Weakly guiding waveguides

13-5 Fundamental modes of waveguides of arbitrary cross-section 285 13-6 Polarization corrections to the scalar propagation constant 286 13-7 Higher-order modes of circular fibers 287 

13-8 Higher-order modes of waveguides of arbitrary cross-section 289 

13-9 Higher-order modes of nearly circular fibers 289 

13-12 Modes of anisotropic waveguides 13-13 Single-mode single-polarization fibers 

13-14 Leaky single-mode single-polarization fibers 

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Waveguides that are nearly isotropic can be treated by the perturbation methods of Chapter 18. Weakly guiding waveguides are discussed in Section 18-9, and waveguides with arbitrarily varying refractive-index profiles are discussed in Section 18-21. 

We now show how to construct the modal fields of weakly guiding waveguides using simple physical arguments based on the above insight. These fields can also be formally derived by applying perturbation methods to Maxwell s equations, as we show in Chapter 32. 

Thus the variation in on weakly guiding waveguides is very narrow, and for all modes we may assume... 

We showed above that the modes of weakly guiding waveguides are approximately TEM waves, with fields e = e, h S h, and h, related to e, by Eq. (13-1). In an exact analysis, the spatial dependence of e,(x,y) requires solution of Maxwell s equations, or, equivalently, the vector wave equation, Eq. (1 l-40a). However, when A 1, polarization effects due to the waveguide structure are small, and the cartesian components of e, are approximated by solutions of the scalar wave equation. The justification in Section 13-1 is based on the fact that the waveguide is virtually homogeneous as far as polarization effects are concerned when A 1. As we showed in Section 11-16, these effects... 

If we are to account for waveguide polarization properties in the propagation constant, we must add a correction dp to the scalar propagation constant p.To determine Sp exactly we would have to solve the vector wave equation. However, the V, Inn term on the right of Eq. (ll-40a) is small for weakly guiding waveguides, so we use simple perturbation methods in Section 32-4. From Eq. (32-24) we have... 

Table 13-1 Boond-mode flelds of weakly guiding waveguides. The form of the transverse electric field depends on the shape of the waveguide cross-section. Vector operators are defined in Table 30-1, page S92, and parameters are defined inside the back cover.

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. 

We have shown that the modes of weakly guiding waveguides are approximately TEM waves, with transverse field components e, and h,. However, the exact modal fields have longitudinal components. For the weakly guiding waveguide these components are very small, and are expressible approximately in terms of e, and h,. From Eq. (32-18) we have... 

Table 13-2 Properties of bound inodes on weakly guiding waveguides. Parameters are defined inside the back cover. The modal amplitude a depends on the source of illumination, and A o is the core cross-section. We assume e, and F are real on nonabsorbing... 

Now we modify the early sections of this chapter to include weakly guiding waveguides constructed from the anisotropic material discussed above, i.e. = = = n. To do this, we first recall from Section 11-23 that the two... 

Characteristics of leaky modes 24-4 Modal parameters 24-5 Modal fields 24-6 Radiation caustic 24-7 Classification of leaky modes 24-8 Plane-wave decomposition 24-9 Weakly guiding waveguides 24-10 Number of leaky modes... 

This chapter shows how radiation modes are used to construct the total radiation fields. We first establish the general properties of radiation modes on arbitrary waveguides and then parallel Chapter 13 with a discussion of radiation modes on weakly guiding waveguides. Finally, we give examples of the application of radiation modes to complement the Green s function solutions given in earlier chapters. 

In Chapter 13 we showed how the bound-mode fields of weakly guiding waveguides can be constructed from solutions of the scalar wave equation. With slight modification, the same procedure applies to the radiation-mode fields as well [4]. However, while the bound modes are approximately TEM waves because j8 = = kn, the radiation modes are not close to being... 

The discussion of bound modes in Section 13-3 applies equally to radiation modes on weakly guiding waveguides, except that the fields are no longer predominantly perpendicular to the waveguide axis. However, the cartesian components of the transverse electric field of Eq. (13-7) are still solutions of the scalar wave equation. Thus, if Vj denotes e j or e j, then... 

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