HELIX WAVEGUIDE 1355 



field patterns may be quite different when the jacket is lossy. This sys- 

 tem is not completely unambiguous, because as will appear in Section 

 IV the mode designations thus obtained are not always unique. However 

 it is a satisfactory way to identify the modes so long as the jacket con- 

 ductivity is high enough for the loss to be treated as a perturbation. 

 Approximations derived on this basis are presented in the next section. 



III. APPROXIMATE EXPRESSIONS FOR PROPAGATION CONSTANTS 



If the jacket were perfectly conducting, the helix waveguide modes 

 would be the same as in an ideal circular waveguide, with propagation 

 constants given by 



where 



V = Xo/Xc = p\o/2Tra 



p = ??i*^ zero of Jnix) for TM„m mode, or rn^^ zero of Jn(x) for TE„m 

 mode 



If the jacket conductivity is sufficiently large, approximate solutions 

 of (6) may be found by replacing Hn'\^2a) and Hn'^'i^iO) with their 

 asymptotic expressions, and expanding Jni^ia) or Jn'(^ia) in a Taylor 

 series near a particular zero. This calculation is carried out in the ap- 

 pendix. The propagation constant may be written in the form 



ih = a + i{^nm + A|S) 

 where to first order the perturbation terms are 

 TM„„ modes 



a + m = ,\ ^ \„ rXT-^r-, (7a) 



a(l — v-y^ 1 -\- tan^ \p 



TE„TO modes 



a 4- i\B - ^ + ''^ ^V" [tan ^ - n(l - vyVyvf . , . 

 ^'^^~a(l -, 2)1/2 ^^^7^2 1 -f tan^ ^ ^'^^ 



and 



? + ^> = (e' - ie'T'" 

 e = e/eo , e = cr/coeo 



The approximations made in deriving (7) are discussed in the appen- 



