Nonuniform fibers

5-3 Example Step-profile with variable core index 

5-5 Ray transit time 5-6 Example Clad power-law profiles 5-7 Small-amplitude nonuniformities 5-8 Example Slight core-radius variations 5-9 Example Slight exponent variations 5-10 Adiabatic invariant 5-11 Example Step profile 5-12 Example Clad power-law profiles 5-13 Radiation loss 

5-14 Example Variable core radius-step profile 

5-15 Example Variable core radius-clad power-law profiles 

5-21 Distribution of bound-ray power 5-22 Example Diffusion equation 5-23 Example Rayleigh scattering 

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The properties of a composite are dictated by the intrinsic properties of the constituents which may be summarized as fiber architecture and fiber-matrix interface (Fowler et al. 2006). The reinforcing efficiency of natural fibers depends on their physical, chemical, and mechanical properties. Major shortcomings of natural plant fibers include fiber nonuniformity, property variation even between individual plants, low degradation temperature, low microbial resistance, and susceptibility to rotting. In addition to naturally occurring nonuniformity, fiber extraction and processing techniques also have major impacts on final fiber quality, not to mention fiber costs and yield (Munder et al. 2005). 

For an arbitrary nonuniform fiber it is necessary to know the ray path r = r(z) explicitly, in order to determine the transit time. However, the step-profile fiber of constant core... 

The transit time for the nonuniform fiber in Fig. 5-3 (a) is the sum of the transit times for each uniform section in Fig. 5-3(b). If tp(z) denotes the transit time over the uniform section of length Zp(z), then from Eq. (2-32) we have... 

The pulse spread for the uniform clad power-law profiles is discussed in Section 3-2. If we repeat the analysis using the transit time of Eq. (5-27) for the nonuniform fiber, we find that the minimum transit time occurs approximately at (z) = of Eq. (3-6)... 

When the nonuniform fiber has a step profile, the core index n z) depends only on z, and the upper limit of integration r,p (z) in Eq. (5-41) is replaced by the core radius p(z). If the path makes angle 0 (z) with the axial direction, then with the help of Eq. (5-2) we have... 

We next consider nonuniform fibers with clad power-law profiles. These profiles have the form given by Eq. (5-15), with A and p dependent on z, but we assume the exponent q remains constant. Substituting into Eq. (5-41) we have... 

On a uniform fiber, a ray which is bound at the beginning of the fiber remains bound along the fiber. However, on a nonuniform fiber, a ray which is initially bound may become a leaky ray over part of its trajectory and lose power by radiation, if the variation in the parameter /S(z) of Eq. (5-2) is sufficiently large. Otherwise it will remain a bound ray over the length of the fiber even though (z) varies. Here we give a simple upper bound for the radiation loss. 

Following the above discussion, the loss of leaky-ray power on noncircular and nonuniform fibers is described by local transmission coefficients for... 

Fig. 19-1 (a) A nonuniform fiber varies along its length and has refractive-index profile n (x,y,z) and (b) the approximate model is a series of sections, where denotes the center of each section and dz is the length of a particular section. 

Although the fields expressed by Eq. (19-1) vary as the profile varies from section to section, the power of a local mode must be conserved along the nonuniform fiber. This requirement is automatically satisfied if we use the orthonormal forms of Eq. (11-15) for the fields in each section, i.e. replace e and hy by e-and hy, respectively, in Eq. (19-1). 

As a local mode propagates, its phase increases across each section by the product of j8y(zj and the section length 5z. Consequently, the phase at an arbitrary position along the nonuniform fiber is a sum of such products. However, the slow variation of the fiber means that the propagation constant /Sy (Zj) varies only slightly from one section to the next Hence we can accurately... 

In nonuniform fibers many problems of practical interest can be easily solved by using local modes, as we demonstrate in the examples below. However, the locahmode fields will be an accurate approximation to the exact fields only if the nonuniformities vary sufficiently slowly along the fiber. Since the localmode fields are constructed from the modal fields of the locally equivalent, cylindrically symmetric fiber, the appropriate slowness condition is determined by the largest distance over which the total field of the cylindrically symmetric fiber changes significantly due to phase differences between the various modes. 

Using the expressions derived above, we can give qualitative criteria for the validity of local-mode solutions on a nonuniform fiber with refractive-index profile n x,y,z). Over distance z, the change in profile varies as (d /dz)Zb, whence we deduce from Eqs. (19-3) and (19-4) that the slowness criteria are... 

The modelling of nonuniform fibers by current sources in the following chapter leads to the tubular current source. This source, which is depicted in Fig. 21-3, consists of a current distribution of density J on the cylindrical surface, or tube, r = Tq defined by [2]... 

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