Abstract

A general expression for a nonlinear polarization in fibers, in which the different symmetry rules for the electronic and Raman contributions to the susceptibility are taken into account, is derived. The contribution of the Raman effect to the total nonlinearity is calculated for different polarizations of a pump and the Stokes wave in polarization-maintaining fibers as well as in fibers with randomly varying birefringence. The error in the measurements of the n2 coefficient caused by the different symmetry rules for the Raman and electronic contributions is shown to depend on germanium concentration and a frequency shift varying from 3.3% in polarization-maintaining fibers to 10% in fibers with random birefringence. The ratio between cross-phase modulation and self-modulation strengths strongly depends on a frequency shift and the polarization state of the interacting waves and varies from 1.2 to 2 in fibers with germanium concentration from 0 to 18 mol.%.

© 2001 Optical Society of America

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References

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  1. A. R. Chraplyvy, “Limitations on lightwave communications imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8, 1548–1557 (1990).
    [CrossRef]
  2. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
    [CrossRef]
  3. K. S. Kim, R. H. Stolen, W. A. Reed, and K. W. Quoi, “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers,” Opt. Lett. 19, 257–259 (1994).
    [CrossRef] [PubMed]
  4. R. H. Stolen, W. A. Reed, K. S. Kim, and G. T. Harvey, “Measurement of the nonlinear refractive index of long dispersion-shifted fibers by self-phase modulation at 1.55 μm,” J. Lightwave Technol. 16, 1006–1012 (1998).
    [CrossRef]
  5. T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, “Measurement of the nonlinear refractive index in optical fiber by the cross-phase-modulation method with depolarized pump light,” Opt. Lett. 20, 988–990 (1995).
    [CrossRef] [PubMed]
  6. T. Kato, Y. Suetsugu, and M. Nishimura, “Estimation of nonlinear refractive index in various silica-based glasses for optical fibers,” Opt. Lett. 20, 2279–2281 (1995).
    [CrossRef] [PubMed]
  7. S. V. Chernikov and J. R. Taylor, “Measurement of normalization factor of n2 for random polarization in optical fibers,” Opt. Lett. 21, 1559–1561 (1996).
    [CrossRef] [PubMed]
  8. A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μm,” Opt. Lett. 21, 1966–1968 (1996).
    [CrossRef] [PubMed]
  9. Y. Namihira, A. Miyata, and N. Tanahashi, “Nonlinear coefficient measurements for dispersion shifted fibres using self-phase modulation method at 1.55 μm,” Electron. Lett. 30, 1171–1172 (1994).
    [CrossRef]
  10. D. Monzon-Hernandez, A. N. Starodumov, Yu. O. Barmenkov, I. Torres-Gomez, and F. Mendoza-Santoyo, “Continuous-wave measurement of the fiber nonlinear refractive index,” Opt. Lett. 23, 1274–1276 (1998).
    [CrossRef]
  11. L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by four-wave mixing,” IEEE Photon. Technol. Lett. 5, 1092–1095 (1993).
    [CrossRef]
  12. R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
    [CrossRef]
  13. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  14. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  15. A. Höök, “Influence of stimulated Raman scattering oncross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115–117 (1992).
    [CrossRef]
  16. R. H. Stolen and E. P. Ippen, “Raman gain in optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
    [CrossRef]
  17. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  18. E. A. Golovchenko and A. N. Pilipetskii, “Unified analysis of four-photon mixing, modulational instability, and stimulated Raman scattering under various polarization conditions in fibers,” J. Opt. Soc. Am. B 11, 92–101 (1994).
    [CrossRef]
  19. S. Trillo and S. Wabnitz, “Parametric and Raman amplification in birefringent fibers,” J. Opt. Soc. Am. B 9, 1061–1082 (1992).
    [CrossRef]
  20. C. Headley III and G. P. Agrawal, “Unified description of ultrafast stimulated Raman scattering in optical fibers,” J. Opt. Soc. Am. B 13, 2170–2177 (1996).
    [CrossRef]
  21. C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent fibers,” Opt. Lett. 16, 566–568 (1991).
    [CrossRef] [PubMed]
  22. S. V. Chernikov and P. V. Mamyshev, “Effect of polarization on Raman scattering in optical fibers,” Sov. Lightwave Commun. 1, 301–312 (1991).
  23. A. N. Starodumov, Yu. O. Barmenkov, A. Martínez, I. Torres, and L. A. Zenteno, “Experimental demonstration of Raman effect-based optical transistor,” Opt. Lett. 23, 352–354 (1998).
    [CrossRef]
  24. S. A. E. Lewis, S. V. Chernikov, and J. R. Taylor, “Multi-wavelength pumped silica-fibre Raman amplifiers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999).
  25. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).
  26. N. Tang and R. L. Sutherland, “Time-domain theory for pump-probe experiments with chirped pulses,” J. Opt. Soc. Am. B 14, 3412–3423 (1997).
    [CrossRef]
  27. R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  28. F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
    [CrossRef]
  29. S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
    [CrossRef]
  30. T. Nakashima, S. Seikai, and M. Nakazawa, “Dependence of Raman gain on relative index difference for GeO2-doped single-mode fibers,” Opt. Lett. 10, 420–422 (1985).
    [CrossRef] [PubMed]
  31. D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995).
    [CrossRef] [PubMed]
  32. D. Mahgerefteh, D. L. Butler, J. Goldhar, B. Rosenberg, and G. L. Burdge, “Technique for measurement of the Raman gain coefficient in optical fibers,” Opt. Lett. 21, 2026–2028 (1996).
    [CrossRef] [PubMed]
  33. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, (2) 148–157 (1996).
    [CrossRef]
  34. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10(1), 28–34 (1992).
    [CrossRef]
  35. R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
    [CrossRef]
  36. E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers,” Opt. Lett. 21, 1117–1119 (1996).
    [CrossRef] [PubMed]
  37. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

1998 (3)

1997 (1)

1996 (6)

1995 (3)

1994 (3)

1993 (1)

L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by four-wave mixing,” IEEE Photon. Technol. Lett. 5, 1092–1095 (1993).
[CrossRef]

1992 (3)

1991 (2)

C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent fibers,” Opt. Lett. 16, 566–568 (1991).
[CrossRef] [PubMed]

S. V. Chernikov and P. V. Mamyshev, “Effect of polarization on Raman scattering in optical fibers,” Sov. Lightwave Commun. 1, 301–312 (1991).

1990 (1)

A. R. Chraplyvy, “Limitations on lightwave communications imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8, 1548–1557 (1990).
[CrossRef]

1989 (3)

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

1985 (1)

1979 (1)

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
[CrossRef]

1978 (2)

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

1977 (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

1975 (1)

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[CrossRef]

Agrawal, G. P.

Ainslie, B. J.

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Barmenkov, Yu. O.

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10(1), 28–34 (1992).
[CrossRef]

Boskovic, A.

Boyd, R. W.

Buckland, E. L.

Burdge, G. L.

Butler, D. L.

Cherlow, J.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Chernikov, S. V.

Chraplyvy, A. R.

A. R. Chraplyvy, “Limitations on lightwave communications imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8, 1548–1557 (1990).
[CrossRef]

Davey, S. T.

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Dougherty, D. J.

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10(1), 28–34 (1992).
[CrossRef]

Galeener, F. L.

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

Geils, R. H.

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

Goldhar, J.

Golovchenko, E. A.

Gordon, J. P.

Gruner-Nielsen, L.

Hamaide, J.-P.

L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by four-wave mixing,” IEEE Photon. Technol. Lett. 5, 1092–1095 (1993).
[CrossRef]

Harvey, G. T.

Haus, H. A.

Headley III, C.

Hellwarth, R.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Höök, A.

Ippen, E. P.

Islam, M. N.

Kärtner, F. X.

Kato, T.

Kim, K. S.

Levring, O. A.

Lin, C.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Mahgerefteh, D.

Mamyshev, P. V.

S. V. Chernikov and P. V. Mamyshev, “Effect of polarization on Raman scattering in optical fibers,” Sov. Lightwave Commun. 1, 301–312 (1991).

Martínez, A.

Mendoza-Santoyo, F.

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, (2) 148–157 (1996).
[CrossRef]

C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent fibers,” Opt. Lett. 16, 566–568 (1991).
[CrossRef] [PubMed]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

Mikkelsen Jr., J. C.

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

Miyata, A.

Y. Namihira, A. Miyata, and N. Tanahashi, “Nonlinear coefficient measurements for dispersion shifted fibres using self-phase modulation method at 1.55 μm,” Electron. Lett. 30, 1171–1172 (1994).
[CrossRef]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10(1), 28–34 (1992).
[CrossRef]

Monzon-Hernandez, D.

Mosby, W. J.

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

Nakashima, T.

Nakazawa, M.

Namihira, Y.

Y. Namihira, A. Miyata, and N. Tanahashi, “Nonlinear coefficient measurements for dispersion shifted fibres using self-phase modulation method at 1.55 μm,” Electron. Lett. 30, 1171–1172 (1994).
[CrossRef]

Nishimura, M.

Pilipetskii, A. N.

Prigent, L.

L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by four-wave mixing,” IEEE Photon. Technol. Lett. 5, 1092–1095 (1993).
[CrossRef]

Quoi, K. W.

Reed, W. A.

Rosenberg, B.

Rothwell, W. J. M.

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Sasaoka, E.

Seikai, S.

Starodumov, A. N.

Stolen, R. H.

Suetsugu, Y.

Sutherland, R. L.

Takagi, M.

Tanahashi, N.

Y. Namihira, A. Miyata, and N. Tanahashi, “Nonlinear coefficient measurements for dispersion shifted fibres using self-phase modulation method at 1.55 μm,” Electron. Lett. 30, 1171–1172 (1994).
[CrossRef]

Tang, N.

Taylor, J. R.

Tomlinson, W. J.

Torres, I.

Torres-Gomez, I.

Trillo, S.

Wabnitz, S.

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, (2) 148–157 (1996).
[CrossRef]

Wakefield, B.

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Williams, D. L.

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Yang, T.-T.

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Zenteno, L. A.

Appl. Phys. Lett. (2)

R. H. Stolen and E. P. Ippen, “Raman gain in optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[CrossRef]

F. L. Galeener, J. C. Mikkelsen, Jr., R. H. Geils, and W. J. Mosby, “The relative Raman cross sections of vitreous SiO2, GeO2, B2O3 and P2O5,” Appl. Phys. Lett. 32(1), 34–36 (1978).
[CrossRef]

Electron. Lett. (1)

Y. Namihira, A. Miyata, and N. Tanahashi, “Nonlinear coefficient measurements for dispersion shifted fibres using self-phase modulation method at 1.55 μm,” Electron. Lett. 30, 1171–1172 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[CrossRef]

R. H. Stolen, “Polarization effects in fiber Raman and Brillouin lasers,” IEEE J. Quantum Electron. QE-15, 1157–1160 (1979).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Prigent and J.-P. Hamaide, “Measurement of fiber nonlinear Kerr coefficient by four-wave mixing,” IEEE Photon. Technol. Lett. 5, 1092–1095 (1993).
[CrossRef]

J. Lightwave Technol. (4)

A. R. Chraplyvy, “Limitations on lightwave communications imposed by optical-fiber nonlinearities,” J. Lightwave Technol. 8, 1548–1557 (1990).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, (2) 148–157 (1996).
[CrossRef]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol. 10(1), 28–34 (1992).
[CrossRef]

R. H. Stolen, W. A. Reed, K. S. Kim, and G. T. Harvey, “Measurement of the nonlinear refractive index of long dispersion-shifted fibers by self-phase modulation at 1.55 μm,” J. Lightwave Technol. 16, 1006–1012 (1998).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Lett. (13)

A. N. Starodumov, Yu. O. Barmenkov, A. Martínez, I. Torres, and L. A. Zenteno, “Experimental demonstration of Raman effect-based optical transistor,” Opt. Lett. 23, 352–354 (1998).
[CrossRef]

D. Monzon-Hernandez, A. N. Starodumov, Yu. O. Barmenkov, I. Torres-Gomez, and F. Mendoza-Santoyo, “Continuous-wave measurement of the fiber nonlinear refractive index,” Opt. Lett. 23, 1274–1276 (1998).
[CrossRef]

E. L. Buckland and R. W. Boyd, “Electrostrictive contribution to the intensity-dependent refractive index of optical fibers,” Opt. Lett. 21, 1117–1119 (1996).
[CrossRef] [PubMed]

S. V. Chernikov and J. R. Taylor, “Measurement of normalization factor of n2 for random polarization in optical fibers,” Opt. Lett. 21, 1559–1561 (1996).
[CrossRef] [PubMed]

A. Boskovic, S. V. Chernikov, J. R. Taylor, L. Gruner-Nielsen, and O. A. Levring, “Direct continuous-wave measurement of n2 in various types of telecommunication fiber at 1.55 μm,” Opt. Lett. 21, 1966–1968 (1996).
[CrossRef] [PubMed]

D. Mahgerefteh, D. L. Butler, J. Goldhar, B. Rosenberg, and G. L. Burdge, “Technique for measurement of the Raman gain coefficient in optical fibers,” Opt. Lett. 21, 2026–2028 (1996).
[CrossRef] [PubMed]

A. Höök, “Influence of stimulated Raman scattering oncross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115–117 (1992).
[CrossRef]

D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995).
[CrossRef] [PubMed]

T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimura, “Measurement of the nonlinear refractive index in optical fiber by the cross-phase-modulation method with depolarized pump light,” Opt. Lett. 20, 988–990 (1995).
[CrossRef] [PubMed]

T. Kato, Y. Suetsugu, and M. Nishimura, “Estimation of nonlinear refractive index in various silica-based glasses for optical fibers,” Opt. Lett. 20, 2279–2281 (1995).
[CrossRef] [PubMed]

T. Nakashima, S. Seikai, and M. Nakazawa, “Dependence of Raman gain on relative index difference for GeO2-doped single-mode fibers,” Opt. Lett. 10, 420–422 (1985).
[CrossRef] [PubMed]

C. R. Menyuk, M. N. Islam, and J. P. Gordon, “Raman effect in birefringent fibers,” Opt. Lett. 16, 566–568 (1991).
[CrossRef] [PubMed]

K. S. Kim, R. H. Stolen, W. A. Reed, and K. W. Quoi, “Measurement of the nonlinear index of silica-core and dispersion-shifted fibers,” Opt. Lett. 19, 257–259 (1994).
[CrossRef] [PubMed]

Phys. Rev. A (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1978).
[CrossRef]

Phys. Rev. B (1)

R. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Proc. IEEE (1)

S. T. Davey, D. L. Williams, B. J. Ainslie, W. J. M. Rothwell, and B. Wakefield, “Optical gain spectrum of GeO2-SiO2 Raman fiber amplifiers,” Proc. IEEE 136, 301–305 (1989).
[CrossRef]

Prog. Quantum Electron. (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Sov. Lightwave Commun. (1)

S. V. Chernikov and P. V. Mamyshev, “Effect of polarization on Raman scattering in optical fibers,” Sov. Lightwave Commun. 1, 301–312 (1991).

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

S. A. E. Lewis, S. V. Chernikov, and J. R. Taylor, “Multi-wavelength pumped silica-fibre Raman amplifiers,” in Optical Fiber Communication Conference, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, New York, 1990).

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992).

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Figures (4)

Fig. 1
Fig. 1

Raman gain spectra of silica-based fibers for 0, 18, and 100 mol.% GeO2 concentration.

Fig. 2
Fig. 2

Spectra of the Raman nonlinear coefficient in silica fibers for 0, 18, and 100 mol.% GeO2 concentration.

Fig. 3
Fig. 3

Raman contribution to n2 (solid curve) and error in percent introduced by the difference in the averaging factor for the Kerr and Raman contributions (dashed curve) as a function of a GeO2 concentration.

Fig. 4
Fig. 4

Ratio between the XPM and the SPM terms for copolarized pump and Stokes fields as a function of the frequency shift.

Tables (1)

Tables Icon

Table 1 Kerr and Raman Nonlinear Contributions to SPM, XPM, and Raman Gain for the Different States of Polarization

Equations (69)

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

Ei(r, z, t)=12(Eωp, i(r, z, t) exp {i[βi(ωP)z-ωPt]}+EωS,i(r, z, t)×exp {i[βi(ωS)z-ωSt]})+c.c.,
i=x, y,
Pω,iNL(t)=ε0p4jkl -+dt1-+dt2×-+dt3Rijkl(3)(t-t1,t-t2,t-t3)×Eω1,j(t1)Eω2,k(t2)Eω3,l(t3)×exp [iω1(t-t1)+iω2(t-t2)+iω3(t-t3)+iβj(ω1)z+iβk(ω2)z+iβ1(ω3)z],
Pω,iNL(t)=ε0p4jklχijkl(3)×(-ω;ω1, ω2, ω3)Eω1,j(t)Eω2,k(t)Eω3,l(t)×exp[iβj(ωl)z+iβk(ω2)z+iβl(ω3)z],
χijkl(3)(-ω;ω1, ω2, ω3)=-+dt1-+dt2×-+dt3Rijkl(3)(t1, t2,t3)×exp (iω1t1+iω2t2+iω3t3).
Rijklelectronic(t1, t2,t3)=σijklδ(t1)δ(t2)δ(t3).
RijklRaman(t1, t2,t3)=δ(t1-t2)hijklR(t2)δ(t3)+δ(t1)δ(t2-t3)hijklR(t3)+hijklR(t1)δ(t2)δ(t1-t3),
χijkl(3)(-ω; ω1, ω2, ω3)=σijkl+χijklR(ω1+ω2)+χijklR(ω2+ω3)+χijklR(ω1+ω3),
PωS,iNL(t)=3ε04 jkl{χijkl(3)×(-ωS;ωS, ωS,-ωS)EωS,j(t)EωS,l*(t)×exp[i(βjs+βks-βls)z]+2χijkl(3)×(-ωS;ωS, ωP,-ωP)EωS,j(t)EωP,k(t)×EωP,l*(t) exp [i(βjS+βkP-βlP)z]},
PωSxNL=2ε0nx(ωS)[n2e+n2R(0)]|EωSx|2+23n2e+2n2R(0)|EωSy|2EωSx exp(iβxSz)+13n2e+n2R(0)-2n2R(0)EωSyEωSx* exp(-iΔβSz)×EωSy exp(iβySz)+[2n2e+n2R(0)+n2R(Δω)]|EωPx|2+23n2e+n2R(0)+n2R(Δω)|EωPy|2EωSx exp(iβxSz)+23n2e+12n2R(Δω)-12n2R(0)-2n2R(Δω)×EωPxEωPy* exp(iΔβPz)EωSy exp(iβySz)+23n2e+12n2R(0)-12n2R(0)+n2R(Δω)EωPyEωPx* exp(-iΔβPz)EωSy
×exp(iβySz)-i2kS[gR(Δω)|EωPx|2+gR(Δω)|EωP|2]EωSx exp(iβxSz)-i2kS{[gR(Δω)-2gR(Δω)]EωPxEωPy* exp(iΔβPz)}EωSy×exp(iβySz)-i2kS[gR(Δω)EωPxEωPx*×exp(-iΔβPz)]EωSy exp(iβySz),
LˆSAS=z AS=ikS56n2e+12n2R(0)+n2R(0)S0SAS
+16n2e+12n2R(0)-n2R(0)(S1Sσ3+S2Sσ1-S3Sσ2)AS
+43n2e+12n2R(0)+12n2R(Δω)+12n2R(0)+12n2R(Δω)S0PAS
+23n2e+12n2R(0)+12n2R(Δω)-12n2R(0)-12n2R(Δω)(S1Pσ3+S2Pσ1)AS
+12n2R(Δω)-32n2R(Δω)S3Pσ2AS
+12 gR(Δω)+gR(Δω)2S0PAS+12 gR(Δω)-gR(Δω)2(S1Pσ3+S2Pσ1)AS
+12 gR(Δω)-3gR(Δω)2S3Pσ2AS
σ1=0110,σ2=0-ii0,σ3=100-1.
S0S=ASAS,S1S=ASσ3AS,
S2S=ASσ1AS,S3S=ASσ2AS,
S0P=APAP,S1P=APσ3AP,
S2P=APσ1AP,S3P=APσ2AP,
CR,SiO2GeO2=CR,SiO21-Xmol%100+CR,GeO2 Xmol%100,
n2R,SiO2GeO2(0)=n2R,SiO2(0)(1+0.0519587Xmol%)×10-16 cm2/W,
LˆxAωs,x(z, t)=ikS{[n2e+n2R(0)]IS,x+[2n2e+n2R(0)+n2R(Δω)]IP,x}Aωs,x(z, t)+12gR(Δω)IP,xAωs,x(z, t).
LˆyAωs,y(z, t)=ikS[n2e+n2R(0)]IS,y+23n2e+n2R(0)+n2R(Δω)IP,xAωs,y(z, t)+12gR(Δω)IP,xAωs,y(z, t).
S1Sσ3AS=S2Sσ1AS=S3Sσ2AS=13S0SAS,
LˆSAS=ikS89n2e+23n2R(0)+23n2R(0)ISAS
+ikS43n2e+12n2R(0)+12n2R(Δω)+12n2R(0)+12n2R(Δω)IPAS
+12 gR(Δω)+gR(Δω)2IPAS,
n2linear-98n2random=14n2R(0)-34n2R(0)532n2R(0).
χijkl(3)[-ωS; ωS(j), ωS(k),-ωS(l)]
=σijkl+χijklR[ωS(k)-ωS(l)]+χijklR[ωS(j)-ωS(l)],
χijkl(3)[-ωS; ωS(j), ωP(k),-ωP(l)]
=σijkl+χijklR[ωP(k)-ωP(l)]+χijklR[ωS(j)-ωP(l)],
χijkl(3)[-ωS; ωS(j), ωS(k),-ωS(l)]
=σijkl+2χijklR[ωS(k)-ωS(l)].
χxxxx(3)=χxxyy(3)+χxyxy(3)+χxyyx(3)
χxyyxR(ωS-ωS)=χxxxxR(ωS-ωS)-2χxxyyR(ωS-ωS).
σijkl=σ3(δijδkl+δikδjl+δilδjk),
χijklR[ωP(k)-ωP(l)]=χijklR[ωP(l)-ωP(k)],
χijklR[ωS(j)-ωP(l)]=χilkjR[ωS(l)-ωP(j)],
χxyxyR(ωP-ωP)
=(1/2)[χxxxxR(ωP-ωP)-χxxyyR(ωP-ωP)],
χxyxyR(ωS-ωP)
=χxxxxR(ωS-ωP)-2χxxyyR(ωS-ωP).
χijkl(3)=χxxyy(3)δijδkl+χxyxy(3)δikδjl+χxyyx(3)δilδjk,
χijkl(3)(-ωS; ωS, ωS,-ωS)
=σ3+2χR(0)(δijδkl+δikδjl)+σ3+2χR(0)-4χR(0)δilδjk,
χijkl(3)(-ωS;ωS,ωP,-ωP)
=σ3+χR(0)+χR(-Δω)δijδkl+σ3+12χR(0)+χR(-Δω)-12χR(0)-2χR(-Δω)δikδjl+σ3+12χR(0)-12χR(0)+χR(-Δω)δilδjk.
Δnx(ωs)38 [σ+2χR(0)]|EωS,x|2nx(ωS).
n2SPM=n2e+n2R(0)=38nx(ωS)σ+34nx(ωS)χR(0),
n2R(Δω)=34nx(ωS)χR(Δω);
n2R(Δω)=34nx(ωS)χR(Δω).
gR,Stokes(Δω)=gR(Δω)=3ωS2cnx(ωS) Im[χR(Δω)],
gR,Stokes(Δω)=gR(Δω)=3ωS2cnx(ωS) Im[χR(Δω)].
EωS,x(r, z, t)=FxS(x, y)AωS,x(z, t)exp(iβxSz),
EωS,y(r, z, t)=FyS(x, y)AωS,y(z, t)exp(iβySz).
z+β1xS t+i2β2xS 2t2+12αxSAωS,x(z, t)
=iμ0ωS22βxSPωSxNL(r, z, t)exp(-iβxSz),
|EωS,x, y|2=f1SS|AωS,x, y|2;
EωSy*EωS,x=f1SSAωS,y*AωS,x,
|EωP,x, y|2=f2PS|AωP,x, y|2;
EωPy*EωPx=f2PSAωPy*AωPx;
EωPx*EωPy=f2PSAωPx*AωPy,
f1SS=|FS(x, y)|4dxdy[|FS(x, y)|2dxdy]2;
f2PS=|FP(x, y)|2|F2(x, y)|2dxdy[|FS(x, y)|2dxdy]2.

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