CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1211 



ciated with each normal mode, and the currents and voltages satisfy an 

 infinite set of generalized telegraphist's equations. The coupling terms 

 in these equations depend upon the curvature of the guide axis and upon 

 the variation of dielectric permittivity over the cross section. In the 

 present application, the distribution of dielectric is taken to be independ- 

 ent of distance along the bend.* 



The generahzed telegraphist's equations for all modes in a curved 

 circular waveguide containing an inhomogeneous dielectric are set up in 

 Section 1.1. As a special case one has the equations for an air-filled 

 curved guide, or for a straight guide with an inhomogeneous dielectric. 

 In Section 1.2 we transform from current and voltage amplitudes to the 

 amplitudes of forward and backward traveling waves on a system of 

 coupled transmission lines and in Section 1.3 we work out in some detail 

 the coupling coefficients which involve the TEoi mode. An approximate 

 formula for dielectric loss in a compensated bend, which is valid at least 

 in the important practical case when the relative permittivity of the 

 dielectric differs but little from unity, is given in Section 1.4. 



Part II applies the foregoing theory to the design of bend compen- 

 sators for the TEoi mode. In a well-designed compensator the ampli- 

 tudes of the backward (reflected) waves are very low, so we shall neglect 

 reflections. The amplitudes of the spurious forward waves should also be 

 low compared to TEoi , so that we may consider them one at a time. We 

 assume that the TEoi mode crosstalks independently into each spurious 

 mode, and represent the interaction between modes by a pair of linear, 

 first-order, differential equations in the wave amplitudes. Miller's treat- 

 ment* of these equations is reviewed in Section 2.1, and applied in Sec- 

 tion 2.2 to TEoi-TMii coupling in plain and compensated bends. Some 

 results of the Jouguet-Rice theory' ' ^ for plain bends are confirmed by 

 coupled-line theory. The condition for decoupling TEoi and TMn in a 

 compensated bend is written down, and the consequences of imperfect 

 decoupling are discussed. 



Three different compensator designs are described in Section 2.3, and 

 evaluated with regard to mode conversions and approximate dielectric 

 losses. In the first case, which may be called the "geometrical optics" 

 solution, the permittivity is supposed to vary continuously in such a way 

 that a bundle of parallel rays entering the bend would be bent into co- 

 axial circular arcs all of the same optical length. This is not a perfect 

 solution of the problem if the wavelength is finite, but it is of some 



* We shall not consider the effects of random inhomogeneities, such as bub- 

 bles in polystyrene foam, although these might conceivably add to the mode 

 conversion if their dimensions were comparable to the operating wavelength. 



