796 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



vanish; others may be small. In isotropic media, /i„„ = n^^ = i^zz = 

 M, «uu = fill. = Uz = (■ and the mutual coefficients vanish. In gyromag- 

 netic media subjected to a strong magnetic field in the 2-direction, the 

 permeability coefficients of superposed ax: fields are^ 



iiuu 



l^vv = M, 



Mru = 



"Mur, 



Hzu = yiz 



= 0. 



(48) 



If the entire waveguide is filled with such a medium, assumed to be 

 homogeneous, equations (43) and (44) become 



Iz.[n]j(^l^zzX[n] II T[„]Tlm]dS = — T 

 Vz.(n)j^^X(n) 11 T (_n)T (_„,-, dS = — 7(, 



[rn] , 



(49) 



In view of the orthogonality of the ^-functions and the normalization 

 conditions (13), we have 



z,[m] — 



X[m] 



V 



[m] 



V 



z, (m) 



X(m) J 



(m) , 



(50) 



where the summation convention is waived. In this case all the transfer 

 coefficients in equations (30) vanish, 



1 (.m)(.n) — J (m)[n] — i [m][»] — i [m](n) — i (m)(.n) — i (m)[nl \0i.) 



= T[„,](n) = T[m]ln] = 0. 



The transfer impedances and admittances are 

 ^(m)(n) = 0, ii n 9^ m, 



2 

 1 X(.m) -r 



= jo:tx + -r- , ii n = m; 

 jcoe 



<(m)[n] — —Ji^f^uv 



II 



dT[n] dT(m) , dT 



d du d du e 



T[n-\ dT(m) 

 ■2 dV 62 dv J 



6162 du dv; 



Y(m){n) = 0, \in 9^ m, 



= jwe, if n = m; 

 Y(m)[n] = 0, all m, n; 



dT[n] dT[m] _ dT[n] dT[m] 



du dv 



Z[m][n] = juHuv 11 



dv du 



= join, if n = m; 



(52) 

 du dv, \i n 9^ m, 



8 C. L. Hogan, "The Ferromagnetic Faraday Effect at Microwave Frequencies 

 and Its Applications — The Microwave Gyrator, Bell System Tech. J., Jan. 1952, 

 p. 9. 



