Planar waveguides weak-guidance approximation

The transmission coefficient T is found by using the local plane-wave description of a ray. We regard the local plane wave as part of an infinite plane-wave incident on a planar interface between unbounded media, whose refractive indices coincide with the core and cladding indices and of the waveguide, as shown in Fig. l-3(b). For the step interface, Tis identical to the Fresnel transmission coefficient for plane-wave reflection at a planar dielectric interface [6]. In the weak-guidance approximation, when s n, the transmission coefficient is independent of polarization, and is derived in Section 35-6. From Eq. (35-20) we have [7]... 

The simplest example of a noncircular waveguide is the planar waveguide of Chapter 12, whose modes are either TE or TM, as explained in Section 11-16. For each TE mode the electric field lies in the cross-section and is uniformly polarized. Consequently the weak-guidance solution is identical to the exact solution for the field ey and the propagation constant. Both satisfy the scalar wave equation of Eq. (12-16), and examples with analytical solutions are given in Table 12-7, page 264. Within the weak-guidance approximation the... 

The eigenvalue equations for leaky TE and TM modes on the step-profile planar waveguide in the weak-guidance approximation are obtained by setting IF = — t Q in the TE mode eigenvalue equations of Table 12-2, page 243. Hence... 

If we substitute for T from Eq. (7-19b) and for Zp from Table 2-1, page 40, and note that p = n in the weak-guidance approximation, it is readily verified that the mode and ray attenuation coefficients are identical [7]. A similar, analytical comparison can be made for refracting leaky modes and rays on the step-profile planar waveguide [8]. 

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