Modal methods for the scalar wave equation

33-2 Orthogonality and normalization 33-3 Integral expressions for the propagation constant 33-4 Phase and group velocities 33-5 Reciprocity relations 

Radiation inodes of the scalar wave equation 33-7 Orthogonality and normalization 33-8 Leaky modes 

33-9 Modal fields and propagation constants 33-10 Slightly perturbed waveguides 

We showed in Section 13-3 that the cartesian components of the transverse electric field are solutions of the scalar wave equation. If denotes either component of Eq. (13-7), and P is the scalar propagation constant, then 

The continuity properties of Eq. (33-1) follow by reductio ad absurdum. Suppose P is not continuous, so that somewhere it has a step, or jump in value. The first derivatives will then contain a singularity described by the Dirac delta function, and the second derivatives will contain the derivative of the delta function. Since the profile (x,y) contains, at most, a step discontinuity, it is clear that Eq. (33-1) cannot be satisfied. Consequently T must be continuous. A similar argument is used to prove that the first derivatives are also continuous. It is not necessary that the second derivatives be continuous. For example, if n(x,y) is a step profile, then at least one of the second derivatives of F must be discontinuous. Finally, since is constructed from solutions of the scalar wave equation, it must have the same continuity properties as F. 

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

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. 

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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