410 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1957 



Z VARIATION 



*>X 



X VARIATION 



Fig. 1 — Rectangular waveguide mode in isotropic medium for cutotT guide. 



position of plane waves of dependence c"'*'^. If we represent k 

 cartesian frame, the wave equation is satisfied for the condition 



ni a 



h' = ki + hi + k; = 



cc'ns 



where n and f are the permeabiUt}' and permittivity respectively of 

 the medium. Satisfaction of wall boundary condition requires that k^ 

 and /;, be real and that each be of the order of the reciprocal of the 

 transverse guide dimensions. Small transverse dimensions thus cause ky~ 

 to be negative, driving the waveguide into a cutoff condition. 



We shall now find that birefringence permits another class of modes 

 in the small size ferrite loaded waveguide. Letting the magnetic axis be in 

 the z direction, it will be shown in the text that corresponding to any 

 mode of the guide ky and k, are unique. Birefringence generally requires 

 that two different magnitudes of k occur simultaneously, causing two 

 different values of A-^ to appear. In particular, let us postulate that both 

 these values of A-;^ are imaginary. Given two exponentials, it is possible 

 now to satisfy the requirements of electric field nulls at either side wall, 

 as shown in Fig. 2. At the other side wall we shall show that the ex- 

 ponentials decay so fast as to effectively cause the field to vanish there. 

 Since A-^i, o are now negative ciuantities, there is no contradiction in pre- 

 suming that ky maj'' now be positive, thus permitting propagation in an 

 arbitrarily small size waveguide. 



The effect of birefringence may then be that of transforming a class 

 of longitudinal!}' cutoff modes into another class that propagates longitu- 

 dinally but cuts off transversely. The condition of this occurrence will 

 be shown to be that for which the diagonal term of the Polder tensor, ^t, 

 is positive and is less in magnitude than the magnitude of the off" diagonal 

 term k. In the case of a small rectangular guide, propagation occurs 

 anomalously for negative values of n, as well but in a manner not as 



