Slowly varying fibers

Here we examine a second case of practical importance for which the transit time can be evaluated approximately. We assume that the change in the nonuniformity along the fiber is so slow that the refractive-index profile is virtually uniform, i.e. independent of z, over the distance Zp required for a ray to undergo a half-period, i.e. Zp fln(r,z)/flz 1. An example of a slowly varying fiber is shown in Fig. 5-2. The nonuniformities may be arbitrarily large, e.g. a tapered fiber whose core radius at one end is many multiples of the radius at the other end, provided the taper angle is everywhere small. 

The ray transit time along a slowly varying fiber is expressed by Eq. (5-11). We can expand L z) and Zp(z) in powers of k(z) in a similar manner to Eq. (5-21). On substituting these expansions into Eq. (5-11), we obtain correct to second... 

This is identical in form to the uniform fiber transit time of Table 2-1, page 40, with p — (0) and q replaced by q. Thus a slight variation in exponent on a slowly varying fiber in the weak-guidance approximation is equivalent to a uniform fiber with a different exponent. If we substitute Eq. (5-31) into Eq. (5-34), we find that correct to second order... 

We now introduce an alternative approach to the description of slowly varying fibers which makes contact with other areas of physics. 

The approximate ray invariant for a slowly varying step-profile taper is expressed by Eq. (5-60). This relationship is accurate provided that the change 6p z) in taper radius over the local ray half-period is small. If the taper and fiber are weakly guiding then 0 (z) 1, and by generalizing the expression for Zp in Table 2-1, page 40, to slowly varying fibers, we deduce that Zp(z) = 2p(z)/0j(z) for meridional rays. In terms of the local taper angle Q(z), the slow-variation... 

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... 

In this chapter we study the phenomenon of optical cross-talk between pairs and between arrays of cylindrically symmetric or slowly varying fibers. Crosstalk arises because the fields of a fiber extend indefinitely into the cladding and interact with any other fiber which may be present. This interaction excites the fields of the second fiber, which in turn interact with the fields of the first fiber. Consequently there will be an exchange of power between the two fibers as the fields propagate. The amount of cross-talk, or power exchange, depends on the overlap of the fields of the two fibers. In ray language, cross-talk is associated with frustrated internal reflection or, equivalently, optical tunneling. 

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]... 

When the two slowly varying fibers are identical, as in Fig. 19-3(a), then and... 

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. 

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