GUIDED-WAVE rKOPAOATKIX TIIIIOTCH CVUOMAOXETir ArKDIA 020 



all when 



i, 



v> \/i-% 



If To is greater than u^ , the (Oi')t curve has appeared. A new region of 

 like signs of the G's arises between it and (/i')r , see Fig. 9(e), and con- 

 tains a sokition curve. This ends at ctq , Xio and begins at cr = at a value 

 of Xi pertaining to the TlMn-mode. Thus, it is clear that as ro passed 

 11-2 , the end-point of the TMn curve jumped discontinuously from 

 {I a)t — h to (To , Xm . This jump is anticipated as ro approaches V2; the 

 (3' — a curve first bulges beyond (J a')t — h towards its later course and 

 returns to that point with positive slope. As ro increases further no change 

 occurs in the qualitative behavior of the mode. It may be noted that 

 above No the mode exists for all p. 



Be^'ond ro = ih , at least part of the area betAveen 7i and {Ia)t is an 

 achnissible region and does in fact contain the TE12 solution curve. It 

 begins at cr = and Xi given by that solution of eqn. (39) which is, in 

 the limit p = 0, the TE12 solution. It is cut off A^ith /3^ = at {Ia)t 

 — I\ , the end point relinquished by the TMn-solution curve. As ro 

 passes j-i , the TE12 solution retains its cut-off point, but, beyond ro = Us , 

 it will transfer this point discontinuously to <to , Xio . Thereafter its course 

 remains essentially unaltered. Tables I, II and III show the progression 

 of cut-off points of the various modes. 



It may be recalled that in the analysis of Ui < ro < ji , the modes in 

 (T < — (To followed essentially the same course as in a > ao ■ This is also 

 true of their progress with changing radius and of the escape process. 

 The singular character of the (Oo)t curve and the presence of Ib lead to 

 some local changes in the progress of the modes but have no effect on 

 their more salient features in this particular range of a. The scheme of 

 progression of the end points is shown in Table I. 



In contrast with the state of affairs in the region just discussed, the 

 mode structure in the area between a = and a = — ao is verj^ mark- 

 edly affected by the presence of (Oo')r and Ib'- 



When ro < Vi , a solution curve exists betw^een a = —I — p, and 

 a = — (To . It starts with /3' = at the intersection of (/..Or and {Ib') with 

 a slope given by (41). For sufficiently small ro , 13 tends to infinity as 

 (T — > cTo , since the solution curve approaches the line 0/ or (0^)r • Its 

 shape is then given by (52), see Section 4.17. As ro increases, Oo falls 

 steadily. Eventually, for sufficiently large p, its minimum falls below 



See footnote on page 628. 



