The Character of Waveguide Modes in Gyromagnetic Media

409 410 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH ] 957 Fig. 1 -- Rectangular waveguide mode in isotropic medium for cutoff guide. position of plane waves of dependence c~ . If we represent k in a cartesian frame, the wave equation is satisfied for the condition k = kj~ + k ~ + k ~ = unc where /z and e are the permeability and permittivity respectively of the medium. Satisfaction of wall boundary condition requires that k and k be real and that each be of the order of the reciprocal of the transverse guide dimensions. Small transverse dimensions thus cause k 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 k and 1c are unique. Birefringence generally requires that two different magnitudes of k occur simultaneously, causing two different values of Jc to appear. In particular, let us postulate that both these values of k 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 exponentials decay so fast as to effectively cause the field to vanish there. Since kli, are now negative quantities, there is no contradiction in presuming that k may now be positive, thus permitting propagation in an arbitrarily small size waveguide.