Coupled local-mode equations coupling coefficients

If we substitute the coupling coefficient of Eq. (28-4) into Eq. (28-8a) and assume bi(0) = 1, the resulting expression for b j is identical with that of Eq. (22-35). The latter was derived using induced currents to represent the slight mismatch between the local-mode fields and the exact fields of the waveguide. Thus we deduce the equivalence of the first iteration of the coupled local-mode equation and the induced-current representation. 

We derived the set of coupled local-mode equations for arbitrary waveguides in Section 31-14. In the weak-guidance approximation, the modal fields in the coupling coefficients of Eq. (31-65c) have only transverse components. If we use Table 13-1, page 288, to relate these components to the corresponding normalized solutions of the scalar wave equation of Eq. (33-45), we find with the help of Eq. (33-48b) that... 

In general, the local-mode propagation constants are contained implicitly within the local eigenvalue equation, and their precise values must be obtained numerically. However, analytical expressions can be derived for the coupling coefficients, as we show in the example below. 

In Sects. 5.5 and 5.6, we have introduced two 2-DOF models for the lead screw drives. In this section, the equations of motion of these models are transformed into matrix form and linearized with respect to their respective steady-sliding equilibrium point. These equations are then used in the next sections to study the local stability of the equilibrium point and the role of mode coupling instability mechanism. In this chapter, for simplicity, the coefficient of friction, n, is taken as a constant. 

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