Abstract

For weakly guiding noncircular single-mode waveguides, we definethe generalized frequency v and find that the fourparameters [v, b(v), d(vb)/dv, vd2(vb)/dv2]play an important role in the theory of noncircular waveguides. Wealso derive the relations between the Laplacian mode-field half-width(MFHW) and the propagation constant, normalized propagationconstant, group delay time/unit length, composite materialdispersion, composite waveguide dispersion, composite profiledispersion, modal birefringence, beat length, polarization-modedispersion, Rayleigh scattering loss, and MFHWW. We also derive Brown’s identity by anew and simple method. Finally, we use our analytical theory todesign a new periodic fiber system.

© 1997 Optical Society of America

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  3. K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
    [Crossref]
  4. A. H. Liang, C. C. Fan, “Mode radius of noncircular field single-mode fiber: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
    [Crossref]
  5. A. H. Liang, “Microbending losses for noncircular weakly-guiding single-mode fibers,” IEEE J. Quantum Electron. 28, 1429–1434 (1992).
    [Crossref]
  6. A. H. Liang, “Relation between transition loss and mode-field half-width for curved noncircular weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 31, 2075–2077 (1995).
    [Crossref]
  7. A. H. Liang, “The Rayleigh scattering loss of isotropic weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 28, 1844–1847 (1992).
    [Crossref]
  8. A. H. Liang, H. K. Tsang, “Transverse offset loss between two identical noncircular single-mode waveguides,” Appl. Opt. 34, 5343–5350 (1995).
    [Crossref] [PubMed]
  9. C. C. Fan, A. H. Liang, “Splice loss between different Gaussian-elliptic-field single-mode fibres,” J. Lightwave Technol. 8, 173–176 (1990).
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  10. A. H. Liang, C. C. Fan, “Mode field radius definition for single-mode planar waveguide with arbitrary index profile and its application to splice loss evaluation,” Electron. Lett. 24, 1011–1013 (1988).
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  11. A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
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  12. G. A. Evans, J. M. Hammer, Surface Emitting Semiconductor Lasers and Arrays (Academic, Boston, 1993).
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  15. A. H. Liang, “New diffraction-limited criterion,” submitted to IEEE J. Quantum Electron.
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    [Crossref]
  21. R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
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  22. V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
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  23. A. Bjarklev, Optical Fiber Amplifiers: Design and System Applications (Artech House, Norwood, Mass., 1993).
  24. B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. 4, 967–979 (1986).
    [Crossref]
  25. C. Saravanos, R. S. Lowe, “The measurement of non-Gaussian mode-fields by the far-field axial scanning technique,” J. Lightwave Technol. 4, 1563–1566 (1986).
    [Crossref]
  26. D. Marcuse, “Single-channel operation in very long nonlinear fibers with optical amplifiers at zero dispersion,” J. Lightwave Technol. 9, 356–361 (1991).
    [Crossref]
  27. N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
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  28. A. W. Snyder, F. Ruhl, “Novel polarization phenomena on anisotropic multimode fibres,” Electron. Lett. 19, 401–402 (1983).
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  29. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chaps. 18, 19, 28, and 29.
  30. C. D. Hussey, “Field to dispersion relationships in single-mode fibers,” Electron. Lett. 20, 1051–1052 (1984).
    [Crossref]
  31. C. D. Hussey, F. Martinez, “Approximate analytic forms for propagation characteristics of single-mode optical fibers,” Electron. Lett. 21, 1103–1104 (1985).
    [Crossref]
  32. D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
    [Crossref] [PubMed]
  33. W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).
  34. A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
    [Crossref]
  35. M. Nakazawa, “Theory of backward Rayleigh scattering in polarization-maintaining single-mode fibers and its application to polarization optical time domain reflectometry,” IEEE J. Quantum Electron. 19, 854–861 (1983).
    [Crossref]
  36. K. Petermann, R. Kruhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4, 2–7 (1986).
    [Crossref]
  37. J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).
  38. N. A. Semenov, “Transmitted power in a dielectric waveguide,” Telecommun. Radio Eng. (USSR) Part 2 21, 85–89 (1966).
  39. K. M. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13, 360–361 (1972).
    [Crossref]
  40. S. Kawakami, “Relation between dispersion and power flow distribution in a dielectric waveguide,” J. Opt. Soc. Am. 65, 41–45 (1975).
    [Crossref]
  41. S. Nemoto, T. Makimoto, “A relationship between phase and group indices of guided modes in dielectric waveguide,” Int. J. Electron. 40, 187–190 (1976).
    [Crossref]

1996 (1)

1995 (3)

A. H. Liang, H. K. Tsang, “Transverse offset loss between two identical noncircular single-mode waveguides,” Appl. Opt. 34, 5343–5350 (1995).
[Crossref] [PubMed]

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

A. H. Liang, “Relation between transition loss and mode-field half-width for curved noncircular weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 31, 2075–2077 (1995).
[Crossref]

1993 (1)

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

1992 (2)

A. H. Liang, “Microbending losses for noncircular weakly-guiding single-mode fibers,” IEEE J. Quantum Electron. 28, 1429–1434 (1992).
[Crossref]

A. H. Liang, “The Rayleigh scattering loss of isotropic weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 28, 1844–1847 (1992).
[Crossref]

1991 (3)

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

D. Marcuse, “Single-channel operation in very long nonlinear fibers with optical amplifiers at zero dispersion,” J. Lightwave Technol. 9, 356–361 (1991).
[Crossref]

A. H. Liang, “Relation between three mode-field radius definitions,” Appl. Opt. 30, 5017–5018 (1991).
[Crossref] [PubMed]

1990 (1)

C. C. Fan, A. H. Liang, “Splice loss between different Gaussian-elliptic-field single-mode fibres,” J. Lightwave Technol. 8, 173–176 (1990).
[Crossref]

1989 (1)

A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
[Crossref]

1988 (2)

A. H. Liang, C. C. Fan, “Mode field radius definition for single-mode planar waveguide with arbitrary index profile and its application to splice loss evaluation,” Electron. Lett. 24, 1011–1013 (1988).
[Crossref]

A. H. Liang, C. C. Fan, “Mode radius of noncircular field single-mode fiber: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[Crossref]

1986 (5)

K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
[Crossref]

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. 4, 967–979 (1986).
[Crossref]

C. Saravanos, R. S. Lowe, “The measurement of non-Gaussian mode-fields by the far-field axial scanning technique,” J. Lightwave Technol. 4, 1563–1566 (1986).
[Crossref]

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

K. Petermann, R. Kruhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4, 2–7 (1986).
[Crossref]

1985 (1)

C. D. Hussey, F. Martinez, “Approximate analytic forms for propagation characteristics of single-mode optical fibers,” Electron. Lett. 21, 1103–1104 (1985).
[Crossref]

1984 (1)

C. D. Hussey, “Field to dispersion relationships in single-mode fibers,” Electron. Lett. 20, 1051–1052 (1984).
[Crossref]

1983 (3)

A. W. Snyder, F. Ruhl, “Novel polarization phenomena on anisotropic multimode fibres,” Electron. Lett. 19, 401–402 (1983).
[Crossref]

K. Petermann, “Constraints for fundamental mode spot size for broad and dispersion-compensated single-mode fibers,” Electron. Lett. 19, 712–713 (1983).
[Crossref]

M. Nakazawa, “Theory of backward Rayleigh scattering in polarization-maintaining single-mode fibers and its application to polarization optical time domain reflectometry,” IEEE J. Quantum Electron. 19, 854–861 (1983).
[Crossref]

1980 (2)

D. Krumbholz, E. Brinkmeyer, E.-G. Neumann, “Core/cladding power distribution, propagation constant, and group delay: simple relation for power-law graded-index fibers,” J. Opt. Soc. Am. 70, 179–183 (1980).
[Crossref]

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

1979 (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).

1976 (2)

K. Petermann, “Theory of microbending loss in monomode fibers with arbitrary refractive index profile,” Asia Electron Union 30, 337–342 (1976).

S. Nemoto, T. Makimoto, “A relationship between phase and group indices of guided modes in dielectric waveguide,” Int. J. Electron. 40, 187–190 (1976).
[Crossref]

1975 (1)

1972 (1)

K. M. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13, 360–361 (1972).
[Crossref]

1971 (1)

1966 (2)

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

N. A. Semenov, “Transmitted power in a dielectric waveguide,” Telecommun. Radio Eng. (USSR) Part 2 21, 85–89 (1966).

Abiru, T.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Ainslie, B. J.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. 4, 967–979 (1986).
[Crossref]

Aoki, Y.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

Bjarklev, A.

A. Bjarklev, Optical Fiber Amplifiers: Design and System Applications (Artech House, Norwood, Mass., 1993).

Bogatyrev, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Brinkmeyer, E.

Brown, J.

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

Bubnov, M. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Case, K. M.

K. M. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13, 360–361 (1972).
[Crossref]

Chan, L. Y.

Chernikov, S. V.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Daikoku, K.

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

Day, C. R.

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. 4, 967–979 (1986).
[Crossref]

Devyatykh, G. G.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Dianov, E. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

DiNatale, W. F.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Evans, G. A.

G. A. Evans, J. M. Hammer, Surface Emitting Semiconductor Lasers and Arrays (Academic, Boston, 1993).

Fan, C. C.

C. C. Fan, A. H. Liang, “Splice loss between different Gaussian-elliptic-field single-mode fibres,” J. Lightwave Technol. 8, 173–176 (1990).
[Crossref]

A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
[Crossref]

A. H. Liang, C. C. Fan, “Mode field radius definition for single-mode planar waveguide with arbitrary index profile and its application to splice loss evaluation,” Electron. Lett. 24, 1011–1013 (1988).
[Crossref]

A. H. Liang, C. C. Fan, “Mode radius of noncircular field single-mode fiber: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[Crossref]

Fukuda, O.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Gambling, W. A.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).

Gloge, D.

Gur’yanov, A. N.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Hammer, J. M.

G. A. Evans, J. M. Hammer, Surface Emitting Semiconductor Lasers and Arrays (Academic, Boston, 1993).

Hayata, K.

K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
[Crossref]

Henmi, N.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

Huang, W. B.

A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
[Crossref]

Hussey, C. D.

C. D. Hussey, F. Martinez, “Approximate analytic forms for propagation characteristics of single-mode optical fibers,” Electron. Lett. 21, 1103–1104 (1985).
[Crossref]

C. D. Hussey, “Field to dispersion relationships in single-mode fibers,” Electron. Lett. 20, 1051–1052 (1984).
[Crossref]

Imoto, N.

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

Inada, K.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Kawakami, S.

Kintzer, E. S.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Koshiba, K.

K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
[Crossref]

Kruhne, R.

K. Petermann, R. Kruhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4, 2–7 (1986).
[Crossref]

Krumbholz, D.

Kurkov, A. S.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Laing, A. H.

Liang, A. H.

A. H. Liang, H. K. Tsang, “Transverse offset loss between two identical noncircular single-mode waveguides,” Appl. Opt. 34, 5343–5350 (1995).
[Crossref] [PubMed]

A. H. Liang, “Relation between transition loss and mode-field half-width for curved noncircular weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 31, 2075–2077 (1995).
[Crossref]

A. H. Liang, “The Rayleigh scattering loss of isotropic weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 28, 1844–1847 (1992).
[Crossref]

A. H. Liang, “Microbending losses for noncircular weakly-guiding single-mode fibers,” IEEE J. Quantum Electron. 28, 1429–1434 (1992).
[Crossref]

A. H. Liang, “Relation between three mode-field radius definitions,” Appl. Opt. 30, 5017–5018 (1991).
[Crossref] [PubMed]

C. C. Fan, A. H. Liang, “Splice loss between different Gaussian-elliptic-field single-mode fibres,” J. Lightwave Technol. 8, 173–176 (1990).
[Crossref]

A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
[Crossref]

A. H. Liang, C. C. Fan, “Mode field radius definition for single-mode planar waveguide with arbitrary index profile and its application to splice loss evaluation,” Electron. Lett. 24, 1011–1013 (1988).
[Crossref]

A. H. Liang, C. C. Fan, “Mode radius of noncircular field single-mode fiber: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[Crossref]

A. H. Liang, “New diffraction-limited criterion,” submitted to IEEE J. Quantum Electron.

Liau, Z. L.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Livas, L. C.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chaps. 18, 19, 28, and 29.

Lowe, R. S.

C. Saravanos, R. S. Lowe, “The measurement of non-Gaussian mode-fields by the far-field axial scanning technique,” J. Lightwave Technol. 4, 1563–1566 (1986).
[Crossref]

Makimoto, T.

S. Nemoto, T. Makimoto, “A relationship between phase and group indices of guided modes in dielectric waveguide,” Int. J. Electron. 40, 187–190 (1976).
[Crossref]

Mamyshev, P. V.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Marcuse, D.

D. Marcuse, “Single-channel operation in very long nonlinear fibers with optical amplifiers at zero dispersion,” J. Lightwave Technol. 9, 356–361 (1991).
[Crossref]

Martinez, F.

C. D. Hussey, F. Martinez, “Approximate analytic forms for propagation characteristics of single-mode optical fibers,” Electron. Lett. 21, 1103–1104 (1985).
[Crossref]

Matsumura, H.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).

Miroshnichenko, S. I.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Missaggia, L. J.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Miya, T.

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

Miyamoto, M.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Mull, D. E.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Nakaya, S.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

Nakazawa, M.

M. Nakazawa, “Theory of backward Rayleigh scattering in polarization-maintaining single-mode fibers and its application to polarization optical time domain reflectometry,” IEEE J. Quantum Electron. 19, 854–861 (1983).
[Crossref]

Nemoto, S.

S. Nemoto, T. Makimoto, “A relationship between phase and group indices of guided modes in dielectric waveguide,” Int. J. Electron. 40, 187–190 (1976).
[Crossref]

Neumann, E.-G.

Nishide, K.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Ogata, T.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

Ohashi, T.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Petermann, K.

K. Petermann, R. Kruhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4, 2–7 (1986).
[Crossref]

K. Petermann, “Constraints for fundamental mode spot size for broad and dispersion-compensated single-mode fibers,” Electron. Lett. 19, 712–713 (1983).
[Crossref]

K. Petermann, “Theory of microbending loss in monomode fibers with arbitrary refractive index profile,” Asia Electron Union 30, 337–342 (1976).

Prokhorov, A. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Ragdale, C. M.

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).

Ruhl, F.

A. W. Snyder, F. Ruhl, “Novel polarization phenomena on anisotropic multimode fibres,” Electron. Lett. 19, 401–402 (1983).
[Crossref]

Rumyantsev, S. D.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Saito, T.

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

Saravanos, C.

C. Saravanos, R. S. Lowe, “The measurement of non-Gaussian mode-fields by the far-field axial scanning technique,” J. Lightwave Technol. 4, 1563–1566 (1986).
[Crossref]

Semenov, N. A.

N. A. Semenov, “Transmitted power in a dielectric waveguide,” Telecommun. Radio Eng. (USSR) Part 2 21, 85–89 (1966).

Semenov, S. L.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Semenov, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Snyder, A. W.

A. W. Snyder, F. Ruhl, “Novel polarization phenomena on anisotropic multimode fibres,” Electron. Lett. 19, 401–402 (1983).
[Crossref]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chaps. 18, 19, 28, and 29.

Sugimura, A.

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

Suzuki, M.

K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
[Crossref]

Sysoliatin, A. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

Tsang, H. K.

Tsao, C.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, New York, 1992), pp. 757–761.

Walpole, J. N.

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Welch, D. F.

D. F. Welch, “Progress in high power semiconductor lasers,” in Proceedings of the 14th International Semiconductor Laser Conference, Maui, Haw. (1994) pp. 3–5.

Yamauchi, R.

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

Appl. Opt. (3)

Asia Electron Union (1)

K. Petermann, “Theory of microbending loss in monomode fibers with arbitrary refractive index profile,” Asia Electron Union 30, 337–342 (1976).

Electron. Lett. (7)

K. Petermann, “Constraints for fundamental mode spot size for broad and dispersion-compensated single-mode fibers,” Electron. Lett. 19, 712–713 (1983).
[Crossref]

K. Hayata, K. Koshiba, M. Suzuki, “Mode spot size of axially nonsymmetric fibers,” Electron. Lett. 22, 127–129 (1986).
[Crossref]

A. H. Liang, C. C. Fan, “Mode radius of noncircular field single-mode fiber: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[Crossref]

A. H. Liang, C. C. Fan, “Mode field radius definition for single-mode planar waveguide with arbitrary index profile and its application to splice loss evaluation,” Electron. Lett. 24, 1011–1013 (1988).
[Crossref]

C. D. Hussey, “Field to dispersion relationships in single-mode fibers,” Electron. Lett. 20, 1051–1052 (1984).
[Crossref]

C. D. Hussey, F. Martinez, “Approximate analytic forms for propagation characteristics of single-mode optical fibers,” Electron. Lett. 21, 1103–1104 (1985).
[Crossref]

A. W. Snyder, F. Ruhl, “Novel polarization phenomena on anisotropic multimode fibres,” Electron. Lett. 19, 401–402 (1983).
[Crossref]

IEEE J. Microwave, Opt. Acoust. (1)

W. A. Gambling, H. Matsumura, C. M. Ragdale, “Mode dispersion, material dispersion and profile dispersion in graded-index single-mode fibers,” IEEE J. Microwave, Opt. Acoust. 3, 239–246 (1979).

IEEE J. Quantum Electron. (5)

A. Sugimura, K. Daikoku, N. Imoto, T. Miya, “Wavelength dispersion characteristics of single-mode fibers in low-loss region,” IEEE J. Quantum Electron. 16, 215–225 (1980).
[Crossref]

M. Nakazawa, “Theory of backward Rayleigh scattering in polarization-maintaining single-mode fibers and its application to polarization optical time domain reflectometry,” IEEE J. Quantum Electron. 19, 854–861 (1983).
[Crossref]

A. H. Liang, “Microbending losses for noncircular weakly-guiding single-mode fibers,” IEEE J. Quantum Electron. 28, 1429–1434 (1992).
[Crossref]

A. H. Liang, “Relation between transition loss and mode-field half-width for curved noncircular weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 31, 2075–2077 (1995).
[Crossref]

A. H. Liang, “The Rayleigh scattering loss of isotropic weakly-guiding single-mode waveguides,” IEEE J. Quantum Electron. 28, 1844–1847 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Z. L. Liau, J. N. Walpole, L. C. Livas, E. S. Kintzer, D. E. Mull, L. J. Missaggia, W. F. DiNatale, “Fabrication of two-sided anamorphic microlenses and direct coupling of tapered high power diode laser to single-mode,” IEEE Photon. Technol. Lett. 7, 1315–1317, (1995).
[Crossref]

Int. J. Electron. (1)

S. Nemoto, T. Makimoto, “A relationship between phase and group indices of guided modes in dielectric waveguide,” Int. J. Electron. 40, 187–190 (1976).
[Crossref]

J. Lightwave Technol. (9)

C. C. Fan, A. H. Liang, “Splice loss between different Gaussian-elliptic-field single-mode fibres,” J. Lightwave Technol. 8, 173–176 (1990).
[Crossref]

A. H. Liang, W. B. Huang, C. C. Fan, “Relation between three mode-field half-width definitions for single-mode planar waveguide and application to splice-loss evaluation,” J. Lightwave Technol. 7, 2046–2051 (1989).
[Crossref]

K. Petermann, R. Kruhne, “Upper and lower limits for the microbending loss in arbitrary single-mode fibers,” J. Lightwave Technol. 4, 2–7 (1986).
[Crossref]

R. Yamauchi, M. Miyamoto, T. Abiru, K. Nishide, T. Ohashi, O. Fukuda, K. Inada, “Design and performance of Gaussian-profile disperion-shifted fibers manufactured by VAD process,” J. Lightwave Technol. 4, 997–1003 (1986).
[Crossref]

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. 9, 561–566 (1991).
[Crossref]

B. J. Ainslie, C. R. Day, “A review of single-mode fibers with modified dispersion characteristics,” J. Lightwave Technol. 4, 967–979 (1986).
[Crossref]

C. Saravanos, R. S. Lowe, “The measurement of non-Gaussian mode-fields by the far-field axial scanning technique,” J. Lightwave Technol. 4, 1563–1566 (1986).
[Crossref]

D. Marcuse, “Single-channel operation in very long nonlinear fibers with optical amplifiers at zero dispersion,” J. Lightwave Technol. 9, 356–361 (1991).
[Crossref]

N. Henmi, Y. Aoki, T. Ogata, T. Saito, S. Nakaya, “A new design arrangement of transmission fiber dispersion for suppressing nonlinear degradation in long-distance optical transmission systems with optical repeater amplifiers,” J. Lightwave Technol. 11, 1615–1621 (1993).
[Crossref]

J. Math. Phys. (1)

K. M. Case, “On wave propagation in inhomogeneous media,” J. Math. Phys. 13, 360–361 (1972).
[Crossref]

J. Opt. Soc. Am. (2)

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

Proc. IEEE (1)

J. Brown, “Electromagnetic momentum associated with waveguide modes,” Proc. IEEE 113, 27–34 (1966).

Telecommun. Radio Eng. (USSR) Part 2 (1)

N. A. Semenov, “Transmitted power in a dielectric waveguide,” Telecommun. Radio Eng. (USSR) Part 2 21, 85–89 (1966).

Other (7)

A. Bjarklev, Optical Fiber Amplifiers: Design and System Applications (Artech House, Norwood, Mass., 1993).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chaps. 18, 19, 28, and 29.

E.-G. Neumann, Single-Mode Fibers (Springer-Verlag, Berlin, 1988), pp. 93–104, 228, 348–368.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford U. Press, New York, 1992), pp. 757–761.

A. H. Liang, “New diffraction-limited criterion,” submitted to IEEE J. Quantum Electron.

G. A. Evans, J. M. Hammer, Surface Emitting Semiconductor Lasers and Arrays (Academic, Boston, 1993).

D. F. Welch, “Progress in high power semiconductor lasers,” in Proceedings of the 14th International Semiconductor Laser Conference, Maui, Haw. (1994) pp. 3–5.

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Figures (1)

Fig. 1
Fig. 1

Mode-field half-width changes along distance z: (a) in our new periodical fiber system, (b) in the conventional periodical fiber system.

Equations (49)

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W x 2 = - + - + E 2 x ,   y d x d y - + - + E x ,   y x 2 d x d y 1 / 2 W y 2 = - + - + E 2 x ,   y d x d y - + - + E x ,   y y 2 d x d y 1 / 2
- + - + E 2 ( x ,   y ) d x d y = 1.
2 E x ,   y x 2 + 2 E x ,   y y 2 + k 0 2 n 2 x ,   y - β 2 E x ,   y = 0 ,
- + - + f ( x ,   y ) E 2 ( x ,   y ) d x d y = - + - + f ( x ,   y ) E 2 ( x ,   y ) d x d y | m = m co - m co m 1 2 m Δ n 1 2 m 1 W x 2 2 + 1 W y 2 2 d m ,
- + - + f x ,   y E 2 ( x ,   y ) d x d y | m = m co = 1 - 1 2 m Δ n 1 2 1 W x 2 2 + 1 W y 2 2 m = m co .
v = k 0 ( n 1 2 - n 2 2 ) 1 / 2 ,   w = ( β 2 - k 0 2 n 2 2 ) 1 / 2 , u = ( k 0 2 n 1 2 - β 2 ) 1 / 2 ,   b = w 2 / v 2 .
V = k 0 a ( n 1 2 - n 2 2 ) 1 / 2 = a v ,   W = a ( β 2 - k 0 2 n 2 2 ) 1 / 2 = a w , U = a ( k 0 2 n 1 2 - β 2 ) 1 / 2 = a u ,   b = W 2 / V 2 = w 2 / v 2 ,
- + - + f ( x ,   y ) E 2 ( x ,   y ) d x d y = 1 - 1 v 2 1 W x 2 2 + 1 W y 2 2 - 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v .
β 2 = k 0 2 n 2 2 + 2 v 2 v co v 1 v 3 ( 1 W x 2 2 + 1 W x 2 2 ) d v ,
b = 2 v co v 1 v 3 ( 1 W x 2 2 + 1 W x 2 2 ) d v ,
β 2 = k 0 2 n 2 2 + 4 V 2 V co V 1 V 3 W L 2 d V ,
b = 4 V co V a 2 V 3 W L 2 d V ,
b 1 = d ( v b ) d v = 2 v 2 1 W x 2 2 + 1 W y 2 2 + b .
τ g = 1 c n 2 n g 2 + ( n 1 n g 1 - n 2 n g 2 ) × b + v 2 d b d v / n 2 2 + ( n 1 2 - n 2 2 ) b 1 / 2 ,
τ g 1 c n g 2 + ( n g 1 - n g 2 ) d ( v b ) d v 1 c n g 2 + n 2 Δ d ( v b ) d v = 1 c ( n g 2 + n 2 Δ b 1 ) = 1 c n g 2 + λ 2 4 π 2 n 2 1 W x 2 2 + 1 W y 2 2 + v 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v .
b 2 = v d 2 ( v b ) d v 2 = d d v 2 v 1 W x 2 2 + 1 W y 2 2 .
D = D cmd + D cwd + D cpd ,
D cmd = - λ n 1 c n 2 d 2 n 1 d λ 2 H ( v ) + [ 1 - H ( v ) ] λ c d 2 n 2 d λ 2 ( 1 + 2 Δ 2 b ) 1 / 2 ,
D cwd = n 2 Δ 2 B ( n ) b 2 c λ ( 1 + 2 Δ 2 b ) 1 / 2 + n 2 Δ 2 2 c λ ( 1 + 2 Δ 2 b ) 3 / 2 × [ B ( n ) ] 1 / 2 - λ 2 Δ 2 d Δ 2 d λ 2 ( b 1 - b ) 2 ,
D cpd = n 2 ( d Δ 2 / d λ ) C ( n ) c ( 1 + 2 Δ 2 b ) 1 / 2 b 2 + b 1 - b 1 + 2 Δ 2 b - n 2 3 ( λ d Δ 2 / d λ ) 2 H ( v ) c n 1 2 λ ( 1 + 2 Δ 2 b ) 3 / 2 ( 1 - b ) ,
H v = 1 2 b 1 + b = 1 v 2 1 W x 2 2 + 1 W y 2 2 + b = 1 v 2 1 W x 2 2 + 1 W y 2 2 + 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v ,
B ( n ) = 1 - λ n 2 d n 2 d λ 2 = n g 2 2 n 2 2 ,
C ( n ) = 1 - λ n 2 d n 2 d λ - λ 4 Δ 2 d Δ 2 d λ , Δ 2 = n 1 2 n 2 2 Δ = n 1 2 - n 2 2 2 n 2 2 .
D cmd - λ c d 2 n 1 d λ 2 H ( v ) + [ 1 - H ( v ) ] d 2 n 2 d λ 2 ,
D cwd - n 2 Δ 2 B n c λ b 2 = - n 2 Δ 2 c λ 1 - λ n 2 d n 2 d λ 2 × d d v 2 v 1 W x 2 2 + 1 W y 2 2 ,
D cpd n 2 C n c b 2 + b 1 - b d Δ 2 d λ = 2 n 2 c v 1 - λ n 2 d n 2 d λ - λ 4 Δ 2 d Δ 2 d λ × d d v 1 W x 2 2 + 1 W y 2 2 d Δ 2 d λ .
D cmd D m = - λ c d 2 n 1 d λ 2 ,
D cwd D w = - n 1 - n 2 c λ v d 2 ( v b ) d v 2 = - n 1 - n 2 c λ b 2 ,
D cpd D p = - n 2 c d Δ d λ v d 2 ( b v ) d v 2 + d ( b v ) d v - b ,
D w = - n 1 - n 2 c λ d d λ 2 v 1 W x 2 2 + 1 W y 2 2 .
B m = | β x - β y | k 0 | ( β x ) 2 - ( β y ) 2 | 2 k 0 2 n 2 x = 1 2 k 0 2 n 2 x - + - + { [ k 0 n x ( x ,   y ) E x ( x ,   y ) ] 2 - [ k 0 n y ( x ,   y ) E y ( x ,   y ) ] 2 } d x d y - 1 ( W x 2 x ) 2 + 1 ( W y 2 x ) 2 - 1 ( W x 2 y ) 2 + 1 ( W y 2 y ) 2 ,
B m = - + - + [ n x ( x ,   y ) - n y ( x ,   y ) ] [ E x ( x ,   y ) ] 2 d x d y .
Δ τ g = | τ g x - τ g y | 1 c n g 2 x + λ 2 4 π 2 n 2 x 1 ( W x 2 x ) 2 + 1 ( W y 2 x ) 2 + ( v x ) 2 - + 1 ( v x ) 3 1 ( W x 2 x ) 2 + 1 ( W y 2 x ) 2 d v x - n g 2 y - λ 2 4 π 2 n 2 y 1 ( W x 2 y ) 2 + 1 ( W y 2 y ) 2 + ( v y ) 2 - + 1 ( v y ) 3 1 ( W x 2 y ) 2 + 1 ( W y 2 y ) 2 d v y ,
α s = 1 λ 4 ( n 2 2 - 1 ) C 1 core E 2 ( x ,   y ) d x d y + C 2 cladd E 2 ( x ,   y ) d x d y + C 1 λ 2 4 π 2 1 W x 2 2 + 1 W y 2 2 + 2 v 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v ( dB / km ) ,
α s = C 1 λ 4 ( n 2 2 - 1 ) + λ 2 4 π 2 1 W x 2 2 + 1 W y 2 2 + 2 v 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v .
α s = C 1 λ 4 ( n 2 2 - 1 ) + λ 2 2 π 2 W L 2 + λ 2 V 2 π 2 V co V 1 V 3 W L 2 d V .
α s C 1 λ 4 ( n 2 2 - 1 ) + 1 4 π 2 λ 2 W x 2 2 + λ 2 W y 2 2 + 2 λ λ co λ W x 2 2 + λ W y 2 2 d λ ,
α s C 1 λ 4 ( n 2 2 - 1 ) + λ 2 2 π 2 W L 2 + 1 π 2 λ λ co λ W L 2 d λ ,
B ( z ,   λ ) = 5   log   P ( 0 ) - α ( λ ) z - 0.25 α s ( λ ) τ v g + 5   log   S ( λ ) ,
S ( λ ) = 3 2 β 2 W x G W y G 3 2 ( k 0 n 2 ) 2 W x G W y G ,
B ( z ,   λ 2 ) = 1 λ 2 4 λ 1 4 B ( z ,   λ 1 ) + 5 ( λ 2 4 - λ 1 4 ) log   P ( 0 ) - [ λ 2 4 α ( λ 2 ) - λ 1 4 α ( λ 1 ) ] z - 0.25 [ λ 2 4 α s ( λ 2 ) - λ 1 4 α ( λ 1 ) ] τ v g + 5   log S λ 2 4 ( λ 2 ) S λ 1 4 ( λ 1 ) = 1 λ 2 4 λ 1 4 B ( z ,   λ 1 ) + 5 ( λ 2 4 - λ 1 4 ) log   P ( 0 ) - [ λ 2 4 α ( λ 2 ) - λ 1 4 α ( λ 1 ) ] z - 0.25 C 1 τ v g [ n 2 2 ( λ 2 ) - n 2 2 ( λ 1 ) ] + 1 4 π 2 × λ 2 W x 2 2 + λ 2 W y 2 2 λ 1 λ 2 + 1 2 π 2 λ 2 λ 1 λ 2 W x 2 2 + λ 2 W y 2 2 d λ + 5   log S λ 2 4 ( λ 2 ) S λ 1 4 ( λ 1 ) .
γ = W x 2 2 - W y 2 2 W x 2 2 .
W = 2 ( β 2 - k 0 2 n 2 2 ) 1 / 2 = 2 w .
W = ( 2 ) 1 / 2 v 2 v co v 1 v 3 1 W x 2 2 + 1 W y 2 2 d v - 1 / 2 .
W = V 2 V co V 1 V 3 W L 2 d V - 1 / 2 .
- + - + f ( x ,   y ) E 2 ( x ,   y ) d x d y = 1 - 0.5 ( b 1 + b ) ,
- + - + n 2 ( x ,   y ) E 2 ( x ,   y ) d x d y = n 2 2 + v 2 k 2 2 ( b 1 + b ) 2 .
d β 2 d k 0 2 = d d k 0 2 ( b v 2 + k 0 2 n 2 2 ) = n 2 2 + v 2 2 k 0 2 ( b 1 + b ) .
1 v p v g = d β 2 d ω 2 = 1 c 2 - + - + n 2 ( x ,   y ) E 2 ( x ,   y ) d x d y - + - + E 2 ( x ,   y ) d x d y ,

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