CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1229 



mode. The complex constants 70 and 71 may be regarded as (modified) 

 propagation constants; note that because of the coupling they are not 

 necessarily equal to the propagation constants of the uncoupled modes. 

 The coupling coefficient is denoted by k; it will be real if the coupling 

 mechanism is lossless, but is not required to be so in the general mathe- 

 matical solution. 



We are interested in the case in which line contains unit power at 

 2 = 0, and line 1 contains no power. Subject to the initial conditions 



ao(0) = 1, a:(0) = 0, 



(59) 



mj 2 



(60) 



mi = M-(7o + 7i) + r], 

 W2 = |[-(7o + 7i) - r]. 



(61) 



(62) 



For the case of two propagating modes without loss, k is real and Ave 

 may write 



so that 



7o = «/3o , 7i = i^i , 



r = iVi^o - i3i)2 + 4k^. 



(63) 



(64) 



A straightforward calculation now gives the power in each line at any 

 point : 





1 



4k^ 



(/So - ^1)^ + 4k 

 4k' 



-smhA((3o - 13^ + 4/c'JX 



(65) 



shr mo - iSiY + 4Kr'z 



{(3o - /3i)2 + 4«2 

 Hence the maximum power transferred from line to line 1 is 



4k' [2k/(/?o - ^1)]' 



V-'^l/max 



(^0 - l3iY + 4k-2 1 + [2k/ {^0- /3i)P' 



(66) 



