GUIDED-WAVE PROPAGATION THROUGH GYROMAGNETIC MEDIA 595 



(Ai - A.,)i2//, = 



( 1 + PePh — ) {i\.2VEPE — 0) — VeA2(pb + Ph) 



A VeVh/ 



(22b) 



VeM ( 1 + PePh — ) — (p£ + P/f)(AoI/£Pfc. — 13) V*>/'1 



\ VeVh/ J 



minus the same expression with suffixes 1 and 2 interchanged. 



Equations (22a) and (221)) may be written in a ^■ariet3' of equivalent 

 forms by making use of the relations V)etween Ai and Ao . The manipula- 

 tions which have been used in deriving (22a) and (22b) assume the use 

 of rectangular coordinates, but the results are \'alid in polar coordinates 



— , — - ) . That this is the case may 

 dr r dd 



be seen from the consideration that the rotation, —d, which carries the 



vector {Ex , Ey) into the vector {Er , Ee) also transforms ( — , — ) into 



\dx dy/ 



d 1 d 



dr r dd^ 



3.2 The characteristic equation 



The boundar}^ conditions of the problem are that E, = and Eg = 

 at r = ro , the radius of the guide. E^ is given by [see (18)]. 



(A.> - Ai)^., = [A2.4:J.(xir) - AiA2Jn(x2r)W"\ (23) 



and vanishes at r = ro if 



. /n(X2''o) . ./n(Xl/'()) 



Ai = ; A2 = . 



A2 Ai 



Hence the relations hold: 



'/'i,2 = -— /n(x2.iro)J'„(xi.2r)e^"^ 



A2,l 



From (22a) it follows that 

 Jnix2ro)e 



inO 



d2 



- S ( 1 4- PhPn — ) (/3A2 — PhVb) 



r \\ VeVh/ 



(Ai - \2)^Ee 



+ VnipH + Pe) \JniX\r) + U/3A2 — PhVh){pe + pa) 



-\- Vh\\ + PePh — — ) ( XvL'iXir) 

 \ VeVh ' J 



* In Ai)i)en(lix III the field components in polar coordinates are written out 

 fully for the ferrite and })la.sniu cases with some changes in notation which are 

 introduced in Sections 4.11 and 4.2. 



