Abstract

A three-dimensional analysis of bending losses in dielectric optical waveguides is presented. It constitutes a nontrivial generalization of previous two- and three-dimensional studies by other authors. Our analysis is based on homogeneous integral equations for the total radiation field and suitable asymptotic approximations for Green’s functions. A key role is played by a new three-dimensional approximation for a relevant Bessel function with large order and argument (the former being larger than the latter). A nontrivial check of the consistency of all those approximations is given. General formulas are presented for the radiated field and the energy flow and for a bending-loss coefficient in three dimensions. Numerical results are also given, in order to assess the difference between the results of other authors and ours. Such a difference is rather small for monomode behavior near cutoff, increases as the behavior of the waveguide changes from monomode to multimode, and decreases as the parameter V increases for a given core radius and propagation mode.

© 1987 Optical Society of America

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  1. E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).
  2. S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
    [CrossRef]
  3. L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
    [CrossRef]
  4. J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).
  5. E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
    [CrossRef]
  6. D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).
  7. J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
    [CrossRef]
  8. D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
    [CrossRef]
  9. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
    [CrossRef]
  10. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
    [CrossRef] [PubMed]
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.
  12. D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
    [CrossRef]
  13. Y. Takuma, M. Miyagi, S. Kawakami, “Bent asymmetric dielectric slab waveguides,” Appl. Opt. 20, 2291–2298 (1981); M. Miyagi, “Bending losses in hollow and dielectric tube leaky waveguides,” Appl. Opt. 20, 1221–1229 (1981).
    [CrossRef] [PubMed]
  14. A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
    [CrossRef]
  15. J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
    [CrossRef]
  16. M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).
  17. V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
    [CrossRef]
  18. W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
    [CrossRef]
  19. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.
  20. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.
  21. M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
    [CrossRef]
  22. P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.
  23. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).
  24. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

1986

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

1982

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
[CrossRef] [PubMed]

1981

1980

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

1976

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
[CrossRef]

1975

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

1974

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

1973

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

1970

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

1969

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

1956

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Alvarez-Estrada, R. F.

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

Calvo, M. L.

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Chang, D. C.

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

Erdelyi, A.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

Felsen, L. B.

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

Feschbach, H.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

Gambling, W. A.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Gloge, D.

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

Kawakami, S.

Kimura, T.

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Kuester, E. F.

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

Lewin, L.

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Marcatili, E. A. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

Marcuse, D.

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
[CrossRef] [PubMed]

D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,”J. Opt. Soc. Am. 66, 311–320 (1976).
[CrossRef]

D. Marcuse, “Curvature loss formula for optical fibers,”J. Opt. Soc. Am. 66, 216–220 (1976).
[CrossRef]

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.

Matsumura, H.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Maurer, S. J.

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

Miller, M. A.

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Mitchell, D. J.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

Miyagi, M.

Morse, P. M.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Sakai, J.

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Sakai, J. I.

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

Shevchenko, V. V.

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

Takuma, Y.

Talanov, V. I.

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

White, J. A.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

E. A. J. Marcatili, “Bends in optical dielectric guides,” Bell Syst. Tech. J. 48, 2103–2132 (1969); E. A. J. Marcatili, S. E. Miller, “Improved relations describing directional control in electromagnetic wave guidance,” Bell Syst. Tech. J. 48, 2161–2187 (1969).

J. A. Arnaud, “Transverse coupling in fiber optics III: bending losses,” Bell Syst. Tech. J. 53, 1379–1394 (1974).

D. Marcuse, “Bent optical waveguides with lossy jacket,” Bell Syst. Tech. J. 53, 1079–1101 (1974).

Electron. Lett.

A. W. Snyder, J. A. White, D. J. Mitchell, “Radiation from bent optical waveguides,” Electron. Lett. 11, 332–333 (1975); J. A. White, “Radiation from bends in optical waveguides: the volume current method,” Microwave Opt. Acoust. 3, 186–188 (1979).
[CrossRef]

W. A. Gambling, D. N. Payne, H. Matsumura, “Radiation from curved single-mode fibers,” Electron. Lett. 12, 567–569 (1976); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Field deformation in a curved single-mode fiber,” Electron. Lett. 14, 130–132 (1978); W. A. Gambling, H. Matsumura, C. M. Ragdale, “Curvature and microbending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

IEEE J. Quantum Electron.

J. I. Sakai, “Microbending loss evaluation in arbitrary-index single-mode optical fibers,” IEEE J. Quantum Electron. QE-16, 36–49 (1980).
[CrossRef]

E. F. Kuester, D. C. Chang, “Surface wave radiation loss from curved dielectric slabs and fibers,” IEEE J. Quantum Electron. QE-11, 903–907 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

S. J. Maurer, L. B. Felsen, “Ray methods for trapped and slightly leaky modes in multilayered or multiwave regions,” IEEE Trans. Microwave Theory Tech. MTT-18, 584–595 (1970).
[CrossRef]

L. Lewin, “Radiation from curved dielectric slabs and fibers,” IEEE Trans. Microwave Theory Tech. MTT-22, 718–727 (1974).
[CrossRef]

D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave Theory Tech. MTT-23, 106–120 (1975); “Bending losses in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D

M. L. Calvo, R. F. Alvarez-Estrada, “Neutron fibres II: some improving alternatives and analysis of bending losses,”J. Phys. D 19, 957–973 (1986).
[CrossRef]

Radio Sci.

J. Sakai, T. Kimura, “Analytical bending loss formula of optical fibers with field deformation,” Radio Sci. 17, 21–29 (1982).
[CrossRef]

Radiophys. Quantum Electron.

V. V. Shevchenko, “Radiation losses in bent waveguides for surface waves,” Radiophys. Quantum Electron. 14, 607–614 (1973) (Russian original 1971).
[CrossRef]

Z. Tekh. Fiz.

M. A. Miller, V. I. Talanov, “Electromagnetic surface waves guided by a boundary with small curvature,”Z. Tekh. Fiz. 26, 2755 (1956).

Other

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Sec. 9.6.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 23.

P. M. Morse, H. Feschbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7, Sec. 2.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965).

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), p. 36.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Secs. 2.5 and 4.6.

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