GUIDED WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA. II 953 



Thus, immediately within the ferrite, 



1 dEy . ///A 



■Liy OX \-/v J/ /external 



It' the two faces of the ferrite are x = —.To and x = .To we tlieu have 

 (1 f^) = ^ cot V^T^T^. .,, - ,,^ = A, 



\Jby ax /x=-Xq fXO 



and 



(^ ^-f) = -^ cot V^^^-^ X, - p.^ = B. 



\hy dx A=xo Mo 



If we write 



1 dE 



^. „ = Vl3r - /3" tan (V/3r -/32 a;), 

 iiy do; 



and make use of the boundary conditions we obtain 



tan2V^V^=..^ Vg3J;U-^.) 



(13) 



The non-reciprocity is clearly contained in the odd power of /3 in A 

 and B. 



For small thickness we replace the tangent by its argument, and, sub- 

 stituting for A and B on one side of the equation, obtain 



^ [cot V/3o- - ^- ^1 + cot V/3o' - iS^ rcs] = 2.ro[iS/' - /3' - AB], 

 Mo 



or 



sin ViSo^ - /32 (xi + a^s) 



(14) 



= 2a;o — sin V)8o' - /3^ Xi sin V/3o'' - jS^ rczLS/ - jS' - AB]. 



Meff 



Since the guide is almost empty, we may write 



ViSo' - /32(a;i + 0:2) = TT - 5, 

 where 5 is small. Or, 



27r5 



^ = '^i + FT 1—^2 ' 



i3i(a;i -f 0:2)2 



where ft^ = /3o^ - 7rV(a;i + X2)\ 



