (1985).
[CrossRef]

IEEE J. Quantum Electron.

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

J. Anal. At. Spectrom.

C. Schnürer-Patschan, A. Zybin, H. Groll, K. Niemax, “Improvement in detection limit in graphite furnace diode laser atomic absorption spectrometry by wavelength modulation technique,” J. Anal. At. Spectrom. 8, 1103–1107 (1993).
[CrossRef]

A. Zybin, C. Schnürer-Patschan, K. Niemax, “Wavelength modulation diode laser atomic spectrometry in modulated low-pressure helium plasmas for element-selective detection in gas chromatography,” J. Anal. At. Spectrom. 10, 563–567 (1995).
[CrossRef]

J. Appl. Phys.

O. E. Myers, E. J. Putzer, “Measurement broadening in magnetic resonance,” J. Appl. Phys. 30, 1987–1991 (1959).
[CrossRef]

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.

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 spectrometry,” J. Geophys. Res. 91, 5371–5378 (1986).
[CrossRef]

J. Opt. Soc. Am. B

Microchem. J.

A. W. Mantz, “A review of the applicability of tunable diode-laser spectroscopy at high sensitivity,” Microchem. J. 50, 351–364 (1994).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

A. M. Russel, D. A. Torchia, “Harmonic analysis in systems using phase sensitive detectors,” Rev. Sci. Instrum. 33, 442–444 (1962).
[CrossRef]

Spectrochim. Acta Part A

P. Werle, “Spectroscopic trace gas analysis using semiconductor diode lasers,” Spectrochim. Acta Part A 52, 805–822 (1996).
[CrossRef]

Spectrochim. Acta Part B

V. Liger, A. Zybin, Y. Kuritsyn, K. Niemax, “Diode-laser atomic-absorption spectrometry by the double-beam–double-modulation technique,” Spectrochim. Acta Part B 52, 1125–1138 (1997).
[CrossRef]

D. Rojas, P. Ljung, O. Axner, “An investigation of the 2f-wavelength modulation technique for detection of atoms under optically thin as well as thick conditions,” Spectrochim. Acta Part B 52, 1663–1686 (1997).
[CrossRef]

J. Gustafsson, D. Rojas, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb atoms in atmospheric pressure atomizers by the 2f-wavelength modulation technique,” Spectrochim. Acta Part B 52, 1937–1953 (1997).
[CrossRef]

J. Gustafsson, O. Axner, “The influence of hyperfine structure and isotope shift on the detection of Rb by 2f-wavelength modulation diode laser absorption spectrometry—experimental verification of simulations,” Spectrochim. Acta Part B 53, 1895–1905 (1998).
[CrossRef]

J. Gustafsson, O. Axner, “Theoretical investigation of the temperature dependence of the 2f-wavelength modulated diode laser absorption signal,” Spectrochim. Acta Part B 53, 1827–1846 (1998).
[CrossRef]

Spectrochim. Acta Rev.

T. Imasaka, “Analytical molecular spectroscopy with diode lasers,” Spectrochim. Acta Rev. 15, 329–348 (1993).

J. Franzke, A. Schnell, K. Niemax, “Spectroscopic properties of commercial laser diodes,” Spectrochim. Acta Rev. 15, 379–395 (1993).

K. Niemax, H. Groll, C. Schnürer-Patschan, “Element analysis by diode laser spectroscopy,” Spectrochim. Acta Rev. 15, 349–377 (1993).

Other

P. C. D. Hobbs, “Shot noise limited optical measurements at baseband with noisy lasers,” in Laser Noise, R. Roy, ed., Proc. SPIE1376, 216–221 (1990).
[CrossRef]

J. Gustafsson, N. Chekalin, D. Rojas, O. Axner have submitted a paper to be called “Extension of the dynamic range of the 2f-wavelength modulated diode laser absorption spectrometry technique—detection of atoms under optically thick conditions” to Spectrochim. Acta Part B.

An alternative definition of the 2f and the 4f WM-signal fractions, which can be considered more convenient under some experimental conditions, is to relate the actually measured nth signal component to the zeroth signal component rather than to the unmodulated signal, as was done in Eqs. (18)–(21), i.e., as SAS,neven(ν̅d, ν̅a) = ΓAS,neven(ν̅d, ν̅a)SAS,0even(ν̅d, ν̅a). This will, however, give the same expression for the nf WM-signal fractions as do expressions (48) and (49) under the conditions considered [i.e., when T0e ≫ Tkk>0e and IL,0e > IL,1e ≫ IL,2e].

Our choice of expression for the modulated laser intensity [Eq. (6)] however, suggests that the linear intensity modulation, given by IL,0(νc)κ1νa cos(ϕ1), should depend on the laser intensity at the center frequency. This is, however, not the case in reality but rather an artifact from our method of writing the intensity modulation. To correct for this artifact when calculating the nf WM spectra, we use the following expression for the wavelength dependence of the intensity-modulation coefficient: κ1(νd) = κ1(0)/[1 + κ1(0)νd].

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Figures (4)

Fig. 1
Fig. 1

Even second (2nd) and fourth (4th) harmonic analytical signal fractions (a) Γ AS , 2 even and (b) Γ AS , 4 even from a single-peak Lorentzian shaped transition as functions of normalized detuning ν̅ d . Curves a in both graphs represent the case without any intensity modulation, whereas curves b–e display the situation in the presence of an intensity modulation for four different normalized modulation amplitudes ν̅ a . The linear intensity-modulation coefficient κ1 is taken according to the discussion in the text, whereas the phase shift is π/15. The width-normalized modulation amplitudes ν̅ a for curves b–e are 2.2, 3.9, 7.4, and 11, respectively; the analyte absorbers have a Δν p of 3 GHz.

Fig. 2
Fig. 2

WM background signals (2f and 4f) from an etalon created between two uncoated surfaces (thus with reflectivities of 4%) as functions of the optical cavity length. The laser light has a linear intensity-modulation coefficient κ1, as in curves b–e in Fig. 1. The width-normalized modulation amplitudes ν̅ a are 2.2 and 3.9 for the 2f and the 4f techniques, respectively. The width of the analyte absorbers is the same as in Fig. 1.

Fig. 3
Fig. 3

(a) Background S BG , 2 even (ν̅ d ) and (b) total S 2 even (ν̅ d ) 2f WM signals represented by curves a and b, respectively, from a short and a long etalon, respectively, created from two uncoated surfaces. The total signal is the sum of the analytical signal from a sample with an optical thickness of 0.1 and the background signal. The FSR’s of the etalons are 50 and 0.5 GHz for (a) and (b), respectively. The analyte absorbers have a Δν p of 3 GHz, and the normalized FM amplitude is 2.2.

Fig. 4
Fig. 4

(a) Background S BG , 4 even (ν̅ d ) and (b) total S 4 even (ν̅ d ) 4f WM signals for a situation corresponding to that of the 2f technique of Fig. 3. All the data are the same except for the normalized FM amplitude, which is 3.9 (i.e., the optimum for the 4f technique).

Tables (1)

Tables Icon

Table 1 Components A n , … , D n of the Expression for the Fourier Components of a Lorentzian Absorption Profile [Eq. (A1)] for the Five Lowest Harmonics (n = 0–4)

Equations (56)

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S ν = β T ν I L ν exp - α 0 χ ¯ ν S BG ν + S AS ν ,
S BG ν = β T ν I L ν = β I ν ,
S AS ν = - β α 0 χ ¯ ν T ν I L ν = - β α 0 χ ¯ ν I ν ,
ν t = ν c + ν a   cos 2 π f m t ,
ν ¯ t = ν ¯ d + ν ¯ a   cos 2 π f m t ,
I L t = I L , 0 ν c 1 + κ 1 ν a   cos 2 π f m t + ϕ 1 + κ 2 ν a 2   cos 2 π 2 f m t + ϕ 2 ,
I t = I 0 1 + M 2 / 2 + 2 M   cos 2 π f m t + M 2 / 2   cos 2 π 2 f m t .
S k ν ¯ d ,   ν ¯ a = 2 - δ k 0 τ 0 τ   S ν ¯ d ,   ν ¯ a ,   t cos 2 π kf m t - θ k d t ,
S k ν ¯ d ,   ν ¯ a = S k even ν ¯ d ,   ν ¯ a cos   θ k + S k odd ν ¯ d ,   ν ¯ a sin   θ k ,
S k even ν ¯ d ,   ν ¯ a = 2 - δ k 0 τ 0 τ   S ν ¯ d ,   ν ¯ a ,   t cos 2 π kf m t d t ,
S k odd ν ¯ d ,   ν ¯ a = 2 τ 0 τ   S ν ¯ d ,   ν ¯ a ,   t sin 2 π kf m t d t .
χ ¯ ν ¯ d ,   ν ¯ a ,   t = n = 0   χ ¯ n e ν ¯ d ,   ν ¯ a cos 2 π nf m t ,
I ν c ,   ν a ,   t = n = 0   I n e ν c ,   ν a cos 2 π nf m t + n = 0   I n o ν c ,   ν a sin 2 π nf m t ,
S BG , k even ν ¯ d ,   ν ¯ a = β I k e ν c ,   ν a ,
S BG , k odd ν ¯ d ,   ν ¯ a = β I k o ν c ,   ν a ,
S AS , k even ν ¯ d ,   ν ¯ a = - β α 0 1 + δ k 0 2 m = 0 k   χ ¯ k - m e ν ¯ d ,   ν ¯ a I m e ν c ,   ν a + 2 - δ k 0 4 m = 0   χ ¯ k + m e ν ¯ d ,   ν ¯ a I m e ν c ,   ν a + 2 - δ k 0 4 m = 0   χ ¯ m e ν ¯ d ,   ν ¯ a I k + m e ν c ,   ν a
S AS , k odd ν ¯ d ,   ν ¯ a = - β α 0 1 2 m = 0 k   χ ¯ k - m e ν ¯ d ,   ν ¯ a I m o ν c ,   ν a - 1 2 m = 0   χ ¯ k + m e ν ¯ d ,   ν ¯ a I m o ν c ,   ν a + 1 2 m = 0   χ ¯ m e ν ¯ d ,   ν ¯ a I k + m o ν c ,   ν a .
S BG , 2 even ν ¯ d ,   ν ¯ a = β I ν c I ¯ 2 e ν c ,   ν a = Γ BG , 2 even ν c ,   ν a S BG ν c ,
S BG , 2 odd ν ¯ d ,   ν ¯ a = β I ν c I ¯ 2 o ν c ,   ν a = Γ BG , 2 odd ν c ,   ν a S BG ν c ,
S AS , 2 even ν ¯ d ,   ν ¯ a = - β α 0 I ν c χ ¯ 2 e I ¯ 0 e + 1 2 χ ¯ 1 e + χ ¯ 3 e I ¯ 1 e + χ ¯ 0 e + χ ¯ 4 e 2 I ¯ 2 e + 1 2 χ ¯ 1 e + χ ¯ 5 e I ¯ 3 e + 1 2 χ ¯ 2 e + χ ¯ 6 e I ¯ 4 e + + = Γ AS , 2 even ν ¯ d ,   ν ¯ a S AS ν c ,
S AS , 2 odd ν ¯ d ,   ν ¯ a = - β α 0 ν c 1 2 χ ¯ 1 e - χ ¯ 3 e I ¯ 1 o + χ ¯ 0 e - χ ¯ 4 e 2 I ¯ 2 o + 1 2 χ ¯ 1 e - χ ¯ 5 e I ¯ 3 o + 1 2 χ ¯ 2 e - χ ¯ 6 e I ¯ 4 o + + = Γ AS , 2 odd ν ¯ d ,   ν ¯ a S AS ν c ,
S BG , 4 even ν ¯ d ,   ν ¯ a = β I ν c I ¯ 4 e ν c ,   ν a = Γ BG , 4 even ν c ,   ν a S BG ν c ,
S BG , 4 odd ν ¯ d ,   ν ¯ a = β I ν c I ¯ 4 o ν c ,   ν a = Γ BG , 4 odd ν c ,   ν a S BG ν c ,
S AS , 4 even ν ¯ d ,   ν ¯ a = - β α 0 I ν c χ ¯ 4 e I ¯ 0 e + 1 2 χ ¯ 3 e + χ ¯ 5 e I ¯ 1 e + 1 2 χ ¯ 2 e + χ ¯ 6 e I ¯ 2 e + 1 2 χ ¯ 1 e + χ ¯ 7 e I ¯ 3 e + χ ¯ 0 e + χ ¯ 8 e 2 I ¯ 4 e + + = Γ AS , 4 even ν ¯ d ,   ν ¯ a S AS ν c ,
S AS , 4 odd ν ¯ d ,   ν ¯ a = - β α 0 I ν c 1 2 χ ¯ 3 e - χ ¯ 5 e I ¯ 1 o +   χ ¯ 2 e - χ ¯ 6 e 2 I ¯ 2 o + 1 2 χ ¯ 1 e - χ ¯ 7 e I ¯ 3 e + χ ¯ 0 e + χ ¯ 8 e 2 I ¯ 4 e + + = Γ AS , 4 odd ν ¯ d ,   ν ¯ a S AS ( ν c )
I ν c ,   ν a ,   t = T ν c ,   ν a ,   t I L ν c ,   ν a ,   t .
I k e ν c ,   ν a = 1 + δ k 0 2 m = 0 k   T k - m e ν c ,   ν a I L , m e ν c ,   ν a + 2 - δ k 0 4 m = 0   T k + m e ν c ,   ν a I L , m e ν c ,   ν a + 2 - δ k 0 4 m = 0   T m e ν c ,   ν a I L , k + m e ν c ,   ν a ,
I k o ν c ,   ν a = 1 2 m = 0 k   T k - m e ν c ,   ν a I L , m o ν c ,   ν a - 1 2 m = 0   T k + m e ν c ,   ν a I L , m o ν c ,   ν a + 1 2 m = 0   T m e ν c ,   ν a I L , k + m o ν c ,   ν a .
I 0 e ν c ,   ν a = T 0 e I L , 0 e + 1 2 T 1 e I L , 1 e + T 2 e I L , 2 e ,
I 1 e ν c ,   ν a = T 1 e I L , 0 e + 1 2 2 T 0 e + T 2 e I L , 1 e + T 1 e + T 3 e I L , 2 e ,
I 1 o ν c ,   ν a = 1 2 2 T 0 e - T 2 e I L , 1 o + T 1 e - T 3 e I L , 2 o ,
I 2 e ν c ,   ν a = T 2 e I L , 0 e + 1 2 T 1 e + T 3 e I L , 1 e + 2 T 0 e - T 4 e I L , 2 e ,
I 2 o ν c ,   ν a = 1 2 T 1 e - T 3 e I L , 1 o + 2 T 0 e - T 4 e I L , 2 o ,
I 4 e ν c ,   ν a = T 4 e I L , 0 e + 1 2 T 3 e + T 5 e I L , 1 e + T 2 e + T 6 e I L , 2 e ,
I 4 o ν c ,   ν a = 1 2 T 3 e - T 5 e I L , 1 o + T 2 e - T 6 e I L , 2 o .
χ ¯ t = 1 1 + ν ¯ 2 t ,
χ ¯ n ν ¯ d ,   ν ¯ a = 2 - δ n 0 2 × 1 - i ν ¯ d 2 + ν ¯ a 2 1 / 2 - 1 - i ν ¯ d n ν ¯ a n 1 - i ν ¯ d 2 + ν ¯ a 2 1 / 2 × i n + c . c . ,
I L , 0 e ν c ,   ν a = I L , 0 ν c ,
I L , 1 e ν c ,   ν a = I L , 0 ν c κ 1 ν a   cos   ϕ 1 ,
I L , 1 o ν c ,   ν a = - I L , 0 ν c κ 1 ν a   sin   ϕ 1 ,
I L , 1 e ν c ,   ν a = I L , 0 ν c κ 2 ν a 2   cos   ϕ 2 ,
I L , 2 o ν c ,   ν a = - I L , 0 ν c κ 2 ν a 2   sin   ϕ 2 .
T ν ,   t = 1 1 + F   sin 2 Φ ν ,   t 2 ,
T ν ,   t 1 - F 2 + 3 F 2 8 + F - F 2 2 cos Φ ν ,   t + F 2 8 cos 2 Φ ν ,   t .
T 0 e ν ˜ c FSR ,   ν ˜ a FSR = 1 - F 2 + 3 F 2 8 + F - F 2 2 cos 2 π ν ˜ c FSR J 0 2 π ν ˜ a FSR + F 2 8 cos 4 π ν ˜ c FSR J 0 4 π ν ˜ a FSR ,
T n e ν ˜ c FSR ,   ν ˜ a FSR = - 1 n + 1 / 2 F - F 2 sin 2 π ν ˜ c FSR × J n 2 π ν ˜ a FSR + F 2 4 sin 4 π ν ˜ c FSR × J n 4 π ν ˜ a FSR ,   n 1 ,   n   odd ,
T n e ν ˜ c FSR ,   ν ˜ a FSR = - 1 n / 2 F - F 2 cos 2 π ν ˜ c FSR × J n 2 π ν ˜ a FSR + F 2 4 cos 4 π ν ˜ c FSR × J n 4 π ν ˜ a FSR ,     n 1 ,     n   even ,
Γ AS , 2 even ν ¯ d ,   ν ¯ a - χ ¯ 2 e + 1 2 χ ¯ 1 e + χ ¯ 3 e κ 1 ν a   cos   ϕ 1 + χ ¯ 0 e + χ ¯ 4 e 2 κ 2 ν a 2   cos   ϕ 2 + 1 2 χ ¯ 1 e + χ ¯ 3 e T ¯ 1 e + χ ¯ 0 e + χ ¯ 4 e 2 T ¯ 2 e + 1 2 χ ¯ 1 e + χ ¯ 5 e T ¯ 3 e + + ,
Γ AS , 4 even ν ¯ d ,   ν ¯ a - χ ¯ 4 e + 1 2 χ ¯ 3 e + χ ¯ 5 e κ 1 ν a   cos   ϕ 1 + 1 2 χ ¯ 0 e + χ ¯ 6 e κ 2 ν a 2   cos   ϕ 2 + 1 2 χ ¯ 3 e + χ ¯ 5 e T ¯ 1 e + 1 2 χ ¯ 2 e + χ ¯ 6 e T ¯ 2 e + 1 2 χ ¯ 1 e + χ ¯ 7 e T ¯ 3 e + + ,
Γ BG , 2 even ν ¯ d ,   ν ¯ a = T ¯ 2 e + T ¯ 1 e + T ¯ 3 e 2   κ 1 ν a   cos   ϕ 1 + 1 + T ¯ 4 e 2 κ 2 ν a 2   cos   ϕ 2 ,
Γ BG , 4 even ν ¯ d ,   ν ¯ a = T ¯ 4 e + T ¯ 3 e + T ¯ 5 e 2   κ 1 ν a   cos   ϕ 1 + T ¯ 2 e + T ¯ 6 e 2   κ 2 ν a 2   cos   ϕ 2 .
L n ν ¯ a ,   ν ¯ d = A n ν ¯ a n B n + C n S + + D n S - 2   R ,
S + = R + M 1 / 2 ,
S - = R - M 1 / 2 ,
R = M 2 + r ν ¯ d 2 1 / 2 ,
M = 1 + ν ¯ a 2 - ν ¯ d 2 ,

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