Abstract

A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption and depletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limit of the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solution is shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abrams and Lind [Opt. Lett. 2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal intensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Comparing the results of the numerical calculation with those of the analytical solution, we discuss the validity of using the analytical solution in practical experiments and show the saturation behavior that is due to the strong probe beam. The line shapes under the various ratios among the input beams and absorption parameters are also obtained and discussed.

© 1999 Optical Society of America

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References

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  1. See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983) and H. J. Eichler, P. Gunter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Academic, New York, 1984).
  2. P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
    [CrossRef]
  3. I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
    [CrossRef]
  4. R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
    [CrossRef] [PubMed]
  5. K. Kohse-Hinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
    [CrossRef]
  6. A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
    [CrossRef]
  7. G. Meijer and D. W. Chandler, “Degenerate four-wave mixing on weak transitions in the gas phase using a tunable eximer laser,” Chem. Phys. Lett. 192, 1–4 (1992).
    [CrossRef]
  8. H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
    [CrossRef]
  9. K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
    [CrossRef]
  10. 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: Photophys. Laser Chem. 60, 11–18 (1995).
    [CrossRef]
  11. B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
    [CrossRef]
  12. K. Nyholm, “Two-dimensional imaging of OH in a flame by using degenerate four-wave mixing in a forward geometry,” Appl. Phys. B 64, 707–712 (1997).
    [CrossRef]
  13. W. P. Brown, “Absorption and depletion effects on degenerate four-wave mixing in homogeneously broadened absorbers,” J. Opt. Soc. Am. 73, 629–634 (1983).
    [CrossRef]
  14. B. Ai and R. J. Knize, “Degenerate four-wave mixing in two-level saturable absorbers,” J. Opt. Soc. Am. B 13, 2408–2419 (1996).
    [CrossRef]
  15. G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Physique 47, 617–630 (1986).
    [CrossRef]
  16. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
    [CrossRef]
  17. See, for example, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

1997 (2)

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

K. Nyholm, “Two-dimensional imaging of OH in a flame by using degenerate four-wave mixing in a forward geometry,” Appl. Phys. B 64, 707–712 (1997).
[CrossRef]

1996 (1)

1995 (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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

1994 (3)

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

K. Kohse-Hinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[CrossRef]

K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
[CrossRef]

1993 (2)

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
[CrossRef]

1992 (2)

G. Meijer and D. W. Chandler, “Degenerate four-wave mixing on weak transitions in the gas phase using a tunable eximer laser,” Chem. Phys. Lett. 192, 1–4 (1992).
[CrossRef]

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[CrossRef] [PubMed]

1989 (1)

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

1986 (1)

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

1984 (1)

P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
[CrossRef]

1983 (1)

Ai, B.

B. Ai and R. J. Knize, “Degenerate four-wave mixing in two-level saturable absorbers,” J. Opt. Soc. Am. B 13, 2408–2419 (1996).
[CrossRef]

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Aminoff, C. G.

K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
[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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Attal-Tretout, B.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

Bervas, H.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

Biaggio, I.

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Bloch, D.

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

Brown, W. P.

Chandler, D. W.

G. Meijer and D. W. Chandler, “Degenerate four-wave mixing on weak transitions in the gas phase using a tunable eximer laser,” Chem. Phys. Lett. 192, 1–4 (1992).
[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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Ducloy, M.

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

Farrow, R. L.

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
[CrossRef]

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[CrossRef] [PubMed]

Grynberg, G.

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

Gustafson, T. K.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Hellwarth, R. W.

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Kaivola, M.

K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
[CrossRef]

Kelley, P.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

Klein, A.

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

Knize, R. J.

B. Ai and R. J. Knize, “Degenerate four-wave mixing in two-level saturable absorbers,” J. Opt. Soc. Am. B 13, 2408–2419 (1996).
[CrossRef]

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Kohse-Hinghaus, K.

K. Kohse-Hinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[CrossRef]

Labrunie, L.

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

Le Boiteux, S.

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

LeBoiteux, S.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

Lucht, R. P.

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
[CrossRef]

Meijer, G.

G. Meijer and D. W. Chandler, “Degenerate four-wave mixing on weak transitions in the gas phase using a tunable eximer laser,” Chem. Phys. Lett. 192, 1–4 (1992).
[CrossRef]

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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Nyholm, K.

K. Nyholm, “Two-dimensional imaging of OH in a flame by using degenerate four-wave mixing in a forward geometry,” Appl. Phys. B 64, 707–712 (1997).
[CrossRef]

K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
[CrossRef]

Oria, M.

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

Partanen, J. P.

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Pinard, M.

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

Rakestraw, D. J.

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
[CrossRef]

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[CrossRef] [PubMed]

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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Taran, J. P.

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

Verkerk, P.

G. Grynberg, M. Pinard, and P. Verkerk, “Saturation in degenerate four-wave mixing: theory for a two-level atom,” J. Physique 47, 617–630 (1986).
[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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Yeh, P.

P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
[CrossRef]

Appl. Phys. B (1)

K. Nyholm, “Two-dimensional imaging of OH in a flame by using degenerate four-wave mixing in a forward geometry,” Appl. Phys. B 64, 707–712 (1997).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem. (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: Photophys. Laser Chem. 60, 11–18 (1995).
[CrossRef]

Chem. Phys. Lett. (1)

G. Meijer and D. W. Chandler, “Degenerate four-wave mixing on weak transitions in the gas phase using a tunable eximer laser,” Chem. Phys. Lett. 192, 1–4 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

B. Ai and R. J. Knize, “Degenerate four-wave mixing in two-level saturable absorbers,” J. Opt. Soc. Am. B 13, 2408–2419 (1996).
[CrossRef]

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 19, 1508–1520 (1993).
[CrossRef]

J. Phys. B (1)

B. Attal-Tretout, H. Bervas, J. P. Taran, S. LeBoiteux, P. Kelley, and T. K. Gustafson, “Saturated FDFWM lineshapes and intensities: theory and application to quantitative measurements in flames,” J. Phys. B 30, 497–522 (1997).
[CrossRef]

J. Physique (1)

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

Mol. Phys. (1)

H. Bervas, S. Le Boiteux, L. Labrunie, and B. Attal-Tretout, “Rotational line strengths in degenerate four wave mixing,” Mol. Phys. 79, 911–941 (1993).
[CrossRef]

Nature (London) (1)

I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapor,” Nature (London) 371, 318–320 (1994).
[CrossRef]

Opt. Commun. (3)

P. Yeh, “Scalar phase conjugator for polarization correction,” Opt. Commun. 51, 195–197 (1984).
[CrossRef]

K. Nyholm, M. Kaivola, and C. G. Aminoff, “Detection of C2 and temperature measurement in a flame by using degenerate four-wave mixing in a forward geometry,” Opt. Commun. 107, 406–410 (1994).
[CrossRef]

A. Klein, M. Oria, D. Bloch, and M. Ducloy, “Saturation behavior and dynamic stark splitting of nearly degenerate four-wave and multiwave mixing in a forward boxcar configuration,” Opt. Commun. 73, 111–116 (1989).
[CrossRef]

Prog. Energy Combust. Sci. (1)

K. Kohse-Hinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[CrossRef]

Science (1)

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[CrossRef] [PubMed]

Other (2)

See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983) and H. J. Eichler, P. Gunter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Academic, New York, 1984).

See, for example, W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, New York, 1992).

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Figures (7)

Fig. 1
Fig. 1

Phase-matching configurations for DFWM: (a) DFWM with phase-conjugate geometry, (b) DFWM with forward geometry.

Fig. 2
Fig. 2

FDFWM signal intensity versus total input intensity for various intensity distributions. αIL=1.0, and δ=0.

Fig. 3
Fig. 3

(a) FDFWM efficiency versus pump beam intensity for various probe-to-pump-beam intensity ratios in logarithmic scale. The efficiency is defined to be the signal-to-probe-beam intensity ratio. αIL=1.0, δ=0, and the two input pump beams are set to have the same intensity. (b) Weak beam intensity regime of (a) in normal scale.

Fig. 4
Fig. 4

FDFWM signal intensity versus total input intensity for various absorption parameters αIL. δ=0, and I1=I2=I3.

Fig. 5
Fig. 5

FDFWM signal intensity versus total input beam intensity for various detunings. I1=I2=I3.

Fig. 6
Fig. 6

FDFWM line shapes for various pump beam intensities. I3=0.01IS0.

Fig. 7
Fig. 7

FDFWM line shapes for various ratios between the two pump beam intensities. All the intensities are normalized to the maximum value of the four cases considered. The addition of two input pump beam intensities is fixed to 2IS0, I3=0.01IS0, and αIL=0.1.

Equations (18)

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

E(r, t)=E(r)exp(-iωt)=j=14Ej(r)exp(ikj·r-iωt).
j=14i2kj·Ej(r)exp(ikj·r)=-4πk2P(r),
P(r)=20α0ki(1-iδ)1+δ2E(r)1+|E(r)|2/ES2(δ),
ES2(δ)=η2(1+δ2)T1T2μ2,
i2k·Ej(r)=Pˆj(r)(j=1,2,3,4).
Pˆj(r)=-4πk2l1l2-l1/2l1/2dx-l2/2l2/2dy exp(-ikj·r)×P(r+r)(j=1,2,3,4).
l1=λ2 sin θ12,l2=λ2 sin θ22,
ddzE3(z)=-αE3(z)+κ*E4*(z),
ddzE4*(z)=-α*E4*(z)+κE3(z).
α=α0 1-iδ1+δ21+(I1+I2)/IS{[1+(I1+I2)/IS]2-4I1I2/IS2}3/2,
κ=α0 1-iδ1+δ22(I1I2)1/2/IS{[1+(I1+I2)/IS]2-4I1I2/IS2}3/2,
E4*(z)=cosh(γz)+i αImγsinh(γz)E4*(0)+κγsinh(γz)E3(0)exp(-αRez),
E4*(L)=exp(-αReL)κγsinh(γL)E3(0),
tanh(γz)=γ/αRe.
0<I1/IS0<δ(1+δ2)(δ+1+δ2)/2.
ηij|(Ei+Ej)2-(Ei-Ej)2|2IiIj,
(i=1,j=3ori=2,j=3),
I4η13I2+η23I1I1I2I3.

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