1210 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



In the first type the TEoi mode, which is not itself a normal mode of 

 the curved guide, is deliberately converted to a combination of TEoi 

 and TMii which is a normal mode, or to a particular polarization of TMu 

 which is another normal mode of the curved guide. After traversing the 

 bend the energy is reconverted to TEoi • A disadvantage of the normal- 

 mode approach is that the mode conversions necessary at the ends of 

 the bend are frequency sensitive, so that bandwidths appear to be 

 limited to the order of 10 per cent. 



A second approach to the bend problem is to break up, by some modi- 

 fication of the guide, the degeneracy which exists between the propaga- 

 tion constants of the TEoi and TMn modes in a perfectly conducting 

 straight pipe. The two modes are still coupled by the curvature of the 

 guide, but as Miller has shown, the maximum power transfer will be small 

 if there is sufficient difference between the phase constants or between the 

 attenuation constants of the coupled modes. A difference in phase con- 

 stants may be provided, for example, by circular corrugations in the 

 waveguide wall, or perhaps most easily by applying a thin layer of dielec- 

 tric to the inner surface of the guide. Differential attenuation may be 

 introduced into the TMu mode by a number of methods, in particular 

 by making the guide out of spaced copper rings or a closely-wound wire 

 helix surrounded by a lossy sheath.^ Unfortunately, the larger the guide 

 diameter in wavelengths the more difficult it is to get the separation of 

 propagation constants necessary to negotiate a bend of given radius 

 satisfactorily. 



Still another solution of the bend problem is to decouple the TEoi 

 and TMii modes in a curved guide by partially filling the cross section 

 of the curved guide with dielectric material. The dielectric must be 

 arranged to produce coupling between the TEoi and TMu modes which 

 is equal and opposite to the coupling produced by the curvature of the 

 guide. This condition may be satisfied in a great variety of ways; but it 

 is not the only requirement for a good bend compensator. Practical re- 

 strictions are that the power levels of all other modes which are coupled 

 to TEoi by the dielectric-compensated bend must be kept low, and of 

 course dielectric losses in the compensator must not be excessive. 



Part I of this paper treats the general problem of a curved circular 

 waveguide containing an inhomogeneous dielectric. A convenient formu- 

 lation of the problem is provided by S. A. Schelkunoff's generalized tele- 

 graphist's equations for Avaveguides.^ The field at any cross section of 

 the dielectric-compensated curved circular guide is represented as a 

 superposition of the fields of the normal modes of an air-filled straight 

 circular guide. A current amplitude and a voltage amplitude are asso- 



