CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1233 



The coupling coefficient between the TEoi and TINrn modes in a plain 

 bend is given by equation (46) as 



Ciu) = 13a/ V2kioi]h = 0.18454 /3a/6 = 1.1595a/X6. (86) 



The (smallest) critical distance for maximum power transfer is given by 

 (67) ^^•ith /3o = /8i , namely 



_ T k[oi] h\ _ 1.3o47bX 



^"o ~ 2c(n) = 2V2a " a ' ^^^^ 



and the critical angle i?co is 



^co = -co/b = 1.3547 \/a radians = 77.62 X/a degrees. (88) 



This expression agrees, as it must, with that obtained by Jouguet and 

 Rice. (We write ■^cq for the uncompensated bend in order to reserve t?r 

 for a bend with dielectric loading.) 



It should be pointed out that c^i) is not necessarily the largest of the 

 coupling coefficients due to curvature. In a guide sufficiently far above 

 cutoff, it appears from (49) that C[n] is approximately equal to C(u) , 

 and C[i2] is one and two-thirds times as large as Cm) . If two transmission 

 lines are coupled over a distance z which is small compared to the dis- 

 tance required for maximum power transfer, then by (65) the relative 

 power transferred to line 1 is 



Pi{z) ^ Kz\ (89) 



which is proportional to the square of the coupling coefficient. It follows 

 that for a sufficiently small bending angle the largest amount of power 

 will go into the mode which has the largest coupling coefficient to TEoi 

 (in the above example, TE12). Each coupling coefficient, however, is 

 proportional to 1/6, and can be made as small as desired by increasing 

 the radius of curvature of the bend. Since the phase constants are unal- 

 tered to first approximation by the curvature, the maximum power trans- 

 ferred tends to zero with 1/6^ for every mode whose unperturbed phase 

 constant differs from A[oi] . The only mode with finite power transfer for 

 a finite bending angle with an arbitrarily large bending radius is TAIu , 

 since /?(n) = /i[0]] . For the present we shall assume a bending radius so 

 large that power transfer to modes other than TMn is negligible. Com- 

 plete power transfer from TEoi to TAIn will then take place in a plain 

 bend at odd multiples of the critical bending angle t^^o • 



We now consider a dielectric-loaded bend in which the permittivity e 

 is a function of the transverse coordinates (p, <p), but does not vary 



