Abstract

We investigate and discuss stimulated Brillouin scattering in optical fibers for the generation of millimeter waves in theory and experiment. With a derivation of the responsible differential equation system we show that the phases of two independently amplified sidebands of a frequency comb will remain the same. Neither third-order nonlinear effects like self- and cross-phase modulation nor the Brillouin amplification has an influence on the phases. We verify our theoretical predictions with phase noise measurements of the generated millimeter-wave signal. The results show that the generated phase noise is in fact very low.

© 2006 Optical Society of America

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  1. A. Hirata, M. Harada, and T. Nagatsuma, "120-GHz wireless link using photonic techniques for generation, modulation, and emission of millimeter-wave signals," J. Lightwave Technol. 21, 2145-2153 (2003).
    [Crossref]
  2. P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).
    [Crossref] [PubMed]
  3. E. E. Narimanov and P. Mitra, "The channel capacity of a fiber optics communication system: perturbation theory," J. Lightwave Technol. 20, 530-537 (2002).
    [Crossref]
  4. C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
    [Crossref]
  5. S. A. Pappert, R. Helkey, and R. Logan, Jr., "Photonics link techniques for microwave frequency conversion," in RF Photonics Technology in Optical Fiber Links, W.Chang, ed. (Cambridge U. Press, 2002), Chap. 10.
    [Crossref]
  6. W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.
  7. K. Noguchi, O. Mitomi, and H. Miyazawa, "Millimeter-wave Ti:LiNbO3 optical modulators," J. Lightwave Technol. 16, 615-619 (1998).
    [Crossref]
  8. G. J. Meslener, "Chromatic dispersion induced distortion of modulated monochromatic light employing direct detection," IEEE J. Quantum Electron. QE-20, 1208-1216 (1984).
    [Crossref]
  9. H. Sotobayashi and K. Kitayama, "Cancellation of the signal fading for 60GHz subcarrier multiplexed optical DSB signal transmission in nondispersion shifted fiber using midway optical phase conjugation," J. Lightwave Technol. 17, 2488-2497 (1999).
    [Crossref]
  10. J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).
  11. G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
    [Crossref]
  12. X. S. Yao, "High-quality microwave signal generation by use of Brillouin scattering in optical fibers," Opt. Lett. 22, 1329-1331 (1997).
    [Crossref]
  13. X. S. Yao, "Brillouin selective sideband amplification of microwave photonic signals," IEEE Photon. Technol. Lett. 10, 138-140 (1998).
    [Crossref]
  14. X. S. Yao, "Phase-to-amplitude modulation conversion using Brillouin selective sideband amplification," IEEE Photon. Technol. Lett. 10, 264-266 (1998).
    [Crossref]
  15. L. A. Johansson and A. J. Seeds, "Generation and transmission of millimeter-wave data-modulated optical signals using an optical injection phase-lock loop," J. Lightwave Technol. 21, 511-520 (2003).
    [Crossref]
  16. Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
    [Crossref]
  17. R. T. Ramos and A. J. Seeds, "Fast heterodyne optical phase-lock loop using double quantum well laser diodes," Electron. Lett. 28, 82-83 (1992).
    [Crossref]
  18. D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
    [Crossref]
  19. L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
    [Crossref]
  20. R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
    [Crossref]
  21. R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
    [Crossref]
  22. D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
    [Crossref]
  23. T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
    [Crossref]
  24. T. Tanemura, Y. Takushima, and K. Kikuchi, "Narrowband optical filter, with a variable transmission spectrum, using stimulated Brillouin scattering in optical fiber," Opt. Lett. 27, 1552-1554 (2002).
    [Crossref]
  25. D. Cotter, "Stimulated Brillouin scattering in monomode optical fiber," J. Opt. Commun. 4, 10-19 (1983).
    [Crossref]
  26. A. R. Chraplyvy, "Limitations on lightwave communications imposed by optical-fiber Nonlinearities," J. Lightwave Technol. 8, 1548-1557 (1990).
    [Crossref]
  27. R. W. Tkach and A. R. Chraplyvy, "Fibre Brillouin amplifiers," Opt. Quantum Electron. 21, S105-S112 (1989).
    [Crossref]
  28. N. A. Olsson and J. P. van der Ziel, "Characterization of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth," J. Lightwave Technol. LT-5, 147-153 (1987).
    [Crossref]
  29. M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
    [Crossref]
  30. Y. Aoki and K. Tajima, "Stimulated Brillouin scattering in a long single-mode fiber excited with a multimode pump laser," J. Opt. Soc. Am. B 5, 358-363 (1988).
    [Crossref]
  31. Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
    [Crossref]
  32. T. Horiguchi, M. Tateda, N. Shibata, and Y. Azuma, "Brillouin gain variation due to a polarization-state change of the pump or Stokes fields in standard single-mode fibers," Opt. Lett. 14, 329-331 (1989).
    [Crossref] [PubMed]
  33. G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).
  34. C. L. Tang, "Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process," J. Appl. Phys. 37, 2945-2955 (1966).
    [Crossref]
  35. T. Schneider, Nonlinear Optics in Telekommunications (Springer, 2004).
  36. A. Loayssa, D. Benito, and M. J. Garde, "Applications of optical carrier Brillouin processing to microwave photonics," Opt. Fiber Technol. 8, 24-42 (2002).
    [Crossref]
  37. Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
    [Crossref]
  38. X. S. Yao, "Polarization insensitive antenna remoting link with frequency conversion gain," IEEE Photon. Technol. Lett. 12, 1382-1384 (2000).
    [Crossref]
  39. M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
    [Crossref]

2006 (1)

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

2005 (1)

Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
[Crossref]

2004 (1)

T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
[Crossref]

2003 (2)

2002 (3)

2001 (1)

P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).
[Crossref] [PubMed]

2000 (1)

X. S. Yao, "Polarization insensitive antenna remoting link with frequency conversion gain," IEEE Photon. Technol. Lett. 12, 1382-1384 (2000).
[Crossref]

1999 (1)

1998 (5)

K. Noguchi, O. Mitomi, and H. Miyazawa, "Millimeter-wave Ti:LiNbO3 optical modulators," J. Lightwave Technol. 16, 615-619 (1998).
[Crossref]

Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
[Crossref]

X. S. Yao, "Brillouin selective sideband amplification of microwave photonic signals," IEEE Photon. Technol. Lett. 10, 138-140 (1998).
[Crossref]

X. S. Yao, "Phase-to-amplitude modulation conversion using Brillouin selective sideband amplification," IEEE Photon. Technol. Lett. 10, 264-266 (1998).
[Crossref]

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

1997 (2)

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[Crossref]

X. S. Yao, "High-quality microwave signal generation by use of Brillouin scattering in optical fibers," Opt. Lett. 22, 1329-1331 (1997).
[Crossref]

1996 (1)

C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
[Crossref]

1995 (2)

D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[Crossref]

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

1994 (1)

M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
[Crossref]

1993 (1)

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

1992 (2)

R. T. Ramos and A. J. Seeds, "Fast heterodyne optical phase-lock loop using double quantum well laser diodes," Electron. Lett. 28, 82-83 (1992).
[Crossref]

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

1990 (1)

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

1989 (2)

1988 (2)

Y. Aoki and K. Tajima, "Stimulated Brillouin scattering in a long single-mode fiber excited with a multimode pump laser," J. Opt. Soc. Am. B 5, 358-363 (1988).
[Crossref]

Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
[Crossref]

1987 (1)

N. A. Olsson and J. P. van der Ziel, "Characterization of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth," J. Lightwave Technol. LT-5, 147-153 (1987).
[Crossref]

1984 (1)

G. J. Meslener, "Chromatic dispersion induced distortion of modulated monochromatic light employing direct detection," IEEE J. Quantum Electron. QE-20, 1208-1216 (1984).
[Crossref]

1983 (2)

L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[Crossref]

D. Cotter, "Stimulated Brillouin scattering in monomode optical fiber," J. Opt. Commun. 4, 10-19 (1983).
[Crossref]

1966 (1)

C. L. Tang, "Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process," J. Appl. Phys. 37, 2945-2955 (1966).
[Crossref]

Ahmed, Z.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[Crossref]

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

Ammann, M. J.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

Aoki, Y.

Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
[Crossref]

Y. Aoki and K. Tajima, "Stimulated Brillouin scattering in a long single-mode fiber excited with a multimode pump laser," J. Opt. Soc. Am. B 5, 358-363 (1988).
[Crossref]

Azuma, Y.

Benito, D.

A. Loayssa, D. Benito, and M. J. Garde, "Applications of optical carrier Brillouin processing to microwave photonics," Opt. Fiber Technol. 8, 24-42 (2002).
[Crossref]

Bowers, J. E.

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Braun, R.-P.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

Chang, D. H.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Chen, K.

Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
[Crossref]

Chraplyvy, A. R.

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

R. W. Tkach and A. R. Chraplyvy, "Fibre Brillouin amplifiers," Opt. Quantum Electron. 21, S105-S112 (1989).
[Crossref]

Cotter, D.

D. Cotter, "Stimulated Brillouin scattering in monomode optical fiber," J. Opt. Commun. 4, 10-19 (1983).
[Crossref]

Dagenais, M.

Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
[Crossref]

Dalton, L. R.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Davies, P. A.

D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[Crossref]

Derickson, D. J.

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Erlig, H.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Fan, Z. F.

Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
[Crossref]

Ferreira, M. F.

M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
[Crossref]

Fetterman, H. R.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Garde, M. J.

A. Loayssa, D. Benito, and M. J. Garde, "Applications of optical carrier Brillouin processing to microwave photonics," Opt. Fiber Technol. 8, 24-42 (2002).
[Crossref]

Goldberg, L.

L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[Crossref]

Grosskopf, G.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

Hannover, D.

T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
[Crossref]

Harada, M.

He, G. S.

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).

Heidemann, R.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

Heim, P. J.

Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
[Crossref]

Helkey, R.

S. A. Pappert, R. Helkey, and R. Logan, Jr., "Photonics link techniques for microwave frequency conversion," in RF Photonics Technology in Optical Fiber Links, W.Chang, ed. (Cambridge U. Press, 2002), Chap. 10.
[Crossref]

Helkey, R. J.

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Hirata, A.

Hofstetter, R.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

Horiguchi, T.

Johansson, L. A.

Junker, M.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
[Crossref]

Kikuchi, K.

Kitayama, K.

Klinger, J.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

Lane, P. M.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

Lauterbach, K. U.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

Lima, C. R.

D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[Crossref]

Liu, S. H.

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).

Loayssa, A.

A. Loayssa, D. Benito, and M. J. Garde, "Applications of optical carrier Brillouin processing to microwave photonics," Opt. Fiber Technol. 8, 24-42 (2002).
[Crossref]

Logan, R.

S. A. Pappert, R. Helkey, and R. Logan, Jr., "Photonics link techniques for microwave frequency conversion," in RF Photonics Technology in Optical Fiber Links, W.Chang, ed. (Cambridge U. Press, 2002), Chap. 10.
[Crossref]

Mars, A.

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Meslener, G. J.

G. J. Meslener, "Chromatic dispersion induced distortion of modulated monochromatic light employing direct detection," IEEE J. Quantum Electron. QE-20, 1208-1216 (1984).
[Crossref]

Mito, I.

Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
[Crossref]

Mitomi, O.

Mitra, P.

Mitra, P. P.

P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).
[Crossref] [PubMed]

Miyazawa, H.

Nagatsuma, T.

Narimanov, E. E.

Noguchi, K.

Novak, D.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[Crossref]

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

Oh, M.-C.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Olsson, N. A.

N. A. Olsson and J. P. van der Ziel, "Characterization of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth," J. Lightwave Technol. LT-5, 147-153 (1987).
[Crossref]

Orazi, R. J.

C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
[Crossref]

O'Reilly, J. J.

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

Pappert, S. A.

C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
[Crossref]

S. A. Pappert, R. Helkey, and R. Logan, Jr., "Photonics link techniques for microwave frequency conversion," in RF Photonics Technology in Optical Fiber Links, W.Chang, ed. (Cambridge U. Press, 2002), Chap. 10.
[Crossref]

Pinto, J. L.

M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
[Crossref]

Ramos, R. T.

R. T. Ramos and A. J. Seeds, "Fast heterodyne optical phase-lock loop using double quantum well laser diodes," Electron. Lett. 28, 82-83 (1992).
[Crossref]

Rocha, J. F.

M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
[Crossref]

Rohde, D.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

Schmidt, F.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

Schneider, T.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
[Crossref]

T. Schneider, Nonlinear Optics in Telekommunications (Springer, 2004).

Schwarzbacher, A. T.

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

Seeds, A. J.

Shen, Y.

Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
[Crossref]

Shibata, N.

Smith, G. H.

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[Crossref]

Sotobayashi, H.

Stark, J. B.

P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).
[Crossref] [PubMed]

Steier, W. H.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Sun, C. K.

C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
[Crossref]

Szep, A.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Tajima, K.

Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
[Crossref]

Y. Aoki and K. Tajima, "Stimulated Brillouin scattering in a long single-mode fiber excited with a multimode pump laser," J. Opt. Soc. Am. B 5, 358-363 (1988).
[Crossref]

Takushima, Y.

Tanemura, T.

Tang, C. L.

C. L. Tang, "Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process," J. Appl. Phys. 37, 2945-2955 (1966).
[Crossref]

Tateda, M.

Taylor, H. F.

L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[Crossref]

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, "Fibre Brillouin amplifiers," Opt. Quantum Electron. 21, S105-S112 (1989).
[Crossref]

Tsap, B.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Tucker, R. S.

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

van der Ziel, J. P.

N. A. Olsson and J. P. van der Ziel, "Characterization of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth," J. Lightwave Technol. LT-5, 147-153 (1987).
[Crossref]

Wake, D.

D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[Crossref]

Wasserbauer, J. G.

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Waterhouse, R. B.

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

Weller, J. F.

L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[Crossref]

Yao, X. S.

X. S. Yao, "Polarization insensitive antenna remoting link with frequency conversion gain," IEEE Photon. Technol. Lett. 12, 1382-1384 (2000).
[Crossref]

X. S. Yao, "Brillouin selective sideband amplification of microwave photonic signals," IEEE Photon. Technol. Lett. 10, 138-140 (1998).
[Crossref]

X. S. Yao, "Phase-to-amplitude modulation conversion using Brillouin selective sideband amplification," IEEE Photon. Technol. Lett. 10, 264-266 (1998).
[Crossref]

X. S. Yao, "High-quality microwave signal generation by use of Brillouin scattering in optical fibers," Opt. Lett. 22, 1329-1331 (1997).
[Crossref]

Zhang, C.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Zhang, H.

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

Zhang, X.

Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
[Crossref]

Electron. Lett. (4)

J. J. O'Reilly, P. M. Lane, R. Heidemann, and R. Hofstetter, "Optical generation of very narrow linewidth millimeter wave signals," Electron. Lett. 28, 2024-2025 (1992).

R. T. Ramos and A. J. Seeds, "Fast heterodyne optical phase-lock loop using double quantum well laser diodes," Electron. Lett. 28, 82-83 (1992).
[Crossref]

L. Goldberg, H. F. Taylor, and J. F. Weller, "Microwave signal generation with injection-locked laser diodes," Electron. Lett. 19, 491-493 (1983).
[Crossref]

T. Schneider, M. Junker, and D. Hannover, "Generation of millimeter wave signals by stimulated Brillouin scattering for radio over fibre systems," Electron. Lett. 40, 1500-1502 (2004).
[Crossref]

IEEE J. Quantum Electron. (1)

G. J. Meslener, "Chromatic dispersion induced distortion of modulated monochromatic light employing direct detection," IEEE J. Quantum Electron. QE-20, 1208-1216 (1984).
[Crossref]

IEEE Photon. Technol. Lett. (7)

C. K. Sun, R. J. Orazi, and S. A. Pappert, "Efficient microwave frequency conversion using photonic link signal mixing," IEEE Photon. Technol. Lett. 8, 154-156 (1996).
[Crossref]

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, "Low-phase-noise millimeter-wave generation at 64GHz and data transmission using optical sideband injection locking," IEEE Photon. Technol. Lett. 10, 728-730 (1998).
[Crossref]

X. S. Yao, "Brillouin selective sideband amplification of microwave photonic signals," IEEE Photon. Technol. Lett. 10, 138-140 (1998).
[Crossref]

X. S. Yao, "Phase-to-amplitude modulation conversion using Brillouin selective sideband amplification," IEEE Photon. Technol. Lett. 10, 264-266 (1998).
[Crossref]

Z. F. Fan, P. J. S. Heim, and M. Dagenais, "Highly coherent RF signal generation by optical phase locking of external cavity semiconductor lasers," IEEE Photon. Technol. Lett. 10, 719-721 (1998).
[Crossref]

Y. Shen, X. Zhang, and K. Chen, "Optical single sideband modulation of 11-GHz RoF system using stimulated Brillouin scattering," IEEE Photon. Technol. Lett. 17, 1277-1279 (2005).
[Crossref]

X. S. Yao, "Polarization insensitive antenna remoting link with frequency conversion gain," IEEE Photon. Technol. Lett. 12, 1382-1384 (2000).
[Crossref]

IEEE Trans. Microwave Theory Tech. (4)

M. Junker, M. J. Ammann, A. T. Schwarzbacher, J. Klinger, K. U. Lauterbach, and T. Schneider, "A comparative test of Brillouin amplification and erbium doped fiber amplification for the generation of millimeter waves with low phase noise properties," IEEE Trans. Microwave Theory Tech. 54, 1576-1581 (2006).
[Crossref]

D. Novak, Z. Ahmed, R. B. Waterhouse, and R. S. Tucker, "Signal generation using pulsed semiconductor lasers for application in millimeter-wave wireless links," IEEE Trans. Microwave Theory Tech. 43, 2257-2262 (1995).
[Crossref]

G. H. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion effects in fiber wireless systems incorporating external modulators," IEEE Trans. Microwave Theory Tech. 45, 1410-1415 (1997).
[Crossref]

D. Wake, C. R. Lima, and P. A. Davies, "Optical generation of millimeter-wave signals for fiber-radio systems using a dual-mode DFB semiconductor laser," IEEE Trans. Microwave Theory Tech. 43, 2270-2276 (1995).
[Crossref]

J. Appl. Phys. (1)

C. L. Tang, "Saturation and spectral characteristics of the Stokes emission in the stimulated Brillouin process," J. Appl. Phys. 37, 2945-2955 (1966).
[Crossref]

J. Lightwave Technol. (8)

Y. Aoki, K. Tajima, and I. Mito, "Input power limits of single-mode optical fibers due to stimulated Brillouin scattering in optical communication systems," J. Lightwave Technol. 6, 710-719 (1988).
[Crossref]

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

N. A. Olsson and J. P. van der Ziel, "Characterization of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth," J. Lightwave Technol. LT-5, 147-153 (1987).
[Crossref]

L. A. Johansson and A. J. Seeds, "Generation and transmission of millimeter-wave data-modulated optical signals using an optical injection phase-lock loop," J. Lightwave Technol. 21, 511-520 (2003).
[Crossref]

A. Hirata, M. Harada, and T. Nagatsuma, "120-GHz wireless link using photonic techniques for generation, modulation, and emission of millimeter-wave signals," J. Lightwave Technol. 21, 2145-2153 (2003).
[Crossref]

H. Sotobayashi and K. Kitayama, "Cancellation of the signal fading for 60GHz subcarrier multiplexed optical DSB signal transmission in nondispersion shifted fiber using midway optical phase conjugation," J. Lightwave Technol. 17, 2488-2497 (1999).
[Crossref]

E. E. Narimanov and P. Mitra, "The channel capacity of a fiber optics communication system: perturbation theory," J. Lightwave Technol. 20, 530-537 (2002).
[Crossref]

K. Noguchi, O. Mitomi, and H. Miyazawa, "Millimeter-wave Ti:LiNbO3 optical modulators," J. Lightwave Technol. 16, 615-619 (1998).
[Crossref]

J. Opt. Commun. (1)

D. Cotter, "Stimulated Brillouin scattering in monomode optical fiber," J. Opt. Commun. 4, 10-19 (1983).
[Crossref]

J. Opt. Soc. Am. B (1)

Microwave Opt. Technol. Lett. (1)

R. J. Helkey, D. J. Derickson, A. Mars, J. G. Wasserbauer, and J. E. Bowers, "Millimeter-wave signal generation using semiconductor diode lasers," Microwave Opt. Technol. Lett. 6, 1-5 (1993).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, "Nonlinear limits to the information capacity of optical fibre communications," Nature 411, 1027-1030 (2001).
[Crossref] [PubMed]

Opt. Fiber Technol. (1)

A. Loayssa, D. Benito, and M. J. Garde, "Applications of optical carrier Brillouin processing to microwave photonics," Opt. Fiber Technol. 8, 24-42 (2002).
[Crossref]

Opt. Lett. (3)

Opt. Quantum Electron. (2)

M. F. Ferreira, J. F. Rocha, and J. L. Pinto, "Analysis of the gain and noise characteristics of fibre Brillouin amplifiers," Opt. Quantum Electron. 26, 35-44 (1994).
[Crossref]

R. W. Tkach and A. R. Chraplyvy, "Fibre Brillouin amplifiers," Opt. Quantum Electron. 21, S105-S112 (1989).
[Crossref]

Other (4)

G. S. He and S. H. Liu, Physics of Nonlinear Optics (World Scientific, 1999).

T. Schneider, Nonlinear Optics in Telekommunications (Springer, 2004).

S. A. Pappert, R. Helkey, and R. Logan, Jr., "Photonics link techniques for microwave frequency conversion," in RF Photonics Technology in Optical Fiber Links, W.Chang, ed. (Cambridge U. Press, 2002), Chap. 10.
[Crossref]

W. H. Steier, M.-C. Oh, H. Zhang, A. Szep, L. R. Dalton, C. Zhang, H. R. Fetterman, D. H. Chang, H. Erlig, and B. Tsap, "Recent advances in low voltage, high frequency polymer electro-optic modulators," in Optical Fiber Communications Conference, Vol. 54 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2001), MJ1-1-MJ1-3.

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Figures (5)

Fig. 1
Fig. 1

Principle setup for the generation of mm waves with Brillouin scattering (SSME, standard single-mode fiber; C, circulator).

Fig. 2
Fig. 2

Arrangement of the different frequencies and wave vectors.

Fig. 3
Fig. 3

Normalized real and imaginary parts of the Brillouin gain depending on the frequency shift between the sideband and the maximum of the Brillouin gain ( ω S 1 , 2 ω S 1 , 2 max ) 2 π for v a = 5.96 km s and α a 4027 m 1 .

Fig. 4
Fig. 4

Generated 39.199989 GHz signal at the output of the photodiode measured with an electrical spectrum analyzer. The inset shows the power variation of the signal over a time of 4 h .

Fig. 5
Fig. 5

Measured phase noise of the electrical generator at 10 GHz and of the generated 39.199989 GHz mm wave.

Equations (40)

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

f S max = 2 f P n v a c ,
E ( z , t ) = 1 2 [ i = 1 , 2 E P i ( z ) e j ( ω P i t k P i z ) + c.c. + E S i ( z ) e j ( ω S i t + k S i z ) + c.c. ] ,
ω a = ω P 1 ω S 1 = ω P 2 ω S 2 .
k a = k P 1 k S 1 = k P 2 k S 2
ρ ( z , t ) = 1 2 [ Q ( z ) e j ( ω a t k a z ) + c.c. ] .
2 E z 2 n 2 c 2 2 E t 2 α n c E t = 1 ϵ 0 c 2 2 P NL t 2 ,
P NL = P NL SBS + P NL χ ( 3 ) = ϵ 0 γ e ρ 0 ρ E + ϵ 0 χ ( 3 ) E E E ,
2 E z 2 n 2 c 2 2 E t 2 α n c E t = 1 c 2 2 t 2 ( γ e ρ 0 ρ E + χ ( 3 ) E E E ) .
2 ρ z 2 α a v a ρ t 1 v a 2 2 ρ t 2 = ϵ 0 γ e 2 v a 2 2 z 2 ( E ) 2 ,
E P 1 z + α 2 E P 1 = j C P 1 Q E S 1 e j Δ k 1 z j D P 1 [ SXPM P 1 + FWM P 1 ] ,
E S 1 z α 2 E S 1 = j C S 1 Q * E P 1 e j Δ k 1 z + j D S 1 [ SXPM S 1 + FWM S 1 ] ,
E P 2 z + α 2 E P 2 = j C P 2 Q E S 2 e j Δ k 2 z j D P 2 [ SXPM P 2 + FWM P 2 ] ,
E S 2 z α 2 E S 2 = j C S 2 Q * E P 2 e j Δ k 2 z + j D S 2 [ SXPM S 2 + FWM S 2 ] ,
SXPM P 1 , 2 = ( E P 1 , 2 2 + 2 E P 2 , 1 2 + 2 E S 1 2 + 2 E S 2 2 ) E P 1 , 2 ,
SXPM S 1 , 2 = ( E S 1 , 2 2 + 2 E S 2 , 1 2 + 2 E P 1 2 + 2 E P 2 2 ) E S 1 , 2 .
FWM P 1 , S 2 = 2 E S 1 E P 2 E S 2 , P 1 * e j Δ k FWM z ,
FWM P 2 , S 1 = 2 E P 1 E S 2 E S 1 , P 2 * e j Δ k FWM z .
Δ k 1 , 2 = k P 1 , 2 + k S 1 , 2 k a ,
Δ k FWM = k S 1 k P 2 k S 2 + k P 1 .
Δ k FWM = n c ( ω S 1 ω S 2 + ω P 1 ω P 2 ) = 2 n c ω RF .
Q z + α 2 Q = j C 3 ( E P 1 E S 1 * e j Δ k 1 z + E P 2 E S 2 * e j Δ k 2 z ) ,
Q ( z ) = j C 3 [ 0 z E P 1 ( z ) E S 1 * ( z ) e j Δ k 1 z α a ( z z ) 2 d z + 0 z E P 2 ( z ) E S 2 * ( z ) e j Δ k 2 z α a ( z z ) 2 d z ] + Q ( 0 ) e α a z 2 .
Q ( z ) = j C 3 [ E P 1 E S 1 * 1 α a 2 j Δ k 1 ( e j Δ k 1 z e α a z 2 ) + E P 2 E S 2 * 1 α a 2 j Δ k 2 ( e j Δ k 2 z e α a z 2 ) ] + Q ( 0 ) e α a z 2 .
Q ( z ) = j C 3 ( Δ k e 1 E P 1 E S 1 * e j Δ k 1 z + Δ k e 2 E P 2 E S 2 * e j Δ k 2 z ) ,
E P 1 z + α 2 E P 1 = C P 1 C 3 [ Δ k e 1 E P 1 E S 1 * 2 + Δ k e 2 E P 2 E S 2 * E S 1 e j ( Δ k 1 Δ k 2 ) z ] j D P 1 [ SXPM P 1 ] .
g B P 1 = ω P 1 2 γ e 2 k a 4 n c 3 ρ 0 v a 2 k p ,
γ P 1 = 3 χ ( 3 ) ω P 1 4 c 2 ε 0 n 2 A eff .
E P 1 z = [ g B 2 A eff Δ k e R 1 P S 1 + α 2 ] E P 1
j { g B 2 A eff Δ k e I 1 P S 1 + γ ( P P 1 + 2 P S 1 + 2 P P 2 + 2 P S 2 ) } E P 1 ,
E S 1 z = [ g B 2 A eff Δ k e R 1 P P 1 α 2 ] E S 1
+ j { g B 2 A eff Δ k e I 1 P P 1 + γ ( P S 1 + 2 P P 1 + 2 P P 2 + 2 P S 2 ) } E S 1 ,
E P 2 z = [ g B 2 A eff Δ k e R 2 P S 2 + α 2 ] E P 2
j { g B 2 A eff Δ k e I 2 P S 2 + γ ( P P 2 + 2 P P 1 + 2 P S 1 + 2 P S 2 ) } E P 2 ,
E S 2 z = [ g B 2 A eff Δ k e R 2 P P 2 α 2 ] E S 2
+ j { g B 2 A eff Δ k e I 2 P P 2 + γ ( P S 2 + 2 P P 1 + 2 P P 2 + 2 P S 1 ) } E S 2 ,
Δ k e R 1 , 2 = α a 2 ( α a 2 ) 2 + ( Δ k 1 , 2 ) 2 ,
Δ k e I 1 , 2 = Δ k 1 , 2 ( α a 2 ) 2 + ( Δ k 1 , 2 ) 2 .
Δ k 1 , 2 = k P 1 , 2 + k S 1 , 2 k a 2 ω P 1 , 2 n c ω P 1 , 2 ω ¯ S 1 , 2 v a = 1 v a [ ω S 1 , 2 ω S 1 , 2 max ] .
Δ k e R 1 , 2 = v a 2 α a 2 ( v a α a 2 ) 2 + ( ω S 1 , 2 ω S 1 , 2 max ) 2
Δ k e I 1 , 2 = v a ( ω S 1 , 2 ω S 1 , 2 max ) ( v a α a 2 ) 2 + ( ω S 1 , 2 ω S 1 , 2 max ) 2

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