WAVEGUIDE MODES IN GYROMACxNETIC MEDIA 417 



mode by equating the propagation constant. Therefore, 



cos ^2 = M ' COS^l (16) 



where cos v?2 is imaginary for propagation. Propagation may therefore 

 occur for n > and cos <pi imaginarj^ and/or, ju < and cos <pi real. 



Boundarj' conditions require Ey and E, to vanish at both guide side 

 walls. Four equations result which may be satisfied, in turn, by a super- 

 position of four transverse waves involving kj^ , — AVi , Ax2 , and —K^ 

 corresponding to ^'alues d=^i,2 . For n > both of the birefringent rays 

 have transverse decay. Since the magnitudes of fcj, , are large in small 

 size guide (see Introduction) boundary conditions need be satisfied for 

 practical purposes at only a single wall. We are then left with the sim- 

 plification of only two equations in two unknowns. 



Setting X = in (14) and (15) and taking equation (16) into account, 

 we ha^•e the boundary conditions 



— sin v^i + i cos (pi I + BlijT' cos ^i] = (17) 



^[m~^] + 5 = (18) 



With the result that 



cot^Pi = -1^ = (pL] (19) 



Choosing ky positi\e real, A^i is positive imaginaiy for k positive and 

 negative imaginary for k negative. The rf field therefore hugs the right 

 wall for K > and the left for k < 0, or, alternatively, switches sides in 

 the change from a forward to backward direction of propagation. 

 Equation (19) may be written equivalently as 



2 



cos' <p, = ^A—. (20) 



II- — K- 



Propagation, occurring for imaginary values of cos tp and /x > 0, is ob- 

 tained for 1 ju ! < I K |. 



Let us now analj^ze, the possibility of small guide propagation for 

 M < 0. We find, from (16), that cos 9?i is real for this case. Two cases 

 arise; the first for which | cos ^i | < 1 and the second for the reverse 

 situation. 



Let us first consider the case of | cos ^i | < 1. From (14), ^^i is real 

 whereas from (15) l\^ is imaginary. Let us associate wave amplitudes 



