Slowly varying waveguides

19-1 Fields of local modes 19-2 Criterion for slow variation 19-3 Example Nonuniform core radius 19-4 Example Twisted elliptical fibers 

19-6 Example Cross-talk between identical fibers 

19-9 Criterion for slow variation Local modes of bent fibers Multimode fibers and rays References 

Fibers which change along their length are not translationally invariant, 

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 

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A common way to treat the problem of a picosecond pulse propagation in regular dielectric waveguide with Kerr nonlinearity was to solve the nonlinear Schrodinger equation (NLSE) for the slowly varying temporal amplitude of electrical field ... 

Here complex amplitude of the pulse envelope E x, z, t) is also a slowly varying function of z and t. Spatiotemporal distribution of the electric field is described by E x,z,t) = E x,z,t)ex-p[iujQt— i/Sz), f3 being the longitudinal wavenumber of the waveguide mode at the pulse peak. 

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. 

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... 

An important application of the solution of the coupled local-mode equations for weak power transfer determines how slowly a waveguide must vary along its length in order that an individual local mode can propagate with negligible variation in its power. If we assume the /th local mode alone is initially excited with unit power, i.e. h((0) = 1, then the fraction of power excited in the jth... 

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... 

Propagation along slowly varying, tapered couplers of the type illustrated in Fig. 28-2 (a) can be described by the local modes of the composite waveguide, as discussed in... 

The construction of ray paths within the core of the step-profile waveguides of Chapters 1 and 2 is based on straight-line trajectories, which are solutions of the ray-path equation of Eq. (1-18) in a uniform medium. When the core is graded, the cartesian component equations of the ray-path equation follow directly, as in Eqs. (1-19) and (2-49). Here we derive the corresponding component equations in directions defined by the cylindrical polar coordinates (r, 0, z) of Fig. 2-1, for application to fibers with graded profiles n(r) in Chapter 2, and, by simple generalization, to slowly varying fibers with profiles n(r, z) in Chapter 5. 

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. 

In Chapter 19 we introduced local modes to describe the fields of waveguides with large nonuniformities that vary slowly along their length. As an individual local mode is only an approximation to the exact fields, it couples power with other local modes as it propagates. Our purpose here is to derive the set of coupled equations which determines the ampUtude of each mode [10]. First, however, we require the relationships satisfied by the fields of such waveguides. 

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