Slowly varying fibers local modes

The slowly varying fiber in Fig. 19-1 (a) has the z-dependent refractive-index profile n x,y,z). To construct its local mode fields, we approximate the fiber by the series of cylindrical sections in Fig. 19-1 (b) [1]. The profile is independent... 

Our derivation of the local-mode fields is sometimes called the adiabatic approximation, since it assumes all changes in profile occur over such large distances that there is a negligible change in the power of the local mode [2]. Thus, although a local mode is an excellent approximation for a slowly varying fiber, it is not an exact solution. The small correction to the local-mode fields is determined by the methods of coupled local modes in Chapter 28 or by the induced current method of Section 22-10. 

One immediate consequence of using local modes for pairs of identical, slowly varying fibers is a simple description of power transfer due to cross-talk between fibers. If fiber 1 in Fig. 19-3(a) is initially illuminated with unit power and fiber 2 with zero power, the distribution of power along the composite waveguide is given by a simple modification to the corresponding problem for cylindrically symmetric fibers in Section 18-13. We... 

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). 

The coupling of local modes on composite waveguides, such as two parallel, slowly varying fibers, is described by the results of this chapter, provided we... 

The coupled local-mode equations of Section 31-14 apply to local modes of the same fiber, and are therefore inappropriate for describing coupling between modes of the two fibers, for reasons given in Section 29-2. Instead we can generalize the derivation of the coupled equations of Eq. (29-4) to slowly varying fibers, and deduce that [8]... 

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. 

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. 

We now consider couplers consisting of pairs of nonidentical, single-mode fibers, such as those illustrated in Fig. 19-4. These couplers are of great practical importance [3], and their properties are readily explained in terms of local modes provided only that the couplers are sufficiently well separated and slowly varying. 

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. 

Although these equations do not have a simple analytical solution, we can obtain an asymptotic solution for large spatial frequencies [4]. At low frequencies, when the sinusoid varies slowly along the fiber, we can use the local-mode description of Chapters 19 and 28. 

To complement the analysis of cross-talk between cylindrically symmetric fibers, we now consider pairs of fibers which vary slowly along their length, such as the identical fibers of Fig. 19-3(a) and the tapered coupler of Fig. 19-4(a). Propagation along these systems was described in Chapter 19 using the local modes of the composite waveguide. Our purpose here is to describe cross-talk in terms of the coupUng of the local modes of each fiber in isolation of the other. 

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