CURVED WAVE GUIDES 3 



can be derived from functions Jnix^) cos }up. In these functions, n is 

 called the aximuthal index because it indicates the type of symmetry around 

 the circumference of the wave guide. When these characteristic functions 

 are multiplied by the distortion factor cosine (p, the resulting expressions 

 are proportional to the sum of cosine (w -{- I) (p and cosine (n — 1) tp. This 

 means that the bending of the wave guide couples mainly those modes which 

 dififer by ±1 in azimuth index. Since the TEoi mode has the azimuthai 

 index 0, it is coupled to all modes of the type TEi™ and TMi^ . 



In the above qualitative discussion we have claimed that coupling exists 

 without defining the physical coupling parameters and their effects. We 

 must now supply this definition and show that the TEoi mode is particularly 

 susceptible to coupling losses. 



^2 



a-COUPLED TRANSMISSION LINES 



Fig- 2 



b-COUPLED RESONATORS 



Our investigation is guided by S. A. ScheUcunoff's statement^ that a wave 

 guide mode has the same equation of propagation as a high-pass transmission 

 line. Schelkunoff further points out^ that the high-pass character of circular 

 wave guide modes can be interpreted as the effect of interfering plane waves 

 whose directions of propagation deviate from the wave guide axis by a 

 constant slanting angle. 



We therefore approach the problem of coupled wave guide modes by 

 studying the behavior of two coupled transmission lines such as shown on 

 Fig. 2a. Each transmission line is schematically shown as an array of small 

 ladder sections. The series impedances per unit length of the lines are Zi 

 and 22 ; their shunt admittances per unit length, yi and y2 . The two 

 lines are loosely coupled by small mutual series impedances per unit length 

 (Zm) and by small mutual shunt admittances per unit length (>»„,)• 



A network of coupled ladder sections is more tractable than a wave guide 

 structure, but still somewhat complicated. Let us therefore carry the 

 analogy one step further. Figure 2b shows two resonant circuits, each 



2 Ref. 4, pp. 378 and 381 of the book. 

 « Ref. 4, p. 410 of the book. 



