HELIX WAVEGUIDE 



1353 



Attention is restricted to waves traveling in the positive ^-direction, 

 which are represented by the factor exp { — ihz), where /i ( = /3 — ia) is 

 the complex phase constant. However it is necessary to consider both 

 right and left circularly polarized waves; this accounts for the use of 

 both positive and negative values of n. 



Substitution of (2), (3), and (4) into the boundary conditions (1) 

 leads to the following set of equations: 



V 2 , , hn 

 fi tan \l/ — — 

 a 



Jn(^ia)an + i(^iJ.(ihJn'{tia)hn = 



^2 tan yp — — 

 a 





(5) 



•to;eofiJ«'(fia)a„* + 



. 2 . , hn 

 fi tan yp — — 

 a 



Jn(^ia)hn 



(2)', 



+ (o- + icoe)^2Hn ' {^2a)an 



[ 



> 2 + , hn 

 ti tan \l/ — — 

 a 



Hr.''\ha)h: = 



If the conductivity of the exterior region is infinite, it is possible to 

 satisfy the boundary conditions with only one of the amplitude coeffi- 

 cients different from zero; for example 



hn = a«' = 6„* = 



a„ 







or 



dn — CLn = bn = 



bj 9^ 



Jni^xO) = Jn'iria) = 



The first case corresponds to TM modes and the second to TE modes 

 in a perfectly conducting circular guide. Linearlj^ polarized modes may 

 be represented as combinations of terms in a,/ and a-n\ or bn and 6_„*. 

 If the exterior region is not perfectly conducting, one can still find 

 solutions having the fields confined to the interior region by propedy 

 choosing the angle of the perfectly conducting helical sheath. For exam- 

 ple, it is easy to verify that equations (5) are satisfied under the follow- 

 ing conditions: 



ttn 



an = bn = 



bj 9^ 



tan yp = 



hn 



Jn'i^ia) = 



