HELIX WAVEGUIDE 1379 



Referring to the values of aa listed in Table I, we see that the un- 

 Iwanted mode attenuations can be made to exceed the TEoi attenuation 

 [by factors of from several hundred to several hundred thousand in the 

 [large helix guide. The attenuation ratios are somewhat smaller in the 

 [smaller guide sizes. 



The attenuation versus conductivity plots of Fig. 3 show that for 

 I many of the modes there is a value of jacket conductivity, depending on 

 ■the mode, the value of ;Soa, and the jacket permittivity, which maximizes 

 the attenuation constant. Since one is accustomed to think of the at- 

 tenuation constant of a waveguide as an increasing function of frequency 

 for all sufficiently high frec^uencies (except for circular electric waves), 

 or as an increasing function of wall resistance, it is worth while to see 

 why one should really expect the attenuation constant to pass through 

 a maximum as the frequency is increased indefinitely in an ordinary 

 metallic guide, or as the wall resistance is increased at a fixed frequency. 

 The argument runs as follows: 



Guided waves inside a cylindrical pipe may be expressed as bundles of 

 plane waves repeatedly reflected from the cylindrical boundary." The 

 angle which the wave normals make with the guide axis decreases as the 

 frequency increases farther above cutoff; and the complementary angle, 

 which is the angle of incidence of the waves upon the boundary, ap- 

 proaches 90°. If the walls are imperfectly conducting, the guided wave is 

 attenuated because the reflection coefficient of the component waves at 

 the boundary is less than unity. The theory of reflection at an imper- 

 fectly conducting surface shows that the reflection coefficient of a plane 

 wave polarized with its electric vector in the plane of incidence first 

 decreases with increasing angle of incidence, then passes through a deep 

 minimum, and finally increases to unity at strictly grazing incidence. ^^ 

 For a metallic reflector, the angle of incidence corresponding to minimum 

 reflection is very near 90°. Inasmuch as all modes in circular guide except 

 for the circular electric family have a component of E in the plane of 

 incidence (the plane 6 = constant), one would expect the attenuation 

 constant of each mode to pass through a maximum at a sufficiently high 

 frequency. For example, the TMoi mode in a 2-inch copper guide should 

 have maximum attenuation at a free-space wavelength in the neighbor- 

 hood of 0.1 mm (100 microns), assuming the dc value for the conductivity 

 of copper. To find the actual maximum, of course, would require the 

 solution of a transcendental equation as in Section IV. 



The circular electric waves all have E normal to the plane of incidence. 



" Reference 9, pp. 411-412. 

 12 Reference 7, pp. 507-509. 



