Abstract

A technique is presented for determining the modal propagation properties of a homogeneous cylindrical dielectric waveguide of arbitrary cross sectional shape and index <i>n</i><sub>1</sub>, embedded in a medium of index <i>n</i><sub>2</sub>. Both the weakly guiding case in which n<sub>1</sub> ≈ <i>n</i><sub>2</sub> and the general case of arbitrary index difference are discussed theoretically. In both cases the approach is to derive integral representations for appropriate components of <b>E</b> and <b>B</b>. These satisfy the appropriate Helmholtz equations inside and outside the guide and also guarantee that the boundary conditions are satisfied. On expansion of the components in certain sets of basis functions, the representations become a set of linear equations. The vanishing of the determinant of this set yields the propagation constants of the various mqdes. Numerical results are given for weakly guiding fibers of various shapes. Among these are rectangles and ellipses, which make comparisons with previous work possible.

© 1979 Optical Society of America

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  1. D. Gloge, "Weakly guiding fibers," Appl. Opt. 10, 2252–2258 (1971).
  2. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Sys. Tech. J. 48, 2133–2160 (1969).
  3. E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Sys. Tech. J. 48, 2071–2102 (1969).
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  5. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  6. L. Eyges, "Solution of Schrodinger and related equations for irregular and composite regions," Ann. Phys. 81, 567–590 (1973).
  7. J. D. Love and A. W. Snyder, "Ray analysis of multimode optical fibres," Ann. Telecommun. 32, 109–114 (1977); P. di Vita, "Theory of propagation in optical fibres: Ray Approach," Ann. Telecommun. 32, 115–134 (1977).
  8. L. Eyges and P. D. Gianino, "Polarizabilities of rectangular dielectric cylinders and of a cube," IEEE Trans. Ant. & Prop., July (1979).
  9. A. E. Nelson and L. Eyges, "Electromagnetic scattering from dielectric rods of arbitrary cross section," J. Opt. Soc. Am. 66, 254–259 (1976).
  10. P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45,1417–1429 (1969) and, "Symmetry unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825–839 (1971).
  11. L. Eyges, "Fiber optic guides of noncircular cross section," Appl. Opt. 17, 1673–1674 (1978).
  12. J. Allard, "Notes on squares and cubes," Math. Mag. 37, 210–214 (1964).
  13. M. Gardner, "Mathematical games," Sci. Am. 213, No. 3, 222–232 (1965).
  14. F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956) p. 330.
  15. R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975) p. 65.
  16. C. Yeh, "Modes in weakly guiding elliptical optical fibres," Opt. and Quant. Elect. 8, 43–47 (1976).

1978 (1)

1977 (1)

J. D. Love and A. W. Snyder, "Ray analysis of multimode optical fibres," Ann. Telecommun. 32, 109–114 (1977); P. di Vita, "Theory of propagation in optical fibres: Ray Approach," Ann. Telecommun. 32, 115–134 (1977).

1976 (2)

A. E. Nelson and L. Eyges, "Electromagnetic scattering from dielectric rods of arbitrary cross section," J. Opt. Soc. Am. 66, 254–259 (1976).

C. Yeh, "Modes in weakly guiding elliptical optical fibres," Opt. and Quant. Elect. 8, 43–47 (1976).

1973 (1)

L. Eyges, "Solution of Schrodinger and related equations for irregular and composite regions," Ann. Phys. 81, 567–590 (1973).

1971 (1)

1969 (2)

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Sys. Tech. J. 48, 2133–2160 (1969).

E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Sys. Tech. J. 48, 2071–2102 (1969).

1965 (1)

M. Gardner, "Mathematical games," Sci. Am. 213, No. 3, 222–232 (1965).

1964 (1)

J. Allard, "Notes on squares and cubes," Math. Mag. 37, 210–214 (1964).

Allard, J.

J. Allard, "Notes on squares and cubes," Math. Mag. 37, 210–214 (1964).

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Eyges, L.

L. Eyges, "Fiber optic guides of noncircular cross section," Appl. Opt. 17, 1673–1674 (1978).

A. E. Nelson and L. Eyges, "Electromagnetic scattering from dielectric rods of arbitrary cross section," J. Opt. Soc. Am. 66, 254–259 (1976).

L. Eyges, "Solution of Schrodinger and related equations for irregular and composite regions," Ann. Phys. 81, 567–590 (1973).

L. Eyges and P. D. Gianino, "Polarizabilities of rectangular dielectric cylinders and of a cube," IEEE Trans. Ant. & Prop., July (1979).

Gardner, M.

M. Gardner, "Mathematical games," Sci. Am. 213, No. 3, 222–232 (1965).

Gianino, P. D.

L. Eyges and P. D. Gianino, "Polarizabilities of rectangular dielectric cylinders and of a cube," IEEE Trans. Ant. & Prop., July (1979).

Gloge, D.

Goell, J. E.

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Sys. Tech. J. 48, 2133–2160 (1969).

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956) p. 330.

Hornbeck, R. W.

R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975) p. 65.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Love, J. D.

J. D. Love and A. W. Snyder, "Ray analysis of multimode optical fibres," Ann. Telecommun. 32, 109–114 (1977); P. di Vita, "Theory of propagation in optical fibres: Ray Approach," Ann. Telecommun. 32, 115–134 (1977).

Marcatili, E. A. J.

E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Sys. Tech. J. 48, 2071–2102 (1969).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Nelson, A. E.

Snyder, A. W.

J. D. Love and A. W. Snyder, "Ray analysis of multimode optical fibres," Ann. Telecommun. 32, 109–114 (1977); P. di Vita, "Theory of propagation in optical fibres: Ray Approach," Ann. Telecommun. 32, 115–134 (1977).

Waterman, P. C.

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45,1417–1429 (1969) and, "Symmetry unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825–839 (1971).

Yeh, C.

C. Yeh, "Modes in weakly guiding elliptical optical fibres," Opt. and Quant. Elect. 8, 43–47 (1976).

Ann. Phys. (1)

L. Eyges, "Solution of Schrodinger and related equations for irregular and composite regions," Ann. Phys. 81, 567–590 (1973).

Ann. Telecommun. (1)

J. D. Love and A. W. Snyder, "Ray analysis of multimode optical fibres," Ann. Telecommun. 32, 109–114 (1977); P. di Vita, "Theory of propagation in optical fibres: Ray Approach," Ann. Telecommun. 32, 115–134 (1977).

Appl. Opt. (2)

Bell Sys. Tech. J. (2)

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Sys. Tech. J. 48, 2133–2160 (1969).

E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Sys. Tech. J. 48, 2071–2102 (1969).

J. Opt. Soc. Am. (1)

Math. Mag. (1)

J. Allard, "Notes on squares and cubes," Math. Mag. 37, 210–214 (1964).

Opt. and Quant. Elect. (1)

C. Yeh, "Modes in weakly guiding elliptical optical fibres," Opt. and Quant. Elect. 8, 43–47 (1976).

Sci. Am. (1)

M. Gardner, "Mathematical games," Sci. Am. 213, No. 3, 222–232 (1965).

Other (6)

F. B. Hildebrand, Introduction to Numerical Analysis, (McGraw Hill, New York, 1956) p. 330.

R. W. Hornbeck, Numerical Methods, (Quantum, New York, 1975) p. 65.

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45,1417–1429 (1969) and, "Symmetry unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 3, 825–839 (1971).

L. Eyges and P. D. Gianino, "Polarizabilities of rectangular dielectric cylinders and of a cube," IEEE Trans. Ant. & Prop., July (1979).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

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