Abstract

We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ D. Bloch et al., Phys. Rev. Lett. 49, 719 ( 1982)].

© 1983 Optical Society of America

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Corrections

Daniel Bloch and Martial Ducloy, "Theory of saturated line shapes in phase-conjugate emission by resonant degenerate four-wave mixing in Doppler-broadened three-level systems: errata," J. Opt. Soc. Am. 73, 1844-1845 (1983)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-73-12-1844

References

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  1. Opt. Eng. 21, 155–283 (1982) (special issue on phase-conjugation).
  2. M. Ducloy, “Nonlinear optical phase-conjugation,” in Festkörperprobleme—Advances in Solid State Physics (Viegweg, Braunschweig, 1982), Vol. XXII, pp. 35–60.
    [Crossref]
  3. R. A. Fisher, Nonlinear Optical Phase-Conjugation (Academic, New York, 1982).
  4. B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).
  5. J. P. Woerdman, “Formation of a transient free carrier hologram in Si,” Opt. Commun. 2, 212–214 (1970).
    [Crossref]
  6. R. W. Hellwarth, “Generation of time-reversed wavefronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [Crossref]
  7. D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
    [Crossref]
  8. D. G. Steel and R. C. Lind, “Multiresonant behavior in nearly degenerate four-wave mixing: the ac Stark effect,” Opt. Lett. 6, 587–589 (1981).
    [Crossref] [PubMed]
  9. S. M. Wandzura, “Effects of atomic motion on wave-front conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Lett. 4, 208–210 (1979).
    [Crossref] [PubMed]
  10. 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]
  11. P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
    [Crossref]
  12. D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
    [Crossref]
  13. L. M. Humphrey, J. P. Gordon, and P. F. Liao, “Angular dependence of line shape and strength of degenerate four-wave mixing in a Doppler-broadened system with optical pumping,” Opt. Lett. 5, 56–58 (1980).
    [Crossref] [PubMed]
  14. J. P. Woerdman and M. F. H. Schuurmans, “Effect of saturation on the spectrum of degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 6, 239–241 (1981).
    [Crossref] [PubMed]
  15. D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
    [Crossref]
  16. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata 3, 205 (1978). In the optically thin sample approximation, the emission intensity line shape is proportional to |κ|2, in the notation of Abrams and Lind.
    [Crossref] [PubMed]
  17. D. J. Harter and R. W. Boyd, in “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980)], have actually shown that, in nearly degenerate four-wave mixing, when the pump-probe frequency detuning is varied, the ac Stark splitting is responsible for the appearance of structures in the emission line shape. Their theory, which does not take into account the atomic motion, cannot apply to DFWM experiments in which the common frequency of pump and probe beams is scanned through the atomic resonance (see also Ref. 8).
    [Crossref]
  18. D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
    [Crossref]
  19. A coupled three-level system is particularly adequate to describe a J= 1 → J= 0 transition with cross-polarized pumps.15
  20. In all the calculation, one neglects the cascade effects induced by spontaneous emission among |b〉, |a〉, and |c〉 so that Λ-type and V-type three-level systems are equivalent.
  21. For sake of simplicity, the choice of the space and time origins is such that the phase term is eliminated in the definition of the incident fields: One can easily verify that the phase of the probe field is sign reversed in the component that we are calculating.
  22. In four-wave mixing, the emission of one photon in the PC field occurs simultaneously with the emission of one photon in the probe field, at the expense of absorption of one photon of each pump field. A too-high increase in probe intensity does not help one to increase the reemitted intensity, so the PC reflectivity RPC(as an efficiency rate) decreases.
  23. S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969); B. J. Feldman and M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375–1396 (1970).
    [Crossref]
  24. Actually, the formalism is not specific to the set of polarizations discussed at the beginning of Section 2. It can describe a two-level system if the polarization of EB is identical with the one of EF and EP(e.m. field coupling between |a〉 and |b〉 only; no coupling between |c〉 and |b〉). The quantity of interest becomes βab(0)〉 instead of 〈βcb(0)〉, and minor and obvious modifications must be done to Eqs. (7)–(13).
  25. The backward pump is too weak to modify the population in level |c〉: αcc(−1)= 0, and the equations yielding αcb(−2)and βac(3)are decoupled.
  26. D. Bloch and M. Ducloy, “Polarization selection rules and disorienting collision effects in resonant degenerate four-wave mixing,” J. Phys. B 14, L-471–L476 (1981).
    [Crossref]
  27. M. Ducloy, R. K. Raj, and D. Bloch, “Polarization characteristics of phase-conjugate mirrors obtained by resonant degenerate four-wave mixing,” Opt. Lett. 7, 60–62 (1982).
    [Crossref] [PubMed]
  28. J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
    [Crossref]
  29. D. Bloch, “Conjugaison de phase dans les milieux gazeux. Spectroscopie de saturation hétérodyne,” Thèse de Troisième Cycle (Université Paris-Nord, Paris, 1980).
  30. J. H. Shirley, “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett. 7, 537–539 (1982).
    [Crossref] [PubMed]
  31. D. Bloch and M. Ducloy, unpublished report (Université Paris-Nord, Paris, 1983).
  32. In the case of weak saturation, an approximate justification for the single-lifetime model can be found if θ is small but not negligible (θ≫ γi/ku, θ~ γij/ku). The effective grating lifetime is essentially shortened by the thermal motion and is almost independent of the relaxation processes (see Ref. 27). For strong saturation, the situation is complicated by the homogeneous broadening of the relaxation processes.
  33. These structures are predicted only for a high-saturation parameter, so that the conditions for infinite Doppler-width approximation may no longer be fulfilled. This could explain why we were unable to observe these types of structure experimentally, whereas all the other line-shape features calculated in the present paper have been observed experimentally.15
  34. These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
    [Crossref]
  35. For SF= 1, the width of the absorption line shape is about 2.5 times the width at SF= 0 (natural width).
  36. B. Couillaud and A. Ducasse, “Les lasers à colorant continus. Leur application à la spectroscopie d’absorption saturée,” Thèse d’Etat (Université de Bordeaux I, Bordeaux, France, 1978).
  37. Let us give a numerical estimation of R0, the limit value of the reflectivity for infinite backward saturation. From Eq. (31), one finds that, at saturation, the maximum value of |χNL/χ(3)(0)|2is 0.7 SB−1for δ= 3ΩB/4. Let us consider transitions having oscillator strengths equal to unity and wavelength λ = 0.6 μ m, an atomic density of 1011atoms/cm3(corresponding to p= 3.10−6Torr at 300 K), an interaction length L= 1 cm, and a mean thermal velocity u= 500 m/sec. With these values, one deduce from Eqs. (26) and (28) that R0/SF= 0.32.
  38. A similar saturation anisotropy was recently observed in SF6[G. P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982)].
    [Crossref] [PubMed]

1982 (5)

1981 (5)

D. Bloch and M. Ducloy, “Polarization selection rules and disorienting collision effects in resonant degenerate four-wave mixing,” J. Phys. B 14, L-471–L476 (1981).
[Crossref]

J. P. Woerdman and M. F. H. Schuurmans, “Effect of saturation on the spectrum of degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 6, 239–241 (1981).
[Crossref] [PubMed]

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
[Crossref]

D. G. Steel and R. C. Lind, “Multiresonant behavior in nearly degenerate four-wave mixing: the ac Stark effect,” Opt. Lett. 6, 587–589 (1981).
[Crossref] [PubMed]

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]

1980 (2)

D. J. Harter and R. W. Boyd, in “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980)], have actually shown that, in nearly degenerate four-wave mixing, when the pump-probe frequency detuning is varied, the ac Stark splitting is responsible for the appearance of structures in the emission line shape. Their theory, which does not take into account the atomic motion, cannot apply to DFWM experiments in which the common frequency of pump and probe beams is scanned through the atomic resonance (see also Ref. 8).
[Crossref]

L. M. Humphrey, J. P. Gordon, and P. F. Liao, “Angular dependence of line shape and strength of degenerate four-wave mixing in a Doppler-broadened system with optical pumping,” Opt. Lett. 5, 56–58 (1980).
[Crossref] [PubMed]

1979 (3)

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[Crossref]

S. M. Wandzura, “Effects of atomic motion on wave-front conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Lett. 4, 208–210 (1979).
[Crossref] [PubMed]

These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
[Crossref]

1978 (3)

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
[Crossref]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata 3, 205 (1978). In the optically thin sample approximation, the emission intensity line shape is proportional to |κ|2, in the notation of Abrams and Lind.
[Crossref] [PubMed]

1977 (1)

1976 (1)

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

1971 (1)

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).

1970 (1)

J. P. Woerdman, “Formation of a transient free carrier hologram in Si,” Opt. Commun. 2, 212–214 (1970).
[Crossref]

1969 (1)

S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969); B. J. Feldman and M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375–1396 (1970).
[Crossref]

Abrams, R. L.

Agrawal, G. P.

Aubourg, P.

Bloch, D.

M. Ducloy, R. K. Raj, and D. Bloch, “Polarization characteristics of phase-conjugate mirrors obtained by resonant degenerate four-wave mixing,” Opt. Lett. 7, 60–62 (1982).
[Crossref] [PubMed]

D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
[Crossref]

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (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]

D. Bloch and M. Ducloy, “Polarization selection rules and disorienting collision effects in resonant degenerate four-wave mixing,” J. Phys. B 14, L-471–L476 (1981).
[Crossref]

D. Bloch and M. Ducloy, unpublished report (Université Paris-Nord, Paris, 1983).

D. Bloch, “Conjugaison de phase dans les milieux gazeux. Spectroscopie de saturation hétérodyne,” Thèse de Troisième Cycle (Université Paris-Nord, Paris, 1980).

Bloom, D. M.

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
[Crossref]

Boulnois, J. L.

Boyd, R. W.

D. J. Harter and R. W. Boyd, in “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980)], have actually shown that, in nearly degenerate four-wave mixing, when the pump-probe frequency detuning is varied, the ac Stark splitting is responsible for the appearance of structures in the emission line shape. Their theory, which does not take into account the atomic motion, cannot apply to DFWM experiments in which the common frequency of pump and probe beams is scanned through the atomic resonance (see also Ref. 8).
[Crossref]

Couillaud, B.

B. Couillaud and A. Ducasse, “Les lasers à colorant continus. Leur application à la spectroscopie d’absorption saturée,” Thèse d’Etat (Université de Bordeaux I, Bordeaux, France, 1978).

Ducasse, A.

B. Couillaud and A. Ducasse, “Les lasers à colorant continus. Leur application à la spectroscopie d’absorption saturée,” Thèse d’Etat (Université de Bordeaux I, Bordeaux, France, 1978).

Ducloy, M.

M. Ducloy, R. K. Raj, and D. Bloch, “Polarization characteristics of phase-conjugate mirrors obtained by resonant degenerate four-wave mixing,” Opt. Lett. 7, 60–62 (1982).
[Crossref] [PubMed]

D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
[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]

D. Bloch and M. Ducloy, “Polarization selection rules and disorienting collision effects in resonant degenerate four-wave mixing,” J. Phys. B 14, L-471–L476 (1981).
[Crossref]

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
[Crossref]

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

D. Bloch and M. Ducloy, unpublished report (Université Paris-Nord, Paris, 1983).

M. Ducloy, “Nonlinear optical phase-conjugation,” in Festkörperprobleme—Advances in Solid State Physics (Viegweg, Braunschweig, 1982), Vol. XXII, pp. 35–60.
[Crossref]

Economou, N. P.

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
[Crossref]

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

Feld, M. S.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

Fisher, R. A.

R. A. Fisher, Nonlinear Optical Phase-Conjugation (Academic, New York, 1982).

Giuliano, C. R.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[Crossref]

Gordon, J. P.

Harter, D. J.

D. J. Harter and R. W. Boyd, in “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980)], have actually shown that, in nearly degenerate four-wave mixing, when the pump-probe frequency detuning is varied, the ac Stark splitting is responsible for the appearance of structures in the emission line shape. Their theory, which does not take into account the atomic motion, cannot apply to DFWM experiments in which the common frequency of pump and probe beams is scanned through the atomic resonance (see also Ref. 8).
[Crossref]

Hellwarth, R. W.

Humphrey, L. M.

Ivakin, E. V.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).

Lam, J. F.

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[Crossref]

Lamb, W. E.

S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969); B. J. Feldman and M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375–1396 (1970).
[Crossref]

Leite, J. R. R.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

Liao, P. F.

L. M. Humphrey, J. P. Gordon, and P. F. Liao, “Angular dependence of line shape and strength of degenerate four-wave mixing in a Doppler-broadened system with optical pumping,” Opt. Lett. 5, 56–58 (1980).
[Crossref] [PubMed]

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
[Crossref]

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

Lind, R. C.

Meisel, E.

These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
[Crossref]

Peng, K. S.

D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
[Crossref]

Raj, R. K.

D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
[Crossref]

M. Ducloy, R. K. Raj, and D. Bloch, “Polarization characteristics of phase-conjugate mirrors obtained by resonant degenerate four-wave mixing,” Opt. Lett. 7, 60–62 (1982).
[Crossref] [PubMed]

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
[Crossref]

Ritze, H. H.

These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
[Crossref]

Rubanov, A. S.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).

Schuurmans, M. F. H.

Sharma, R. D.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

Sheffield, R. L.

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

Shirley, J. H.

Snyder, J. J.

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
[Crossref]

Steel, D. G.

D. G. Steel and R. C. Lind, “Multiresonant behavior in nearly degenerate four-wave mixing: the ac Stark effect,” Opt. Lett. 6, 587–589 (1981).
[Crossref] [PubMed]

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[Crossref]

Stenholm, S.

S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969); B. J. Feldman and M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375–1396 (1970).
[Crossref]

Stepanov, B. I.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).

Stert, V.

These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
[Crossref]

Van Lerberghe, A.

Wandzura, S. M.

Woerdman, J. P.

Appl. Phys. Lett. (2)

P. F. Liao, D. M. Bloom, and N. P. Economou, “Cw optical wave-front conjugation by saturated absorption in atomic sodium vapor,” Appl. Phys. Lett. 32, 813–815 (1978).
[Crossref]

D. G. Steel, R. C. Lind, J. F. Lam, and C. R. Giuliano, “Polarization-rotation and thermal-motion studies via resonant degenerate four-wave mixing,” Appl. Phys. Lett. 35, 376–379 (1979).
[Crossref]

IEEE J. Quantum Electron. (1)

D. J. Harter and R. W. Boyd, in “Nearly degenerate four-wave mixing enhanced by the ac Stark effect,” IEEE J. Quantum Electron. QE-16, 1126–1131 (1980)], have actually shown that, in nearly degenerate four-wave mixing, when the pump-probe frequency detuning is varied, the ac Stark splitting is responsible for the appearance of structures in the emission line shape. Their theory, which does not take into account the atomic motion, cannot apply to DFWM experiments in which the common frequency of pump and probe beams is scanned through the atomic resonance (see also Ref. 8).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. (Paris) (1)

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)

D. Bloch and M. Ducloy, “Polarization selection rules and disorienting collision effects in resonant degenerate four-wave mixing,” J. Phys. B 14, L-471–L476 (1981).
[Crossref]

J. Phys. Lett. (Paris) (1)

D. Bloch, R. K. Raj, J. J. Snyder, and M. Ducloy, “Heterodyne detection of phase-conjugate emission in an Ar discharge with a low-power c.w. laser,” J. Phys. Lett. (Paris) 42, L31–L34 (1981).
[Crossref]

Opt. Commun. (2)

J. P. Woerdman, “Formation of a transient free carrier hologram in Si,” Opt. Commun. 2, 212–214 (1970).
[Crossref]

These narrow structures have been observed in recent experiments in neon (see Ref. 15). It is also probable that the peculiar line shapes observed in the saturated regime of polarization spectroscopy [H. H. Ritze, V. Stert, and E. Meisel, “High resolution polarization spectroscopy in the strong saturation regime,” Opt. Commun. 29, 51–56 (1979)] are somehow related to the structures observed here.
[Crossref]

Opt. Eng. (1)

Opt. Eng. 21, 155–283 (1982) (special issue on phase-conjugation).

Opt. Lett. (9)

D. M. Bloom, P. F. Liao, and N. P. Economou, “Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 2, 158–160 (1978).
[Crossref]

D. G. Steel and R. C. Lind, “Multiresonant behavior in nearly degenerate four-wave mixing: the ac Stark effect,” Opt. Lett. 6, 587–589 (1981).
[Crossref] [PubMed]

S. M. Wandzura, “Effects of atomic motion on wave-front conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Lett. 4, 208–210 (1979).
[Crossref] [PubMed]

L. M. Humphrey, J. P. Gordon, and P. F. Liao, “Angular dependence of line shape and strength of degenerate four-wave mixing in a Doppler-broadened system with optical pumping,” Opt. Lett. 5, 56–58 (1980).
[Crossref] [PubMed]

J. P. Woerdman and M. F. H. Schuurmans, “Effect of saturation on the spectrum of degenerate four-wave mixing in atomic sodium vapor,” Opt. Lett. 6, 239–241 (1981).
[Crossref] [PubMed]

J. H. Shirley, “Modulation transfer processes in optical heterodyne saturation spectroscopy,” Opt. Lett. 7, 537–539 (1982).
[Crossref] [PubMed]

M. Ducloy, R. K. Raj, and D. Bloch, “Polarization characteristics of phase-conjugate mirrors obtained by resonant degenerate four-wave mixing,” Opt. Lett. 7, 60–62 (1982).
[Crossref] [PubMed]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata 3, 205 (1978). In the optically thin sample approximation, the emission intensity line shape is proportional to |κ|2, in the notation of Abrams and Lind.
[Crossref] [PubMed]

A similar saturation anisotropy was recently observed in SF6[G. P. Agrawal, A. Van Lerberghe, P. Aubourg, and J. L. Boulnois, “Saturation splitting in the spectrum of resonant degenerate four-wave mixing,” Opt. Lett. 7, 540–542 (1982)].
[Crossref] [PubMed]

Phys. Rev. (1)

S. Stenholm and W. E. Lamb, “Semiclassical theory of a high-intensity laser,” Phys. Rev. 181, 618–635 (1969); B. J. Feldman and M. S. Feld, “Theory of a high-intensity gas laser,” Phys. Rev. A 1, 1375–1396 (1970).
[Crossref]

Phys. Rev. A (1)

J. R. R. Leite, R. L. Sheffield, M. Ducloy, R. D. Sharma, and M. S. Feld, “Theory of coherent three-level beats,” Phys. Rev. A 14, 1151–1168 (1976).
[Crossref]

Phys. Rev. Lett. (1)

D. Bloch, R. K. Raj, K. S. Peng, and M. Ducloy, “Dispersive character and directional anisotropy of saturated susceptibilities in resonant backward four-wave mixing,” Phys. Rev. Lett. 49, 719–722 (1982).
[Crossref]

Sov. Phys. Dokl. (1)

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “Recording two-dimensional and three-dimensional dynamic holograms in transparent substances,” Sov. Phys. Dokl. 16, 46–48 (1971).

Other (15)

M. Ducloy, “Nonlinear optical phase-conjugation,” in Festkörperprobleme—Advances in Solid State Physics (Viegweg, Braunschweig, 1982), Vol. XXII, pp. 35–60.
[Crossref]

R. A. Fisher, Nonlinear Optical Phase-Conjugation (Academic, New York, 1982).

A coupled three-level system is particularly adequate to describe a J= 1 → J= 0 transition with cross-polarized pumps.15

In all the calculation, one neglects the cascade effects induced by spontaneous emission among |b〉, |a〉, and |c〉 so that Λ-type and V-type three-level systems are equivalent.

For sake of simplicity, the choice of the space and time origins is such that the phase term is eliminated in the definition of the incident fields: One can easily verify that the phase of the probe field is sign reversed in the component that we are calculating.

In four-wave mixing, the emission of one photon in the PC field occurs simultaneously with the emission of one photon in the probe field, at the expense of absorption of one photon of each pump field. A too-high increase in probe intensity does not help one to increase the reemitted intensity, so the PC reflectivity RPC(as an efficiency rate) decreases.

D. Bloch, “Conjugaison de phase dans les milieux gazeux. Spectroscopie de saturation hétérodyne,” Thèse de Troisième Cycle (Université Paris-Nord, Paris, 1980).

Actually, the formalism is not specific to the set of polarizations discussed at the beginning of Section 2. It can describe a two-level system if the polarization of EB is identical with the one of EF and EP(e.m. field coupling between |a〉 and |b〉 only; no coupling between |c〉 and |b〉). The quantity of interest becomes βab(0)〉 instead of 〈βcb(0)〉, and minor and obvious modifications must be done to Eqs. (7)–(13).

The backward pump is too weak to modify the population in level |c〉: αcc(−1)= 0, and the equations yielding αcb(−2)and βac(3)are decoupled.

D. Bloch and M. Ducloy, unpublished report (Université Paris-Nord, Paris, 1983).

In the case of weak saturation, an approximate justification for the single-lifetime model can be found if θ is small but not negligible (θ≫ γi/ku, θ~ γij/ku). The effective grating lifetime is essentially shortened by the thermal motion and is almost independent of the relaxation processes (see Ref. 27). For strong saturation, the situation is complicated by the homogeneous broadening of the relaxation processes.

These structures are predicted only for a high-saturation parameter, so that the conditions for infinite Doppler-width approximation may no longer be fulfilled. This could explain why we were unable to observe these types of structure experimentally, whereas all the other line-shape features calculated in the present paper have been observed experimentally.15

For SF= 1, the width of the absorption line shape is about 2.5 times the width at SF= 0 (natural width).

B. Couillaud and A. Ducasse, “Les lasers à colorant continus. Leur application à la spectroscopie d’absorption saturée,” Thèse d’Etat (Université de Bordeaux I, Bordeaux, France, 1978).

Let us give a numerical estimation of R0, the limit value of the reflectivity for infinite backward saturation. From Eq. (31), one finds that, at saturation, the maximum value of |χNL/χ(3)(0)|2is 0.7 SB−1for δ= 3ΩB/4. Let us consider transitions having oscillator strengths equal to unity and wavelength λ = 0.6 μ m, an atomic density of 1011atoms/cm3(corresponding to p= 3.10−6Torr at 300 K), an interaction length L= 1 cm, and a mean thermal velocity u= 500 m/sec. With these values, one deduce from Eqs. (26) and (28) that R0/SF= 0.32.

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Figures (7)

Fig. 1
Fig. 1

Principle of a backward four-wave mixing experiment.

Fig. 2
Fig. 2

Schematic of the three-level atom.

Fig. 3
Fig. 3

Theoretical absorption and dispersion line shapes for various backward saturation parameters. (Relaxations are described by a single parameter, γ.) The quantities represented on the vertical axis are Re ( S B χ NL ) and Im ( S B χ NL ), which are proportional to the PC e.m. field if the probe and forward pump intensities are kept constant. The same (arbitrary) units apply to all vertical scales.

Fig. 4
Fig. 4

Amplitude of the PC field versus frequency detuning for high backward saturation. (A) and (D) are the absorption component [ Im S B χ NL)] and the dispersion component [ Re S B χ NL)], respectively. Note the sign reversal of the absorption and the change in vertical scale between (A) and (D).

Fig. 5
Fig. 5

Absorption and dispersion line shapes in the case of forward saturation. (Conditions similar to the ones of Fig. 3.) Note the change in the vertical scale for SF = 4, 20.

Fig. 6
Fig. 6

Emission-intensity line shape (S|χNL|2 versus δ) for various backward (B) and forward (F) saturation parameters.

Fig. 7
Fig. 7

Peak reflectivity versus saturation parameter: comparison between forward and backward saturation. The peak reflectivity RPC(m) is defined as the maximum value in the curves of Fig. 6. R0 is the asymptotic value reached by RPC(m) for infinite backward saturation (cf. Ref. 37).

Equations (45)

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χ γ - i δ γ 2 + Ω 2 + δ 2 ,
F ( v ) = π - 3 / 2 u - 3 exp ( - v 2 / u 2 ) .
E ( r , t ) = μ = F , B , P E μ ( r , t ) = μ 1 2 { E μ exp [ i ( ω μ t - k μ · r ) ] + c . c . } ,
P PC ( r , t ) = 1 2 { P PC exp [ i ( ω t + Kr ) ] + c . c . } .
E PC = - i k L 2 0 P P C .
P PC ( r , t ) = μ b c ρ ˆ c b ( r , t ) + c . c .
ρ ( r , t ) = d 3 v F ( v ) ρ ( r , v , t ) ,
ρ ˙ a a = γ a ( n a - ρ a a ) - i 2 Ω F ρ a b exp [ - i ( ω t - k · r ) ] + i 2 Ω F ρ b a exp [ i ( ω t - k · r ) ] - i 2 Ω P ρ a b exp [ - i ( ω t - K · r ) ] + i 2 Ω P ρ b a exp [ i ( ω t - K · r ) ] , ρ ˙ b b = γ b ( n b - ρ b b ) + i 2 Ω F ρ a b exp [ - i ( ω t - k · r ) ] - i 2 Ω F ρ b a exp [ i ( ω t - k · r ) ] + i 2 Ω B ρ c b exp [ - i ( ω t + k · r ) ] - i 2 Ω B ρ b c exp [ i ( ω t + k · r ) ] + i 2 Ω P ρ a b × exp [ - i ( ω t - K · r ) ] - i 2 Ω P ρ b a exp [ i ( ω t - K · r ) ] , ρ ˙ c c = γ c ( n c - ρ c c ) - i 2 Ω B ρ c b exp [ - i ( ω t + k · r ) ] + i 2 Ω B ρ b c exp [ i ( ω t + k · r ) ] , ρ ˙ a b = ( - γ a b + i ω a b ) ρ a b + i 2 Ω F exp [ i ( ω t - k · r ) ] ( ρ b b - ρ a a ) - i 2 Ω B exp [ i ( ω t + k · r ) ] ρ a c + i 2 Ω P exp [ i ( ω t - K · r ) ] ( ρ b b - ρ a a ) , ρ ˙ b c = ( - γ c b - i ω c b ) ρ b c + i 2 Ω F exp [ - i ( ω t - k · r ) ] ρ a c + i 2 Ω B exp [ - i ( ω t + k · r ) ] ( ρ c c - ρ b b ) + i 2 Ω P exp [ - i ( ω t - K · r ) ] ρ a c , ρ ˙ a c = ( - γ a c + i ω a c ) ρ a c + i 2 Ω F exp [ i ( ω t - k · r ) ] ρ b c - i 2 Ω B exp [ - i ( ω t + k · r ) ] ρ a b + i 2 Ω P exp [ i ( ω t - K · r ) ] ρ b c ,
ρ ˙ = ( t + v · ) ρ ,
ρ = n ρ ( n ) ,
ρ ( 1 ) j j = n { α j ( n ) ( v ) exp [ - i ( n k + K ) · r ] + β j ( n ) ( v ) exp [ i ( - n k + K ) · r }             ( odd n ) , ρ ( 1 ) a c = n { α a c ( n ) ( v ) exp [ - i ( n k + K ) · r ] + β a c ( n ) ( v ) exp [ i ( - n k + K ) · r ] }             ( odd n ) , ρ ( 1 ) a b = n { α a b ( n ) ( v ) exp i [ ω t - ( n k + K ) · r ] + β a b ( n ) ( v ) exp i [ ω t - ( n k - K ) · r ] }             ( even n ) , ρ ( 1 ) c b = n { α c b ( n ) ( v ) exp i [ ω t - ( n k + K ) · r ] + β c b ( n ) ( v ) exp i [ ω t - ( n k - K ) · r ] }             ( even n ) ,
E PC = - i k L 0 μ b c β c b ( 0 ) ,
ρ ( 0 ) j j = ( even n ) ρ ˜ ( 0 ) j j ( n ) ( v ) exp ( i n k · r ) , ρ ( 0 ) a c = ( even n ) ρ ˜ ( 0 ) a c ( n ) ( v ) exp ( i n k · r ) , ρ ( 0 ) a b = ( odd n ) ρ ˜ ( 0 ) a b ( n ) ( v ) exp [ i ( ω t + n k · r ) ]
[ γ a - i ( n k + K ) · v ] α a ( n ) = - i 2 Ω F α a b ( n + 1 ) + i 2 Ω F β a b ( - n + 1 ) * + i Ω P 2 [ ρ ˜ 0 a b ( n ) ] * , [ γ b - i ( n k + K ) · v ] α b ( n ) = i 2 Ω F α a b ( n + 1 ) - i 2 Ω F β a b ( - n + 1 ) * + i 2 Ω B α c b ( n - 1 ) - i 2 Ω B β c b ( - n - 1 ) * - i 2 Ω P [ ρ ˜ ( 0 ) a b ( n ) ] * , [ γ c - i ( n k + K ) · v ] α c ( n ) = - i 2 Ω B α c b ( n - 1 ) + i 2 Ω B β c b ( - n - 1 ) * , [ γ a b + i ( ω - ω a b ) - i ( n k + K ) · v ] α a b ( n ) = i 2 Ω F [ α b ( n - 1 ) - α a ( n - 1 ) ] - i 2 Ω B α a c ( n + 1 ) + i 2 Ω P [ ρ ˜ ( 0 ) b b ( - n ) - ρ ˜ ( 0 ) a a ( - n ) ] , [ γ a b + i ( ω - ω a b ) - i ( n k - K ) · v ] β a b ( n ) = i 2 Ω F [ α b ( - n + 1 ) * - α a ( - n + 1 ) * ] - i 2 Ω B β a c ( n + 1 ) , [ γ c b + i ( ω - ω c b ) - i ( n k + K ) · v ] α c b ( n ) = - i 2 Ω F β a c ( - n + 1 ) * + i 2 Ω B [ α b ( n + 1 ) - α c ( n + 1 ) ] - i 2 Ω P [ ρ ˜ ( 0 ) a c ( n ) ] * , [ γ c b + i ( ω - ω c b ) - i ( n k - K ) · v ] β c b ( n ) = - i 2 Ω F α a c ( - n + 1 ) * + i 2 Ω B [ α b ( - n - 1 ) * - α c ( - n - 1 ) * ] , [ γ a c - i ω a c - i ( n k + K ) · v ] α a c ( n ) = i 2 Ω F β c b ( - n + 1 ) * - i 2 Ω B α a b ( n - 1 ) + i 2 Ω P [ ρ ˜ ( 0 ) c b ( n ) ] * , [ γ a c - i ω a c - i ( n k - K ) · v ] β a c ( n ) = i 2 Ω F α c b ( - n + 1 ) * - i 2 Ω B β a b ( n - 1 ) .
ρ ˜ ( 0 ) a b ( - 1 ) ( v ) = - i 2 n b a Ω f γ a b - i ( δ - k · v ) γ a b 2 ( 1 + S F ) + ( δ - k · v ) 2 [ ρ ˜ ( 0 ) a a ( 0 ) - ρ ˜ ( 0 ) b b ( 0 ) ] ( v ) = n b a γ a b 2 + ( δ - k · v ) 2 γ a b 2 ( 1 + S F ) + ( δ - k · v ) 2 ρ ˜ ( 0 ) c b ( 1 ) ( v ) = - i 2 Ω B γ a c + 2 i k · v { [ γ c b + i ( δ + k · v ) ] [ γ a c + 2 i k · v ] + γ γ a b S F 4 } × { n b c - S F [ ρ ˜ ( 0 ) a a - ρ ˜ ( 0 ) b b ] γ γ a b 2 2 γ b [ γ a b 2 + ( δ - k · v ) 2 ] - S F [ ρ ˜ ( 0 ) a a - ρ ˜ ( 0 ) b b ] γ γ a b 4 ( γ a c - 2 i k · v ) ( γ a b + i ( δ - k · v ) ) } , ρ ˜ ( 0 ) a c ( - 2 ) ( v ) = γ c b - i ( δ + k · v ) { ( γ a c - 2 i k · v ) [ γ c b - i ( δ + k · v ) ] + γ γ a b S F 4 } × { - n b a Ω F Ω B 4 [ γ a b - i ( δ - k · v ) ] γ a b 2 ( 1 + S F ) + ( δ - k · v ) 2 + Ω F Ω B n c b 4 [ γ c b - i ( δ + k · v ) ] + Ω F 3 Ω B n b a γ a b 8 γ b [ γ c b - i ( δ + k · v ) ] [ γ a b 2 ( 1 + S F ) + ( δ - k · v ) 2 ] } ,
S F = Ω F 2 γ γ a b ,
β c b ( 0 ) ( v ) = [ γ b c + i ( δ + K · v ) + Ω F 2 4 L a c ] - 1 { i 2 Ω P [ ρ ˜ ( 0 ) a a - ρ ˜ ( 0 ) b b ] Ω F Ω B L a c 4 + Ω F Ω B Λ b 4 [ 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) ] γ a b - i ( δ - K · v ) + Ω F 2 ( Λ a + Λ b ) 4 [ 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) ] + Ω F Ω P 4 ρ ˜ ( 0 ) a b ( - 1 ) ( Λ a + Λ b ) Ω F Ω B L a c 4 + Ω F Ω B Λ B / 4 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) [ 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) ] [ γ a b - i ( δ - K · v ) + Ω F 2 ( Λ a + Λ b ) / 4 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) ] - Ω F Ω P 4 L a c ρ ˜ ( 0 ) c b ( - 1 ) - Ω B Ω P Λ b ρ ˜ a b ( - 1 ) 4 [ 1 + Ω F 2 4 L a b ( δ ) ( Λ a + Λ b ) ] } ,
L a c = [ γ a c + i ( k + K ) v ] - 1 , L a b ( δ ) = { γ a b + i [ δ - ( 2 k - K ) v ] } - 1 , Λ i = [ γ i - i ( k - K ) v ] - 1             ( i = a , b ) .
β c b ( 0 ) = i Ω F Ω B Ω P 8 d v exp ( - v 2 / u 2 ) u π × n b a [ ( ( γ a c + 2 i k v ) 2 γ a b γ b + γ a b + i ( δ - k v ) ) ( γ a b 2 + ( δ - k v ) 2 - γ γ a b S F / 4 [ γ b c + i ( δ + k v ) ] ( γ a c + 2 i k v ) + γ γ a b S F 4 ) { [ γ b c + i ( δ + k v ) ] ( γ a c + 2 i k v ) + γ γ a b S F 4 } [ γ a b 2 ( 1 + S F ) + ( δ - k v ) 2 ] 2 + n b c ( γ a c + 2 i k v ) { [ γ b c + i ( δ + k v ) ] ( γ a c + 2 i k v ) + γ γ a b S F 4 } 2 ] .
β c b ( 0 ) ( v ) = γ b c ( 1 + S B 2 ) - i ( δ + k v ) γ b c 2 ( 1 + S B ) + ( δ + k v ) 2 × { i Ω B 2 Ω P 4 γ ρ ˜ ( 0 ) a c [ γ b c ( 1 + S B 2 ) - i ( δ + k v ) ] + Ω B Ω F ρ ˜ ( 0 ) c b 4 ( γ a c + 2 i k v ) × [ - 1 + S B γ γ c b ( γ a c + 2 i k v ) / 4 [ γ a b - i ( δ - k v ) ] ( γ a c + 2 i k v ) + S B γ γ c b 4 × ( 1 γ b + 1 γ a c + 2 i k v - S B γ b c 2 γ b [ γ b c ( 1 + S B 2 ) - i ( δ + k v ) ] ) ] + Ω B Ω F ρ ˜ ( 0 ) a b γ b ( - 1 + S B γ c b / 2 γ c b ( 1 + S B 2 ) - i ( δ + k v ) ) - i Ω F Ω B Ω P [ ρ ˜ ( 0 ) b b - ρ ˜ ( 0 ) a a ] ( γ a c + 2 i k v ) 8 { [ γ a b - i ( δ - k v ) ] [ γ a c + 2 i k v ] + S B γ γ b c 4 } × ( 1 γ b + 1 γ a c + 2 i k v - S B γ b c 2 γ b [ γ b c ( 1 + S B 2 ) - i ( δ + k v ) ] ) } .
[ γ b c + i ( δ + k v ) ] ( γ a c + 2 i k v ) + S F γ γ a b 4 .
[ γ a b 2 ( 1 + S F ) + ( δ - k v ) 2 ] - 2 .
v = - δ - i γ a b 1 + S F k .
β c b ( 0 ) = i π k u Ω F Ω B Ω P 8 n b a × ( 1 f 1 f 2 + S F γ γ a b 4 [ - 1 2 + 1 2 1 + S F - S F 4 ( 1 + S F ) 3 / 2 + γ a b S F γ b ( 1 + S F ) + f 2 ( 2 + S F ) 2 γ b ( 1 + S F ) 3 / 2 ] - S F ( f 1 f 2 + S F γ γ a b 4 ) 2 × { γ γ a b ( 1 - 1 + S F ) 8 1 + S F + γ a b ( 1 - 1 + S F ) ( 2 f 1 + f 2 ) 4 ( 1 + S F ) + f 2 [ γ 4 1 + S F + γ a b ( 2 f 1 + f 2 ) 2 γ b ( 1 + S F ) ] } ) ,
f 1 = γ c b + γ a b 1 + S F + 2 i δ , f 2 = γ a c + 2 γ a b 1 + S F + 2 i δ .
v = - δ - i γ c b 1 + S B k .
[ γ a b + i ( δ - k v ) ] ( γ a c - 2 i k v ) + S B γ γ c b 4 .
v = - δ + i γ c b 1 + S B k .
[ γ c b ( 1 + S B 2 ) - i ( δ + k v ) ] - 1
β c b ( 0 ) = - i Ω F Ω B Ω P 8 π k u { 1 F 1 * F 2 * + S B γ γ c b 4 × ( n b c { S B γ γ c b [ γ b ( 1 - 1 + S B ) 2 + ( 1 - 1 + S B ) F 2 * ] - γ b 2 ( 1 - 1 + S B ) 3 F 1 * 8 γ b 2 γ c b ( 1 + S B ) 3 / 2 + γ ( 1 - 1 + S B ) 8 γ b ( 1 + S B ) - F 1 * ( 4 + S B - 4 1 + S B ) 8 γ c b ( 1 + S B ) + γ S B 8 γ b 2 ( 1 + S B ) [ F 2 * + γ b + 2 γ c b ( 1 + S B - 1 ) ] } - n b a 2 1 + S B [ ( 1 - 1 + S B ) ( F 1 * γ b + 1 ) + S B 2 ] ) + 1 F 1 F 2 + S B γ γ c b 4 [ n b c ( γ S B γ c b ( 1 + S B ) 3 / 2 [ 2 γ b c γ b 2 1 + S B F 2 - 2 γ c b S B γ b + 2 ( 1 - 1 + S B ) F 1 γ ] + S B 8 ( 1 + S B ) { γ 1 + S B γ b - γ γ b 2 [ F 2 + 2 γ c b ( 1 + S B + 1 ) ] + F 1 γ c b + 1 + S B - 1 } ) + n b a [ S B 4 1 + S B - ( 1 + 1 + S B ) F 2 2 γ b 1 + S B ] ] + n b c ( F 1 * F 2 * + S B γ γ c b 4 ) 2 { S B γ γ c b ( 2 F 1 * + F 2 * ) 8 γ b 2 ( 1 + S B ) [ ( 1 - 1 + S B ) 2 γ b + ( 1 - 1 + S B ) F 2 * ] + ( 1 - 1 + S B ) 3 32 ( 1 + S B ) ( γ γ c b - 8 F 1 * 2 ) } + n b c ( F 1 F 2 + S B γ γ c b 4 ) 2 × { γ S B ( 2 F 1 + F 2 ) 16 ( 1 + S B ) [ 2 γ c b 1 + S B F 2 γ b 2 - 2 γ c b S B γ b + 2 ( 1 - 1 + S B ) F 1 γ ] } } ,
F 1 = γ a b + γ c b 1 + S B + 2 i δ , F 2 = γ a c + 2 γ c b 1 + S B + 2 i δ .
γ i j = γ j = γ .
P PC = 0 4 χ NL ( E μ ) E F E B E P .
E PC = - i k L 8 χ NL E F E B E P
R PC = | E PC E P | 2 = | k L 2 γ 2 8 μ a b μ b c | 2 χ NL 2 S F S B .
χ NL = γ χ ( 3 ) ( 0 ) 2 + S F 2 ( 1 + S F ) 3 / 2 γ 1 + i δ ( γ 2 + i δ ) 2 ,
χ ( 3 ) ( 0 ) = i N μ a b 2 μ b c 2 0 γ 2 3 π k u ,
γ 1 = γ 4 ( 1 + 6 1 + S F 2 + S F ) ,
γ 2 = γ 4 ( 1 + 3 1 + S F ) .
χ NL γ χ ( 3 ) ( 0 ) 7 Ω B + 8 i δ ( 3 Ω B + 4 i δ ) 2 - 7 Ω B - 8 i δ ( 3 Ω B - 4 i δ ) 2 - Ω B 9 Ω B 2 - 24 i Ω B δ - 64 δ 2 ( 3 Ω B - 4 i δ ) 4 .
γ 1 = γ a 4 + γ a + 2 γ b 2 1 + S F 2 + S F ,
γ 2 = γ a 4 + γ a + 2 γ b 4 1 + S F .
χ NL γ χ ( 3 ) ( 0 ) 2 γ Ω F 4 i δ ( 3 Ω F + 4 i δ ) 2 ,
χ NL 2 Ω F 2 4 γ 4 χ ( 3 ) ( 0 ) 2 16 δ 2 ( 9 Ω F 2 + 16 δ 2 ) 2 .
γ a + 2 γ b 2 S F = γ a + 2 γ b 2 γ a γ b Ω F .