Propagation constant perturbed fibers

When the perturbed and unperturbed fibers are weakly guiding, the variation in both profiles n and n is small. The modal fields can then be constructed from solutions of the scalar wave equation, as described in Chapter 13. If T and jS denote the unknown solution and propagation constant for the perturbed fiber. 

For slight perturbations, it is normally sufficient to assume that the transverse, or X, y dependence of the modal fields on the perturbed and unperturbed fibers is similar, i.e. e S e, h = h, and consequently T =. An exception is the calculation of the polarization corrections to the scalar propagation constant, discussed in the following section, for which higher-order corrections to are required. These can be obtained using either eigenfunction expansions, as outlined in Sections 33-9 and 33-10, or Green s functions, as discussed in Chapter 34. 

The polarization corrections, and SPy, to the scalar propagation constant P for the Xq- and yo-polarized modes on the perturbed, noncircular fiber are in general unequal, and their difference describes the anisotropic, or birefringent, nature of propagation. This is of basic interest for the two fundamental modes on single-mode fibers. The calculation of the corrections from the formula in Table 13-1, page 288, requires first-order corrections to the approximation We derive these corrections for the slightly elliptical fiber in Section 18-10. 

Another example of a three-region fiber is illustrated in Fig. 18-l(b). A thin concentric ring of width 6p and uniform refractive index n is introduced at the interface of the unperturbed fiber. Hence = n - in this region, and the perturbed propagation constant follows from Eq. (18-7) as... 

Consider a step-profile fiber with core and cladding indices and and let the core radius change from p to p + 5p. This perturbation is identical to the pertubation in Fig. 18-l(b) if we set o = , Hence, we deduce from Eq. (18-12) that the perturbed propagation constant is given by... 

The combination of solutions of the scalar wave equation for the transverse fields of weakly guiding fibers of circular cross-section are given in Table 13-1, page 288. As we showed in the previous section, these combinations can be derived using perturbation theory. In this section we show how the combinations can be deduced using only symmetry arguments [2]. We start with the four vector solutions constructed from the solutions of the scalar wave equation with the common propagation constant P, and denote them by... 

The expression for the modal power attenuation coefficient used above is derived by perturbation methods from the eigenvalue equation for the step-profile fiber. We can quantify the agreement between the modal and ray power attenuation coefficients by solving the eigenvalue equation numerically for the complex values of the propagation constant. This solution was discussed in Section 24-19. We find that there is good agreement for those modes with the smallest attenuation coefficients, while, for modes... 

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