II.
Prof. L. D. Smullin
Prof. H. A. Haus
Prof. A. Bers
A.
MICROWAVE ELECTRONICS
Prof. L.
J.
R. P. Porter
H. M. Schneider
R. S. Smith
Chu
R. J. Briggs
P. A. Mandics
WAVES IN A BRILLOUIN-FOCUSED ELECTRON-BEAM
The only known wave solution for the Brillouin-focused
I
WAVEGUIDE
electron beam are the c-
independent slow waves that were found from an approximate (quasi-static and weak
2
We shall give here the exact solution for all waves.
space-charge) analysis.
PERFECT
ELECTRIC
CONDUCTOR
b
L
'
Bo
ELECTRON
Fig. II-1.
Brillouin-focused,
z
BEAM
circular electron-beam waveguide.
Consider a nonrelativistic Brillouin-focused electron-beam waveguide (Fig. II-1).
In the unperturbed state this beam is characterized by a uniform velocity vo along the
applied magnetic field B , a constant angular velocity equal to the Larmor frequency
and the Brillouin-focusing condition
WL = -eBo/2m,
2
1
O L = _T
where
2
p,
2= e p /meo is the plasma frequency.
The small-signal equations of this system are most conveniently written in a coordinate system that is translating with velocity v 0 and rotating with angular velocity WL.
Quantities relating to this coordinate system will be designated by double primes. In
this frame Maxwell's equations take the simple form
VXE" = -jw
OH"
(2)
This work was supported in part by Purchase Order DDL B-00368 with Lincoln
Laboratory, a center for research operated by Massachusetts Institute of Technology
with the joint support of the U. S. Army, Navy, and Air Force under Air Force Contract
AFl 9(604) -7400.
QPR No. 69
(II.
MICROWAVE ELECTRONICS)
p
H" = jWEo
VX
-
2
E"
(3)
which corresponds exactly to the equations that govern an isotropic plasma waveguide.
The complete solutions for waves in an isotropic plasma waveguide are known.3 Hence
the complete solutions for the waves in a Brillouin-focused electron-beam waveguide
can readily be obtained by a proper transformation of the isotropic plasma waveguide
solutions.
1.
Dispersion Characteristics
The dispersion characteristic for the isotropic plasma waveguide is shown 3 in
Fig. II-2. Here we shall consider only those branches that contain slow waves (P>w/c).
For w < co (the cutoff frequency of the empty waveguide) the branches that contain only
fast waves occur for w > wp, and are not shown in Fig. II-2. The dispersion characteristics shown are for the field solutions with dependence exp j(w"t" - m"" - P"t").
The dispersion characteristics in a coordinate system (denoted by single primes)
that is rotating at a constant angular velocity wL are readily obtained through the nonrelativistic transformations
nt= U' - mWL
(4)
= m'
m"
PI =
(5)
p1.
(6)
The results are shown in Fig. 11-3.
For most electron-beam systems of interest, rwL
c and the nonrelativistic transformations of Eqs. 4 and 5 apply. The first-order corrections to these transformations can be obtained from general relativity, 4 and are
('-m'w L)Y
("
mf
(7)
,
(8)
where
S2 2 -1/2
y
=
1
r2 2
1 +-----
c
2c
(9)
The dispersion characteristics in the laboratory frame, which is translating at the
constant velocity vo with respect to the primed frame, are obtained through the nonrelativistic transformation
w' =
QPR No. 69
-
voo
(10)
--
~IYi
~Cbl
-- --
Fig. I-2. Dispersion characteristics in the rotating and translating frame-plasma frame.
Fig. II-3. Dispersion characteristics in the frame rotating with respect to the
plasma
frame (translating with respect to the laboratory frame). (a) m = 0, (b) m =
±1, (c) m = ±2, (d) m = ±3.
i/
Fig. 11-4.
7'a'
,.
Dispersion characteristics in the laboratory frame.
waves), (b) m = ±1, (c) m = ±2, (d) m = ±3.
Fig. 11-5.
QPR No. 69
(a) m = 0 (no backward
Dispersion characteristics in the laboratory frame
for m = 0 waves containing a backward-wave region
(Eq. 15).
(II.
MICROWAVE ELECTRONICS)
1'= P.
(11)
The results are shown in Fig. 11-4.
is valid for v
The nonrelativistic transformation of Eqs. 10 and 11
<<c and for slow waves, p > w/c.
o
The relativistically correct transfor-
mations are readily obtained from special relativity,
4
and are
(12)
' = (w-pv)yO
' =
(
YO,
-
(13)
c
where
S=
(14)
In Figs. II-4b-d we note that backward-wave characteristics (positive group velocity
and negative phase velocity) are obtained for all waves with m * 0.
For the waves with
circular symmetry, m = 0, Fig. II-3a and Eq. 10 or 12 show that for certain beam
velocities, v o , backward-wave characteristics must also be possible.
The condition
for this possibility is
v
o
<c 1 +
kpb
I (k b)
b) In
-1/2
(kpb) Ina
where I and I1 are the modified Bessel functions, kp = p /c,
(15)
and c is the velocity of
light in free space.
Unlike the case for a confined flow beam, B = oo, Eq. 15 is easily
satisfied for most beams that can be produced in practice. Under the conditions of
Eq. 15, the dispersion characteristic of Fig. II-4a changes to that shown in Fig. 11-5.
2.
Energy-Power Characteristics
The time-averaged, small-signal energy per unit length of waveguide, W", in the
a
3
isotropic plasma waveguide described by Eqs. 2 and 3, is positive definite.
Hence all
of the waves in the double-prime frame are passive waves.
The power flow in this
frame, P", is entirely electromagnetic.
In the coordinate system rotating at the angular frequency wL with respect to the
5
double-prime frame, the nonrelativistic transformations of energy and power give
W'
a
W"
a
W =
P'
P"
z_ w'v'g
g
g
QPR No. 69
(16)
(17)
(II.
MICROWAVE ELECTRONICS)
where vg is the group velocity (energy velocity) for the wave.
The transformations in
Eqs. 4 and 16 show how the active (negative energy) waves appear in the prime frame,
Fig. 11-3.
The group velocity of the waves remains unchanged.
The transformations of energy and power to the laboratory frame 6 are
W
a
-
P
z
wv-
g
W'
a
w
(18)
P'
z
w'V'
(19)
g
The appearance of active waves can now be traced to the transformations in Eqs. 10 and
18. We note the existence of active waves with phase velocities that may be either slower
or faster than the velocity of light and v .
A. Bers, R. S. Smith
References
1. L. Brillouin, A theorem of Larmor and its importance for electrons in magnetic
fields, Phys. Rev. 67, 260-266 (1945).
2. W. W. Rigrod and J. A. Lewis, Wave propagation along a magnetically focused
cylindrical electron beam, Bell System Tech. J. 33, 399-416 (1954); G. R. White,
Space charge waves in relativistic Brillouin beams, Mikrowellenrohren, Vortrage der
Internationalen Tagung in Miinchen, 7-11 June 1960; NTF, Vol. 22, pp. 271-278, 1961.
3. W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas
(The M. I. T. Press, Cambridge, Mass., 1963).
4.
C. MOller, The Theory of Relativity (Oxford University Press, London, 1955).
5. R. J. Briggs, H. A. Haus, and A. Bers, Small-Signal Energy, Power and
Momentum Relations, Internal Memorandum, Research Laboratory of Electronics,
M.I.T., 1963.
6. P. A. Sturrock, In what sense do slow waves carry negative energy?,
Phys. 31, 2052-2056 (1960).
QPR No. 69
J. Appl.