900 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 



TRANSMISSION LOSS OF A WAVEGUIDE WITH A SPURIOUS MODE RESONATING 

 REGION 



Let US consider a single-mode waveguide connected to another of 

 different cross-section that admits two modes. Since these two modes 

 are orthogonal, the junctions may be considered as made of three single- 

 mode lines connected together, provided we define the elements of the 

 scattering matrix properly. The three modes, or lines in which they 

 travel, are indicated by the subscripts 0, 1, and 2, as shown in Fig. 1. 

 If ao , tti , Qi and 6o , &i , ?>2 are the complex amplitudes of the electric 

 field of the incident and reflected waves respectively, then 



where 



Too Foi ro2 

 [S] = Toi Tu ri2 (1) 



_ro2 Flo r22_ 



is the scattering matrix.^ 



This specific type of change of cross section may be treated as a 

 three-port junction. 



Now, if a length / of a two mode waveguide is terminated sym- 

 metrically at both ends mth a single mode waveguide (Fig. 2), each 

 joint is described by the same matrix (1), and the connecting two mode 

 wave guide has the following scattering matrix: 



(10 



in which 



je, = y,(. = («i +JiSl)f 

 je2 = 72^ = (a2 + jS-^t, 



7i and 72 are the propagation constants of modes 1 and 2. 



• N. Marcuvitz, Waveguide Handbook, 10, M.I.T., Rad. Lab. Series, McGraw- 

 Hill, New York, 1951, pp. 107-8. 



