(10) 



CIRCULAR ELECTRIC WAVE IN SERPENTINE BENDS 1283 



^1,2(2) = ± jAt sin 2cS exp ± 2^^{z - 2 f sin^ cS dz\ . 



As long as we have the condition / | 11,2(2') | dz' « 1, approximate 



sohitions of (9) can be written down which proceed essentially in powers 

 of ^1,2 as follows, 



w^i) = wi(0) - W2{0) [ ^i{z) dz 



Jo 



+ wM r ^i(/) r uz") dz" dz', 



Jo Jo 



woiz) = W2(0) — Wi{0) I ^2(2') dz' 



Jo 



+ 1^2(0) r U^) [ 6(2") dz" dz'. 

 Jo Jo 



The new coordinates are related to the wave amplitudes by: 



(11) 



Eiiz) = -e ^" iwj{z) exp 



+ jw-ziz) exp 



— A7 ( 2 — 2 / sin^ Cod dz' J 

 Ay iz — 2 / sin' Co9 dz 



sin Cod dz' ] cos Cod 

 sin Cod>, 



1 



E2{z) = 2^ "' y^i(^) ^^P 



— A7 



+ Wiiz) exp 



A7 



[z — 2 \ sn\ Cod dz j 

 [z — 2 \ sin" Cod dz j 



(12) 



sin Ci)d 



cos Cod >. 



The solution (12) in combination with (11) is general and may be 

 applied to any form of curvature as long as the converted power remains 

 small compared to the original power in either of the modes. 



III. -WAVE PROPAGATION IN SERPENTINE BENDS 



If we apply (12) to a section of a serpentine bend with the length I we 

 have 6(1) = 0. The output amplitudes are related to the input ampli- 

 tudes of both modes by a transmission matrix 



EiH) = II r II Em. 



The elements of \\ T \\ are obtained from (11) and (12): 



(13) 



