WAVEGUIDE MODES IN GYROMAGNETIC MEDIA 419 



Only ill the limiting condition of a infinitely greater than h do all modes 

 (m, n) propagate in the negative region of /x. In this particular case, 

 sin ^1 = and we find from (21a) and (21b) that C = D = 0. Thus we 

 find a situation in which the guide boundary conditions are satisfied by 

 but a single class rays of the two classes available. 



This result is entirely comprehensible if we observe the wave number 

 relationship obe3^ed by ki and ko . Employing the definition of p which 

 states that k' = kf/p, and using (11a) and (lib), we have 



k,' + ky,' + ki/n = (26a) 



kr' + kyi + k: = (26b) 



As stated in the Introduction, it is an entirel}^ consistent procedure to 

 satisfy boundary requirements with real wa^-e numbers over the negative 

 range of ^ using the class of rays indicated for (26a) above. 



More generally, (25) shows a complex relationship of the ordering 

 of propagation modes by n and w, for finite a, for a given negati^'e value 

 of fi. In contrast to the /z > case, propagation may possibly not occur 

 for a range of lower order integral values of m. As n becomes increasingly 

 large in magnitude, m must likewise take on increasingly higher values 

 for transmission to occur. 



The case of cos <pi real and greater, in magnitude, than unity, leads 

 to triA'ial result. Both partial waves have imaginaiy values of A'x , for 

 this case, and the far wall receives essentially no coupling. Anal3^sis 

 simply repeats the result of (20) and we find that | /x | > \ k\ and /x < 0. 

 If the Polder tensor components given in (2a) and (2b) are plotted (see 

 Fig. 5). We find that this last set of inecjualities form an impossible 

 combination. 



Summarizing we find that a rectangular waveguide of any dimension 

 (and, in particular a guide of arbitrarily small dimensions), filled with a 

 lossless transversely magnetized ferrite medium, will support an infinite 

 number of freely propagating modes at any frecjuency for which | m | < 

 \ k\. The character of these modes differs considerably in the two regions 

 of M < and /x > and somewhat different viewpoints of propagation 

 must be taken. We shall find similar results relating to the longitudinally 

 magnetized ferrite filled circular waveguide in the following section. 



III. ANALYSIS OF LONGITUDINALLY MAGNETIZED FERRITE IN CIRCULAR 

 GUIDE 



We now proceed to a second structural geometry in which an anoma- 

 lous behavior occurs attributable to the birefringence of the medium. 



