NORMAL MODE BENDS FOR CIRCULAR ELECTRIC WAVES 1293 



coated waveguide has shown/ this dielectric layer changes the phase 

 constant of the TMn wave without appreciably affecting the propaga- 

 tion characteristics of the TEoi mode. 



With removal of the TEoi-TMn degeneracy the mode conversion 

 which occurs when a TEoi wave passes through a bend of constant curva- 

 ture may be considerably reduced, but it will not be completely elimi- 

 nated. To design a bend with still lower mode conversion losses, we con- 

 sider the effect of tapering the curvature along the guide. 



Guided wave propagation is most easily explained in terms of normal 

 modes. Normal modes are solutions of the wave equation in a particular 

 waveguide structure, and represent waves propagating without loss of 

 poAver except for dissipation. In the straight waveguide the normal mode, 

 in which we are mainly interested here is the TEoi mode. The normal 

 modes of the curved section are not as simple as the straight guide modes, 

 but they can be expressed as the sum of the normal modes in the straight 

 waveguide. Here the mode of our main concern is the one which, when 

 represented as a sum of straight guide modes, has most of the power in 

 the TEoi part of the sum; in other words, is most similar to the TEoi mode 

 of the straight waveguide. 



At a transition from a straight waveguide to a curved waATguide the 

 normal (TEoi) mode of the straight waveguide will certainly excite this 

 normal mode in the curved waveguide but it will also excite a series of 

 other normal modes. All these modes propagate in the curved section, 

 and at the other transition to the straight guide excite not only the TEoi 

 mode but a series of other normal modes of the straight waveguide. All 

 the power in the other normal modes represents mode conversion loss 

 of the bend. 



A transition which transforms the normal (TEoi) mode of the straight 

 waveguide into only one of the normal modes of the curved guide and 

 vice versa will avoid all mode conversion losses. vSuch a transition can 

 be realized approximately by tapering the curvature. Beginning with 

 zero curvature at the straight guide end of the taper, the curvature is 

 increased gradually along the taper to the finite value of the bend. The 

 normal (TEoi) mode mcident from the straight guide will be gradually 

 transformed into the particular normal mode of the bent guide which is 

 most similar in field configuration to the circular electric wave. At each 

 point along the taper there is essentially only one local normal mode cor- 

 responding in its configuration to the value of curvature at that point. 



This taper, unless it is infinitel}^ long, is however only an approxima- 

 tion of the ideal normal mode transition. There will still be other modes 

 excited with a very low power level. In the next section we shall analyze 



