1272 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



order for all other modes. Thus, the degeneracy of equal phase constants 

 of circular electric waves and TMim waves can be removed quite effec- 

 tively. The additional attenuation to TEoi wave as caused by the dielec- 

 tric loss in the coat and the increased wall current loss remains small as 

 long as the coat is thin. 



The dielectric-coated guide may be used for negotiating intentional 

 bends or for avoiding extreme straightness requirements on normally 

 straight sections. For intentional bends of as small a radius as possible, 

 an optimum thickness of the coat makes the mode-conversion losses to 

 TMii and TE12 modes equal and minimizes the total conversion loss. 

 For random deviations from straightness, an average radius of curva- 

 ture is defined. For this average radius an optimum thickness of the coat 

 minimizes the additional TEm attenuation as caused by curvature and 

 dielectric coat. In random deviations from straightness, propagation of 

 only one normal mode is maintained as long as only the rate of change 

 of curvature is small compared to the square of the difference in phase 

 constant between TEoi and any coupled mode. 



Serpentine bends, caused by equally spaced supports and the associ- 

 ated elastic deformation of the pipe, increase the TEoi attenuation sub- 

 stantially at certain critical frequencies, when the supporting distance 

 is a multiple of the beat wavelength. The lowest critical frequencies of 

 TEoi-TMii coupling corresponding to a beat wavelength, which is equal, 

 to the supporting distance or half of it, have to be avoided by choosing} 

 the proper coat thickness. 



At transitions from plain waveguide to dielectric-coated guide higher 

 circular electric waves are excited by the TEoi wave. However, the power 

 level of these spurious modes is low for a thin dielectric coat. ■ 



ACKNOWLEDGMENTS \ 



The dielectric-coated waveguide is the subject of two patents. Some 

 of its useful properties were brought to the writer's attention by a com- 

 munication between the Standard Telecommunication Laboratory, Ltd., 

 England, and S. E. Miller. For helpful discussions the writer is indebted j 

 to E. A. J. Marcatili, S. E. Miller, and D. H. Ring. 



APPENDIX I 



Approximate Solutions of the Characteristic Equation 

 In the following calculation we will use the definitions: 



R„{x, px) = Jn{x) N„-i{px) — J„_i(p.-c)A'„(a;), 



Sn{x, px) = Jn-l{x) Nnipx) - J u(pX^ A'„_i(.t), 



