1300 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



I 



By the law of reciprocity the same conversion loss occurs in the second 

 transition. Both conversion parts phase with each other. To get an aver- 

 age total conversion we may add them in quadrature and the total con- 

 version loss to one of the unwanted modes is, expressed in nepers, 



Under most unfavorable phase conditions the conversion loss may be 

 twice this value, but it is veiy unlikely that such phase conditions will 

 be satisfied for all coupled modes simultaneously. 



Besides mode conversion loss the local normal mode suffers attenuation 

 in the bend. This attenuation is larger than the straight guide attenua- 

 tion. Each straight guide mode contained in the local normal mode of 

 the bend causes an increase in attenuation. From (15) the loss caused by 

 one of these straight guide modes is: 



J 1.1 2 

 T^ ^^- (21) 



Ap^ 



Where aim is the attenuation constant of the TEi^ and TMn waves 

 respectively in the dielectric coated waveguide. 



Introducing the trapezoidal curvature function of Fig. 1 into (21) 

 we get 



A, = 1^, (aim - «oi) (l-^ 2i). (22) 



The loss caused by the TEoi attenuation in the straight dielectric coated 

 guide is from (17) and (18) 



As = aoil 



1 + (e - 1)^5^ 



VOm~ 



+ ^ ^!^ 8\ (23) 



3 1 — foi 



The total bend loss is finally obtained by summing up all the terms of 

 (20), (22), and (23), 



A = As-^ZA, + T.Ae. (24) 



The summation signs indicate that all coupled modes (TMn and TEi,;,) 

 have to be taken into account. 



The effectiveness of the normal mode bend is best demonstrated by a 

 practical example. A copper pipe now in experimental use at Bell Tele- 

 phone Laboratories for circular electric Avave transinission near 5.4 mm 

 wavelength has 2-inch I. D. and 2f-inch CD. Suppose we want to change 

 the direction of a waveguide fine with this copper pipe bn an angle ^o • 

 We can do this most easily by inserting a dielectric-coated section, which 



