CIRCULAR WAVEGUIDE WITH INHOMOGENEOUS DIELECTRIC 1239 



To get an idea of tolerable bending radii with a geometrical optics 

 compensator, we shall calculate the radius at which the maximum mode 

 conversion loss from TEoi into TE:2 (the worst spurious mode) is 0.1 

 db. Setting Pi = 0.02276, Ave find from (68) that the minimum bending 

 radius for a |-inch guide is 5.69 inches, and for a 2-inch guide, 12.95 

 inches, both at a wavelength of 5.4 mm. It is worth noting that if | dm] \ 

 is increased by 5 per cent of its theoretical value, the minimum bending 

 radius becomes 7.89 inches for the |-inch guide and 39.2 inches for the 

 2-inch guide. (This assumes that the TMn mode is still properly de- 

 coupled and that TE12 is still the worst spurious mode.) 



Dielectric losses are likely to be a serious problem in a geometrical 

 optics compensator, inasmuch as the whole volume of the bent guide has 

 to be filled with dielectric. Relatively large values of 8 are required to 

 negotiate bends as sharp as those just discussed. For example, if 6 = 13a, 

 in a practical case 8 might range from 0.058 at the inner surface of the 

 bend to 0.250 at the center of the guide to 0.442 at the outer surface 

 (referred to eo as the permittivity of free space). The loss tangent of 

 present-day dielectrics in this range is approximately 2 X 10~ . A large 

 (i.e., far above cutoff) waveguide filled with a dielectric of relative per- 

 mittivity 1.25 and loss tangent 2 X 10"* will show a dielectric loss of 

 about 1.13 db/meter or 0.34 db/ft at 5.4 mm. The dielectric loss in a 

 90° bend with a bending radius of 1 foot would be about 0.54 db, and for 

 other bends the loss would be directly proportional to the length of the 

 bend, and to the loss tangent if different from 2 X 10~^ It is true that 

 loss tangents as low as 5 X 10~^ may be obtained with lower values of 

 permittivity, say 5 = 0.033; but with such a small 8 the bend radius 

 must be proportionately larger, and the total dielectric losses in a bend of 

 given angle would be larger than with a higher permittivity material. 



We proceed now to demonstrate the assertion made earlier that a per- 

 fect bend compensator does not exist. More precisely, we shall show that 

 it is impossible to compensate the bend with an isotropic medium whose 

 permittivity and permeability are everywhere finite, but otherwise 

 arbitrary, in such a way that there is no conversion from TEoi to any 

 other mode at any point at any frequency. 



If there is no mode conversion at any point of the bend then the fields 

 at all points must be those of the TEoi mode, referred to the bent cylindri- 

 cal coordinate system described at the beginning of Section 1.1. In other 

 words, we have prescribed the electromagnetic field and are asking 

 whether it is possible to choose the permeability and permittivity so 

 that the given field will satisfy Maxwell's equations. Usually the 



