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.
9 Figures and Tables
Fig. 1. Singly and doubly twisted optical fibers.
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. 4. Normalized trajectory of the beam center in the straight section for the divergent case of 9 = A I and 9 d l A1 + A21, where
Fig. 5. Normalized trajectory of the beam center in the straight section for the divergent case of 9 s IA1 I and .9 = IA1 + A21, where A1/9 = 0.01, A2/9 = 0.99, and r/r 2 = 3 are assumed with the input
Fig. 7. Normalized trajectory of the beam center in the circulatory bent section for the divergent case of 9 = I Ai and 9 IA1 + A21, where A1/9 = 10, A2 /9 = -1.10, r/r2 = 3, and Ao 2 R/(g2 r2 ) = 0.025 are assumed with the input conditions
Fig. 8. Normalized trajectory of the beam center in the circularly bent section for the divergent case of 9 # I Ai and = IA1 + A21, where AX/9 = 0.01, A2/9 = 0.99, r/r2 = 3, and A0 2 R/92r2 =
Fig. 9. Maximum total displacement of the beam center normalized by r, for light beams in the straight section with the matched input conditions of Eq. (28).
Table I. Numerical Examples of the Pitches d, and d, Satisfying the Divergent Condition
Table II. Numerical Examples of the Total Displacement of the Beam Trajectory 6 (1)M for Nondivergent Cases
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