Local modes scalar

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

On a weakly guiding, nonuniform waveguide the total scalar field d>(x, y, z) satisfies the three-dimensional scalar wave equation, as expressed by Eq. (33-36). We express d> as a summation over orthonormal local modes... 

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... 

Since the limit cycle is invariant with respect to a phase shift, one of the Floquet exponents is zero and the asymptotic stability of the limit cycle implies that the other Floquet exponents have all negative real parts. Thus the amplitude of all modes decay in time and the uniform oscillatory state is stable to small localized perturbations (within the filament model and in the linear regime to which Eq. (8.6) applies). The slowest decaying mode (i.e. for p = 0 and k = 0) decays on average as exp(—At), that is the same as the decay rate of a passive scalar inhomogeneity in the filament model. 

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