V.
MICROWAVE ELECTRONICS
A.
D. Parker
A. Poeltinger
Prof. A. Bers
Prof. L. J. Chu
R. J. Briggs
Prof. L. D. Smullin
Prof. H. A. Haus
TRANSVERSE FREE WAVES ON ELECTRON BEAMS
The transverse waves that propagate on a one-dimensional electron beam in a finite
magnetic field are being studied. The general dielectric tensor for an unbounded, relativistic electron beam is presented and a detailed discussion of the waves along the magnetic field is given. The results are useful in understanding the instabilities that arise
(See Section VIII-C of this report. )
when such a beam passes through a plasma.
1.
General Dielectric Tensor
We assume that the electrons
The system to be investigated is shown in Fig. V-l.
A plane wave of the form
are cold and are neutralized by ions of infinite mass.
exp[j(wt-k - F)] is assumed, and the vector k is chosen to lie in the x-z plane. If we
use the relativistic force equation and Maxwell's equations to solve for the current in
terms of the electric field, we can define a dielectric tensor for the medium and write
Maxwell's equations as
kXE=
(1)
LoH
(2)
kXH=-EEoK.E
Here,
1-
r
J
-j 22
2
w.kw
e
2
Wpt rWc
2_
22 2
2r
Wpt r c
2
2(2
2 2
Wpt r
1-
2
r
c
WPtkxVoO
e
2)
2
c
2
2
pt x or
2(
2
pt x o r
2
2 2
Wpt r
2
r
2)
c
Wpt x 0oc
-j W2(w2
\r
)2
c)
i
Z
W. 2
222
pt kxo 0
2r- W2
c)
r
This work was supported in part by the U. S. Navy (Office of Naval Research) under
Contract Nonr-1841(49); and 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
AFI9(604)-7400.
QPR No. 67
(V.
MICROWAVE ELECTRONICS)
w
r
=w-k v
zo
lel B0
c
mt
2
ep 0
pt-
2
pi
and
(4)
Comt
epo
-
E mp
po is the charge density of the beam as measured by a stationary observer.
The
transverse and longitudinal masses are defined by
mt = Yomo
m
3
= ymo ,
where
v2/c2
yo= (Iand
to
m0
is
include
the rest
a
1/2
mass.
multiparticle
The
dielectric
system by
tensor can easily be
summing
over
all
manner.
X
k
z
B0
k= k i +k
xx
Fig. V-1.
QPR No. 67
i
Z Z
Electron-beam system.
species
generalized
in
the
usual
(V.
2.
MICROWAVE ELECTRONICS)
Waves along the Magnetic Field
For the waves along the magnetic field, k x = 0 and the dielectric tensor assumes
the form
K
-K
I.
0
x
SKx
K
0
0
0
K
where the elements are easily obtained from the general tensor.
types (in the rest of this report,
The waves are of two
k z = k):
(a) Longitudinal oscillations with H = 0 and E = E z
; K 11 = 0.
These are the well-
known space-charge waves and will not be discussed further in this investigation.
(b) Transverse waves that are right or left circularly polarized with
2
2
k=--
K
for left polarization (clockwise)
(6)
c
2
2
2
k
c
(7)
2_Kr for right polarization (counterclockwise)
where
K =K
2
Wpt r
2
- jKx = 1
(8)
2
pt r
Kr
K. + jK x = 1
2
(9)
r
c
The solutions of the dispersion equations are complicated because of their cubic nature.
The propagating modes are most easily obtained by a transformation into a reference
frame moving with the beam because the dispersion equation is then quadratic (Fig. V-2).
The transformation of (w, k) is
w= Yo(w' + k'v
o
)
k = -o(k' + u'vo/c
QPR
No. 67
(10)
2
).
(11)
(V.
MICROWAVE ELECTRONICS)
XI
x
V
Z
y
CO-MOVING FRAME
LABORATORY FRAME
Fig. V-2.
Co-moving reference frames.
In the primed reference frame, the dispersion equations are
wo2
k' 2
2
p
-
-
(left)
(+co
c
(12)
co
where
epo
2
p
2
E0
pt
00
(14)
eB
W
co
and
p'
=-
is
m
o
= lYW
oc
the charge density as measured in the co-moving reference
It should be noted that Eqs.
The dis-
the left and right circularly polarized waves in an electron plasma.
persion of the right polarized
left polarized case;
frame.
12 and 13 are just the dispersion relations for
waves can be obtained by letting
w - -w
in the
thus a plot of w-k for positive and negative frequencies
for the left polarized waves will cover both cases in the primed reference
frame.
If we have
reasonably low densities
dispersion equation is
of the
and high magnetic
form sketched in Fig. V-3.
fields
(
<<w co), the
The transformation
to the laboratory frame gives the dispersion curve shown in Fig. V-4.
We
notice that the left polarized wave has a low-frequency backward wave under
QPR No. 67
Fig. V-3.
Dispersion of left polarized wave in
co-moving frame (w <<wco).
p co
I
LJ
V
0
Ld
C
LEFT - POLARIZED
RIGHT -POLARIZED
V
Fig. V-4.
QPR
No. 67
Electron-beam dispersion (wp<<w ).
pc
W Vo
Pt
c
2
o c
c
Fig. V-5.
Detail of low-frequency, backward wave
valid if -wC
c
1 .<
380
>
- 0
s 3O
REAL k
Y
(kCOMPLEX k
RICHT - POLARIZED
2
Fig. V-6.
QPR No. 67
Dispersion for
2
Vo
= Wco
coo c
p
0.5.
MICROWAVE ELECTRONICS)
(V.
in
This branch
conditions.
these
the next
phase
velocities
is
active
an
c.
less than
backward
wave
a
Therefore,
negative
that propagates
negative
of the
portion
It
Fig. V-5.
wave carries
left polarized
that the
section
detail in
shown in
is
will be
energy
low-frequency
energy opposite
shown
for
all
branch
to the beam
velocity.
LEFT- POLARIZED
2
0
2
\
3
4
co0
(k
___Re
>
COVFLEX
3
RIGHT-
<
POLAFIZED
2
0
I
2_
w-k plots
approximate
formation
to
analysis
gives
obtain the
root.
These
for
at least
complex
are
some
are
also
k
4
3
Dispersion for
Fig. V-7.
The
_
co
2
= 10
p
2
o
- 0. 5.
o ,
co c
cases
that
representative
shown in
Figs. V-6, V-7,
k
for
all
values from Eqs. 6
and
7
one
real
shown in
root
of
Figs. V-6, V-7,
and
are
not covered
and V-8.
w, it
is
Since
a
by factoring
V-8.
It
by this
the trans-
simple
out the
can be
matter
real
shown that
the low-frequency backward wave will no longer exist when (wp/wc)(Vo/c) exceeds
a certain critical value of the order of unity.
V-7, and V-8.
QPR No. 67
This is
also evident from Figs. V-6,
(V.
MICROWAVE ELECTRONICS)
0
>3
0
°
3
LEFT - POLARIZED
.025
0.5
0
.05
Wco
3
-0.52
REAL k
-_
---
Re (k)
Re- (k)
Im (k)J
---
COMPLEX k
RIGHT - POLARIZED
> 03
01 3
Dispersion for w
Fig. V-8.
3.
2
p
10ow
2
CO
,
Vo
C
0. 1.
Energy and Power
The application of Sturrock's criterion would show that the left polarized waves with
phase velocity less than c carry negative energy, whereas the right polarized waves
carry positive energy.1
This can also be shown from an explicit evaluation of the
energy density in the laboratory frame.
such a medium is
W = 1/4
i
2
The total energy density of a plane wave in
' 3
=22
+ 1/4
-*
E
-
0
=
(wK)E.
-
(15)
It can be shown that the right and left polarized waves are orthogonal, and that the
energy density of the left polarized wave is
QPR No. 67
(V.
W = 1/4
io Ti 2
MICROWAVE ELECTRONICS)
+ 1/4 E I1 28 (wK2 )
2
Wpt c
1/4
=
Eo0
2 2
w(w-kv +
For a slow wave, w/k < c, it can be shown that for propagating waves the wave number
must lie in the range
w+w
<k
c
<
o
o
By using the disperion relations, the energy density can be written as
2
+
2w(w-kvo)
W
=-
1/4 EIE12
+ 2kcw(1 -
(w-kc)
w(kv o - w) (w+wc-kv o
which is negative definite for slow waves.
)
The energy density of the right polarized
wave is
Wr = 1/4
Wr Eo
22 +
o(o
kptw
c)2
which is positive definite.
R. J. Briggs, A. Bers
References
1. P. A. Sturrock, In what sense do slow
Phys. 31, 2052-2056 (1960).
2. A. Bers, Properties of waves in timeProgress Report No. 65, Research Laboratory
pp. 89-93.
3. A. Bers, Energy and power in media
Quarterly Progress Report No. 66, Research
July 15, 1962, pp. 111-116.
QPR No. 67
waves carry negative energy? J. Appl.
and space-dispersive media, Quarterly
of Electronics, M. I. T., April 15, 1962,
with temporal and spacial dispersion,
Laboratory of Electronics, M.I. T.,