Abstract

Many skew rays within a circular dielectric rod that geometric optics predicts are trapped by total internal reflection are in fact leaky. By finding the complex roots of the eigenvalue equation, we derive a concise analytic expression for the loss of all weakly attenuated rays. The solution is uniformly valid for leaky as well as those refracted rays that obey Fresnel’s laws. The results provide a unified theory of light transmission within fibers and represent the necessary generalization of Fresnel’s laws for cylindrical structures. A weakly leaky mode is formed by a family of leaky rays. The number of weakly leaky modes is determined.

© 1974 Optical Society of America

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References

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  1. A. W. Snyder, D. J. Mitchell, and C. Pask, J. Opt. Soc. Am. 64, 608 (1974).
    [CrossRef]
  2. R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11.
  3. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chs. 2 and 3.
  4. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Ch. 8.
  5. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
    [CrossRef]
  6. T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
    [CrossRef]
  7. J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 349.
  8. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [CrossRef]
  9. Handbook of Mathematical Functions, edited by H. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), Ch. 9. In particular Eqs. (9.1.3), (9.1.15), (9.3.2), (9.3.3), (9.3.35), and (9.3.36).
  10. D. Ludwig, Studies in Applied Mathematics 6, edited by D. Ludwig and F. W. J. Olver (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1970).
  11. A. W. Snyder and C. Pask, J. Opt. Soc. Am. 63, 806 (1973).
    [CrossRef]
  12. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  13. A. W. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972), Appendix A.
    [CrossRef]

1974 (1)

1973 (1)

1972 (1)

1971 (2)

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
[CrossRef]

1969 (1)

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

1963 (1)

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chs. 2 and 3.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11.

Gloge, D.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chs. 2 and 3.

Ludwig, D.

D. Ludwig, Studies in Applied Mathematics 6, edited by D. Ludwig and F. W. J. Olver (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1970).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Ch. 8.

Mitchell, D. J.

Oliner, A. A.

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

Pask, C.

Snyder, A. W.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 349.

Tamir, T.

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
[CrossRef]

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

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

T. Tamir and A. A. Oliner, Proc. IEEE 51, 317 (1963).
[CrossRef]

Other (6)

J. A. Stratton, Electromagnetic Theory (McGraw–Hill, New York, 1941), p. 349.

Handbook of Mathematical Functions, edited by H. Abramowitz and I. A. Stegun. Natl. Bur. Stand. (U.S.) Appl. Math. Ser. 55 (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965), Ch. 9. In particular Eqs. (9.1.3), (9.1.15), (9.3.2), (9.3.3), (9.3.35), and (9.3.36).

D. Ludwig, Studies in Applied Mathematics 6, edited by D. Ludwig and F. W. J. Olver (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1970).

R. E. Collin, Field Theory of Guided Waves (McGraw–Hill, New York, 1960), Ch. 11.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chs. 2 and 3.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Ch. 8.

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Figures (5)

Fig. 1
Fig. 1

Infinite cylinder of radius ρ and index of refraction n1 surrounded by an unbounded medium of refractive index n2.

Fig. 2
Fig. 2

Field of a leaky mode on a circular optical waveguide of radius 2ρ. For r < rtp, the field resembles a bound mode. For r > rtp, energy is radiated. Ψ is the direction of radiation in the far field. At r = rtp, the wave vector k 2 = β z ˆ + ( Q / ρ ) Φ ˆ, where z ˆ and Φ ˆ are the unit vectors in the axial and azimuthal directions, respectively.

Fig. 3
Fig. 3

Geometrical interpretation of the fields inside and outside of the fiber. Cin is the caustic within the fiber, related to the asymptotic representation of J1(Ur/ρ); Cout is the caustic outside the fiber, related to the asymptotic representation of Hl(Qr/ρ). The shaded regions correspond to exponentially small fields. The unshaded regions correspond to rays, i.e., oscillatory fields given by the Debye expansion. Cin is located at a radius r related to ρl = Ur and Cout to ρl = Qr.

Fig. 4
Fig. 4

Illustration of angles defined at incidence on fiber boundary. P is the point of incidence. 0 is the center of the circular cross section. θN is the angle between the normal to the boundary, given by line 0P and the incident-ray direction k ˆ. 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. 5
Fig. 5

Sketch of the U plane for leaky (LR) and refracting (RR) rays.

Equations (84)

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ψ = J l ( U r ρ ) e i ( ω t - l ϕ - β z ) ,             r < ρ
ψ = H l ( Q r ρ ) e i ( ω t - l ϕ - β z ) ,             r > ρ
U 2 = ( 2 π ρ λ n 1 ) 2 - ( ρ β ) 2 ,
Q 2 = ( 2 π ρ λ n 2 ) 2 - ( ρ β ) 2 .
V 2 = ( 2 π ρ λ n 1 ) 2 sin 2 θ c = U 2 - Q 2 ,
sin θ c = [ 1 - ( n 2 n 1 ) 2 ] 1 2 .
θ c [ 1 - ( n 2 n 1 ) 2 ] 1 2 1.
G l ( U ) = U J l ( U ) J l - 1 ( U ) - Q H l ( Q ) H l - 1 ( Q ) = 0 ,
0 U < V ,
0 < Q V .
V Re U 2 π ρ n 1 / λ ,
0 Re Q < 2 π ρ n 2 / λ .
G ( U ) = G ( U r ) + i U i d G ( U r ) d U = 0.
G r ( U r ) - U i d G i d U ( U r ) = 0 ,
G i ( U r ) + U i d G r d U ( U r ) = 0 ,
U i = G r ( U r ) d G i ( U r ) / d U = - G i ( U r ) d G r ( U r ) / d U .
d G ( U ) d U V 2 U | H l ( Q ) H l - 2 ( Q ) H l - 1 2 ( Q ) | .
Im G ( U r ) = Q Im { H l ( Q ) H l - 1 ( Q ) }
= 2 π | 1 H l - 1 ( Q ) | 2 .
U i = 2 π U V 2 [ 1 H l ( Q ) H l - 2 ( Q ) ] ,
H l ( Q ) = [ J l 2 ( Q ) + Y l 2 ( Q ) ] 1 2 .
J l ( u r ) r - i l ϕ = [ 1 2 π ( u 2 r 2 - l 2 ) 1 2 ] 1 2 × { exp [ i ξ + ( r , ϕ ) ] + exp [ i ξ - ( r , ϕ ) ] } ,
ξ ± ( r , ϕ ) = - l ϕ ± { ( u 2 r 2 - l 2 ) 1 2 - l cos - 1 ( l u r ) - π 4 } .
k ± = z ˆ β + l ρ ϕ ˆ ± ( U 2 - l 2 ) 1 2 ( 1 ρ ) r ˆ ,
x = z z ˆ + ρ ϕ ϕ ˆ + ρ r ˆ ,
cos θ ϕ = ϕ · k t k t = l U ,
cos θ z = z ˆ · k k = β k ,
cos θ N = r ˆ · k k = ( U 2 - l 2 ) 1 2 ρ k ,
= sin θ ϕ sin θ z ,
U = V ( sin θ z sin θ c ) ,
l = V ( sin θ z sin θ c ) cos θ ϕ ,
Q = V [ ( sin θ z sin θ c ) 2 - 1 ] 1 2 ,
0 θ z < θ c
π 2 θ z > θ c ,
π 2 θ N > π 2 - θ c .
θ N < ( π 2 - θ c )
sin θ c = sin θ z sin θ ϕ .
N = S ( 2 π ) 2 A ( k x , k y ) × 2
= ρ 2 2 π A ( k x , k y ) ,
A ( k x , k y ) = k x k y d ξ x d ξ y ,
= 1 ρ 1 θ ϕ U ξ d ξ d θ ,
N tm = V 2 2 ,
A ( k x , k y ) = ( 4 ρ 2 ) θ c π / 2 ( V 2 2 sin 2 θ ϕ - V 2 2 ) d θ ϕ
= 2 V 2 2 { ctn θ c + θ c - π 2 } ,
N wlm = V 2 π { ctn θ c + θ c - π 2 }
V 2 π { 1 θ c - π 2 } ,             θ c 1.
N = k 2 2 = V 2 2 sin 2 θ c
1 2 ( V θ c ) 2 ,             θ c 1.
N slm = N - ( N tm + N wlm )
= V 2 { 1 2 ( 1 sin 2 θ c ) - ( ctn θ c π ) - θ c π }
V 2 { 1 2 θ c 2 - 1 π θ c } ,             θ c 1.
P ( z ) = P ( 0 ) e - α z / ρ ,
α = - 2 ρ β i ,
β i β r = - U r U i = Q r Q i .
ρ β i = - U i tan θ z ,
α = 2 U i tan θ z .
α = 4 π ( 1 V ) ( sin 2 θ z θ c cos θ z ) [ 1 H l - 1 ( Q ) H l + 1 ( Q ) ] ,
l = V ( sin θ z θ c ) cos θ ϕ
Q = V [ ( sin θ z θ c ) 2 - 1 ] 1 2 ,
V = ( 2 π ρ λ ) n 1 θ c ,
θ c = [ 1 - ( n 2 n 1 ) 2 ] 1 2 ,
H l ( Q ) 2 2 π ( Q 2 - l 2 ) 1 2 ,
α = 2 ( sin 2 θ z θ c cos θ z ) [ ( sin θ z sin θ ϕ θ c ) 2 - 1 ] 1 2 .
H l ( Q ) 2 0.800 l 2 3 ,
α = 1.59 ( θ c cos θ z ) ( 1 V ) 1 3 ( sin θ z θ z ) 2 [ ( sin θ z θ c ) 2 - 1 ] 1 3 .
H l ( Q ) 2 { 2 π ( l 2 - Q 2 ) 1 2 } × exp [ 2 l cosh - 1 l Q - 2 ( l 2 - Q 2 ) 1 2 ] ,
α 2 ( sin θ c cos θ z ) 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 ) ] ,
H l ( Q ) 2 4 ( 1 l ) 2 3 | 4 ξ 1 - ( Q / l ) 2 | 1 2 Ai ( e 2 π i / 3 l 2 3 ξ ) 2 ,
2 3 ( - ξ ) 3 2 = [ Q 2 l - 1 ] 1 2 - cos - 1 ( l Q ) ,
[ ( n 1 / n 2 ) 2 J l ( U ) U J l ( U ) - H l ( Q ) Q H l ( Q ) ] [ J l ( U ) U J l ( U ) - H l ( Q ) Q H l ( Q ) ] = ( l β k 2 ) 2 ( V U Q ) 4 ,
J l ( U ) U J l ( U ) = H l ( Q ) Q H l ( Q ) = ± l β k 2 ( V U Q ) 2 .
J l - 1 ( U ) J l ( U ) - ( U Q ) H l - 1 ( Q ) H l ( Q ) = - U l ( V U Q ) 2 { ± β k 2 + 1 } ,
U J l ( U ) J l - 1 ( U ) = Q H l ( Q ) H l - 1 ( Q ) .
| U l ( V U γ ) 2 { ± β k 2 - 1 } | < 1 - cos θ z [ ( sin θ z / θ c ) 2 - 1 ] 1 2 .
G l ( U ) = d d U { U J l ( U ) J l - 1 ( U ) } - U Q d d Q { Q H l ( Q ) H l - 1 ( Q ) } .
d d U { U J l ( U ) J l - 1 ( U ) } = U { 1 - ( Q U ) 2 H l ( Q ) H l - 2 ( Q ) H l - 1 2 ( Q ) }
U 2 d d Q { Q H l ( Q ) H l - 1 ( Q ) } = U { 1 - H l ( Q ) H l - 2 ( Q ) H l - 1 2 ( Q ) } .
G l ( U ) = ( V 2 U ) { H l ( Q ) H l - 2 ( Q ) H l - 1 2 ( Q ) } .
T = 1 - power of the reflected wave power of incident wave ,
= 4 cos θ N { cos 2 θ N - sin 2 θ c } 1 2 [ cos θ N + { cos 2 θ N - sin 2 θ c } 1 2 ] 2 ,
n T = Δ P / P = α Δ z / ρ ,
α = T ρ / Δ z R .
Δ z R = 2 ρ sin θ ϕ cot θ z ,
α = 2 sin 2 θ z { cos 2 θ N - sin 2 θ c } 1 2 cos θ z [ cos θ N + { cos 2 θ N - sin 2 θ c } 1 2 ] 2 .