CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1235 



many modes have phase velocities close to TEoi • In the next section we 

 shall compute the coupling to various modes for a number of cases. 



Mention may be made here of the effect of imperfect decoupling. If 

 we could satisfy the decoupling condition (93) exactly, then to first 

 order there would be no transfer of power from TEoi to TMn in a bend 

 of any angle. In practice we cannot expect to satisfy (93) exactly, on 

 account of uncertainties in the permittivity and dimensions of the 

 compensator. If the coupling coefficients are constant along the bend, the 

 effect of reducing K{u) by making d^u) nearly equal and opposite to C(n) 

 is to increase the distance required for maximum power transfer to 

 TMn to take place. In the simple case where /3(u) = i3[oi] , as, for example, 

 when the compensator is made of a bent half-cylinder of dielectric, the 

 critical angle for a fixed bend radius is inversely proportional to k^d • 

 The critical angle i^c for an imperfectly compensated bend, in terms of 

 the critical angle &co for an uncompensated bend, is 



C(ll) 



^c = ^i^ ^.0 . (94) 



C(ii) -r «(ii) 



If (i(u) can be made equal and opposite to C(ii) within 1 per cent, say, 

 then ??c = 100 dc^ . The power transferred in a bend of angle ■& is simply 

 proportional to sin (??/t?c) . 



2,3 Various Compensator Designs 



From now on we shall consider a compensated bend in which the 

 TEoi and TMn modes are completely decoupled, and we shall investigate 

 the coupling between TEoi and spurious modes. (By "spurious mode" 

 we mean any mode of the straight round guide except TEoi or TMn •) 

 We shall assume that the power in all spurious modes is everywhere 

 low compared to the power level of TEoi . To first order, therefore, we 

 may compute the power abstracted from TEoi by mode conversion by 

 assuming that the TEoi mode crosstalks into each spurious mode inde- 

 pendently. In practice the crosstalk is greatest into modes whose phase 

 velocities are nearest to the phase velocity of TEoi ; and it seems more 

 than adequate, at least for foam dielectric compensators, to consider 

 about a dozen modes. 



From (68), the maximum relative power (assumed small) which cross- 

 talks from TEoi into a given spurious mode is 



(Pl)max ^ [2k/(^0 - /3l)]', (95) 



where k is the total coupling coefficient. For all spurious modes we 

 assume that the difference in phase constants is not much changed by 



