1294 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



the propagation in the normal mode taper. We will find that the taper 

 should have a certain minimum length to work properly. Usually it has 

 to be long compared to the beat wavelength between the TEoi normal 

 mode and any of the other normal modes into which power may be con- 

 verted. 



In the plain waveguide the degeneracy between TEoi and TMn causes 

 an infinitely long beat wavelength. Hence, the normal mode taper would 

 not work there. A nondegenerate waveguide is an essential condition 

 for the normal mode bend. 



We shall confine our attention to the linear taper. This is not the 

 optimum taper form, but it is most easily built. 



The residual mode conversion in the bend is to be accounted for as 

 bend loss. This loss and the loss caused by the normal mode attenuation 

 in the bend add up to a total bend loss. We shall evaluate the total bend 

 loss for bend configurations which might be useful in circular electric 

 wave transmission. For specified waveguide dimensions the total bend 

 loss can be minimized by choosing the proper bend geometry. 



The normal mode bend is an hiherently broadband device. The total 

 bend loss shows the same order of frequency dependence as the loss in 

 the straight waveguide. 



II. ANALYSIS OF THE NORMAL MODE TAPER 



In the curved waveguide, wave propagation can be described in terms 

 of the normal modes of the straight w^aveguide. The relation between 

 these modes is then given by an infinite system of simultaneous first 

 order linear differential equations. It represents the mutual coupling of 

 the straight guide modes in the curved waveguide. We are interested 

 mainly in TEoi propagation and shall restrict ourselves to a low 

 power level in all other modes. Consequently, an adequate procedure is 

 to consider only coupling between TEoi and one of the coupled modes at 

 a time. Thus, the infinite system of equations reduces to the well known 

 coupled line equations: 



dz 



dEj 

 dz 



(1) 



= jcEi — 72£'2, 



in which 



Ei,2{z) = wave amplitudes in mode 1 (here always TEoi) and 

 mode 2 (TMn or one of the TEim), respectively; 



