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June 1980 / Vol. 5, No. 6 / OPTICS LETTERS 273

Bending-induced birefringence in single-mode fibersR. Ulrich, S. C. Rashleigh,* and W. Eickhoff

Max-Planck-Institut for Festkorperforschung, 7000Stuttgart 80, Federal Republic of GermanyReceived February 19, 1980

In a bent optical fiber, lateral internal stress induces a birefringence, Ob. At X = 0.633 gm in a silica fiber, fb = 7.7X 107 r2 /R2 deg/m, where r is the fiber radius and R is the bending radius.

in single-mode optical fibers is known(1) from deviations of the core geometry fromcircular cylinder and (2) from me-stress through the elasto-optic effect. Thus1 in fibers2 and in twisted fibers.3 In allfirst-order electromagnetic perturbation theorythe well-known first-order elastic stress fields givesolutions. For bending-induced birefrin-

d3b however, no such simple explanation is possi-Several attempts to explain /b gave results too4In this Letter, we point out that bending-inducedessentially a stress effect. It resultslateral, compressive stress -aox that builds up

The well-known dominant stress component 5bent fiber (Fig. 1) is o- = KEx, where K = 1hR is theis Young's modulus. Across the xyu, is a tensile stress in outer (x > 0) layers, butr- is compressive in all inner (x < 0) layers. Conse--aox in thedirection on the inner layers. This pressurehighest on the fiber axis and causes the birefringence,

2The dominant stress a, does not contribute directlybirefringence: No stress effect proportional tocan exist because o- is an odd function of x, whereaselectromagnetic wave functions of the two orthog-

11 modes are even functions.first-order perturbation integrals3 vanish.no geometry effect proportional to K exists,order of approximation the core defor-fourfold symmetry.It is not surprising, therefore, that a second-orderfb -K 2, has been observed. 4 A full the-

theory as well as the evaluationthe elastic stress up to terms -K 2. Of the largeof terms in such an analysis, our present studyAs we find it in good agreement withFor a calculation of the lateral stress in the crosswe postulate the fiber as elastically ho-and isotropic, and we consider the limit of

weak curvature, Kr

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274 OPTICS LETTERS / Vol. 5, No. 6 / June 1980indexn, the birefringencebecomes fib -x-ky t k (On,- 5ny), where k, and ky are the propagation constantsof the HE11 mode polarized along the x and y axes, re-spectively, and k = 27/Xvac. The elasto-optic indexchanges bnx and 5ny are evaluated from the stress oa (0)at the fiber axis by6 6n, = - W/2) 1Pije The Pij de-note the strain-optical coefficients, and the ejare thecomponents of strain. Using Poisson's ratio v, we haveCx Cy= (oa - oy)(I + v)/E,and we arrive at

fib = 0.25kn3(pl - P12)(1 + V)K2r 2 (4)The rigorous integration of Eq. (2) given in the Ap-pendix happens to give exactly the same result.Equation (4) predicts correctly the observed4 variationfib K 2. Moreover, it shows that fib is proportional tothe square of the outer fiber diameter 2r. Our mea-surements on three fibers of different diameters confirmthis dependence (Fig.2).These measurements were performed at -1-2-m-longsections of weakly birefringent single-mode fibers fromvarious manufacturers. To exclude any other stresseffects in the measurements, the fibers were laid out inone or more circular turns on a flat surface and helddown only with high-viscosity oil. He-Ne laser lightwas coupled into one fiber end at 450azimuth relativeto the plane, and the state of polarization at the outputend was analyzed with a Soleil-Babinet compensatorand Glan-Thompson analyzer.Inserting into Eq. (4) the published7 material con-stants of fused silica [n = 1.46;v = 0.17; (P11 - P12) =-0.15], and with X = 0.633 sum,we obtain

fb = -7.7 X 107 K2r2 deg/m. (5)This result should hold universally for weakly guidingsilica fibers, regardless of their core diameters and indexprofiles. The theoreticalprediction is supported by thegood coincidence of our measurements (Fig. 2) with the

rT-'10'0asV 0

m

-z

wCDzIL

mco

0w0CDz3Zzco in

10 3 3.J10-3Fig. 2. Bending birefringence of single-mode silica fibers.The solid line represents the calculated birefringence (Eq. (5)].The points are measurements at 0.633 and 0.676Am usingthree fibers of different origins.

line representing Eq. (5). The birefringence reportedin Ref. 4 seemsto disagree with Eq. (5), but actually theagreement is excellent if the correct 8 fiber diameter, 115/Im, is taken into account.The negative sign in Eq. (5) indicates that the fastaxis of the bending birefringence coincides with theradius of curvature. The slow axis is normal to theplane of curvature. These directions agree with ourobservations.In order to illustrate the absolute magnitude of thebending-induced birefringence, we consider as an ex-ample a fiber of 2r = 100-Arm iameter coiled on a drumof 2R = 4-cm diameter. With K = 0.5 cm-, we find thatfb = 481 deg/m, corresponding to a polarization beatlength of 75 cm. Thus we see that for small coils thebending-induced birefringencebecomescomparable tothe intrinsic birefringence of typical present-day sin-gle-mode fibers.In conclusion, we have explained the bending-in-duced birefringence as a second-order stress effect andhave calculated and measured its magnitude. The re-sult is useful in the design of fiber-optical interferom-eters.AppendixIn the limit Kr

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June 1980 / Vol. 5, No. 6 / OPTICS LETTERS 275* Present address, U.S. Naval Research Laboratory,Washington, D.C. 20375. S. C. Rashleigh was anAlexander von Humboldt Fellow.

References1. D. MNarcuse,Theory of Optical Waveguides (Academic,

New York, 1974); J. D. Love, R. A. Sammut, and A. W.Snyder, Electron. Lett. 15, 615 (1979).2. Y. Namihira, M. Kudo, and Y. Mushiako, Trans. Inst.Chem. Eng. 60 C, 391 (1977).3. R. Ulrich and A. Simon, Appl. Opt. 18,-2241 (1979).

4. A. M. Smith, presented at the Optical CommunicationConference, Amsterdam, The Netherlands, September1979, paper 10.2.5. N. I. Muskhelishvili, "Einige Grundaufgaben zur mathe-matischen Elastizitatstheorie" (Carl Hanser Verlag, Mu-nich, 1971).6. J. F. Nye, Physical Properties of Crystals (Clarendon,Oxford, UK, (1969).7. D. E. Gray, ed., American Institute of Physics Handbook,3rd ed. (McGraw-Hill, New York, 1972).8. A. M. Smith, Central Electricity Research Laboratories,Leatherhead, Surrey KT22 7SE, England, personal com-munication.