Elliptical fibers

This scheme assumes that the density of all hair fibers is the same. It requires a minimum of manipulations and is an excellent averaging technique for dry state dimensions of hair fibers. Cross-sectional area and volume estimates for circular and elliptical fibers should be relatively accurate, as well as diameter and radius for round fibers. This method does not provide an indication of ellipticity but provides an average diameter with respect to length as well as to cross section (average diameter, not maximum or minimum diameter). The deviation of fiber diameter with increasing ellipticity is described in Table 8-9. 

Besides circular and elliptical fiber cross sections, that work investigated the effects of more complex, realistic shapes depicted in Fig. 5.3 below... 

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. 

Plane-wave decomposition of the modal fields 12-13 Multilayered and elliptical fibers... 

The step-profile waveguide has a core of uniform refractive index n, surrounded by a cladding of uniform refractive index n, which is assumed unbounded. Thus the only variation in profile is a step, or jump, discontinuity at the core-cladding interface in Fig. 11-1 (a). This profile has exact analytical solutions for the modal fields on planar waveguides, circularly symmetric fibers and elliptical fibers. 

Fig. 16-1 Contours of constant refractive index for the elliptical fiber of infinite parabolic profile defined in Eq. (16-1).

Table 16-1 Fundamental modes of the elliptical fiber of infinite parabolic profile.

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... 

When IAI < 1, the modal fields behave like the fields of the circular fiber, i.e. the linear combinations of Table 14-1, page 304, and when A > 1 the modal fields are uniformly polarized on the elliptical fiber. Since A 1 for a practical fiber, it is clear that only the slightest asymmetry is necessary for the modes to be uniformly polarized, and thus the circular fiber is an ideal which requires high precision to be realized. 

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],... 

By analogy with the fundamental-mode solution for the elliptical fiber of infinite parabolic profile given in Table 16-1, page 356, we assume that the solution 4 (x, y) of the scalar wave equation of Eq. (16-3) can be approximated by setting [1, 2]... 

In the cross-section of an elliptical fiber, contours of constant refractive index are closed ellipses. If the contours all have the same eccentricity, the fiber... 

To relate the elliptical fiber parameters to the circular fiber parameters, we assume the profile of Eq. (17-8) is clad and has the same profile volume 2 as the corresponding circular fiber. Substituting into Eq. (14—42) and integrating over the elliptical core xjp Y + (ylPyY 1, we find with the help of Eq. (17-9) that... 

If we fix Py and let p - co, the elliptical fiber is transformed into the symmetric planar waveguide of Fig. 12-1. In this limit, the spot size oo, but the normalized spot size A = ajp -r 0. See, for example, the expression for A in Eq. (17-5). Hence both a and b of Eq. (17-11) become unbounded. If we substitute the asymptotic forms of Eq. (37-88) for the modified Bessel functions into the second equation in Eq. (17-20) and retain lowest-order terms, we deduce... 

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. 

A step-profile, circular fiber with core and cladding indices n and is deformed into the slightly elliptical fiber of Fig. 18-2(a) in such a way that the core cross-sectional area is unchanged. If the semi-major and semi-minor axes of the elliptical fiber have lengths Px und Py, and e is the eccentricity, then... 

For the fundamental modes on a circular fiber, is independent of 0. Consequently —tiy and p = p in Eq. (18-21). In other words, the propagation constants for the circular and elliptical fibers are identical for slight eccentricity, provided the core areas are equal [6]. The latter condition is equivalent to requiring equal profile volumes, as is clear from Eq. (17-13). Hence the present result is consistent with the more general result of Section 17-3, which showed that, within the Gaussian approximation, P = on an arbitrary, elliptical-profile fiber of slight eccentricity, provided the profile volumes are equal. 

The approximation V = P is inadequate for determining the vector corrections to the scalar propagation constant on the elliptical fiber. In Section 34-8 we use Green s functions to show that correct to second order in eccentricity is given by... 

By symmetry the fundamental-mode fields of the elliptical fiber are polarized along the X- and y-axes in Fig. 18-2(a). On a step profile, the corrections Sfi, and Sfi, to the scalar propagation constant p can be obtained from Eq. (32-26) by setting e, = 4 x and e, = y, respectively, where x and y are unit vectors parallel to the axes. To second order in eccentricity, the birefringence is given by the line integral. 

We next consider the modes of the elliptical fiber which correspond to the / = 1 modes of the circular fiber. The two solutions of the scalar wave equation for the latter are given by... 

Fields of local modes 19-2 Criterion for slow variation 19-3 Example Nonuniform core radius 19-4 Example Twisted elliptical fibers... 

Consider a single-mode, elliptical fiber whose refractive-index profile rotates along its length, as shown in Fig. 19-2. We recall from Section 13-5 that in the weak-guidance approximation one fundamental mode of the cylindrically symmetric, elliptical fiber is plane polarized with its transverse electric field parallel to the x-axis in Fig. 19-2(a) and has propagation constant The other fundamental mode s field is parallel to the y-axis... 

Fig. 19-2 The cross-section of an elliptical fiber rotates along its length. Axes X and y are fixed along the major and minor axes, and axes Xq and yo are fixed in space.

Example Slightly elliptical fibers 34-9 Example Far-field corrections... 

---