Local modes nonuniform fibers

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

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 local-mode concept also applies to slowly varying composite waveguides, such as the two identical fibers in Fig. 19-3(a) and the pairs of nonidentical fibers in Fig. 19-4, and is therefore a powerful method for studying the properties of nonuniform couplers. 

At each position z along a nonuniform, multimode fiber, a high-order local mode is equivalent to a single family of rays, as is clear from Section 36-2. Each ray follows a path which changes slowly over the local half-period Zp(z) of Eq. (5-12). This is the ray analogue of the multimode-fiber discussion in Section 19-2. Furthermore, the equivalence of mode and ray transit times, which is demonstrated in Section 36-9, is readily extended to slowly varying fibers, for which the transit time is given by Eq. (5-11). 

In Chapter 19 we introduced the concept of local modes to describe propagation on fibers with arbitrary nonuniformities. It is clear from the method of construction in Section 19-1 that the local-mode fields are an accurate approximation to the exact fields of the fiber provided the nonuniformities vary sufficiently slowly with z, as discussed in Section 19-2. Nevertheless, the local-mode fields are not an exact solution of Maxwell s equations, and the slight error can be described by induced currents. 

The set of coupled equations for local modes can be derived intuitively, using a differential section of the nonuniform fiber [12]. Here we give a brief description of the method. A section of fiber of length dz is shown in Fig. 31-2. The jth forward-propagating local mode is incident on the section at z with amplitude Uj z). The variation in bj(z) of Eq. (31-61) across the section is given by... 

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