Download AMPLIFICATION AND GENERATION OF HIGH

SPACE-FREQUENCY MODEL FOR PULSED BEAM FREEELECTRON LASER OPERATING IN THE SPACE-CHARGE
(COLLECTIVE) DOMINATED REGIME
Y. Pinhasi and Yu. Lurie
Dept. of Electrical and Electronic Engineering, Faculty of Engineering
Ariel University Center of Samaria, Ariel, Israel
Intense radiation devices such as microwave tubes, free-electron lasers (FELs) and
masers, utilize distributed interaction between an electron beam and the
electromagnetic field. In these devices the electron beam serves as the gain medium
generating radiation, which is coupled out of the device through a transmission
system. We developed a three-dimensional, space-frequency theory for the analysis
and simulation of radiation excitation and propagation in electron devices and freeelectron lasers operating in millimeter wavelengths and in the Tera-Hertz
frequencies. The total electromagnetic field (radiation and space-charge waves) is
presented in the frequency domain as an expansion in terms of transverse eigenmodes of the (cold) cavity, in which the field is excited and propagates. The mutual
interaction between the electron beam and the electromagnetic field is fully described
by coupled equations, expressing the evolution of mode amplitudes and electron beam
dynamics. The approach is applied in a numerical particle code WB3D, simulating
wide-band interaction of a free-electron laser operating in the linear and non-linear
regimes. The model is utilized to study spontaneous and super-radiant emissions
radiated by a an electron bunch at the sub-millimeter regime, taking into account
three dimensional space-charge effects emerging in such ultra short bunches.
1. INTRODUCTION
Electron devices such as microwave tubes and free-electron lasers (FELs) utilize
distributed interaction between an electron beam and electromagnetic radiation. Many
models have been developed to describe the mutual interaction between the gain
medium (electron beam) and the excited radiation. These models are based on a
solution of Maxwell equations and the Lorenz force equation in the time domain.
Contrary to space-time models, formulation of the electromagnetic excitation
equations in the frequency domain inherently takes into account dispersive effects
arising from the cavity and the gain medium. Moreover, it facilitates consideration of
the statistical features of the electron beam and the excited radiation, necessary for the
study of spontaneous emission, synchrotron amplified spontaneous emission (SASE),
super-radiance and noise.
In this paper we develop a space-frequency model, which describes broadband
phenomena occurring in electron devices, masers and FELs and characterized by a
continuum of frequencies. The total electromagnetic field is presented in the
frequency domain as a summation of transverse eigen-modes of the cavity, in which it
is excited and propagates. A set of coupled excitation equations, describing the
evolution of each transverse mode, is solved self-consistently with beam dynamics
equations. This coupled-mode model is employed in a three-dimensional numerical
simulation WB3D [1,2]. The code was used to study the statistical and spectral
characteristics of the radiation generated in a pulsed beam free-electron laser,
operating in the millimeter and sub-millimeter wavelengths.
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2. PRESENTATION OF THE ELECTROMAGNETIC FIELD IN THE
FREQUENCY DOMAIN
The electromagnetic field in the time domain is described by the space-time electric
Er, t  and magnetic Hr, t  signal vectors. r stands for the x, y, z  coordinates,
where x, y  are the transverse coordinates and z is the axis of propagation. The
Fourier transform of the electric field is defined by:

Er, f    Er, t  e j 2 f t d t
(1)

where f denotes the frequency. Similar expression is defined for the Fourier
transform Hr, f  of the magnetic field. Since the electromagnetic signal is real (i.e.
E r, t   Er, t  ), its Fourier transform satisfies E r, f   Er, f  .
Analytic representation of the signal is given by the complex expression [2]:
~
ˆ r, t 
(2)
Er, t   Er, t   j E
where
Er, t '
d t'
  t  t '
ˆ r, t   1
E


(3)
is the Hilbert transform of Er, t . Fourier transformation of the analytic
~
representation (2) results in a “phasor-like” function Er, f  defined in the positive
frequency domain and related to the Fourier transform by:
2 Er, f  f  0
~
Er, f   2 Er, f   u  f   
(4)
0
f 0

The Fourier transform can be decomposed in terms of the “phasor like” functions
according to:
1~
1~
Er, f   Er, f   E r, f 
(5)
2
2
and the inverse Fourier transform is then:

 ~

Er, t    Er, f  e j 2 f t d f  Re   Er, f  e j 2 f t df 
(6)

0

3. THE WIENER-KHINCHINE AND PARSEVAL THEOREMS FOR
ELECTROMAGNETIC FIELDS
The cross-correlation function of the time dependent electric Er, t  and magnetic
Hr, t  fields is given by:
REH  z ,      Er, t    Hr, t  zˆ dx dy d t

(7)

Note that for finite energy signals, the total energy carried by the electromagnetic
field is given by W z   REH z,0 .
According to the Wiener-Khinchine theorem, the spectral density function of the
electromagnetic signal energy S EH z, f  is related to the Fourier transform of the
cross-correlation function REH z,  through the Fourier transformation:
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S EH z, f  

 R  z ,  e
 j 2 f 
EH



d   Er, f  H r, f   zˆ dx dy



~
~
1
(8)


r, f   zˆ dx dy
E
r
,
f

H
f 0

4


1
~
~
  Er, f  H r, f   zˆ dx dy f  0
4
Following Parseval theorem, the total energy carried by the electromagnetic field
can also be calculated by integrating the spectral density S EH z, f  over the entire
frequency domain:





~
~
1

W z    S EH z, f  d f     Re Er, f   H r, f   zˆ dx dy  df
2


0
We identify:
(9)
d W z  1
~
~
 Re  Er, f  H r, f   zˆ dx dy
(10)
df
2
as the spectral energy distribution of the electromagnetic field (over positive
frequencies).
 


4. MODAL PRESENTATION OF ELECTROMAGNETIC FIELD IN THE
FREQUENCY DOMAIN
The “phasor like” quantities defined in (4) can be expanded in terms of transverse
eigenmodes of the medium in which the field is excited and propagates [3]-[5]. The
perpendicular component of the electric and magnetic fields are given in any crosssection as a linear superposition of a complete set of transverse eigenmodes:
~
 jk z
 jk z ~
E r, f    C q z, f  e zq  C q z, f  e zq Eq  x, y 

~
H r, f    C
q
 jk
 q z, f  e

zq
z
 C q z, f  e

H~
 j k zq z
q   x, y 
(11)
q
C q z, f  and C q z, f  are scalar amplitudes of the qth forward and backward
~
~
modes respectively with electric field Eq  x, y  and magnetic field H q  x, y 
profiles and axial wavenumber:
 k2  k 2
k  k q (propagati ng modes)

q
k zq  
(12)
2
2
j
k

k
k

k
(cut
off
modes)


q

q

Expressions for the longitudinal component of the electric and magnetic fields are
obtained after substituting the modal representation (11) of the fields into Maxwell's
~
equations, where source of electric current density J r, f  is introduced:
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

~
 jk z
 jk z ~
Ez r, f    C q z, f  e zq  C q z, f  e zq Eq z x, y  
q
~
J z r, f 
j 2 f  0
~

1
E z r , f


(13)

~
 jk z
 jk z ~
H z r, f    C q z, f  e zq  C q z, f  e zq H q z x, y 
q
The evolution of the amplitudes of the excited modes is described by a set of
coupled differential equations:
 ~
d
1  j kzq z  Z q ~
~
ˆ




C q  z , f   
e
J
r
,
f

z
J
r
,
f

  Eq x, y  dx dy (14)

z
  Z q
dz
Nq


The normalization of the field amplitudes of each mode is made via each mode's
complex Poynting vector power:
~ 
~
N q   Eq   x, y   H q   x, y   ẑ dx dy
(15)


and the mode impedance is given by:
 0 k
2 f  0

for TE modes

kzq
  0 kzq
Zq  
(16)
kzq
 0 k z q

for TM modes
 
k
2

f

0
0

Substituting the expansion (11) in (10) results in an expression for the spectral
energy distribution of the electromagnetic field (over positive frequencies) as a sum of
energy spectrum of the excited modes:
2
2
d W z 
1
 
C q z, f   C q z, f   Re N q
df
2
q
Propagating
(17)

  Im C q z, f  C q z, f   Im N q



 
  
q
Cut - off
5. THE LONGITUDINAL SPACE-CHARGE FIELD
The expressions for the perpendicular and longitudinal components of the electric
field include full three-dimensional description of space-charge forces due to
collective effects in the electron beam. In electron devices and free-electron lasers, the
electron beam is usually magnetically confined in the transverse direction. This allows
a common approximation in which transverse variation of the current density is
~
~
neglected, i.e.    J    J z /  z . In this case, the longitudinal electric field (13)
gives rise to the dominant part of space-charge forces along the e-beam pulse. Modal
~
description of E z r , f  consists of a summation over TM modes of the waveguide and
~
~
the additional term Ez r, f   J z r, f  / j 2 f  0  identified as the longitudinal
space-charge field in a 1D description. In the time domain, the term corresponds to
the field given by:
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
  r     r 
1
~
~
Ez r, t   
J z r, f  e  j 2 f t d f   
2 0
 0 j 2 f  0

(18)
where
  r  

~
 J r,  d
z
(19)
t
and
t
  r  
~
 J r,  d
z
(20)

are surface charge densities associated with the longitudinal component of the beam
current.
A one-dimensional description of the space-charge field in electron devices
becomes inadequate in the case of a finite cross-section beam, where there is fringing
field near the edges of the beam. Fringing is increased further by the presence of
conducting walls of a waveguide. In such cases full 3D space-charge model is
required, by considering in Eq. (13) the full summation over TM modes.
6. THE ELECTRON BEAM DYNAMICS
The state of the particle i is described by a six-component vector, which consists
of the particle's position coordinates ri  xi , yi , zi  and velocity vector v i . The
velocity of each particle, in the presence of electric Er, t  field and magnetic
induction Br, t   0 Hr, t , is found from the Lorentz force equation:

d vi 1 
 e 1
Eri , t   vi  Bri , t   vi d  i 
 
(21)
d z i 
dz 
 m vz i

where e and m are the electron charge and mass respectively. The fields represent the
total (DC and AC) forces operating on the particle, and include also the self-field due
to space-charge. The Lorentz relativistic factor  i of each particle is found from the
equation for kinetic energy:
d i
e 1

v i  Eri , t 
(22)
dz
m c 2 vzi
where c is the velocity of light.
The time it takes a particle to arrive at a position z, is a function of the time t0i
when the particle entered at z=0, and its instantaneous longitudinal velocity v z i z 
along the path of motion:
z
ti  z   t 0 i  
0
1
v z i  z '
d z'
(23)
The current distribution is determined by the position and the velocity of the
particles in the beam:
Q N v 
J r, t      i   x  xi   y  yi  t  ti z 
(24)
N i 1  vz i 
Here Q  I 0 T is the total charge of the e-beam pulse with DC current I 0 and
temporal duration T. The “phasor like” current density is given by:
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
~
J r, f   2 u  f   J r, t  e  j 2 f t d t

Q N v
 2 u  f   i  x  xi   y  yi  e  j 2 f ti  z 
N i1 vzi
(25)
and it’s substitution into the excitation equation (14) leads to:
d
C q  z , f  
dz
 j 2 f ti  z  k zq z 
1 Q N  Z q 1
~ 
~ 

v i  Eq  xi , yi   Eq z xi , yi  e
 

N q N i1  Z q vzi

(26)
The resulted expression, together with the beam dynamics equations (21)-(23), forms
a close set of equations, enabling a self-consistent solution of the electromagnetic
fields (radiation and space-charge waves) in electron devices and free-electron lasers.
Note that in the particle presentation of the current density, the expression for the
one-dimensional longitudinal space-charge field becomes:
1 Q N
~
Ez r, t  
(27)
  x  xi   y  yi  sgn t  ti z 
2 0 N i1
 1 t  0
where sgn t   
is the sign function.
 1 t  0
7. NUMERICAL RESULTS
When the electron beam is pre-bunched to short pulses, the fields excited by the
electrons become correlated and coherent summation of radiation fields from
individual particles occurs. All electrons radiate in phase with each other in this
situation, and the generated radiation is termed as super-radiant emission [6-8].
It was shown that energy flux of super-radiant emission is proportional to the
square of the total charge Qb of the driving electron pulse and to the square of the
interaction length Lw. However if the charge density is increased, collective effects
force the bunch to expand along the interaction region, causing the particles to lose
their initial phase coherence. As the temporal duration of the electron pulse
approaches the period of the emitted radiation, the super-radiance effect is suppressed
and the spontaneous emission becomes dominant. The energy flux of spontaneous
emission is proportional to the bunch charge Qb.
We use WB3D numerical code [1, 2] to simulate pulsed beam FEL radiation in
the sub-millimeter wavelengths. The parameters of the FEL are summarized in Table
1. Trajectories of electrons obtained in the simulations with bunch charges of
Qb=30pC and 0.5nC are demonstrated in the Fig. 1. At the entrance to interaction
region, the pulse Tb is relatively short so as f s Tb  1 ( f s is the frequency of the
emitted radiation at synchronism), resulting in a strong super-radiant emission. As the
electron bunch propagates along the wiggler, its temporal duration increases due to
the space-charge forces, as clearly seen in the Fig. 1. When the temporal bunch width
becomes f s Tb  1 , only spontaneous emission takes place in the following stages of
the interaction.
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Table 1. Parameters of pulsed beam FEL operating in the sub-millimeter band.
Accelerator
Electron beam energy:
Initial pulse duration:
Ek  5.5 MeV
Tb  0.05 pS
Qb  30 pC  1 nC
Pulse charge:
Wiggler
Magnetic induction:
Period:
Bw  2000 G
w  2.5 cm
N w  20
Number of periods:
Waveguide
Rectangular waveguide:
15  10 mm 2
Qb =500 pC
50
45
45
40
40
35
35
30
30
z [cm]
z [cm]
Qb =30 pC
50
25
25
20
20
15
15
10
10
5
5
0
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0
-3
0.5
 t [pSec]
-2
-1
0
1
2
3
4
 t [pSec]
Fig. 1. Electron trajectories along the interaction region relative to the bunch center in
simulations with bunch charges Qb=30 pC (left) and 0.5 nC (right). Red lines show
RMS pulse widths.
Qb =30 pC
Qb =500 pC
2500
80
Nw = 5
70
Nw = 10
2000
Nw = 15
Nw = 20
60
dW/df [pJ/GHz]
dW/df [pJ/GHz]
Nw = 5
Nw = 10
50
40
30
Nw = 15
Nw = 20
1500
1000
20
500
10
0
2500
2600
2700
2800
2900
3000
3100
3200
3300
0
3400
Frequency [GHz]
2500
2600
2700
2800
2900
3000
3100
3200
3300
Frequency [GHz]
Fig. 2. Evolution of energy flux spectral density for Qb=30 pC (left) and 0.5 nC
(right).
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3400
The situation is illustrated in Fig. 2, describing the evolution of the energy flux
spectral density of the radiation emitted along the interaction region. When the bunch
charge is Qb=30 pC, the expansion of the bunch due to space-charge forces is
relatively small, and the cumulative contribution of a longer interaction path to the
radiation is considerable. However, as the charge of the bunch is increased (to Qb=1
nC), space-charge forces give rise to a fast expansion of the electron pulse so the most
part of the radiation is emitted only at the first stages of the wiggler.
In order to measure the phase coherence between the particles, we define a
bunching factor:
1 N
b  z   e j 2 f s t i  z    e j 2  f s t i  z 
(28)
N i 1
and draw it along the interaction region, as shown in Fig. 3. If low-charge electron
bunches are applied, the bunching factor remains relatively high along the most part
of its path of propagation, generating an intensified super-radiant emission. The total
energy flux Wtot grows fast along the wiggler. Increasing of the bunch charge leads to
a fast reduction of the bunching factor even at the very first stage of the radiation
buildup process, what results in a destruction of the emission coherence and a low
energy flux.
1
Qb = 30 pC
b(z)
Qb = 500 pC
Qb = 1 nC
0.5
Wtot / Qb2 [J/(nC)2]
0
0
5
10
15
20
5
10
15
20
25
30
35
40
45
50
25
30
35
40
45
50
-5
2
x 10
1.5
1
0.5
0
0
z [cm]
Fig. 3. Bunching factor bz  and normalized total energy flux for different pulse
charges.
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[4]
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[6]
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