Exact modal solutions

The equations (4.84, 85) and their solutions are based on the approximation (4.83) which is valid for a uni-modal probability distribution only. Contrary to this assumption, however, any initially uni-modal distribution can be expected to develop first into the doubled-peaked quasi-stationary solution Pqs (n) shown in Fig. 4.10 and then to go over finally to the exact stationary solution, that of an extinct population. Consequently deviations of the true time paths of (n), and o t) from those described by (4.84, 85) are to be expected. A calculation of the development of a model population with time with the exact master equation (4.63) and with the parameters A = 0.5, n = 0.2, and bi = 0.01 confirms this expectation (Figs. 4.11, 12). The Fig. 4.11 shows the exactly calculated change with time of a distribution which starts as normal distribution but soon develops into the form of the bimodal quasi-stationary distribution Pqs(n). In Fig. 4.12 and for the same model parameters the exact paths of the mean value (n)(and the variance a t) are compared with the paths obtained by solving the approximate equations (4.84, 85). 

The step-profile waveguide has a core of uniform refractive index n, surrounded by a cladding of uniform refractive index n, which is assumed unbounded. Thus the only variation in profile is a step, or jump, discontinuity at the core-cladding interface in Fig. 11-1 (a). This profile has exact analytical solutions for the modal fields on planar waveguides, circularly symmetric fibers and elliptical fibers. 

Eq. (d) is plotted in Fig. PI2.24b, where it is compared with the exact solution by classical modal analysis. The static correction method provides reasonable results the error is primarily because the second mode has significant dynamic effects, as indicated by R 2 1-14 being significantly larger than one. 

Comparison of Eqs. (h) and (q) with the exact solutions, Eq. (r), indicates that Rayleigh-Ritz method using two force-dependent Ritz vetors gives a roughly similar accuracy as modal analysis with two modes for responses 2> 3 U4, and us. However, the error in uj determined by modal analysis is large. This becomes obvious from the following table ... 

The basic properties of bound modes on optical waveguides were given in the previous chapter. In this chapter we display these properties explicitly for those few profiles which have exact solutions of Maxwell s equations. Our primary objective is to derive analytical expressions for the modal vector fields, which contain all the polarization properties of the waveguide discussed in Section 11-16. We pay particular attention to fundamental modes, since these... 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

In Section 11-13 we showed that the exact propagation constant is given explicitly in terms of integrals over the vector modal fields. Here we derive the analogous expression for the scalar propagation constant in terms of scalar solutions of the scalar wave equation. Starting with Eq. (33-1), we multiply by P and integrate over the infinite cross-section to obtain... 

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