Abstract

The HE<sub>11</sub> fields of weakly guiding fibers with graded refractive-index profiles are nearly identical to the fields of a step-index fiber with dimensionless frequency V¯ and radius ρ¯, where V¯ and ρ¯ are found by an elementary variational method. The results are remarkably accurate, with errors of a fraction of a percent at most, so that a simple, closed-form expression for the HE<sub>11</sub> fields of graded fibers is now available. The step-fiber approximation is more widely applicable than the Gaussian field approximation which is inadequate for small <i>V</i> and for describing the evanescent field. However, the variational procedure also complements the Gaussian field approximation by providing analytical expressions for the spot size and propagation constant in several profiles of interest.

© 1980 Optical Society of America

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  1. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  2. A. Ankiewicz and C. pask, "Geometric optics approach to light acceptance and propagation in graded-index fibres," Opt. Quantum Electron. 9, 87–109 (1977).
  3. W. Streiffer and C.N. Kurtz, "Scalar analysis of radially inhomogeneous guiding media," J. Opt. Soc. Am. 57, 779–786 (1967)
  4. D. Gloge and E. A. J. Marcatili, "Multimode theory of graded core fibres," Bell Syst. Tech. J. 52, 1563–1578 (1973).
  5. A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibres with absorbing cladding," Electron. Lett. 12, 255–257 (1976).
  6. A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnelling leaky rays in graded fibers," Electron. Lett. 12, 324–326 (1976).
  7. A. W. Snyder and W. R. Young, "Modes of optical waveguides," J. Opt. Soc. Am. 68, 297–309 (1978).
  8. R. Yamada, T. Meiri, and N. Okamoto, "Guided waves along an optical fiber with parabolic index profile," J. Opt. Soc. Am. 67, 96–103 (1977).
  9. W. A. Gambling and H. Matsumura, "Propagation in radially-inhomogeneous single-mode fibre," Opt. Quantum Electron. 10, 31–40 (1978).
  10. J. D. Love, "Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent," Opt. Quantum Electron. 11, September, 1979.
  11. W. A. Gambling and H. Matsumura, "Simple characterization factor for practical single-mode fibres," Electron. Lett. 13, 691–693 (1977).
  12. D. Marcuse, "Loss analysis of single-mode fiber splices," Bell Syst. Tech. J. 56, 703–718 (1977).
  13. D. Marcuse, "Gaussian approximation of the fundamental modes of graded-index fibers," J. Opt. Soc. Am. 68, 103–109 (1978).
  14. M. O. Vassell, "Calculation of propagating modes in a graded-index optical fibre," Opto-electronics 6, 271–286 (1974); see also references cited therein.
  15. M. Matsuhara, "Analysis of TEM modes in dielectric waveguides, by a variational method," J. Opt. Soc. Am. 63, 1514–1517 (1973).
  16. T. Okoshi and K. Okamoto, "Analysis of wave propagation in inhomogeneous optical fibers using a variational method," IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
  17. K. Okamoto and T. Okoshi, "Vectorial wave analysis of inhomogeneous optical fibers using finite element method," IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).
  18. S. E. Miller, "Light propagation in generalized lens-like media," Bell Syst. Tech. J. 44, 2017–2064 (1965).
  19. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.
  20. The HE11 mode is the mode with the largest value of β, or equivalently, the smallest value of U.
  21. A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
  22. C. Pask and R. A. Sammut, "Developments in the theory of fibre optics," Proc. IREE (Aust.) 40, 89–101 (1979).
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  24. A. W. Snyder, "Mode propagation in an optical waveguide," Electron. Lett. 6, 561–562 (1970).

1978 (4)

W. A. Gambling and H. Matsumura, "Propagation in radially-inhomogeneous single-mode fibre," Opt. Quantum Electron. 10, 31–40 (1978).

K. Okamoto and T. Okoshi, "Vectorial wave analysis of inhomogeneous optical fibers using finite element method," IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).

D. Marcuse, "Gaussian approximation of the fundamental modes of graded-index fibers," J. Opt. Soc. Am. 68, 103–109 (1978).

A. W. Snyder and W. R. Young, "Modes of optical waveguides," J. Opt. Soc. Am. 68, 297–309 (1978).

1977 (4)

R. Yamada, T. Meiri, and N. Okamoto, "Guided waves along an optical fiber with parabolic index profile," J. Opt. Soc. Am. 67, 96–103 (1977).

W. A. Gambling and H. Matsumura, "Simple characterization factor for practical single-mode fibres," Electron. Lett. 13, 691–693 (1977).

D. Marcuse, "Loss analysis of single-mode fiber splices," Bell Syst. Tech. J. 56, 703–718 (1977).

A. Ankiewicz and C. pask, "Geometric optics approach to light acceptance and propagation in graded-index fibres," Opt. Quantum Electron. 9, 87–109 (1977).

1976 (2)

A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibres with absorbing cladding," Electron. Lett. 12, 255–257 (1976).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnelling leaky rays in graded fibers," Electron. Lett. 12, 324–326 (1976).

1974 (2)

M. O. Vassell, "Calculation of propagating modes in a graded-index optical fibre," Opto-electronics 6, 271–286 (1974); see also references cited therein.

T. Okoshi and K. Okamoto, "Analysis of wave propagation in inhomogeneous optical fibers using a variational method," IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).

1973 (2)

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded core fibres," Bell Syst. Tech. J. 52, 1563–1578 (1973).

M. Matsuhara, "Analysis of TEM modes in dielectric waveguides, by a variational method," J. Opt. Soc. Am. 63, 1514–1517 (1973).

1970 (1)

A. W. Snyder, "Mode propagation in an optical waveguide," Electron. Lett. 6, 561–562 (1970).

1969 (1)

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

1967 (1)

1965 (1)

S. E. Miller, "Light propagation in generalized lens-like media," Bell Syst. Tech. J. 44, 2017–2064 (1965).

Abramowitz, M.

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

Ankiewicz, A.

A. Ankiewicz and C. pask, "Geometric optics approach to light acceptance and propagation in graded-index fibres," Opt. Quantum Electron. 9, 87–109 (1977).

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

Gambling, W. A.

W. A. Gambling and H. Matsumura, "Propagation in radially-inhomogeneous single-mode fibre," Opt. Quantum Electron. 10, 31–40 (1978).

W. A. Gambling and H. Matsumura, "Simple characterization factor for practical single-mode fibres," Electron. Lett. 13, 691–693 (1977).

Gloge, D.

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded core fibres," Bell Syst. Tech. J. 52, 1563–1578 (1973).

Kurtz, C. N.

Love, J. D.

J. D. Love, "Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent," Opt. Quantum Electron. 11, September, 1979.

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnelling leaky rays in graded fibers," Electron. Lett. 12, 324–326 (1976).

A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibres with absorbing cladding," Electron. Lett. 12, 255–257 (1976).

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded core fibres," Bell Syst. Tech. J. 52, 1563–1578 (1973).

Marcuse, D.

D. Marcuse, "Gaussian approximation of the fundamental modes of graded-index fibers," J. Opt. Soc. Am. 68, 103–109 (1978).

D. Marcuse, "Loss analysis of single-mode fiber splices," Bell Syst. Tech. J. 56, 703–718 (1977).

Matsuhara, M.

Matsumura, H.

W. A. Gambling and H. Matsumura, "Propagation in radially-inhomogeneous single-mode fibre," Opt. Quantum Electron. 10, 31–40 (1978).

W. A. Gambling and H. Matsumura, "Simple characterization factor for practical single-mode fibres," Electron. Lett. 13, 691–693 (1977).

Meiri, T.

Miller, S. E.

S. E. Miller, "Light propagation in generalized lens-like media," Bell Syst. Tech. J. 44, 2017–2064 (1965).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

Okamoto, K.

K. Okamoto and T. Okoshi, "Vectorial wave analysis of inhomogeneous optical fibers using finite element method," IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).

T. Okoshi and K. Okamoto, "Analysis of wave propagation in inhomogeneous optical fibers using a variational method," IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).

Okamoto, N.

Okoshi, T.

K. Okamoto and T. Okoshi, "Vectorial wave analysis of inhomogeneous optical fibers using finite element method," IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).

T. Okoshi and K. Okamoto, "Analysis of wave propagation in inhomogeneous optical fibers using a variational method," IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).

pask, C.

A. Ankiewicz and C. pask, "Geometric optics approach to light acceptance and propagation in graded-index fibres," Opt. Quantum Electron. 9, 87–109 (1977).

C. Pask and R. A. Sammut, "Developments in the theory of fibre optics," Proc. IREE (Aust.) 40, 89–101 (1979).

Sammut, R. A.

C. Pask and R. A. Sammut, "Developments in the theory of fibre optics," Proc. IREE (Aust.) 40, 89–101 (1979).

Snyder, A. W.

A. W. Snyder and W. R. Young, "Modes of optical waveguides," J. Opt. Soc. Am. 68, 297–309 (1978).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnelling leaky rays in graded fibers," Electron. Lett. 12, 324–326 (1976).

A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibres with absorbing cladding," Electron. Lett. 12, 255–257 (1976).

A. W. Snyder, "Mode propagation in an optical waveguide," Electron. Lett. 6, 561–562 (1970).

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

Stegun, I. A.

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

Streiffer, W.

Vassell, M. O.

M. O. Vassell, "Calculation of propagating modes in a graded-index optical fibre," Opto-electronics 6, 271–286 (1974); see also references cited therein.

Yamada, R.

Young, W. R.

Bell Syst. Tech. J. (3)

D. Gloge and E. A. J. Marcatili, "Multimode theory of graded core fibres," Bell Syst. Tech. J. 52, 1563–1578 (1973).

D. Marcuse, "Loss analysis of single-mode fiber splices," Bell Syst. Tech. J. 56, 703–718 (1977).

S. E. Miller, "Light propagation in generalized lens-like media," Bell Syst. Tech. J. 44, 2017–2064 (1965).

Electron. Lett. (4)

A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibres with absorbing cladding," Electron. Lett. 12, 255–257 (1976).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunnelling leaky rays in graded fibers," Electron. Lett. 12, 324–326 (1976).

A. W. Snyder, "Mode propagation in an optical waveguide," Electron. Lett. 6, 561–562 (1970).

W. A. Gambling and H. Matsumura, "Simple characterization factor for practical single-mode fibres," Electron. Lett. 13, 691–693 (1977).

IEEE Trans. Microwave Theory Tech. (3)

A. W. Snyder, "Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides," IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).

T. Okoshi and K. Okamoto, "Analysis of wave propagation in inhomogeneous optical fibers using a variational method," IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).

K. Okamoto and T. Okoshi, "Vectorial wave analysis of inhomogeneous optical fibers using finite element method," IEEE Trans. Microwave Theory Tech. MTT-26, 109–114 (1978).

J. Opt. Soc. Am. (5)

Opt. Quantum Electron. (3)

W. A. Gambling and H. Matsumura, "Propagation in radially-inhomogeneous single-mode fibre," Opt. Quantum Electron. 10, 31–40 (1978).

J. D. Love, "Power series solutions of the scalar wave equation for cladded, power-law profiles of arbitrary exponent," Opt. Quantum Electron. 11, September, 1979.

A. Ankiewicz and C. pask, "Geometric optics approach to light acceptance and propagation in graded-index fibres," Opt. Quantum Electron. 9, 87–109 (1977).

Opto-electronics (1)

M. O. Vassell, "Calculation of propagating modes in a graded-index optical fibre," Opto-electronics 6, 271–286 (1974); see also references cited therein.

Other (5)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 1112.

The HE11 mode is the mode with the largest value of β, or equivalently, the smallest value of U.

C. Pask and R. A. Sammut, "Developments in the theory of fibre optics," Proc. IREE (Aust.) 40, 89–101 (1979).

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

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

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