CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1243 



three-sector compensator which will not couple either TE21 or TEm to 



TEoi . 



A three-sector compensator is shown schematically in Fig. 2(c). The 

 angle of the center sector is called di and the angle of each outer sector 

 62 . Each outer sector makes an angle \p, measured between center planes, 

 with the center sector. 



The condition for decoupling TEoi from TMn with the three-sector 

 compensator is 



^ 1.5295 a , . 



(2 cos V' sin 62/2 + sin di/2) h' ^ '^ 



If 8 and a/h are given, two additional conditions may be imposed upon 

 61 , 62 , and \}/. The conditions are taken to be: 



2 cos 2\l/ sin 62 + sin di = 0, 



(108) 

 2 cos 3i/' sm 3^2 + sin 30i/2 = 0. 



If equations (108) are satisfied, the compensator does not couple TEoi 

 to any modes of the TE2m , TEs^ , TM2m , or TM^m families. 



To design a three-sector compensator with given values of a/b and 

 8, one can solve (107) and (108) numerically for di , 62 , and xp. However, 

 if &i is small (^20°, for example), a simpler design procedure may be 

 used. The following equations are approximately true: 



126.8a , 

 di = — 5g— clegees, 



.. = 0.618<,.. »»«' 



rP = 72°. 



It should perhaps be pointed out that the precision of di , 62 , and xp 

 individually is not critical, since there is no necessity for the coupling 

 to TE21 and TEai to be exactly zero, so long as it is reasonably small. 

 One should, however, strive to make the TEoi-TMn coupling as nearly 

 zero as possible, and it is therefore important to satisfy (107) with the 

 greatest possible precision. 



Expressions for the dielectric coupling coefficients due to the three- 

 sector compensator may be obtained from those for the single-sector 

 compensator if we merely replace sin nd/2, wherever it occurs, by 

 sin ndi/2 + 2 cos nxp sin n02/2, where w is the angular mode index, as 

 usual. If ^1 , 02 , and \l/ satisfy (108), then the coupling to TEo,,, , TEsm , 

 TM2,,, , and TMsm is zero, and the mode having the largest value of 

 2(c -|- d)/{ho — hi) is TE12 . As noted earlier, since the fields of TMu 



