11G4 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



first order in Itr , 



1 + Pr = Trl + Tra 



1 _ ^Uo ( . (27) 



i- — Pr — —, ^Trl — Tr2) 



PrfJ'r+ 



where 



Mr+ = MO I 1 



1 rr \ 



Jr) 



Similarly at the output plane, z = d, the transmitted amplitude Xrs 

 satisfies 



TrZ = Trie + TrfC , 



/3^Mo , -,e ,-9x (28) 



Tr3 = —, (Trie — Tr2e ), 



where = ^r^ d. Comparison of equations (27) and (28) ^^dth the cor- 

 responding equations for the infinite slab shows that they have the 

 same form; however, equations (27) and (28) hold only for small mag- 

 netization. Hence only those results for the slab that relate to small 

 magnetization can be generalized to the present case. One such result is 

 that for the rotation gain. If we re-identify variables by 



2 ^r 2 /3, ' 



^0 — ^ Or = /3r d, 

 a^ ttr = ^rl^'r , 



we obtain, for the rotation gain, 



cosh ^r sec Or — — - — - sinh ^^ 



Qr = 



(29) 



1 + cosh2 rPr tan2 dr 

 where 



e*"- = ar = fir/^'r • 



The formulae and conditions for maximum and minimum rotation gain 

 derived in the previous section, apply here also, in terms of the re-defined 

 variables. 



The conversion of part of the incident mode to the .s^ mode in the 

 transmitted wave can be examined by multiplying the matching equa- 



