Abstract

When a tunneling leaky ray propagates along the core of a nonabsorbing, multilayered, dielectric waveguide of slab or cylindrical geometry, the tunneling coefficient describing the power loss at a reflection or turning point is a simple product of fundamental quantities: (i) a WKB integration over the continuous part of the refractive index profile in the evanescent region between the reflection or turning point and the position at which the ray reappears; and (ii) a factor |<i>T</i><sub><i>F</i></sub>| at each jump in the refractive index profile between adjacent layers in the evanescent region, where <i>T</i><sub><i>F</i></sub> is the analytic continuation of the Fresnel power transmission coefficient as defined between two half-spaces of constant refractive indices corresponding to the values on either side of the jump. The ray power attenuation coefficient is equal to the tunneling coefficient divided by the distance along the waveguide axis between successive reflections or turning points at which power is lost.

© 1978 Optical Society of America

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  1. A. W. Snyder and J. D. Love, "Tunneling leaky modes on optical waveguides," Opt. Commun. 12, 325–328 (1974).
  2. M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).
  3. J. D. Love and A. W. Snyder, "Fresnel's and Snell's laws for the multimode optical waveguide of circular cross section," J. Opt. Soc. Am. 65, 1241–1247 (1975).
  4. A. W. Snyder and J. D. Love, "Reflection at a curved dilectric interface-electromagnetic tunneling," IEEE Trans. MTT-13, 134–141 (1975).
  5. A. W. Snyder and D. J. Mitchell, "Generalized Fresnel's laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438–459 (1975).
  6. A. W. Snyder and J. D. Love, "Attenuation coefficient for tunneling leaky rays in graded fibers," Electron. Lett. 12, 324 (1975).
  7. W. J. Stewart, "A new technique for determining the V values and refractive index profiles of optical fibers," paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).
  8. K. Petermann, "The mode attenuation in general graded core multimode fibers," Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).
  9. J. D. Love and A. W. Snyder, "Generalized Fresnel's laws for a curved absorbing interface," J. Opt. Soc. Am. 65, 1072–1074 (1975).
  10. C. Pask and A. W. Snyder, "Multimode optical fibers: interplay of absorption and radiation losses," Appl. Opt. 15, 1295–1298 (1976).
  11. A. W. Snyder and J. D. Love, "Attenuation coefficient for rays in graded fibers with absorbing cladding," Electron. Lett. 12, 255–257 (1976).
  12. J. D. Love and C. Winkler, "The effects of material absorption on ray power attenuation in multilayered optical waveguides" (unpublished).
  13. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.
  14. A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599–607 (1974).
  15. H. Bremmer, "The W. K. B. approximation as the first term of a geometric-optical series," in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).
  16. A. W. Snyder and J. D. Love, "Goos-Hanchen shift," Appl. Opt. 15, 236–238 (1976).
  17. D. Gloge and E. A. J. Marcatili, "Multimode theory of graded fibers," Bell Syst. Tech. J. 52, 1563–1578 (1973).
  18. M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).
  19. L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

1976 (4)

M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).

A. W. Snyder and J. D. Love, "Goos-Hanchen shift," Appl. Opt. 15, 236–238 (1976).

C. Pask and A. W. Snyder, "Multimode optical fibers: interplay of absorption and radiation losses," Appl. Opt. 15, 1295–1298 (1976).

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

1975 (7)

J. D. Love and A. W. Snyder, "Generalized Fresnel's laws for a curved absorbing interface," J. Opt. Soc. Am. 65, 1072–1074 (1975).

J. D. Love and A. W. Snyder, "Fresnel's and Snell's laws for the multimode optical waveguide of circular cross section," J. Opt. Soc. Am. 65, 1241–1247 (1975).

M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).

A. W. Snyder and J. D. Love, "Reflection at a curved dilectric interface-electromagnetic tunneling," IEEE Trans. MTT-13, 134–141 (1975).

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel's laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438–459 (1975).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunneling leaky rays in graded fibers," Electron. Lett. 12, 324 (1975).

K. Petermann, "The mode attenuation in general graded core multimode fibers," Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

1974 (2)

A. W. Snyder and J. D. Love, "Tunneling leaky modes on optical waveguides," Opt. Commun. 12, 325–328 (1974).

A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599–607 (1974).

1973 (1)

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

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Adams, M. J.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).

M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).

Born, M.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

Bremmer, H.

H. Bremmer, "The W. K. B. approximation as the first term of a geometric-optical series," in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).

Gloge, D.

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

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

Love, J. D.

A. W. Snyder and J. D. Love, "Goos-Hanchen shift," Appl. Opt. 15, 236–238 (1976).

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

A. W. Snyder and J. D. Love, "Reflection at a curved dilectric interface-electromagnetic tunneling," IEEE Trans. MTT-13, 134–141 (1975).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunneling leaky rays in graded fibers," Electron. Lett. 12, 324 (1975).

J. D. Love and A. W. Snyder, "Generalized Fresnel's laws for a curved absorbing interface," J. Opt. Soc. Am. 65, 1072–1074 (1975).

J. D. Love and A. W. Snyder, "Fresnel's and Snell's laws for the multimode optical waveguide of circular cross section," J. Opt. Soc. Am. 65, 1241–1247 (1975).

A. W. Snyder and J. D. Love, "Tunneling leaky modes on optical waveguides," Opt. Commun. 12, 325–328 (1974).

J. D. Love and C. Winkler, "The effects of material absorption on ray power attenuation in multilayered optical waveguides" (unpublished).

Marcatili, E. A. J.

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

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel's laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438–459 (1975).

A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599–607 (1974).

Pask, C.

Payne, D. H.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).

Payne, D. N.

M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).

Petermann, K.

K. Petermann, "The mode attenuation in general graded core multimode fibers," Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

Sladen, F. M. E.

M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).

M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).

Snyder, A. W.

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

C. Pask and A. W. Snyder, "Multimode optical fibers: interplay of absorption and radiation losses," Appl. Opt. 15, 1295–1298 (1976).

A. W. Snyder and J. D. Love, "Goos-Hanchen shift," Appl. Opt. 15, 236–238 (1976).

J. D. Love and A. W. Snyder, "Generalized Fresnel's laws for a curved absorbing interface," J. Opt. Soc. Am. 65, 1072–1074 (1975).

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunneling leaky rays in graded fibers," Electron. Lett. 12, 324 (1975).

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel's laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438–459 (1975).

A. W. Snyder and J. D. Love, "Reflection at a curved dilectric interface-electromagnetic tunneling," IEEE Trans. MTT-13, 134–141 (1975).

J. D. Love and A. W. Snyder, "Fresnel's and Snell's laws for the multimode optical waveguide of circular cross section," J. Opt. Soc. Am. 65, 1241–1247 (1975).

A. W. Snyder and J. D. Love, "Tunneling leaky modes on optical waveguides," Opt. Commun. 12, 325–328 (1974).

A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical fibers," J. Opt. Soc. Am. 64, 599–607 (1974).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

Stewart, W. J.

W. J. Stewart, "A new technique for determining the V values and refractive index profiles of optical fibers," paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).

Winkler, C.

J. D. Love and C. Winkler, "The effects of material absorption on ray power attenuation in multilayered optical waveguides" (unpublished).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

Appl. Opt. (2)

Arch. Elektron. Ubertragungsteck (1)

K. Petermann, "The mode attenuation in general graded core multimode fibers," Arch. Elektron. Ubertragungsteck 29, 345–348 (1975).

Bell Syst. Tech. J. (1)

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

Electron. Lett. (3)

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

A. W. Snyder and J. D. Love, "Attenuation coefficient for tunneling leaky rays in graded fibers," Electron. Lett. 12, 324 (1975).

M. J. Adams, D. N. Payne, and F. M. E. Sladen, "Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles," Electron. Lett. 11, 238–240 (1975).

IEEE Trans. (1)

A. W. Snyder and J. D. Love, "Reflection at a curved dilectric interface-electromagnetic tunneling," IEEE Trans. MTT-13, 134–141 (1975).

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

A. W. Snyder and J. D. Love, "Tunneling leaky modes on optical waveguides," Opt. Commun. 12, 325–328 (1974).

M. J. Adams, D. H. Payne, and F. M. E. Sladen, "Lengthdependent effects due to leaky modes on multimode gradedindex optical fibers," Opt. Commun. 17, 204–209 (1976).

Optik (1)

A. W. Snyder and D. J. Mitchell, "Generalized Fresnel's laws for determining radiation loss from optical waveguides and curved dielectric structures," Optik 40, 438–459 (1975).

Other (6)

W. J. Stewart, "A new technique for determining the V values and refractive index profiles of optical fibers," paper presented at OSA/IEEE meeting on Optical Fiber Transmission, Williamsburg, Virginia (1975).

L. D. Landau and E. M. Lifshitz, Electrodynamics of continuous media (Pergamon, New York, 1963), p. 286.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), pp. 446–449.

J. D. Love and C. Winkler, "The effects of material absorption on ray power attenuation in multilayered optical waveguides" (unpublished).

M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), p. 40.

H. Bremmer, "The W. K. B. approximation as the first term of a geometric-optical series," in The Theory of Electromagnetic Waves, edited by M. Kline (Dover, New York, 1951).

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