Waveguides with exact solutions

12-8 Modal fields 12-9 Propagation constants 12-10 Modal properties 12-11 Weakly guiding fiber 

12-12 Plane-wave decomposition of the modal fields 12-13 Multilayered and elliptical fibers 

12-19 Step-profile planar waveguide 12-20 Step-profile uniaxial fiber 

The basic properties of bound modes on optical waveguides were given in the previous chapter. In this chapter we display these properties explicitly for those few profiles which have exact solutions of Maxwell s equations. Our primary objective is to derive analytical expressions for the modal vector fields, which contain all the polarization properties of the waveguide discussed in Section 11-16. We pay particular attention to fundamental modes, since these 

The step-profile waveguide has a core of uniform refractive index n, surrounded by a cladding of uniform refractive index n, which is assumed unbounded. Thus the only variation in profile is a step, or jump, discontinuity at the core-cladding interface in Fig. 11-1 (a). This profile has exact analytical solutions for the modal fields on planar waveguides, circularly symmetric fibers and elliptical fibers. 

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A significant contribution was made in a series of papers by Vesnitskii and co-authors. In 1969 [40] he gave an exact solution for the problem of a rectangular waveguide with a uniformly moving lateral wall, specifically, for the equation and the boundary conditions... 

The exact number of bound modes which propagate on a waveguide can be found by counting the discrete solutions of the eigenvalue equation. In general this is a cumbersome procedure, but simple expressions are available for multimode waveguides with F > 1. Examples are given in Sections 36-12 and 36-13. 

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 radiation-mode fields are solutions of the same equations satisfied by the bound-mode fields, so that whenever an exact solution exists for bound modes, a corresponding solution for radiation modes exists. We showed in Chapter 12 that, for waveguides with arbitrary variation in profile, there are few known profiles for which exact solutions of Maxwell s equations can be obtained analytically. Even in these cases, the expressions for the radiation-mode fields are generally more complex than those for the bound-mode fields. In the following section we consider the step-profile fiber. The radiation-mode fields of the step-profile planar waveguide can be derived similarly. 

It has been shown that by irradiation of a PMMA polymer substrate with ions waveguide structures can be produced. Detailed results of the ion-induced chemical changes that lead to increased index of refraction have been reported elsewhere [156, 158]. A possible solution of the problem of exact coupling between the optical fiber and the device has been suggested. 

We showed above that the modes of weakly guiding waveguides are approximately TEM waves, with fields e = e, h S h, and h, related to e, by Eq. (13-1). In an exact analysis, the spatial dependence of e,(x,y) requires solution of Maxwell s equations, or, equivalently, the vector wave equation, Eq. (1 l-40a). However, when A 1, polarization effects due to the waveguide structure are small, and the cartesian components of e, are approximated by solutions of the scalar wave equation. The justification in Section 13-1 is based on the fact that the waveguide is virtually homogeneous as far as polarization effects are concerned when A 1. As we showed in Section 11-16, these effects... 

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