Rays and asymptotic modal methods

The main purpose of this chapter is to demonstrate the equivalence of the ray and asymptotic modal descriptions of multimode waveguides. We begin by deriving the asymptotic relationship between a mode of the waveguide and the equivalent family of 

The trigonometrical dependence of the core fields for the modes of the step-profile planar waveguide in Table 12-1, page 242, can be decomposed into two components. For example, the electric field for the even TE modes is expressible as 

In contrast to the planar waveguide, the core fields of the circular fiber, which are given in terms of Bessel functions in Table 12-3, page 250, do not represent a single family of rays [1]. However, the fields of higher-order modes, which have C/ 1, are asymptotically equivalent to a single family. To demonstrate this, we first use the recurrence relations of Eq. (37-72) to express i in Table 12-3 in terms of J, and its derivative. Then, provided 1/ 1 and 1/ — v s we can substitute the far Debye approximations of Eq. (37-89). The azimuthal dependence on sin(v ) or cos(v ) is 

If we include the z-dependence of Eq. (12-10), then the accumulated phase follows as 

We define plane-wave vectors k with components relative to the radial, azimuthal and longitudinal directions in Fig. 12-3 given by 

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