Smooth magnetic cusp profile calculation for axis-encircling electron
beam generation
Y. Yi n ,
a)
B. Wang, H. L. Li, and L. Meng
School of Physical Electronics, University of Electronic Science and Technology of China,
North Jianshe Road 2-4, Chengdu, Sichuan 610054, China
(Received 13 January 2012; accepted 19 March 2012; published online 6 April 2012)
The calculation method of a smooth magnetic cusp profile has been introduced to obtain an axis-
encircling electron beam in a fixed electrostatic field distribution. Using the calculated smooth
magnetic cusp profile, an axis-encircling electron beam of current 1 A and voltage 30 kV can be
obtained with a velocity ratio 1.8 and low beam ripples. Theoretical calculation and 3D simulations
agree well, which show that this method can be widely used in other axis-encircling electron gun
designing.
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2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.3701557]
I. INTRODUCTION
Recently, there has been a great demand for axis-
encircling (large orbit) electron beam
1–3
generation to drive
different kinds of gyrodevices, such as the gyrotron traveling
wave amplifier (Gyro-TWA),
4,5
gyrotron backward wave
oscillators (Gyro-BWO),
6
and harmonic gyro-peniotron oscil-
lators.
7
These kinds of gyrodevices are advantageous to oper-
ate at harmonic of the electron cyclotron frequency, which
allows for a substantially lower magnetic field strength than
for the gyrotrons and conventional magnetrons that are oper-
ated at the base frequency. And the use of the axis-encircling
electron beam can also mitigate the problem of parasitic oscil-
lation because the mode selectivity property of such a beam
(harmonic number s) has relations to the cavity mode (TE
mn
,
m and n are azimuthal and radial mode indexes, respectively).
Since the investigation of the relativistic electron
dynamics in a symmetric and an asymmetric magnetic
cusp,
8,9
many attempts have been made to improve the per-
formance of a cusp gun. Traditionally, the large orbit elec-
tron beam was produced by passing a hollow non-rotating
beam through a narrow magnetic cusp, where a large cusp
amplitude was needed due to the fully accelerated electron
momentum and the small radial position of the beam. This
would require many coils and even magnetic material inside
the electron gun to achieve the required sharp cusp. A more
promising way is to use a smooth cusp, which can be formed
by several coils or the permanent magnet system without any
magnetic shaping poles inside the electron gun.
Although the theoretical, simulation, and experimental
studies about the smooth cusp gun have been carried out, the
trajectory of the axis-encircling electron beam can be solved
by the Lagrange equations, and the design and optimization
of this kind of electron gun through numerical simulation
code, such as
MAGIC
,
10
agree well with the experimental
results. But the optimizing of the cusp gun includes various
configurations,
11,12
such as the length of the cathode-anode
gap, the shape of the anode tip, the tilt angle of the cathode.
This is very time consuming, and actually, this method is to
choose the best accelerating electric field distribution at the
given magnetic field. This paper provides a method to calcu-
late the azimuthally symmetric magnetic field profile at the
given electric field. In addition, this method can provide the
way to adjust the magnetic shape in experiment. Simulation
results using the calculated smooth magnetic cusp profile are
presented in comparison to the theoretical calculation.
II. THEORETICAL STUDY AND NUMERICAL METHOD
The Lagrangian for the motion of relativistic electrons
in the presence of external axis symmetric electric and mag-
netic fields can be written as
L ¼Àm
e
c
2
1 À
_
r
2
þ r
2
_
h
2
þ _z
2
c
2
!
1=2
þ eu À e
r
2
2
_
hB
z
; (1)
where e and m
e
are the electric charge and rest mass of an
electron, r, h, and z are the radial, azimuthal, and axis coordi-
nates, respectively. The upper point denotes the derivative of
these parameters against time. The u and B
z
are the electro-
static potentials and magnetic fields along axis, respectively.
The Lagrange equations for the r, h, and z coordinates
can be easily obtained through derivation. The conservation
of the electron canonical angular momentum gives raise to
Eq. (2) relating to the electron emit location at the cathode
and the final electron beam parameters in a uniform mag-
netic field.
P
h
¼Àer
2
c
B
c
=2 ¼ cðz;rÞm
e
r
2
ðzÞ
_
hðzÞÀer
2
ðzÞB
c
f ðzÞ=2
¼ c
0
m
e
r
2
0
_
h
0
À er
2
0
B
0
=2; (2)
where c is the relativistic factor of the electron beam, which
is a function of z and r. Subscripts “c” and “0” denote the
cathode and final region, respectively, The function f(z)is
the magnetic cusp profile to be calculated, and its value is 1
and B
0
/B
c
at the cathode and final region, respectively. B
c
f(z)
equals to B
z
.
The motion equation of the electrons can be easily
deduced by the Lagrangian shown in Eq. (1) as follows:
a)
Author to whom correspondence should be addressed. Electronic mail:
yin.yong@ 163.com. Telephone: 96-28-83206782.
1070-664X/2012/19(4)/043102/5/$30.00
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2012 American Institute of Physics19, 043102-1
PHYSICS OF PLASMAS 19, 043102 (2012)