CIRCULu\R WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1213 



The appendix contains brief descriptions of three dielectric compen- 

 sators which can be inserted in a straight section of guide adjacent to a 

 bend. The first two are transducers which convert TEox to a normal mode 

 of the curved guide; they are subject to bandwidth limitations as men- 

 tioned by Miller.^ The third type merely takes the output mixture of 

 TEoi and TMn from a plain bend with a pure TEoi input, and reconverts 

 it all to TEoi ; it is essentially a broadband device. The spurious modes 

 generated by a bend plus compensator have not been calculated, but it 

 is very unlikely that a smaller bending radius will be permitted when the 

 compensator is outside the bend than when it is inside. 



I. THEORY 



1 .1 Generalized Telegraphist's Equations 



To describe electromagnetic fields in a curved circular waveguide one 

 is naturally led to use "bent cylindrical coordinates" (p, <p, z), ' ' in 

 which the longitudinal coordinate z is distance measured along the 

 curved axis of the guide, w^hile p and <p are polar coordinates in a plane 

 normal to the axis of the guide, with origin at the guide axis. The fines 

 ^ = and ^ = TT lie in the plane of the bend. The radius of the guide 

 is denoted by a and the radius of the bend (i.e., the radius of curvature 

 of the guide axis) is denoted by b. The coordinate system is showii in 

 Fig. 1. 



For the moment regarding (p, (p, z) as general orthogonal curvilinear 

 coordinates {u, v, w), we let 



u = p, V = <p, w = z. (1) 



The element of length in this system is 



ds^ = exdxi + e-idv + e{dw^, (2) 



where 



ei = 1, ^2 = P, 63 = 1 + $, (3) 



and 



^ = (p/6) cos if. (4) 



IVIaxwell's equations for a field with time dependence e'"' may be 



