624 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1054 



as the earlier (37) except that the smallest root of F(z) = — 1 replaces 

 that of F{z) = 1. The remaining solution curves confined to the region 

 bounded by Oo', Oc , (Ia)t portray 4he incipient modes already encoun- 

 tered in the first quadrant. Their behavior near o- = — 1 also follows 

 (58), associated with the higher roots of F(z) = —1. Their end points, 

 the intersections of /,/ with (/x)r, are still given by the parametric 

 representation (46-8), due regard being paid to the signs of o-„ and p. 



The remaining branch of the TEn-limit mode, lying above a = — o-q , 

 is found in the triangle between (I a) t , o- = and Ib . Its end points are 

 given by (39) with p negative and by the intersection of Ib' with {I„)t 

 which is o- = — (1 + p), X2 = — 1, Xi = 0. Thus the cut-off in contrast 

 to the analogous branch for a > 0, is given by jS" = 0, a = — (1 + p). 

 (When p < —1, the branch does not exist at all.) We note that a left- 

 circular plane wave is cut off at exactly the same value of a as the TE- 

 mode is in this particular case (see, however, the following sections). 

 The slope at cut-off is determined b}^ expanding the G functions near 

 their infinities at /„ and Ib' and utihzing the Polder relation. The slope 

 is found to be 



da p(l - F{ro)) ' ^ ^ 



A further solution curve lies in the region between Oc and (Oo)r for 

 0- > — o-Q . It has no analogue in a guide with isotropic material and 

 will be discussed later. 



In the discussion of the mode spectrum for radii between tii and ji 

 three distinct types of cut-off point have alread}^ been encountered. When 

 larger radii are treated it is found that no other types arise.* In Section 

 4.17 formulas rele^'ant to the three types are given. An examination 

 of the field components in the neighborhood of the cut-off points is of 

 some interest. Cut-off points of type one (intersections of /„ and (Ia')t 

 or In and {Ia)t), at which jS" = 0, have Ez = and the field is of a 

 pure TE-type. The medium behaves transversely as though it had a 

 permeability, /x — k /n. Although the field is purely TE at cut-off the 

 mode terminating at such a point may in the limit of vanishing magne- 

 tization be either a TE- or a TM-mode. This impartiality extends to cut- 

 off points of the other types. Cut-off points of type II [(O,/)?- — 0^ or 

 (On)r — OcT occur at 0" = ±0-0 , where n = and here /S" does not van- 

 ish. In such cases one of the x's is finite and the corresponding contri- 

 butions to the field pattern quite normal. The other, however, tends 



* There is an exception to this statement. This is the type designated in Sec- 

 tion 4.17 as 2o=o which cuts off an isolated mode having no TE or TM analogue. 



