Circular fibers weakly guiding

The fundamental modes of weakly guiding circular fibers are sometimes designated LPq, modes [3] rather than HE,i modes. The nomenclature is intended to emphasize that the modal fields are everywhere polarized in the same direction, i.e. the fields are plane, or uniformly, polarized. 

Cj = e , and is expressible in terms of simple integrals in Table 14-1, page 304. Examples of modes on weakly guiding circular fibers are given in the following chapter. 

Modes of weakly guiding waveguides obey the fundamental properties of modes delineated in Chapter 11, and mainly because of the approximate TEM nature of the modal fields, these properties have the simpler forms of TaHe 13—2. The expressions in the first column are in terms of the transverse electric field e, and apply to all weakly guiding waveguides. Those in the second column are for waveguides which are sufficiently noncircular that e, can be replaced by either of the two fields for noncircular waveguides in Table 13-1, while the third column is for circular fibers only, when e, is replaced by any one of the four linear combinations Ct, for circular cross-sections in Table 13-1. We emphasize that Table 13-2 applies to all modes. 

The small polarization correction to the scalar propagation constant due to structural anisotropy is given by Eq. (13-11). For an isotropic fiber of circular cross-section, the corrections for the two fundamental modes are identical, i.e. SPx = Py This is not the case for the anisotropic fiber, since the parameters in Eq. (13-19) depend on polarization. However, SP — SPy is small compared to the difference in propagation constants in Eq. (13-20), Px — Py, since the fiber is weakly guiding, and can be ignored. 

If the principal axes of the anisotropic material are parallel to the optical axes of a fiber of noncircular cross-section, it is intuitive that the two fundamental modes must also be polarized along these axes. We then have a situation identical to the circular cross-section, discussed above, except that (r) in Eq. (13-20) is replaced by P,(x,y) and P,(r) by P,(x,y). Thus, all results for weakly guiding isotropic waveguides apply to weakly guiding anisotropic waveguides by following the simple substitution discussed above. 

Table 14-1 Modal fields of weakly guiding circular fibers. Subscripts t and z denote transverse and longitudinal components. Unit vectors x, y and z are parallel to the... 

To summarize these two sections, the complete modal fields and corrected propagation constants for all bound modes of the weakly guiding circular fiber are given in Table 14-1. Consequently, for each profile considered below, we need only determine the solution of Eq. (14-4). 

Table 15-1 Gaussian approximation for circular fibers. Gaussian approximation for the fundamental-mode fields of weakly guiding, circular fibers. Parameters are defined inside the back cover and coordinates are illustrated in Fig. 14-1...

The Gaussian approximation was introduced in Chapter 15 to provide simple, but accurate, analytical expressions for fundamental-mode quantities of interest on circular fibers of arbitrary profile. Here we show how to generalize this approximation and describe fundamental-mode propagation on weakly guiding fibers of arbitrary cross-section. 

The efficiency with which beams excite the fundamental modes of circular fibers, with the fields of Eq. (13-9), is of particular interest when the fiber is single moded. In order to account for weakly guiding fibers of otherwise arbitrary profile, when analytical solutions of the scalar wave equation for Fo (r) are not available, we use the Gaussian approximation of Chapter 15. The radial dependence of the fundamental-mode transverse fields is approximated in Eq. (15-2) by setting... 

When the fiber is weakly guiding, the approximations of Chapter 13 can be used to simplify Eq. (21-2). If for convenience we assume a circular fiber with current sources parallel to the x-axis in the cross-section, then we deduce from Tables 13-2, page 292, and 14-1, p>age 304, and Eq. (21-3) that the power in each mode is given by... 

To demonstrate the usefulness of the induced-current representation, we determine, through examples, the power radiated from the fundamental mode of a weakly guiding circular fiber due to slight nonuniformities of various kinds. 

The results of this section apply to all fundamental and higher-order modes on weakly guiding waveguides of arbitrary cross-section, and parallel the results of Sections 13-5 and 13-8 derived by physical arguments. In addition they apply to the fundamental modes of fibers with circular cross-sections, discussed in the next section. 

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... 

Our discussions of weakly guiding circular fibers both in this chapter, and in Chapters 13 and 14, have tacitly assumed that the combinations in Table 13-1, page 288, form the most basic set of modes for describing approximate solutions of the vector wave equation. For example, we might have considered the set of four right- and left-handed, circularly polarized fields which have the complex forms... 

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