Abstract

We analyze the stability of two recently demonstrated photorefractive resonator circuits. The analysis is based on single-mode models of the multimode circuits. The flip-flop, which consists of two competitively coupled rings, is considered in the two limits where the rings share or have separate gain volumes. Both configurations are found to be stable for typical experimental conditions. The feature extractor consists of two rings with a shared gain volume. It is found to be unconditionally stable. The results are discussed in the context of the experimental demonstrations.

© 1995 Optical Society of America

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  1. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
    [CrossRef]
  2. S.-K. Kwong and A. Yariv, "One-way, real time wave front converters," Appl. Phys. Lett. 48, 564–566 (1986).
    [CrossRef]
  3. S. Weiss and B. Fischer, "Photorefractive saturable absorptive and dispersive optical bistability," Opt. Commun. 70, 515–521 (1989).
    [CrossRef]
  4. D. Z. Anderson, "Coherent optical eigenstate memory," Opt. Lett. 11, 56–58 (1986).
    [CrossRef] [PubMed]
  5. D. Z. Anderson and M. C. Erie, "Resonator memories and optical novelty filters," Opt. Eng. 26, 434–444 (1987).
    [CrossRef]
  6. L.-S. Lee, H. M. Stoll, and M. C. Tackitt, "Continuous-time optical neural network associative memory," Opt. Lett. 14, 162–164 (1989).
    [CrossRef] [PubMed]
  7. D. M. Lininger, P. J. Martin, and D. Z. Anderson, "Bistable ring resonator utilizing saturable photorefractive gain and loss," Opt. Lett. 14, 697–699 (1989).
    [CrossRef] [PubMed]
  8. D. Z. Anderson, C. Benkert, B. Chorbajian, and A. Hermanns, "Photorefractive flip-flop," Opt. Lett. 16, 250–252 (1991).
    [CrossRef] [PubMed]
  9. C. Benkert and D. Z. Anderson, "Controlled competitive dynamics in a photorefractive ring oscillator: winner-takesall and the 'voting paradox' dynamics," Phys. Rev. A 44, 4633–4638 (1991).
    [CrossRef] [PubMed]
  10. M. Saffman, C. Benkert, and D. Z. Anderson, "Selforganizing photorefractive frequency demultiplexer," Opt. Lett. 16, 1993–1995 (1991).
    [CrossRef] [PubMed]
  11. D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.
  12. M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).
  13. N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
    [CrossRef]
  14. D. Z. Anderson, M. Saffman, and A. Hermanns, "Manipulating the information carried by an optical beam with reflexive photorefractive beam coupling," J. Opt. Soc. Am. B 12, 117–123 (1995).
    [CrossRef]
  15. D. Z. Anderson and J. Feinberg, "Optical novelty filters," IEEE J. Quantum Electron. 25, 635–647 (1989).
    [CrossRef]
  16. A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447–452 (1989).
    [CrossRef]
  17. N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Sov. Tech. Phys. Lett. 2, 438–440 (1976).
  18. M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
    [CrossRef]
  19. D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
    [CrossRef]
  20. M. Saffman and D. Z. Anderson, "Mode multiplexing and holographic demultiplexing communication channels on a multimode fiber," Opt. Lett. 16, 300–302 (1991).
    [CrossRef] [PubMed]

1995 (1)

1991 (4)

1990 (1)

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

1989 (6)

L.-S. Lee, H. M. Stoll, and M. C. Tackitt, "Continuous-time optical neural network associative memory," Opt. Lett. 14, 162–164 (1989).
[CrossRef] [PubMed]

D. M. Lininger, P. J. Martin, and D. Z. Anderson, "Bistable ring resonator utilizing saturable photorefractive gain and loss," Opt. Lett. 14, 697–699 (1989).
[CrossRef] [PubMed]

M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
[CrossRef]

D. Z. Anderson and J. Feinberg, "Optical novelty filters," IEEE J. Quantum Electron. 25, 635–647 (1989).
[CrossRef]

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447–452 (1989).
[CrossRef]

S. Weiss and B. Fischer, "Photorefractive saturable absorptive and dispersive optical bistability," Opt. Commun. 70, 515–521 (1989).
[CrossRef]

1987 (1)

D. Z. Anderson and M. C. Erie, "Resonator memories and optical novelty filters," Opt. Eng. 26, 434–444 (1987).
[CrossRef]

1986 (2)

S.-K. Kwong and A. Yariv, "One-way, real time wave front converters," Appl. Phys. Lett. 48, 564–566 (1986).
[CrossRef]

D. Z. Anderson, "Coherent optical eigenstate memory," Opt. Lett. 11, 56–58 (1986).
[CrossRef] [PubMed]

1982 (1)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

1977 (1)

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
[CrossRef]

1976 (1)

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Anderson, D. Z.

D. Z. Anderson, M. Saffman, and A. Hermanns, "Manipulating the information carried by an optical beam with reflexive photorefractive beam coupling," J. Opt. Soc. Am. B 12, 117–123 (1995).
[CrossRef]

M. Saffman and D. Z. Anderson, "Mode multiplexing and holographic demultiplexing communication channels on a multimode fiber," Opt. Lett. 16, 300–302 (1991).
[CrossRef] [PubMed]

C. Benkert and D. Z. Anderson, "Controlled competitive dynamics in a photorefractive ring oscillator: winner-takesall and the 'voting paradox' dynamics," Phys. Rev. A 44, 4633–4638 (1991).
[CrossRef] [PubMed]

M. Saffman, C. Benkert, and D. Z. Anderson, "Selforganizing photorefractive frequency demultiplexer," Opt. Lett. 16, 1993–1995 (1991).
[CrossRef] [PubMed]

D. Z. Anderson, C. Benkert, B. Chorbajian, and A. Hermanns, "Photorefractive flip-flop," Opt. Lett. 16, 250–252 (1991).
[CrossRef] [PubMed]

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

D. M. Lininger, P. J. Martin, and D. Z. Anderson, "Bistable ring resonator utilizing saturable photorefractive gain and loss," Opt. Lett. 14, 697–699 (1989).
[CrossRef] [PubMed]

D. Z. Anderson and J. Feinberg, "Optical novelty filters," IEEE J. Quantum Electron. 25, 635–647 (1989).
[CrossRef]

D. Z. Anderson and M. C. Erie, "Resonator memories and optical novelty filters," Opt. Eng. 26, 434–444 (1987).
[CrossRef]

D. Z. Anderson, "Coherent optical eigenstate memory," Opt. Lett. 11, 56–58 (1986).
[CrossRef] [PubMed]

M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

Benkert, C.

C. Benkert and D. Z. Anderson, "Controlled competitive dynamics in a photorefractive ring oscillator: winner-takesall and the 'voting paradox' dynamics," Phys. Rev. A 44, 4633–4638 (1991).
[CrossRef] [PubMed]

D. Z. Anderson, C. Benkert, B. Chorbajian, and A. Hermanns, "Photorefractive flip-flop," Opt. Lett. 16, 250–252 (1991).
[CrossRef] [PubMed]

M. Saffman, C. Benkert, and D. Z. Anderson, "Selforganizing photorefractive frequency demultiplexer," Opt. Lett. 16, 1993–1995 (1991).
[CrossRef] [PubMed]

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

Chorbajian, B.

Cronin-Golomb, M.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Crouch, D. D.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

Erie, M. C.

D. Z. Anderson and M. C. Erie, "Resonator memories and optical novelty filters," Opt. Eng. 26, 434–444 (1987).
[CrossRef]

Feinberg, J.

D. Z. Anderson and J. Feinberg, "Optical novelty filters," IEEE J. Quantum Electron. 25, 635–647 (1989).
[CrossRef]

Fischer, B.

S. Weiss and B. Fischer, "Photorefractive saturable absorptive and dispersive optical bistability," Opt. Commun. 70, 515–521 (1989).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Hebler, V.

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

Hermanns, A.

Jang, J.-S.

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
[CrossRef]

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Kwong, S.-K.

S.-K. Kwong and A. Yariv, "One-way, real time wave front converters," Appl. Phys. Lett. 48, 564–566 (1986).
[CrossRef]

Lee, L.-S.

Lininger, D. M.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

D. M. Lininger, P. J. Martin, and D. Z. Anderson, "Bistable ring resonator utilizing saturable photorefractive gain and loss," Opt. Lett. 14, 697–699 (1989).
[CrossRef] [PubMed]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Martin, P. J.

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

D. M. Lininger, P. J. Martin, and D. Z. Anderson, "Bistable ring resonator utilizing saturable photorefractive gain and loss," Opt. Lett. 14, 697–699 (1989).
[CrossRef] [PubMed]

Montgomery, D.

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Saffman, M.

D. Z. Anderson, M. Saffman, and A. Hermanns, "Manipulating the information carried by an optical beam with reflexive photorefractive beam coupling," J. Opt. Soc. Am. B 12, 117–123 (1995).
[CrossRef]

M. Saffman and D. Z. Anderson, "Mode multiplexing and holographic demultiplexing communication channels on a multimode fiber," Opt. Lett. 16, 300–302 (1991).
[CrossRef] [PubMed]

M. Saffman, C. Benkert, and D. Z. Anderson, "Selforganizing photorefractive frequency demultiplexer," Opt. Lett. 16, 1993–1995 (1991).
[CrossRef] [PubMed]

M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

Stoll, H. M.

Tackitt, M. C.

Tikhonchuk, V. T.

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447–452 (1989).
[CrossRef]

M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
[CrossRef]

Weiss, S.

S. Weiss and B. Fischer, "Photorefractive saturable absorptive and dispersive optical bistability," Opt. Commun. 70, 515–521 (1989).
[CrossRef]

White, J. O.

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Yariv, A.

S.-K. Kwong and A. Yariv, "One-way, real time wave front converters," Appl. Phys. Lett. 48, 564–566 (1986).
[CrossRef]

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

Zhanuzakov, M. G.

M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
[CrossRef]

Zozulya, A. A.

M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
[CrossRef]

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447–452 (1989).
[CrossRef]

M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).

Appl. Phys. Lett. (2)

J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv, "Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3," Appl. Phys. Lett. 40, 450–452 (1982).
[CrossRef]

S.-K. Kwong and A. Yariv, "One-way, real time wave front converters," Appl. Phys. Lett. 48, 564–566 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Z. Anderson and J. Feinberg, "Optical novelty filters," IEEE J. Quantum Electron. 25, 635–647 (1989).
[CrossRef]

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

J. Sov. Laser Res. (1)

M. G. Zhanuzakov, A. A. Zozulya, and V. T. Tikhonchuk, "Stability and bistability of stationary states in four-wave interaction in a photorefractive medium," J. Sov. Laser Res. 11, 59–68 (1989).
[CrossRef]

Opt. Commun. (3)

D. M. Lininger, D. D. Crouch, P. J. Martin, and D. Z. Anderson, "Theory of bistability and self pulsing in a ring resonator with saturable photorefractive gain and loss," Opt. Commun. 76, 89–96 (1990).
[CrossRef]

S. Weiss and B. Fischer, "Photorefractive saturable absorptive and dispersive optical bistability," Opt. Commun. 70, 515–521 (1989).
[CrossRef]

N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, "Transient energy transfer during hologram formation in LiNbO3 in external electric field," Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Opt. Eng. (1)

D. Z. Anderson and M. C. Erie, "Resonator memories and optical novelty filters," Opt. Eng. 26, 434–444 (1987).
[CrossRef]

Opt. Lett. (6)

Phys. Lett. A (1)

A. A. Zozulya and V. T. Tikhonchuk, "Investigation of stability of four-wave mixing in photorefractive media," Phys. Lett. A 135, 447–452 (1989).
[CrossRef]

Phys. Rev. A (1)

C. Benkert and D. Z. Anderson, "Controlled competitive dynamics in a photorefractive ring oscillator: winner-takesall and the 'voting paradox' dynamics," Phys. Rev. A 44, 4633–4638 (1991).
[CrossRef] [PubMed]

Sov. Tech. Phys. Lett. (1)

N. V. Kukhtarev, "Kinetics of hologram recording and erasure in electrooptic crystals," Sov. Tech. Phys. Lett. 2, 438–440 (1976).

Other (2)

D. Z. Anderson, C. Benkert, V. Hebler, J.-S. Jang, D. Montgomery, and M. Saffman, "Optical implementation of a self-organizing feature extractor," in Advances in Neural-Information Processing Systems IV, J. E. Moody, S. J. Hanson, and R. P. Lippmann, eds. (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 821–828.

M. Saffman, D. Montgomery, A. A. Zozulya, and D. Z. Anderson, "Topology preserving mappings in a self-imaging photorefractively pumped ring resonator," Chaos Solitons Fractals (to be published).

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Figures (3)

Fig. 1
Fig. 1

Photorefractive flip-flop: (a) separate gain media and (b) shared gain media. R is the cavity reflectivity and α is the fraction of the oscillating energy that is diverted into the loss pumps.

Fig. 2
Fig. 2

Feature extractor: (a) experimental geometry with reflexive coupling in each ring and (b) simplified model.

Fig. 3
Fig. 3

Single ring pumped by two signals.

Equations (164)

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r x = ν p ,
p x = ν * r ,
( τ t + 1 ) ν = Γ 2 r p * I T ,
| r 0 | out 2 = | r 0 | in 2 M ,
| p 0 | out 2 = | p 0 | in 2 M exp ( Γ ) ,
M = 1 + y y + exp ( Γ ) ,
r ( x , t ) = r 0 ( x ) + Re [ δ r ( x ) exp ( f t ) ] ,
p ( x , t ) = p 0 ( x ) + Re [ δ p ( x ) exp ( f t ) ] ,
[ δ r δ p ] out = T ( 2 ) [ δ r δ p ] in ,
T 11 ( 2 ) = exp ( Γ / 2 ) [ 1 + y exp ( Γ ) ] M × { y exp ( Γ / 2 ) + exp [ 1 1 + τ f ( ln M Γ 2 ) ] } ,
T 12 ( 2 ) = exp ( Γ / 2 ) [ 1 + y exp ( Γ ) ] M y × { exp ( Γ / 2 ) exp [ 1 1 + τ f ( ln M Γ 2 ) ] } ,
T 21 ( 2 ) = exp ( Γ / 2 ) [ 1 + y exp ( Γ ) ] M y × { 1 exp [ Γ 2 + 1 1 + τ f ( ln M Γ 2 ) ] } ,
T 22 ( 2 ) = exp ( Γ / 2 ) [ 1 + y exp ( Γ ) ] M × { 1 + y exp [ Γ 2 + 1 1 + τ f ( ln M Γ 2 ) ] } .
r 1 x = ν 1 p ,
r 2 x = ν 2 p ,
p x = ν 1 * r 1 ν 2 * r 2 ,
( τ t + 1 ) ν 1 = Γ 2 r 1 p * I T ,
( τ t + 1 ) ν 2 = Γ 2 r 2 p * I T .
| r 10 | out 2 = | r 10 | in 2 M ,
| r 20 | out 2 = | r 20 | in 2 M ,
| p 0 | out 2 = | p 0 | in 2 M exp ( Γ ) ,
M = 1 + y y + exp ( Γ ) ,
y = y 1 + y 2 , ( 7 e)
[ δ r 1 δ r 2 δ p ] out = T ( 3 ) [ δ r 1 δ r 2 δ p ] in ,
T 11 ( 3 ) = y 1 y T 11 ( 2 ) ( y ) + y 2 y exp [ 1 2 ( 1 + τ f ) ln M ] ,
T 22 ( 3 ) = y 2 y T 11 ( 2 ) ( y ) + y 1 y exp [ 1 2 ( 1 + τ f ) ln M ] ,
T 33 ( 3 ) = T 22 ( 2 ) ( y ) ,
T 12 ( 3 ) = y 1 y 2 y { T 11 ( 2 ) ( y ) exp [ 1 2 ( 1 + τ f ) ln M ] } ,
T 21 ( 3 ) = T 12 ( 3 ) ,
T 23 ( 3 ) = y 1 y 2 T 12 ( 2 ) ( y ) ,
T 31 ( 3 ) = ( y 1 / y ) T 21 ( 2 ) ( y ) ,
T 32 ( 3 ) = ( y 2 / y ) T 21 ( 2 ) ( y ) .
y L 1 = ( 1 α ) G ( 1 ) y G 1 α G ( 2 ) y G 2 ,
y L 2 = ( 1 α ) G ( 2 ) y G 2 α G ( 1 ) y G 1 .
y G 1 = G ( 1 ) L ( 1 ) R ( 1 α ) y G 1 ,
y G 2 = G ( 2 ) L ( 2 ) R ( 1 α ) y G 2 ,
y G 1 = R ( 1 α ) exp ( Γ G ) ,
y G 2 = 0 ,
y G 1 = y G 2 = R ( 1 α ) α exp ( Γ L ) exp ( Γ G ) .
y G = exp ( Γ G ) 1 z z ,
z = 1 2 ( 1 α ) q × ( q 1 ± { [ 1 + ( 1 2 α ) q ] ( q 3 + 2 α ) 1 2 α } 1 / 2 ) ,
q = R ( 1 α ) α exp ( Γ G Γ L ) ,
min [ 1 , α ( 3 2 α ) ] < R ( 1 α ) exp ( Γ G Γ L ) < max [ 1 , α ( 3 2 α ) ] .
δ r 1 ( before α ) = T 11 ( 2 ) ( G , 1 ) R × [ T 11 ( 2 ) ( L , 1 ) 1 α δ r 1 ( before α ) + T 12 ( 2 ) ( L , 1 ) α δ r 2 ( before α ) ] .
δ r 2 ( before α ) = T 11 ( 2 ) ( G , 2 ) R × [ T 11 ( 2 ) ( L , 2 ) 1 α δ r 2 ( before α ) + T 12 ( 2 ) ( L , 2 ) α δ r 1 ( before α ) ] .
[ 1 R ( 1 α ) T 11 ( 2 ) ( G , 1 ) T 11 ( 2 ) ( L , 1 ) ] × [ 1 R ( 1 α ) T 11 ( 2 ) ( G , 2 ) T 11 ( 2 ) ( L , 2 ) ] = R α T 11 ( 2 ) ( G , 1 ) T 11 ( 2 ) ( G , 2 ) T 12 ( 2 ) ( L , 1 ) T 12 ( 2 ) ( L , 2 ) .
T 11 ( 2 ) ( G , 1 ) = T 11 ( 2 ) ( G , 2 ) T 11 ( 2 ) ( G ) ,
T 11 ( 2 ) ( L , 1 ) = T 11 ( 2 ) ( L , 2 ) T 11 ( 2 ) ( L ) ,
T 12 ( 2 ) ( L , 1 ) = T 12 ( 2 ) ( L , 2 ) T 12 ( 2 ) ( L ) ,
1 R ( 1 α ) T 11 ( 2 ) ( G ) T 11 ( 2 ) ( L ) = ± R α T 11 ( 2 ) ( G ) T 12 ( 2 ) ( L ) ,
R ( 1 α ) exp ( Γ G 1 + τ G f ) = 1 .
f τ G ( 1 ) = Γ G ln [ R ( 1 α ) ] .
R ( 1 α ) exp { Γ G 2 [ 1 + τ G ( 2 ) f Γ L 2 [ 1 + τ L ( 2 ) f } = 1 .
Γ L + C > Γ G ,
τ G τ L > Γ G C Γ L + C ,
Γ L > τ L τ G ( Γ G C ) C
R ( 1 α ) exp [ Γ G 2 ( 1 + τ G f ) ] = 1 .
q exp [ τ G f 1 + τ G f ( Γ G 2 ln q ) ] = ( 1 2 α ) + 2 exp [ τ L f 1 + τ L f ( Γ L 2 + ln α ) ] ,
y L 1 = ( 1 α ) y G 1 α y G 2 ,
y L 2 = ( 1 α ) y G 2 α y G 1 ,
y G 1 = y G 2 = 1 2 [ R ( 1 α ) α exp ( Γ L ) exp ( Γ G ) ] ,
[ 1 R ( 1 α ) T 11 ( 3 ) T 11 ( 2 ) ( L , 1 ) R α T 21 ( 3 ) T 12 ( 2 ) ( L , 1 ) ] × [ 1 R ( 1 α ) T 22 ( 3 ) T 11 ( 2 ) ( L , 2 ) R α T 12 ( 3 ) T 12 ( 2 ) ( L , 2 ) ] = [ R ( 1 α ) T 12 ( 3 ) T 11 ( 2 ) ( L , 1 ) + R α T 22 ( 3 ) T 12 ( 2 ) ( L , 1 ) ] × [ R ( 1 α ) T 21 ( 3 ) T 11 ( 2 ) ( L , 2 ) R α T 11 ( 3 ) T 12 ( 2 ) ( L , 2 ) ] .
1 = R T 11 ( 2 ) ( G ) [ 1 α T 11 ( 2 ) ( L ) + α T 12 ( 2 ) ( L ) ]
1 = R exp [ 1 2 ( 1 + τ G f ) ln G ] × [ 1 α T 11 ( 2 ) ( L ) α T 12 ( 2 ) ( L ) ]
exp [ τ G f 1 + τ G f ( Γ G 2 ln q ) ] = ( 1 2 α ) + 2 exp [ τ L f 1 + τ L f ( Γ L 2 ln α ) ] ,
exp [ τ G f 1 + τ G f ( Γ G 2 ln q ) + 1 1 + τ L f Γ L 2 ] = 1 ,
r 11 x = ν 11 p 11 + ν 12 p 21 ,
r 12 x = ν 11 p 12 + ν 12 p 22 ,
p 11 x = ν 11 * r 11 ,
p 12 x = ν 11 * r 12 ,
p 21 x = ν 12 * r 11 ,
p 22 x = ν 12 * r 12 ,
( τ t + 1 ) ν 11 = Γ 2 I T ( r 11 p 11 * + r 12 p 12 * ) ,
( τ t + 1 ) ν 12 = Γ 2 I T ( r 11 p 21 * + r 12 p 22 * ) ,
p 11 , in = p 11 , in ( 0 ) ,
p 22 , in = p 22 , in ( 0 ) ,
p 12 , in = p 21 , in = 0 ,
r 11 , out R = r 11 , in ,
r 12 , out R = r 12 , in ,
| r 11 | out 2 = | r 11 | in 2 G ,
| p 11 | out 2 = | p 11 | in 2 G exp ( Γ n ) ,
G = 1 + y y + exp ( Γ n ) ,
Γ n = Γ 1 + y 1 + y + m ,
y = R exp ( Γ n ) .
a ( x , t ) = a ( 0 ) ( x ) + Re [ δ a ( x ) exp ( f t ) ] .
d δ r 11 d x = δ ν 11 p 11 ( 0 ) + ν 11 ( 0 ) δ p 11 ,
d δ p 11 d x = δ ν 11 r 11 ( 0 ) ν 11 ( 0 ) δ r 11 ,
δ ν 11 = Γ 2 I T ( 0 ) ( 1 + τ f ) × [ r 11 ( 0 ) δ p 11 + p 11 ( 0 ) δ r 11 r 11 ( 0 ) p 11 ( 0 ) δ I T I T ( 0 ) ] ,
δ p 11 , in = 0 ,
δ r 11 , out R = δ r 11 , in ,
d δ r 12 d x = δ ν 12 p 22 ( 0 ) + ν 11 ( 0 ) δ p 12 ,
d δ p 12 d x = ν 11 ( 0 ) δ r 12 ,
d δ p 21 d x = δ ν 12 r 11 ( 0 ) ,
δ ν 12 = Γ 2 I T ( 0 ) ( 1 + τ f ) [ δ r 12 p 22 ( 0 ) + r 11 ( 0 ) δ p 21 ] ,
δ p 12 , in = δ p 21 , in = 0 ,
δ r 12 , out R = δ r 12 , in .
δ r 11 , out = T δ r 11 , in ,
T = exp ( Γ n / 2 ) G [ 1 + y exp ( Γ n ) ] [ y exp ( Γ n / 2 ) + B ] + 2 m 1 + m y B I ,
B = exp [ 1 1 + τ f ( ln G Γ n / 2 ) ] ,
I = Γ n exp ( Γ n / 2 ) 2 ( 1 + τ f ) G 0 1 d x exp ( τ f 1 + τ f × { Γ n 2 x + ln [ y + exp ( Γ n x ) ] } ) .
τ f = 2 Γ n ( Γ n + ln R ) .
r = δ r 12 / r 11 ( 0 ) ,
z 1 = 1 1 + τ f [ r p 22 ( 0 ) 2 + δ p 21 p 22 ( 0 ) ] ,
z 2 = r p 11 ( 0 ) 2 δ p 12 p 11 ( 0 ) ,
d r d x = Γ 2 I T ( 0 ) ( z 1 z 2 ) ,
d z 1 d x = Γ 2 I T ( 0 ) ( 1 + τ f ) { [ r 11 ( 0 ) 2 p 22 ( 0 ) 2 ] z 1 + p 22 ( 0 ) 2 z 2 } ,
d z 2 d x = Γ 2 I T ( 0 ) { p 11 ( 0 ) 2 z 1 [ p 11 ( 0 ) 2 + r 11 ( 0 ) 2 ] z 2 } .
r out r in = 0 1 d x ( z 1 z 2 ) ,
( z 1 z 2 ) = ( z 1 z 2 ) in exp ( Γ 2 1 + y m 1 + y + m x ) .
r i j x = k = 1 , 2 ν i k p k j ( i , j = 1 , 2 ) ,
p i j x = k = 1 , 2 ν k i * r k j ( i , j = 1 , 2 ) ,
( τ t + 1 ) ν i j = Γ 2 I T k = 1 , 2 r i k p j k * ( i , j = 1 , 2 ) ,
p 11 , in = p 11 , in ( 0 ) ,
p 22 , in = p 22 , in ( 0 ) ,
p 12 , in = p 21 , in = 0 ,
r i j , out R i = r i j , in ( i , j = 1 , 2 ) ,
c 1 = r 11 2 + r 21 2 + p 11 2 + p 21 2 ,
c 2 = r 12 2 + r 22 2 + p 12 2 + p 22 2 ,
c 3 = r 11 r 12 + r 21 r 22 + p 11 p 12 + p 21 p 22 ,
d r 11 d x = Γ 2 I T r 11 p 11 2 ,
d r 21 d x = Γ 2 I T r 21 p 11 2 ,
d p 11 d x = Γ 2 I T ( r 11 2 + r 21 2 ) p 11 .
r 11 , out = r 11 , in G ,
r 21 , out = r 21 , in G ,
G = 1 y 11 + y 21 + exp ( Γ n ) ,
y i j = ( r i j / p 11 ) in 2 ,
Γ n = Γ 1 1 + m ,
m = ( p 22 / p 11 ) in 2 .
d δ r 21 d x = δ ν 21 p 11 ( 0 ) ,
δ ν 21 = Γ 2 I T ( 0 ) ( 1 + τ f ) δ r 21 p 11 ( 0 ) ,
δ r 21 , out R 2 = δ r 21 , in ,
d δ r 22 d x = δ ν 22 p 22 ( 0 ) ,
δ ν 22 = Γ 2 I T ( 0 ) ( 1 + τ f ) δ r 22 p 22 ( 0 ) ,
δ r 22 , out R 2 = δ r 22 , in ,
τ f = ln ( 1 / R 2 ) ln ( R 1 / R 2 ) .
τ f = m Γ n ln ( 1 / R 2 ) ln ( 1 / R 2 ) .
r j j , out 2 = r j j , in 2 G j ( j = 1 , 2 ) ,
p j j , out 2 = p j j , in 2 G j exp ( Γ j ) ( j = 1 , 2 ) ,
G j = 1 y j + exp ( Γ j ) ( j = 1 , 2 ) ,
Γ 1 = Γ 1 1 + m ,
Γ 2 = Γ m 1 + m ,
y j = R j exp ( Γ j ) .
d δ r j j d x = δ ν j j p j j ( 0 ) + ν j j ( 0 ) δ p j j ,
d δ p j j d x = δ ν j j r j j ( 0 ) ν j j ( 0 ) δ r j j ,
δ ν j j = Γ 2 I T ( 0 ) ( 1 + τ f ) × [ r j j ( 0 ) δ p j j + p j j ( 0 ) δ r j j r j j ( 0 ) p j j ( 0 ) δ I T I T ( 0 ) ] .
d δ r 12 d x = δ ν 12 p 22 ( 0 ) + ν 11 ( 0 ) δ p 12 ,
d δ r 21 d x = δ ν 21 p 11 ( 0 ) + ν 22 ( 0 ) δ p 21 ,
d δ p 12 d x = δ ν 21 r 22 ( 0 ) ν 11 ( 0 ) δ r 12 ,
d δ p 21 d x = δ ν 12 r 11 ( 0 ) ν 22 ( 0 ) δ r 21 ,
δ ν i j = Γ 2 ( 1 + τ f ) [ r i i ( 0 ) δ p j i + δ r i j p j j ( 0 ) ] .
[ δ r 11 δ r 22 ] out = T [ δ r 11 δ r 22 ] in ,
T j j = exp ( Γ j / 2 ) [ 1 + y j exp ( Γ j ) ] G j [ y j exp ( Γ j / 2 ) + B j ] + 2 m 2 j 1 + m y j B j I j ,
T i j ( i j ) = 2 m 1 + m y 1 y 2 B i I i ,
B j = exp [ 1 1 + τ f ( ln G j Γ j 2 ) ] ,
I j = Γ j exp ( Γ j / 2 ) 2 ( 1 + τ f ) G j 0 1 d x × exp ( τ f 1 + τ f { Γ j 2 x + ln [ y j + exp ( Γ j ) ] } ) .
( R 1 T 11 1 ) ( R 2 T 22 1 ) R 1 R 2 T 12 T 21 = 0 .
τ f j = 2 Γ j [ Γ j + ln ( R j ) ] .
r 12 r 11 = r 21 r 22 = c ,
y 1 = δ r 12 r 11 ( 0 ) δ r 21 r 22 ( 0 ) ,
y 2 = δ r 12 r 11 ( 0 ) + δ r 21 r 22 ( 0 ) ,
r 11 δ r 12 + r 22 δ r 21 + p 11 δ p 12 + p 22 δ p 21 = const . ,
d y 1 d x = Γ 2 I T [ ( p 22 2 p 11 2 ) y 2 τ f p 2 y 1 ] ,
d y 2 d x = Γ 2 I T [ y 1 , in ( p 22 , in 2 p 11 , in 2 ) + 2 r in 2 y 2 , in 2 r 2 y 2 ] ,
τ f = Γ 2 4 ln ( 1 / R ) ( p 22 , in 2 p 11 , in 2 I T ) 2 .

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