Abstract

A theoretical description of Doppler-broadened noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) under optically saturated conditions is presented. Expressions for the strength and shape of the Doppler-broadened NICE-OHMS signals are given for both the absorption and the dispersion phase, in the Voigt regime as well as in the Doppler limit. It is shown that Doppler-broadened NICE-OHMS is affected less by optical saturation than other cavity-enhanced techniques; in the Doppler limit the absorption signal decreases by a factor of (1+G±1)12, where G±1 is the degree of saturation for one of the frequency modulation sidebands, whereas the dispersion signal is virtually unaffected by optical saturation. In the Voigt regime both signals show additional dependence on optical saturation. The concept of saturation-insensitive detection is introduced and its conditions are identified.

© 2008 Optical Society of America

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  1. J. Ye, L. S. Ma, and J. L. Hall, “Sub-Doppler optical frequency reference at 1.064μm by means of ultrasensitive cavity-enhanced frequency modulation spectroscopy of a C2HD overtone transition,” Opt. Lett. 21, 1000-1002 (1996).
    [CrossRef] [PubMed]
  2. J. Ye, L. S. Ma, and J. L. Hall, “Ultrastable optical frequency reference at 1.064μm using a C2HD molecular overtone transition,” IEEE Trans. Instrum. Meas. 46, 178-182 (1997).
    [CrossRef]
  3. J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: Demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6-15 (1998).
    [CrossRef]
  4. L. S. Ma, J. Ye, P. Dube, and J. L. Hall, “Ultrasensitive frequency-modulation spectroscopy enhanced by a high-finesse optical cavity: Theory and application to overtone transitions of C2H2 and C2HD,” J. Opt. Soc. Am. B 16, 2255-2268 (1999).
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    [CrossRef]
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    [CrossRef]
  7. J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
    [CrossRef]
  8. M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
    [CrossRef]
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    [CrossRef]
  10. N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
    [CrossRef]
  11. J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
    [CrossRef] [PubMed]
  12. F. M. Schmidt, A. Foltynowicz, W. Ma, and O. Axner, “Fiber-laser-based noise-immune cavity-enhanced optical heterodyne molecular spectrometry for Doppler-broadened detection of C2H2 in the parts per trillion range,” J. Opt. Soc. Am. B 24, 1392-1405 (2007).
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    [CrossRef]
  17. We have, for clarity, represented complex entities with a tilde, e.g., Ẽ(ωc,t), and used cc to denote complex conjugate.
  18. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Oritz, “Frequency modulation (FM) spectroscopy: Theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145-152 (1983).
    [CrossRef]
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    [CrossRef]
  20. This expression is valid for small intracavity absorption (∣δ0−δ±1∣ and ∣ϕ0−ϕ±1∣≪1) and small modulation index (so that terms of the order of β2 can be neglected).
  21. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).
  22. Since the mode Ẽ−1(z,t) is fully out of phase with Ẽ0(z,t) and Ẽ1(z,t) (in the absence of absorbers), the factor Jj(β)/J∣j∣(β), which is equal to the sign of the mode number, i.e., 1 for j=0 and 1, and −1 for j=−1, has been included in the ansatz for ρ̃21,j for symmetry reasons.
  23. M. Sargent III, M. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).
  24. C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
    [CrossRef]
  25. J. Ye and T. W. Lynn, “Applications of optical cavities in modern atomic, molecular, and optical physics,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2003), pp. 1-83.
  26. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).
  27. L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
    [CrossRef]
  28. H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. Como. 2, 545-557 (1927).
  29. R. L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 545-557 (1926).

2008 (2)

2007 (2)

2006 (1)

J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
[CrossRef] [PubMed]

2004 (3)

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

N. J. van Leeuwen and A. C. Wilson, “Measurement of pressure-broadened, ultraweak transitions with noise-immune cavity-enhanced optical heterodyne molecular spectroscopy,” J. Opt. Soc. Am. B 21, 1713-1721 (2004).
[CrossRef]

2003 (1)

J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
[CrossRef]

1999 (3)

1998 (2)

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: Demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6-15 (1998).
[CrossRef]

1997 (1)

J. Ye, L. S. Ma, and J. L. Hall, “Ultrastable optical frequency reference at 1.064μm using a C2HD molecular overtone transition,” IEEE Trans. Instrum. Meas. 46, 178-182 (1997).
[CrossRef]

1996 (1)

1983 (1)

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

1975 (1)

C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
[CrossRef]

1927 (1)

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. Como. 2, 545-557 (1927).

1926 (1)

R. L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 545-557 (1926).

Axner, O.

Bjorklund, G. C.

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

Bood, J.

J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
[CrossRef] [PubMed]

J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
[CrossRef]

Borde, C. J.

C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
[CrossRef]

Brown, L. R.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Camy-Peyret, C.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Cannon, B. D.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

Chance, K. V.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Dana, V.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Dube, P.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Edwards, D. P.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Flaud, J. M.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Foltynowicz, A.

Fox, R. W.

Gamache, R. R.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Gianfrani, L.

Goldman, A.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Hall, J. L.

Hollberg, L.

Howard, D. L.

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

Hummer, D. G.

C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
[CrossRef]

Ishibashi, C.

C. Ishibashi and H. Sasada, “Highly sensitive cavity-enhanced sub-Doppler spectroscopy of a molecular overtone band with a 1.66μm tunable diode laser,” Jpn. J. Appl. Phys., Part 1 38, 920-922 (1999).
[CrossRef]

Jucks, K. W.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Kjaergaard, H. G.

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

Kramers, H. A.

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. Como. 2, 545-557 (1927).

Kronig, R. L.

R. L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 545-557 (1926).

Kunasz, C. V.

C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
[CrossRef]

Lamb, W. E.

M. Sargent III, M. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Lenth, W.

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

Levenson, M. D.

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

Lock, T.

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

Lynn, T. W.

J. Ye and T. W. Lynn, “Applications of optical cavities in modern atomic, molecular, and optical physics,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2003), pp. 1-83.

Ma, L. S.

Ma, W.

Mandin, J. Y.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Massie, S. T.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

McCann, A.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

McIlroy, A.

J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
[CrossRef] [PubMed]

J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Myers, T. L.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

Nemtchinov, V.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Oritz, C.

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

Osborn, D. L.

J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
[CrossRef] [PubMed]

J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
[CrossRef]

Perrin, A.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Rinsland, C. P.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Rothman, L. S.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Sargent, M.

M. Sargent III, M. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Sasada, H.

C. Ishibashi and H. Sasada, “Highly sensitive cavity-enhanced sub-Doppler spectroscopy of a molecular overtone band with a 1.66μm tunable diode laser,” Jpn. J. Appl. Phys., Part 1 38, 920-922 (1999).
[CrossRef]

Schmidt, F. M.

Schroeder, J.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Scully, M.

M. Sargent III, M. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

Taubman, M. S.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

van Leeuwen, N. J.

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

N. J. van Leeuwen and A. C. Wilson, “Measurement of pressure-broadened, ultraweak transitions with noise-immune cavity-enhanced optical heterodyne molecular spectroscopy,” J. Opt. Soc. Am. B 21, 1713-1721 (2004).
[CrossRef]

Varanasi, P.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Wattson, R. B.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Williams, R. M.

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

Wilson, A. C.

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

N. J. van Leeuwen and A. C. Wilson, “Measurement of pressure-broadened, ultraweak transitions with noise-immune cavity-enhanced optical heterodyne molecular spectroscopy,” J. Opt. Soc. Am. B 21, 1713-1721 (2004).
[CrossRef]

Ye, J.

L. S. Ma, J. Ye, P. Dube, and J. L. Hall, “Ultrasensitive frequency-modulation spectroscopy enhanced by a high-finesse optical cavity: Theory and application to overtone transitions of C2H2 and C2HD,” J. Opt. Soc. Am. B 16, 2255-2268 (1999).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: Demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6-15 (1998).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “Ultrastable optical frequency reference at 1.064μm using a C2HD molecular overtone transition,” IEEE Trans. Instrum. Meas. 46, 178-182 (1997).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “Sub-Doppler optical frequency reference at 1.064μm by means of ultrasensitive cavity-enhanced frequency modulation spectroscopy of a C2HD overtone transition,” Opt. Lett. 21, 1000-1002 (1996).
[CrossRef] [PubMed]

J. Ye, “Ultrasensitive high resolution laser spectroscopy and its application to optical frequency standards,” Ph.D. dissertation (University of Colorado, 1997).

J. Ye and T. W. Lynn, “Applications of optical cavities in modern atomic, molecular, and optical physics,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2003), pp. 1-83.

Yoshino, K.

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Appl. Phys. B (1)

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

Atti. Congr. Int. Fis. Como. (1)

H. A. Kramers, “La diffusion de la lumière par les atomes,” Atti. Congr. Int. Fis. Como. 2, 545-557 (1927).

IEEE Trans. Instrum. Meas. (1)

J. Ye, L. S. Ma, and J. L. Hall, “Ultrastable optical frequency reference at 1.064μm using a C2HD molecular overtone transition,” IEEE Trans. Instrum. Meas. 46, 178-182 (1997).
[CrossRef]

J. Chem. Phys. (1)

J. Bood, A. McIlroy, and D. L. Osborn, “Measurement of the sixth overtone band of nitric oxide, and its dipole moment function, using cavity-enhanced frequency modulation spectroscopy,” J. Chem. Phys. 124, 084311 (2006).
[CrossRef] [PubMed]

J. Mol. Spectrosc. (1)

N. J. van Leeuwen, H. G. Kjaergaard, D. L. Howard, and A. C. Wilson, “Measurement of ultraweak transitions in the visible region of molecular oxygen,” J. Mol. Spectrosc. 228, 83-91 (2004).
[CrossRef]

J. Opt. Soc. Am. B (7)

F. M. Schmidt, A. Foltynowicz, W. Ma, and O. Axner, “Fiber-laser-based noise-immune cavity-enhanced optical heterodyne molecular spectrometry for Doppler-broadened detection of C2H2 in the parts per trillion range,” J. Opt. Soc. Am. B 24, 1392-1405 (2007).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “Ultrasensitive detections in atomic and molecular physics: Demonstration in molecular overtone spectroscopy,” J. Opt. Soc. Am. B 15, 6-15 (1998).
[CrossRef]

L. S. Ma, J. Ye, P. Dube, and J. L. Hall, “Ultrasensitive frequency-modulation spectroscopy enhanced by a high-finesse optical cavity: Theory and application to overtone transitions of C2H2 and C2HD,” J. Opt. Soc. Am. B 16, 2255-2268 (1999).
[CrossRef]

L. Gianfrani, R. W. Fox, and L. Hollberg, “Cavity-enhanced absorption spectroscopy of molecular oxygen,” J. Opt. Soc. Am. B 16, 2247-2254 (1999).
[CrossRef]

O. Axner, W. Ma, and A. Foltynowicz, “Sub-Doppler dispersion and noise-immune cavity-enhanced optical heterodyne molecular spectroscopy revised,” J. Opt. Soc. Am. B 25, 1166-1177 (2008).
[CrossRef]

N. J. van Leeuwen and A. C. Wilson, “Measurement of pressure-broadened, ultraweak transitions with noise-immune cavity-enhanced optical heterodyne molecular spectroscopy,” J. Opt. Soc. Am. B 21, 1713-1721 (2004).
[CrossRef]

A. Foltynowicz, W. Ma, F. M. Schmidt, and O. Axner, “Doppler-broadened noise-immune cavity-enhanced optical heterodyne molecular spectroscopy signals from optically saturated transitions under low pressure conditions,”J. Opt. Soc. Am. B 25, 1156-1165 (2008).
[CrossRef]

J. Opt. Soc. Am. Rev. Sci. Instrum. (1)

R. L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 545-557 (1926).

J. Quant. Spectrosc. Radiat. Transf. (1)

L. S. Rothman, C. P. Rinsland, A. Goldman, S. T. Massie, D. P. Edwards, J. M. Flaud, A. Perrin, C. Camy-Peyret, V. Dana, J. Y. Mandin, J. Schroeder, A. McCann, R. R. Gamache, R. B. Wattson, K. Yoshino, K. V. Chance, K. W. Jucks, L. R. Brown, V. Nemtchinov, and P. Varanasi, “The HITRAN molecular spectroscopic database and HAWKS (HITRAN Atmospheric Workstation): 1996 edition,”J. Quant. Spectrosc. Radiat. Transf. 60, 665-710 (1998).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (1)

C. Ishibashi and H. Sasada, “Highly sensitive cavity-enhanced sub-Doppler spectroscopy of a molecular overtone band with a 1.66μm tunable diode laser,” Jpn. J. Appl. Phys., Part 1 38, 920-922 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

C. J. Borde, J. L. Hall, C. V. Kunasz, and D. G. Hummer, “Saturation absorption line shape: Calculation of the transit-time broadening by a perturbation approach,” Phys. Rev. A 14, 236-263 (1975).
[CrossRef]

Proc. SPIE (1)

J. Bood, A. McIlroy, and D. L. Osborn, “Cavity-enhanced frequency modulation absorption spectroscopy of the sixth overtone band of nitric oxide,” Proc. SPIE 4962, 89-100 (2003).
[CrossRef]

Spectrochim. Acta, Part A (1)

M. S. Taubman, T. L. Myers, B. D. Cannon, and R. M. Williams, “Stabilization, injection and control of quantum cascade lasers, and their application to chemical sensing in the infrared,” Spectrochim. Acta, Part A 60, 3457-3468 (2004).
[CrossRef]

Other (9)

A. Foltynowicz, F. M. Schmidt, W. Ma, and O. Axner, “Noise-immune cavity-enhanced optical heterodyne molecular spectroscopy: Current status and future potential,” (accepted for publication in Appl. Phys. B).

J. Ye, “Ultrasensitive high resolution laser spectroscopy and its application to optical frequency standards,” Ph.D. dissertation (University of Colorado, 1997).

We have, for clarity, represented complex entities with a tilde, e.g., Ẽ(ωc,t), and used cc to denote complex conjugate.

This expression is valid for small intracavity absorption (∣δ0−δ±1∣ and ∣ϕ0−ϕ±1∣≪1) and small modulation index (so that terms of the order of β2 can be neglected).

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Since the mode Ẽ−1(z,t) is fully out of phase with Ẽ0(z,t) and Ẽ1(z,t) (in the absence of absorbers), the factor Jj(β)/J∣j∣(β), which is equal to the sign of the mode number, i.e., 1 for j=0 and 1, and −1 for j=−1, has been included in the ansatz for ρ̃21,j for symmetry reasons.

M. Sargent III, M. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, 1974).

J. Ye and T. W. Lynn, “Applications of optical cavities in modern atomic, molecular, and optical physics,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2003), pp. 1-83.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

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Figures (5)

Fig. 1
Fig. 1

Influence of optical saturation on Doppler-broadened f m -NICE-OHMS line-shape functions. (a)–(d) absorption mode of detection; (e)–(h) dispersion mode of detection. The four pairs of panels, (a) and (e), (b) and (f), (c) and (g), and (d) and (h), represent y = 0 , 0.003, 0.01, and 0.1, respectively. Each panel displays six curves, corresponding to saturation parameters, G + , of 0, 1, 3, 10, 30, and 100 counted from the uppermost curve in each panel. The inset in each lower panel shows a zoom of the dispersion signal line shape around the positive peak.

Fig. 2
Fig. 2

(a)–(d) Polar plots of the peak-to-peak value of the Doppler-broadened f m -NICE-OHMS line-shape function as a function of detection phase for y = 0 , 0.003, 0.01, and 0.1, respectively. The six contours in each panel show various degrees of saturation, corresponding to values of G + of 0, 1, 3, 10, 30, and 100. The phases 0 ° ( 180 ° ) and 90 ° ( 270 ° ) correspond to pure absorption and dispersion phases, respectively.

Fig. 3
Fig. 3

(a) The SIP detection phase θ f m SIP and (b) the peak-to-peak value of the Doppler-broadened f m -NICE-OHMS line-shape function at the SIP phase θ f m SIP as a function of y.

Fig. 4
Fig. 4

Peak-to-peak values of the Doppler-broadened (a) absorption and (b) dispersion line-shape functions (in terms of χ 0 ) as a function of Doppler-width normalized homogeneous linewidth in the absence of saturation, y, and saturation parameter, G + . The lower data sets (planes) in each panel represent the full expressions, given by either Eq. (30) or (33), whereas the upper sets (planes) correspond to the simplified expressions, valid in the Doppler limit, Eq. (36) or (37).

Fig. 5
Fig. 5

Illustration of the influence of saturating light on the velocity distribution of molecules. The dashed-dotted curve in (a) shows the thermal population difference between the two states of a Maxwellian velocity distribution of molecules in the absence of saturation, which is proportional to f ( v ) . The other two curves (solid and dotted, where the solid curve represents a smaller pressure broadening as compared to the dotted curve) represent the population difference in the presence of a saturating mode of light, which burns a Bennet hole around the velocity corresponding to the detuning v = ( Δ ω + j ω m ) k j . The two curves in (b) represent the molecular velocity distribution that contributes to the saturation and is given by the product of the second and third factors in the integrals in Eqs. (42, 43). The curves in (c) and (d) correspond to conventional absorption and dispersion functions, representing the first factors in Eqs. (42, 43), respectively. The influence of optical saturation on absorption is given by the area of the product of a curve in (b) and one in (c), which takes a nonzero (positive) value. The influence of saturation on the dispersion mode, given by the area of a product of a curve in (b) and one in (d), becomes significantly smaller since the influence on dispersion of the low and high velocity molecules in the Bennet hole cancels to a large extent.

Equations (43)

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E ̃ ( ω c , t ) = ε ̂ E 0 2 e i [ ω c t + β sin ( ω m t ) ] = ε ̂ E 0 2 j = J j ( β ) e i ( ω c + j ω m ) t = j = E ̃ j ( ω c , t ) ,
E ̃ j ( ω c , z , t ) = ε ̂ E 0 2 J j ( β ) e i ( ω c , j t k j 0 z ) ,
E ̃ j A ( Δ ω , z , t ) = ε ̂ E 0 2 J j ( β ) e i [ ω c , j t k ̃ j ( Δ ω ) z ] ,
E ̃ j A ( Δ ω , z , t ) = T ̃ j ( Δ ω ) E ̃ j ( ω c , z , t ) ,
T ̃ j ( Δ ω ) = e δ j ( Δ ω ) i ϕ j ( Δ ω ) ,
S f m n o ( Δ ω , θ f m ) = η f m 4 F π J 0 ( β ) J 1 ( β ) P 0 { [ δ 1 ( Δ ω ) δ 1 ( Δ ω ) ] cos θ f m + [ ϕ 1 ( Δ ω ) 2 ϕ 0 ( Δ ω ) + ϕ 1 ( Δ ω ) ] sin θ f m } ,
δ j ( Δ ω ) = ω c , j L 2 c χ j ( Δ ω ) ω c L 2 c χ j ( Δ ω ) ,
ϕ j ( Δ ω ) = ω c , j L 2 c χ j ( Δ ω ) ω c L 2 c χ j ( Δ ω ) ,
P ̃ j ( Δ ω , z , t ) = ε 0 χ ̃ j ( Δ ω ) E ̃ j ( ω c , z , t ) ,
P ̃ j ( Δ ω , z , t ) = n A p ̃ j ( Δ ω , v , z , t ) f ( v ) d v .
p ̃ j ( Δ ω , v , z , t ) = μ ρ ̃ 21 , j ( Δ ω , v , z , t ) ,
ρ ̃ 21 , j ( Δ ω , v , z , t ) = J j ( β ) J j ( β ) ρ ̃ 21 , j 0 ( Δ ω , v , z , t ) e i ( ω c , j t k ̃ j 0 z ) ,
χ ̃ j ( Δ ω , z , t ) = 2 n A μ ε ̂ E 0 π u ε 0 E 0 2 J j ( β ) ρ ̃ 21 , j 0 ( Δ ω , v , z , t ) e v 2 u 2 d v ,
f ( v ) = 1 π u e v 2 u 2 ,
( t + v z ) ρ 11 , j ( z , t ) = γ 1 [ ρ 11 , j ( z , t ) ρ 11 0 ] i [ μ E ̃ j ( z , t ) ρ ̃ 12 , j ( z , t ) c c ] ,
( t + v z ) ρ 22 , j ( z , t ) = γ 2 [ ρ 22 , j ( z , t ) ρ 22 0 ] + i [ μ E ̃ j ( z , t ) ρ ̃ 12 , j ( z , t ) c c ] ,
( t + v z ) ρ ̃ 21 , j ( z , t ) = ( i ω 0 + γ 12 ) ρ ̃ 21 , j ( z , t ) + i μ E ̃ j ( z , t ) [ ρ 11 , j ( z , t ) ρ 22 , j ( z , t ) ] ,
ρ ̃ 12 , j ( z , t ) = ρ ̃ 21 , j * ( z , t ) ,
ρ ̃ 21 , j 0 ( Δ ω , v , G j ) = μ ε ̂ E 0 2 J j ( β ) Δ ρ 11 , 22 0 ( Δ ω + j ω m k j v ) i γ 12 ( Δ ω + j ω m k j v ) 2 + γ 12 2 ( 1 + G j ) .
G j = I + I sat J j 2 ( β ) = G + J j 2 ( β ) ,
I sat = 3 c ε 0 2 γ 1 γ 2 μ 2 γ 12 γ 1 + γ 2 ,
I sat = 3 c ε 0 2 2 μ 2 ( γ t t + 2 π B p ) 2 .
I sat = 3 c ε 0 2 γ t t 2 2 μ 2 ,
G + = 2 μ 2 I + 3 c ε 0 2 γ t t 2 = 32 μ 2 w 2 I + 3 π 2 c ε 0 2 u 2 .
S = ω μ 2 6 ε 0 c 2 Δ ρ 11 , 22 0 ,
χ ̃ j ( Δ ω , G j ) = S n A 2 c π k u ( Δ ω + j ω m k j v ) i γ 12 ( Δ ω + j ω m k j v ) 2 + γ 12 2 ( 1 + G j ) e v 2 u 2 d v ,
δ j ( Δ ω , G j ) = S n A L 2 χ 0 ω c π c γ 12 ( Δ ω + j ω m k j v ) 2 + γ 12 2 ( 1 + G j ) e v 2 u 2 d v = S n A L 2 χ ̂ j abs ( Δ ω , G j ) ,
ϕ j ( Δ ω , G j ) = S n A L 2 χ 0 ω c π c Δ ω + j ω m k j v ( Δ ω + j ω m k j v ) 2 + γ 12 2 ( 1 + G j ) e v 2 u 2 d v = S n A L 2 χ ̂ j disp ( Δ ω , G j ) ,
S f m n o ( Δ ω , θ f m , G + ) = η f m 2 F π J 0 ( β ) J 1 ( β ) P 0 S n A L × { [ χ ̂ 1 abs ( Δ ω , G 1 ) χ ̂ 1 abs ( Δ ω , G 1 ) ] cos θ f m + [ χ ̂ 1 disp ( Δ ω , G 1 ) 2 χ ̂ 0 disp ( Δ ω , G 0 ) + χ ̂ 1 disp ( Δ ω , G 1 ) ] sin θ f m } ,
χ ̂ j abs ( Δ ω , G j ) = χ 0 1 1 + G j 1 π y j e s 2 ( x j s ) 2 + y j 2 d s , = χ 0 1 1 + G j Re [ w ( x j + i y j ) ] ,
w ( z ) = e z 2 [ 1 + i erfi ( z ) ] ,
erfi ( z ) = i erfi ( z ) = 2 π 0 z e s 2 d s ,
χ ̂ j disp ( Δ ω , G j ) = χ 0 1 π ( x j s ) e s 2 ( x j s ) 2 + y j 2 d s = χ 0 Im [ w ( x j + i y j ) ] .
χ ̂ j , u n s abs ( Δ ω ) = χ 0 Re [ w ( x j + i y ) ] ,
χ ̂ j , u n s disp ( Δ ω ) = χ 0 Im [ w ( x j + i y ) ] ,
χ ̂ j abs ( Δ ω , G j ) Δ ω D γ 12 = χ 0 1 1 + G j e x j 2 ,
χ ̂ j disp ( Δ ω , G j ) Δ ω D γ 12 = 2 π χ 0 e x j 2 0 x j e s 2 d s ,
χ ̂ j , u n s abs ( Δ ω ) Δ ω D γ 12 = χ 0 e x j 2 ,
S f m n o ( Δ ω p , θ f m , G + ) G + θ f m SIP 0 ,
S f m n o ( Δ ω , θ f m , G + ) Δ ω Δ ω p = 0 .
θ f m SIP arctan { G + [ χ ̂ 1 abs ( Δ ω p , G 1 ) χ ̂ 1 abs ( Δ ω p , G 1 ) ] G + [ χ ̂ 1 disp ( Δ ω p , G 1 ) 2 χ ̂ 0 disp ( Δ ω p , G 0 ) + χ ̂ 1 disp ( Δ ω p , G 1 ) ] } .
Δ χ ̂ j abs ( Δ ω , G j ) = χ 0 G j ω π c γ 12 ( Δ ω + j ω m k v ) 2 + γ 12 2 × γ 12 2 ( Δ ω + j ω m k v ) 2 + γ 12 2 ( 1 + G j ) e v 2 u 2 d v ,
Δ χ ̂ j disp ( Δ ω , G j ) = χ 0 G j ω π c ( Δ ω + j ω m k v ) ( Δ ω + j ω m k v ) 2 + γ 12 2 × γ 12 2 ( Δ ω + j ω m k v ) 2 + γ 12 2 ( 1 + G j ) e v 2 u 2 d v ,

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