Abstract

We report for the first time, to our knowledge, a technique that has the capability to measure both the Raman gain coefficient and the nonlinear refractive index of an optical fiber, using the same experimental setup. This measurement utilizes the induced-grating autocorrelation (IGA) technique, which is based upon time-delayed four-beam coupling in a photorefractive crystal. The standard IGA trace, which is based upon two-beam coupling, fits a simple model based on pure self-phase modulation (SPM). We demonstrate that, in the negligible-dispersion regime of an optical fiber, the addition of stimulated Raman scattering (SRS) leads to a measurable distortion of the standard (pure SPM) IGA trace. We have developed a new IGA model from the analytical solution of the coupled-amplitude nonlinear Schrodinger equation. This new model successfully accounts for the effect of SRS on the IGA trace in the negligible-dispersive regime of the fiber and allows the direct determination of the Raman gain coefficient and the nonlinear refractive index from the fit of the SRS distorted IGA trace. The measured nonlinear refractive index and the Raman gain coefficient are in good agreement with published results.

© 2005 Optical Society of America

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2003 (1)

2002 (4)

2001 (1)

2000 (1)

C. Vinegoni, M. Wegmuller, and N. Gisin, "Determination of nonlinear coefficient (n2/Aeff) using self-aligned interferometer and Faraday mirror," Electron. Lett. 26, 886-888 (2000).
[Crossref]

1998 (4)

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[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]

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

1996 (4)

1995 (3)

1994 (2)

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, 262-264 (1994).
[Crossref]

A. M. Levine, E. Ozizmir, R. Trebino, C. C. Hayden, A. M. Johnson, and K. L. Tokuda, "Induced grating autocorrelation of ultrashort pulses in slowly responding medium," J. Opt. Soc. Am. B 11, 1609-1618 (1994).
[Crossref]

1993 (1)

L. Prigent and J. P. Hamaide, "Measurement of fiber nonlinear Kerr coefficient by four-wave mixing," IEEE Photonics Technol. Lett. 5, 1092-1095 (1993).
[Crossref]

1992 (1)

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

1991 (1)

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, "Effects of fiber nonlinearities on a long-distance transmission," J. Lightwave Technol. 9, 121-128 (1991).
[Crossref]

1990 (2)

1989 (1)

1987 (3)

J. Manassah and O. Cockings, "Time domain characterization of a Raman pulse in the presence of a pump," Appl. Opt. 26, 3749-3752 (1987).
[Crossref] [PubMed]

M. Monerie and Y. Durtestse, "Direct interferometric measurement of nonlinear refractive index of optical fibers by cross-phase modulation," Electron. Lett. 23, 961-963 (1987).
[Crossref]

E. Desurvire, A. Imamoglu, and H. Shaw, "Low-threshold synchronously pumped all-fiber ring Raman laser," J. Lightwave Technol. 5, 89-96 (1987).
[Crossref]

1986 (2)

R. Trebino, E. K. Gustafson, and A. E. Siegman, "Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources," J. Opt. Soc. Am. B 3, 1295-1304 (1986).
[Crossref]

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

1985 (2)

1984 (1)

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

1981 (1)

N. Shibata, M. Horigudhi, and T. Edahiro, "Raman spectra of binary high-silica glasses and fibers containing GeO2,P2O5 and B2O3," J. Non-Cryst. Solids 45, 115-126 (1981).
[Crossref]

1978 (3)

Y. Y. Huang and A. Sarkar, "Relationship between composition, density, and refractive index for germaina silica glasses," J. Non-Cryst. Solids 27, 29-37 (1978).
[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, 34-36 (1978).
[Crossref]

R. H. Stolen and C. Lin, "Self-phase modulation in silica optical fibers," Phys. Rev. A 4, 1448-1453 (1978).
[Crossref]

1977 (2)

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

1972 (2)

1970 (1)

M. Hass, "Raman spectra of vitreous silica, germania and sodium silicate glasses," J. Phys. Chem. Solids 31, 415-422 (1970).
[Crossref]

Abramov, A. A.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Aquari, V. M.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Artiglia, M.

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[Crossref]

M. Artiglia, E. Ciaramella, and B. Sordo, "Using modulation instability to determine Kerr coefficient in optical fibers," Electron. Lett. 31, 1012-1013 (1995).
[Crossref]

Barmenkov, Yu. O.

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[Crossref]

Bergano, N. S.

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

Boskovic, A.

Bromage, J.

J. Bromage, K. Rottwitt, and M. E. Lines, "A method to predict the Raman gain spectra of germanosilicate fbers with arbitrary index profile," IEEE Photonics Technol. Lett. 14, 24-26 (2002).
[Crossref]

Burdge, G. L.

Butler, D. L.

Caponi, R.

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[Crossref]

Chenikov, S. V.

Chernikov , S. V.

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, "Effects of fiber nonlinearities on a long-distance transmission," J. Lightwave Technol. 9, 121-128 (1991).
[Crossref]

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

Ciaramella, E.

M. Artiglia, E. Ciaramella, and B. Sordo, "Using modulation instability to determine Kerr coefficient in optical fibers," Electron. Lett. 31, 1012-1013 (1995).
[Crossref]

Cisterninno, F.

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[Crossref]

Cockings, O.

Cohen, L. G.

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

Desurvire, E.

E. Desurvire, A. Imamoglu, and H. Shaw, "Low-threshold synchronously pumped all-fiber ring Raman laser," J. Lightwave Technol. 5, 89-96 (1987).
[Crossref]

Dianov, E. M.

DiGiovanni, D. J.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Dougherty, D. J.

Durtestse, Y.

M. Monerie and Y. Durtestse, "Direct interferometric measurement of nonlinear refractive index of optical fibers by cross-phase modulation," Electron. Lett. 23, 961-963 (1987).
[Crossref]

Edahiro, T.

N. Shibata, M. Horigudhi, and T. Edahiro, "Raman spectra of binary high-silica glasses and fibers containing GeO2,P2O5 and B2O3," J. Non-Cryst. Solids 45, 115-126 (1981).
[Crossref]

Eggleton, B. J.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Espindola, R. P.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Evangelides, S. G.

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

French, W. G.

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[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, 34-36 (1978).
[Crossref]

Garcia, H.

Gavrilovic, P.

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[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, 34-36 (1978).
[Crossref]

Gisin, N.

C. Vinegoni, M. Wegmuller, and N. Gisin, "Determination of nonlinear coefficient (n2/Aeff) using self-aligned interferometer and Faraday mirror," Electron. Lett. 26, 886-888 (2000).
[Crossref]

Glass, A. M.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Goldhar, J.

Gordon, J. P.

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

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]

Gruner-Nielsen, L.

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

Gruner-Nielson, L.

Gustafson, E. K.

Hamaide, J. P.

L. Prigent and J. P. Hamaide, "Measurement of fiber nonlinear Kerr coefficient by four-wave mixing," IEEE Photonics Technol. Lett. 5, 1092-1095 (1993).
[Crossref]

Hansen, P. B.

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Harvey, G. T.

Hass, M.

M. Hass, "Raman spectra of vitreous silica, germania and sodium silicate glasses," J. Phys. Chem. Solids 31, 415-422 (1970).
[Crossref]

Haus, H. A.

Hayden, C. C.

Heinz, T. F.

Heritage, J. P.

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

Hill, K. O.

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

Horigudhi, M.

N. Shibata, M. Horigudhi, and T. Edahiro, "Raman spectra of binary high-silica glasses and fibers containing GeO2,P2O5 and B2O3," J. Non-Cryst. Solids 45, 115-126 (1981).
[Crossref]

Huang , Y. Y.

Y. Y. Huang and A. Sarkar, "Relationship between composition, density, and refractive index for germaina silica glasses," J. Non-Cryst. Solids 27, 29-37 (1978).
[Crossref]

Imamoglu, A.

E. Desurvire, A. Imamoglu, and H. Shaw, "Low-threshold synchronously pumped all-fiber ring Raman laser," J. Lightwave Technol. 5, 89-96 (1987).
[Crossref]

Ippen, E. P.

Jacobovitz-Veselka, G.

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

Johnson, A. M.

Johnson, D. C.

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

Karasek , M.

Kartner, F. X.

Kato, D.

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

Kato, T.

Kawasaki, B. S.

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

Kim, K. S.

Kortan, A. R.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Levine, A. M.

Levring, O. A.

Lin, C.

R. H. Stolen and C. Lin, "Self-phase modulation in silica optical fibers," Phys. Rev. A 4, 1448-1453 (1978).
[Crossref]

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

Lines, M. E.

J. Bromage, K. Rottwitt, and M. E. Lines, "A method to predict the Raman gain spectra of germanosilicate fbers with arbitrary index profile," IEEE Photonics Technol. Lett. 14, 24-26 (2002).
[Crossref]

Mahberefteh, D.

Manassah , J.

Marcuse, D.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, "Effects of fiber nonlinearities on a long-distance transmission," J. Lightwave Technol. 9, 121-128 (1991).
[Crossref]

Matson, D. W.

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

Menif, M.

Mikkelsen, 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, 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, 262-264 (1994).
[Crossref]

Mollenauer, L. F.

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

Monerie , M.

M. Monerie and Y. Durtestse, "Direct interferometric measurement of nonlinear refractive index of optical fibers by cross-phase modulation," Electron. Lett. 23, 961-963 (1987).
[Crossref]

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, 34-36 (1978).
[Crossref]

Naddeo, C.

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[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, 262-264 (1994).
[Crossref]

Newbury, N. R.

Nielsen, T. N.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Nishimira, M.

Oguama, F. A.

Ozizmir, E.

Palfrey , S. L.

Pedrazzani, J. R.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

Philpotts, J. A.

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

Prigent , L.

L. Prigent and J. P. Hamaide, "Measurement of fiber nonlinear Kerr coefficient by four-wave mixing," IEEE Photonics Technol. Lett. 5, 1092-1095 (1993).
[Crossref]

Reed, W. A.

Roccato, D.

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[Crossref]

Rosenberg, B.

Rottwitt, K.

J. Bromage, K. Rottwitt, and M. E. Lines, "A method to predict the Raman gain spectra of germanosilicate fbers with arbitrary index profile," IEEE Photonics Technol. Lett. 14, 24-26 (2002).
[Crossref]

Roush, T. L.

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

Sarkar, A.

Y. Y. Huang and A. Sarkar, "Relationship between composition, density, and refractive index for germaina silica glasses," J. Non-Cryst. Solids 27, 29-37 (1978).
[Crossref]

Sasaoka, E.

Seikai, S.

Sharma, S. K.

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

Shaw, H.

E. Desurvire, A. Imamoglu, and H. Shaw, "Low-threshold synchronously pumped all-fiber ring Raman laser," J. Lightwave Technol. 5, 89-96 (1987).
[Crossref]

Shibata, N.

N. Shibata, M. Horigudhi, and T. Edahiro, "Raman spectra of binary high-silica glasses and fibers containing GeO2,P2O5 and B2O3," J. Non-Cryst. Solids 45, 115-126 (1981).
[Crossref]

Siegman, A. E.

Simpson, W. M.

Slusher, R. E.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Smith, R. G.

Sordo, B.

M. Artiglia, E. Ciaramella, and B. Sordo, "Using modulation instability to determine Kerr coefficient in optical fibers," Electron. Lett. 31, 1012-1013 (1995).
[Crossref]

Starodmov, A. N.

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[Crossref]

Stentz, A. J.

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Stolen, R. H.

Stolen , R. H.

R. H. Stolen and C. Lin, "Self-phase modulation in silica optical fibers," Phys. Rev. A 4, 1448-1453 (1978).
[Crossref]

Stolen, R. H.

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

Stolen , R. H.

R. H. Stolen and E. P. Ippen, "Raman gain in glass optical waveguides," Appl. Phys. Lett. 22, 276-279 (1972).
[Crossref]

Strasser, T. A.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Suetsugu, Y.

Suzuki, H.

Takachio , N.

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, 262-264 (1994).
[Crossref]

Tasker, G. W.

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

Taylor, J. R.

Thurston, R.

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, "Effects of fiber nonlinearities on a long-distance transmission," J. Lightwave Technol. 9, 121-128 (1991).
[Crossref]

Tokuda, K. L.

Tomlinson, W. J.

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]

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

Torres, I.

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[Crossref]

Trebino, R.

Trivedi, S.

Vinegoni, C.

C. Vinegoni, M. Wegmuller, and N. Gisin, "Determination of nonlinear coefficient (n2/Aeff) using self-aligned interferometer and Faraday mirror," Electron. Lett. 26, 886-888 (2000).
[Crossref]

Wegmuller, M.

C. Vinegoni, M. Wegmuller, and N. Gisin, "Determination of nonlinear coefficient (n2/Aeff) using self-aligned interferometer and Faraday mirror," Electron. Lett. 26, 886-888 (2000).
[Crossref]

Weiner, A. M.

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

White, A. E.

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Zenteno, L. A.

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

I. Torres, A. N. Starodmov, Yu. O. Barmenkov, L. A. Zenteno, and P. Gavrilovic, "Raman effect based modulators for high power fiber lasers," Appl. Phys. Lett. 72, 401-403 (1998).
[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, 34-36 (1978).
[Crossref]

R. H. Stolen and E. P. Ippen, "Raman gain in glass optical waveguides," Appl. Phys. Lett. 22, 276-279 (1972).
[Crossref]

Bell Syst. Tech. J. (1)

A. M. Glass, D. J. DiGiovanni, T. A. Strasser, A. J. Stentz, R. E. Slusher, A. E. White, A. R. Kortan, and B. J. Eggleton, "Advances in fiber optics," Bell Syst. Tech. J. 2000, 168-187.

Electron. Lett. (6)

P. B. Hansen, G. Jacobovitz-Veselka, L. Gruner-Nielsen, and A. J. Stentz, "Raman amplification for loss compensation in dispersion compensating fibre modules," Electron. Lett. 34, 1136-1137 (1998).
[Crossref]

D. C. Johnson, K. O. Hill, B. S. Kawasaki, and D. Kato, "Tunable Raman fiber-optic laser," Electron. Lett. 13, 53 (1977).
[Crossref]

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, 262-264 (1994).
[Crossref]

M. Monerie and Y. Durtestse, "Direct interferometric measurement of nonlinear refractive index of optical fibers by cross-phase modulation," Electron. Lett. 23, 961-963 (1987).
[Crossref]

M. Artiglia, E. Ciaramella, and B. Sordo, "Using modulation instability to determine Kerr coefficient in optical fibers," Electron. Lett. 31, 1012-1013 (1995).
[Crossref]

C. Vinegoni, M. Wegmuller, and N. Gisin, "Determination of nonlinear coefficient (n2/Aeff) using self-aligned interferometer and Faraday mirror," Electron. Lett. 26, 886-888 (2000).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picosecond pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. QE-22, 682-685 (1986).
[Crossref]

IEEE Photonics Technol. Lett. (3)

J. Bromage, K. Rottwitt, and M. E. Lines, "A method to predict the Raman gain spectra of germanosilicate fbers with arbitrary index profile," IEEE Photonics Technol. Lett. 14, 24-26 (2002).
[Crossref]

L. Prigent and J. P. Hamaide, "Measurement of fiber nonlinear Kerr coefficient by four-wave mixing," IEEE Photonics Technol. Lett. 5, 1092-1095 (1993).
[Crossref]

T. N. Nielsen, P. B. Hansen, A. J. Stentz, V. M. Aquari, J. R. Pedrazzani, A. A. Abramov, and R. P. Espindola, "8×10Gb/s 1.3-µm unrepeated transmission over a distance of 141 km with Raman post- and pre-amplifiers," IEEE Photonics Technol. Lett. 10, 1492-1494 (1998).
[Crossref]

J. Lightwave Technol. (8)

E. M. Dianov, "Advances in Raman fibers," J. Lightwave Technol. 20, 1457-1462 (2002).
[Crossref]

M. Karasek and M. Menif, "Channel addition/removal response in Raman fiber amplifiers: modeling and experimentation," J. Lightwave Technol. 20, 1680-1687 (2002).
[Crossref]

N. Takachio and H. Suzuki, "Application of Raman-distributed amplification to WDM transmission systems using 1.55-µm dispersion-shifted fibers," J. Lightwave Technol. 19, 60-69 (2001).
[Crossref]

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, "Effects of fiber nonlinearities on a long-distance transmission," J. Lightwave Technol. 9, 121-128 (1991).
[Crossref]

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

E. Desurvire, A. Imamoglu, and H. Shaw, "Low-threshold synchronously pumped all-fiber ring Raman laser," J. Lightwave Technol. 5, 89-96 (1987).
[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]

S. G. Evangelides, Jr., L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, "Polarization multiplexing with solitons," J. Lightwave Technol. 10, 28-35 (1992).
[Crossref]

J. Non-Cryst. Solids (3)

S. K. Sharma, D. W. Matson, J. A. Philpotts, and T. L. Roush, "Raman study of the structure of glasses along the joint SiO2-GeO2," J. Non-Cryst. Solids 68, 99-114 (1984).
[Crossref]

Y. Y. Huang and A. Sarkar, "Relationship between composition, density, and refractive index for germaina silica glasses," J. Non-Cryst. Solids 27, 29-37 (1978).
[Crossref]

N. Shibata, M. Horigudhi, and T. Edahiro, "Raman spectra of binary high-silica glasses and fibers containing GeO2,P2O5 and B2O3," J. Non-Cryst. Solids 45, 115-126 (1981).
[Crossref]

J. Opt. Soc. Am. B (4)

J. Phys. Chem. Solids (1)

M. Hass, "Raman spectra of vitreous silica, germania and sodium silicate glasses," J. Phys. Chem. Solids 31, 415-422 (1970).
[Crossref]

Opt. Commun. (1)

C. Lin, L. G. Cohen, R. H. Stolen, G. W. Tasker, and W. G. French, "Near-infrared sources in the 1-1.3-µm region by efficient stimulated Raman emission in glass fibers," Opt. Commun. 20, 426-428 (1977).
[Crossref]

Opt. Fiber Technol. (1)

M. Artiglia, R. Caponi, F. Cisterninno, C. Naddeo, and D. Roccato, "A new method for the measurement of the nonlinear refractive index of optical fiber," Opt. Fiber Technol. 2, 75-79 (1996).
[Crossref]

Opt. Lett. (9)

T. Kato, Y. Suetsugu, M. Takagi, E. Sasaoka, and M. Nishimira, "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]

H. Garcia, A. M. Johnson, F. A. Oguama, and S. Trivedi, "A new approach to the measurement of the nonlinear refrac-tive index of short ( <25 m) lengths of silica and erbium-doped fibers," Opt. Lett. 28, 1796-1798 (2003).
[Crossref] [PubMed]

R. Trebino, C. C. Hayden, A. M. Johnson, W. M. Simpson, and A. M. Levine, "Chirp and self-phase modulation in induced-grating autocorrelation measurements of ultrashort pulses," Opt. Lett. 15, 1079-1081 (1990).
[Crossref] [PubMed]

D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, "Measurement of Raman gain spectrum of optical fibers," Opt. Lett. 20, 31-33 (1995).
[Crossref] [PubMed]

D. Mahberefteh, 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).
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N. R. Newbury, "Raman gain:pump-wavelength dependence in single mode fiber," Opt. Lett. 27, 1232-1234 (2002).
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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).
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A. Boskovic, S. V. Chenikov, J. R. Taylor, L. Gruner-Nielson, and O. A. Levring, "Direct measurement of n2 in various types of telecommunication fiber at 1.55 µm," Opt. Lett. 21, 1966-1968 (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]

Phys. Rev. A (1)

R. H. Stolen and C. Lin, "Self-phase modulation in silica optical fibers," Phys. Rev. A 4, 1448-1453 (1978).
[Crossref]

Other (15)

F. A. Oguama, A. Tchouassi, and A. M. Johnson, "Influence of stimulated Raman scattering and high GeO2 doping on the induced grating autocorrelation measurements in Er-Al-Ge doped single mode fibers," presented at Annual Conference of the National Society of Black Physicists, Atlanta, Ga., February 12-15, 2003.

Y. R. Shen and G. Z. Yang, "Theory of self-phase modulation and spectral broadening," in The Supercontinuum Laser Source , R. R. Alfano, ed. (Springer-Verlag, Berlin, 1989), pp. 1-32.

F. A. Oguama, A. M. Johnson, and W. Reed, "Measurement of the nonlinear coefficient of Er-Al-Ge doped fibers as a function of the doping profile, using the photorefractive beam coupling technique," J. Opt. Soc. Am. B, submitted for publication.

F. A. Oguama, A. Tchouassi, and A. M. Johnson, "Effect of high germania content and stimulated Raman scattering on n2 measurements in erbium-doped single mode fibers," in OSA Annual Meeting (Optical Society of America, Washington, D.C., 2002).

F. A. Oguama, "Measurement of the nonlinear refractive index and stimulated Raman scattering in optical fibers as a function of germania content, using the photorefractive beam-coupling technique," Ph.D. thesis (New Jersey Institute of Technology, Newark, N.J., August 2003), Chap. 6, pp. 101-136, http://www.library.njit.edu/etd/index.cfm.

G. P. Agrawal, Nonlinear Fiber Optics - Optics and Photonics , 3rd ed. (Academic, New York, 2001).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

N. Newbury, "Full wavelength dependence of Raman gain in optical fibers: measurement using a single pump laser," in Optical Fiber Communication Conference , Vol. 86 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper WB5.

J. Cordina and C. R. S. Flunger, "Changes in Raman gain coefficient with pump wavelength in modern transmission fibers," in Optical Amplifiers and their Applications , Vol. 92 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper OMC3.

P. V. Mamyshev, "Fibre nonlinearities," in Laser Sources and Applications , A. M. Miller and D. M. Finlayson, eds. (SUSSP Publications, London, 1997), p. 369.

R. H. Stolen, W. A. Reed, K. S. Kim, and K. W. Quoi, "Measurement of optical nonlinearity of transmission fibers," in Technical Digest - Symposium on Optical Fiber Measurements, 1992 , NIST Special Publication 839 (National Institute of Standards and Technology, Boulder, Colo., 1992), pp. 71-75.

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications , I. P. Kaminow and T. L. Koch, eds. (Academic, San Diego, California, 1997), Vol. IIIA, pp. 196-264.

Fiber provided by Dr. Jake Bromage of OFS Laboratories, Holmdel, N.J.

N. Newbury, "Full wavelength dependence of Raman gain in optical fibers: measurement using a single pump laser," in Optical Fiber Communication Conference , Vol. 86 of 2003 OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), paper WB5.

R. H. Stolen, "Nonlinear properties of optical fibers," in Optical Fiber Telecommunications , S. E. Miller and A. G. Chynoweth, eds. (Academic, San Diego, Calif., 1979), pp. 125-150.

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Figures (9)

Fig. 1
Fig. 1

Typical measured IGA trace for a self-phase-modulated Gaussian pulse (after propagation through a 20-m length of fiber, with Aeff8.8 µm2) fitted to the standard IGA model.

Fig. 2
Fig. 2

Typical measured SRS distorted IGA trace at 15% Raman conversion, fitted to the standard IGA model for pure SPM. The experimental data showed clear deviations from the pure SPM model [Er-Al-Ge doped fiber with high (∼28% M) GeO2 content].

Fig. 3
Fig. 3

Experimental setup and beam geometry for the photorefractive four-beam coupling IGA technique.

Fig. 4
Fig. 4

Typical spectra of the output pulse from some of the tested fibers. (a) Fiber B, Er–Al–Ge-doped fiber with high (∼28%M) GeO2 (Leff20 m, Ppeak24 W); (b) fiber C, high Ge-doped fiber with ∼30%M GeO2 (Leff20 m, Ppeak55 W); (c) fiber A, silica core fiber (Leff20 m, Ppeak55 W); and (d) fiber A, silica core fiber (Leff98 m, Ppeak48 W).

Fig. 5
Fig. 5

Four-beam coupling in a photorefractive media. Beams E1 and E3 are coherent, with each other and beam E2 is coherent with beam E4. The index grating consists of two contributions with a common wave vector.

Fig. 6
Fig. 6

Typical measured SRS distorted IGA trace fitted to the modified IGA model [Eq. (23)] for fiber B (Er–Al–Ge doped) with 28%M GeO2; Leff20 m.

Fig. 7
Fig. 7

Typical measured SRS distorted IGA trace fitted to the modified IGA model that accounts for SPM and SRS [Eq. (23)]. Fiber A, silica core fiber; Leff98 m.

Fig. 8
Fig. 8

Plot of the (RGP) versus peak power for fiber B (Er–Al–Ge-doped) with ∼28% M GeO2; Leff20 m. Slope of graph is proportional to gR.

Fig. 9
Fig. 9

Plot of the SPMP versus peak power for fiber B (Er–Al–Ge-doped) with ∼28% M GeO2; Leff20 m. Slope of graph is proportional to n2.

Tables (2)

Tables Icon

Table 1 Values of the Raman Gain Coefficient (gR), the Nonlinear Refractive Index (n2), and the Walk-off Parameter (d) Determined from IGA Measurements at 1064 nm

Tables Icon

Table 2 Literature Values of the Raman Gain Coefficient (gR) and the Nonlinear Refractive Index (n2)

Equations (43)

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χ(3)(Δω)=χNR(3)+χR(3)(Δω).
gR(Δω)=Im[χR(3)(Δω)] 4πωocno.
n2=2πno{χNR(3)+Re[χR(3)(0)]}.
|κ(τ)|2=-E(t)E*(t+τ)dt2.
E(t)=E0 exp-2 ln 2tτp2-i[ω0t+ϕ(t)].
ϕ(t)=ω0τpQ exp-4 ln 2tτp2,
Apz+i2β2p 2ApT2+αp2Ap
=iγp[|Ap|2+(2-fR)|As|2]Ap-gp2|As|2Ap,
Asz+i2β2s 2AsT2+αs2As-d AsT
=iγs[|As|2+(2-fR)|Ap|2]As+gs2|Ap|2As,
Apz=iγp[|Ap|2+(2-fR)|As|2]Ap-gp2|As|2Ap,
Asz-d AsT
=iγs[|As|2+(2-fR)|Ap|2]As+gs2|Ap|2As.
Apz=iγp|Ap|2Ap,
Asz-d AsT=γs(2-fR)|Ap|2As+gs2|Ap|2As.
Ap(z, T)=Ap(0, T)exp[iγp|Ap(0, T)|2z],
As(z, T)=As(0, T+zd)×expgs2+iγs(2-fR)ψ(z, T),
ψ(z, T)=0z|Ap(0, T+zd-zd)|2dz.
Ap(0, T)=Pm exp-2 ln 2T2τp2-i(ωt+ϕ),
ψ(z, T)=14Pmπln 21/2zδ{erf[2ln 2(τ+δ)]-erf[2ln 2(τ)]},
Ej=Ej(t)exp[i(ωjt-kjr)],
ω1=ω3,ω2=ω4,
k1-k3=k4-k2=K,
G(t, τ)-tE1(t)E3*(t+τ)dt+-tE2(t)×E4*(t+τ)dt,
EdifG(t, τ)Epr(t+τ),
Epr(t+τ)=Ep(t+τ)+ER(t+τ),
Wdet(τ)-|Edet(t+τ)|2dt.
Edet(t+τ)E(t)+Edif(t, τ),
E(t)=Ep(t)+ER(t).
Wdet(τ)-{|E(t)|2+|Edif(t)|2+2 Re[E*(t)Edif(t, τ)]}dt.
Wdet(τ)-2 Re[E*(t)Edif(t, τ)]dt.
Wdet(τ)-2 Re[Ep*(t)+ER*(t)]-t[Ep(t)Ep*(t+τ)+ER(t)ER*(t+τ)]dt[Ep(t+τ)+ER(t+τ)]dt
Wdet(τ)-Ep(t)Ep*(t+τ)dt2+-ER(t)ER*(t+τ)dt2+-ER(t)ER*(t+τ)dt×-Ep(t)Ep*(t+τ)dt+2 Re-Ep(t)ER(t+τ)dt×-Ep(t)Ep*(t+τ)dt+-Ep*(t)ER(t+τ)dt×-ER(t)ER*(t+τ)dt.
E(z, t)=[exp i(ω0t-k0r)]F(x, y)A(z, t),
SPMP=γpz|Ap(0, T)|2,
RGP=gz|Ap(0, T)|2,
WOTδ=zdτp,
E02=8πI0nc,
Pm=2ln 2π1/2 PavgtRτp,
n2(esu)=Aeff ncλLeff 32π2ln 2π1/2×107-1 SPMPPpeak,
gR(esu)=Aeff4π·ncLeff×32π2ln 2π1/2×107-1 RGPPpeak,
gR(λ, λs)=λpλgR(λp, λs),
gR(cm/W)=0.94×10-11λp(1+80Δ),

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