1234 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



from one cross section to another. The permeability /x is assumed to be 

 constant. Thus we shall write, as in (25), 



M = Mo , 



(90) 

 € = eo[l + 8{p, <p)], 



and assume that 



Ifj^ldS « 1, (91) 



where S is the cross-sectional area of the guide. The inequality (91) im- 

 plies either that e does not vary much over the cross section, or that it 

 varies extensively over only a small part of the cross section. The first 

 alternative corresponds to a dielectric whose relative permittivity dif- 

 fers but little from unity, and is the most likely case in practice. If, on 

 the other hand, 5 is large in a small region (a thin sliver of high-permit- 

 tivity material), the dielectric coupling coefficients will not diminish 

 very rapidly with increasing mode number. The TEoi mode will be appre- 

 ciably coupled to a large number of modes, and it may not be safe to 

 assume that the total power converted into spurious modes is small just 

 because the conversion into any given mode is small. We shall not try 

 to decide here what maximum value of | 5 | is practicable. 



If the distribution of dielectric is symmetric with respect to the plane 

 of the bend, the dielectric couples the TEoi mode to a definite polarization 

 of each spurious mode, and in particular to the same polarization of the 

 TMii mode that is coupled by curvature alone.* The dielectric coupling 

 coefficient between TEoi and TMn is, from (50) and (51), 



dai) = /o ^7 r277 V / 8{p,<p)Ji\xm]p) ■ f/*5. (92) 

 V27ra/C[oi]Jo {tC[oi]) Js P 



It is obvious that the decoupling condition, namely: 



K(ii) = C(ii) -F d(n) = 0, (93) 



may be satisfied by an infinite number of different distributions of per- 

 mittivity. Ingenuity is required, however, to find a configuration which 

 is easy to fabricate and which does not couple the TEoi mode too strongly 

 to any other mode in the guide. The spurious mode problem is quite 

 serious when the diameter of the guide (in wavelengths) is so large that 



* A dielectric insert which is not symmetric with respect to the plane of the 

 bend will couple TEoi to the other polarization of TMn , with potentially^ complete 

 power transfer to this mode on account of the equality of phase velocities. A small 

 accidental lack of symmetry should not lead to serious mode conversion in a bend 

 of moderate angle. 



