1240 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



answer to this question will be "No." The Maxwell equations, 



V X E = —iwfiH, 



V X H = iooeE, 



(100) 



are equivalent to six scalar equations, and if the components of E and 

 H are prescribed, one cannot in general satisfy all these equations by 

 merely adjusting the two scalar functions n and e. 



It is particularly easy to see the difficulty for the TEoi mode in a 

 curved guide. Recall that the TEoi mode fields are independent of the 

 coordinate cp, and that the only non- vanishing field components are 

 E^{p, z), Hp(p, z), and H,{p, z). The fourth of equations (5) is: 



1 



A([i + (p/b) cos <p]H:) -~{pH,) 



p[l + (p/6) cos ip] \_dip dz 



(101) 



= icoeEo . 



p 



Since Ep = 0, the right side of the eciuation is zero for any finite value of 

 € at any finite frequency, and the whole equation reduces to 



H.sm^ =0, (102) 



6 + p cos (p 



which can be true for all values of p and (p only if the radius of curvature 

 of the bend is infinite. Hence a perfect compensator cannot be designed 

 with any value of e. j 



The practical importance of this result does not appear to be great, j 

 since theoretically the geometrical optics solution would provide an 

 extraordinarily good compensator. Until one has a dielectric whose 

 permittivity is continuously variable and precisely controllable, and 

 whose loss tangent is very low, even this solution is of only academic 

 interest. 



2.3.2 The Single- Sector Compensator 



In practice the simplest way to compensate a bend is to fill part of 

 the cross section of the guide with homogeneous dielectric material of 

 relative permittivity (1 + 8), and leave the remainder empty. In many 

 cases a suitable shape for the cross section of the dielectric is a sector ot 

 a circle, inserted on the side of the guide nearest the center of curvature 

 of the bend, and symmetrically placed with respect to the plane of the 

 bend. Such a sector, of total angle d, is shown in Fig. 2(b). We shall now 

 discuss the properties of a single-sector compensator. 



i 



