Abstract

By expressing Maxwell’s equations in a linear-operator formalism, it is shown that the orthogonality and normalization properties of the continuous spectrum of radiation modes in a dielectric waveguide with arbitrary refractiveindex profile and cross-sectional shape can be established directly from the properties of much simpler free-space fields.

© 1982 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).
  2. D. P. Nyquist, D. R. Johnson, and S. V. Hsu, "Orthogonality and amplitude spectrum of radiation modes along open-boundary waveguides," J. Opt. Soc. Am. 71, 49–54 (1981).
  3. C. Vassallo, "Orthogonality and amplitude spectrum of radiation modes along open-boundary waveguides: comment," J. Opt. Soc. Am. 71, 1282 (1981).
  4. J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189–195 (1955).
  5. A. W. Snyder, "Continuous modes of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
  6. B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956), pp. 238–241, 251–252.
  7. This result has been mentioned previously in R. A. Sammut, "Orthogonality of ITM and ITE continuous modes," Proc. IEE 122, 1376 (1975), but the details are contained in an unpublished Ph.D. thesis. The original assertion of its validity for the stepindex case was made in Ref. 5.
  8. A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).
  9. Note that, although gz, is singular in the six-space in which the vectors Φ are defined, in four-space gz-1 = gz The factor -iζgz, must be introduced in Eq. (17) because it is clear from Eq. (13) that the operator L - βgz is not invertible.
  10. R. A. Sammut, C. Pask, and A. W. Snyder, "Excitation and power of the unbound modes within a circular dielectric waveguide," Proc. IEE 122, 25–33 (1975).

1981

1971

A. W. Snyder, "Continuous modes of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

1958

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).

1955

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189–195 (1955).

Bresler, A. D.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).

Friedman, B.

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956), pp. 238–241, 251–252.

Hsu, S. V.

Johnson, D. R.

Joshi, G. H.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).

Marcuvitz, N.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).

Nyquist, D. P.

Pask, C.

R. A. Sammut, C. Pask, and A. W. Snyder, "Excitation and power of the unbound modes within a circular dielectric waveguide," Proc. IEE 122, 25–33 (1975).

Sammut, R. A.

R. A. Sammut, C. Pask, and A. W. Snyder, "Excitation and power of the unbound modes within a circular dielectric waveguide," Proc. IEE 122, 25–33 (1975).

This result has been mentioned previously in R. A. Sammut, "Orthogonality of ITM and ITE continuous modes," Proc. IEE 122, 1376 (1975), but the details are contained in an unpublished Ph.D. thesis. The original assertion of its validity for the stepindex case was made in Ref. 5.

Snyder, A. W.

A. W. Snyder, "Continuous modes of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

R. A. Sammut, C. Pask, and A. W. Snyder, "Excitation and power of the unbound modes within a circular dielectric waveguide," Proc. IEE 122, 25–33 (1975).

Vassallo, C.

Wait, J. R.

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189–195 (1955).

Can. J. Phys.

J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Can. J. Phys. 33, 189–195 (1955).

IEEE Trans. Microwave Theory Tech.

A. W. Snyder, "Continuous modes of a circular dielectric rod," IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

J. Appl. Phys.

A. D. Bresler, G. H. Joshi, and N. Marcuvitz, "Orthogonality properties for modes in passive and active uniform waveguides," J. Appl. Phys. 29, 794–799 (1958).

J. Opt. Soc. Am.

Other

Note that, although gz, is singular in the six-space in which the vectors Φ are defined, in four-space gz-1 = gz The factor -iζgz, must be introduced in Eq. (17) because it is clear from Eq. (13) that the operator L - βgz is not invertible.

R. A. Sammut, C. Pask, and A. W. Snyder, "Excitation and power of the unbound modes within a circular dielectric waveguide," Proc. IEE 122, 25–33 (1975).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, Princeton, N.J., 1972).

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956), pp. 238–241, 251–252.

This result has been mentioned previously in R. A. Sammut, "Orthogonality of ITM and ITE continuous modes," Proc. IEE 122, 1376 (1975), but the details are contained in an unpublished Ph.D. thesis. The original assertion of its validity for the stepindex case was made in Ref. 5.

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