Weakly guiding waveguides fields

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. 

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... 

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... 

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... 

In Chapter 13 we showed how to construct the fields of bound modes on weakly guiding waveguides using simple physical arguments, and then, in Chapter 25, we extended the method to include radiation modes. To complement the physical approach, we now give the formal mathematical derivation using perturbation theory on the vector wave equation. 

If we put the above results together, the fields of bound modes on weakly guiding waveguides have the expansions... 

Table 32-1 Governing equations for modal field expansions. Equations for the lowest and first-order transverse electric fields of a weakly guiding waveguide. Remaining field components correct to order are given explicitly in terms of the solutions of the two equations.

In Section 32-2 we derived an expansion for the bound-mode fields of weakly guiding waveguides. The terms in this expansion depend in turn on the expansion of the modal parameter U in Eq. (32-8). Here we show how to find these terms. 

We recall from Chapters 11 and 12 that modes with = 0 everywhere are TE modes. Table 32-1 shows that the term V, -e, in Eq. (32-22) is proportional to on weakly guiding waveguide. Consequently V,-e, = 0, and the scalar propagation constant is identical to the exact propagation constant. This is in keeping with Chapter 12, where we showed that the exact TE mode fields are derivable from the scalar wave equation. 

In Section 13-5 we used physical arguments to show that the transverse electric field must be polarized along the optical axes of the weakly guiding waveguide. If and yo are unit vectors parallel to the optical axes, then the two polarizations are expressible as... 

For virtually all applications to weakly guiding waveguides in Part II, it is sufficient to know the lowest order modal fields e, and h,. These fields with the - omitted were discussed at length in the previous five sections. We showed in Section 32-2 that the first corrections are the longitudinal fields and which are given explicitly in terms... 

We showed how to determine the radiation modes of weakly guiding waveguides in Sections 25-9 and 25-10, starting with the transverse electric field e, which is constructed from solutions of the scalar wave equation. However, unlike bound modes, the corresponding magnetic field h, of Eq. (25-23b) does not satisfy the scalar wave equation. This means that the orthogonality and normalization of the radiation modes differ in form from that of the bound modes in Table 13-2, page 292, as we now show. 

Thus, the discrete values of P for the bound inodes of Eq. (33-1) are replaced by a continuum of values for P(Q). We explained in Chapter 25 why it is more convenient to work with the radiation mode parameter Q, which is defined inside the back cover. We are also reminded that both the electric and magnetic transverse fields, e, and h, of the vector bound modes of weakly guiding waveguides are solutions of the scalar wave equation. However, only e Q) of the vector radiation modes satisfies the scalar wave equation, as we showed in Chapter 25. 

The radiation field of the scalar wave equation can be represented by the continuum of scalar radiation modes discussed above, or by a discrete summation of scalar leaky modes and a space wave. This is clear by analogy with the discussion of vector radiation and leaky modes for weakly guiding waveguides in Chapters 25 and 26. Scalar leaky modes have solutions P of Eq. (33-1) below their cutoff values when P becomes complex. Many of the properties of bound modes derived in this chapter also apply to leaky modes. For example, the orthogonality condition of Eq. (33-5a) applies to leaky modes, provided only that the cross-sectional area A. is replaced by the complex area A of Section 24-15 to ensure that the line integral of Eq. (33-4) vanishes. 

In Chapter 32 we derived the governing equation which relates the total electric field E of a weakly guiding waveguide to current sources of density J within the waveguide. Using the waveguide parameter definition inside the back cover, Eq. (32-52) is expressible as... 

To simplify the algebra, and remain consistent with Chapters 6 to 9, we determine the transmission coefficients for weakly guiding waveguides only. This means that the expressions we derive are independent of the direction of the incident electric field. 

The fields of modes of weakly guiding waveguides are constructed from solutions of the scalar wave equation, as explained in Chapter 13. However, when the waveguide... 

For weakly guiding structures, the second term can be neglected, and we obtain the standard Helmholtz equation in which individual components of the electric field intensity vector E remain uncoupled. For high contrast waveguides this is clearly not the case. The second term in Eq. (2) in which the transversal electric field components are mutually coupled must be retained. 

If the waveguide is weakly guiding, there is only a slight variation in its profile and A < 1. We then assume that we can express the transverse electric field e, as a power series in A, treating V and A as independent parameters [1-3]. Thus... 

On a weakly guiding, nonuniform waveguide the total scalar field d>(x, y, z) satisfies the three-dimensional scalar wave equation, as expressed by Eq. (33-36). We express d> as a summation over orthonormal local modes... 

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