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CHAPTER 13   Guided Waves              481

                     E and H.For example, (77a) and (78b) can be combined, eliminating E ys ,togive

                                               − j     ∂ H zs  ∂E zs
                                          H xs =    β     − ω                       (79a)
                                                κ 2   ∂x       ∂y
                     Then, using (76b) and (77a), eliminate E xs between them to obtain
                                               − j     ∂ H zs  ∂E zs
                                          H ys =    β     + ω                       (79b)
                                                κ 2   ∂y       ∂x
                     Using the same equation pairs, the transverse electric field components are then found:

                                               − j     ∂E zs  ∂ H zs
                                         E xs =    β     + ωµ                       (79c)
                                               κ 2    ∂x       ∂y

                                               − j     ∂E zs  ∂ H zs
                                         E ys =  2  β    − ωµ                       (79d)
                                               κ      ∂y       ∂x
                     κ is defined in the same manner as in the parallel-plate guide [Eq. (35)]:

                                                       2
                                                 κ =  k − β 2                        (80)
                               √
                     where k = ω µ .In the parallel-plate geometry, we found that discrete values of κ
                     and β resulted from the analysis, which we then subscripted with the integer mode
                     number, m (κ m and β m ). The interpretation of m was the number of field maxima that
                     occurred between plates (in the x direction). In the rectangular guide, field variations
                     will occur in both x and y, and so we will find it necessary to assign two integer
                     subscripts to κ and β, thus leading to


                                                       2
                                                            2
                                               κ mp =  k − β mp                      (81)
                     where m and p indicate the number of field variations in the x and y directions. The
                     form of Eq. (81) suggests that plane wave (ray) theory could be used to obtain the
                     mode fields in the rectangular guide, as was accomplished in Section 13.3 for the
                     parallel-plate guide. This is, in fact, the case, and is readily accomplished for cases
                     in which plane wave reflections occur between only two opposing boundaries (either
                     top to bottom or side to side), and this would be true only for certain TE modes. The
                     method becomes complicated when reflections occur at all four surfaces; but in any
                     case, the interpretation of κ mp is the transverse (xy plane) component of the plane
                     wave-vector k, while β mp is the z component, as before.
                         The next step is to solve the wave equation for the z components of E and H,
                     from which we will find the fields of the TM and TE modes.

                     13.5.2 TM Modes
                     Finding the TM modes begins with the wave equation [Eq. (59)], in which derivatives
                     with respect to z are equivalent to multiplying by jβ.We write the equation for the z
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