GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA 657 



or 



2 



X„, = !^!L__i (90) 



The corresponding o-„s are given by 



1 — Xns u 



2 



1 — a-„sX„s r(? 



(91) 



and are all positive. Thus all saddle points lie in the first quadrant. At a 

 saddle point G = —ro/u„ and therefore it is the intersection of two 

 contours G = —To /u„ . Forn > 1, one of these obeys the asymptotic 

 formula (83), the other is asymptotic to 7„ and /„_i (see Fig. 5), and 

 obe3's (84), with n and n — I, near those curves. For n = 1, one of them 

 still follows formula (83), but the two "arms" of the other are asymp- 

 totic to 0- = 1/X and 7i , and so follow (85) and (84) with ri = 1 re- 

 spectivel3^ 



Three further facts useful in the construction of G^-diagrams are: 



1 — X^ Un^ r\? 



Along a curve r = — 2", G equals — — 2; thus the zero curves of F 



1 — (tX ro Un 



are contours of constant G. 



Along X = +1, 



(92) 



As 0-, X — > — 1 along (o- + 1) = «(X + 1); 



« + 1 



(93) 



G^ - 



('•" Z^) 



As for G(T(\), cr, To) we have, in addition to the asymptotic formulas 

 in the table : 



Ib' transforms into = 1 + — 7—- . 



0- + 1 



The intersection of (/„)r or (/„')r with cr = is given by 



% 



X., = p±y/i-4! 



