CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1225 



coupling between TEoi and TMi,„ . Referring to (31), we have 



»-(lm)[01] = A[01](lm) 



= f ^(grad T(i„o) • (flux T[oi]) dS 



^ f'' r '\/2./i(x[oi]p)-^i(x(im)p) cos^y (45) 



^0 Jo Trahk(im)Jo{k[Ol])Jo(,k(im)) 



\a/\/2k[oi]h if m = 1, 



[O if 7W F^ 1. 



Hence the only transverse magnetic mode coupled to TEoi by the bend 

 is TMn , and from (41) the forward coupling coefficient is: 



Cai)[oii = cfoiKU) = l3a/\/2k[oi]b = 0.18454,Sa/6. (46) 



To obtain the coupling between TEoi and TEi;„ , we must evaluate two 

 integrals. From (28), the first is 



^[01] [Im] = ^[lOT][01] 



X[01]X[lm] / ^T[oi]T[im] dS 



^ Z"^" r \/2X[lm]pVo(x[01lp)'/l(x[]m]p) cos' (p , ^"1") 



Jo Jo Trab{k[un]^ — l)Uo(fc[oi])/i(/c[im]) 

 _ ' \/2a A;[im](/v[oi] + ^'[im]v 



and from (31), the second is 



A[01][]m] — A[lm][01] 



= f Kgrad Tio,]) • (grad l\un]) dS 



•Is 



'S 



^ ^ ^.T p \/2X[lm]P^Jl(xi0l]P)JliX[lm]p) cos' V? (-IS) 



Jo Jo ■7rab(k[lm]^ — l)'Joik[Ol])Jl{k[lm]) 



_ 2\/2a k[oi]k[im]' 



A short table of numerical values of the above two integrals follows: 



