HELIX WAVEGUIDE 



1383 



where 



^ + iv = 



1 - 



1 - 



2 -] 



ie" 



1/2 



It follows from (A5) and (A7) that for TM modes, 



a + iA^ = 



a(l - u') 



2 \ 1 2 



1+1 



___| tan V_ 



where ^ + irj is given by (A8). 



For the TE„„ mode, let p be the m*^ root of J,/ ; then 



/n'(fia) = Jn'iV + X) = ^""^ ~/^'' Mp) 



y 



Equation (A4) yields 



X 



ip V 



tan rp 



nil - v ') 

 pv 



2\l/2-|2 



(^ + irj) 



(p^ — n^) 



1+ i-e^' r^"'* 



and, using (A5), we have for TE modes, 



a + 2A/3 



V 



tan \p — 



n(l - / ) 



2\l/2-12 



(^ + t^) 



(p- - 71^) a(l - 1^2)1/2 



1+1 



7>^tan lA 



t€ 



where ^ + z?? is given by (AS). 



In view of (A5), the condition that | a; | <3C 1 is equivalent to 



^^ I aa + lA^a \ « 1 



(AS) 



(A9) 



(AlO) 



(All) 



In all the numerical cases treated in the present paper, the approximate 

 formulas agree well with the exact ones provided that the left side of 

 (All) is not greater than about 0.1. 



A condition Avhich is usually satisfied in practice, although not strictly 

 a consequence of the assumptions (A3) or (All), is 



1 



«1 



^e 



