Abstract

We derive the gain for mixed Stokes and anti-Stokes waves propagating in a birefringent fiber in the presence of an intense pump wave, under the combined action of parametric four-photon mixing and Raman scattering, for different types of phase matching. We discuss the conditions for the suppression or the enhancement of the Raman gain as a function of the linear mismatch and of the sideband detuning, owing to coupling between Stokes and anti-Stokes components of the growing wave. We confirm the analytical predictions of the linearized analysis with the numerical solution of the coupled nonlinear Schrödinger equations, including the contribution of the tensorial Raman response of the fiber.

© 1992 Optical Society of America

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  1. R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
    [CrossRef]
  2. R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
    [CrossRef]
  3. R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
    [CrossRef]
  4. R. H. Stolen, M. A. Bosch, and C. Lin, Opt. Lett. 6, 213 (1981).
    [CrossRef] [PubMed]
  5. K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
    [CrossRef]
  6. K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
    [CrossRef]
  7. R. K. Jain and K. Stenersen, Appl. Phys. B 35, 49 (1984).
    [CrossRef]
  8. H. G. Park, J. D. Park, and S. S. Lee, Appl. Opt. 26, 2974 (1987).
    [CrossRef] [PubMed]
  9. M. Nakazawa, T. Nakashima, and S. Seikai, J. Opt. Soc. Am. B 2, 515 (1985).
    [CrossRef]
  10. A. B. Grudinin, E. M. Dianov, and D. V. Khaidarov, Sov. J. Tech. Phys. 57, 788 (1987).
  11. R. Schulz, M. Kuckartz, and H. Harde, Opt. Commun. 70, 239 (1989).
    [CrossRef]
  12. J. K. Chee and J. M. Liu, Opt. Lett. 14, 820 (1989).
    [CrossRef] [PubMed]
  13. J. K. Chee and J. M. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
    [CrossRef]
  14. J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
    [CrossRef] [PubMed]
  15. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
    [CrossRef]
  16. T. Yang and P. Gao, Opt. Lett. 15, 1002 (1990).
    [CrossRef] [PubMed]
  17. D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
    [CrossRef]
  18. R. H. Stolen, IEEE J. Quantum Electron. QE-15, 1157 (1979).
    [CrossRef]
  19. K. O. Hill, B. S. Kawasaki, and D. C. Johnson, Appl. Phys. Lett. 29, 181 (1976);R. K. Jain, C. Lin, R. H. Stolen, W. Pleibel, and P. Kaiser, Appl. Phys. Lett. 30, 162 (1977).
    [CrossRef]
  20. M. Nakazawa, K. Suzuki, and H. A. Haus, Phys. Rev. A 38, 5193 (1988);IEEE J. Quantum Electron. 25, 2036 (1990);M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1990).
    [CrossRef] [PubMed]
  21. M. C. Tobin and T. Baak, J. Opt. Soc. Am. 58, 1459 (1968);M. Haas, J. Phys. Chem. Solids 31, 415 (1970).
    [CrossRef]
  22. R. H. Stolen and E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
    [CrossRef]
  23. R. Hellwarth, J. Cherlow, and T. T. Yang, Phys. Rev. B 11, 964 (1975);R. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
    [CrossRef]
  24. Y. R. Shen and N. Bloembergen, Phys. Rev. 6A, 1787 (1965);N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), pp. 110–119.
    [CrossRef]
  25. E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).
  26. A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 1062 (1980).
  27. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
    [CrossRef]
  28. K. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
    [CrossRef]
  29. S. Wabnitz, Phys. Rev. A 38, 2018 (1988);S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
    [CrossRef] [PubMed]
  30. R. Hawkins, Lawrence Livermore National Laboratory; Livermore, Calif. 94550 (personal communication).
  31. G. Cappellini and S. Trillo, J. Opt. Soc. Am. B 8, 824 (1991);Opt. Lett. 16, 895 (1991).
    [CrossRef]
  32. S. Wabnitz, S. Trillo, E. M. Wright, and G. I. Stegeman, J. Opt. Soc. Am. B 8, 602 (1991).
    [CrossRef]
  33. A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.
  34. E. A. Golovchenko and A. N. Pilipotskii, Sov. Lightwave Commun. 1, 271 (1991).

1991 (4)

D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
[CrossRef]

G. Cappellini and S. Trillo, J. Opt. Soc. Am. B 8, 824 (1991);Opt. Lett. 16, 895 (1991).
[CrossRef]

S. Wabnitz, S. Trillo, E. M. Wright, and G. I. Stegeman, J. Opt. Soc. Am. B 8, 602 (1991).
[CrossRef]

E. A. Golovchenko and A. N. Pilipotskii, Sov. Lightwave Commun. 1, 271 (1991).

1990 (4)

J. K. Chee and J. M. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

T. Yang and P. Gao, Opt. Lett. 15, 1002 (1990).
[CrossRef] [PubMed]

1989 (5)

R. Schulz, M. Kuckartz, and H. Harde, Opt. Commun. 70, 239 (1989).
[CrossRef]

J. K. Chee and J. M. Liu, Opt. Lett. 14, 820 (1989).
[CrossRef] [PubMed]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

K. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).

1988 (2)

M. Nakazawa, K. Suzuki, and H. A. Haus, Phys. Rev. A 38, 5193 (1988);IEEE J. Quantum Electron. 25, 2036 (1990);M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1990).
[CrossRef] [PubMed]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988);S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef] [PubMed]

1987 (2)

A. B. Grudinin, E. M. Dianov, and D. V. Khaidarov, Sov. J. Tech. Phys. 57, 788 (1987).

H. G. Park, J. D. Park, and S. S. Lee, Appl. Opt. 26, 2974 (1987).
[CrossRef] [PubMed]

1985 (1)

1984 (2)

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

R. K. Jain and K. Stenersen, Appl. Phys. B 35, 49 (1984).
[CrossRef]

1982 (2)

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
[CrossRef]

1981 (1)

1980 (1)

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 1062 (1980).

1979 (1)

R. H. Stolen, IEEE J. Quantum Electron. QE-15, 1157 (1979).
[CrossRef]

1976 (1)

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, Appl. Phys. Lett. 29, 181 (1976);R. K. Jain, C. Lin, R. H. Stolen, W. Pleibel, and P. Kaiser, Appl. Phys. Lett. 30, 162 (1977).
[CrossRef]

1975 (2)

R. Hellwarth, J. Cherlow, and T. T. Yang, Phys. Rev. B 11, 964 (1975);R. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
[CrossRef]

1974 (1)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

1973 (1)

R. H. Stolen and E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

1968 (1)

1965 (1)

Y. R. Shen and N. Bloembergen, Phys. Rev. 6A, 1787 (1965);N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), pp. 110–119.
[CrossRef]

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

Baak, T.

Belotitskii, V. I.

D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
[CrossRef]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

Bloembergen, N.

Y. R. Shen and N. Bloembergen, Phys. Rev. 6A, 1787 (1965);N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), pp. 110–119.
[CrossRef]

Blow, K.

K. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

Bosch, M. A.

Brinkman, W. F.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 1062 (1980).

Cappellini, G.

Chee, J. K.

J. K. Chee and J. M. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

J. K. Chee and J. M. Liu, Opt. Lett. 14, 820 (1989).
[CrossRef] [PubMed]

Cherlow, J.

R. Hellwarth, J. Cherlow, and T. T. Yang, Phys. Rev. B 11, 964 (1975);R. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

Cosentino, A.

A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.

Dianov, E. M.

E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).

A. B. Grudinin, E. M. Dianov, and D. V. Khaidarov, Sov. J. Tech. Phys. 57, 788 (1987).

Drummond, P. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Gao, P.

Golovchenko, E. A.

E. A. Golovchenko and A. N. Pilipotskii, Sov. Lightwave Commun. 1, 271 (1991).

E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).

Gordon, J. P.

Grudinin, A. B.

A. B. Grudinin, E. M. Dianov, and D. V. Khaidarov, Sov. J. Tech. Phys. 57, 788 (1987).

Harde, H.

R. Schulz, M. Kuckartz, and H. Harde, Opt. Commun. 70, 239 (1989).
[CrossRef]

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Hasegawa, A.

A. Hasegawa and W. F. Brinkman, IEEE J. Quantum Electron. QE-16, 1062 (1980).

Haus, H. A.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

M. Nakazawa, K. Suzuki, and H. A. Haus, Phys. Rev. A 38, 5193 (1988);IEEE J. Quantum Electron. 25, 2036 (1990);M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1990).
[CrossRef] [PubMed]

Hawkins, R.

R. Hawkins, Lawrence Livermore National Laboratory; Livermore, Calif. 94550 (personal communication).

Hellwarth, R.

R. Hellwarth, J. Cherlow, and T. T. Yang, Phys. Rev. B 11, 964 (1975);R. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

Hill, K. O.

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, Appl. Phys. Lett. 29, 181 (1976);R. K. Jain, C. Lin, R. H. Stolen, W. Pleibel, and P. Kaiser, Appl. Phys. Lett. 30, 162 (1977).
[CrossRef]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

Jain, R. K.

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

R. K. Jain and K. Stenersen, Appl. Phys. B 35, 49 (1984).
[CrossRef]

Johnson, D. C.

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, Appl. Phys. Lett. 29, 181 (1976);R. K. Jain, C. Lin, R. H. Stolen, W. Pleibel, and P. Kaiser, Appl. Phys. Lett. 30, 162 (1977).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, B. S. Kawasaki, and D. C. Johnson, Appl. Phys. Lett. 29, 181 (1976);R. K. Jain, C. Lin, R. H. Stolen, W. Pleibel, and P. Kaiser, Appl. Phys. Lett. 30, 162 (1977).
[CrossRef]

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Khaidarov, D. V.

A. B. Grudinin, E. M. Dianov, and D. V. Khaidarov, Sov. J. Tech. Phys. 57, 788 (1987).

Kitayama, K.

K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
[CrossRef]

Kuckartz, M.

R. Schulz, M. Kuckartz, and H. Harde, Opt. Commun. 70, 239 (1989).
[CrossRef]

Kuzin, E. A.

D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
[CrossRef]

Lee, S. S.

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, Opt. Commun. 78, 137 (1990).
[CrossRef]

Lin, C.

Liu, J. M.

J. K. Chee and J. M. Liu, IEEE J. Quantum Electron. 26, 541 (1990).
[CrossRef]

J. K. Chee and J. M. Liu, Opt. Lett. 14, 820 (1989).
[CrossRef] [PubMed]

Mamyshev, P. V.

E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).

Nakashima, T.

Nakazawa, M.

M. Nakazawa, K. Suzuki, and H. A. Haus, Phys. Rev. A 38, 5193 (1988);IEEE J. Quantum Electron. 25, 2036 (1990);M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1990).
[CrossRef] [PubMed]

M. Nakazawa, T. Nakashima, and S. Seikai, J. Opt. Soc. Am. B 2, 515 (1985).
[CrossRef]

Ovsyannikov, D. V.

D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
[CrossRef]

Park, H. G.

Park, J. D.

Petrov, M. P.

D. V. Ovsyannikov, E. A. Kuzin, M. P. Petrov, and V. I. Belotitskii, Opt. Commun. 82, 80 (1991).
[CrossRef]

Pilipetskii, A. N.

E. A. Golovchenko, E. M. Dianov, P. V. Mamyshev, and A. N. Pilipetskii, Pis′ma Zh. Eksp. Teor. Fiz. 50, 170 (1989) [JETP Lett. 50, 190 (1990)];IEEE J. Quantum Electron. 26, 1815 (1990).

Pilipotskii, A. N.

E. A. Golovchenko and A. N. Pilipotskii, Sov. Lightwave Commun. 1, 271 (1991).

Romagnoli, M.

A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.

Rothenberg, J. E.

J. E. Rothenberg, Phys. Rev. A 42, 682 (1990).
[CrossRef] [PubMed]

Schulz, R.

R. Schulz, M. Kuckartz, and H. Harde, Opt. Commun. 70, 239 (1989).
[CrossRef]

Seikai, S.

M. Nakazawa, T. Nakashima, and S. Seikai, J. Opt. Soc. Am. B 2, 515 (1985).
[CrossRef]

K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
[CrossRef]

Shen, Y. R.

Y. R. Shen and N. Bloembergen, Phys. Rev. 6A, 1787 (1965);N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), pp. 110–119.
[CrossRef]

Stegeman, G. I.

Stenersen, K.

R. K. Jain and K. Stenersen, Appl. Phys. B 35, 49 (1984).
[CrossRef]

K. Stenersen and R. K. Jain, Opt. Commun. 51, 121 (1984).
[CrossRef]

Stolen, R. H.

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, J. Opt. Soc. Am. B 6, 1159 (1989).
[CrossRef]

R. H. Stolen and J. E. Bjorkholm, IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef]

R. H. Stolen, M. A. Bosch, and C. Lin, Opt. Lett. 6, 213 (1981).
[CrossRef] [PubMed]

R. H. Stolen, IEEE J. Quantum Electron. QE-15, 1157 (1979).
[CrossRef]

R. H. Stolen, IEEE J. Quantum Electron. QE-11, 100 (1975).
[CrossRef]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

R. H. Stolen and E. P. Ippen, Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, and H. A. Haus, Phys. Rev. A 38, 5193 (1988);IEEE J. Quantum Electron. 25, 2036 (1990);M. Nakazawa, K. Suzuki, H. Kubota, and H. A. Haus, IEEE J. Quantum Electron. 25, 2045 (1990).
[CrossRef] [PubMed]

Tobin, M. C.

Tomlinson, W. J.

Trillo, S.

S. Wabnitz, S. Trillo, E. M. Wright, and G. I. Stegeman, J. Opt. Soc. Am. B 8, 602 (1991).
[CrossRef]

G. Cappellini and S. Trillo, J. Opt. Soc. Am. B 8, 824 (1991);Opt. Lett. 16, 895 (1991).
[CrossRef]

A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.

Uchida, N.

K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
[CrossRef]

Vozzella, R.

A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.

Wabnitz, S.

S. Wabnitz, S. Trillo, E. M. Wright, and G. I. Stegeman, J. Opt. Soc. Am. B 8, 602 (1991).
[CrossRef]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988);S. Trillo and S. Wabnitz, J. Opt. Soc. Am. B 6, 238 (1989).
[CrossRef] [PubMed]

A. Cosentino, M. Romagnoli, S. Trillo, R. Vozzella, and S. Wabnitz, in Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper M63;in Conference on Lasers and Electro-Optics, Vol. 7 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QWD34.

Wood, D.

K. Blow and D. Wood, IEEE J. Quantum Electron. 25, 2665 (1989).
[CrossRef]

Wright, E. M.

Yang, T.

Yang, T. T.

R. Hellwarth, J. Cherlow, and T. T. Yang, Phys. Rev. B 11, 964 (1975);R. Hellwarth, Prog. Quantum Electron. 5, 1 (1977).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

R. K. Jain and K. Stenersen, Appl. Phys. B 35, 49 (1984).
[CrossRef]

Appl. Phys. Lett. (4)

K. Kitayama, S. Seikai, and N. Uchida, Appl. Phys. Lett. 41, 322 (1982);K. Kitayama and M. Ohashi, Appl. Phys. Lett. 41, 619 (1982).
[CrossRef]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Appl. Phys. Lett. 24, 308 (1974).
[CrossRef]

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R. Hawkins, Lawrence Livermore National Laboratory; Livermore, Calif. 94550 (personal communication).

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Figures (31)

Fig. 1
Fig. 1

Measured curves of the Raman susceptibility. (a) Real part of both the F1111(Ω[cm−1]) = A(Ω) + B(Ω) (solid curve) and 2F1221(Ω) = B(Ω) (dashed curve) components; (b) corresponding imaginary parts (Raman gain).

Fig. 2
Fig. 2

Raman gain G from Eq. (20) versus linear wave-vector mismatch among the pump, Stokes, and anti-Stokes waves, normalized to the gain value for long fibers.

Fig. 3
Fig. 3

Intensity ratio of Stokes and anti-Stokes components of the growing eigenmode with gain as in Fig. 2.

Fig. 4
Fig. 4

Combined Raman and parametric gain versus linear wave-vector mismatch, with the same polarization for pump and sidebands.

Fig. 5
Fig. 5

Intensity ratio of Stokes and anti-Stokes components associated with the growing mode with gain as in Fig. 4.

Fig. 6
Fig. 6

Same as Fig. 4 but with a pump aligned with one fiber axis (say, x), whereas the Stokes and anti-Stokes components have the same polarization on the orthogonal (say, y) axis.

Fig. 7
Fig. 7

Intensity ratio of Stokes and anti-Stokes components associated with the growing mode with gain as in Fig. 6.

Fig. 8
Fig. 8

Spectral dependence of the parametric gain G of orthogonal sidebands versus the fiber birefringence δβ when the pump is coupled to the x mode: (a) normal dispersion, βx < βy; (b) anomalous dispersion, βx < βy; (c) normal dispersion βx > βy.

Fig. 9
Fig. 9

Dispersion relationship in the anomalous-dispersion regime, with equal axis excitation by the pump, and two values of the group-velocity mismatch 5. (a) δ = 0, (b) δ = 1. The solid (dashed) curves denote real (imaginary) parts of the wave number, which correspond to sideband oscillation (growth). Here all quantities are given in terms of dimensionless variables.

Fig. 10
Fig. 10

Spectrum of the (dimensionless) gains G1 and G2 of the two sideband eigenmodes with a mixed-mode pump propagating in the anomalous-dispersion regime versus the group-velocity mismatch δ.

Fig. 11
Fig. 11

Fractional contribution (in intensity) of the four sideband component modes to the first growing eigenmode in Fig. 10. Here Sx,y are the intensities in the Stokes waves in the x and y axes, and Ax,y are the anti-Stokes components.

Fig. 12
Fig. 12

Same as Fig. 11 but for the second eigenmode in Fig. 10.

Fig. 13
Fig. 13

Same as Fig. 10 but including the Raman effect.

Fig. 14
Fig. 14

Gain spectrum with a two-sideband approximation to the first growing eigenmode whose gain is reported in Fig. 10: (a) without the Raman effect, (b) including the Raman effect, (c) fraction of intensity in the Stokes component (on the fast y axis) of the growing eigenmode as in (b).

Fig. 15
Fig. 15

Spectral dependence of the gain of the growing eigenmode with the mixed-mode pump in the normal-dispersion regime.

Fig. 16
Fig. 16

Fractional intensity in the x-polarized Stokes and anti-Stokes components of the unstable eigenmode of Fig. 15. The y-polarized components are obtained simply by an interchange of the Stokes and anti-Stokes waves.

Fig. 17
Fig. 17

Gain spectrum of the two growing eigenmodes with a mixed-mode pump in the normal-dispersion regime with the Raman effect.

Fig. 18
Fig. 18

Gain spectrum with a two-sideband approximation to the growing eigenmode with gain G2 in Fig. 17: (a) without the Raman effect, (b) with the Raman effect, (c) fractional intensity in the Stokes component (on the slow axis) of the growing eigenmode whose gain is reported in (b).

Fig. 19
Fig. 19

Same as Fig. 4 but with a pump oriented at 45° between the axes. The two-sideband approximation is used here.

Fig. 20
Fig. 20

Intensity ratio between the anti-Stokes and Stokes components in correspondence with Fig. 19.

Fig. 21
Fig. 21

Analytic approximations of the frequency dependence of the measured tensorial Raman complex susceptibility in Fig. 1. (a) Real part of the parallel (solid curve) and orthogonal (dashed curve) Raman susceptibility whose Fourier transform is reported in Eqs. (34). (b) Imaginary part of the parallel (solid curve) and orthogonal (dashed curve) Raman gain.

Fig. 22
Fig. 22

Calculated pulse intensities and spectral intensities emerging from the slow and the fast modes of the fiber. Here the input pump pulse power is 10 kW, and its wavelength is 532 nm. The initial polarization of the pump is rotated by 8° from the slow axis, the input pulse full width is tfwhm = 1 ps, the fiber length is 1.5 m, and the birefringence is δn = 2.6 × 10−5.

Fig. 23
Fig. 23

Same as Fig. 22, but here the input pump power is 2.3 kW, the input pulse width is 1.7 ps, the linear polarization of the pump pulse is rotated by 8° from the slow axis of the fiber, the fiber length is 4.6 m, and the birefringence is δn = 8.8 × 10−6.

Fig. 24
Fig. 24

Same as Fig. 22, but here the total input peak power is 172 W (86 W on each axis), the linear input polarization of the pump pulse is oriented at 45° between the axes, the temporal width of the input pulse is 9 ps, the mean wavelength is 532 nm, the fiber length is 10.5 m, and the linear birefringence is δn = 4.6 × 10−4.

Fig. 25
Fig. 25

Same as Fig. 24 but with a total input peak power of the pump of 306 W.

Fig. 26
Fig. 26

Same as Fig. 24 but with a total input peak power of 460 W.

Fig. 27
Fig. 27

Same as Fig. 22, but here the input peak power is equal to 1.1 kW on each axis, the initial time width of the pulses is 1.7 ps at the wavelength of 532 nm, the fiber length is 1.5 m, and the linear birefringence of the fiber is δn = 1.5 × 10−3.

Fig. 28
Fig. 28

Same as Fig. 27, but here the fiber length is 4.6 m.

Fig. 29
Fig. 29

Same as Fig. 27, but here the input peak power is 2 kW on each mode and the total fiber length is 30 cm.

Fig. 30
Fig. 30

Same as Fig. 29, but here the fiber length is 1.5 m.

Fig. 31
Fig. 31

Same as Fig. 27, but here the input pulse width is 850 fs, the input peak power is 8.7 kW in each mode, and the total length of the fiber is 40 cm.

Equations (37)

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E ( r , z , s ) = x E x + y E y .
P i ( 3 ) ( r , z , s ) = + d s 1 + d s 2 + d s 3 χ ijkl ( 3 ) ( s , s 1 , s 2 , s 3 ) × E j ( r , s 1 ) E k ( r , s 2 ) E l ( r , s 3 ) .
P i ( 3 ) ( r , z , s ) = σ ijkl E j ( s ) E k ( s ) E l ( s ) + E j ( s ) s d s 1 f ijkl ( s s 1 ) E k ( s 1 ) E l ( s 1 ) .
f ijkl ( s ) = a ( s ) δ ij δ kl + ½ b ( s ) ( δ il δ jk + δ ik δ jl ) ,
σ ijkl = σ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) .
E j ( r , z , s ) = ½ { E ̂ j ( r , z , s ) exp [ i β j ( ω 0 ) z i ω 0 s ] + c . c . } , P j ( r , z , s ) = ½ { P ̂ j ( r , z , s ) exp [ i β j ( ω 0 ) z i ω 0 s ] + c . c . } , j = x , y ,
P ̂ x NR = 3 σ 4 ( | E ̂ x | 2 E ̂ x + 2 3 | E ̂ y | 2 E ̂ x + 1 3 E ̂ y 2 E ̂ x * × exp { 2 i [ β y ( ω 0 ) β x ( ω 0 ) ] z } ) .
P ̂ x R ( s ) = 1 2 ( E ̂ x ( s ) s d s 1 ( a + b ) ( s s 1 ) | E ̂ x ( s 1 ) | 2 + E ̂ x ( s ) s d s 1 a ( s s 1 ) | E ̂ y ( s 1 ) | 2 + E ̂ y ( s ) s d s 1 b ( s s 1 ) 2 ( E ̂ x ( s 1 ) E ̂ y * ( s 1 ) + E ̂ y ( s 1 ) E ̂ x * ( s 1 ) exp { 2 i [ β y ( ω 0 ) β x ( ω 0 ) ] z } ) .
E ( r , z , s ) = ½ x { g s ( r ) E sx exp [ i β x ( ω s ) z i ω s s ] + g a ( r ) E ax exp [ i β x ( ω a ) z i ω a s ] + g p ( r ) E px exp [ i β x ( ω p ) z i ω p s ] + c . c . } = ½ y { g s ( r ) E sy exp [ i β y ( ω s ) z i ω s s ] + g a ( r ) E ay exp [ i β y ( ω a ) z i ω a s ] + g p ( r ) E py exp [ i β y ( ω p ) z i ω p s ] + c . c . } ,
F ijkl ( ω ) = A ( ω ) δ ij δ kl + ½ B ( ω ) ( δ ik δ jl + δ il δ jk ) ,
A ( ω ) + a ( s ) exp ( i ω s ) d s , B ( ω ) + b ( s ) exp ( i ω s ) d s .
F 1111 ( ω ) = A ( ω ) + B ( ω ) , F 1221 ( ω ) = F 1212 ( ω ) = B ( ω ) 2 , F 1122 ( ω ) = A ( ω ) .
χ 1111 R ( Ω ) = χ 1111 R ( ω s ; ω s , ω p , ω p ) = A ( 0 ) + B ( 0 ) + A ( Ω ) + B ( Ω ) 4 , χ 1212 R ( Ω ) = χ 1212 R ( ω s ; ω s , ω p , ω p ) = A ( Ω ) 4 + B ( 0 ) 8 , χ 1122 R ( Ω ) = χ 1122 R ( ω s ; ω s , ω p , ω p ) = A ( 0 ) 4 + B ( Ω ) 8 , χ 1221 R ( Ω ) = χ 1221 R ( ω s ; ω s , ω p , ω p ) = B ( 0 ) + B ( Ω ) 8 ,
χ 1111 NR = 3 χ 1212 NR = 3 χ 1122 NR = 3 χ 1221 NR = 3 σ / 4 .
i d E px d z = 2 π ω p 2 c 2 k p A p ( { [ χ 1111 NR + A ( 0 ) + B ( 0 ) 2 ] | E px | 2 + [ 2 χ 1122 NR + A ( 0 ) 2 + B ( 0 ) 4 ] | E py | 2 } E px + [ χ 1221 NR + B ( 0 ) 4 ] E py 2 E px * exp ( i Δ k 1 z ) ) ,
i d E sx d z = 2 π ω s 2 c 2 k s ( 2 A s 1 { [ χ 1111 NR + χ 1111 R ( Ω ) ] | E px | 2 + [ χ 1122 NR + χ 1122 R ( Ω ) ] | E py | 2 } E sx + 1 A s 2 { [ χ 1111 NR + A ( Ω ) + B ( Ω ) 2 ] E px 2 E ax * exp ( i Δ k 2 z ) + [ χ 1221 NR + B ( Ω ) 4 ] E py 2 E ax * exp ( i Δ k 3 z ) + [ 2 χ 1212 NR + B ( Ω ) 4 + A ( Ω ) 2 ] E px E py E ay * exp ( i Δ k 4 z ) } + 2 A s 1 { [ χ 1221 NR + χ 1221 R ( Ω ) ] E px * E py exp ( i Δ k 5 z ) + [ χ 1212 NR + χ 1212 R ( Ω ) ] E px E py * exp ( i Δ k 6 z ) } E sy ) .
Δ k 2 = 2 β x ( ω p ) β x ( ω a ) β x ( ω s ) , Δ k 3 = 2 β y ( ω p ) β x ( ω a ) β x ( ω s ) , Δ k 4 = β x ( ω p ) + β y ( ω p ) β y ( ω a ) β x ( ω s ) , Δ k 5 = β y ( ω s ) + β y ( ω p ) β x ( ω p ) β x ( ω s ) , Δ k 6 = β y ( ω s ) + β x ( ω p ) β y ( ω p ) β x ( ω s ) .
Δ k 2 = α x Ω 2 , Δ k 3 = 2 ( β y β x ) α x Ω 2 , Δ k 4 = Ω ( 1 V gx 1 V gy ) α x + α y 2 Ω 2 , Δ k 5 = 2 ( β y β x ) Ω ( 1 V gy 1 V gx ) + α y α x 2 Ω 2 , Δ k 6 = Ω ( 1 V gx 1 V gy ) + α y α x 2 Ω 2 .
d U d z = i κ V exp ( i η z ) , d V d z = i κ U exp ( i η z ) .
U E sx , V E ax * , κ P [ R i g 0 ( Ω ) 2 ] , η 2 RP g 0 ( Ω ) P Δ k 2 .
G 2 μ / Re [ g 0 ( Ω ) ] P .
γ 2 Δ k / Re [ g 0 ( Ω ) ] P .
U E sy , V E ay * , κ P [ R 3 i g 0 ( Ω ) 2 ] , η 2 π ω p 2 B ( 0 ) c 2 k s A p 2 RP 3 i g 0 ( Ω ) P Δ k 3 , 2 RP 3 i g 0 ( Ω ) P Δ k 3 .
δ Ω L c ( Δ k 3 ) Ω | Δ k 3 = 0 = 2 π ,
L c = π δ Ω 1 [ 2 | ( β x β y ) α | ] 1 / 2 .
e jk = ( R t 0 2 / | α | ) 1 / 2 exp ( i β z ) E jk .
t = ( s z / V ) / t 0 ζ = | α | z / t 0 2 ,
K 2 = B + A 2 ± [ ( B + A 2 ) 2 + C 2 ( B A 2 ) 2 ] 1 / 2 ,
A = Ω δ , B = ± Ω 2 ( 1 2 ± Ω 2 4 ) , C = ± Ω 2 3 .
B = ± Ω 2 [ 1 i g ( Ω ) 2 ± Ω 2 2 ] , c = 1 2 i g ( Ω ) 2 .
χ RES ( Ω ) = ρ τ 1 2 + τ 2 2 τ 1 2 + τ 2 2 ( 1 + i τ 1 Ω ) ,
U E sy , V E ax * , κ P [ R 3 i g 0 ( Ω ) 4 + i g 0 ( Ω ) 4 ] , η π ω p 2 B ( 0 ) 2 c 2 k s A p + RP i g 0 ( Ω ) P 2 i g 0 ( Ω ) P 2 Δ k 4 R P i g 0 ( Ω ) P 2 i g 0 ( Ω ) P 2 Δ k 4 .
L c = 2 π c / δ n δ Ω .
i ( e x ζ + δ e x t ) ± 1 2 2 e x t 2 + ( 1 ρ ) × [ ( | e x | 2 + 2 3 | e y | 2 ) e x + 1 3 e y 2 e x * exp ( 2 i δ β ζ ) ] + ρ [ e x ( t ) t d s ( a + b ) ( t s ) | e x ( s ) | 2 + e x ( t ) t d s a ( t s ) × | e y ( s ) | 2 + e y ( t ) t d s b ( t s ) 2 ( e x ( s ) e y * ( s ) + e y ( s ) e x * ( s ) exp { 2 i [ β y ( ω 0 ) ] ζ } ) ] = 0 ,
f ( t ) = a ( t ) + b ( t ) = τ 1 2 + τ 2 2 τ 1 τ 2 2 exp ( t / τ 2 ) sin ( t / τ 1 ) , g ( t ) = b ( t ) 2 = r τ 2 exp ( t / τ 2 ) .
r = 4 χ ( τ 1 2 + τ 2 2 ) τ 1 χ ( τ 2 2 + 4 τ 1 2 )
e x ( ζ = 0 , t ) = A x sech ( t ) + n x ( t ) , e y ( ζ = 0 , t ) = A y sech ( t ) + n y ( t ) ,

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