Coupled local-mode equations derivation

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. 

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 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... 

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 Chapter 19 we introduced local modes to describe the fields of waveguides with large nonuniformities that vary slowly along their length. As an individual local mode is only an approximation to the exact fields, it couples power with other local modes as it propagates. Our purpose here is to derive the set of coupled equations which determines the ampUtude of each mode [10]. First, however, we require the relationships satisfied by the fields of such waveguides. 

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... 

The set of coupled equations for local modes can be derived intuitively, using a differential section of the nonuniform fiber [12]. Here we give a brief description of the method. A section of fiber of length dz is shown in Fig. 31-2. The jth forward-propagating local mode is incident on the section at z with amplitude Uj z). The variation in bj(z) of Eq. (31-61) across the section is given by... 

Another approach towards a thermodynamics of steady-state systems is presented by Santamaria-Holek et al.193 In this formulation a local thermodynamic equilibrium is assumed to exist. The probability density and associated conjugate chemical potential are interpreted as mesoscopic thermodynamic variables from which the Fokker-Planck equation is derived. Nonequilibrium equations of state are derived for a gas of shearing Brownian particles in both dilute and dense states. It is found that for low shear rates the first normal stress difference is quadratic in strain rate and the viscosity is given as a simple power law in the strain rate, in contrast to standard mode-coupling theory predictions (see Section 6.3). 

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