846 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



This information suggests that the growing-noise mechanism is really 

 a two-stage process: amplification of a broad band of microwave fre- 

 quencies, located far above the observation frequency, followed by a 

 transfer of noise energy to lower frequencies. 



(a) First stage 



At low magnetic fields, there are few ripples per unit length, but their 

 amplitude is usually large; whereas at moderate to large magnetic fields, 

 the ripple amplitude is small, but so is the ripple wavelength. In either 

 case, the bandwidth of RBA is large, usually many thousands of mega- 

 cycles. 



The increase of both bandwidth and gain per scallop with ripple 

 amplitude has been explained by Pierce by analogy between the gain 

 band of a rippled beam and the stop band of a transmission line filter: a 

 sharply varying periodic disturbance on the latter will reflect short as 

 well as long waves, whereas smoother perturbations will not reflect the 

 shorter waves to any extent. 



Another way to study the amplification bandwidth is to derive the 

 equations for RF current in a one-dimensional beam with sinusoidal varia- 

 tion of the reduced plasma wave number, j3q = p-^p, as Hefl'ner,'^ 

 Bloom,^^ and others^^' ^^ have done. This leads to a Mathieu equation, 

 whose solutions may be studied on the Mathieu stability plot (A, q): 



V^ + (A - 2? cos 2oc)I = 0. (6) 



Here / is the RF current, q a measure of the perturbation amplitude, 

 X = Tz/L, and A = (2L/Xg) , where L is the ripple wavelength and X, 

 the reduced plasma wavelength. Bloom has shown that, if n is the in- 

 tegral number of scallop wavelengths between initial and final planes, 

 the greater the product nq, the greater the total amplification or deam- 

 plification, and the less critical the phase relation between RF standing 

 wave and ripple for amplification. Ultimately, for very large nq, ampli- 

 fication will take place for all values of this phase angle. 



At higher magnetic fields, both the ripple amplitude and ripple wave- 

 length are decreased. This means that q is reduced, but /i increased over 

 any fixed span. This combination usually tends to increase the product 

 nq up to some fairly high field, after which it may decline. More impor- 

 tant, the reduction in ripple wavelength shifts the band of amplification 

 to higher frequencies, and greatly increases the frequency band. This 

 occurs because the "resonant" space-charge wavelength is shorter. 



