Abstract

An expression for the peak-to-peak sub-Doppler optical phase shift of two counterpropagating modes of light, to which the noise-immune cavity-enhanced optical heterodyne molecular spectroscopy (NICE-OHMS) dispersion signal is proportional, valid for arbitrary degree of saturation, is derived. For low degrees of saturation it agrees with the expression for weakly saturating (ws) conditions, [(1+S)12(1+2S)12]α02, where S is the degree of saturation and α0 is the unsaturated peak absorption. However, the new expression, which can be written as 0.45S(1+S)1α02, does not predict a distinct maximum as the ws expression does; instead it predicts an optical phase shift that increases monotonically with S and levels off to 0.45α02 for large S. This alters the optimum conditions for the sub-Doppler NICE-OHMS technique and improves its shot-noise-limited detectability. The new expression is based upon the same explicit assumptions as the ws expression but not the Kramers–Kronig relations, which are not valid for nonlinear responses, and is supported by experimental results up to S=100. The new expression is expected to be valid for all techniques measuring sub-Doppler dispersion signals.

© 2008 Optical Society of America

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References

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  1. J. Ye and J. L. Hall, “Absorption detection at the quantum limit: Probing high-finesse cavities with modulation techniques,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds. (Academic, 2002), pp. 83-127.
  2. 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.
  3. A. FoltynowiczF. 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).
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. J. Ye, L. S. Ma, and J. L. Hall, “High-resolution frequency standards at 1030nm for Yb:YAG solid-state lasers,” J. Opt. Soc. Am. B 17, 927-931 (2000).
    [CrossRef]
  9. C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultrasensitive molecular spectroscopy,” Proc. SPIE 4634, 58-69 (2002).
    [CrossRef]
  10. M. L. Silva, “Spectroscopic Investigation of the X and A state dynamics of C213H2,” Ph.D. dissertation (Massachusetts Institute of Technology, 2002).
  11. 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]
  12. C. Ishibashi and H. Sasada, “Near-infrared laser spectrometer with sub-Doppler resolution, high sensitivity, and wide tunability: A case study in the 1.65-μm region of CH3I spectrum,” J. Mol. Spectrosc. 200, 147-149 (2000).
    [CrossRef]
  13. 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]
  14. D. R. Weise, “Preparation and highly sensitive detection of ultracold molecules,” Ph.D. dissertation (University of Konstanz, 2004).
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. F. M. Schmidt, A. Foltynowicz, W. Ma, T. Lock, and O. Axner, “Doppler-broadened fiber-laser-based NICE-OHMS--improved detectability,” Opt. Express 15, 10822-10831 (2007).
    [CrossRef]
  21. W. Ma, A. Foltynowicz, and O. Axner, “Theoretical description of Doppler-broadened noise-immune cavity-enhanced optical heterodyne molecular spectroscopy under optically saturated conditions,” J. Opt. Soc. Am. B 25, 1144-1155 (2008).
    [CrossRef]
  22. 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]
  23. J. Ye, “Ultrasensitive high resolution laser spectroscopy and its application to optical frequency standards,” Ph.D. dissertation (University of Colorado, 1997).
  24. H. A. Kramers, “La diffusion de la lumiere par les atomes,” Atti. Congr. Int. Fis. Como. 2, 545-557 (1927).
  25. R. L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12, 545-557 (1926).
    [CrossRef]
  26. W. Demtröder, Laser Spectroscopy, 2nd ed. (Springer Verlag, 1996).
  27. A. Foltynowicz, W. Ma, and O. Axner are preparing a paper to be called “Fiber-laser-based sub-Doppler NICE-OHMS for quantitative trace species detection.”

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]

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]

2002 (1)

C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultrasensitive molecular spectroscopy,” Proc. SPIE 4634, 58-69 (2002).
[CrossRef]

2000 (2)

C. Ishibashi and H. Sasada, “Near-infrared laser spectrometer with sub-Doppler resolution, high sensitivity, and wide tunability: A case study in the 1.65-μm region of CH3I spectrum,” J. Mol. Spectrosc. 200, 147-149 (2000).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “High-resolution frequency standards at 1030nm for Yb:YAG solid-state lasers,” J. Opt. Soc. Am. B 17, 927-931 (2000).
[CrossRef]

1999 (3)

1998 (1)

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)

1926 (1)

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

Axner, O.

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]

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]

Demtröder, W.

W. Demtröder, Laser Spectroscopy, 2nd ed. (Springer Verlag, 1996).

Dube, P.

Foltynowicz, A.

Fox, R. W.

Gianfrani, L.

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]

Ishibashi, C.

C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultrasensitive molecular spectroscopy,” Proc. SPIE 4634, 58-69 (2002).
[CrossRef]

C. Ishibashi and H. Sasada, “Near-infrared laser spectrometer with sub-Doppler resolution, high sensitivity, and wide tunability: A case study in the 1.65-μm region of CH3I spectrum,” J. Mol. Spectrosc. 200, 147-149 (2000).
[CrossRef]

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]

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 lumiere 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. 12, 545-557 (1926).
[CrossRef]

Lock, T.

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.

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]

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]

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]

Sasada, H.

C. Ishibashi and H. Sasada, “Near-infrared laser spectrometer with sub-Doppler resolution, high sensitivity, and wide tunability: A case study in the 1.65-μm region of CH3I spectrum,” J. Mol. Spectrosc. 200, 147-149 (2000).
[CrossRef]

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.

Silva, M. L.

M. L. Silva, “Spectroscopic Investigation of the X and A state dynamics of C213H2,” Ph.D. dissertation (Massachusetts Institute of Technology, 2002).

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 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]

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]

Weise, D. R.

D. R. Weise, “Preparation and highly sensitive detection of ultracold molecules,” Ph.D. dissertation (University of Konstanz, 2004).

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 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]

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]

Ye, J.

C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultrasensitive molecular spectroscopy,” Proc. SPIE 4634, 58-69 (2002).
[CrossRef]

J. Ye, L. S. Ma, and J. L. Hall, “High-resolution frequency standards at 1030nm for Yb:YAG solid-state lasers,” J. Opt. Soc. Am. B 17, 927-931 (2000).
[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]

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]

J. Ye and J. L. Hall, “Absorption detection at the quantum limit: Probing high-finesse cavities with modulation techniques,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds. (Academic, 2002), pp. 83-127.

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.

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]

J. Mol. Spectrosc. (2)

C. Ishibashi and H. Sasada, “Near-infrared laser spectrometer with sub-Doppler resolution, high sensitivity, and wide tunability: A case study in the 1.65-μm region of CH3I spectrum,” J. Mol. Spectrosc. 200, 147-149 (2000).
[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]

J. Opt. Soc. Am. (1)

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

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

W. Ma, A. Foltynowicz, and O. Axner, “Theoretical description of Doppler-broadened noise-immune cavity-enhanced optical heterodyne molecular spectroscopy under optically saturated conditions,” J. Opt. Soc. Am. B 25, 1144-1155 (2008).
[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]

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]

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]

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]

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, “High-resolution frequency standards at 1030nm for Yb:YAG solid-state lasers,” J. Opt. Soc. Am. B 17, 927-931 (2000).
[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]

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)

Proc. SPIE (1)

C. Ishibashi, J. Ye, and J. L. Hall, “Issues and applications in ultrasensitive molecular spectroscopy,” Proc. SPIE 4634, 58-69 (2002).
[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)

D. R. Weise, “Preparation and highly sensitive detection of ultracold molecules,” Ph.D. dissertation (University of Konstanz, 2004).

M. L. Silva, “Spectroscopic Investigation of the X and A state dynamics of C213H2,” Ph.D. dissertation (Massachusetts Institute of Technology, 2002).

J. Ye and J. L. Hall, “Absorption detection at the quantum limit: Probing high-finesse cavities with modulation techniques,” in Cavity-Enhanced Spectroscopies, R.D.van Zee and J.P.Looney, eds. (Academic, 2002), pp. 83-127.

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.

A. FoltynowiczF. 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).

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

W. Demtröder, Laser Spectroscopy, 2nd ed. (Springer Verlag, 1996).

A. Foltynowicz, W. Ma, and O. Axner are preparing a paper to be called “Fiber-laser-based sub-Doppler NICE-OHMS for quantitative trace species detection.”

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Figures (7)

Fig. 1
Fig. 1

Theoretical sub-Doppler optical phase shifts of the carrier, ϕ 00 , sub , in terms of α 0 2 , equal to sub-Doppler f m -NICE-OHMS dispersion signals, S 00 f m , disp , in terms of η f m ( 4 F π ) J 0 ( β ) J 1 ( β ) P 0 α 0 2 , as given by Eq. (14). The four panels correspond to y values of (a) 0.0003, (b) 0.001, (c) 0.003, and (d) 0.01, respectively, whereas the various curves in each panel represent degrees of saturation of 0.01, 0.03, 0.1, 0.3, 1, 3, 10, 30, and 100, respectively. The left and right insets illustrate degrees of saturation of 1 and 100, respectively, where the solid and dashed curves (overlapping almost completely) are the sub-Doppler optical phase shift calculated according to Eq. (14) and a fitted Lorentzian dispersion line-shape function, respectively.

Fig. 2
Fig. 2

p p -sub-Doppler optical phase shift, i.e., ϕ 00 , sub p p , as a function of the degree of saturation, G 0 , calculated from Eqs. (17, 14) for the same four y values as given in Figs. 1a, 1b, 1c, 1d. Solid curves: Fits of Eq. (22) to each set of data points. Dashed-dotted curve: ϕ 00 , sub p p given by Eq. (1). The inset shows the data and the theoretical expressions for the low degree of saturation.

Fig. 3
Fig. 3

Doppler-width normalized homogenous linewidth of the sub-Doppler optical phase shift, ϕ 00 , sub , as a function of the degree of saturation for the same four y values as given in Figs. 1a, 1b, 1c, 1d. The solid curves are plots of the expression y ( 1 + G 0 ) 1 2 .

Fig. 4
Fig. 4

Typical experimental f m -NICE-OHMS dispersion signals from a partial pressure of C 2 H 2 of 10 μ Torr and an intracavity power of 4.6 W for four different total intracavity pressures. (a)–(d) represent total pressures of 10, 20, 50 and 200 mTorr , which correspond to y values of 0.00074, 0.0011, 0.0018 and 0.006, and degrees of saturation of 99, 53, 16, and 1.5, respectively. Dashed curves: Fits of Doppler-broadened f m -NICE-OHMS line shapes. The residuals to the fits are shown below each panel.

Fig. 5
Fig. 5

Typical sub-Doppler f m -NICE-OHMS dispersion signals for the same experimental conditions as in Fig. 4. Dashed curves: Fits of a Lorentzian dispersion line-shape function. The residuals to the fits are shown below each panel.

Fig. 6
Fig. 6

p p -sub-Doppler f m -NICE-OHMS phase shift as a function of degree of saturation for a set of intracavity pressures, from 10 to 300 mTorr , labeled according to the figure legend. Upper solid curve: Eq. (22); lower solid curve: Fit of Eq. (26); dashed-dotted curve: Eq. (1). The inset shows the data and theoretical expressions for low degrees of saturation.

Fig. 7
Fig. 7

Homogenous linewidth (HWHM) of the sub-Doppler f m -NICE-OHMS dispersion signals as a function of degree of saturation for a set of intracavity pressures, from 10 to 200 mTorr , labeled according to the figure legend. The solid curves show fits of the form γ 12 ( 1 + G 0 ) 1 2 , while the dashed curves show fits of the phenomenologically extended expression γ 12 ( 1 + a G 0 ) 1 2 .

Equations (28)

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

ϕ 00 , sub p p = δ 00 , sub p p = Δ α 2 = ( 1 1 + S 1 1 + 2 S ) α 0 2 ,
S D b f m ( 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 ) = 2 S n A L δ j ( Δ ω , G j ) = ω S n A c χ j ( Δ ω , G j ) ,
χ ̂ j disp ( Δ ω , G j ) = 2 S n A L ϕ j ( Δ ω , G j ) = ω S n A c χ j ( Δ ω , G j ) ,
χ ̃ j ( Δ ω , G j ) = 2 n A μ ε ̂ E 0 π u ε 0 E 0 2 J j ( β ) ρ ̃ 21 , j 0 ( Δ ω , v , G j ) e v 2 u 2 d v ,
S 00 f m , disp ( Δ ω , θ f m , G 0 ) = η f m F π J 0 ( β ) J 1 ( β ) P 0 S n A L × 2 [ χ ̂ 00 + , disp ( Δ ω , G 0 ) + χ ̂ 00 , disp ( Δ ω , G 0 ) ] sin θ f m = η f m 2 F π J 0 ( β ) J 1 ( β ) P 0 2 [ ϕ 00 + ( Δ ω , G 0 ) + ϕ 00 ( Δ ω , G 0 ) ] sin θ f m ,
E ̃ ( z , t ) = E 0 2 J 0 ( β ) [ e i ( ω c t k c z ) + e i ( ω c t + k c z ) ] ,
ρ ̃ 21 , 0 ( Δ ω , v , G 0 , z , t ) = ρ ̃ 21 , 0 0 , + ( Δ ω , v , G 0 , z , t ) e i ( ω c t k c z ) + ρ ̃ 21 , 0 0 , ( Δ ω , v , G 0 , z , t ) e i ( ω c t + k c z ) .
ρ ̃ 21 , 0 0 , ± ( Δ ω , v , G 0 ) = μ ε ̂ E 0 2 J 0 ( β ) ( Δ ω k v ) i γ 12 ( Δ ω k v ) 2 + γ 12 2 Δ ρ 11 , 22 0 1 + G 0 [ L 0 ± ( Δ ω , v ) + L 0 ( Δ ω , v ) ] ,
L 0 ± ( Δ ω , v ) = γ 12 2 ( Δ ω k v ) 2 + γ 12 2 .
χ ̂ 00 ± , disp ( Δ ω , G 0 ) = χ 0 ω π c Δ ω k v ( Δ ω k v ) 2 + γ 12 2 e v 2 u 2 1 + G 0 [ L 0 ± ( Δ ω , v ) + L 0 ( Δ ω , v ) ] d v ,
χ ̂ 00 , sub disp ( x 0 , y , G 0 ) = χ 0 1 π ( x 0 s ) ( x 0 s ) 2 + y 2 e s 2 1 + G 0 L 0 + ( x 0 , y , s ) d s + χ 0 1 π ( x 0 s ) ( x 0 s ) 2 + y 2 e s 2 1 + G 0 [ L 0 + ( x 0 , y , s ) + L 0 ( x 0 , y , s ) ] d s ,
χ ̂ 0 disp ( Δ ω , G 0 ) = χ 0 Im [ w ( x 0 + i y 0 ) ] ,
ϕ 00 , sub ( x 0 , y , G 0 ) = e y 2 1 erf ( y ) { Im [ w ( x 0 + i y 0 ) ] + 1 π ( x 0 s ) [ ( x 0 + s ) 2 + y 2 ] e s 2 ( x 0 2 s 2 ) 2 + 2 ( 1 + G 0 ) ( x 0 2 + s 2 ) y 2 + ( 1 + 2 G 0 ) y 4 d s } α 0 2 ,
Im [ w ( x 0 + i y 0 ) ] 2 π e x 0 2 0 x 0 e s 2 d s ,
ϕ 00 , sub ( x 0 , y , G 0 ) = { 2 π e x 0 2 0 x 0 e s 2 d s + 1 π ( x 0 s ) [ ( x 0 + s ) 2 + y 2 ] e s 2 ( x 0 2 s 2 ) 2 + 2 ( 1 + G 0 ) ( x 0 2 + s 2 ) y 2 + ( 1 + 2 G 0 ) y 4 d s } α 0 2 .
ϕ 00 , sub p p ( y , G 0 ) = max [ ϕ 00 , sub ( x 0 , y , G 0 ) ] min [ ϕ 00 , sub ( x 0 , y , G 0 ) ] ,
Υ = S 00 , sub f m , disp , p p ( Δ ω , G 0 ) S D b f m , disp , p p ( Δ ω , G + ) = 2 ϕ 00 , sub p p ( y , G 0 ) e y 2 [ 1 erf ( y ) ] { Im [ w ( x 1 + i y 1 ) ] 2 Im [ w ( x 0 + i y 0 ) ] + Im [ w ( x 1 + i y 1 ) ] } p p 2 α 0 ,
ϕ 00 , sub p p ( y , G 0 ) = { Im [ w ( x 1 + i y 1 ) ] 2 Im [ w ( x 0 + i y 0 ) ] + Im [ w ( x 1 + i y 1 ) ] } p p 2 Υ α 0 2 .
χ ̂ 00 , sub disp ( x 0 , y , G 0 ) = χ 0 1 π ( x 0 s ) e s 2 ( x 0 s ) 2 + y 2 [ 1 1 + G 0 L 0 + 1 1 + G 0 ( L 0 + + L 0 ) ] d s χ 0 1 π ( x 0 s ) e s 2 ( x 0 s ) 2 + y 2 G 0 L 0 d s χ 0 G 0 y 2 2 π ( x 0 s ) ( x 0 s ) 2 + y 2 y e s 2 ( x 0 s ) 2 + y 2 d s = χ 0 G 0 y 2 x 0 x 0 2 + y 2 ,
ϕ 00 , sub p p ( G 0 ) = A [ 1 1 1 + B G 0 ] α 0 2 = A B G 0 1 + B G 0 α 0 2 ,
ϕ 00 , sub p p ( G 0 ) = 0.45 G 0 1 + G 0 α 0 2 .
ϕ 00 , sub p p ( G 0 ) = A G 0 + C G 0 2 1 + B G 0 + D G 0 2 α 0 2
G 0 = P c + P sat J 0 2 ( β ) ,
I sat = C s ( Γ t t + B p ) 2 ,
ϕ ¯ 00 , sub p p ( G 0 , w ) = 0 ϕ 00 , sub p p ( G 0 , r , w ) e 2 ( r w ) 2 2 π r d r 0 e 2 ( r w ) 2 2 π r d r = 4 w 2 0 ϕ 00 , sub p p ( G 0 , r , w ) e 2 ( r w ) 2 r d r ,
ϕ min = π 2 F 1 J 0 ( β ) J 1 ( β ) e B w η P 0 ,
n A , min = χ 0 S L α 0 , min 2 0.45 χ 0 S L ϕ min = 1 0.45 J 0 ( β ) J 1 ( β ) π F χ 0 S L e B w η P 0 ,

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