Scalar radiation modes

Sometimes it is useful to know the normalization of the scalar radiation modes, as discussed in Section 33-7. By substituting from Table 25-4 into Eq. (33-31), we obtain, in an analogous manner to the normalization in Table 25-2, the scalar normalization... 

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... 

We define Fj(Q) to be the normalization of each scalar radiation mode, given by... 

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 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 used the polarization properties of the waveguide to determine the direction of e, and the correction S j to the scalar propagation constant Pj. However, the propagation constant p for radiation modes takes any value in the range 0 < jS < kn y and is therefore a continuous variable independent of waveguide polarization. Consequently, higher-order correc-... 

The scalar propagation constant Pj is determined from the eigenvalue equation. We have ignored the continuum of radiation modes in Eq. (27-1) since we shall be considering coupling between bound modes only. 

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. 

Consider two distinct radiation modes with electric fields Cj = ej(Q) and = e iQ ), and form the triple scalar product Cj xh z, where denotes complex conjugate and z is the unit vector parallel to the waveguide axis. This product requires only the transverse components of the fields. Thus, if we substitute for h, using Eq. (25-23bfand rearrange using Eqs. (37-24) and (25-23a), then... 

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. 

For a Gaussian beam, the fields of the radiating electric and magnetic multipoles satisfy the same boundary conditions (vanishing faster than 1/p as p oo) so that the fields in the plane(s) defined by the transverse E (H) field and the optical axis are symmetric. It is difficult to generate a balanced hybrid mode in conventional smooth-walled metallic waveguide instead, one may use a component called a scalar horn. 

Scalar Mode. The PDR-77 can be run in the scalar mode to increase the sensitivity of the probe. The probe will average the readings over a 5-minute period to give a better indication of the radiation level. 

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

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