Abstract

The operation of a new all-optical switch potentially capable of ultrafast exchange of information between two orthogonally polarized light beams with identical or slightly different wavelength and comparable intensity are described in terms of two interacting nonlinear couplers. We consider the four lowest-order modes (including polarization) of a dual-core optical-fiber nonlinear coupler. We show how the output power ratio between two modes with identical linear polarization propagating in parallel cores can be controlled in an incoherent manner by the presence of another orthogonally polarized light beam by virtue of the optical Kerr effect. An interesting new feature is that, owing to the spatial instability of the coupled-mode equations, when the fiber is sufficiently long (e.g., two linear coupling lengths), a relatively weak pulse is capable of completely switching out a stronger (ideally square) orthogonally polarized pulse. We find that nonlinear coupling may lead to spatial stochasticity and chaotic power-dependent switching whenever the input conditions are given in the vicinity of the normal (or array) modes of the coupler. Conversely, when the modes of the individual cores are excited, the solutions turn out to be regular for almost any input power level.

© 1988 Optical Society of America

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  18. B. Crosignani, S. Piazzolla, P. Spano, and P. Di Porto, Opt. Lett. 10, 89 (1985).
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  39. G. I. Stegeman, E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Commun. 63, 281 (1987).
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  40. G. Metz, J. R. Dunphy, W. W. Morey, and E. Snitzer, Appl. Opt. 22, 464 (1983).
    [CrossRef]
  41. K. Kitayama and Y. Ishida, J. Opt. Soc. Am. A 2, 90 (1985).
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  42. K. Kitayama, N. Shibata, and M. Ohashi, J. Opt. Soc. Am. A 2, 84 (1985).
    [CrossRef]
  43. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), Chap. 8.
  44. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983), pp. 42 ff.
  45. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  46. B. Crosignani, C. H. Papas, and P. Di Porto, Opt. Lett. 6, 61 (1981).
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  47. B. Crosignani, A. Cutolo, and P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
    [CrossRef]
  48. D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
    [CrossRef]
  49. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 570 ff.
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  51. A. Hardy, S. Shakir, and W. Streifer, Opt. Lett. 5, 324 (1986).
    [CrossRef]
  52. A. Hardy, W. Streifer, and M. Osinski, Opt. Lett. 11, 742 (1986).
    [CrossRef] [PubMed]
  53. P. D. Maker and R. W. Terhune, Phys. Rev. A 137, 801 (1965).
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    [CrossRef]
  55. S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. (to be published).
  56. A. M. Weiner, J. P. Heritage, and R. N. Thurston, Opt. Lett. 11, 153 (1986).
    [CrossRef]

1987 (10)

N. J. Halas, D. Krokel, and D. Grischkowsy, Appl. Phys. Lett. 50, 886 (1987).
[CrossRef]

T. Morioka, M. Saruwatari, and A. Takada, Electron. Lett. 23, 453 (1987).
[CrossRef]

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, in Appl. Phys. Lett. 51, 1135 (1987).
[CrossRef]

G. I. Stegeman, C. T. Seaton, C. N. Ironside, T. Cullen, and A. C. Walker, Appl. Phys. Lett. 50, 1035 (1987).
[CrossRef]

Y. Silberberg and G. I. Stegeman, Appl. Phys. Lett. 50, 801 (1987).
[CrossRef]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, Appl. Phys. Lett. 51, 293 (1987).
[CrossRef]

G. I. Stegeman, E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Commun. 63, 281 (1987).
[CrossRef]

S. Wabnitz, Phys. Rev. Lett. 58, 1415 (1987).
[CrossRef] [PubMed]

A. Mecozzi, S. Trillo, S. Wabnitz, and B. Daino, Opt. Lett. 12, 275 (1987).
[CrossRef] [PubMed]

E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Lett. 12, 1044 (1987).
[CrossRef] [PubMed]

1986 (12)

H. G. Winful, Opt. Lett. 11, 33 (1986).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, Opt. Lett. 11, 42 (1986).
[CrossRef]

A. M. Weiner, J. P. Heritage, and R. N. Thurston, Opt. Lett. 11, 153 (1986).
[CrossRef]

F. Matera and S. Wabnitz, Opt. Lett. 11, 467 (1986).
[CrossRef] [PubMed]

H. G. Winful and A. Hu, Opt. Lett. 11, 668 (1986).
[CrossRef] [PubMed]

A. Hardy, W. Streifer, and M. Osinski, Opt. Lett. 11, 742 (1986).
[CrossRef] [PubMed]

S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, Appl. Phys. Lett. 49, 11 (1986).
[CrossRef]

S. Trillo and S. Wabnitz, Appl. Phys. Lett. 49, 752 (1986).
[CrossRef]

A. Hardy, S. Shakir, and W. Streifer, Opt. Lett. 5, 324 (1986).
[CrossRef]

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, Appl. Phys. Lett. 49, 1224 (1986).
[CrossRef]

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

1985 (12)

H. G. Winful, Appl. Phys. Lett. 47, 213 (1985).
[CrossRef]

P. Li, Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, Electron. Lett. 21, 26 (1985).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 623 (1985).
[CrossRef]

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
[CrossRef]

K. Kitayama, N. Shibata, and M. Ohashi, J. Opt. Soc. Am. A 2, 84 (1985).
[CrossRef]

K. Kitayama and Y. Ishida, J. Opt. Soc. Am. A 2, 90 (1985).
[CrossRef]

A. Hardy and W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

B. Crosignani, S. Piazzolla, P. Spano, and P. Di Porto, Opt. Lett. 10, 89 (1985).
[CrossRef] [PubMed]

I. Tomazeau, J. Etchepare, G. Grillon, and A. Migus, Opt. Lett. 10, 223 (1985).
[CrossRef]

J. P. Heritage, A. M. Weiner, and R. N. Thurston, Opt. Lett. 10, 609 (1985).
[CrossRef] [PubMed]

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 317 (1985).
[CrossRef]

1983 (4)

A. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 1718 (1983).
[CrossRef]

K. Kitayama and S. Wang, Appl. Phys. Lett. 43, 17 (1983).
[CrossRef]

G. Metz, J. R. Dunphy, W. W. Morey, and E. Snitzer, Appl. Opt. 22, 464 (1983).
[CrossRef]

B. Nikolaus, D. Grischkowsky, and A. C. Balant, Opt. Lett. 8, 189 (1983).
[CrossRef] [PubMed]

1982 (4)

B. Crosignani, A. Cutolo, and P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
[CrossRef]

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 158 (1982).

A. A. Maier, Sov. J. Quantum Electron. 12, 1490 (1982); Sov. J. Quantum Electron. 14, 101 (1984); Sov. J. Quantum Electron. 15, 1016 (1985); Sov. J. Quantum Electron. 16, 892 (1986).
[CrossRef]

R. H. Stolen, J. Botineau, and A. Ashkin, Opt. Lett. 7, 512 (1982).
[CrossRef] [PubMed]

1981 (2)

1980 (1)

W. Gadomski and M. Roman, Opt. Commun. 33, 331 (1980).
[CrossRef]

1975 (1)

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
[CrossRef]

1973 (1)

R. H. Stolen and A. Ashkin, Appl. Phys. Lett. 22, 294 (1973).
[CrossRef]

1969 (1)

M. A. Duguay and J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[CrossRef]

1965 (1)

P. D. Maker and R. W. Terhune, Phys. Rev. A 137, 801 (1965).

Andrejco, M. J.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, in Appl. Phys. Lett. 51, 1135 (1987).
[CrossRef]

Ashkin, A.

Assanto, G.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, and G. I. Stegeman, Appl. Phys. Lett. 49, 1224 (1986).
[CrossRef]

Balant, A. C.

Bennion, I.

M. J. Goodwin, C. Edge, and I. Bennion, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper THO3.

Botineau, J.

Caglioti, E.

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, Appl. Phys. Lett. 51, 293 (1987).
[CrossRef]

G. I. Stegeman, E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Commun. 63, 281 (1987).
[CrossRef]

E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Lett. 12, 1044 (1987).
[CrossRef] [PubMed]

Crosignani, B.

Cullen, T.

G. I. Stegeman, C. T. Seaton, C. N. Ironside, T. Cullen, and A. C. Walker, Appl. Phys. Lett. 50, 1035 (1987).
[CrossRef]

Cutolo, A.

Daino, B.

A. Mecozzi, S. Trillo, S. Wabnitz, and B. Daino, Opt. Lett. 12, 275 (1987).
[CrossRef] [PubMed]

E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, Appl. Phys. Lett. 51, 293 (1987).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, Opt. Lett. 11, 42 (1986).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Di Porto, P.

Dianov, E. M.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
[CrossRef]

Duguay, M. A.

M. A. Duguay and J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[CrossRef]

Dunphy, J. R.

Dziedzic, J. M.

Edge, C.

M. J. Goodwin, C. Edge, and I. Bennion, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper THO3.

Etchepare, J.

Friberg, S. R.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, in Appl. Phys. Lett. 51, 1135 (1987).
[CrossRef]

S. R. Friberg and P. W. Smith, IEEE J. Quantum Electron. (to be published).

Gadomski, W.

W. Gadomski and M. Roman, Opt. Commun. 33, 331 (1980).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1950), Chap. 8.

Goodwin, M. J.

M. J. Goodwin, C. Edge, and I. Bennion, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper THO3.

Gregori, G.

G. Gregori and S. Wabnitz, Phys. Rev. Lett. 56, 600 (1986).
[CrossRef] [PubMed]

B. Daino, G. Gregori, and S. Wabnitz, Opt. Lett. 11, 42 (1986).
[CrossRef]

B. Daino, G. Gregori, and S. Wabnitz, J. Appl. Phys. 58, 4512 (1985).
[CrossRef]

Grillon, G.

Grischkowsky, D.

Grischkowsy, D.

N. J. Halas, D. Krokel, and D. Grischkowsy, Appl. Phys. Lett. 50, 886 (1987).
[CrossRef]

Gusovskii, D. D.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
[CrossRef]

Halas, N. J.

N. J. Halas, D. Krokel, and D. Grischkowsy, Appl. Phys. Lett. 50, 886 (1987).
[CrossRef]

Hansen, J. W.

M. A. Duguay and J. W. Hansen, Appl. Phys. Lett. 15, 192 (1969).
[CrossRef]

Hardy, A.

A. Hardy, W. Streifer, and M. Osinski, Opt. Lett. 11, 742 (1986).
[CrossRef] [PubMed]

A. Hardy, S. Shakir, and W. Streifer, Opt. Lett. 5, 324 (1986).
[CrossRef]

A. Hardy and W. Streifer, IEEE J. Lightwave Technol. LT-3, 1135 (1985).
[CrossRef]

Hasegawa, A.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

Haus, H. A.

A. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 1718 (1983).
[CrossRef]

M. Shirasaki, H. A. Haus, and D. Liu Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), p. 284.

Heritage, J. P.

Hu, A.

Ippen, E. P.

A. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 1718 (1983).
[CrossRef]

Ironside, C. N.

G. I. Stegeman, C. T. Seaton, C. N. Ironside, T. Cullen, and A. C. Walker, Appl. Phys. Lett. 50, 1035 (1987).
[CrossRef]

Ishida, Y.

Jensen, S. M.

S. M. Jensen, IEEE J. Quantum Electron. QE-18, 158 (1982).

Jewell, J. L.

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Appl. Phys. Lett. 49, 236 (1986).
[CrossRef]

Kimura, Y.

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 317 (1985).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 623 (1985).
[CrossRef]

Kitayama, K.

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 623 (1985).
[CrossRef]

K. Kitayama, Y. Kimura, and S. Seikai, Appl. Phys. Lett. 46, 317 (1985).
[CrossRef]

K. Kitayama and Y. Ishida, J. Opt. Soc. Am. A 2, 90 (1985).
[CrossRef]

K. Kitayama, N. Shibata, and M. Ohashi, J. Opt. Soc. Am. A 2, 84 (1985).
[CrossRef]

K. Kitayama and S. Wang, Appl. Phys. Lett. 43, 17 (1983).
[CrossRef]

Krokel, D.

N. J. Halas, D. Krokel, and D. Grischkowsy, Appl. Phys. Lett. 50, 886 (1987).
[CrossRef]

Lattes, A.

A. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 1718 (1983).
[CrossRef]

Leonberger, F. J.

A. Lattes, H. A. Haus, F. J. Leonberger, and E. P. Ippen, IEEE J. Quantum Electron. QE-19, 1718 (1983).
[CrossRef]

Li, P.

P. Li, Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, Electron. Lett. 21, 26 (1985).
[CrossRef]

Lichtenberg, A. J.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983), pp. 42 ff.

Lieberman, M. A.

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983), pp. 42 ff.

Liu Wong, D.

M. Shirasaki, H. A. Haus, and D. Liu Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), p. 284.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), pp. 570 ff.

Maier, A. A.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
[CrossRef]

A. A. Maier, Sov. J. Quantum Electron. 12, 1490 (1982); Sov. J. Quantum Electron. 14, 101 (1984); Sov. J. Quantum Electron. 15, 1016 (1985); Sov. J. Quantum Electron. 16, 892 (1986).
[CrossRef]

Maker, P. D.

P. D. Maker and R. W. Terhune, Phys. Rev. A 137, 801 (1965).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 54, 985 (1975).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Mason, N. J.

P. Li, Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, Electron. Lett. 21, 26 (1985).
[CrossRef]

Matera, F.

Mecozzi, A.

Metz, G.

Migus, A.

Morey, W. W.

Morioka, T.

T. Morioka, M. Saruwatari, and A. Takada, Electron. Lett. 23, 453 (1987).
[CrossRef]

Neustruev, V. B.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
[CrossRef]

Nikolaus, B.

Ohashi, M.

Oliver, M. K.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, in Appl. Phys. Lett. 51, 1135 (1987).
[CrossRef]

Osinski, M.

Papas, C. H.

Piazzolla, S.

Roberts, J. S.

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D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, Sov. J. Quantum Electron. 15, 1523 (1985).
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S. Trillo and S. Wabnitz, Appl. Phys. Lett. 49, 752 (1986).
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G. I. Stegeman, C. T. Seaton, C. N. Ironside, T. Cullen, and A. C. Walker, Appl. Phys. Lett. 50, 1035 (1987).
[CrossRef]

Y. Silberberg and G. I. Stegeman, Appl. Phys. Lett. 50, 801 (1987).
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E. Caglioti, S. Trillo, S. Wabnitz, B. Daino, and G. I. Stegeman, Appl. Phys. Lett. 51, 293 (1987).
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J. Opt. Soc. Am. (1)

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

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G. I. Stegeman, E. Caglioti, S. Trillo, and S. Wabnitz, Opt. Commun. 63, 281 (1987).
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Phys. Rev. A (1)

P. D. Maker and R. W. Terhune, Phys. Rev. A 137, 801 (1965).

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M. Shirasaki, H. A. Haus, and D. Liu Wong, in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), p. 284.

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Figures (10)

Fig. 1
Fig. 1

Reshaping of an input Gaussian pulse detected beyond an analyzer measuring the x-polarized component from the output end of fiber core 1: fiber length is z = Lb; normalized peak powers are px = 1.5 and py = 0.1; mode excitation conditions are S1x = S1y = 0, S2x = S2y = 0.98, and S3x = −S3y = (1 − S2x2)1/2 ≅ 0.2. An instantaneous response is assumed.

Fig. 2
Fig. 2

Same as in Fig. 1, except that z = 3Lb.

Fig. 3
Fig. 3

Same as in Fig. 1, except that z = 6Lb.

Fig. 4
Fig. 4

Power-dependent transmission measured beyond an analyzer as in Fig. 1 for a fiber length z = 6Lb. Excitation conditions are the same as in Fig. 1, but now all the input power is coupled to the two LP01x modes (r = 0).

Fig. 5
Fig. 5

Same as in Fig. 4, except that r = 0.1.

Fig. 6
Fig. 6

Power-dependent transmission measured at the output as in Fig. 1 when all the incident power is coupled to fiber core 1. The r = 0 line is obtained for an x-polarized input beam, and the r = 1 line is obtained for equal excitation of LP01x,y modes of core 1. Fiber length z = Lc.

Fig. 7
Fig. 7

Transmitted pulses measured as in Fig. 6, with peak input power px = 0.6 The dashed line is obtained for r = 0; the solid line is for r = 1. The fiber length is z = Lc = Lb/2.

Fig. 8
Fig. 8

As in Fig. 6, with fiber length z = Lb.

Fig. 9
Fig. 9

Transmission as in Fig. 6. The dashed line is obtained for px = 0.99, the solid line is for px = 0.98, and the dotted line is for px = 0.97. The fiber length is z = Lb.

Fig. 10
Fig. 10

Pulse reshaping measured as in Fig. 6, with px = 0.98. The dashed line is for r = 0; the solid line is for r = 0.08. Fiber length is z = Lb.

Equations (23)

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E t ( r , z , t ) = Re { [ a 1 x ( z ) exp ( i β 1 x z ) ψ 1 x ( r ; ω ) + a 2 x ( z ) exp ( i β 2 x z ) ψ 2 x ( r ; ω ) ] x × exp ( - i ω t ) + [ a 1 y ( z ) exp ( i β 1 y z ) ψ 1 y ( r ; ω ) + a 2 y ( z ) exp ( i β 2 y z ) ψ 2 y ( r ; ω ) ] y exp ( - i ω t ) } .
z × ( H ω / z ) = - i ω [ ( r ) : E ω + P ω ( 3 ) ] , z × ( E ω / z ) = i ω μ 0 H ω ,
P ω ( 3 ) = 3 [ 2 χ x x y y ( ω , ω , - ω ) ( E ω · E ω * ) E ω + χ x y y x ( ω , ω , - ω ) ( E ω · E ω ) E ω * + 2 χ x x y y ( ω , ω , - ω ) ( E ω · E ω * ) E ω + 2 χ x y x y ( ω , ω , - ω ) ( E ω · E ω * ) E ω + 2 χ x y y x ( ω , ω , - ω ) ( E ω · E ω ) E ω * ] .
- + z · [ ψ j m * ( r , ω ) × ρ k n ( r , ω ) + ψ k n ( r , ω ) × ρ j m * ( r , ω ) ] d x d y = 4 Q ω δ j k δ m n , δ j k = { 0 if j k 1 if j = k ,
- i d a 1 x / d z = γ 1 x a 1 x + κ 1 x exp [ i ( β 2 x - β 1 x ) z ] a 2 x + ( R 1 x x x a 1 x 2 + R 1 x y y a 1 y 2 ) a 1 x + δ ω ω R 1 y y x a 1 y 2 a 1 x * exp [ 2 i ( β 1 y - β 1 x ) z ] , - i d a 2 x / d z = γ 2 x a 2 x + κ 2 x exp [ i ( β 1 x - β 2 x ) z ] a 1 x + ( R 2 x x x a 2 x 2 + R 2 x y y a 2 y 2 ) a 2 x + δ ω ω R 2 y y x a 2 y 2 a 2 x * exp [ 2 i ( β 2 y - β 2 x ) z ] .
γ j x = ( ω / 4 Q ω ) - + ψ j x * ( r , ω ) Δ j x x ( r ) ψ j x ( r , ω ) d x d y ,
κ j x = ( ω / 4 Q ω ) - + ψ j x * ( r , ω ) Δ j x x ( r ) ψ ( 3 - j ) x ( r , ω ) d x d y ,
R j x x x = ( 3 χ x x x x ( ω , ω , - ω ) ω / 4 Q ω ) - + ψ j x ( r , ω ) 4 d x d y ,
R j x y y = ( 6 χ x x y y ( ω , ω , - ω ) ω / 4 Q ω ) - + ψ j x ( r , ω ) 2 × ψ j y ( r , ω ) 2 d x d y ,
R j y y x = ( 3 χ x y y x ( ω , ω , - ω ) ω / 4 Q ω ) - + ψ j x ( r , ω ) 2 × ψ j y ( r , ω ) 2 d x d y ,             j = 1 , 2.
- i d A 1 x / d z = ( β 1 x + γ 1 x ) A 1 x + κ x A 2 x + R [ A 1 x 2 + ( 2 / 3 ) A 1 y 2 ] A 1 x , - i d A 2 x / d z = ( β 2 x + γ 2 x ) A 2 x + κ x A 1 x + R [ A 2 x 2 + ( 2 / 3 ) A 2 y 2 ] A 2 x ,
A eff = [ ψ ( r ) 2 d x d y ] 2 ψ ( r ) 4 d x d y .
d I 1 x / d z = - H / ϕ 1 x , d ϕ 1 x / d z = H / I 1 x , d I 2 x / d z = - H / ϕ 2 x , d ϕ 2 x / d z = H / I 2 x ;
H x = ( R / 2 ) ( I 1 x 2 + I 2 x 2 ) + [ ( β 1 x + γ 1 x ) I 1 x + ( β 2 x + γ 2 x ) I 2 x ] + 2 κ x ( I 1 x I 2 x ) 1 / 2 cos ( ϕ 1 x - ϕ 2 x ) , H x y = ( 2 R / 3 ) ( I 1 x I 1 y + I 2 x I 2 y ) ,
F = J x ( ϕ 1 x - ϕ 2 x ) + J y ( ϕ 1 y - ϕ 2 y ) + C x ( ϕ 1 x + ϕ 2 x ) + C y ( ϕ 1 y + ϕ 2 y ) ,
I 1 x = J x + C x , I 2 x = C x - J x , I 1 y = J y + C y , I 2 y = C y - J y .
θ x = F / J x = ϕ 1 x - ϕ 2 x , θ y = F / J y = ϕ 1 y - ϕ 2 y , φ x = F / C x = ϕ 1 x + ϕ 2 x , φ y = F / C y = ϕ 1 y + ϕ 2 y .
H x = R J x 2 + 2 Δ x J x + 2 κ x ( C x 2 - J x 2 ) 1 / 2 cos θ x , H x y = 4 R J x J y / 3.
H nlc ( q , θ ) = p q 2 + ( 1 - q 2 ) 1 / 2 cos θ + δ q ,
d q x / d ζ = - H t / θ x , d θ x / d ζ = H t / q x , r d q y / d ζ = - H t / θ y , d θ y / d ζ = H t / ( r q y ) ,
r p y / p x , H t = H nlc ( q x , θ z ) + r H nlc ( q y , θ y ) + r H int ( q x , q y ) , H int ( q x , q y ) = 4 p x q x q y / 3.
S 1 x = q x , S 2 x = ( 1 - q x 2 ) 1 / 2 cos θ x , S 3 x = ( 1 - q x 2 ) 1 / 2 sin θ x .
d S 1 x / d ζ = S 3 x , d S 2 x / d ζ = - 2 p x S 1 x S 3 x - ( 4 / 3 ) p y S 1 y S 3 x , d S 3 x / d ζ = - S 1 x + 2 p x S 1 x S 2 x + ( 4 / 3 ) p y S 1 y S 2 x .

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