Scalar wave equation modal methods

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. 

Section 33—1 Modal methods for the scalar wave equation 641... 

In Section 23-10, we discussed the shift of the modal fields due to bending of a fiber. If the bending radius is large compared to the core radius, we can use perturbation methods to describe this shift. We show that the effect of the bend can be described by a straight fiber with an effective refractive-index profile [14]. The fields of the bent fiber can then be found by solving the scalar wave equation for the straight fiber with the effective profile. 

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