Abstract

Maxwell’s equations are reformulated to describe electromagnetic propagation along weakly guiding fibers of arbitrary refractive-index profile. The reformulation is particularly powerful for solving radiation from sources or scatterers within the fiber, leading to the most concise expressions possible. For example, radiation from sources in weakly guiding fibers is found by an elementary modification of the standard method for determining radiation from sources in free space. When the source has circular symmetry, the radiation field equals the free-space result multiplied by a factor that accounts for the presence of the fiber. Detailed examples are given for point and tubular sources.

© 1980 Optical Society of America

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References

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  1. A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
    [Crossref]
  2. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  3. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  4. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  5. H. G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).
  6. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).
  7. A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [Crossref]
  8. (a)A. W. Snyder and R. A. Sammut, “Radiation modes of optical waveguides,” Electron. Lett. 15, 4–5 (1979). (b)A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).
  9. A. W. Snyder and R. A. Sammut, “Radiation from optical waveguides: leaky mode interpretation,” Electron. Lett. 15, 58–60 (1979).
    [Crossref]
  10. S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).
  11. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).
  12. J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).
  13. A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys. 4, 273–298 (1974).
    [Crossref]
  14. A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibres,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
    [Crossref]
  15. E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surfaces,” Appl. Opt. 13, 2270–2275 (1974).
    [Crossref]
  16. P. J. B. Clarricoats, “Optical fibre waveguides—a review” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976).
  17. I. A. White and A. W. Snyder, “Radiation from dielectric optical waveguides: a comparison of techniques,” Appl. Opt. 16, 1470–1472 (1977).
    [Crossref] [PubMed]
  18. To do this we assume ψ(β) = ψFS(β+ Δβ), where β= kc1cosθ, i.e., we assume the radiation at observation angle θ equals that of free-space radiation at angle θ+ Δθ. Now β enters Eq. (14) in the combination k2(r) − β2, whereas in free space it appears as kc12−β2. Thus for the step fiber, we rewrite k2(r) − β2as kc12−{β2+kc12−kc02}calling the bracketed quantity (β+ Δβ)2or β˜for short. By replacing β in the free-space radiation expressions with (β2+kc12−kc02)1/2, or replacing Q by U we account for the shift. Equivalently, we can replace θ in the free-space radiation expressions by θ˜=θ+Δθ, where cos2θ≅cos2θ−θc2(1+θc2)provided cos2θ>θc2(1+θc2)and 0 < r′≤ ρ.
  19. I. A. White, Radiation Losses in Dielectric Optical Waveguides, Ph.D. thesis, Australian National University, 1977 (unpublished).
  20. G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

1979 (2)

(a)A. W. Snyder and R. A. Sammut, “Radiation modes of optical waveguides,” Electron. Lett. 15, 4–5 (1979). (b)A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, “Radiation from optical waveguides: leaky mode interpretation,” Electron. Lett. 15, 58–60 (1979).
[Crossref]

1978 (1)

1977 (1)

1974 (2)

E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surfaces,” Appl. Opt. 13, 2270–2275 (1974).
[Crossref]

A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys. 4, 273–298 (1974).
[Crossref]

1971 (1)

1970 (2)

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibres,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

1969 (1)

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

Clarricoats, P. J. B.

P. J. B. Clarricoats, “Optical fibre waveguides—a review” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976).

Gloge, D.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

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

Mathews, J.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

Ramo, S.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

Rawson, E. G.

E. G. Rawson, “Analysis of scattering from fiber waveguides with irregular core surfaces,” Appl. Opt. 13, 2270–2275 (1974).
[Crossref]

Sammut, R. A.

(a)A. W. Snyder and R. A. Sammut, “Radiation modes of optical waveguides,” Electron. Lett. 15, 4–5 (1979). (b)A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, “Radiation from optical waveguides: leaky mode interpretation,” Electron. Lett. 15, 58–60 (1979).
[Crossref]

Snyder, A. W.

(a)A. W. Snyder and R. A. Sammut, “Radiation modes of optical waveguides,” Electron. Lett. 15, 4–5 (1979). (b)A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, “Radiation from optical waveguides: leaky mode interpretation,” Electron. Lett. 15, 58–60 (1979).
[Crossref]

A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
[Crossref]

I. A. White and A. W. Snyder, “Radiation from dielectric optical waveguides: a comparison of techniques,” Appl. Opt. 16, 1470–1472 (1977).
[Crossref] [PubMed]

A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys. 4, 273–298 (1974).
[Crossref]

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibres,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

Unger, H. G.

H. G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).

Van Duzer, T.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

Walker, R. L.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

Whinnery, J. R.

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

White, I. A.

I. A. White and A. W. Snyder, “Radiation from dielectric optical waveguides: a comparison of techniques,” Appl. Opt. 16, 1470–1472 (1977).
[Crossref] [PubMed]

I. A. White, Radiation Losses in Dielectric Optical Waveguides, Ph.D. thesis, Australian National University, 1977 (unpublished).

Yip, G. L.

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

Young, W. R.

Appl. Opt. (3)

Appl. Phys. (1)

A. W. Snyder, “Leaky-ray theory of optical waveguides of circular cross section,” Appl. Phys. 4, 273–298 (1974).
[Crossref]

Electron. Lett. (2)

(a)A. W. Snyder and R. A. Sammut, “Radiation modes of optical waveguides,” Electron. Lett. 15, 4–5 (1979). (b)A. W. Snyder, “Continuous mode spectrum of a circular dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-19, 720–727 (1971).

A. W. Snyder and R. A. Sammut, “Radiation from optical waveguides: leaky mode interpretation,” Electron. Lett. 15, 58–60 (1979).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

A. W. Snyder, “Radiation losses due to variations of radius on dielectric or optical fibres,” IEEE Trans. Microwave Theory Tech. MTT-18, 608–615 (1970).
[Crossref]

A. W. Snyder, “Asymptotic expressions for eigenfunctions and eigenvalues of dielectric or optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[Crossref]

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. Microwave Theory Tech. MTT-18, 1033–1041 (1970).

J. Opt. Soc. Am. (1)

Other (10)

To do this we assume ψ(β) = ψFS(β+ Δβ), where β= kc1cosθ, i.e., we assume the radiation at observation angle θ equals that of free-space radiation at angle θ+ Δθ. Now β enters Eq. (14) in the combination k2(r) − β2, whereas in free space it appears as kc12−β2. Thus for the step fiber, we rewrite k2(r) − β2as kc12−{β2+kc12−kc02}calling the bracketed quantity (β+ Δβ)2or β˜for short. By replacing β in the free-space radiation expressions with (β2+kc12−kc02)1/2, or replacing Q by U we account for the shift. Equivalently, we can replace θ in the free-space radiation expressions by θ˜=θ+Δθ, where cos2θ≅cos2θ−θc2(1+θc2)provided cos2θ>θc2(1+θc2)and 0 < r′≤ ρ.

I. A. White, Radiation Losses in Dielectric Optical Waveguides, Ph.D. thesis, Australian National University, 1977 (unpublished).

P. J. B. Clarricoats, “Optical fibre waveguides—a review” in Progress in Optics, edited by E. Wolf (North-Holland, Amsterdam, 1976).

S. Ramo, J. R. Whinnery, and T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965).

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965).

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1965).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

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

H. G. Unger, Planar Optical Waveguides and Fibres (Clarendon, Oxford, 1977).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976).

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Figures (3)

FIG. 1
FIG. 1

Tubular currents J within the core of an optical fiber of graded refractive index. The currents have cos/ϕ symmetry and arbitrary z variation as defined by Eq. (9).

FIG. 2
FIG. 2

Correction factor CF as defined by Eq. (20a) for a step profile. It fully accounts for the modification of free-space radiation due to the light trapping property of an optical fiber for the current J of Eq. (9). θ is the observation angle of Fig. 1(b).

FIG. 3
FIG. 3

Radiation due to currents at the core-cladding boundary of a step-profile fiber when the currents have azimuthal symmetry and an eiαzsinΩz dependence as discussed in Sec. III E. The radiation occurs at one angle θ = θ0 only, where θ0 is given by Eq. (21). The numerical results are from Eq. (22) with l = 0 and r = ρ. The free-space radiated power PFS is κ(1 + cos2θ0) J 0 2(Q0) which equals 2κ at θ0 = 0.

Equations (55)

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[ 2 + k 2 ( x , y ) ] E + [ E · ln ε ( x , y ) ] = i ω μ J + 1 i ω [ · J / ε ( x , y ) ]
k 2 ( x , y ) = ω 2 μ ε ( x , y ) = [ 2 π n ( x , y ) / λ ] 2
n co = ( ε co / ε 0 ) 1 / 2 = maximum refractive index of the core ,
n cl = ( ε cl / ε 0 ) 1 / 2 = uniform cladding refractive index .
[ 2 + k 2 ( x , y ) ] E = i ω μ J + 1 i ω ε cl ( · J ) .
μ H = × A ,
E = i ω A 1 i ω μ ε cl ( · A ) ,
[ 2 + k 2 ( x , y ) ] A = μ J .
E ( x , y , z ) = a n E n ( x , y , z ) = a n e n ( x , y ) e i β n z ,
[ t 2 + k 2 ( x , y ) ] e n t = β n 2 e n t .
a n = ( μ ε cl ) 1 / 2 υ J · e n * e i β n z d υ / 2 A | e n t | 2 d υ ,
J = j ˆ g ( z ) cos l ( ϕ ϕ ) δ ( r r ) / 2 π r ,
E ( r , ϕ , θ ) = C ( θ ) E F S ( r , ϕ , θ ) ,
S = | C | 2 S F S .
P = A s r ˆ s · S r s 2 d Ω ,
C ( θ ) = ψ / ψ F S ,
[ t 2 + k 2 ( r ) β 2 ] ψ = cos l ( ϕ ϕ ) δ ( r r ) / 2 π r .
C ( θ ) = J l ( U r / ρ ) W l ( Q , Q ) / J l ( Q r / ρ ) W l ( U , Q )
J l ( U r / ρ ) / J l ( Q r / ρ ) when U Q 1.
W l ( U , Q ) = U J l + 1 ( U ) H l ( 1 ) ( Q ) Q J l ( U ) H l + 1 ( 1 ) ( Q )
= 2 i / π , U = Q ,
Q = ρ k cl sin θ = ( U 2 V 2 ) 1 / 2 ,
V = ρ ( k co 2 k cl 2 ) = ρ k co sin θ c ,
θ c sin θ c = [ 1 ( n cl / n co ) 2 ] 1 / 2 .
C F = W l ( Q , Q ) / W l ( U , Q ) , fiber guiding component ,
C S = J l ( U r / ρ ) / J l ( Q r / ρ ) , source resonance component .
θ 0 = cos 1 [ ( α Ω / k cl ) ] .
P = κ | C ( θ 0 ) | 2 ( 1 + cos 2 θ 0 ) J l 2 ( Q 0 r / ρ )
κ ( 1 + cos 2 θ 0 ) J l 2 ( U 0 r / ρ ) ; U 0 Q 0 1 ,
P = κ | C ( θ ) | 2 sin 2 θ 0 J l 2 ( Q 0 r / ρ )
κ sin 2 θ 0 J l 2 ( U 0 r / ρ ) ; U 0 Q 0 1.
F l ( U r / ρ ) = ( 2 π r X ( r ) ) 1 / 2 cos ( π 4 r i t r X ( ξ ) d ξ ) ,
X ( ξ ) = [ k 2 ( ξ ) β 2 ( 1 / ξ ) 2 ] 1 / 2 ,
J ( r , ϕ , z ) = j ˆ g ( z ) f ( r , ϕ ) ,
J = j ˆ g ( z ) cos l ( ϕ ϕ ) δ ( r r ) / 2 π r
E = l = 0 r d r c l ( r ) 0 2 π d ϕ f ( r , ϕ ) × E l F S ( r , r , ϕ , ϕ ) .
J = j ˆ δ ( z z 0 ) δ ( r r 0 ) δ ( ϕ ϕ 0 ) / r 0 ,
E = l = c l ( r 0 ) E l F S ,
E ˜ = i ω A ˜ + ( i ω μ ε cl ) 1 ( t + i β z ˆ ) [ t · A ˜ t + i β A ˜ z ] ,
[ t 2 + k 2 ( x , y ) β 2 ] A ˜ = μ J ˜ ,
E ˜ ( β ) = E ( z ) e i β z d z ; E ( z ) = 1 2 π E ˜ ( β ) e i β z d β .
J ˜ l = j ˆ g ˜ ( β ) cos l ( ϕ ϕ ) δ ( r r ) / 2 π r ,
A ˜ l ( r , ϕ ) = C l ( β ) A ˜ l F S ( r , ϕ ) ,
E ˜ l = C l E ˜ l F S ,
E l = 1 2 π C l ( β ) E ˜ l F S ( β ) e i β z d β .
β = k cl cos θ ,
E l = C l ( β ) E l F S ,
C l = ψ l / ψ l F S ,
[ t 2 + k 2 ( r ) β 2 ] ψ l = δ ( r r ) cos l ( ϕ ϕ ) / 2 π r ,
( ψ l / r ) | r ε r + ε = cos l ( ϕ ϕ ) / 2 π r ,
ψ l = J l ( U r / ρ ) H l ( 1 ) ( Q r / ρ ) cos l ( ϕ ϕ ) / 2 π W l ( U , Q ) ,
U 2 = ρ 2 ( k co β 2 ) ( V / θ c ) 2 ( ρ β ) 2
Q 2 = ρ 2 ( k c 1 2 β 2 ) ( V / θ c ) 2 ( 1 θ c 2 ) ( ρ β ) 2 .
β k co k cl ,
U Q ,