CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 



1223 



In these definitions the plus signs are taken together, likewise the minus 

 signs. The factors of i are introduced in order that the k's may be real for 

 pi'opagating modes in a lossless guide. 



For the small-coupling case discussed at the end of the preceding sec- 

 tion, it is convenient to separate a typical coupling coefficient into two 

 parts; thus, 



K = c + d. (39) 



Here c is the coupling coefficient due to curvature and is zero unless the 

 angular indices of the two modes involved differ by unity. The coupling 

 coefficient d is due to the dielectric. All c^'s vanish if the dielectric is 

 homogeneous; otherwise particular symmeti'ies may cause certain classes 

 of d's to be zero. The c's and d's may be expressed in terms of integrals 

 written down in the preceding section if we substitute for the F's and 

 Z's in (38) their definitions according to (30). 



The k''"'s which have equal subscripts {n){n) or [n][n] may be regarded 

 as phase constants (of particular TM or TE modes) which have been 

 modified by the presence of the dielectric. For the modified phase 

 constants we introduce the symbols /3(„) and /?[„] ; thus, 



o _ + J, _i_ X(rt)'5(„)(„) -f /l(„)2A („)(„) 



2/i( 



n) 



^W 



K-[n\[n] — H[n] -p 



(40) 



2h[n 



[n] 



The general expressions for the coupling coefficients between any two 

 different modes are as follows: 



C{m)(n) 



d(m) (re) 



C(m)[7i] 

 d(m)[n] 

 C[m] (n) 



1 



2 



vh(_m)h(n) i^(m)(n) ± 



^ 'B(m)(n) — X(m)X{n)^(m)(n) 

 X(m)X(n)5{m)(n) 



1 /7 7 X(m)X(n)6(7re)(n) 



^5 V«'(m)'i(n) '^(m)(n) ± /r 7 j 



- L V ll(m)H(n) _ 



2 /3H(m)[n] [\/h(m)/h[n] ± \/ h[n\/h^,n)\ , 

 2 ^^{m)ln] V /i(m)//l[n]) 



I /^H[,„](«)[V/t(n)//l(m] ± V'h[,n]/h(n)]j 



d[m]U') — 



Clm][n] 



d[,n][n] 



2 /3A[„,](„) y/h(n)/h[m] , 



/3^S[OT][n] — X[m]X[n]^M[n] 



i[ 



'\/h[m]h[n] 



>^[m][n] 'Vh[,n]h[n] , 



0'^lm]ln] 

 2 '\/h[m]h[n] 



(41) 



