Abstract

Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Pender and L. Hesselink, “Phase conjugation in a flame,” Opt. Lett. 10, 264–266 (1985).
    [CrossRef] [PubMed]
  2. P. Ewart and S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
    [CrossRef] [PubMed]
  3. 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]
  4. T. Dreier and D. J. Rakestraw, “Degenerate four-wave mixing diagnostics on OH and NH radicals in flames,” Appl. Phys. B 50, 479–485 (1990).
    [CrossRef]
  5. P. Ewart, P. Snowdon, and I. Magnusson, “Two-dimensional phase-conjugate imaging of atomic distributions in flames by degenerate four-wave mixing,” Opt. Lett. 14, 563–565 (1989).
    [CrossRef] [PubMed]
  6. D. J. Rakestraw, R. L. Farrow, and T. Dreier, “Two-dimensional imaging of OH in flames by degenerate four-wave mixing,” Opt. Lett. 15, 709–711 (1990).
    [CrossRef] [PubMed]
  7. 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]
  8. M. Winter and P. P. Radi, “Nearly degenerate four-wave mixing using phase conjugate pump beams,” Opt. Lett. 17, 320–322 (1992).
    [CrossRef] [PubMed]
  9. R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, “High-resolution investigation of degenerate four-wave mixing in the γ(0, 0) band of nitric oxide,” in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1993), pp. 1653–1659.
  10. P. Ewart and P. Snowdon, “Multiplex degenerate four-wave mixing in a flame,” Opt. Lett. 15, 1403–1405 (1990).
    [CrossRef] [PubMed]
  11. 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]
  12. M. S. Brown, L. A. Rahn, and T. Dreier, “High-resolution degenerate four-wave mixing spectral profiles for OH,” Opt. Lett. 17, 76–78 (1992).
    [CrossRef] [PubMed]
  13. H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
    [CrossRef]
  14. 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 10, 1770–1777 (1992).
    [CrossRef]
  15. 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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
    [CrossRef]
  16. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum 3, 205 (1978).
    [CrossRef] [PubMed]
  17. J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a Doppler-broadened resonant medium,” J. Opt. Soc. Am. 71, 180–183 (1981).
    [CrossRef]
  18. M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened systems. I. Angular dependence of intensity and lineshape of phase-conjugate emission,” J. Phys. (Paris) 42, 711–721 (1981).
    [CrossRef]
  19. D. Bloch and M. Ducloy, “Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems,” J. Opt. Soc. Am. 73, 635–646 (1983); erratum 73, 1844–1845 (1983).
    [CrossRef]
  20. M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
    [CrossRef] [PubMed]
  21. S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
    [CrossRef]
  22. G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
    [CrossRef]
  23. G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
    [CrossRef]
  24. P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
    [CrossRef] [PubMed]
  25. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), p. 191.
  26. R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
    [CrossRef] [PubMed]
  27. A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
    [CrossRef]
  28. M.-S. Chou, A. M. Dean, and D. Stern, “Laser-induced fluorescence and absorption measurements of NO in NH3/O2and CH4/air flames,” J. Chem. Phys. 78, 5962–5970 (1983).
    [CrossRef]
  29. R. P. Lucht and R. L. Farrow, “Calculation of saturation line shapes and intensities in coherent anti-Stokes Raman scattering spectra of nitrogen,” J. Opt. Soc. Am. B 5, 1243–1252 (1988).
    [CrossRef]
  30. R. P. Lucht and R. L. Farrow, “Saturation effects in coherent anti-Stokes Raman scattering spectra of hydrogen,” J. Opt. Soc. Am. B 6, 2313–2325 (1989).
    [CrossRef]
  31. T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
    [CrossRef]
  32. J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
    [CrossRef] [PubMed]
  33. S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
    [CrossRef]

1992 (8)

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]

M. S. Brown, L. A. Rahn, and T. Dreier, “High-resolution degenerate four-wave mixing spectral profiles for OH,” Opt. Lett. 17, 76–78 (1992).
[CrossRef] [PubMed]

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (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 10, 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]

M. Winter and P. P. Radi, “Nearly degenerate four-wave mixing using phase conjugate pump beams,” Opt. Lett. 17, 320–322 (1992).
[CrossRef] [PubMed]

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

1990 (4)

1989 (3)

1988 (1)

1987 (2)

T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
[CrossRef]

P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef] [PubMed]

1986 (3)

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
[CrossRef]

P. Ewart and S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
[CrossRef] [PubMed]

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

1985 (2)

J. Pender and L. Hesselink, “Phase conjugation in a flame,” Opt. Lett. 10, 264–266 (1985).
[CrossRef] [PubMed]

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

1984 (1)

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
[CrossRef]

1983 (2)

1981 (2)

J. Nilsen and A. Yariv, “Nondegenerate four-wave mixing in a Doppler-broadened resonant medium,” J. Opt. Soc. Am. 71, 180–183 (1981).
[CrossRef]

M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened systems. I. Angular dependence of intensity and lineshape of phase-conjugate emission,” J. Phys. (Paris) 42, 711–721 (1981).
[CrossRef]

1978 (1)

1972 (1)

A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
[CrossRef]

Abrams, R. L.

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum 3, 205 (1978).
[CrossRef] [PubMed]

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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
[CrossRef]

Alber, G.

J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Attal-Tretout, B.

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
[CrossRef]

Bervas, H.

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
[CrossRef]

Bloch, D.

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

D. Bloch and M. Ducloy, “Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems,” J. Opt. Soc. Am. 73, 635–646 (1983); erratum 73, 1844–1845 (1983).
[CrossRef]

M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened systems. I. Angular dependence of intensity and lineshape of phase-conjugate emission,” J. Phys. (Paris) 42, 711–721 (1981).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), p. 191.

Brown, M. S.

Charlton, A.

J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Chou, M.-S.

M.-S. Chou, A. M. Dean, and D. Stern, “Laser-induced fluorescence and absorption measurements of NO in NH3/O2and CH4/air flames,” J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Cooper, J.

J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Danehy, P. M.

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]

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]

de Oliveira, F. A. M.

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

Dean, A. M.

M.-S. Chou, A. M. Dean, and D. Stern, “Laser-induced fluorescence and absorption measurements of NO in NH3/O2and CH4/air flames,” J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Dreier, T.

Ducloy, M.

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

D. Bloch and M. Ducloy, “Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems,” J. Opt. Soc. Am. 73, 635–646 (1983); erratum 73, 1844–1845 (1983).
[CrossRef]

M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened systems. I. Angular dependence of intensity and lineshape of phase-conjugate emission,” J. Phys. (Paris) 42, 711–721 (1981).
[CrossRef]

Ewart, P.

Farmer, A. J. D.

A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
[CrossRef]

Farrow, R. L.

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 10, 1770–1777 (1992).
[CrossRef]

D. J. Rakestraw, R. L. Farrow, and T. Dreier, “Two-dimensional imaging of OH in flames by degenerate four-wave mixing,” Opt. Lett. 15, 709–711 (1990).
[CrossRef] [PubMed]

R. P. Lucht and R. L. Farrow, “Saturation effects in coherent anti-Stokes Raman scattering spectra of hydrogen,” J. Opt. Soc. Am. B 6, 2313–2325 (1989).
[CrossRef]

R. P. Lucht and R. L. Farrow, “Calculation of saturation line shapes and intensities in coherent anti-Stokes Raman scattering spectra of nitrogen,” J. Opt. Soc. Am. B 5, 1243–1252 (1988).
[CrossRef]

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, “High-resolution investigation of degenerate four-wave mixing in the γ(0, 0) band of nitric oxide,” in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1993), pp. 1653–1659.

Green, D. S.

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

Grynberg, G.

P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef] [PubMed]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Hanson, R. K.

Hasson, V.

A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
[CrossRef]

Hesselink, L.

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]

Imajo, T.

T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
[CrossRef]

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]

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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
[CrossRef]

Le Boiteaux, S.

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
[CrossRef]

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
[CrossRef]

Lind, R. C.

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum 3, 205 (1978).
[CrossRef] [PubMed]

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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
[CrossRef]

Lucht, R. P.

Magnusson, I.

Meacher, D. R.

J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Nicholls, R. W.

A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
[CrossRef]

Nilsen, J.

O’Leary, S. V.

Obi, K.

T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
[CrossRef]

Pender, J.

Pinard, M.

P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef] [PubMed]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Radi, P. P.

Rahn, L. A.

M. S. Brown, L. A. Rahn, and T. Dreier, “High-resolution degenerate four-wave mixing spectral profiles for OH,” Opt. Lett. 17, 76–78 (1992).
[CrossRef] [PubMed]

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

Rakestraw, D. J.

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 10, 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]

T. Dreier and D. J. Rakestraw, “Degenerate four-wave mixing diagnostics on OH and NH radicals in flames,” Appl. Phys. B 50, 479–485 (1990).
[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]

D. J. Rakestraw, R. L. Farrow, and T. Dreier, “Two-dimensional imaging of OH in flames by degenerate four-wave mixing,” Opt. Lett. 15, 709–711 (1990).
[CrossRef] [PubMed]

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, “High-resolution investigation of degenerate four-wave mixing in the γ(0, 0) band of nitric oxide,” in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1993), pp. 1653–1659.

Sethuraman, S.

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

Shibuya, K.

T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
[CrossRef]

Simoneau, P.

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[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. Fisher, ed. (Academic, New York, 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/O2and CH4/air flames,” J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

Taran, J.-P.

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
[CrossRef]

Trebino, R.

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

Vander Wal, R. L.

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 degenerate four-wave mixing in the γ(0, 0) band of nitric oxide,” in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1993), pp. 1653–1659.

Verkerk, P.

P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef] [PubMed]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
[CrossRef]

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Williams, S.

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

Winter, M.

Yariv, A.

Yip, B.

Zare, R. N.

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

Appl. Phys. B (1)

T. Dreier and D. J. Rakestraw, “Degenerate four-wave mixing diagnostics on OH and NH radicals in flames,” Appl. Phys. B 50, 479–485 (1990).
[CrossRef]

Chem. Phys. Lett. (2)

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]

T. Imajo, K. Shibuya, and K. Obi, “Rotational energy transfer in the NO A2Σ+(υ′= 0) state with He and Ar,” Chem. Phys. Lett. 137, 139–143 (1987).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Le Boiteaux, P. Simoneau, D. Bloch, F. A. M. de Oliveira, and M. Ducloy, “Saturation behavior of resonant degenerate four-wave and multiwave mixing in the Doppler-broadened regime: experimental analysis on a low pressure Ne discharge,” IEEE J. Quantum Electron. QE-22, 1229–1247 (1986).
[CrossRef]

J. Am. Chem. Soc. (1)

S. Williams, D. S. Green, S. Sethuraman, and R. N. Zare, “Detection of trace species in hostile environments using degenerate four-wave mixing: CH in an atmospheric pressure flame,” J. Am. Chem. Soc. 114, 9122–9130 (1992).
[CrossRef]

J. Chem. Phys. (1)

M.-S. Chou, A. M. Dean, and D. Stern, “Laser-induced fluorescence and absorption measurements of NO in NH3/O2and CH4/air flames,” J. Chem. Phys. 78, 5962–5970 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (3)

J. Phys. (Paris) (2)

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Phys. (Paris) 47, 617–630 (1986).
[CrossRef]

M. Ducloy and D. Bloch, “Theory of degenerate four-wave mixing in resonant Doppler-broadened systems. I. Angular dependence of intensity and lineshape of phase-conjugate emission,” J. Phys. (Paris) 42, 711–721 (1981).
[CrossRef]

J. Phys. B (1)

H. Bervas, B. Attal-Tretout, S. Le Boiteaux, and J.-P. Taran, “OH detection and spectroscopy by DFWM in flames; comparison with CARS,” J. Phys. B 25, 949–969 (1992).
[CrossRef]

J. Quantum Spectrosc. Radiat. Transfer (1)

A. J. D. Farmer, V. Hasson, and R. W. Nicholls, “Absolute oscillator strength measurements of the (υ″= 0, υ′= 0) bands of the (A2Σ − X2Π) γ-system of nitric oxide,” J. Quantum Spectrosc. Radiat. Transfer 12, 627–633 (1972).
[CrossRef]

Opt. Commun. (1)

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: the dressed atom approach,” Opt. Commun. 50, 261–264 (1984).
[CrossRef]

Opt. Lett. (10)

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum 3, 205 (1978).
[CrossRef] [PubMed]

M. Winter and P. P. Radi, “Nearly degenerate four-wave mixing using phase conjugate pump beams,” Opt. Lett. 17, 320–322 (1992).
[CrossRef] [PubMed]

P. Ewart and P. Snowdon, “Multiplex degenerate four-wave mixing in a flame,” Opt. Lett. 15, 1403–1405 (1990).
[CrossRef] [PubMed]

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]

M. S. Brown, L. A. Rahn, and T. Dreier, “High-resolution degenerate four-wave mixing spectral profiles for OH,” Opt. Lett. 17, 76–78 (1992).
[CrossRef] [PubMed]

P. Ewart, P. Snowdon, and I. Magnusson, “Two-dimensional phase-conjugate imaging of atomic distributions in flames by degenerate four-wave mixing,” Opt. Lett. 14, 563–565 (1989).
[CrossRef] [PubMed]

D. J. Rakestraw, R. L. Farrow, and T. Dreier, “Two-dimensional imaging of OH in flames by degenerate four-wave mixing,” Opt. Lett. 15, 709–711 (1990).
[CrossRef] [PubMed]

J. Pender and L. Hesselink, “Phase conjugation in a flame,” Opt. Lett. 10, 264–266 (1985).
[CrossRef] [PubMed]

P. Ewart and S. V. O’Leary, “Detection of OH in a flame by degenerate four-wave mixing,” Opt. Lett. 11, 279–281 (1986).
[CrossRef] [PubMed]

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]

Phys. Rev. A (4)

M. Ducloy, F. A. M. de Oliveira, and D. Bloch, “Theory of resonant Doppler-broadened backward four-wave mixing in the pump saturation regime,” Phys. Rev. A 32, 1614–1623 (1985).
[CrossRef] [PubMed]

P. Verkerk, M. Pinard, and G. Grynberg, “Backward saturation in four-wave mixing in neon: case of cross-polarized pumps,” Phys. Rev. A 35, 4679–4695 (1987).
[CrossRef] [PubMed]

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

J. Cooper, A. Charlton, D. R. Meacher, P. Ewart, and G. Alber, “Revised theory of resonant degenerate four-wave mixing with broad-bandwidth lasers,” Phys. Rev. A 40, 5705–5715 (1989).
[CrossRef] [PubMed]

Other (3)

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), p. 191.

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. Fisher, ed. (Academic, New York, 1983), pp. 211–284.
[CrossRef]

R. L. Vander Wal, R. L. Farrow, and D. J. Rakestraw, “High-resolution investigation of degenerate four-wave mixing in the γ(0, 0) band of nitric oxide,” in Twenty-Fourth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1993), pp. 1653–1659.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (20)

Fig. 1
Fig. 1

Geometry for the DFWM calculations.

Fig. 2
Fig. 2

Time dependence of DFWM signals for two different dephasing rates at low laser intensity. The DFWM resonance was homogeneously broadened, and the laser frequency was tuned to line center. For both dephasing rates (γ21 = 5 and 25 ns−1), Ipump = 0.0023Isat and Ipr = Ipump.

Fig. 3
Fig. 3

Time dependence of DFWM signals for two different dephasing rates at high laser intensity. The DFWM resonance was homogeneously broadened, and the laser frequency was tuned to line center. For both dephasing rates (γ21 = 5 and 25 ns−1), Ipump = 14Isat and Ipr = Ipump/4.

Fig. 4
Fig. 4

Time dependence of the DFWM signals for a collision-and Doppler-broadened resonance. The laser frequency was tuned to line center. For dephasing rates γ21 = 5 ns−1, Ipump = 14Isat and Ipr = Ipump. Thirty-one velocity groups with a spacing of 0.01 cm−1 and 500 spatial grid points were included in the calculations.

Fig. 5
Fig. 5

Time dependence of the excited-state population ρ22 for specific velocity groups at different levels of saturation. Calculations are shown for a, Δωvel = 0 cm−1, Ipump = 15Isat; b, Δωvel = 0.02 cm−1, Ipump = 0.6Isat; c, Δωvel = 0.02 cm−1, Ipump = 15Isat. The laser detuning ΔωL, = ω0ωL = 0.05 cm−1, the dephasing rate γ21 = 5 ns−1, and Ipr = Ipump/4 for all three calculations.

Fig. 6
Fig. 6

Comparison of line shapes from perturbation theory and direct numerical integration of the density matrix equations for low laser intensity. For the direct numerical integration solution, Ipump = 0.0028Isat and Ipr = Ipump. The perturbation solution was generated by a computer code developed by Vander Wal et al.9

Fig. 7
Fig. 7

Comparison of experimental line shapes and the results of nonperturbative calculations for the NO O12(2) rotational transition in the A2Σ+X2Π (0, 0) band. The cell contained ~25 mTorr of NO. The laser pulse energy was kept constant as the He pressure in the cell was varied. The probe-laser intensity was equal to 1/4 of the pump laser intensity in both the experiment and the calculations.

Fig. 8
Fig. 8

Normalized DFWM line shapes as a function of laser intensity for a weak probe (Ipr = Isat/10) and a homogeneously broadened line. For the DFWM resonance, ΔωC = 0.053 cm−1 and Γ21 = γ21.

Fig. 9
Fig. 9

Normalized DFWM line shapes as a function of laser intensity for a strong probe (Ipr = Ipump) and a homogeneously broadened line. For the DFWM resonance, ΔωC = 0.053 cm−1 and Γ21 = γ21.

Fig. 10
Fig. 10

Normalized DFWM line shapes as a function of laser intensity for a strong-probe case (Ipr = Isat/4) and a line with both Doppler and collisional broadening. For the DFWM resonance, ΔωD = 0.1 cm−1, ΔωC = 0.053 cm−1, and Γ21 = γ21.

Fig. 11
Fig. 11

Normalized DFWM line shapes as a function of laser intensity for a weak-probe case Ipr = Isat/10) and a line with both Doppler and collisional broadening. For the DFWM resonance, ΔωD = 0.1 cm−1, ΔωC = 0.053 cm−1, and Γ21 = γ21.

Fig. 12
Fig. 12

Normalized DFWM line shapes as a function of laser intensity for a strong-probe case (Ipr = Ipump) and a line with both Doppler and collisional broadening. For the DFWM resonance, ΔωD = 0.1 cm−1, ΔωC = 0.053 cm−1, and Γ21 = γ21.

Fig. 13
Fig. 13

Calculated spatial profiles of the excited-state population along the z axis for the zero-velocity group for three different values of the probe intensity: a, Ipr = 0; b, Ipr = Isat/10; c, Ipr = 2.5Isat. For all three calculations, Ipump = 10Isat and ΔωC = 0.053 cm−1.

Fig. 14
Fig. 14

Calculated and experimental9 pressure dependence of the peak (line center) DFWM signal for the NO O12(2) transition. For the NO in He buffer gas the collisional width ΔωC = 0.053 cm−1 for a He pressure of 100 Torr. The Doppler widith ΔωD = 0.1 cm−1. The probe-laser intensity was equal to 1/4 of the pump-laser intensity in both the experiment and the calculations (Ωpr = Ωpump/2).

Fig. 15
Fig. 15

Data of Fig. 14 replotted with an expanded vertical scale. The position on each constant pump intensity curve where Ipump = 2Isat is indicated by an inverted triangle.

Fig. 16
Fig. 16

Calculated saturation curves for DFWM resonances with different ratios of Doppler to collisional broadening. The curves are normalized so that the peak DFWM signal S4 = 1 for Ipump = Isat for each curve. The Doppler width ΔωD = 0.1 cm−1 for each curve. The probe-laser intensity was equal to 1/4 of the pump-laser intensity in the calculations.

Fig. 17
Fig. 17

Calculated saturation curves for three different values of probe-laser intensity. The peak DFWM signal is normalized such that S4 = 1 for Ipump = Isat, for the curve corresponding to Ipr = Ipump.

Fig. 18
Fig. 18

Calculated reflectivity versus pump intensity for three different values of probe-laser intensity. The reflectivity is given in terms of arbitrary units and corresponds only to relative efficiency.

Fig. 19
Fig. 19

Calculated saturation curves for three different values of the electronic quenching rate Γ21. The collisional width ΔωC = 0.053 cm−121 = 5 ns−1) and the Doppler width ΔωD = 0.1 cm−1 are the same for each curve. The probe-laser intensity was equal to 1/4 of the pump-laser intensity in the calculations. The pump-laser intensity is normalized so that Ipump = 1 at the saturation intensity for the case where Γ21 = γ21. The saturation intensities for Γ21 = γ21, Γ21 = γ21/2, and Γ21 = γ21/5 are indicated by the arrows.

Fig. 20
Fig. 20

Calculated laser intensity dependence of the normalized DFWM signal for three different values of the electronic quenching rate Γ21. The collisional width ΔωC = 0.053 cm−121 = 5 ns−1) and the Doppler width ΔωD = 0.1 cm−1 are the same for each curve. The probe-laser intensity was equal to 1/4 of the pump-laser intensity in the calculations.

Equations (24)

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

ρ 11 ( z , t ) t = i ћ ( V 12 ρ 21 ρ 12 V 21 ) + Γ 21 ρ 22 ,
ρ 21 ( z , t ) t = ρ 21 ( i ω 21 + γ 21 ) i ћ V 21 ( ρ 11 ρ 22 ) ,
V 21 = μ 21 · E ( z , t ) = μ 21 [ E 1 ( z , t ) + E 2 ( z , t ) + E 3 ( z , t ) ] ,
E n ( z , t ) = 1 2 A n ( t ) exp { i [ k n ( cos θ n ) z ω n t ] } + c . c ., n = 1 , 2 , 3.
α 21 ( z , t ) t = α 21 γ 21 β 21 ( ω 1 ω 21 ) ( ρ 11 ρ 22 ) W ( z , t ) ,
β 21 ( z , t ) t = β 21 γ 21 + α 21 ( ω 1 ω 21 ) + ( ρ 11 ρ 22 ) U ( z , t ) ,
ρ 11 ( z , t ) t = 2 α 21 W ( z , t ) + 2 β 21 U ( z , t ) + ρ 22 Γ 21 ,
U ( z , t ) = Ω 1 ( t ) cos ( Φ 1 z ) + Ω 2 ( t ) cos ( Φ 2 z ) cos [ ( ω 1 ω 2 ) t ] Ω 2 ( t ) sin ( Φ 2 z ) sin [ ( ω 1 ω 2 ) t ] + Ω 3 ( t ) cos ( Φ 3 z ) cos [ ( ω 1 ω 3 ) t ] Ω 3 ( t ) sin ( Φ 3 z ) sin [ ( ω 1 ω 2 ) t ] ,
W ( z , t ) = Ω 1 ( t ) sin ( Φ 1 z ) + Ω 2 ( t ) cos ( Φ 2 z ) sin [ ( ω 1 ω 2 ) t ] + Ω 2 ( t ) sin ( Φ 2 z ) cos [ ( ω 1 ω 2 ) t ] + Ω 3 ( t ) cos ( Φ 3 z ) sin [ ( ω 1 ω 3 ) t ] + Ω 3 ( t ) sin ( Φ 3 z ) cos [ ( ω 1 ω 2 ) t ] .
Ω n ( t ) = μ 21 A n ( t ) 2 ћ
P ( z , t ) = μ 21 [ ρ 12 ( z , t ) + ρ 21 ( z , t ) ] = μ 21 [ σ 12 exp ( + i ω 1 t ) + σ 21 exp ( i ω 1 t ) ] ,
P ( z , t ) = P 4 ( z , t ) + P other ( z , t ) .
P 4 ( z , t ) = P 40 ( t ) exp [ i ( k 4 z ω 4 t ) ] .
A 4 ( t ) = A 4 ( z M , t ) = m = 1 m = M p ( z m , t ) exp [ i ( k 4 z m ω 4 t ) ] .
A 4 ( t ) = μ 21 m = 1 m = M { [ α 21 ( z m , t ) cos ( k 4 z m ) + β 21 ( z m , t ) sin ( k 4 z m ) ] + i [ α 21 ( z m , t ) sin ( k 4 z m ) + β 21 ( z m , t ) cos ( k 4 z m ) ] } .
S 4 ( ω L ) = 0 4 t pulse { [ u z A 4 i ( t ) ] 2 + [ u z A 4 r ( t ) ] 2 } d t ,
P 4 L ( z , t ) = P 4 L 0 ( t ) exp [ i ( k L z ω L t ) ] ,
P 4 L ( z , t ) = P 40 ( t ) exp [ i ( k L z ω L t ) ] = { P 4 ( z , t ) exp [ i ( k 4 z ω 4 t ) ] } exp [ i ( k L z ω L t ) ] ,
P L ( z , t ) = P 4 L ( z , t ) + P other L ( z , t ) = { P ( z , t ) exp [ i ( k 4 z ω 4 t ) ] } exp [ i ( k L z ω L t ) ] .
Ω sat = ( γ 21 Γ 21 ) 1 / 2 / 2
Ω sat = ( γ 21 Γ 21 ) 1 / 2 / 4 π c
α 21 ( z , t ) t = β 21 ( z , t ) t = ρ 22 ( z , t ) δ t = 0 ,
μ 21 = ( 3 ћ e 2 f 12 / 2 m e ω 21 ) 1 / 2 = 4.9 × 10 20 g 1 / 2 cm 5 / 2 s 1 ,
I pump ( theor ) = n c 8 π A pump 2 = n c 8 π ( 4 π c ћ Ω pump / μ 21 ) 2 = 4.0 MW / cm 2 .

Metrics