Local-mode coupling

Kellman, M. E. (1983), Dynamical Symmetries in a Unitary Algebraic Model of Coupled Local Modes of Benzene, Chem. Phys. Lett. 103,40. 

It was further demonstrated that the dynamics of the modulation by the coupled local modes could be very complex, in which additional quasi-periodic and chaotic relaxation oscillations have been observed. In addition, there were also periodic short-time quasi-(2-switched spikes, which were attributed to the presence of foreign inclusions with SA properties likely at the grain boundaries [295]. Experimental results indicated that such local modes are present in coarse-grained ceramics and absent in fine-grained samples. 

Otsuka K, Narita T, Miyasaka Y, Lin CC, Ko JY, Chu SC (2006) Nonlinear dynamics in thin-slice Nd YAG ceramic lasers coupled local-mode laser model. Appl Phys Lett 89 081117... 

Our derivation of the local-mode fields is sometimes called the adiabatic approximation, since it assumes all changes in profile occur over such large distances that there is a negligible change in the power of the local mode [2]. Thus, although a local mode is an excellent approximation for a slowly varying fiber, it is not an exact solution. The small correction to the local-mode fields is determined by the methods of coupled local modes in Chapter 28 or by the induced current method of Section 22-10. 

However, before we begin, we emphasize that coupled local-mode theory is particularly useful in situations when there is a large transfer of power between local modes. Apart from such exceptional situations, power transfer is slight and-the induced-current methods of Sections 22-10 and 22-11 are sufficient. 

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)... 

In Section 27-1 we showed that in applying the coupled mode equations there is an intrinsic restriction to weakly guiding waveguides. This restriction does not occur in the application of the coupled local-mode equations. A general solution of the coupled local-mode equations is derived for weak power transfer in the following section, and the radiation modes are discussed later in the chapter. 

The coupled local-mode equations can be solved approximately when only a small fraction of the total power of the perturbed waveguide is transferred between modes. We show that the first-order solution is identical to the induced-current solution of Chapter 22. For convenience we assume that only the Ith forward-propagating local mode is excited at z = 0. To lowest order we ignore coupling to all other modes. The solution of Eq. (28-2) is then... 

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. 

An important application of the solution of the coupled local-mode equations for weak power transfer determines how slowly a waveguide must vary along its length in order that an individual local mode can propagate with negligible variation in its power. If we assume the /th local mode alone is initially excited with unit power, i.e. h((0) = 1, then the fraction of power excited in the jth... 

When this condition is satisfied, we can neglect coupling between the local modes and the radiation field, and the coupled local-mode equations of Eq. (28-2a) reduce to [S]... 

The coupled local-mode equations discussed in Section 28-1 implicitly include coupling to the radiation field. In keeping with the concept of local modes. 

The inclusion of the local radiation modes in Eq. (28-1) follows by analogy with the coupled mode equations derived in Section 31-11. For example, the amplitude and phase dependence bj (z, Q) of the j th forward-propagating local radiation mode satisfies the coupled local-mode equation... 

The coupled local-mode equations of Section 31-14 apply to local modes of the same fiber, and are therefore inappropriate for describing coupling between modes of the two fibers, for reasons given in Section 29-2. Instead we can generalize the derivation of the coupled equations of Eq. (29-4) to slowly varying fibers, and deduce that [8]... 

Fields of z-dependent waveguides 31-14 Coupled local-mode equations 31-15 Alternative form of the coupling coeflScients 31-16 Physical derivation of the coupled equations... 

Relationship with the coupled local-mode equations for arbitrary waveguides... 

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... 

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