42 



BELL SVSTIiM TECIIXICAL JOLRXAL 



amperes per square centimeter rather tlian in amperes per square meter, 

 while / is expressed in megacycles. 



Below a minimum value of \V, which will be called Il'.v/ , there is no gain. 

 W i, is a function of the velocity separation b and of the ratio of the beam 

 radius a to the beam wavelength, X, . A plot of (Il\w/7>)- as a function of 

 (a/Xs) is shown in Fig. 4. 



The variation of gain in the interval, 11 '^ < IT < x , is shown in Fig. 3 

 where "Decibels gain/ wavelength/unit b" is plotted as a function of 

 (W/Wm)'- This is the same curve which was derived in section 2. The 



0.2 



0.4 0.6 0.8 1.0 



^ Fig. 4 — As the ratio of beam radius a to wavelength in the beam, X, , is increased, the 

 critical value of 11', W \f , decreases and less current is needed in order to obtain gain. 

 Here (ll/w/^)', which is called H{a/X,), is plotted vs. (a/X,). 



ratio (U'/Wm)- is the same as the parameter {U/U mY used there, although 

 V and W are not the same. 



The curve in Fig. 3 is useful in that it reduces the interdependence of a 

 large number of parameters to a single curve. However, there are cases as, 

 for example, when one is computing the bandwidth of an amplilier, in which 

 it would be more convenient to have the curve in Fig. 3 broken up into a 

 family of curves. We can do this by the following means: 



We can write the gain in db/'wavelength in the form 



db/wavelength = bF{W/WM)- 



(29) 



