Abstract

We report investigations of degenerate four-wave mixing (DFWM) line intensities in the A2+X2Π electronic transitions of nitric oxide. Contributions from population gratings (spatially varying perturbations in the level populations of absorbing species) and thermal gratings (spatially varying perturbations in the overall density) were distinguished and compared by several experimental and analytical techniques. For small quantities of nitric oxide in a strongly quenching buffer gas (carbon dioxide), we found that thermal-grating contributions dominated at room temperature for gas pressures of ≈0.5 atm and higher. In a nearly nonquenching buffer (nitrogen) the population-grating mechanism dominated at pressures of ≈1.0 atm and lower. At higher temperatures in an atmospheric-pressure methane/air flame, population gratings of nitric oxide also dominated. We propose a simple model for the ratio of thermal- to population-grating scattering intensities that varies as P4T−4.4. Preliminary investigations of the temperature dependence and detailed studies of the pressure dependence are in agreement with this model. Measurements of the temporal evolution and the peak intensity of isolated thermal-grating signals are in detailed agreement with calculations based on a linearized hydrodynamic model [ J. Opt. Soc. Am. B 12, 384 ( 1995)].

© 1995 Optical Society of America

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1995 (5)

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. 102, 2718–2726 (1995).
[CrossRef]

A. Dreizler, T. Dreier, and J. Wolfrum, "Thermal grating effects in infrared degenerate four-wave mixing for trace gas detection," J. Chem. Phys. Lett. 233, 525 (1995).
[CrossRef]

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

P. H. Paul, R. L. Farrow, and P. M. Danehy, "Gas-phase thermal-grating contributions to four-wave mixing," J. Opt. Soc. Am. B 12, 384–392 (1995).
[CrossRef]

E. B. Cummings, I. A. Leyva, and H. G. Hornung, "An expression for laser-induced thermal acoustic (LITA)," Appl. Opt. 34, 3290–3302 (1995).
[CrossRef] [PubMed]

1994 (6)

E. B. Cummings, "Laser-induced thermal acoustics: simple accurate gas measurements," Opt. Lett. 19, 1361–1363 (1994).
[CrossRef] [PubMed]

R. B. Williams, P. Ewart, and A. Dreizler, "Velocimetry of gas flows using degenerate four-wave mixing," Opt. Lett. 19, 1486–1488 (1994).
[CrossRef] [PubMed]

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. II. Strong-field limit," J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. I. Weak-field limit," J. Chem. Phys. 101, 1072–1092 (1994).
[CrossRef]

E. J. Friedman-Hill, L. A. Rahn, and R. L. Farrow, "On the interpretation and rotational assignment of degenerate four-wave mixing spectra: four-photon line strengths for crossover resonances in NO A2Σ+ −X2II," J. Chem. Phys. 100, 4065–4076 (1994).
[CrossRef]

1993 (5)

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460–5468 (1993).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. II. Photodissociation dynamics via photofragment transient gratings," J. Chem. Phys. 98, 5469–5476 (1993).
[CrossRef]

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

1992 (7)

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

J. A. Gray, P. H. Paul, and J. L. Durant, Jr., "Electronic quenching rates for NO (A2Σ+) measured in a shock tube," Chem. Phys. Lett. 190, 266–270 (1992).
[CrossRef]

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

R. L. Farrow, D. J. Rakestraw, and T. Dreier, "Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment," J. Opt. Soc. Am. B 9, 1770–1777 (1992).
[CrossRef]

B. Yip, P. M. Danehy, and R. K. Hanson, "Degenerate four-wave mixing temperature measurements in a flame," Opt. Lett. 17, 751–753 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

1989 (1)

1987 (2)

D. J. McGraw, J. Michaelson, and J. M. Harris, "Anharmonic forced Rayleigh scattering: a technique for study of saturated absorption in liquids," J. Chem. Phys. 86, 2536–2547 (1987).
[CrossRef]

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

1985 (1)

1983 (1)

M.-S. Chou, A. M. Dean, and D. Stern, "Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames," J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

1982 (1)

I. S. McDermid and J. B. Laudenslager, "Radiative lifetimes and electronic quenching rate constants for single-photon-excited rotational levels of NO (A2Σ+, v′ = 0)," J. Quant. Spectrosc. Radiat. Transfer 27, 483–492 (1982).
[CrossRef]

1981 (2)

J. Nilsen and A. Yariv, "A tunable narrowband optical filter via phase conjugation by nondegenerate four-wave mixing in a Doppler-broadened resonant medium," Opt. Commun. 39, 199–204 (1981).
[CrossRef]

W. C. Gardiner, Jr., Y. Hidaka, and T. Tanzawa, "Refractivity of combustion gases," Combust. Flame 40, 213–219 (1981).
[CrossRef]

Abrams, R. L.

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

Astill, A. G.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

Booze, J. A.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Buntine, M. A.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. 102, 2718–2726 (1995).
[CrossRef]

Butenhoff, T. J.

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460–5468 (1993).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. II. Photodissociation dynamics via photofragment transient gratings," J. Chem. Phys. 98, 5469–5476 (1993).
[CrossRef]

Chandler, D. W.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. 102, 2718–2726 (1995).
[CrossRef]

Chang, A. Y.

For the temperature dependence of γ12 we used that reported by A. Y. Chang, M. D. Di Rosa, and R. K. Hanson ["Temperature dependence of collisional broadening and shift in the NO A ← X (0, 0) band in the presence of argon and nitrogen," J. Quant. Spectrosc. Radiat. Transfer 47, 375–390 (1992)] for NO in N2. We assumed the population decay rate Γ0 to be proportional to the hard-sphere collision frequency (αT−0.5) because of a lack of similar information.
[CrossRef]

Chou, M.-S.

M.-S. Chou, A. M. Dean, and D. Stern, "Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames," J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Crim, F. F.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Cummings, E. B.

Danehy, P. M.

P. H. Paul, R. L. Farrow, and P. M. Danehy, "Gas-phase thermal-grating contributions to four-wave mixing," J. Opt. Soc. Am. B 12, 384–392 (1995).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

B. Yip, P. M. Danehy, and R. K. Hanson, "Degenerate four-wave mixing temperature measurements in a flame," Opt. Lett. 17, 751–753 (1992).
[CrossRef] [PubMed]

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

P. M. Danehy and R. L. Farrow, "Gas-phase velocimetry by nearly degenerate four-wave mixing," submitted to Appl. Phys. B.

Dean, A. M.

M.-S. Chou, A. M. Dean, and D. Stern, "Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames," J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Delve, P. A.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Dreier, T.

Dreizler, A.

A. Dreizler, T. Dreier, and J. Wolfrum, "Thermal grating effects in infrared degenerate four-wave mixing for trace gas detection," J. Chem. Phys. Lett. 233, 525 (1995).
[CrossRef]

R. B. Williams, P. Ewart, and A. Dreizler, "Velocimetry of gas flows using degenerate four-wave mixing," Opt. Lett. 19, 1486–1488 (1994).
[CrossRef] [PubMed]

Durant, J. L.

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

J. A. Gray, P. H. Paul, and J. L. Durant, Jr., "Electronic quenching rates for NO (A2Σ+) measured in a shock tube," Chem. Phys. Lett. 190, 266–270 (1992).
[CrossRef]

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

Eichler, H. J.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Ewart, P.

Farrow, R. L.

P. H. Paul, R. L. Farrow, and P. M. Danehy, "Gas-phase thermal-grating contributions to four-wave mixing," J. Opt. Soc. Am. B 12, 384–392 (1995).
[CrossRef]

E. J. Friedman-Hill, L. A. Rahn, and R. L. Farrow, "On the interpretation and rotational assignment of degenerate four-wave mixing spectra: four-photon line strengths for crossover resonances in NO A2Σ+ −X2II," J. Chem. Phys. 100, 4065–4076 (1994).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

R. L. Farrow, D. J. Rakestraw, and T. Dreier, "Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment," J. Opt. Soc. Am. B 9, 1770–1777 (1992).
[CrossRef]

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

R. L. Farrow and D. J. Rakestraw, "Detection of nitric oxide in a methane/air flame by degenerate four-wave mixing," to be submitted to Appl. Phys. B.

P. M. Danehy and R. L. Farrow, "Gas-phase velocimetry by nearly degenerate four-wave mixing," submitted to Appl. Phys. B.

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, "High-resolution investigation of DFWM in the γ(0, 0) band of nitric oxide," in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1992), pp. 1653–1659.
[CrossRef]

Fayer, M. D.

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

Forsman, J. W.

Friedman-Hill, E. J.

E. J. Friedman-Hill, L. A. Rahn, and R. L. Farrow, "On the interpretation and rotational assignment of degenerate four-wave mixing spectra: four-photon line strengths for crossover resonances in NO A2Σ+ −X2II," J. Chem. Phys. 100, 4065–4076 (1994).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

Gardiner, W. C.

W. C. Gardiner, Jr., Y. Hidaka, and T. Tanzawa, "Refractivity of combustion gases," Combust. Flame 40, 213–219 (1981).
[CrossRef]

Govoni, D. E.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Gray, J. A.

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

J. A. Gray, P. H. Paul, and J. L. Durant, Jr., "Electronic quenching rates for NO (A2Σ+) measured in a shock tube," Chem. Phys. Lett. 190, 266–270 (1992).
[CrossRef]

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

Greenhalgh, D. A.

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Hall, G.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Hanson, R. K.

B. Yip, P. M. Danehy, and R. K. Hanson, "Degenerate four-wave mixing temperature measurements in a flame," Opt. Lett. 17, 751–753 (1992).
[CrossRef] [PubMed]

For the temperature dependence of γ12 we used that reported by A. Y. Chang, M. D. Di Rosa, and R. K. Hanson ["Temperature dependence of collisional broadening and shift in the NO A ← X (0, 0) band in the presence of argon and nitrogen," J. Quant. Spectrosc. Radiat. Transfer 47, 375–390 (1992)] for NO in N2. We assumed the population decay rate Γ0 to be proportional to the hard-sphere collision frequency (αT−0.5) because of a lack of similar information.
[CrossRef]

Harris, J. M.

D. J. McGraw, J. Michaelson, and J. M. Harris, "Anharmonic forced Rayleigh scattering: a technique for study of saturated absorption in liquids," J. Chem. Phys. 86, 2536–2547 (1987).
[CrossRef]

Hayden, C. C.

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. 102, 2718–2726 (1995).
[CrossRef]

Hesselink, L.

Hidaka, Y.

W. C. Gardiner, Jr., Y. Hidaka, and T. Tanzawa, "Refractivity of combustion gases," Combust. Flame 40, 213–219 (1981).
[CrossRef]

Holman, J. P.

J. P. Holman, Heat Transfer, 6 ed. (McGraw-Hill, New York, 1986).

Holmes, B. E.

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

Hornung, H. G.

Jefferies, I. P.

I. P. Jefferies, A. J. Yates, and P. Ewart, "Temperature measurement by multiplex degenerate four-wave mixing in OH in methane/air flames," in Coherent Raman Spectroscopy Application and New Development, E. Castellucci, ed. (World Scientific, Singapore, 1992).

Jeffries, J. B.

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

Kaczmarek, M.

Kandel, S. A.

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

Kee, R. J.

R. J. Kee, F. M. Rupley, and J. A. Miller, "The Chemkin thermodynamic data base," Rept. SAND87-8215B (Sandia National Laboratories, Livermore, Calif., 1987).

Lam, J. F.

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

Laudenslager, J. B.

I. S. McDermid and J. B. Laudenslager, "Radiative lifetimes and electronic quenching rate constants for single-photon-excited rotational levels of NO (A2Σ+, v′ = 0)," J. Quant. Spectrosc. Radiat. Transfer 27, 483–492 (1982).
[CrossRef]

Leyva, I. A.

Liao, P. F.

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

Lind, R. C.

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

Lucht, R. P.

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

Magnusson, I.

Mann, B. A.

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

McDermid, I. S.

I. S. McDermid and J. B. Laudenslager, "Radiative lifetimes and electronic quenching rate constants for single-photon-excited rotational levels of NO (A2Σ+, v′ = 0)," J. Quant. Spectrosc. Radiat. Transfer 27, 483–492 (1982).
[CrossRef]

McGraw, D. J.

D. J. McGraw, J. Michaelson, and J. M. Harris, "Anharmonic forced Rayleigh scattering: a technique for study of saturated absorption in liquids," J. Chem. Phys. 86, 2536–2547 (1987).
[CrossRef]

Michaelson, J.

D. J. McGraw, J. Michaelson, and J. M. Harris, "Anharmonic forced Rayleigh scattering: a technique for study of saturated absorption in liquids," J. Chem. Phys. 86, 2536–2547 (1987).
[CrossRef]

Miller, J. A.

R. J. Kee, F. M. Rupley, and J. A. Miller, "The Chemkin thermodynamic data base," Rept. SAND87-8215B (Sandia National Laboratories, Livermore, Calif., 1987).

Neyer, D. W.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Nilsen, J.

J. Nilsen and A. Yariv, "A tunable narrowband optical filter via phase conjugation by nondegenerate four-wave mixing in a Doppler-broadened resonant medium," Opt. Commun. 39, 199–204 (1981).
[CrossRef]

O'Leary, S. V.

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

Paul, P. H.

P. H. Paul, R. L. Farrow, and P. M. Danehy, "Gas-phase thermal-grating contributions to four-wave mixing," J. Opt. Soc. Am. B 12, 384–392 (1995).
[CrossRef]

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

J. A. Gray, P. H. Paul, and J. L. Durant, Jr., "Electronic quenching rates for NO (A2Σ+) measured in a shock tube," Chem. Phys. Lett. 190, 266–270 (1992).
[CrossRef]

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

Pender, J.

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

Rahn, L. A.

E. J. Friedman-Hill, L. A. Rahn, and R. L. Farrow, "On the interpretation and rotational assignment of degenerate four-wave mixing spectra: four-photon line strengths for crossover resonances in NO A2Σ+ −X2II," J. Chem. Phys. 100, 4065–4076 (1994).
[CrossRef]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. I. Weak-field limit," J. Chem. Phys. 101, 1072–1092 (1994).
[CrossRef]

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. II. Strong-field limit," J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

Rakestraw, D. J.

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

R. L. Farrow, D. J. Rakestraw, and T. Dreier, "Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment," J. Opt. Soc. Am. B 9, 1770–1777 (1992).
[CrossRef]

T. Dreier and D. J. Rakestraw, "Measurement of OH rotational temperatures in a flame using degenerate four-wave mixing," Opt. Lett. 15, 72–74 (1990).
[CrossRef] [PubMed]

R. L. Farrow and D. J. Rakestraw, "Detection of nitric oxide in a methane/air flame by degenerate four-wave mixing," to be submitted to Appl. Phys. B.

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, "High-resolution investigation of DFWM in the γ(0, 0) band of nitric oxide," in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1992), pp. 1653–1659.
[CrossRef]

Rohlfing, E. A.

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460–5468 (1993).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. II. Photodissociation dynamics via photofragment transient gratings," J. Chem. Phys. 98, 5469–5476 (1993).
[CrossRef]

Rosa, M. D. Di

For the temperature dependence of γ12 we used that reported by A. Y. Chang, M. D. Di Rosa, and R. K. Hanson ["Temperature dependence of collisional broadening and shift in the NO A ← X (0, 0) band in the presence of argon and nitrogen," J. Quant. Spectrosc. Radiat. Transfer 47, 375–390 (1992)] for NO in N2. We assumed the population decay rate Γ0 to be proportional to the hard-sphere collision frequency (αT−0.5) because of a lack of similar information.
[CrossRef]

Rose, T. S.

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

Rupley, F. M.

R. J. Kee, F. M. Rupley, and J. A. Miller, "The Chemkin thermodynamic data base," Rept. SAND87-8215B (Sandia National Laboratories, Livermore, Calif., 1987).

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Sinha, A.

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

Smith, A. P.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Snowdon, P.

Steel, D. G.

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

Stern, D.

M.-S. Chou, A. M. Dean, and D. Stern, "Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames," J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Tanzawa, T.

W. C. Gardiner, Jr., Y. Hidaka, and T. Tanzawa, "Refractivity of combustion gases," Combust. Flame 40, 213–219 (1981).
[CrossRef]

Thoman, J. W.

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

Vaccaro, P. H.

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

Wäckerle, G.

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

Wal, R. L. Vander

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, "High-resolution investigation of DFWM in the γ(0, 0) band of nitric oxide," in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1992), pp. 1653–1659.
[CrossRef]

Wasserman, T. A. W.

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

Whitaker, B. J.

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Williams, R. B.

Williams, S.

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. II. Strong-field limit," J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. I. Weak-field limit," J. Chem. Phys. 101, 1072–1092 (1994).
[CrossRef]

Wilson, W. L.

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

Wolfrum, J.

A. Dreizler, T. Dreier, and J. Wolfrum, "Thermal grating effects in infrared degenerate four-wave mixing for trace gas detection," J. Chem. Phys. Lett. 233, 525 (1995).
[CrossRef]

Yariv, A.

J. Nilsen and A. Yariv, "A tunable narrowband optical filter via phase conjugation by nondegenerate four-wave mixing in a Doppler-broadened resonant medium," Opt. Commun. 39, 199–204 (1981).
[CrossRef]

Yates, A. J.

I. P. Jefferies, A. J. Yates, and P. Ewart, "Temperature measurement by multiplex degenerate four-wave mixing in OH in methane/air flames," in Coherent Raman Spectroscopy Application and New Development, E. Castellucci, ed. (World Scientific, Singapore, 1992).

Yip, B.

Zare, R. N.

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. I. Weak-field limit," J. Chem. Phys. 101, 1072–1092 (1994).
[CrossRef]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. II. Strong-field limit," J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

S. Williams, L. A. Rahn, P. H. Paul, J. W. Forsman, and R. N. Zare, "Laser-induced thermal grating effects in flames," Opt. Lett. 19, 1681–1683 (1994).
[CrossRef] [PubMed]

Zhang, Q.

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (3)

P. H. Paul, J. A. Gray, J. L. Durant, Jr., and J. W. Thoman, Jr., "A model for temperature-dependent collisional quenching of NO A2Σ+," Appl. Phys. B 57, 249–259 (1993).
[CrossRef]

B. A. Mann, S. V. O'Leary, A. G. Astill, and D. A. Greenhalgh, "Degenerate four-wave mixing in nitrogen dioxide: application to combustion diagnostics," Appl. Phys. B 54, 271–277 (1992).
[CrossRef]

P. M. Danehy, E. J. Friedman-Hill, R. P. Lucht, and R. L. Farrow, "The effects of collisional quenching on degenerate four-wave mixing," Appl. Phys. B 57, 243–248 (1993).
[CrossRef]

Appl. Phys. B. (1)

A. P. Smith, G. Hall, B. J. Whitaker, A. G. Astill, D. W. Neyer, and P. A. Delve, "Effects of inert gases on the degenerate four-wave mixing spectrum of NO2," Appl. Phys. B. 60, 11–18 (1995).
[CrossRef]

Chem. Phys. Lett. (3)

J. A. Gray, P. H. Paul, and J. L. Durant, Jr., "Electronic quenching rates for NO (A2Σ+) measured in a shock tube," Chem. Phys. Lett. 190, 266–270 (1992).
[CrossRef]

D. E. Govoni, J. A. Booze, A. Sinha, and F. F. Crim, "The non-resonant signal in laser-induced grating spectroscopy of gases," Chem. Phys. Lett. 216, 525–529 (1993).
[CrossRef]

R. L. Vander Wal, B. E. Holmes, J. B. Jeffries, P. M. Danehy, R. L. Farrow, and D. J. Rakestraw, "Detection of HF using infrared degenerate four-wave mixing," Chem. Phys. Lett. 191, 251–258 (1992).
[CrossRef]

Combust. Flame (1)

W. C. Gardiner, Jr., Y. Hidaka, and T. Tanzawa, "Refractivity of combustion gases," Combust. Flame 40, 213–219 (1981).
[CrossRef]

J. Chem. Phys. (11)

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. II. Strong-field limit," J. Chem. Phys. 101, 1093–1107 (1994).
[CrossRef]

S. Williams, R. N. Zare, and L. A. Rahn, "Reduction of degenerate four-wave mixing spectra to relative populations. I. Weak-field limit," J. Chem. Phys. 101, 1072–1092 (1994).
[CrossRef]

T. S. Rose, W. L. Wilson, G. Wäckerle, and M. D. Fayer, "Gas phase dynamics and spectroscopy probed with picosecond transient grating experiments," J. Chem. Phys. 86, 5370–5391 (1987).
[CrossRef]

M. A. Buntine, D. W. Chandler, and C. C. Hayden, "Detection of vibrational-overtone excitation in water via laser-induced grating spectroscopy," J. Chem. Phys. 102, 2718–2726 (1995).
[CrossRef]

E. J. Friedman-Hill, L. A. Rahn, and R. L. Farrow, "On the interpretation and rotational assignment of degenerate four-wave mixing spectra: four-photon line strengths for crossover resonances in NO A2Σ+ −X2II," J. Chem. Phys. 100, 4065–4076 (1994).
[CrossRef]

Q. Zhang, S. A. Kandel, T. A. W. Wasserman, and P. H. Vaccaro, "Detection of stimulated emission pumping via degenerate four-wave mixing," J. Chem. Phys. 96, 1640–1643 (1992).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. I. Spectroscopy of predissociating NO2," J. Chem. Phys. 98, 5460–5468 (1993).
[CrossRef]

T. J. Butenhoff and E. A. Rohlfing, "Laser-induced gratings in free jets. II. Photodissociation dynamics via photofragment transient gratings," J. Chem. Phys. 98, 5469–5476 (1993).
[CrossRef]

J. W. Thoman, Jr., J. A. Gray, J. L. Durant, Jr., and P. H. Paul, "Collisional electronic quenching of NO A2Σ+ by N2 from 300 to 4500 K," J. Chem. Phys. 97, 8156–8163 (1992).
[CrossRef]

M.-S. Chou, A. M. Dean, and D. Stern, "Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames," J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

D. J. McGraw, J. Michaelson, and J. M. Harris, "Anharmonic forced Rayleigh scattering: a technique for study of saturated absorption in liquids," J. Chem. Phys. 86, 2536–2547 (1987).
[CrossRef]

J. Chem. Phys. Lett. (1)

A. Dreizler, T. Dreier, and J. Wolfrum, "Thermal grating effects in infrared degenerate four-wave mixing for trace gas detection," J. Chem. Phys. Lett. 233, 525 (1995).
[CrossRef]

J. Opt. Soc. Am. B (2)

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

I. S. McDermid and J. B. Laudenslager, "Radiative lifetimes and electronic quenching rate constants for single-photon-excited rotational levels of NO (A2Σ+, v′ = 0)," J. Quant. Spectrosc. Radiat. Transfer 27, 483–492 (1982).
[CrossRef]

Opt. Commun. (1)

J. Nilsen and A. Yariv, "A tunable narrowband optical filter via phase conjugation by nondegenerate four-wave mixing in a Doppler-broadened resonant medium," Opt. Commun. 39, 199–204 (1981).
[CrossRef]

Opt. Lett. (7)

Other (11)

P. M. Danehy and R. L. Farrow, "Gas-phase velocimetry by nearly degenerate four-wave mixing," submitted to Appl. Phys. B.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, New York, 1986).
[CrossRef]

I. P. Jefferies, A. J. Yates, and P. Ewart, "Temperature measurement by multiplex degenerate four-wave mixing in OH in methane/air flames," in Coherent Raman Spectroscopy Application and New Development, E. Castellucci, ed. (World Scientific, Singapore, 1992).

R. L. Abrams, J. F. Lam, R. C. Lind, D. G. Steel, and P. F. Liao, "Phase conjugation and high-resolution spectroscopy by resonant degenerate four-wave mixing," in Optical Phase Conjugation, R. A. Fischer, ed. (Academic, San Diego, Calif., 1983), pp. 211–284.
[CrossRef]

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, "High-resolution investigation of DFWM in the γ(0, 0) band of nitric oxide," in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1992), pp. 1653–1659.
[CrossRef]

For the temperature dependence of γ12 we used that reported by A. Y. Chang, M. D. Di Rosa, and R. K. Hanson ["Temperature dependence of collisional broadening and shift in the NO A ← X (0, 0) band in the presence of argon and nitrogen," J. Quant. Spectrosc. Radiat. Transfer 47, 375–390 (1992)] for NO in N2. We assumed the population decay rate Γ0 to be proportional to the hard-sphere collision frequency (αT−0.5) because of a lack of similar information.
[CrossRef]

R. L. Farrow and D. J. Rakestraw, "Detection of nitric oxide in a methane/air flame by degenerate four-wave mixing," to be submitted to Appl. Phys. B.

R. J. Kee, F. M. Rupley, and J. A. Miller, "The Chemkin thermodynamic data base," Rept. SAND87-8215B (Sandia National Laboratories, Livermore, Calif., 1987).

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

D. E. Gray, ed., American Institute of Physics Handbook (McGraw-Hill, New York, 1972).

J. P. Holman, Heat Transfer, 6 ed. (McGraw-Hill, New York, 1986).

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Figures (10)

Fig. 1
Fig. 1

Ratio of thermal- to population-grating scattering intensities as a function of temperature and pressure, from Eq. (5). Excitation of the nearly degenerate Q2(18) + Q12(18) transitions of NO in CO2 buffer gas was assumed.

Fig. 2
Fig. 2

Schematic diagram of the experiment. PEM, photoelastic modulator; PMT’s, photomultiplier tubes; PD, photodetector.

Fig. 3
Fig. 3

Experimental DFWM intensities from a fixed density of NO in mixtures of N2 and CO2, as a function of the mole fraction of CO2 (a zero mole fraction of CO2 corresponds to pure N2). Measurements performed in ~100-Torr mixtures are indicated by triangles and in ~300-Torr mixtures by circles.

Fig. 4
Fig. 4

Experimental DFWM intensities from a fixed density of NO in various pressures of CO2 buffer gas, indicated by symbols. Measurements were performed with backward-pump delays of 2 ns (diamonds) and 24 ns (circles) with vertical laser and signal polarizations (VVVV). The triangles indicate measurements with a horizontally polarized probe (HVVH) and a 2-ns delay. The dashed line is based on a power-law fit to the HVVH measurements, increased by a factor of 12.9 as described in the text. Smooth curves have been drawn through the data points to guide the eye.

Fig. 5
Fig. 5

Experimental ratios of thermal- to population-grating scattering intensities (symbols) compared with theoretical predictions (curves) from the excitation of NO in various CO2 buffer pressures. The filled circles were obtained from the measurements shown in Fig. 4, as discussed in the text. The open diamonds were obtained from an independent analysis of NDFWM spectra (see Fig. 6). The solid curve is from Eq. (5). The dashed curve is the ratio of intensities predicted by a hydrodynamics model (for thermal gratings) to those of a moving-absorber perturbation theory (for population gratings).

Fig. 6
Fig. 6

NDFWM spectrum measured by exciting the O12(2) transition of NO with fixed forward-pump and probe frequencies and tuning the backward-pump frequency. The gas mixture was 30 mTorr of NO in 100 Torr of CO2. The central feature occurred when the backward pump resonantly diffracted from the O12(2) population grating excited by the forward-pump and probe beams. The other features resulted from resonances with O12(3) (left) and the P2 + P12 bandhead (right), whose level populations were perturbed by rotational energy transfer from the directly excited O12(2) levels.

Fig. 7
Fig. 7

NDFWM spectrum from an atmospheric-pressure methane/air flame, measured by tuning the backward pump frequency through the region of the Q2(18) + Q12(18) line of NO. The forward pump and probe frequencies were tuned to the center of this line. The strength of the observed peak relative the background shows that the population-grating mechanism dominates in DFWM of NO in this flame.

Fig. 8
Fig. 8

Time evolution of thermal-grating intensities from excitation of NO in various pressures of CO2 buffer gas. Thermal gratings were formed by the probe and forward-pump beams and were probed by stepping the time delay of the backward-pump beam. Each scan has been scaled to 1 and offset from the origin for clarity. The curve through the 100-Torr spectrum is a Gaussian. The other curves are calculations from a hydrodynamics model.

Fig. 9
Fig. 9

Time evolution of thermal-grating intensities from excitation of NO in 1000 Torr of CO2 buffer gas (top panel) and 1000 Torr of N2 (bottom panel). The smooth curves were calculated from a hydrodynamic model.

Fig. 10
Fig. 10

Experimental thermal-grating intensities (circles) compared with theoretical predictions (curve) from the excitation of NO in various CO2 buffer pressures. The long-dashed line shows a P4 dependence and the short-dashed line a P−2 dependence, corresponding to theoretical asymptotes for low and high pressures, respectively.

Equations (16)

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Δ n = Δ n res + Δ n res + Δ n buff , Δ α = Δ α res + Δ α res .
η = I sig I bp ( fp ) = ( π L Δ n λ ) 2 + ( L Δ α 4 ) 2 ,
S PG = τ I 3 ( σ Δ N 0 L I sat ) 2 1 ( 1 + 4 I / I sat ) 3 ,
S TG = 16 τ 3 I 3 3 [ ( n buff ρ ) T ] 2 ( π σ Δ N 0 L λ T c p ) 2 × 1 ( 1 + 4 I / I sat ) 2             ( I / I sat < 0.1 ) ,
S TG S PG = 16 3 [ ( n buff ρ ) T ] 2 ( π τ I sat λ T c p ) 2 × ( 1 + 4 I / I sat )             ( I / I sat < 0.1 ) .
S PG = 0 τ I b η PG ( t ) d t ,
Δ α abs = σ Δ ( Δ N ) ,
Δ N = Δ N 0 1 + I / I sat ,
I sat = ω Γ 0 σ ,
Δ ( Δ N ) = Δ N peak - Δ N valley Δ N 0 I sat ( I peak - I valley ) = Δ N 0 I sat 4 ( I f I b ) 1 / 2 .
S PG = τ I 3 ( σ Δ N 0 L I sat ) 2 ,
Δ n buff = ( n buff ρ ) T Δ ρ + ( n buff T ) ρ Δ T .
n - 1 = ( n 0 - 1 ) ρ ρ 0 ,
Δ T ( t ) = σ Δ N 0 t 4 I ρ c p .
S TG = 16 τ 3 I 3 3 [ ( n buff ρ ) T ] 2 ( π σ Δ N 0 L λ T c p ) 2 .
S TG = 16 τ 3 I 3 3 [ ( n buff ρ ) T ] 2 ( π σ Δ N 0 L λ T c p ) 2 × 1 ( 1 + 4 I / I sat ) 2 .

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