Abstract

In addition to rays that lose energy by undergoing refraction, there is a large class of weakly attenuated rays in circular optical fibers. These leaky rays are incorrectly predicted to be lossless by Fresnel’s laws. Thus, Fresnel’s laws fail for the analysis of long fibers. The significance and properties of leaky rays are discussed. A very simple attenuation coefficient is given, from which the loss of all rays is computed. This attenuation coefficient makes it possible to extend the use of ray tracing and Snell’s laws for analyzing circular optical fibers.

© 1974 Optical Society of America

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References

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  1. N. S. Kapany, Fiber Optics (Academic, New York, 1967), p. 31.
  2. V. Maxia, M. Murgia, and K. Testa, Appl. Opt. 12, 98 (1973).
    [Crossref] [PubMed]
  3. J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
    [Crossref]
  4. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  5. R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), p. 470.
  6. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.
    [Crossref]
  7. A. W. Snyder, IEEE Trans Microwave Theory Tech. 17, 1130 (1969).
    [Crossref]
  8. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.
  9. A. W. Snyder and D. J. Mitchell, Electron. Lett. 9, 437 (1973).
    [Crossref]
  10. A. W. Snyder and C. Pask, J. Opt. Soc. Am. 63, 806 (1973).
    [Crossref]
  11. R. D. Maurer, Proc. IEEE 61, 452 (1973).
    [Crossref]
  12. A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am. 63, 59 (1973).
    [Crossref] [PubMed]
  13. F. G. Reick, Appl. Opt. 4, 1395 (1965).
    [Crossref]
  14. D. Marcuse, J. Opt. Soc. Am. 63, 1372 (1973).
    [Crossref]
  15. A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [Crossref]

1974 (1)

1973 (7)

1969 (1)

A. W. Snyder, IEEE Trans Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

1965 (1)

1961 (1)

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.
[Crossref]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), p. 470.

Dakin, J. P.

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Gambling, W. A.

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.
[Crossref]

N. S. Kapany, Fiber Optics (Academic, New York, 1967), p. 31.

Marcuse, D.

D. Marcuse, J. Opt. Soc. Am. 63, 1372 (1973).
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.

Matsumra, H.

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Maurer, R. D.

R. D. Maurer, Proc. IEEE 61, 452 (1973).
[Crossref]

Maxia, V.

Mitchell, D. J.

Murgia, M.

Pask, C.

Payne, D. N.

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Potter, R. J.

Reick, F. G.

Snyder, A. W.

Sunak, H. R. D.

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Testa, K.

Appl. Opt. (2)

Electron. Lett. (1)

A. W. Snyder and D. J. Mitchell, Electron. Lett. 9, 437 (1973).
[Crossref]

IEEE Trans Microwave Theory Tech. (1)

A. W. Snyder, IEEE Trans Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

J. P. Dakin, W. A. Gambling, H. Matsumra, D. N. Payne, and H. R. D. Sunak, Opt. Commun. 7, 1 (1973).
[Crossref]

Proc. IEEE (1)

R. D. Maurer, Proc. IEEE 61, 452 (1973).
[Crossref]

Other (4)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 305.

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), p. 470.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 7.
[Crossref]

N. S. Kapany, Fiber Optics (Academic, New York, 1967), p. 31.

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Figures (11)

Fig. 1
Fig. 1

Illustration of angles defined at incidence on fiber boundary. P is the point of incidence. O is the center of the circular cross section. θN is the angle between the normal to the boundary, given by line OP, and the incident-ray direction (RD). The angles θN, θz, and θϕ are related by sinθz sinθϕ = cosθN. PR is the projection of the ray direction onto the cylinder cross section.

Fig. 2
Fig. 2

Percent power % PLR carried by the leaky rays within an optical fiber excited by an incoherent or diffuse source S. The total power Ptot is also shown. Upper scale refers to n2/n1.

Fig. 3
Fig. 3

The ray type associated with θϕ and θz for the case when θz and θc are small. LR = leaky rays, TR = trapped rays, and RR = refracted rays.

Fig. 4
Fig. 4

The attenuation coefficient divided by θc vs θz/θc for V = 100. LR = leaky rays, RR = refracted rays. Curves A, B, and C are associated with the approximate expressions, Eqs. (12a), (12b), and (12c), respectively. Curve A is associated with l = 0, i.e., the meridional ray. Curve C is the most skew ray.

Fig. 5
Fig. 5

Same as Fig. 4 with V = 104.

Fig. 6
Fig. 6

Attenuation of rays near the critical angle. θϕ = 25.8° (cosθϕ = 0.9). The V = 104 curve obeys Fresnel’s laws (FL) except close to the critical angle. The critical angle is given by the vertical line. LR is leaky rays, and RR refracting rays.

Fig. 7
Fig. 7

Same as Fig. 6, but with (θz/θc) = 1.35, and with varying θϕ.

Fig. 8
Fig. 8

Attenuation of leaky rays. θϕ = 25.8° (cosθϕ = 0.9). The ray is at the critical angle when θz/θc = 2.294.

Fig. 9
Fig. 9

Attenuation of leaky rays. The heavy dashed curve represents the critical angle.

Fig. 10
Fig. 10

Attenuation of rays when V = 500. The vertical bars on the curves represent the ciritical angle given by θz sinθϕ = θc.

Fig. 11
Fig. 11

Attenuation of rays vs θϕ when V = 500. Vertical bars indicate the critical angle, as in Fig. 10.

Equations (20)

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β = ( 2 π n 1 λ ) cos θ z ,
cos θ c = n 2 n 1 .
P TR = sin 2 θ c ,
sin 2 θ c = 1 - ( n 2 n 1 ) 2 .
P TOT = P TR + P LR = 1 - 2 π [ ( δ - δ 2 ) 1 2 + ( 1 - 2 δ ) cos - 1 δ ] ,
% P LR = 100 × P LR P TR + P LR
θ c sin θ c = [ 1 - ( n 2 n 1 ) 2 ] 1 2 1.
θ z sin θ z 1.
P ( z ) = P ( 0 ) e - α z / ρ ,
α θ c = 4 π ( θ z θ c ) 2 ( 1 V ) | 1 H l - 1 ( Q ) H l + 1 ( Q ) | .
l = ( θ z θ c ) V cos θ ϕ ,
Q = V [ ( θ z θ c ) 2 - 1 ] 1 2 ,
V = ( 2 π ρ n 1 λ ) θ c .
α θ c 2 ( θ z θ c ) 2 [ ( θ z θ c ) 2 - 1 ] 1 2
α θ c 1.59 ( 1 V ) 1 3 ( θ z θ c ) 2 [ ( θ z θ c ) 2 - 1 ] 1 3
α θ c 2 ( θ z θ c ) 2 ( θ z - θ c θ z + θ c ) V [ θ z / θ c ] - 1 e 2 V
α θ c 2 ( θ z θ c ) 2 [ ( θ z θ c ) 2 sin 2 θ ϕ - 1 ] 1 2 .
z ρ α 0.7.
α θ c = 0.7 θ c ( ρ z ) = 0.7 θ c 10 - 8 ,
α θ c 2 R 2 ( 1 - R 2 sin 2 θ ϕ ) 1 2 × exp [ ( - 2 3 V ) ( 1 - R 2 sin 2 θ ϕ ) 3 2 ( R 2 - 1 ) ] ,