1248 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



(117) 



where 



h = h[oi] = h(n) , 



C = C(ii) , 



it is evident that we can add and subtract to get the equivalent pair of 

 equations: 



rf(ao+aO_^ .(^_^^)(^^_^^^) =0, 



(118) 

 + iQi — c)(ao — tti) = 0. 



dz 



d(ao — aO 



dz 



Hence the normal modes of the curved guide are the combinations 

 ao±ai,orTEoi±TM". 



In order to launch only the normal mode TEoi + TMu" at the input 

 of the bend, the amplitude of the other mode must be zero, or ao = ai . 

 Similarly, to launch only the mode TEoi - TMn", the condition is 

 ao = — tti . Hence the output of the normal mode transducer of length /, 

 say, must be 



ao(0 = ±ai(Z). 



(119) 



We return to the solution of the coupled line equations in Section 2.1 

 and write 



K = rf(ii) = d, 



7o = ?'/3[ni] = i^o , 



r = is = Wi^n - )Si)- + 4rf2. 

 Then equations (60) of Section 2.1 become: 



(120) 



ao(0 = 



1 , . (/3o - /3i) . 1 / 

 cos ^si — I sm f Si 



-jt(/3o+^i)' 



ai{i) = —I — sm ^si e 

 s 



(121) 



-\i(?0+^\)l 



Substituting into (119) and equating real and imaginary parts gives: 



/ = xA, (122) 



I /3o - iSi 1 = I 2d I . (123) 



