1232 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



ponents are developed in powers of the small parameter a/6, but the 

 field perturbations are not expressed in terms of the modes of the straight 

 waveguide. We shall now consider propagation in plain and compen- 

 sated bends from the coupled-mode viewpoint. 



Denote the coefficient of coupling between the TEoi mode and the 

 TM„,„ mode or the TE„« mode by 



(81) 



respectively, where as usual we indicate TM modes by enclosing the 

 subscripts in parentheses and TE modes by enclosing the subscripts in 

 brackets. In Part I the coupling coefficients are written with two pairs of 

 subscripts, since in general they may refer to any two modes, but here 

 one pair of subscripts would always be [01] and will be omitted. The co- 

 efficient c represents that part of the coupling (if any) which is due to the 

 curvature of the guide, and d represents the coupling (if any) which is 

 due to the inhomogeneity of the dielectric. We assume that the dielec- 

 tric loading, if present, is symmetric with respect to the plane of the 

 bend, so that coupling to only one polarization of each mode need be 

 considered. 



The phase constants of the TM„,„ and TE,j,„ modes in a straight, empty 

 guide are, respectively, 



/l(nm) = -S/^- — Xinm)"^, h[nm] = a/jS^ — X[nm]^ (§2) 



where 



and 



Also 



X(nOT) = k(nm)/(l, Xl'im] — k[mn]/a, (83) 



/„ (A-(„„o) = 0, J/(A-[,„„]) = 0. (84) 



= 2t/\, (85) 



where X is the free-space wavelength. 



As noted in the preceding section, the presence of coupling may 

 cause the modified phase constants |8(„„o and lS[„m] of the coupled modes 

 to differ slightly from the unperturbed phase constants /i(„m) and h[nm] • 

 In a plain bend (curvature coupling only), the /3's are equal to the h's, 

 and in most cases the effect of a small amount of dielectric coupling on 

 the phase constants may be neglected. Exact values of I3[nm] and I3{nm) 

 may be obtained if necessary from (40). 



