Coupled mode equations derivation

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... 

To solve Eq. (29-2) we need to know . An expression could be obtained by expanding over the complete set of Pj for the bound and radiation modes of the first fiber, and this would, of course, lead to the infinite set of coupled mode equations derived in Section 33-11. The disadvantage of this description is that each mode of the first fiber by itself is a poor approximation to the field within the second fiber. Consequently large numbers of modes are required for accuracy, and. the set of coupled equations is then intractable. 

The set of coupled mode equations satisfied by the modal amplitudes can be derived directly from the scalar wave equation for the perturbed fiber, as we show in Section 33-11, or may be regarded as the weak-guidance limit of the corresponding equations for arbitrary profile fibers, derived in Section 31-11. The latter approach leads to... 

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 mode equations of the previous section can be derived intuitively. This also provides insight into the physical mechanism of the coupling process. Consider a differential section of the perturbed waveguide of length dz, as shown in Fig. 31-2, and its effect on the k th forward-propagating bound mode. The z dependence of the fields, hi(z) of Eq. (31-45), is expressible as... 

For convenience we only include bound modes in our derivation radiation modes are readily incorporated by analogy with the coupled mode equations of Section 31-11. We express the total transverse field of the perturbed waveguide as an expansion over the complete set of forward- and backward-propagating local modes... 

As a first step in derivation of various versions of UPPE, we derive an exact coupled-modes system of equations. This is a well-known textbook formula that can be found in the literature in several different forms. To keep our derivation self-contained, we re-derive the starting formula using a standard approach based on orthogonality relations for modes of an electromagnetic... 

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)... 

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. 

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... 

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... 

The linearized equations of motion for this system were developed in Sect. 7.1.1. and the conditions for mode coupling instability were derived in the previous chapter. Here, we will only focus on the possibility of instability due to the kinematic constraint mechanism in the undamped system. 

Finally we shall derive the equation used by Bixon and Jortner. Suppose that an intramolecular vibrational mode, say Qi, plays a very important role in electron transfer. To this mode, we can apply the strong-coupling approximation (or the short-time approximation). From Eq. (3.40), we have... 

Of more concern are the comments by De Schepper et al. [528] and Resibois and De Leener [490]. They have discussed whether such a fourth-order derivative can have meaning. A mode-coupling theory and a kinetic theory of hard spheres both indicate that the Burnett coefficient diverges at tin. There seems little or no reason for the continued use of the Burnett equation in discussing chemical reaction rates in solution. Other effects are clearly more important and far more reasonable from a theoretical point of view. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

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