CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1247 



transducer about the guide axis to insure that the polarization which 

 enters the bend is TMi/. We shall design such a transducer using a di- 

 electric sector in a straight pipe. 



From Section 2.1, for complete power transfer from TEoi to TMn we 

 must have 



/3[oi] - ^(11) = 0; (111) 



the transfer then takes place in a distance 



I = 7r/2|ic(n)|. (112) 



The modified phase constants /3[oi] and /3(ii) for a single dielectric sector 

 of angle 6 are given by (103) of Section 2.3.2. Substituting these values 

 into (111), we find that the only condition under which it is satisfied is 



sin e ^ 0, 



(113) 



e = 180°. 



The transducer must therefore be a half cylinder. From (104) we have 



K'(u) = c?(ii) = -0.12066^5, (114) 



and so (112) gives for the length of the transducer, 



I = 2.072 \/b. (115) 



The TEoi - TMi/ transducer should be placed on either side of the 

 diametral plane of the straight guide which lies in the plane of the 

 bend. An exactly similar transducer on the other side of the bend will 

 reconvert TMi/ into TEoi • Since TEoi and TMu have the same velocity 

 in a straight guide, the transducer can be made of a number of sections 

 with arbitrary spacing and of total length /; but in practice one \\i\\ not 

 wish to have too long a run of TMn in the empty guide because of the 

 higher heat losses of this mode. 



A.2 The TEoi ± T:\rii" Normal Mode Solidi<m 



If we write the coupled wave ecjuations for TEoi and TMn" in a plain 

 bend in the form 



-r- + ihao + icui = 0, 

 dz 



da ^^^^^ 



icao -\- -r^ -{- ihai = 0, 

 dz 



