Noncircular fibers

Covering power The ability of fibers to occupy space. Noncircular fibers have a greater covering power than circular fibers. 

The cross-sectional shape of a fiber can affect many properties, e.g. luster, density, optical properties, feel of the fabric and an important characteristic called the covering power of a fabric. Covering power is the ability of fibers to occupy space. The reader can easily visualize that fibers having a circular cross-section will have a lesser covering power than fibers having a lobed cross-section. Noncircular fibers can provide a greater density in a fabric than circular fibers. 

This suggests that the calculated circular diameter for Mongolian and Caucasian hair averages approximately 11 and 15%, respectively, from the major and minor axes of noncircular fibers. In most circumstances, the assumption of circularity is an acceptable approximation however, this deviation averages approximately 38% for Ethiopian hair, and, therefore, the assumption of circularity for Ethiopian hair is generally not an acceptable approximation. 

Cross-section. The cross-section of a fiber can be observed using a microscope. It has been found that the cross-sectional shape of a fiber can have a significant effect on its thermal insulation characteristics (Varshney et al., 2011). A fiber s cross-section with more trapped air may provide higher thermal insulation than a perfectly cylindrical fiber. For example, a hollow fiber traps more air inside its structure than a solid circular fiber. This is the reason why hoUow-fiber based fabrics can provide higher thermal insulation than solid circular-fiber based fabrics. In the same fashion, a noncircular fiber, say with a trilobal or scalloped oval surface, can trap more air than a circular fiber, because of its shape. Relatively large amounts of air trapped by noncircular fibers ultimately enhance thermal insulation characteristics (Matsudaira et al., 1993 Murakami etal, 1978). 

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. 

We consider noncircular fibers with profiles expressible in terms of an arbitrary homogeneous function f (x, y) in the form... 

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]. 

The derivation of the transverse fields of the two fundamental modes on a weakly guiding, noncircular fiber was described in Section 13-5. These fields are given in Table 16-1 in terms of the solution T(x,y) of the scalar wave equation, which in cartesian coordinates has the form... 

The procedure for constructing higher-order modes of noncircular fibers was established in Section 13-8. For each mode the transverse fields are identical in direction and form to the fundamental-mode fields of Table 16-1, except that F now denotes the appropriate higher-order solution of the scalar wave equation of Eq. (16-3). Only when the fiber cross-section is sufficiently close to circular is this representation inappropriate, as explained in Section 13-9. We quantify this transition in the following section. 

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],... 

We define a set of infinite power-law profiles for noncircular fibers relative to cartesian axes in the cross-section by... 

Section 17-3 Gaussian approximation for noncircular fibers 371 17-3 Slight eccentricity... 

In many of the examples below, the unperturbed fiber has a circularly symmetric cross-section and profile. For perturbations which maintain this symmetry, the polarization of each mode is unchanged, and obeys the rules laid down in Sections 13-4 and 13-7. However, for perturbations which break circular symmetry, the modes are polarized along the optical axes and y of the perturbed fiber. Only the fundamental and HE, modes remain plane polarized on the circular and noncircular fibers. If the perturbation is sufficiently small, the polarization of all other modes lies in the transition region between the circular and noncircular situations, as discussed in Section 13-9. 

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. 

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