Abstract

The present paper analyzes the propagation behavior of light beams along parabolic index optical fibers for the cases where the center axes of the fibers are deformed along helical bends, which are caused when several optical fibers are twisted into a bundle for the purpose of cabling. The analysis is based on geometrical optics and is limited to the case where the center axes of the fibers are bent along a double helix, which arises when two fibers are twisted into a bundle, and the two bundles thus obtained are entwisted once more into a cable. It is also assumed that the center axis of the cable thus established is curved in a circular bend with a constant curvature. Ray equations for this case are derived, and their solutions are studied in detail theoretically and numerically. As a result, conditions are obtained for the occurrence of the divergence phenomenon of the beam trajectory as well as for the matched incidence of light beams to minimize the undulation amplitude of beam trajectories. Moreover, it is clarified that whether the two helices composing the double helix are twisted in the same or opposite directions has somewhat different effects upon the conditions for the divergence phenomenon and the matched incidence as well as the propagation behavior of light beams. Some problems with the application of the present cabling technique to parabolic index optical fibers are also discussed.

© 1976 Optical Society of America

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  1. P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
    [CrossRef]
  2. H. G. Unger, Arch. Elekt. Übertragung 19, 189 (1965).
  3. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [CrossRef]
  4. K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumitomo, Appl. Opt. 13, 255 (1974).
    [CrossRef]
  5. S. Sawa, Reports of Technical Group on Microwaves, Inst. Elect. Commun. Eng. Jpn. 75, 65 (1975).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  7. T. Kitano, A. Ueki, Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn., No. 993 (1973).
  8. S. Sawa, N. Kumagai, IEEE Trans. Microwave Theory Tech., MTT 24, 441 (1976).
    [CrossRef]
  9. K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
    [CrossRef]
  10. I. Nitta, X-ray Crystallography (Maruzen, Tokyo, 1961).
  11. S. Sawa, IEEE Trans. Microwave Theory Tech. MTT-23, 566 (1975).
    [CrossRef]

1976 (1)

S. Sawa, N. Kumagai, IEEE Trans. Microwave Theory Tech., MTT 24, 441 (1976).
[CrossRef]

1975 (2)

S. Sawa, IEEE Trans. Microwave Theory Tech. MTT-23, 566 (1975).
[CrossRef]

S. Sawa, Reports of Technical Group on Microwaves, Inst. Elect. Commun. Eng. Jpn. 75, 65 (1975).

1974 (2)

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumitomo, Appl. Opt. 13, 255 (1974).
[CrossRef]

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

1973 (1)

T. Kitano, A. Ueki, Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn., No. 993 (1973).

1970 (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

1965 (2)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

H. G. Unger, Arch. Elekt. Übertragung 19, 189 (1965).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Fukuyo, H.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Furukawa, M.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumitomo, Appl. Opt. 13, 255 (1974).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Hata, S.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Iga, K.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Ikeda, Y.

Kato, Y.

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Kitano, I.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumitomo, Appl. Opt. 13, 255 (1974).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Kitano, T.

T. Kitano, A. Ueki, Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn., No. 993 (1973).

Koizumi, K.

K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumitomo, Appl. Opt. 13, 255 (1974).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Kumagai, N.

S. Sawa, N. Kumagai, IEEE Trans. Microwave Theory Tech., MTT 24, 441 (1976).
[CrossRef]

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Nitta, I.

I. Nitta, X-ray Crystallography (Maruzen, Tokyo, 1961).

Sawa, S.

S. Sawa, N. Kumagai, IEEE Trans. Microwave Theory Tech., MTT 24, 441 (1976).
[CrossRef]

S. Sawa, Reports of Technical Group on Microwaves, Inst. Elect. Commun. Eng. Jpn. 75, 65 (1975).

S. Sawa, IEEE Trans. Microwave Theory Tech. MTT-23, 566 (1975).
[CrossRef]

Sumitomo, T.

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

Ueki, A.

T. Kitano, A. Ueki, Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn., No. 993 (1973).

Unger, H. G.

H. G. Unger, Arch. Elekt. Übertragung 19, 189 (1965).

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Appl. Opt. (1)

Arch. Elekt. Übertragung (1)

H. G. Unger, Arch. Elekt. Übertragung 19, 189 (1965).

IEEE J. Quantum Electron. (1)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

S. Sawa, N. Kumagai, IEEE Trans. Microwave Theory Tech., MTT 24, 441 (1976).
[CrossRef]

S. Sawa, IEEE Trans. Microwave Theory Tech. MTT-23, 566 (1975).
[CrossRef]

Inst. Elect. Commun. Eng. Jpn. (1)

S. Sawa, Reports of Technical Group on Microwaves, Inst. Elect. Commun. Eng. Jpn. 75, 65 (1975).

Jpn. J. Appl. Phys. (1)

K. Iga, S. Hata, Y. Kato, H. Fukuyo, Jpn. J. Appl. Phys. 13, 79 (1974).
[CrossRef]

Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn. (1)

T. Kitano, A. Ueki, Natl. Conv. Rec. Inst. Elect. Commun. Eng. Jpn., No. 993 (1973).

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, Proc. IEEE 53, 129 (1965).
[CrossRef]

Other (2)

I. Nitta, X-ray Crystallography (Maruzen, Tokyo, 1961).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (9)

Fig. 1
Fig. 1

Singly and doubly twisted optical fibers.

Fig. 2
Fig. 2

Doubly helical bend of a parabolic index optical fiber, in which the center axis D-D′ of the large helix is assumed to be curved in a circular bend with the constant curvature 1/R.

Fig. 3
Fig. 3

Normalized trajectories of the beam center in the straight section for the nondivergent case of G ≠ |λ1| and G ≠ |λ1 + λ2|, where λ1/ G = 001, λ2/ G = 0.10, and r1/r2 = 3, and the input conditions are taken as (I) ξ(0) = 0, ξ′(0) = 0, η(0) = 0, η′(0) = 0; (II) ξ(0) = ξs(0), ξ′(0) = ξs′(0) = η(0) = ηs(0) = 0, η′(0) = ηs′(0); and (III) ξ(0) = 2ξs(0), ξ′(0) = ξs′(0) = η(0) = ηs(0) = 0, η′(0) = ηs′(0).

Fig. 4
Fig. 4

Normalized trajectory of the beam center in the straight section for the divergent case of G = |λ1| and G ≠ |λ1 + λ2|, where λ1/ G = 1, λ2/ G = −1.1, and r1/r2 = 3. The input conditions are taken as ξ(0) = ξ′(0) = η(0) = η′(0) = 0.

Fig. 5
Fig. 5

Normalized trajectory of the beam center in the straight section for the divergent case of G ≠ |λ1| and G = |λ1 + λ2|, where = λ1/ G = 0.01, λ2/ G = 0.99, and r1/r2 = 3 are assumed with the input conditions ξ(0) = ξ′(0) = η(0) = η(0) = η′(0) = 0.

Fig. 6
Fig. 6

Normalized trajectories of the beam center in the circularly bent section for the nondivergent case of G ≠ |λ1| and G ≠ |λ1 + λ2| where (λ1)/ G = 0.01, (λ2)/ G = 0.10, r1/r2 = 3, and (λ02R)/( G 2r2) = 0.025 are assumed together with the input conditions as (I) ξ(0) = ξ′(0) = η(0) = 0, η′(0) = 0; (II) ξ(0)= ξc(0), ξ′(0) = ξc′(0) = 0, η(0) = ηc(0) = 0, η′(0) = ηc′(0); and (III) ξ(0) = 2ξc(0), ξ′(0) = ξc(0) = 0, η(0) = ηc(0) = 0, η′(0) = ηc′(0).

Fig. 7
Fig. 7

Normalized trajectory of the beam center in the circulatory bent section for the divergent case of G = |λ1| and G ≠ |λ1 + λ2|, where λ1/ G = 1.0, λ2/ G = −1.10, r1/r2 = 3, and λ02R/( G 2r2) = 0.025 are assumed with the input conditions ξ(0) = ξ′(0) = η(0) = η′(0) = 0.

Fig. 8
Fig. 8

Normalized trajectory of the beam center in the circularly bent section for the divergent case of G ≠ |λ1| and G = |λ1 + λ2|, where λ1/ G = 0.01, λ2/ G = 0.99, r1/r2 = 3, and λ02R/ G 2r2 = 0.025 are assumed with the input conditions ξ(0)= ξ′(0) = η(0) = η′(0) = 0.

Fig. 9
Fig. 9

Maximum total displacement of the beam center normalized by r2, for light beams in the straight section with the matched input conditions of Eq. (28).

Tables (2)

Tables Icon

Table I Numerical Examples of the Pitches d1 and d2, Satisfying the Divergent Condition g = |λ1| or g = |λ1 + λ2|

Tables Icon

Table II Numerical Examples of the Total Displacement of the Beam Trajectory δ(l)Max for Nondivergent Cases

Equations (36)

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x P = r 1 cos λ 1 l 1 ,             y P = r 1 sin λ 1 l 1 ,             z P = 0
u Q = r 2 cos λ 2 l 2 ,             v Q = r 2 sin λ 2 l 2 ,             w Q = 0 ,
λ i = [ r i 2 + ( d i / 2 π ) 2 ] - 1 / 2 ( i = 1 , 2 ) .
l 1 / l 2 = d 2 λ 2 / ( 2 π ) .
[ x y z ] = [ T 1 ] [ u v w ] + [ x P y P z P ]
[ u v w ] = [ T 2 ] [ ξ η ζ ] + [ u Q v Q w Q ]
[ T i ] = [ cos λ i l i - sin λ i l i cos α i - sin λ i l i sin α i sin λ i l i cos λ i l i cos α i cos λ i l i sin α i 0 - sin α i cos α i ]             ( 1 = 1 , 2 ) .
α i = tan - 1 ( 2 π r i / d i )             ( i = 1 , 2 ) .
x M = R ( 1 - cos λ 0 l 2 ) ,             y M = 0 ,             z M = R sin λ 0 l 2 ,
λ 0 = d 1 d 2 λ 1 λ 2 4 π 2 R
[ x y z ] = [ T 3 ] [ x y z ] + [ R ( 1 - cos λ 0 l 2 ) 0 R sin λ 0 l 2 ,
[ T 3 ] = [ cos λ 0 l 2 0 sin λ 0 l 2 0 1 0 - sin λ 0 l 2 0 cos λ 0 l 2 ] .
x = [ ( ξ + r 2 ) cos ( λ 1 + λ 2 ) l - η sin ( λ 1 + λ 2 ) l + r 1 cos λ r 1 l ] cos λ 0 l + 2 R sin 2 ( λ 0 l / 2 ) , y = ( ξ + r 2 ) sin ( λ 1 + λ 2 ) l + η cos ( λ 1 + λ 2 ) l + r 1 sin λ 1 l , z = - [ ( ξ + r 2 ) cos ( λ 1 + λ 2 ) l - η sin ( λ 1 + λ 2 ) l + r 1 cos λ 1 l ] sin λ 0 l + R sin λ 0 l ,
d d s ( n d r d s ) = grad n
n = n 0 [ 1 - G 2 2 ( ξ 2 + η 2 ) ] ,
d 2 r d l 2 = grad [ 1 - G 2 2 ( ξ 2 + η 2 ) ] ,
[ d 2 x d l 2 + G 2 ( ξ ξ x + η η x ) ] i + [ d 2 y d l 2 + G 2 ( ξ ξ y + η η y ) ] j + [ d 2 z d l 2 + G 2 ( ξ ξ z + η η z ) ] k = 0 ,
d 2 ξ d l 2 - 2 ( λ 1 + λ 2 ) d η d l + [ G 2 - ( λ 1 + λ 2 ) 2 ] ξ = r 2 ( λ 1 + λ 2 ) 2 + r 1 λ 1 2 cos λ 2 l - λ 0 2 R cos ( λ 1 + λ 2 ) l
d 2 η d l 2 + 2 ( λ 1 + λ 2 ) d ξ d l + [ G 2 - ( λ 1 + λ 2 ) 2 ] η = - r 1 λ 1 2 sin λ 2 l + λ 0 2 R sin ( λ 1 + λ 2 ) l ,
r 1 R ,             r 2 + ξ R ,             η R ,             α i 1             ( i = 1 , 2 ) .
ξ ( l ) = r 2 p 2 G 2 - p 2 + r 1 λ 1 2 G 2 - λ 1 2 cos λ 2 l - λ 0 2 R G 2 cos p l - 1 2 G [ A 1 cos ( G + p ) l + B 1 cos ( G - p ) l - C sin ( G + p ) l + D sin ( G - p ) l ] ,
η ( l ) = - r 1 λ 1 2 G 2 - λ 1 2 sin λ 2 l - λ 0 2 R G 2 sin p l + 1 2 G [ A 1 sin ( G + p ) l - B 1 sin ( G - p ) l + C cos ( G + p ) l + D cos ( G - p ) l ] ,
A 1 = r 1 λ 1 2 G + λ 1 + r 2 p 2 G + p - λ 0 2 R G - ( G - p ) ξ ( 0 ) + η ( 0 ) B 1 = r 1 λ 1 2 G - λ 1 + r 2 p 2 G - p - λ 0 2 R G - ( G + p ) ξ ( 0 ) - η ( 0 ) C = ( G - p ) η ( 0 ) + ξ ( 0 ) ,             D = ( G + p ) η ( 0 ) - ξ ( 0 ) }
p = λ 1 + λ 2 .
{ ξ ( 0 ) = ξ c ( 0 ) r 1 λ 1 2 G 2 - λ 1 2 + r 2 p 2 G 2 - p 2 - λ 0 2 R G 2 , ξ ( 0 ) = ξ c ( 0 ) 0 , η ( 0 ) = η c ( 0 ) 0 , η ( 0 ) = η c ( 0 ) - r 1 λ 1 2 λ 2 G 2 - λ 1 2 + λ 0 2 R G 2 p .
{ ξ ( l ) = r 2 p 2 G 2 - p 2 + r 1 λ 1 2 G 2 - λ 1 2 cos λ 2 l - λ 0 2 R G 2 cos ( λ 1 + λ 2 ) l , η ( l ) = - r 1 λ 1 2 G 2 - λ 1 2 sin λ 2 l + λ 0 2 R G 2 sin ( λ 1 + λ 2 ) l .
ξ ( l ) = r 2 p 2 G 2 - p 2 + r 1 λ 1 2 2 G l sin ( G - p ) l - λ 0 2 R G 2 cos p l - 1 2 G [ A 2 ± cos ( G + p ) l + B 2 ± cos ( G - p ) l - C sin ( G + p ) l + D sin ( G - p ) l ] ,
η ( l ) = ± r 1 λ 1 2 2 G l cos ( G - p ) l + λ 0 2 R G 2 sin p l + 1 2 G [ A 2 ± sin ( G + p ) l - B 2 ± sin ( G - p ) l + C cos ( G + p ) l + D cos ( G - p ) l ] ,
A 2 ± = r 2 p 2 G + p ± r 1 λ 1 2 2 G - λ 0 2 R G - ( G - p ) ξ ( 0 ) + η ( 0 ) , B 2 ± = r 2 p 2 G - p r 1 λ 1 2 2 G - λ 0 2 R G - ( G + p ) ξ ( 0 ) - η ( 0 ) ,
ξ ( l ) = r 1 λ 1 2 G 2 - λ 1 2 cos λ 2 l - λ 0 2 R G 2 cos p l - 1 2 G { [ r 1 λ 1 2 G ± λ 1 + r 2 p 2 2 G ± η ( 0 ) - λ 0 2 R G ] cos 2 G l - ξ ( 0 ) sin 2 G l + r 1 λ 1 2 G λ 1 - r 2 p 2 2 G - λ 0 2 R G - 2 G ξ ( 0 ) η ( 0 ) } ,
η ( l ) = - r 1 λ 1 2 G 2 - λ 1 2 sin λ 2 l r 2 p 2 2 G l + λ 0 2 R G 2 sin p l ± 1 2 G { [ r 1 λ 1 2 G ± λ 1 + r 2 p 2 2 G ± η ( 0 ) - λ 0 2 R G ] sin 2 G l + ξ ( 0 ) cos 2 G l ± 2 G η ( 0 ) - ξ ( 0 ) } ,
{ ξ ( 0 ) = ξ s ( 0 ) r 1 λ 1 2 G 2 - λ 1 2 + r 2 p 2 G 2 - p 2 ' ξ ( 0 ) = ξ s ( 0 ) 0 , η ( 0 ) = η s ( 0 ) 0 ,             η ( 0 ) = η s ( 0 ) - r 1 λ 1 2 λ 2 G 2 - λ 1 2 .
G = λ 1             G = λ 1 + λ 2
d 1 2 π G             d 1 d 2 d 1 + d 2 2 π G
δ ( l ) = [ ( r 2 p 2 G 2 - p 2 ) 2 + ( r 1 λ 1 2 G 2 - λ 1 2 ) 2 + 2 r 1 r 2 p 2 λ 1 2 ( G 2 - p 2 ) ( G 2 - λ 1 2 ) cos λ 2 l ] 1 / 2
δ ( l ) Max = | r 1 λ 1 2 G 2 - λ 1 2 | + | r 2 ( λ 1 + λ 2 ) 2 G 2 - ( λ 1 + λ 2 ) 2 | .

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