Abstract

An exact analysis using the Green’s function formulation of an arbitrarily oriented off-axis dipole in a three-layer cylindrical dielectric waveguide is carried out. The practical significance of such a formulation is illustrated by its immediate application to the problem of radiation due to scattering from an arbitrarily located discrete inhomogeneity in step-index fibers in which the dominant HE11 mode propagates. For small dielectric differences between the adjacent layers, the analysis yields radiation loss that agrees well with that evaluated from the infinite-medium approximation. However, the spatial distribution of the radiation loss differs markedly from that predicted by the infinite-medium approximation.

© 1980 Optical Society of America

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References

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  1. F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).
  2. J. W. Duncan, “The efficiency of excitation of a surface mode on a dielectric cylinder,” IRE Trans. MTT-7, 257–268 (1959).
    [Crossref]
  3. J. Brown and H. Stachera, “The launching of an axial cylindrical surface wave,” Proc. IEEE 109C, 18–25 (1962).
  4. A. W. Snyder, “Excitation and scattering of modes on a dielectric rod or optical fiber,” IEEE Trans. MTT-17, No. 12, 1138–1144 (1969).
  5. P. J. B. Clarricoats and K. B. Chan, “Propagation behavior of cylindrical dielectric-rod waveguides,” Proc. IEEE 120, No. 11, 1371–1378 (1973).
  6. G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. MTT-18, No. 12, 1033–1041 (1970).
  7. G. L. Yip and J. Martucci, “Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. I: radiation loss,” Appl. Opt 17, 2131–2136 (1976).
    [Crossref]
  8. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1968), p. 361.
  9. A. Safaai-Jazi, “A study of mode classification and scattering from an off-axis inhomogeneity in step-index optical fiber,” Doctoral dissertation, Dept. of E. E., McGill University, Ch. 4, May, 1978 (unpublished).
  10. Reference 9, Chap. 5.
  11. A. Safaai-Jazi and G. L. Yip, “Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber,” IEEE-MTT-S International Microwave Symposium, Session on “Fiber and Integrated Optics,” Digest, 113–115 (June, 1978).
  12. S. Kawakami and S. Nishida, “Characteristics of a doubly-clad optical fiber with a low-index inner cladding,” IEEE Trans. J. Quantum Electron. QE-10, 879–887, (1974).
    [Crossref]
  13. A. R. Tynes, A. D. Pearson, and D. L. Bisbee, “Loss mechanisms and measurements in clad glass fibers and bulk glass,” J. Opt. Soc. Am. 61, 143–153, (1971).
    [Crossref]

1976 (1)

G. L. Yip and J. Martucci, “Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. I: radiation loss,” Appl. Opt 17, 2131–2136 (1976).
[Crossref]

1974 (1)

S. Kawakami and S. Nishida, “Characteristics of a doubly-clad optical fiber with a low-index inner cladding,” IEEE Trans. J. Quantum Electron. QE-10, 879–887, (1974).
[Crossref]

1973 (1)

P. J. B. Clarricoats and K. B. Chan, “Propagation behavior of cylindrical dielectric-rod waveguides,” Proc. IEEE 120, No. 11, 1371–1378 (1973).

1971 (1)

1970 (2)

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. MTT-18, No. 12, 1033–1041 (1970).

F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).

1969 (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric rod or optical fiber,” IEEE Trans. MTT-17, No. 12, 1138–1144 (1969).

1962 (1)

J. Brown and H. Stachera, “The launching of an axial cylindrical surface wave,” Proc. IEEE 109C, 18–25 (1962).

1959 (1)

J. W. Duncan, “The efficiency of excitation of a surface mode on a dielectric cylinder,” IRE Trans. MTT-7, 257–268 (1959).
[Crossref]

Bisbee, D. L.

Brown, J.

J. Brown and H. Stachera, “The launching of an axial cylindrical surface wave,” Proc. IEEE 109C, 18–25 (1962).

Chan, K. B.

P. J. B. Clarricoats and K. B. Chan, “Propagation behavior of cylindrical dielectric-rod waveguides,” Proc. IEEE 120, No. 11, 1371–1378 (1973).

Clarricoats, P. J. B.

P. J. B. Clarricoats and K. B. Chan, “Propagation behavior of cylindrical dielectric-rod waveguides,” Proc. IEEE 120, No. 11, 1371–1378 (1973).

Duncan, J. W.

J. W. Duncan, “The efficiency of excitation of a surface mode on a dielectric cylinder,” IRE Trans. MTT-7, 257–268 (1959).
[Crossref]

Kapron, F. R.

F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).

Kawakami, S.

S. Kawakami and S. Nishida, “Characteristics of a doubly-clad optical fiber with a low-index inner cladding,” IEEE Trans. J. Quantum Electron. QE-10, 879–887, (1974).
[Crossref]

Keck, D. E.

F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).

Martucci, J.

G. L. Yip and J. Martucci, “Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. I: radiation loss,” Appl. Opt 17, 2131–2136 (1976).
[Crossref]

Maurer, R. D.

F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).

Nishida, S.

S. Kawakami and S. Nishida, “Characteristics of a doubly-clad optical fiber with a low-index inner cladding,” IEEE Trans. J. Quantum Electron. QE-10, 879–887, (1974).
[Crossref]

Pearson, A. D.

Safaai-Jazi, A.

A. Safaai-Jazi and G. L. Yip, “Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber,” IEEE-MTT-S International Microwave Symposium, Session on “Fiber and Integrated Optics,” Digest, 113–115 (June, 1978).

A. Safaai-Jazi, “A study of mode classification and scattering from an off-axis inhomogeneity in step-index optical fiber,” Doctoral dissertation, Dept. of E. E., McGill University, Ch. 4, May, 1978 (unpublished).

Snyder, A. W.

A. W. Snyder, “Excitation and scattering of modes on a dielectric rod or optical fiber,” IEEE Trans. MTT-17, No. 12, 1138–1144 (1969).

Stachera, H.

J. Brown and H. Stachera, “The launching of an axial cylindrical surface wave,” Proc. IEEE 109C, 18–25 (1962).

Tynes, A. R.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1968), p. 361.

Yip, G. L.

A. Safaai-Jazi and G. L. Yip, “Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber,” IEEE-MTT-S International Microwave Symposium, Session on “Fiber and Integrated Optics,” Digest, 113–115 (June, 1978).

G. L. Yip and J. Martucci, “Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. I: radiation loss,” Appl. Opt 17, 2131–2136 (1976).
[Crossref]

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. MTT-18, No. 12, 1033–1041 (1970).

Appl. Opt (1)

G. L. Yip and J. Martucci, “Scattering from a localized inhomogeneity in a cladded fiber optical waveguide. I: radiation loss,” Appl. Opt 17, 2131–2136 (1976).
[Crossref]

Digest (1)

A. Safaai-Jazi and G. L. Yip, “Scattering from an arbitrarily located off-axis inhomogeneity in a step-index optical fiber,” IEEE-MTT-S International Microwave Symposium, Session on “Fiber and Integrated Optics,” Digest, 113–115 (June, 1978).

IEE Conf. Publ. London (1)

F. R. Kapron, D. E. Keck, and R. D. Maurer, “Radiation losses in glass optical waveguides,” IEE Conf. Publ. London 71, 148–153, (1970).

IEEE Trans. (2)

A. W. Snyder, “Excitation and scattering of modes on a dielectric rod or optical fiber,” IEEE Trans. MTT-17, No. 12, 1138–1144 (1969).

G. L. Yip, “Launching efficiency of the HE11surface wave mode on a dielectric rod,” IEEE Trans. MTT-18, No. 12, 1033–1041 (1970).

IEEE Trans. J. Quantum Electron. (1)

S. Kawakami and S. Nishida, “Characteristics of a doubly-clad optical fiber with a low-index inner cladding,” IEEE Trans. J. Quantum Electron. QE-10, 879–887, (1974).
[Crossref]

IRE Trans. (1)

J. W. Duncan, “The efficiency of excitation of a surface mode on a dielectric cylinder,” IRE Trans. MTT-7, 257–268 (1959).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

J. Brown and H. Stachera, “The launching of an axial cylindrical surface wave,” Proc. IEEE 109C, 18–25 (1962).

P. J. B. Clarricoats and K. B. Chan, “Propagation behavior of cylindrical dielectric-rod waveguides,” Proc. IEEE 120, No. 11, 1371–1378 (1973).

Other (3)

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, Cambridge, England, 1968), p. 361.

A. Safaai-Jazi, “A study of mode classification and scattering from an off-axis inhomogeneity in step-index optical fiber,” Doctoral dissertation, Dept. of E. E., McGill University, Ch. 4, May, 1978 (unpublished).

Reference 9, Chap. 5.

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Figures (9)

FIG. 1
FIG. 1

(a) Geometry of a multilayer cylindrical dielectric waveguide embodying a dipole moment; (b) current density components.

FIG. 2
FIG. 2

Transformation of coordinates in the z = 0 plane.

FIG. 3
FIG. 3

Contour of integration in the complex β ¯ plane.

FIG. 4
FIG. 4

Normalized radiated power P ¯ r versus the normalized frequency V for a cladded fiber with r1 = 2.341, r2 = 2.250, r3 = 1., ρ2/ρ1 = 5, and the HE11 mode incident. — exact, ---- infinite-medium approximation.

FIG. 5
FIG. 5

(A) Normalized radiated power P ¯ r versus ρ0/ρ1 for a cladded fiber with r1 = 2.341, r2 = 2.25, r3 = 1. and ρ2/ρ1 = 5. (a) HE11, (b) HE12, (c) EH11, (d) HE21, (e) TM01 incident. (B) Normalized radiated power P ¯ r versus ρ0/ρ1, (a) for the cladded fiber with ρ2/ρ1 = 5; (b) for the cladded fiber with ρ2/ρ1 = 20; (c) for the infinite-cladding fiber, ρ2/ρ1 = ∞.

FIG. 6
FIG. 6

Radiation patterns for cladded fibers with r1 = 2.341, r2 = 2.25, and r3 = 1., at V= 2, ρ0 = 0, and ϕ0 = 0, for ρ2/ρ1 = 5, (a) P ¯ ( θ , 0 ° ), (b) P ¯ ( θ , 90 ° ), and for ρ2/ρ1 = 20, (c) P ¯ ( θ , 0 ° ), (d) P ¯ ( θ , 90 ° ).

FIG. 7
FIG. 7

(A) Normalized radiated power versus frequency for a W-type fiber with r1 = 2.250, r2 = 1.822, r3 = 2.205, and ρ2/ρ1 = 1.5. ϕ0 = 0° and HE11 mode is incident; (B) radiation patterns for a W-type fiber with r1 = 2.250, r2 = 1.822, r3 = 2.205 and ρ2/ρ1 = 1.5 at V = 10, ϕ0 = 0°, and ρ0 = 0. (i) P ¯ ( θ , 0 ° ), (ii) P ¯ ( θ , 90 ° ).

FIG. 8
FIG. 8

(A) Normalized radiated power versus V for a dielectric tube with r1 = r3 = 2.250, r2 = 2.341, and ρ2/ρ1 = 2, HE11 mode is incident and ϕ0 = 0°. (B) Radiation patterns for a dielectric tube with r1 = r3 = 2.25, r2 = 2.341 and ρ2/ρ1 = 2., at V = 2., ρ0 = 0, ϕ0 = 0°. (i) P ¯ ( θ , 0 ° ), (ii) P ¯ ( θ , 90 ° ).

FIG. 9
FIG. 9

Normalized radiated powers versus ρ0/ρ1. (a) For a cladded fiber with r1 = 2.341, r2 = 2.25, r3 = 1, and ρ2/ρ1 = 20 at V = 2., (b) for a W-type fiber with r1 = 2.25, r2 = 1.822, r3 = 2.205, and ρ2/ρ1 = 1.5 at V = 10., (c) for a tube with r1 = r3 = 2.25, r2 = 2.341, and ρ2/ρ1 = 2., at V = 2. HE11 mode incident and ϕ0 = 0°.

Equations (25)

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J 1 = ( J ρ ρ ˆ 0 + J ϕ ϕ ˆ 0 + J z z ˆ 0 ) · δ ( ρ ρ 0 ) ,
J 1 = { [ J ρ cos ( ϕ ϕ 0 ) + J ϕ sin ( ϕ ϕ 0 ) ] ρ ˆ + [ J ϕ cos ( ϕ ϕ 0 ) J ρ sin ( ϕ ϕ 0 ) ] ϕ ˆ + J z z ˆ } · δ ( ρ ρ 0 ) .
( 2 + k ¯ i 2 ) [ H E ] = { × J j ( k 0 μ r i J + · · J / k 0 r i ) , ρ i 1 < ρ < p i
( 2 + k ¯ p 2 ) [ H E ] = 0 elsewhere ( i p )
ρ = [ ρ 2 + ρ 0 2 2 ρ ρ 0 cos ( ϕ ϕ 0 ) ] 1 / 2 , ϕ = ϕ ϕ 0 + ϕ 1 = π ϕ 2 .
J = μ 0 [ ( J ρ cos ϕ + J ϕ sin ϕ ) ρ ˆ + ( J ϕ cos ϕ J ρ sin ϕ ) ϕ ˆ + J z z ˆ ] δ ( ρ ) δ ( z ) / 2 π ρ .
f ( ρ , ϕ , β ) = F ( ρ , ϕ , z ) e j β z d z .
( t 2 + k i 2 ) ( H z ( ρ , ϕ β ¯ ) E z ( ρ , ϕ , β ¯ ) ) = ( j μ 0 4 π d d ρ ( δ ( ρ ) ρ ) m = 1 , 1 m ( J ρ j m J ϕ ) e j m ϕ μ 0 [ j k i 2 J z 2 π k 0 r i δ ( ρ ) ρ + β ¯ 4 π r i · d d ρ ( δ ( ρ ) ρ ) m = 1 , 1 ( J ρ j m J ϕ ) e j m ϕ ] ) ,
H z i p ( ρ , ϕ , β ¯ ) = T 0 m = 1 , 1 1 2 ( j ρ j m J ϕ ) Y m ( k i ρ ) e j m ϕ ,
E z i p ( ρ , ϕ , β ¯ ) = T 0 ( k i J z k 0 r i Y 0 ( k i ρ ) + m = 1 , 1 j m β ¯ 2 r i ( J ρ j m J ϕ ) Y m ( k i ρ ) e j m ϕ ) ,
e j ν ϕ 1 Z ν ( k i ρ ) = n = Z n + ν ( k i ρ ) J n ( k i ρ 0 ) e j n α ,
E z 1 , n ( ρ , ϕ , β ¯ ) = [ ( R n 1 + a n 1 ) J n ( k 1 ρ ) ] F n H z 1 , n ( ρ , ϕ , β ¯ ) = [ ( S n 1 + b n 1 ) J n ( k 1 ρ ) ] F n 0 ρ ρ 0 ,
E z 1 , n ( ρ , ϕ , β ¯ ) = [ R n 1 Y n ( k 1 ρ ) + a n 1 J n ( k 1 ρ ) ] F n H z 1 , n ( ρ , ϕ , β ¯ ) = [ S n 1 Y n ( k 1 ρ ) + b n 1 J n ( k 1 ρ ) ] F n ρ 0 ρ ρ 1 ;
E z 2 , n ( ρ , ϕ , β ¯ ) = [ a n 2 J n ( k 2 ρ ) + a n 3 Y n ( k 2 ρ ) ] F n H z 2 , n ( ρ , ϕ , β ¯ ) = [ b n 2 J n ( k 2 ρ ) + b n 3 Y n ( k 2 ρ ) ] F n ρ 1 ρ ρ 2 ;
E z 3 , n ( ρ , ϕ , β ¯ ) = [ a n 4 H n ( 1 ) ( k 3 ρ ) ] F n H z 3 , n ( ρ , ϕ , β ¯ ) = [ b n 4 H n ( 1 ) ( k 3 ρ ) ] F n ρ 2 ρ ,
E z 1 , n ( ρ , ϕ , β ¯ ) = [ a n 1 J n ( k 1 ρ ) ] F n H z 1 , n ( ρ , ϕ , β ¯ ) = [ b n 1 J n ( k 1 ρ ) ] F n 0 ρ ρ 1 ;
E z 2 , n ( ρ , ϕ , β ¯ ) = [ ( R n 2 + a n 2 ) J n ( k 2 ρ ) + a n 3 Y n ( k 2 ρ ) ] F n H z 2 , n ( ρ , ϕ , β ¯ ) = [ ( S n 2 + b n ) J n ( k 2 ρ ) + b n 3 Y n ( k 2 ρ ) ] F n ρ 1 ρ ρ 0 ,
E z 2 , n ( ρ , ϕ , β ¯ ) = [ a n 2 J n ( k 2 ρ ) + ( R n 2 + a n 3 ) Y n ( k 2 ρ ) ] F n H z 2 , n ( ρ , ϕ , β ¯ ) = [ b n 2 J n ( k 2 ρ ) + ( S n 2 + b n 3 ) Y n ( k 2 ρ ) ] F n ρ 0 ρ ρ 2 .
f ( ρ , ϕ , z ) = k 0 2 π F ( ρ , ϕ , β ¯ ) e j k 0 β ¯ z d β ¯ ,
= c 1 c 2 + 2 π j residues .
P r = 1 2 μ 0 0 0 2 π 0 π P ( r , ϕ , θ ) r 2 sin θ d ϕ d θ ,
P ( r , ϕ , θ ) = R e [ E θ ( r , ϕ , θ ) H ϕ * ( r , ϕ , θ ) E ϕ ( r , ϕ , θ ) H θ * ( r , ϕ , θ ) ] .
P r = 1 π μ 0 0 n = × β ¯ r β ¯ r [ r 3 | a n 4 ( β ¯ ) | 2 + μ r 3 | b n 4 ( β ¯ ) | 2 ] d β ¯ ( β ¯ r 2 β ¯ 2 ) .
× H = j k 0 r i E + J i = 1 or 2 ,
J = j k 0 Δ r E = j k 0 Δ r Δ υ δ ( ρ ρ 0 ) E i .