10. S.V. Averin and V. A. Popov, Radiotekh. Elektron., 2._22, No. 5, 1057 (1977). 11. V.D. Kaplun, A. D. Plotkin, and V. V. Salamatin, All'Union Scientific Session ...
2. 3. 4.
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
W . V . McLevige, T. Itoh, and R. Mittra, IEEE T r a n s . , MTT-23, 788 (1975). R . M . Knox and P. P. Toulis, P r o e . Symp. Submill. Waves, XXV, April (1970). V . F . Vzyatyshev, B. I. Byabov, and Yu. I. Orekhov, All-Union S y m p o s i u m on Devices, Techniques, and P r o p a g a t i o n of M i l l i m e t e r and S u b m i l l i m e t e r Waves in the A t m o s p h e r e [in Russian], Moscow (1976), p. 111. E.A.J. Marcatili, Bell. Syst. Tech. J., 4__88, 2071 (1969). V . F . Vzyatyshev, G. D. l~ozhkov, and A. N. M e r k u r ' e v , Zarub. Radio6lektron., No. 12, 60 (1970). V . I . Anikin, Zarub. ILudiQ61el~ron., No. 7, 111 (1971). H. Kogelnik, I E E E T r a n s . , MTT-23, 2 (1975). S . P . Sehlesinger and D. D. King, IEEE T r a n s . , M T T - 6 ~ 270 (1958). S . V . Averin and V. A. Popov, Radiotekh. Elektron., 2._22, No. 5, 1057 (1977). V . D . Kaplun, A. D. Plotkin, and V. V. Salamatin, All'Union Scientific Session of the Popov Radio Society [in Russian], Moscow (1974), p. 75. I . E . Goell, R. D. Standley, and T. L i , E l e c t r o n i c s , 2_.0.0, 60 (1970). G. Gloge, Appl. Opt., 1_~1, 2506 (1972). D . B . Keck and A. R. Tynes, Appl. Opt., 1._1.1, 1502 (1972). S . E . Miller, E. A. J. Marcatili, and T. Li, P r o c . IEEE, 61, 1703 (1973).
THEORY
OF
TRANSIENTS
BACKWARD-WAVE
IN RELATIVISTIC
TUBES*
N. S. G i n z b u r g , S. P . a n d T . N. F e d o s e e v a
Kuznetsov,
UDC 621.385.633
The duration of the linear stage of the t r a n s i e n t in r e l a t i v i s t i c BWT is e s t i m a t e d under the a s sumption that the signal a r i s e s f r o m s p e c t r a l components of the radiation of the e l e c t r o n b e a m f r o n t and fluctuations in c h a r g e density (shot effect) in the s y n c h r o n i s m band. It is shown that in the nonlinear stage when the e l e c t r o n c u r r e n t slightly e x c e e d s the s t a r t i n g value, the signal amplitude attains a m a x i m u m and begins to execute attenuating oscillations, this leading to e s t a b l i s h m e n t of a s t e a d y - s t a t e condition. When the c u r r e n t m a r k e d l y exceeds the s t a r t i n g c u r r e n t (by a f a c t o r of 3.25 f o r a BWT with a low efficiency), a m o r e c o m p l e x s e l f - o s c i l l a t o r y mode is established in which the output signal amplitude is modulated at a period c o r r e s p o n d i n g to the t i m e f o r the e l e c t r o n s and wave to t r a V e r s e (with group velocity) the tube length. This modulation, which is initially sinusoidal, c o m e s to r e s e m b l e a sequence of s h a r p peaks as the c u r r e n t i n c r e a s e s , then c e a s e s to be periodic. Since the a p p e a r a n c e of h i g h - c u r r e n t e l e c t r o n a c c e l e r a t o r s [1], t h e r e has been a g r e a t deal of attention given to the c r e a t i o n of BWT with intense r e l a t i v i s t i c e l e c t r o n b e a m s [2, 3]. Since t h e duration of the pulses produced by such a c c e l e r a t o r s is s m a l l (~ 10-100 nsec) and cofnparable to the duration of the oscillation settling p r o c e s s (see Sec. 4), the use of a nonstationary nonlinear theory to evaluate the principal g e n e r a t o r p a r a m e t e r s is of fundamental i m p o r t a n c e . In p a r t i c u l a r , we should note that the duration of the t r a n s i e n t is m a r k e d l y a f fected by the level of the initial "push" leading to the a p p e a r a n c e of s e l f - o s c i l l a t i o n . Evidently, in u l t r a r e l a tivistic BWT this "push" is provided by the radiation of the steep edge of the e l e c t r o n b e a m . F u r t h e r m o r e , analysis of such m a n i f e s t l y nonstatJonary modes of o r d i n a r y weakly r e l a t i v i s t i c BWT as pulse generation and amplification, s u p e r r e g e n e r a t i v e amplification, and so forth, is of some p r a c t i c a l s i g nificance. Finally, we should note that BWT a r e e x a m p l e s of distributed s y s t e m s in which it is t h e o r e t i c a l l y possible to have highly distinctive m o d e s of s e l f - o s c i l l a t i o n together with the trivial s i n g l e - f r e q u e n c y kind (see Sec. 5). The investigation of such modes is of i n t e r e s t both f r o m the standpoint of p o s s i b l e p r a c t i c a l a p p l i c a tions and f o r the theory of distributed s e l f - o s C i l l a t o r y s y s t e m s . *We will c o n s i d e r only O-type BWT in this paper. S c i e n t i f i c - R e s e a r c h Institute of Radiophysics. T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 21, No. 7, pp. 1037-1052, July, 1978. Original a r t i c l e submitted April 14, 1977.
728
0033-8443/78/2107-0728507.50
9
Plenum Publishing C o r p o r a t i o n
w
Statement
of the
Problem;
Basic
Initial
Equations
The BWT model u n d e r investigation (Fig. 1) is a s e g m e n t of periodic waveguide of length L, m a t c h e d a t the ends, which is p e n e t r a t e d by an e l e c t r o n b e a m that has an input v e l o c i t y v 0. The b e a m is d i r e c t e d by a s t r o n g longitudinal m a g n e t i c field, so that t h e r e is no t r a n s v e r s e e l e c t r o n d i s p l a c e m e n t . Effective i n t e r a c t i o n between the b e a m and the field of the b a c k w a r d wave is p o s s i b l e if any spatial h a r m o n i c of the wave h a s a p h a s e v e l o c i t y that is c l o s e to the e l e c t r o n v e l o c i t y . Let us a s s u m e that, in the r e g i o n of the b e a m , the longitudinal e l e c t r i c field of this h a r m o n i c c a n be w r i t t e n in the f o r m E : Re {E (x, t)exp [i 0 ( t - - xlvo)] ],
(1)
w h e r e the f r e q u e n c y [~ is d e t e r m i n e d f r o m the equation Vph(2)
----
re,
(2)
and the slowly v a r y i n g c o m p l e x amplitude of the wave E(x, t) s a t i s f i e s the condition
> 1 ) t h e n o r m a l i z a t i o n [see (13) and (26)] coincides with that used in [2]. w
Linear
Stage
of Transient
in
BWT
We will confine o u r s e l v e s to the case in which the rise time of the beam c u r r e n t I(t) f r o m zero to the s t e a d y - s t a t e value I 0 is small as c o m p a r e d to the total settling time of the oscillations. Using the general asymptotic representation of solution (25), we can obtain a lower bound for the duration of the transient f r o m the formula tq- vo
v h ~, In I F~/Fin I t~t ~ t~t-- ~ c Re
(30)
if we know the level of the s t e a d y - s t a t e mode I F s s I and the initial amplitude of the fundamental mode I Finl. Note that to obtain a s a t i s f a c t o r y estimate for t~e t it suffices to find only the o r d e r of I F s s / F i n l , since this quantity a p p e a r s in (30) as a logarithm. In what follows, therefore, we will take I F s s I ~ 1 (see [7, 2] and the following section). Evidently the m o s t important contribution to IFinl c o m e s f r o m noise radiation of density fluctuations (shot effect) and the radiation of the abrupt front of the e l e c t r o n beam (obviously, the front a r i s e s when the c u r r e n t at the input changes sufficiently rapidly, and propagates along the tube at a velocity close to v0). Let us f i r s t c o m p a r e the significance of both effects. We will a s s u m e that, when self-oscillation a r i s e s , the " p r i m e r " is provided only by the s p e c t r a l radiation components in the s y n c h r o n i s m band Af~ ~ ~2C. In this f r e o?. quency interval the radiation e n e r g y o f the front is proportional to I I f i ( ~ 2 / 2 ~ ) l 2, where Io = J I (t) e - i ~ t d t , while the e n e r g y of n o i s e radiation is (1/~r)eIoAl2 in conformitylwith Schottkyts formula for the mean square o f the noise c u r r e n t . Correspondingly, the radiation of the front predominates over noise if
I Io. I* >
(31)
4,: e l o ~C
As I Fin t we can take the radiation amplitude of the front at the frequency of s y n c h r o n i s m (Appendix 2): V
IFml~C!
--1
io 1
Zo l"
(32)
In the opposite ease, comparing the expressions obtainod above, we find that I]/4,~ e f~C.
(33)
We will give a numerical example, using p a r a m e t e r s typical of u I t r a r e l a t i v i s t i c BWT in a quasioptimal mode with r e s p e c t to efficiency [2]: l = 2.2, v = 1.85 [in the notation of (26), l t = 1.2; I ~ 1]; To = 4, C = 0.06, ~2 = 6 . 1 0 1 ~ sec -1, I 0=104 A, v 0 ~ V g r ~ c . F o r the given l w e h a v e R e ~ l ~ 0.26. We will a s s u m e that the
732
Fig. 2. Electron c u r r e n t pulse. beam c u r r e n t i n c r e a s e s in time, as shown in Fig. 2, with the c u r r e n t jumping by an amount AI over a time that is small as c o m p a r e d to 2v/~2 (the jump can be physically associated with the transition f r o m autoelectronic to explosive emission). A s s u m e , f u r t h e r m o r e , that we have the inequalities 2 ~ / ~
