(66) 



800 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 



Eliminating Fdo] and V'[oi] , we have 



(/3^ — /3Io)2(io] = —jJ0loim] , 



(/3^ - ^oi)/[oi] = jk^lJ[io] . 



Multiplying term by term, we obtain the characteristic equation 



^' - {0lo + ^l)^' + (1 - k')^lo^l = 0. (67) 



Solving, we have 



/3' = K^io + /3oi) ± mlo -/3oi)' + 4:k'l3lo^lf". (68) 



The effect of coupling is to increase the larger phase constant and de- 

 crease the smaller one ; that is, to make the slower wave slower, and the 

 faster wave faster. 



Let us assume a > b; then /3io > /3oi . Taking the upper sign in (68) 

 and substituting in the second equation of the set (66), we have 



(69) 



(70) 



Hence, the ratio of the power carried in the TE[oi] mode to that in the 

 TE[io] mode is 



(71) 



Po\ t^[01]/(011 ^ 



This ratio increases as k increases and p decreases. 



If the phase constants of the vmcoupled modes are equal, then p = 

 and Poi = ^10 for all values of the coupling coefficient. In this case (68) 

 becomes 



iS' = ^Io(l ±k) or /3 = Ao(l ± kf'\ (72) 



In terms of the original constants. 



/5 = 



_(m ± ^ jVx.) (co^e - -^)_ 



ni/2 



(73) 



The cutoff frequencies of both normal modes are seen to be independent 

 of either the transverse permeability or the mutual permeability. Since 



