Modal methods for Maxwell’s equations

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

The weak-guidance approximation, described in Chapter 13, greatly simplifies the determination of the modal fields of optical waveguides, because it depends on solutions of the scalar wave equation, rather than on vector solutions of Maxwell s equations. For circular fibers, with an arbitrary profile, the scalar wave equation must normally be solved by purely numerical methods. We discussed the few profiles that have analytical solutions in Chapter 14. These solutions, including those for profiles of practical interest such as the step and clad power-law profiles, are given in terms of special functions or by series expansions, which usually necessitate tables or numerical evaluation to reveal the physical attributes of the modes. 

There are essentially two methods for deriving the equations satisfied by the bj(z). The more physical approach is to divide the fiber into a series of differential sections, one of which is shown in Fig. 31-2, and then consider the change in each modal amplitude across each section [1]. Details are given in Section 31-16. Alternatively, we substitute Eq. (28-1) into Maxwell s equations and use the orthogonality conditions for local modes to derive the set of coupled local-mode equations [2,3], This approach is presented in Section 31-14, and leads to Eq. (31-65)... 

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