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Universal Journal of Applied Mathematics Vol. 13(1), pp. 14 - 21
DOI: 10.13189/ujam.2025.130102
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Strict First Order Theory of Macrobending Transmission Loss in Step-Index Fiber-Optic Cables

Sujit K. Bose ^*
S. N. Bose National Center for Basic Sciences, Salt Lake City, Kolkata 700106, India

ABSTRACT

Transmission of data by fiber-0ptic cables over long distances in particular, is a preferred channel of communication. However, it is known that bends in such cables lead to loss of power of the carrier light wave traveling towards the desired destination, thus deteriorating the delivery process of the data packets. An analytical theory of this power loss due to the geometric macrobending only, ignoring altogether the microbending processes of microcracking and thermal heating, was presented by this author in a recent paper, in which a simplifying approximation was assumed. A simple, yet important in practice, step-index fiber with a homogeneous core and a cladding of slightly less refractive index was considered. The bend in question was ideally configured as a circular tore, and simple toroidal coordinates were employed to formulate the governing Maxwell equations for the propagating light wave. Moreover, the solution of the Maxwell equations was expressed in terms of the single Hertz vector , and as the propagation takes place in the axial direction, that equation reduces to a single toroidal wave equation for the axial component . Because of the fact that optical fibers are very thin, the toroidal equation was then approximated by a cylindrical wave equation. This lacuna is removed in this paper, and the full toroidal wave equation is treated strictly to the first order in the radial coordinate , superseding the earlier approximate theory. The present extended first order theory however yields exactly the results reported in the earlier approximate theory, in as much as the additional terms in the electric and magnetic intensities of the field, obtained from the solution for do not unexpectedly contribute to the bendloss formulae. Accordingly, as was found in the earlier approximate theory, it means that when the radius of curvature of the bend exceeds a certain critical value , the bendloss of power varies linearly with practically independent of , while if , it varies as , depending on as a factor of the form . In practice therefore, bends in a fiber-optic cable, particularly formation of unnecessary small coils, is best avoided to the extent permitted by the architecture of a given application.

KEYWORDS
Fiber-Optic Cable, Macrobending Loss, Hertz Vector, Simple Toroidal Coordinates

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Sujit K. Bose , "Strict First Order Theory of Macrobending Transmission Loss in Step-Index Fiber-Optic Cables," Universal Journal of Applied Mathematics, Vol. 13, No. 1, pp. 14 - 21, 2025. DOI: 10.13189/ujam.2025.130102.

(b). APA Format:
Sujit K. Bose (2025). Strict First Order Theory of Macrobending Transmission Loss in Step-Index Fiber-Optic Cables. Universal Journal of Applied Mathematics, 13(1), 14 - 21. DOI: 10.13189/ujam.2025.130102.