HELIX WAVEGUIDE 1359 



left circularly polarized waves when xp = 0. Using the relationships 



ha = KM' + (M' (e' - ie" - l)f\ Im^a < 



ha = {%af - (rla)T'^ Im /la < 



it is clear that Fni^a) is an even function of ^a, involving the parame- 

 ters Pott (= 27ra/Xo), e', e", and n. 



When specific values have been assigned to /3oa, e', and e", roots of 

 (10) can be found numerically by the straightforward procedure of 

 evaluating Fni^a) at a regular network of points in the plane of the 

 complex variable ^a, plotting the families of curves Re F„ = and 

 Im Fn = 0, and reading off the values of ^a corresponding to the inter- 

 sections of curves of the two families. 



The procedure just outlined has been applied to the cases n = and 

 n = 1. When n = one can take out of Fo(fia) the factor Jo'(fia), whose 

 roots correspond to the TEom modes; the roots of the other factor are 

 the TMom-limit modes. When n = 1 the function Fi(ha) does not factor, 

 and its roots correspond to both TEi^-limit and TMi^-limit modes. If 

 the jacket conductivity is high it is easy to identify the various limit 

 modes, and a given mode can be traced continuously if the conductivity 

 is decreased in sufficently small steps. 



The numerical calculations were set up, more or less arbitrarily, to 

 cover the region ^ Re ^a ^ 10, —10 ^ Im ^a ^ 10, for each set 

 of parameter values. A few plots of Re Fn and Im Fn made it apparent 

 that for propagating modes the roots in this region are all in the first 

 quadrant and usually near the real axis. The entire process of solution 

 was then programmed by Mrs. F. M. Laurent for automatic execution 

 on an IBM 650 magnetic drum calculator. The calculator first evaluated 

 Fni^ci) at a network of points spaced half a unit apart in both directions, 

 then examined the sign changes of Re F„ and Im Fn around each ele- 

 mentary square. If it appeared that a particular square might contain 

 a root of Fn , the values of Fn at the four corner points were fitted by an 

 interpolating cubic polynomial ° which was then solved. If the cubic 

 had a root inside the given square, this was recorded as an approximate 

 root of Fn . The normalized propagation constant iha = aa -{- i(3a was 

 also recorded for each root. 



The calculated roots ^a and the normalized propagation constants 

 are summarized in Tables 1(a) to 1(f), which relate to the following cases: 



Table 1(a)— /3oa = 29.554, e' = 4, e" variable 



Table 1(b) —/3oa = 29.554, e' = 100, e" variable 



Table 1(c) —/3oa = 29.554, e = e", both variable 



10 A. N. Lowan and H. E. Salzer, Jour. Math, and Phys., 23, p. 157, 1944. 



