GUIDED-WAVE PROPAGATION' THROUGH GYROMAGNETIC MEDIA 023 



from which higher modes are drawn as the guide radius is increased. 

 That the propagation of modes which for larger guide radii correspond 

 to higher TM and TE modes is possible for limited ranges of c might 

 be ascribed to the larger /i-values in those ranges, which cause the wave 

 to see an effectively larger guide. This explanation is convincing only 

 when (T > 1 . When cto < o" < 1 , m is negative, and the propagation must 

 then be the result of an interplay between ju and k. In passing we remark 

 that we are here dealing with the propagation analog of so-called "shape 

 resonances," which physicists sometimes encounter in resonance experi- 

 ments on small spheres of ferrite in cavities. 



We now turn to a discussion of the solution curves for o- < which 

 lie in the third quadrant. Fig. 9(b) shows the partition of the region 

 allowed by the Polder relation (again for p = %) into positive and 

 negative regions by the various /', 0', (Z)r and (0)r curves. Regions in 

 which G(X2 , <y) and G(T{X-i), a) have opposite signs are shaded. For 

 a < — (To , the question whether a given region of like signs is the site 

 of a solution curve may, with one exception, be answered by the same 

 type of geometrical argument as used for a > 0. The singular area is 

 that part of the region bounded by Ib' and Oc in which the G'-functions 

 are both positive. Here both G(X2 , (t) and G(T{\2), <t) are zero on Oc ; 

 G{\2 , <r) goes to 00 on Ib', whereas G{T(\2), a) is finite throughout the 

 region. Xo intersection can be predicted, then, by the earlier argument. 

 It can indeed be shown (for all n) that there is no such intersection. For, 

 in the case p = 0, the solution curves are /„' or 0„' curves as demon- 

 strated in Section (4.11). The region under consideration contains no 

 such curves, and hence no solution curves. Thus, for p = 0, since G{)^2 , <t) 

 goes to infinity on Ib', the surface G(T(\2), a) must lie entirely below 

 the surface ^(Xo , a). Consider now, for fixed a and increasing p, a point 

 on the 6*(T(Xo), a) surface whose height remains unchanged. For such a 

 point T(\2 , p, (t) remains fixed and from the Polder relation this means 

 an increasingly negative X2 . Since it can be shown that G^(X2 , a) goes 

 monotonically from to >: as Xo becomes more negative, it follows that 

 GiTiXo), <t) continues to be below G(\2 , (t) for all p. 



All other regions of common sign do carry solution curves. That 

 corresponding to the TEn-limit mode begins at a- = —1, X; = — 1, 

 passes through the intersection of (Oo)r and Oo' and persists for indefi- 

 nitely large a. The asymptotic formula (38), for /3^ at large a- also holds 

 as 0- ^ — oc , if the signs of both a and p are taken to be negative. The 

 behavior of /3" near a = —I mav he found by the same means used at 



1 

 (7 = 1. The resulting expression* is to order , — - essentially the same 



(T -h 1, 



* See Section 4.17 for a more exact formula. 



