1250 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 M 



provided that* /3o = /3i in the straight guide (this may be arranged, for 

 example, by using a half-cyhnder of dielectric), and provided that the 

 length of the compensator and the coupling coefficient d are properly 

 chosen. The amplitudes of the two modes in the straight guide, in terms 

 of arbitrary initial values ao(0) and ai(0), are 



ao(z) = [ao(0) cos dz — mi(0) sin dz]e~' °% 



_.. (130) 



ai('2) = [ — laoiO) sin dz + ai(0) cos dz]e °\ 



Now suppose that the length of the bend is h and the length of the 

 compensator k , and take the origin of z at the input to the bend. A pure 

 TEfli input is represented by 



(131) 



(132) 

 (133) 



ao(0) = 1, 

 ai(0) = 0, 

 and so, applying (129) and (130) in succession, 



ao(/i) = cos ck e~" \ 



aiih) = —i sin di e~' \ 



aoih + k) = cos (ch + dk)e-'^'''+^'''\ 



aiik + ^2) = -i sin {ch + dl2)e-'^''''^^''''\ 



The condition that all the power be in TEoi at the output of the com- 

 pensator at every frequency is 



di + dh = 0, (134) i 



or 



dh = -0.18454 /3aZi/6, (135) 



on referring to equation (86) of Section 2.2 for the value of c. 



A convenient form of compensator would be a half cylinder of dielec- 

 tric whose diametral plane is perpendicular to the plane of the bend. 

 The coupling coefficient of the half cylinder is given by (104), and the ' 

 condition for complete compensation becomes 



k8 = 1.5295 ka/b. (136) 



The most easily adjustable parameter is probably the length ^2 of the com- 

 pensator. 



* The necessity for /3o = /3i is apparent if we consider that under certain condi- 

 tions the bend may put out a pure TMn" mode, and complete reconversion is pos- 

 sible only if the compensator has /3o = /3i . 



