Noncircular fibers elliptical

Most of the chapter is devoted to the construction of ray paths and their classification on circular fibers with axisymmetric profiles. However, we also consider noncircular fibers since cross-sections can differ from circular symmetry in practice, e.g. elliptical fibers. Finally, since this chapter parallels Chapter 1 to a large extent, it may be helpful to compare the results of corresponding sections. 

When the asymmetry is slight, it is sometimes possible to simplify the above analysis by treating the noncircular fiber as a small perturbation of a circular fiber. Thus, for example, the ray invariant I of the circular fiber can be replaced by an approximate invariant l(z) which varies very slowly along the noncircular fiber. The spatial transient on the elliptical, clad parabolic-profile fiber can be determined within this approximation. Details are given elsewhere [13]. 

We can use the elliptical fiber to quantify the transition from the uniformly polarized modes of the noncircular fiber to the modes of the circular fiber in Table 13-1, page 288, which was discussed qualitatively in Section 13-9. For this purpose we consider the modes corresponding to the 4 21 and 4 12 solutions, which are the successive lowest-order modes after the fundamental modes. Thus Eqs. (16-14) and (37-107) give to within constant multiples... 

The basis of the Gaussian approximation for circular fibers is the observation that the fundamental-mode field distribution on an arbitrary profile fiber is approximately Gaussian. Coupled with the fact that the same field on an ihfinite parabolic-profile fiber is exactly Gaussian, the approximation fits the field of the arbitrary profile fiber to the field of an infinite parabolic-profile fiber. The optimum fit is found by the variational procedure described in Section 15-1. Now in Chapter 16, we showed that the fundamental-mode field distribution on an elliptical fiber with an infinite parabolic profile has a Gaussian dependence on both spatial variables in the cross-section. Accordingly, we fit the field of such a profile to the unknown field of the noncircular fiber of arbitrary profile by a similar variational procedure, as we show below [1, 2],... 

The polarization corrections, and SPy, to the scalar propagation constant P for the Xq- and yo-polarized modes on the perturbed, noncircular fiber are in general unequal, and their difference describes the anisotropic, or birefringent, nature of propagation. This is of basic interest for the two fundamental modes on single-mode fibers. The calculation of the corrections from the formula in Table 13-1, page 288, requires first-order corrections to the approximation We derive these corrections for the slightly elliptical fiber in Section 18-10. 

We showed in Section 2-13 that the transit time for a step-profile fiber is independent of the cross-sectional geometry. Consequently Eqs. (3-2) and (3-3) give the ray dispersion for step-profile fibers of arbitrary cross-section. We also found in Section 2-13 that the ray transit time for the noncircular, clad power-law profiles of Eq. (2-55) is identical to the transit time for the symmetric, clad power-law profiles in Table 2-1, page 40, i.e. dependent on only. Thus Eqs. (3-8) and (3-9) also give the optimum profile and minimum pulse spread for those noncircular profiles [5], which includes the clad parabolic-profile fiber of elliptical cross-section. In other words, ray dispersion on step-profilefibers of arbitrary cross-section and clad power-law profilefibers of noncircular cross-section is also given by the corresponding solutions for planar waveguides. 

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