1298 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



and 



f V . A)8^2i r Co2 /, , . / CozY I 1 . , -1 



2coZ 



4co L^jS2i 'y ' • ' \Al3zi/ ' 2 A/3zi_ 



We introduce (10) and (11) into (7) and take advantage of 



A^ 



«1, 



(11) 



(12) 



which is satisfied for a gentle enough bend. The unwanted mode level 

 is then given by 



3\ 



^2(^1) I = 



Co 



A/S^^i 



(1 - e'"''') 



+ 8 



Co 



A^3 



(13) 



The general restriction (5) for the solution (6) in case of the linear 

 taper is 



2cn 



1 +4 



2 2 N 



CqZ 



A^, 



-3/2 



«1, 



A/3221 



and in view of (12) only 



I A^2i I ^ 1 (14) 



is required. The length of the transition has to be of the order or greater 

 than the largest beat wavelength. 



In addition to mode conversion loss the normal mode suffers heat loss 

 along the taper. From (3) and (6) this heat loss is given by the real part 

 of [yz — p{z)]. If A/3 » Aa we get 



r' c 



R[-yz - p] = aiz -\- Aa -—- dz 



JQ AB^ 



(15) 



It follows from (15) that the attenuation of the coupled straight guide 

 modes should not be too high for the normal mode bend to work properly. 



III. THE TOTAL BEND LOSS 



We will consider bends of the dielectric-coated waveguide only. The 

 normal modes of the dielectric-coated waveguide have phase constants 

 which are slightly different from the phase constants of the modes in 

 the plain guide^ 



5, 



(16) 



/3o„ 



3 1 - 



VQn 



