Scalar wave equation radiation modes

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... 

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. 

Radiation inodes of the scalar wave equation 33-7 Orthogonality and normalization 33-8 Leaky modes... 

The scalar wave equation has both scalar bound and scalar radiation modes. The bound modes are analogous to the elements of a Fourier series, while the radiation modes can be viewed as the elements of a Fourier integral. Both are necessary to form a complete set of modes for representing an arbitrary field. The radiation modes are characterized... 

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. 

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