GinOED WAVE PROPAGATION TIinOT'GII GYROM AGXETlf MEDIA. Ill 1137 



hereafter that Em and Hm pertain to positive j3„ ^'alues. For a given 



iinpei-turl)e(l mode it follows that -^, reverses sign when the direction 



of propagation reverses. Substituting these series for Kt and //, in ecjua- 

 tions (3), one finds 



and 



E 



Z 



<ib„ 



Iz 



+ j^nttn 



+ jlSnhn 



jw Ml 





(4a) 



(4b) 



The orthogonality relation between functions of different 7i has been 

 given by Adler." It is 



f Etn*-ffi,ndS = 0, 



where n 9^ m ; the tilde denotes the complex conjugate and the integra- 

 tion goes over the cross section of the guide. We consider the fields to be 

 un-normalized and write 



Clearly 





Htn dS = Are , 



■EtndS = -Are. 



(5) 



^Multiplying equation (4b) by Hm- and equation (4a) by Em- and inte- 

 grating each expression over the cross section we have 



An 

 An 



'db„ 



+ j^nttn 



dan I -^ , 

 dz 



Ml 



* — dS. 



= - f ArEtndS-^ J^ f Etn-V*~dS, 



J JCO J it, 



= -[BrfftndS - 1 f Htn-V 

 Now we use the identity 



|GrV*Fd^= -f F{Gfds)+ j FV-G*dS 



surface 

 This yields 



boundary surface 



