22 BELL SYSTEM TECHNICAL JOURNAL 



since the inside diameter of the heUx is less than a by the radius of the wire. 

 Further, focusing difficulties preclude attaining a beam radius equal even to 

 the inside radius of the helix. 



Experience indicates that at very short wavelengths (around 6 milli- 

 meters, say) it is extremely important to have a well-focused electron beam 

 with as large a value of r/a as is attainable. 



A characteristic impedance Kt may be defined in terms of a "transverse" 

 voltage Vt, obtained by integrating the peak radial field from a to oo , and 

 from the power flow. In Fig. 3.7, (v/c) Kt is plotted vs. ja. A "longitudinal" 

 characteristic impedance Kf is related to Kt (3.13). For slow waves Kf 

 is nearly equal to Ki. The impedance parameter E~/^~P evaluated at the 

 surface of the cylinder is twice Kf. We see that Ke falls with increasing 

 frequency. 



A simplified approach in analysis of the helically conducting sheet is that 

 of "developing" the sheet; that is, slitting it normal to the direction of con- 

 duction and flattening it out as in Fig. 3.8. The field equations for such a 

 flattened sheet are then solved. For large values of ya the field is concentrated 

 near the helically conducting sheet, and the fields near the developed sheet 

 are similar to the fields near the cylindrical sheet. Thus the dashed line 

 in Fig. 3.7 is for the developed sheet and the solid Hue is for a cylindrical 

 sheet. 



For the developed sheet, the wave always propagates with the speed of 

 light in the direction of conduction. In a plane normal to the direction of 

 conduction, the field may be specified by a potential satisfying Laplace's 

 equation, as in the case, for instance, of a two-wire or coaxial line. Thus, 

 the fields can be obtained by the solution of an electrostatic problem. 



One can develop not only a helically conducting sheet, but an actual 

 helix, giving a series of straight wires, shown in cross-section in Fig. 3.9. 

 In Case I, corresponding to approximately two turns per wavelength, suc- 

 cessive wires are — , +, — , + etc.; in case II, corresponding to approxi- 

 mately four turns per wavelength, successive wires are +,0, — , 0, -f , etc. 



Figures 3.10 and 3.11 illustrate voltages along a developed sheet and a 

 developed helix. 



Figure 3.13 shows the ratio, R^''\ of {E-f^-Py^ on the axis to that for a 

 developed helically conducting sheet, plotted vs. d/p. We see that, for a 

 large wire diameter d, {E?/0^Py'^ may be larger on the axis than for a heli- 

 cally conducting sheet with the same mean radius and hence the same pitch 

 angle and phase velocity. This is merely because the thick wires extend nearer 

 to the axis than does tlie sheet. The actual helix is really inferior to the 

 sheet. 



We see this by noting that the highest value of {E'/ff'Py^ for a helically 

 conducting sheet is that at the sheet {r = a). With a finite wire size, the 



