Local modes coupled equations

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... 

Finally, a few comments shall be made on the concept of local modes as compared to normal modes [3,33-35], The main idea of the local mode model is to treat a molecule as if it were made up of a set of equivalent diatomic oscillators, and the reason for the local mode behavior at high energy (>8000 cm ) may be understood qualitatively as follows. As the stretching vibrations are excited to high energy levels, the anharmonicity term / vq (Equation (2.9)) tends, in certain cases, to overrule the effect of interbond coupling and the vibrations become uncoupled vibrations and occur as local modes. ... 

In contrast to the nonretarded treatment, which is solely based on the electrostatic interaction potential V(r), we equate the electric potential to zero in the retarded case. The electric interaction is completely covered by the vector potential A(r). The transverse modes under investigation enable the Lorentz gauge to be used for vanishing electric potential. This concept turns out especially useful with respect to the Schrodinger formalism presented in the next Chapter. We may ascribe the retarded interaction between electrons located at different particles solely to the vector potential A(r). In addition to providing the proper multipole susceptibilities, quantum theory still has to answer the question regarding statistics. Are the localized modes which are strictly coupled to molecular electron transitions, still Bosons ... 

In polymer systems such a mutually independent dipole orientation is inapplicable because dipole orientation is highly correlated. The very essence of a pd3mer chain generally renders independent orientation of a main chain dipole component impos-rible and frequently, coupling between side chain and main chain modes are involved. For a rigid chain polymer in solution, dipole orientation requires rotatory diffusion of a macromolecule as a whole, and no component due to local modes is involved, but this situation is the exception. Flexible polymers permit polarization by local mode motions as well as rotatory diffusion as illustrated in Fig. 5. Equation (5) is also inapplicable for polymers because of dispersity in molecular weight, since if the relaxation involves molecular wel t dependent modes there will be a tead of relax-... 

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... 

We see from both equations 8.32 and 8.33 that the most unstable mode is the mode and that ai t) = 1 - 1/a is stable for 1 < a < 3 and ai t) = 0 is stable for 0 < a < 1. In other words, the diffusive coupling does not introduce any instability into the homogeneous system. The only instabilities present are those already present in the uncoupled local dynamics. A similar conclusion would be reached if we were to carry out the same analysis for period p solutions. The conclusion is that if the uncoupled sites are stable, so are the homogeneous states of the CML. Now what about inhomogeneous states ... 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

---