1226 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Putting these values into the expression (41) for the forward coupling 

 coefficient, we obtain: 



"0.09319(^a)2 - 0.84204 



C[01][ll] 



C[01][12] 



C[01][13] 



y/h[oi]ah[n]a 

 '0.15575(^a)" - 3.35688 



V/i;oi]a/i[i2]a 

 "0.01376(/3a)2 - 0.60216 



+ 0.09319 VA[oi]a/i[n,a 



+ 0.15575V^[oi]tt/i[i2]a 

 + 0.01376\//i[oi]aA[i3]a 



1 



b' 



1 

 b' 



(49) 



■\/hioi]ah[i3]a 



1.3.2 Coupling Coefficients due to Dielectric 



Depending upon the distribution of the dielectric material over the 

 cross section of the guide, the TEoi mode may be coupled to any mode 

 except those of the TMom family. The dielectric coupling coefficients, as 

 given by equations (41), are 



d[0\](nm) = «(nm) [01] = 2/5^[01] (»'») V ^(nOT)/"[Ol] , 

 ,+ ^"A [01] [nm] 



7+ 



(50) 



dp d(p, 



d[01][nm] — d[„m][01] '— r,^/T T • 



■^ V "[01]«'[nm] 



The A's are obtained from equations (32) ; thus, 



A[oi](nm) = / 5(grad T(„„o)-(flux T[cii]) f/*S 

 •Is 



^ r'" r 8{p,ip)n\/7„JniX(nm)p)Jl{Xm]p) COS thp 

 Jo Jo '7rak^„m)Jn— l{K{nm)) J 0\k [01]) 



A[oi][n,«] = / 6 (grad T [„,„])• (grad T [01]) d*S 

 Js 



^ ^ ^2t -.a g(p^ <p)\/ZX[nm]Jn'(Xlnm]p)Jl(Xm]p) COS mp 

 Jo Jo Tra{k[nrn]- " W") V„(/v[„„j] ) Jo('^[01] ) 



1.4 Dielectric Losses 



To take account of dielectric dissipation we may introduce a complex 

 permittivity, 



(51) 



€ = e' - ie" = e' (1 - i tan if), 



(52) 



where ^ is the loss angle of the dielectric and is not, of course, to be con- 

 fused with the coordinate (p. If we let 



e' = 60 (1 + 5'), 



(53), 



