Abstract

The instability of a plane wave in an optical medium with two-photon absorption is studied. The analysis is based on the modified nonlinear Schrödinger equation. The linearized equation for the modulation is shown to have an exact solution in terms of confluent hypergeometric functions. It is found that the gain spectrum varies with position. This may result in a change of the wave dynamics and in a decrease of the repetition rate of the pulse train developed from the plane wave. The application of the results to the optical pulse propagation in semiconductor gratings and fiber gratings is discussed.

© 2001 Optical Society of America

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References

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP 3, 307–310 (1966).
  2. T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  4. D. Anderson and M. Lisak, “Modulational instability of coherent optical-fiber transmission signals,” Opt. Lett. 9, 468–470 (1984).
    [CrossRef] [PubMed]
  5. A. Hasegawa and K. Tai, “Effects of modulational instability on coherent transmission systems,” Opt. Lett. 14, 512–513 (1989).
    [CrossRef] [PubMed]
  6. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071–2077 (1995).
  7. V. A. Vysloukh and N. A. Sukhostova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
    [CrossRef]
  8. F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
    [CrossRef]
  9. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  10. F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
    [CrossRef]
  11. F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
    [CrossRef]
  12. F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
    [CrossRef]
  13. P. Millar, R. M. De La Rue, T. F. Krauss, J. S. Aitchison, N. G. R. Broderick, and D. J. Richardson, “Nonlinear propagation effects in AlGaAs Bragg grating filter,” Opt. Lett. 24, 685–687 (1999).
    [CrossRef]
  14. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  15. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. 3, pp. 203–260.
  16. Y. Silberberg, “Solitons and two-photon absorption,” Opt. Lett. 15, 1005–1007 (1990).
    [CrossRef] [PubMed]
  17. V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
    [CrossRef]
  18. V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. I. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989).
    [CrossRef] [PubMed]
  19. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972).
  20. L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, London, 1960).
  21. J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
    [CrossRef]
  22. B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]

1999 (1)

1998 (1)

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

1997 (3)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
[CrossRef]

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

1996 (2)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

1995 (2)

V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
[CrossRef]

M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071–2077 (1995).

1994 (1)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

1990 (1)

1989 (3)

1987 (1)

V. A. Vysloukh and N. A. Sukhostova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

1984 (1)

1967 (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP 3, 307–310 (1966).

Abdullaev, F. Kh.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972).

Acevez, A. B.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Afanasjev, V. V.

V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

Aitchison, J. S.

P. Millar, R. M. De La Rue, T. F. Krauss, J. S. Aitchison, N. G. R. Broderick, and D. J. Richardson, “Nonlinear propagation effects in AlGaAs Bragg grating filter,” Opt. Lett. 24, 685–687 (1999).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
[CrossRef]

Anderson, D.

Andrejco, M. J.

Benjamin, T. B.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP 3, 307–310 (1966).

Bischoff, S.

Bishoff, S.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Broderick, N. G. R.

Christiansen, P. L.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Darmanyan, S. A.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

De La Rue, R. M.

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. 3, pp. 203–260.

DeLong, K. W.

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Feir, J. E.

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

Galvao, R. M. O.

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

Gnavi, G.

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

Gomberoff, L.

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

Gratton, F. T.

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

Hasegawa, A.

Hutchings, D. C.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

Kang, J. U.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

Karlsson, M.

Kivshar, Y. S.

V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
[CrossRef]

Kobyakov, A.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

Krauss, T. F.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Lederer, F.

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

Lisak, M.

Millar, P.

Mizrahi, V.

Richardson, D. J.

Saifi, M. I.

Silberberg, Y.

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. 3, pp. 203–260.

Slater, L. J.

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, London, 1960).

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Sørensen, M. P.

F. Kh. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, “Modulational instability of electromagnetic waves in media with varying nonlinearity,” J. Opt. Soc. Am. B 14, 27–33 (1997).
[CrossRef]

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

Stegeman, G. I.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. I. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972).

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Sukhostova, N. A.

V. A. Vysloukh and N. A. Sukhostova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Tai, K.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP 3, 307–310 (1966).

Villeneuve, A.

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

Vysloukh, V. A.

V. A. Vysloukh and N. A. Sukhostova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. 33, 341–348 (1997).
[CrossRef]

J. Fluid Mech. (1)

T. B. Benjamin and J. E. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech. 27, 410–430 (1967).
[CrossRef]

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

JETP (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP 3, 307–310 (1966).

Opt. Commun. (3)

F. Kh. Abdullaev, S. A. Darmanyan, S. Bishoff, P. L. Christiansen, and M. P. Sørensen, “Modulational instability in optical fibers near the zero dispersion point,” Opt. Commun. 108, 60–64 (1994).
[CrossRef]

V. V. Afanasjev, J. S. Aitchison, and Y. S. Kivshar, “Splitting of high-order spatial solitons under the action of two-photon absorption,” Opt. Commun. 116, 331–338 (1995).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, A. B. Acevez, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Opt. Lett. (5)

Phys. Lett. A (1)

F. Kh. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A 220, 213–218 (1996).
[CrossRef]

Phys. Rev. E (1)

F. T. Gratton, G. Gnavi, R. M. O. Galvao, and L. Gomberoff, “Self-modulation of a strong electromagnetic wave in a positron–electron plasma induced by relativistic temperatures and phonon damping,” Phys. Rev. E 55, 3381–3392 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

V. A. Vysloukh and N. A. Sukhostova, “Influence of third-order dispersion on the generation of a train of picosecond pulses in fiber waveguides due to self-modulation instability,” Sov. J. Quantum Electron. 17, 1509–1511 (1987).
[CrossRef]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Wiley, New York, 1972).

L. J. Slater, Confluent Hypergeometric Functions (Cambridge University, London, 1960).

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E. Wolf, ed. (Elsevier, Amsterdam, 1994), Chap. 3, pp. 203–260.

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Figures (6)

Fig. 1
Fig. 1

Dependence of (a) Φ(3/2-iγ, 3, τ) and (b) Ψ(3/2-iγ, 3, τ) on parameters γ and |τ|.

Fig. 2
Fig. 2

Dependence of total gain gtot on Ω for μ2=0.005, 0.01, and 0.02. Points are the result of numerical simulations of Eqs. (3). Solid curves correspond to Eq. (9).

Fig. 3
Fig. 3

Dependence of average gain gav on Ω for μ2=0.01 and L=5 (hollow squares), 10 (hollow triangles), 20 (hollow circles), and 50 (solid circles). Points are the result of numerical simulations of Eqs. (3). Solid curves correspond to Eq. (11).

Fig. 4
Fig. 4

Evolution of a plane wave with different modulations found from numerical simulation of Eq. (1), with μ2=0.02: (a) Ω1=0.849 and (b) Ω2=2Ω1.

Fig. 5
Fig. 5

Evolution of the spectrum from numerical simulation of Eq. (1). Solid (dashed) curves correspond to μ2=0.01 (μ2=0).

Fig. 6
Fig. 6

Turning point as a function frequency for an AlGaAs grating filter13 (solid curve) and for a fiber Bragg grating22 (dashed curve). Horizontal lines show the length of the corresponding gratings.

Equations (15)

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

i uz+β2 2ut2+μ1|u|2u=-iμ2|u|2uR[u],
up(z)=ρ0y(z) expiμ12μ2 log[y(z)]+ϕ0,
dadz=-2μ2ρ2(z)a+βΩ22b,
dbdz=2μ1ρ2(z)-βΩ22a,
d2fdy2+ν2(y)f=0,
ν2(y)=β2Ω416µ22ρ04-βΩ2μ14µ22ρ02 1y-34y2.
zT=12µ2ρ02 2µ1ρ02βΩ2 1+1+3 μ22μ12-1.
zD(βΩ2)-1,zN(μ1ρ02)-1,zA(2μ2ρ02)-1,
d2fdτ2+-14+iγτ-34τ2f=0,
a(z)=11+2μ2ρ02ze-τ/2τ3/2c1Φ32-iγ, 3, τ+c2Ψ32-iγ, 3, τ,
gtot(Ω)log4 zDzN Φ[3/2-iγ, 3, 4iγ]Φ[3/2-iγ, 3, izA/zD],
gtot(Ω)zAzN π2-1-zN2zD-12+sin-11-zN2zD,μ2μ1.
gav(Ω, L)1L log(1+L/zA)×Φ[3/2-iγ, 3, i(zA+L)/zD]Φ[3/2-iγ, 3, izA/zD].
Ωmax(z)=Ωc(z)2ρ0y(z) 2μ1β.
β=(1-v2)3/2κv3V0,μ1=Γ(3-v2)2v,

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