902 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 



From (2), (3), (4) and (5) 



V = 



I r,o |2 



2|ri.2|2(l - cos ^) 



1 + 



T22 



J«l12 



1 - [rooe"' - + Tne"'''] 



(6) 



Avliere 



i 22 1 22 6 



(f = 61 — 02 — <pn ~{~ <P 



22 



111 order to understand this expression physically, let us suppose first 

 that there is no attenuation. The transmission coefl&cient r becomes 

 when the following equations are fulfilled simultaneously 



and 



182^ — <P22 = PT 



(f = (2q -\- l)x 



p = 0, 1,2, 3--- (7) 



5 =^ 0, 1,2, 3 ••• (8) 



The first of these equations states that the line carrying the feebly 

 coupled mode must be at resonance, since this condition is satisfied when 

 the electric length of this line is modified by a multiple of tt radians. The 

 second condition, (8), implies that both paths, in lines 1 and 2, must 

 differ in such a way that electromagnetic waves coming through them 

 must arrive in opposite phase at the end of the two-mode waveguide. 

 This is quite clear if we think that, in order to get complete reflection, 

 signals coming through lines 1 and 2 must recombine again Avith the 

 same intensity and opposite phase. In order to get both modes with the 

 same intensity, the converted mode must be built up through resonance; 

 the opposite phase is obtained by an appropriate electric length adjust- 

 ment. When attenuation is present, F will not be 0, and conditions (7) 

 and (8) for minimum transmission are modified only slightly if the 



0-- 



ap — » D,^-^ bp — » 



•* t)o ^ 32 



* — l--- * 





Fig. 2 — Schematic of a two-mode waveguide terminated symmetrical!}' on 

 each side with a single-mode waveguide. 



