Abstract

The optical bistability of a four-level system with triplet-state absorption typical of organic media is studied. We derive the state equation of optical bistability with triplet-state absorption. Different bistabilities with resonant absorption or dispersion of the triplet state are numerically discussed. The effects of the cooperativity parameter and the atomic- and cavity-detuning parameters on optical bistable behavior are analyzed. It is found that the hysteresis loops decrease with an increase of both the cooperativity parameter and the atomic-detuning parameter of the triplet state and that under some conditions reverse saturable-absorptive media can display bistability.

© 1995 Optical Society of America

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  1. A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
    [CrossRef]
  2. S. L. McCall, "Instabilities in continuous-wave light propagation in absorbing media," Phys. Rev. A 9, 1515–1523 (1974); R. Bonifacio and L. A. Lugiato, "Cooperative effects and bistability for resonance fluorescence," Opt. Commun. 19, 172–176 (1976); J. H. Marburger and F. S. Felber, "Theory of a lossless nonlinear Fabry–Perot interferometer," Phys. Rev. A 17, 335–342 (1978); G. P. Agrawal and H. J. Carmichael, "Optical bistability through nonlinear dispersion and absorption," Phys. Rev. A 19, 2074–2086 (1979).
    [CrossRef]
  3. D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
    [CrossRef]
  4. G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
    [CrossRef] [PubMed]
  5. S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
    [CrossRef] [PubMed]
  6. T. C. Ralph, "Bistability in a four-level laser with a resonant pump mode," Phys. Rev. A 49, 4979–4984 (1994).
    [CrossRef] [PubMed]
  7. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
    [CrossRef]
  8. H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
    [CrossRef]
  9. A. D. Lloyd and B. S. Wherrett, "All-optics bistability in nematic liquid crystals at 20 μW power levels," Appl. Phys. Lett. 53, 460–461 (1988); B. P. Singh and P. N. Prasad, "Optical bistable behavior of a planar quasi-waveguide interferometer mode with a conjugated organic polymer film," J. Opt. Soc. Am. B 5, 453–456 (1988); K. Sasaki, K. Fujii, and T. Tomioka, "All-optical bistabilities of polydiacetylene Langmuir-Blodgett film waveguides," J. Opt. Soc. Am. B 5, 457–461 (1988).
    [CrossRef]
  10. F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
    [CrossRef]
  11. M. Hercher, "An analysis of saturable absorbers," Appl. Opt. 6, 947–954 (1967).
    [CrossRef] [PubMed]
  12. S. Miyanaga, H. Ohtateme, K. Kawano, and H. Fujiwara, "Excited-state absorption and pump propagation effects on optical phase conjugation in a saturable absorber," J. Opt. Soc. Am. B 10, 1069–1076 (1993).
    [CrossRef]

1994 (1)

T. C. Ralph, "Bistability in a four-level laser with a resonant pump mode," Phys. Rev. A 49, 4979–4984 (1994).
[CrossRef] [PubMed]

1993 (2)

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

S. Miyanaga, H. Ohtateme, K. Kawano, and H. Fujiwara, "Excited-state absorption and pump propagation effects on optical phase conjugation in a saturable absorber," J. Opt. Soc. Am. B 10, 1069–1076 (1993).
[CrossRef]

1992 (2)

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
[CrossRef] [PubMed]

1988 (1)

A. D. Lloyd and B. S. Wherrett, "All-optics bistability in nematic liquid crystals at 20 μW power levels," Appl. Phys. Lett. 53, 460–461 (1988); B. P. Singh and P. N. Prasad, "Optical bistable behavior of a planar quasi-waveguide interferometer mode with a conjugated organic polymer film," J. Opt. Soc. Am. B 5, 453–456 (1988); K. Sasaki, K. Fujii, and T. Tomioka, "All-optical bistabilities of polydiacetylene Langmuir-Blodgett film waveguides," J. Opt. Soc. Am. B 5, 457–461 (1988).
[CrossRef]

1981 (1)

D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
[CrossRef]

1979 (1)

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

1976 (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

1974 (1)

S. L. McCall, "Instabilities in continuous-wave light propagation in absorbing media," Phys. Rev. A 9, 1515–1523 (1974); R. Bonifacio and L. A. Lugiato, "Cooperative effects and bistability for resonance fluorescence," Opt. Commun. 19, 172–176 (1976); J. H. Marburger and F. S. Felber, "Theory of a lossless nonlinear Fabry–Perot interferometer," Phys. Rev. A 17, 335–342 (1978); G. P. Agrawal and H. J. Carmichael, "Optical bistability through nonlinear dispersion and absorption," Phys. Rev. A 19, 2074–2086 (1979).
[CrossRef]

1969 (1)

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

1967 (1)

Bava, G. P.

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

Castelli, F.

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

Daneu, V.

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Debernardi, P.

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

Fujiwara, H.

Gibbs, H. M.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

Goldhar, J.

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Gong, S.

S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
[CrossRef] [PubMed]

Gossard, A. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Hercher, M.

Jiang, M.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Kawano, K.

Kumit, N. A.

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Lin, F.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Lloyd, A. D.

A. D. Lloyd and B. S. Wherrett, "All-optics bistability in nematic liquid crystals at 20 μW power levels," Appl. Phys. Lett. 53, 460–461 (1988); B. P. Singh and P. N. Prasad, "Optical bistable behavior of a planar quasi-waveguide interferometer mode with a conjugated organic polymer film," J. Opt. Soc. Am. B 5, 453–456 (1988); K. Sasaki, K. Fujii, and T. Tomioka, "All-optical bistabilities of polydiacetylene Langmuir-Blodgett film waveguides," J. Opt. Soc. Am. B 5, 457–461 (1988).
[CrossRef]

Lugiato, L. A.

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

Luo, T.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

McCall, S. L.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

S. L. McCall, "Instabilities in continuous-wave light propagation in absorbing media," Phys. Rev. A 9, 1515–1523 (1974); R. Bonifacio and L. A. Lugiato, "Cooperative effects and bistability for resonance fluorescence," Opt. Commun. 19, 172–176 (1976); J. H. Marburger and F. S. Felber, "Theory of a lossless nonlinear Fabry–Perot interferometer," Phys. Rev. A 17, 335–342 (1978); G. P. Agrawal and H. J. Carmichael, "Optical bistability through nonlinear dispersion and absorption," Phys. Rev. A 19, 2074–2086 (1979).
[CrossRef]

Miyanaga, S.

Ohtateme, H.

Pan, S.

S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
[CrossRef] [PubMed]

Passner, A.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Qian, Q.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Ralph, T. C.

T. C. Ralph, "Bistability in a four-level laser with a resonant pump mode," Phys. Rev. A 49, 4979–4984 (1994).
[CrossRef] [PubMed]

Steyn-Ross, M. L.

D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
[CrossRef]

Szoke, A.

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

Venkatesan, T. N. C.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

Walls, D. F.

D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
[CrossRef]

Wherrett, B. S.

A. D. Lloyd and B. S. Wherrett, "All-optics bistability in nematic liquid crystals at 20 μW power levels," Appl. Phys. Lett. 53, 460–461 (1988); B. P. Singh and P. N. Prasad, "Optical bistable behavior of a planar quasi-waveguide interferometer mode with a conjugated organic polymer film," J. Opt. Soc. Am. B 5, 453–456 (1988); K. Sasaki, K. Fujii, and T. Tomioka, "All-optical bistabilities of polydiacetylene Langmuir-Blodgett film waveguides," J. Opt. Soc. Am. B 5, 457–461 (1988).
[CrossRef]

Wiegmann, W.

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

Wu, Z.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Xie, Y.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Yang, G.

S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
[CrossRef] [PubMed]

Zeng, H.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Zhao, J.

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

Zoller, P.

D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

A. Szoke, V. Daneu, J. Goldhar, and N. A. Kumit, "Bistable optical element and its applications," Appl. Phys. Lett. 15, 376–379 (1969).
[CrossRef]

H. M. Gibbs, S. L. McCall, T. N. C. Venkatesan, A. C. Gossard, A. Passner, and W. Wiegmann, "Optical bistability in semiconductors," Appl. Phys. Lett. 35, 451–453 (1979).
[CrossRef]

A. D. Lloyd and B. S. Wherrett, "All-optics bistability in nematic liquid crystals at 20 μW power levels," Appl. Phys. Lett. 53, 460–461 (1988); B. P. Singh and P. N. Prasad, "Optical bistable behavior of a planar quasi-waveguide interferometer mode with a conjugated organic polymer film," J. Opt. Soc. Am. B 5, 453–456 (1988); K. Sasaki, K. Fujii, and T. Tomioka, "All-optical bistabilities of polydiacetylene Langmuir-Blodgett film waveguides," J. Opt. Soc. Am. B 5, 457–461 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. F. Walls, P. Zoller, and M. L. Steyn-Ross, "Optical bistability from three-level atoms," IEEE J. Quantum Electron. QE-17, 380–383 (1981).
[CrossRef]

J. Appl. Phys. (1)

F. Lin, J. Zhao, T. Luo, M. Jiang, Z. Wu, Y. Xie, Q. Qian, and H. Zeng, "Optical limitation and bistability in fullerenes," J. Appl. Phys. 74, 2140–2142 (1993).
[CrossRef]

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

Phys. Rev. A (4)

S. L. McCall, "Instabilities in continuous-wave light propagation in absorbing media," Phys. Rev. A 9, 1515–1523 (1974); R. Bonifacio and L. A. Lugiato, "Cooperative effects and bistability for resonance fluorescence," Opt. Commun. 19, 172–176 (1976); J. H. Marburger and F. S. Felber, "Theory of a lossless nonlinear Fabry–Perot interferometer," Phys. Rev. A 17, 335–342 (1978); G. P. Agrawal and H. J. Carmichael, "Optical bistability through nonlinear dispersion and absorption," Phys. Rev. A 19, 2074–2086 (1979).
[CrossRef]

G. P. Bava, F. Castelli, P. Debernardi, and L. A. Lugiato, "Optical bistability in a multiple-quantum-well structure with Fabry–Perot and distributed-feedback resonators," Phys. Rev. A 45, 5180–5192 (1992).
[CrossRef] [PubMed]

S. Gong, S. Pan, and G. Yang, "Optical bistability in a dye-ring cavity," Phys. Rev. A 45, 6655–6658 (1992); "Stability analysis for an optical bistable dye system," Phys. Rev. A 47, 2205–2210 (1993).
[CrossRef] [PubMed]

T. C. Ralph, "Bistability in a four-level laser with a resonant pump mode," Phys. Rev. A 49, 4979–4984 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, "Differential gain and bistability using a sodium-filled Fabry–Perot interferometer," Phys. Rev. Lett. 36, 1135–1138 (1976).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Energy-level diagram. σ0 is the absorption cross section from level 1 to level 2, σT is the absorption cross section from level 3 to level 4, γ is the transition rate of an intersystem crossing, γ34 is the relaxation rate from level 4 to level 3, and Tf and T are the relaxation lifetimes from level 2 and level 3, respectively, to ground level 1.

Fig. 2
Fig. 2

Parameter condition figure for the bistability equation having real roots.

Fig. 3
Fig. 3

Parameter requirement figure for the bistability equation having positive roots.

Fig. 4
Fig. 4

Bistability curves for CT = 0, 0.2, 0.5, 0.7, 10, with C0 = 10 and δ12 = δ34 = Θ = 0.

Fig. 5
Fig. 5

Bistability curves for C0 = 0.2, 6, 10, 15, 20 with CT = 0.2 and δ12 = δ34 = Θ = 0.

Fig. 6
Fig. 6

Bistability curves for |Θ| = 0, 0.5, 1, 1.5, 2, and 2.6, with C0 = 10, CT = 0.2, and δ12 = δ34 = 0.

Fig. 7
Fig. 7

Bistability curves for |δ12| = 0, 1, 3, 5, with C0 = 10, CT = 0.2, and δ34 = Θ = 0.

Fig. 8
Fig. 8

Bistability curves for Θ = −4, −2, 0, 1, 2, with C0 = 10, CT = 11, δ12 = 6, and δ34 = 0.

Fig. 9
Fig. 9

Bistability curves for |δ34| = 0, 1, 3, 5, with C0 = 10, CT = 0.2, and δ12 = Θ = 0.

Fig. 10
Fig. 10

Bistability curves for δ34 = −2.5, −2, −1, −0.5, 0, with C0 = CT = 6, δ12 = 5, and Θ = 0.

Fig. 11
Fig. 11

Bistability curves for (a) δ12 = −2, …, 2, (b) δ12 = − 2, −1, 0, 1, 2. C0 = 10, CT = 0.2, δ34 = 1, and Θ = 0.

Fig. 12
Fig. 12

Bistability curves for CT = 0, 3, 4, 6, 7, 9 with C0 = 6, δ12 = 5, δ34 = −1, and Θ = 0.

Fig. 13
Fig. 13

Bistability curves for Θ = 2, 1, 0, −1, −2, with C0 = 10, CT = 0.2, δ12 = −2 and δ34 = 1.

Fig. 14
Fig. 14

Diagram of the parameters’ effects on the hysteresis loop of the bistability curve. With an increase of the relative parameters along the directions of the arrows, the hysteresis loop increases.

Equations (27)

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

P ( r , t ) = i ( 0 N / k 0 ) [ σ 0 Δ 12 ( ρ 1 - ρ 2 ) + σ T Δ 34 ( ρ 3 - ρ 4 ) ] E ( r , t ) ,
P ( r , t ) = 0 N i k 0 [ σ T ( 1 + i δ 34 ) + σ 0 ( 1 + i δ 12 ) - σ T ( 1 + i δ 34 ) 1 + E 2 / E s 2 ] E ( r , t ) ,
χ ( r ) = - N k 0 ( σ 0 δ 12 + σ T δ 34 E 2 / E s 2 1 + E 2 / E s 2 ) + i N k 0 ( σ 0 + σ T E 2 / E s 2 1 + E 2 / E s 2 ) .
E t = 1 - R 1 - R exp ( i 2 n k 0 L ) E i ,
E t = ( 1 - R ) E i 1 - R ( 1 + i χ k 0 L - i θ ) ,
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 + ( Θ + 2 C 0 δ 12 + 2 C T δ 34 x 2 1 + x 2 ) 2 ] ,
x 2 = E t 2 ( 1 - R ) E s 2 ,             y 2 = E i 2 ( 1 - R ) E s 2 ,
C 0 , T = N L R σ 0 , T 2 ( 1 - R ) ,
Θ = θ R 1 - R .
u Z 3 + 4 w Z + 8 v = 1 ,
Z = x 2 + 1 , u = ( 1 + 2 C T ) 2 + ( Θ + 2 C T δ 34 ) 2 , v = ( C 0 - C T ) 2 + ( C 0 δ 12 - C T δ 34 ) 2 , w = ( C 0 - C T ) ( 3 C T - C 0 + 1 ) + ( C 0 δ 12 - C T δ 34 ) ( 3 C T δ 34 - C 0 δ 12 + Θ ) .
4 w 3 + 27 v 2 u < 0.
( - 4 w / 3 u ) 1 / 2 cos ϕ / 3 > 1 / 2 , o r , ( - 4 w / 3 u ) 1 / 2 cos ( π / 3 - ϕ / 3 ) < - 1 / 2 , o r , ( - 4 w / 3 u ) 1 / 2 cos ( π / 3 + ϕ / 3 ) < - 1 / 2 ,
y = x ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) .
C 0 > 9 C T + 4.
y 2 = x 2 [ Θ 2 + ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 ] ,
0 < Θ 2 < ( 2 C 0 + 1 ) 2 ( C 0 - 4 - 9 C T ) 27 ( C 0 - C T ) ,
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 + ( 2 C 0 δ 12 1 + x 2 ) 2 ] .
y 2 = x 2 [ ( 1 + 2 C 0 ) 2 + ( 2 C 0 δ 12 1 + x 2 ) 2 ] .
δ 12 > 3 3 ( 1 2 C 0 + 1 ) .
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 + ( Θ + 2 C 0 δ 12 1 + x 2 ) 2 ] .
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 + ( 2 C T δ 34 x 2 1 + x 2 ) 2 ] .
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T x 2 1 + x 2 ) 2 + ( Θ + 2 C T δ 34 x 2 1 + x 2 ) 2 ] .
y 2 = x 2 [ ( 1 + 2 C 0 + 2 C T 1 + x 2 ) 2 + ( 2 C T δ 34 ) 2 ] .
( C 0 - C T ) [ - 108 C T 2 ( C 0 - C T ) δ 34 2 + ( 2 C 0 + 1 ) 2 ( C 0 - 4 - 9 C T ) ] > 0.
y 2 = x 2 [ ( 1 + 2 C T ) 2 + ( 2 C 0 δ 12 + 2 C T δ 34 1 + x 2 ) 2 ] .
( δ 12 - δ 34 ) { - 9 δ 34 [ 2 C 0 δ 12 + ( 2 C 0 + 1 ) 3 ] [ 2 C 0 δ 12 - ( 2 C 0 + 1 ) 3 ] + δ 12 [ 2 C 0 δ 12 + 3 ( 2 C 0 + 1 ) 3 ] [ 2 C 0 δ 12 - 3 ( 2 C 0 + 1 ) 3 ] } > 0.

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