Abstract

The amount of light power that is transmitted within a semi-infinite circular optical fiber when it is illuminated obliquely by a coherent beam of light is determined from an electromagnetic-theory analysis. The limit λ→0 is not classical geometric optics, i.e., not that found by tracing all rays along the fiber. Instead, the limit λ→0 corresponds to that of treating all rays as if they were meridional, i.e., as if they cross the fiber axis, ignoring rays skew to the axis. Thus, ray tracing is incorrect for fibers illuminated by coherent light. However, the acceptance property of an optical fiber illuminated by coherent light is very simply found from meridional ray tracing, if the dimensionless quantity 2πρ{n12n22}1/2/λ is much greater than unity, where ρ is the fiber radius, λ the wavelength of light in vacuum, and n1, n2 the refractive indices of the fiber and its surround, respectively.

© 1973 Optical Society of America

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References

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  1. D. Gloge, Proc. Inst. Electr. Eng. A 58, 1513 (1970).
    [Crossref]
  2. N. S. Kapany, Fiber Optics (Academic, New York, 1967).
  3. J. M. Enoch, J. Opt. Soc. Am. 53, 57 (1963).
    [Crossref]
  4. F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).
  5. R. J. Potter, J. Opt. Soc. Am. 51, 1079 (1961).
    [Crossref]
  6. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [Crossref]
  7. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [Crossref]
  8. A. W. Snyder, J. Opt. Soc. Am. 56, 601 (1966).
    [Crossref]
  9. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
    [Crossref]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 292.
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 38.
  12. M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley–Interscience, New York, 1965).

1970 (2)

D. Gloge, Proc. Inst. Electr. Eng. A 58, 1513 (1970).
[Crossref]

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

1969 (2)

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

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

1966 (1)

1963 (1)

J. M. Enoch, J. Opt. Soc. Am. 53, 57 (1963).
[Crossref]

1961 (2)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 38.

Enoch, J. M.

J. M. Enoch, J. Opt. Soc. Am. 53, 57 (1963).
[Crossref]

Gloge, D.

D. Gloge, Proc. Inst. Electr. Eng. A 58, 1513 (1970).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 292.

Kapany, N. S.

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

Kay, I. W.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley–Interscience, New York, 1965).

Kline, M.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley–Interscience, New York, 1965).

Potter, R. J.

Snitzer, E.

Snyder, A. W.

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

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

A. W. Snyder, J. Opt. Soc. Am. 56, 601 (1966).
[Crossref]

Varela, F. G.

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

Wiitanen, W.

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 38.

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

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

J. Gen. Physiol. (1)

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

J. Opt. Soc. Am. (4)

Proc. Inst. Electr. Eng. A (1)

D. Gloge, Proc. Inst. Electr. Eng. A 58, 1513 (1970).
[Crossref]

Other (4)

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 292.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 38.

M. Kline and I. W. Kay, Electromagnetic Theory and Geometric Optics (Wiley–Interscience, New York, 1965).

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Figures (7)

F. 1
F. 1

Ray-optics illustration of light entering a circular fiber of radius ρ and index of refraction n1, surrounded by a medium of refractive index n2<n1. The material in front of the fiber is characterized by a refractive index n0. θ is the angle of incidence and θt the angle of the refracted or transmitted ray in the fiber.

F. 2
F. 2

Light rays entering the fiber of Fig. 1 at different angles In (a) θ produces an angle θt which is less than the critical angle θc necessary for total internal reflection. Based on ray optics, all of the incident light is transmitted within the fiber by total internal reflections. In (b) θ is larger than in (a) and produces an angle of refraction θt that is greater than the critical angle. Therefore, light is lost at each reflection, and eventually, after many reflections, is totally outside the fiber.

F. 3
F. 3

The fraction P(θ)/Pinc of the incident-light power transmitted within an optical fiber, based on geometric optics. The dashed curve is derived from the meridional-ray analysis (MRO) whereas the solid curve is derived from the skew-ray analysis. At α = 1, θt of Fig. 2 equals the critical angle θc.

F. 4
F. 4

The fraction ηi, of light power within the fiber, for each of the six mode types that exist for V<6.38.

F. 5
F. 5

The fraction P(θ)/Pinc of the light power transmitted within an optical fiber of circular cross section when it is illuminated by a uniform beam of coherent light at angle θ confined to the aperture of the fiber. The results are from the electromagnetic analysis. T(θ) is a parameter that accounts for the polarization and reflection of the incident light and is defined by Eq. (7). For the important case of small (less than 20°) angles of light incidence T(θ)≅[2n0/(n0+n1)]2 and is unity when n0n1. V is a dimensionless parameter defined by Eq. (4). α = 1 corresponds to an angle θ of incidence that produces an angle of refraction θt at the critical angle θc within the fiber, i.e., Fig. 2(a) with θt = θc. The step-function curve is the meridional-ray-optics result. The curves are labeled by their V values. To avoid overcrowding, only the V = 1.2, 1.5, 2.4, 3, and 15 curves are continued beyond α = 1.05.

F. 6
F. 6

The individual modes that contribute to the V = 3 curve of Fig. 5. ηiPi is the fraction of incident-light power within the fiber of the individual modes, as shown by Eq. (6). An HE11, HE21, and one of either the TM01 or TE01 modes is excited by the incident light (see Ref. 9). Therefore, the V = 3 curve of Fig. 5 is found by adding the HE21 + (TM01 or TE01) to the HE11 curve.

F. 7
F. 7

Contour for Eq. (A4) in the complex plane. ImU and RetU represent the imaginary and real parts of U, respectively. The crosses on the real axis represent some of the solutions of Eq. (A2). The solid circles are the poles of Pl(U) for the case α<1.

Tables (1)

Tables Icon

Table I Numerical values used to construct Fig. 5

Equations (27)

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n 0 sin θ = n 1 sin θ t ,
sin θ c = [ 1 ( n 2 n 1 ) 2 ] 1 2 .
P ( θ ) T ( θ ) P inc = 1 , θ < θ c
= 1 2 π [ cos 1 ( 1 α ) + 1 α ( 1 1 α 2 ) 1 2 ] , θ > θ c ,
α = n 0 sin θ [ n 1 2 n 2 2 ] 1 2 ,
V = 2 π ρ λ [ n 1 2 n 2 2 ] 1 2 ,
θ t n 0 n 1 sin θ 1 .
P ( θ ) = P inc T ( θ ) i η i P i ( θ ) ,
T ( θ ) = T ( θ ) = ( 2 n 0 cos θ n 1 + n 0 cos θ ) 2 .
T ( θ ) = T ( θ ) = ( 2 n 0 cos θ n 0 + n 1 cos θ ) 2 .
T ( θ ) = T up ( θ ) = 1 2 [ T ( θ ) + T ( θ ) ] .
P ( θ ) = 0 P inc ( ω ) [ P ( θ , ω ) T ( θ ) P inc ( ω ) ] d ω ,
S = l m η l ( U l m ) P l ( U l m ) ,
f l ( U ) = U J l ( U ) W J l 1 ( U ) K l ( W ) K l 1 ( W ) = 0 ,
V 2 = U 2 + W 2 .
I Γ = 1 2 π i Γ f ( U ) f ( U ) η l ( U ) P l ( U ) d U ,
l I Γ = 2 S + 2 l Res at poles of η l ( U ) P l ( U ) ,
η l ( U ) = ( U V ) 2 [ ( W U ) 2 + K l 1 2 ( W ) K l ( W ) K l 2 ]
P l ( U ) = 4 U 2 V 2 × [ α V J l ( α V ) K l 1 ( W ) W K l ( W ) J l 1 ( α V ) ] 2 K l ( W ) K l 2 ( W ) ( U 2 α 2 V 2 ) 2 .
η l ( U ) 1 ,
P l ( U ) 4 U 2 V 2 [ α V J l ( α V ) W J l 1 ( α V ) ] 2 ( U 2 α 2 V 2 ) 2 ,
f l ( U ) U J l ( U ) W J l 1 ( U ) ,
f l ( n ) ( U ) U J l ( n ) ( U ) W J l 1 ( n ) ( U ) ,
Res at α V U α V [ f l ( α V ) f l ( α V ) { f l ( α V ) f l ( α V ) } 2 ] × ( U α V ) 2 P l ( U ) .
l = [ J l ( U ) ] 2 = l = J l ( U ) J l ( U ) = 1 2 ,
l = J l ( U ) J l 1 ( U ) = l = J l ( U ) J l ± 1 ( U ) = 0 ,
l = Res at α V 1 ,