Abstract

A calculation of degenerate four-wave mixing in a homogeneously broadened two-level saturable absorber is presented. An analytical formula for the local electric dipole polarization responsible for degenerate four-wave mixing is derived for arbitrary electric-field strengths. The overall efficiency of the medium is calculated by numerically solving the coupled wave equations with the use of the phase-matching electric dipole polarization. Solutions are presented when the laser is tuned on and off resonance for various input laser intensities and linear absorption coefficients. Optimal conditions for maximum efficiency are discussed.

© 1996 Optical Society of America

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References

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  1. R. W. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1 (1977).
    [CrossRef]
  2. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16 (1977).
    [CrossRef]
  3. See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983), and M. C. Gower, “The physics of phase conjugate mirrors,” Prog. Quantum Electron. 9, 101 (1984).
    [CrossRef]
  4. I. Biaggio, J. P. Partanen, B. Ai, R. J. Knize, and R. W. Hellwarth, “Optical image processing by an atomic vapour,” Nature (London) 371, 318 (1994).
    [CrossRef]
  5. I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
    [CrossRef] [PubMed]
  6. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94 (1978); Erratum, Opt. Lett. 3, 205 (1978).
    [CrossRef] [PubMed]
  7. R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
    [CrossRef]
  8. W. P. Brown, “Absorption and depletion effects on degenerate four-wave mixing in homogeneously broadened absorbers,” J. Opt. Soc. Am. 73, 629 (1983).
    [CrossRef]
  9. M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, “Exact theory of pump wave propagation and its effects on degenerate four-wave mixing,” J. Opt. Soc. Am. B 2, 1117 (1985).
    [CrossRef]
  10. A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).
  11. R. J. Knize, “Efficiency of degenerate four-wave mixing in a two-level saturable absorbing medium,” Opt. Lett. 18, 1606 (1993).
    [CrossRef] [PubMed]
  12. Two definitions for T2 are used in the literature. We use the definition that is in P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990). Another definition, 2T2, was introduced by F. Bloch [F. Bloch, W. W. Hansen, and M. Packard, “Nuclear induction,” Phys. Rev. 70, 960 (1946)].
  13. See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1975), and P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).
  14. See, for example, Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  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, 1993), p. 221.
  16. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971).
    [CrossRef]
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992).
  18. B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
    [CrossRef]
  19. B. Ai, “Degenerate four-wave mixing and image processing in cesium vapors,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1995).

1994 (3)

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

I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
[CrossRef] [PubMed]

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

1993 (1)

1986 (1)

A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).

1985 (1)

1983 (1)

1982 (1)

R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
[CrossRef]

1978 (1)

1977 (2)

Abrams, R. L.

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94 (1978); Erratum, Opt. Lett. 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, 1993), p. 221.

Ai, B.

I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
[CrossRef] [PubMed]

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

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

B. Ai, “Degenerate four-wave mixing and image processing in cesium vapors,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1995).

Allen, L.

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1975), and P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Biaggio, I.

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

I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
[CrossRef] [PubMed]

Boyd, R. W.

A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, “Exact theory of pump wave propagation and its effects on degenerate four-wave mixing,” J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Brown, W. P.

Butcher, P. N.

Two definitions for T2 are used in the literature. We use the definition that is in P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990). Another definition, 2T2, was introduced by F. Bloch [F. Bloch, W. W. Hansen, and M. Packard, “Nuclear induction,” Phys. Rev. 70, 960 (1946)].

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971).
[CrossRef]

Cotter, D.

Two definitions for T2 are used in the literature. We use the definition that is in P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990). Another definition, 2T2, was introduced by F. Bloch [F. Bloch, W. W. Hansen, and M. Packard, “Nuclear induction,” Phys. Rev. 70, 960 (1946)].

Dunning, G. J.

R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
[CrossRef]

Eberly, J. H.

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1975), and P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

Flannery, B. P.

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

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971).
[CrossRef]

Gaeta, A. L.

A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, “Exact theory of pump wave propagation and its effects on degenerate four-wave mixing,” J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Glassner, D. S.

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

Gruneisen, M. T.

A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).

M. T. Gruneisen, A. L. Gaeta, and R. W. Boyd, “Exact theory of pump wave propagation and its effects on degenerate four-wave mixing,” J. Opt. Soc. Am. B 2, 1117 (1985).
[CrossRef]

Hellwarth, R. W.

Knize, R. J.

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

I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
[CrossRef] [PubMed]

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

R. J. Knize, “Efficiency of degenerate four-wave mixing in a two-level saturable absorbing medium,” Opt. Lett. 18, 1606 (1993).
[CrossRef] [PubMed]

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, 1993), p. 221.

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, 1993), p. 221.

Lind, R. C.

R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
[CrossRef]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94 (1978); Erratum, Opt. Lett. 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, 1993), p. 221.

Partanen, J. P.

I. Biaggio, B. Ai, R. J. Knize, J. P. Partanen, and R. W. Hellwarth, “Optical correlator that uses cesium vapor,” Opt. Lett. 19, 1765 (1994).
[CrossRef] [PubMed]

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

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

Pepper, D. M.

Press, W. H.

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

Shen, Y. R.

See, for example, Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Steel, D. G.

R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
[CrossRef]

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, 1993), p. 221.

Teukolsky, S. A.

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

Vetterling, W. T.

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

Yariv, A.

Appl. Phys. Lett. (1)

B. Ai, D. S. Glassner, R. J. Knize, and J. P. Partanen, “A thin atomic vapor as a nonlinear optical medium,” Appl. Phys. Lett. 64, 951 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. L. Gaeta, M. T. Gruneisen, and R. W. Boyd, “Theory of degenerate four-wave mixing in saturable absorbing media with the inclusion of pump propagation effects,” IEEE J. Quantum Electron. QE-22, 1905 (1986).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

Nature (London) (1)

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

Opt. Eng. (1)

R. C. Lind, D. G. Steel, and G. J. Dunning, “Phase conjugation by resonantly enhanced degenerate four-wave mixing,” Opt. Eng. 21, 190 (1982).
[CrossRef]

Opt. Lett. (4)

Other (8)

Two definitions for T2 are used in the literature. We use the definition that is in P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990). Another definition, 2T2, was introduced by F. Bloch [F. Bloch, W. W. Hansen, and M. Packard, “Nuclear induction,” Phys. Rev. 70, 960 (1946)].

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1975), and P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

See, for example, Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

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, 1993), p. 221.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971).
[CrossRef]

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

See, for example, R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983), and M. C. Gower, “The physics of phase conjugate mirrors,” Prog. Quantum Electron. 9, 101 (1984).
[CrossRef]

B. Ai, “Degenerate four-wave mixing and image processing in cesium vapors,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1995).

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Figures (10)

Fig. 1
Fig. 1

Geometry of DFWM in the phase-conjugating configuration and the Cartesian coordinates used in the calculation.

Fig. 2
Fig. 2

Energy diagram of a two-level saturable absorber. ω0 is the resonance frequency of the absorber, and ω is the laser frequency.

Fig. 3
Fig. 3

Efficiency of DFWM for various power distributions in the input beams. The linear intensity absorption is αIL=1.0, and the detuning is δ=0 (on resonance). The total intensity is the sum of the forward pump intensity IF, the backward pump intensity IB, and the probe intensity IP.

Fig. 4
Fig. 4

Resonant DFWM efficiency as a function of the total intensity for various absorptions αIL. The intensities of all three input beams are set equal for maximal efficiency. The numbers next to each curve indicate the values of αIL.

Fig. 5
Fig. 5

Electric-field distribution in the nonlinear medium for a resonant case in which αIL=1.0 and IF=IB=IP = 0.53Is0 (efficiency is maximum). The probe beam has the same field distribution as that of the forward pump beam.

Fig. 6
Fig. 6

Same as Fig. 5, but for αIL=45 and IF=IB=IP = 16.7Is0.

Fig. 7
Fig. 7

DFWM efficiency for various detunings δ at αIL=1.0. All three input beam intensities are set equal for maximum efficiency.

Fig. 8
Fig. 8

Same as Fig. 7, but for αIL=8.0.

Fig. 9
Fig. 9

Electric-field distribution in the nonlinear medium for δ=5, αIL=1.0, and IF=IB=IP14Is0 (efficiency is near the maximum). The forward pump field has the same spatial distribution in the medium as that of the probe beam. The filled squares denote the real part of the electric field; the open squares denote the imaginary part of the electric field, and the solid curves denote the amplitude of the electric field.

Fig. 10
Fig. 10

Same as Fig. 9, but for δ=5, αIL=8.0, and IF=IB = IP87Is0.

Equations (86)

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

(r, t)=12E(r)exp(-iωt)+c.c.,
P(r,t)=12P(r)exp(-iωt)+c.c.=Nμ,
P(r)=i(1-iδ)α(δ)2πkE(r)1+|E(r)|2/Es2(δ),
δ=(ω0-ω)T2,
αI(δ)=2α(δ)=(Na0-Nb0)4πkμ2T2ћ(1+δ2),
Es2(δ)=ћ2(1+δ2)T1T2μ2.
Ei(r)=Ai(r)Es(δ)exp(iki·r),
|k2Ai|kAiz2Aiz2.
E(r)=i Ei(r)=i Ai(r)Es(δ)exp(iki·r).
P(r)=iPi(r)exp(iki·r),
Pi(r)=1V dr P(r+r)exp[-iki·(r+r)].
L1=λ2 sin(θ/2),
L2=λ2 cos(θ/2),
Pi(r)=i(1-iδ)α(δ)2πkEs(δ)Ri(r),
Ri(r)=1VdrjAj(r+r)exp[ikj·(r+r)]1+lAl(r+r)exp[ikl·(r+r)]2×exp[-iki·(r+r)],
2(r,t)-1c22t2(r,t)=4πc22t2P(r,t),
2E(r)+ω2c2E(r)=-4πω2c2P(r).
ki·Ai(r)=-k(1-iδ)α(δ)Ri(r).
Ap(r)z=-(1-iδ)α(δ)Rp(r), (p=1,2),
Aq(r)ξθ=-(1-iδ)α(δ)Rq(r), (q=3,4),
ξθ=zcos θ.
A1(r)z=-α1(r)A1(r)-κ1(r)A2*(r)A3(r)A4(r),
A2(r)z=α2(r)A2(r)+κ2(r)A1*(r)A3(r)A4(r),
A3(r)ξθ=-α3(r)A3(r)-κ3(r)A1(r)A2(r)A4*(r),
A4(r)ξθ=α4(r)A4(r)+κ4(r)A1(r)A2(r)A3*(r),
R1(r)=R1[e1(r),e2(r),e3(r),e4(r)],
ei(r)=|Ai(r)|.
-(1-iδ)α(δ)R1[e1(r),e2(r),e3(r),e4(r)]=-α1(r)e1(r)-κ1(r)e2(r)e3(r)e4(r).
α1(r)=(1-iδ)α(δ){R1[e1(r),e2(r),e3(r),e4(r)]-R1[-e1(r),e2(r),e3(r),e4(r)]}/2e1(r),
κ1(r)=(1-iδ)α(δ){R1[e1(r),e2(r),e3(r),e4(r)]+R1[-e1(r),e2(r),e3(r),e4(r)]}/2e2×(r)e3(r)e4(r).
R4=1V
× drj ej(r+r)exp[ikj·(r+r)]1+i ei(r+r)exp[iki·(r+r)]2
×exp[-ik4·(r+r)].
R4=1V11+i=14 [ei(r)]2 dri=14 εi cos(ki-k4)·r+i i=14 εi sin(ki-k4)·r1+i,j=1(ij)4 εiεj cos(ki-kj)·r,
εiei(r)1+i=14 [ei(r)]2.
Aε4+ε43ε22-ε42ε22,
Bε2+2ε42ε2-ε1ε3ε4ε22-12ε22,
Cε4-ε1ε3ε2,
a1+ε42ε22,
b2ε4ε2,
d1-4(ε12+ε32)(ε22+ε42),
e4[(ε1ε3+ε2ε4)-2(ε1ε4+ε2ε3)×(ε1ε2+ε3ε4)],
f4(ε1ε3-ε2ε4)2.
d-e+f>0,
d+e+f>0.
e2-4df<0,
R4=2πd-e+f11+i=14 ei2×A-B+Ca-bd+e+fd-e+f1/4Kρ1+-Kρ1+11-γ1Ππ2,γ12γ12-1,ρ1×2C/bd+e+fd-e+f1/4d+e+fd-e+f1/2-1+-Kρ111-ϕ1Ππ2,ϕ12ϕ12-1,ρ1×2Bab-Ca2-Ab2ba-b2d+e+fd-e+f1/4d+e+fd-e+f1/2-a+ba-b+11+i=14 ei2Q4e1,e2,e3,e4,
ρ112-d-f2(d+f)2-e21/2,
γ1d+e+f-d-e+fd+e+f+d-e+f,
ϕ1d+e+f-a+ba-bd-e+fd+e+f+a+ba-bd-e+f,
Q4(e1,e2,e3,e4)=12ε4if ε4ε20otherwise.
e2-4df>0,
R4=2πd-f+e2-4df1/211+i=14 ei2×A-B+Ca-bKρ2-2C/bd-f-e2-4dfd-e+f-1×Kρ2+γ22-1Ππ2,γ22γ22-1,ρ2-2Bab-Ca2-Ab2ba-b2d-f-e2-4dfd-e+f-a+ba-b×Kρ2+ϕ22-1Ππ2,ϕ22ϕ22-1,ρ2+11+i=14 ei2Q4e1,e2,e3,e4,
ρ22e2-4dfd-f+e2-4df1/2,
γ21-d-f-e2-4dfd-e+f1/2,
ϕ21-d-f-e2-4dfd-e+fa-ba+b1/2.
e2-4df=0,
R4=1d-e+f×11+i=14 ei2A-B+Ca-bd-e+fd+e+f1/4+CbF4e1,e2,e3,e4+Bab-Ca2-Ab2ba-b2G4e1,e2,e3,e4+11+i=14 ei2Q4e1,e2,e3,e4,
F4e1,e2,e3,e4
=1if e=02d-e+fd+e+f1/41+d+e+fd-e+f1/4otherwise
G4e1,e2,e3,e4=a-ba+b3/2if a+ba-b=d+e+fd-e+f1/22a-ba+b1/2d-e+fd+e+f1/4a+ba-b1/2+d+e+fd-e+f1/4otherwise.
for R1: 41,32,23,14,
for R2: 42,31,24,13,
for R3: 43,34,21,12.
E4(z=L)=0.
A2*(m)=Ψ1+Ψ2,
A1(m+1)=-Ψ1(α2*-a+)exp(a+l)+Ψ2(α2*-a-)exp(a-l)κ2*A3*A4*,
Ψ1=[A2*(m+1)(α2*-a-)+A1(m)×κ2*A3*A4*exp(a-l)]/[(α2*-a-)exp(a+l)-(α2*-a+)exp(a+l)],
Ψ2=-A1(m)κ2*A3*A4*+Ψ1(α2*-a+)α2*-a-,
a±=12(α2*-α1)±(α2*+α1)2-4κ1κ2*A3*A4*2.
η=IsignalItotal=IsignalIF+IB+IP,
u=(k3-k2)·r,
v=(k3-k1)·r,
R4=1(2π)21+i=14 ei2dudv×ζ1(v)+ζ2(v)cos u+ζ3(v)sin uη1(v)+η2(v)cos u+η3(v)sin u,
ζ1(v)=ε4+ε2 cos v,
ζ2(v)=ε1+ε3 cos v,
ζ3(v)=ε3 sin v,
η1(v)=1+2(ε1ε3+ε2ε4)cos v,
η2(v)=2(ε1ε4+ε2ε3)+2(ε1ε2+ε3ε4)cos v,
η3(v)=2(ε3ε4-ε1ε2)sin v.
η1>η2,
η12>η22+η32.
R4=12π1+i=14 ei2dv×1η1(v)2-η2(v)2-η3(v)2×ζ1(v)-ζ2(v)η2(v)+ζ3(v)η3(v)η2(v)2+η3(v)2η1(v)+11+i=14 ei2Q4(e1,e2,e3,e4)=12π1+i=14 ei2 dv×A+B cos v+C cos2 va+b cos v×1d+e cos v+f cos2 v+11+i=14 ei2×Q4(e1,e2,e3,e4),
tanv2=t,
R4=2/π1+i=14 ei20 dtA+B+C+2(A-C)t2+(A-B+C)t4(1+t2)[a+b+(a-b)t2]d+e+f+2(d-f)t2+(d-e+f)t4+11+i=14 ei2Q4(e1,e2,e3,e4).
R4=2/π1+i=14 ei20 dtA-B+Ca-b+2C/b1+t2+2Bab-Ca2-Ab2ba-b2a+ba-b+t2×1d+e+f+2d-ft2+d-e+ft4+11+i=14 ei2Q4e1,e2,e3,e4.

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