International Journal of Enhanced Research Publications, ISSN: XXXX-XXXX
Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com
Self-fields effects on gain in a helical wiggler free
electron laser with ion-channel guiding and axial
magnetic field
Hassan Ehsani Amri1 ; Taghi Mohsenpour2
1
2
Department of Physics, Islamic azad University,nour branch, nour, Iran
Department of Physics, Faculty of Basic Sciences, University of Mazandaran, Babolsar, Iran
Abstract: In this paper, we have investigated the effects of self-fields on gain in a helical wiggler free electron
laser with axial magnetic field and ion-channel guiding. The self-electric and self-magnetic fields of a relativistic
electron beam passing through a helical wiggler are analyzed. The electron trajectories and the small signal gain
are derived. Numerical investigation is shown that for group I orbits, gain decrement is obtained relative to the
absence of the self-fields, while for group II orbit gain enhancement is obtained.
Keywords: Free-electron laser, self-fields, ion-channel, axial magnetic field, gain
Introduction
A Raman free-electron laser (FEL) produces coherent radiation by passage of a cold intense relativistic electron beam
through a static magnetic (wiggler) field which is spatially periodic along the beam axis. In Raman regime, due to the
high density and low energy of the electron beam, an axial magnetic field generated by current in a solenoid is usually
employed to focus on the beam [1-4]. Also the ion-channel guiding is used to focus the electrons against the selfrepulsive electrostatic force generated by the beam itself [5-9].
In Raman regime, equilibrium self-electric and self-magnetic fields, due to the charge and current densities of the beam,
can have considerable effects on the equilibrium orbits. It has been shown that self-field can induce chaos in the singleparticle trajectories [10]. The effects of self-fields on the stability of equilibrium trajectories and gain were studied in a
FEL with a one-dimensional helical wiggler and axial magnetic field [11,12] or ion-channel guiding [13-14].
In recent years, electron trajectories and gain in a planar and helical wiggler FEL with ion-channel guiding and axial
magnetic field were studied, detailed analysis of the stability and negative mass regimes were considered [15-18]. The
purpose of this paper is to study the effects of the self-fields on gain in a FEL with ion-channel guiding and axial
magnetic field. In Sec. 2.1, steady-state trajectories are obtained in the absence of self-fields. In Sec. 2.2, self-fields are
calculated using Poisson,s equation and Ampere,s law. Equilibrium orbits are found under the influence of self-fields. In
Sec. 2.3, the derivation of the gain equation is presented. In Sec. 3, the numerical results are given. In Sec. 4, concluding
remarks are made.
Theory Details
A. Basic Assumption
The evolution of the motion of a single electron in a FEL is governed by the relativistic Lorentz equation
d  m v 
1


 e E i  v  B w  B0 e z 
dt
c


where
and
Ei
Bw
(1)
is the transverse electrostatic field of an ion-channel and it can be written as
E i  2 eni x xˆ  y yˆ 
(2)
is the idealized helical wiggler magnetic field and can be described by
B w  Bw  xˆ cosk w z  yˆ sin k w z 
(3)
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and
B0
is the axial static magnetic field. Here,
number,
w
Bw denotes the wiggler amplitude, k w  2  w  is the wiggler wave
is the wiggler wavelength (period),
ni is the density of positive ions having charge e. The steady-state
velocity of an electron moving in this field is
v 0  v w xˆ cosk w z  yˆ sin k w z  v|| zˆ 
(4)
where v|| is the component in the positive z direction and
vw 
kwv||2 w
i2  kwv|| 0  kwv|| 
is the signed magnitude of the wiggler-induced transverse velocity.
(5)
 i  2 ni e 2  m  , is the ion-channel
12
 mc , are the axial guide and wiggler magnetic field frequencies.
frequency and  0 , w  e B0, w
B. Self-Field Calculation and Steady-State Orbits
The self-electric and self-magnetic fields are induced by the steady-state charge density and current of the non-neutral
electron beam. Solving Poisson’s equation
1 
rEr( s )  4enb r 
r r


(6)
yields the self-electric field in the form,
E s   2 enb r rˆ  2 enb x xˆ  y yˆ 
where
(7)
nb is the number density of the electrons.
The self-magnetic field is determined by Ampere,s law,
4
(8)
Jb
c
where J b  e nb v   v|| eˆ z  is the beam current density and v  is the transverse velocity. In cylindrical
B 
coordinates
r, , z  , Ampere s law may be written in the form,
,
1  s   s 
4 enb r v
(9)
Bz 
B  
cosk w z   
r 
z
c
 s   s 
4 enb r v
(10)
Br  Bz  
sin k w z   
z
r
c
4 enb r v||
1 
1  s 
(11)
r Bs  
Br  
r r
r 
c
where v  vw . The solution of Eqs. (9)-(11) can be obtained by using the methods used in reference 10 (or references


12 and 13). This yields
1
Bs1  Bs||  Bsw
(12)
where
v||
v
r eˆ  2 enb ||  y xˆ  xyˆ 
c
c


4

e
n
r
v
1
b
w
er coskw z     e sin kw z   
Bsw

kw
c2
Bs  2 enb


2b2 v||2 c 2
 2
B
i  k wv|| 0  kwv||  w
and
(13)
(14)
b  2 nb e 2  m  . The first term in Eq. (12) is due to axial velocity; the second term is due to the transverse
12
velocity induced by the wiggler magnetic field. With inclusion of the self-magnetic field, the total magnetic field up to
first-order correction, may be written as
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Vol. 2 Issue 4, April-2013, pp: (1-4), Available online at: www.erpublications.com
B1  Bw  Bs1  B0 eˆ z  1 Bw  2 enb
where

v||
 y xˆ  xyˆ   B0 eˆ z
c

2b2 v||2 c 2
 1 2
i  kwv|| 0  kwv|| 
1
(15)
(16)
is the first-order correction factor for the wiggler magnetic field.
The equation of electron motion in the presence of self-fields may be written (in the scalar form):
d  mvx 
2 nb e 2
ev
eB
 2 e 2 ni  nb x 
v||vz x  z 1 Bw sin k w z  0 vx (17)
2
dt
c
c
c
d  mv y 
2 nbe2
ev
eB
 2 e2 ni  nb  y 
v||vz y  z 1 Bw cos kw z  0 v y (18)
2
dt
c
c
c
v

d  mvz 
e
   1 Bw vx sin kw z  v y cosk w z   2 nb e || vx x  v y y 
dt
c
c

(19)
vx  vw cos kw z and v y  vw sin kw z ,
The steady-state solution of Eqs. (17) and (18) may be expressed in the form,
where
kwv||2 1 w
vw  2
i  b2  || 2  kwv|| 0  kwv|| 
(20)
and vz  v||  constant . Equation (20) describes the transverse velocity in the presence of the self-electric and first
order self-magnetic field. Substituting new electron velocity components into Eq. (8), and using the method applied in
reference 10 (or references 12 and 13), we obtain the wiggler-induced self-magnetic field up to second-order correction
as
B sw2   K 1B w
where
K
and
(21)


2b2 v||2 c 2
i2  kwv|| 0  kwv|| 
 ||  1  v||2 c 2 
1 2
(22)
. Using Eq. (21), the total magnetic field up to second-order correction is given by
B1  2  Bw  2 enb
v||
 y xˆ  xyˆ   B0 eˆ z
c
(23)
where
2   1  K 1 .
(24)
This process may be continued to find the higher order terms. We can write
Bswn   K n 1B w ,
n  3,4,5,
(25)
n  2,3,4,
(26)
where
n   1  K n 1 ,
The total wiggler-induced magnetic field becomes
Bsw  lim Bswn   K  Bw
(27)
n 
where


2b2 v||2 c 2
  lim   lim  K  lim 2
Kn
n 
n 
n     k v   k v 
i 0
i
w ||
0
w ||
n 
n 1
i
(28)
If the absolute value of K is less than one, then the series in Eq. (28) will converge to 1
the right-hand side will goes to zero. In this case, Eq. (28) may be expressed in the form

i2  b2  ||2  kwv|| 0  kwv|| 
i2  b2 1  v||2 c 2   kwv|| 0  kwv|| 
1  K  , and the last
term in
(29)
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Substituting Eq. (29) into Eq. (27) and using Eq. (22), the total wiggler-induced self-magnetic field is given by
B sw 

2b2 v||2 c 2

i2  b2 1  v||2 c 2   k wv|| 0  kwv|| 
(30)
The transverse part of the steady-state helical trajectories of electrons, in the presence of self-fields, can be found as
vw 
k wv||2  w
i2  b2 1  v||2 c 2   kwv|| 0  k wv|| 
(31)
Equation (31) shows resonant enhancement in the magnitude of the transverse velocity when
i2  b2 1  v||2 c2   kwv|| 0  kwv||   0
(32)
Steady-state trajectories may be classified according to the type of guiding. There are three main categories:
1. When the value of axial magnetic field is zero 0  0 ; that is, a FEL with a helical wiggler and an ion-channel
guiding. In this type of guiding Eq. (31) becomes,
vw 
k wv||2  w
i2  b2 1  v||2 c 2   k w2 v||2
2. When the density of positive ions is zero
(33)
i  0 ; that is, a FEL with a helical wiggler and an axial magnetic field.
In this type of guiding Eq. (36) becomes,
vw 
k wv||2  w
 
k wv||  0  k wv||   b2 1  v||2 c 2

(34)
3. When both the ion-channel and the axial magnetic field are present; that is, a helical wiggler FEL with an ion-channel
and an axial magnetic field. When both types of guiding are present, vw is given by Eq. (31). This case contains three
types: (a) two guiding frequencies are taken to be equal (  0
 i ), (b) the ion-channel frequency is taken to be
constant, (c) the axial magnetic field frequency is taken to be constant.
C. Small Signal Gain
Let electromagnetic radiation copropagate with the electron beam in the FEL interaction region, such as below
E r  E r xˆ cos   yˆ sin  

B r  E r xˆ sin   yˆ cos  

(35)

where   k r z   r t   , r  kr c is frequency, and E r is the amplitude of the wave. The electron equation of
motion in the presence of the electromagnetic radiation and self-fields can be written in scalar form
 2 e 2 ni  nb  2 nb e 2 v||2 
eB


  x    x  

x   w  z sin k w z
2
mc
mc c 
mc

(36)
eB0
eEr
1   z cos 

x 
mc
mc
 2 e 2 ni  nb  2 nb e 2 v||2 
eB


  y    y  

y   w  z cos k w z
2
mc
mc c 
mc

(37)
eB0
eEr
1   z sin 

y 
mc
mc
eB
2 nb e 2 v||
 x x   y y 
  z   z   w  y cos k w z   x sin k w z  
mc
mc c
(38)
eEr
 y sin    x cos  

mc
The last term in Eqs. (36) and (37), which represents the transverse optical force acting on the electron. Now, we
consider radiation terms as a small external factor which produces small perturbation in the steady-state velocity
components,
 x   w coskw z  x   x 0  x
(39)
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 y   w sin kw z  y   y 0  y
(40)
 z  || z  z   z 0  z
(41)
Then, rewrite Eqs. (36)–(38) to first order in perturbed quantities and their derivatives,
  x 0   0 x    x 0    0 wz sin kw z   0 r 1   z 0 cos  (42)
  y 0   0 y    y 0    0wz cos kw z   0r 1   z 0 sin  (43)
         cos k z   sin k z    
z0
0
z
0
w
y
w
  y 0 sin    x 0 cos  
x
w
0
r
(44)
eEr
,    0    , and
 mc
       03  x0x   y 0y   z 0z  x x0  y y 0
where  r 

Now, multiply Eqs. (42) and (43) by

(45)
 x 0  0 and  y 0  0 , respectively, and add the results to deduce
 2
   yo2    x0x   y 0 y  r  w 1   z 0 coskw z    (46)
 0 xo
3
The relations                  , which can
x0
x
y0
y
0
z0
z
x
x0
y
y0
 x20   y20  1   z20  1  02 , may be used in Eq. (46) to obtain


1   z20    z 0z  ck w  w  z 0 x sin k w z   y cos k w z 
0
  r  w 1   z 0  cosk w z   
Now the term

y
be obtained from Eq. (45), and
(47)
cos kw z  x sin kw z  may be eliminated between Eqs. (44) and (47) to obtain


  w 1   z20   ck w  w  z20   z   w  z 0  ck w  w  z 0    r  w
0
   w 1   z 0   ck w  w  z 0 cosk w z   
This equation is an important result which relates  to
derived as follows:
(48)
 z in the presence of radiation. Now a relation for  can be
e
β  Er    eEr  w coskw z      0r  w coskw z    (49)
mc
mc
Substitution of this value in Eq. (48) with some algebra will deduce  as
  
z  z   r  w coskw z   
z
(50)
Using a first-order approximation for z  v||t  c||t , Eqs. (49) and (50) may be written in the following form:
   0r  w cost   
(51)
z  z   r  w cost   
(52)
where   kr  k w v||  r .
The phase  in the above equation determines the initial position of the electron relative to the optical wave. Averaging
 over all phases yields
   
1
2

2
0
 d  0
(53)
Therefore, to first order there is no net transfer of power between the electron beam and the optical wave. The secondorder correction will consist of accounting for the fact that as an individual electron (with a phase  ) gains or loses in
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

energy, its position relative to the unperturbed position z  c||t is advanced or retarded. Therefore, the unperturbed
position, z  c||t , must be replaced by
zt   c||t  c   z t dt
t
(54)
0
t
where
 z t    z tdt is the change of  z relative to the unperturbed state. The substitution of z in Eq. (54)
0
will yield
cD
(55)
t sin   cost     cos 

where D  r  w 1   z 0   . Substitution of Eq. (55) in Eq. (49) will yield
z t   c||t 


   r 0  w cost    
r  w D t sin   cost     cos  (56)



w  kwc . Comparison of Eq. (56) with Eq. (51) reveals that the perturbed state consists of a phase slippage,
  r  D t sin   cost     cos 
  w
(57)
where

Since D is proportional to
 1 and Er , 
can be made arbitrarily small. Therefore, expanding the cosine term in Eq.
(55) for    leads to
r  w 



 t sin   cost     cos  
Averaging over phase  yields
  w  t cos t  sin t 
    r  0  w r
2
   r  0  w cost     sin t   
(58)
(59)
Now, integrating Eq. (59) over the electron transit time through the wiggler interaction length yields the average change
in  per electron:
    
T  L c z
0
where
g T  
   dt  r 0  w
r  w DT 3 g T 
2
2  2 cos T  T sin T 
3T 3
(60)
(61)
and where L is the FEL interaction length.
The change in radiation power in one transit is
I
P   mc 2    
(62)
e
2
where I  nb ev|| rb is the average electron beam current. By using Eqs. (60) and (62), under the assumption that the
electrons are near resonance with the wave, i.e.,   kr  k w c||  r  0 , the gain equation in the presence of the
self-fields can be rewritten as
3
 L  2
P
   g T 
Gs 
 b2w 
 c  w ||
P
 || 

(63)

where P  c rb E r 4 .
2
2
In the limit eliminating self-fields, the maximum gain Eq. (63) reduces to
3
 L  2
   g T 
Gs 0   w 
 c  w0 ||0
 ||0 
2
b
(64)
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where
 w0
and  ||0 are the wiggler-induced transverse velocity and the axial velocity in the absence self-fields,
respectively. If
i
and
 0 are both set equal to zero (eliminating the ion-channel and the axial magnetic field), Eq.
(63) reduces to
3
 L  2
 aw  0 g T 
G0   w 
 c 0 
2
b
where a w   eBw
(65)
 m k w c 2 and  0  1  a w2    2  .
12
Numerical Results
A numerical study of the self-fields effects on gain in a helical wiggler free-electron laser with axial magnetic field and
ion-channel guiding have been made. The parameters that are used in this section are k w  3.14 cm
1
,
nb  8  1012 cm 3 , Bw  1kG . The Lorentz relativistic factor  was taken to be 30, and the normalized axial
velocity
 || was taken to be close to 1. The FEL interaction length L was taken to be 100 w .
Figure 1 shows the variation of the normalized gain
Gs
G0  (solid lines) and value of the ratio of gain in the presence
of the self-fields to the gain in the absence of the self-fields
normalized guiding frequency with the two taken to be equal i
 Gs Gs 0  (dotted lines) as a function of either
kwc  0 kwc  , for group I orbits. Fig. 1 shows that
the normalized gain increases with increase the normalized guiding frequency. As shown in Fig. 1,  is less than 1 and
therefore the gain decrement is obtained due to the self-fields. Figure 2 shows the variation of the normalized gain
Gs G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of either normalized guiding
frequency with the two taken to be equal i
kwc  0 kwc  , for group II orbits. Fig. 2 shows that the normalized
kwc  0 kwc   1, the
condition  || close to 1, as required for derivation of the gain equation, breaks down. For group II orbits because  is
gain decreases with increase the normalized guiding frequency. For group II orbits if
i
greater than 1, the gain enhancement is obtained due to the self-fields.
Figure 1. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized axial magnetic field frequency that is equal to the ion-channel frequency, for group I orbits. The parameters are
k w  3.14 cm 1 , nb  8  1012 cm 3 , and Bw  1kG .
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Figure 2. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized axial magnetic field frequency that is equal to the ion-channel frequency, for group II orbits. The parameters are
k w  3.14 cm 1 , nb  8  1012 cm 3 , and Bw  1kG .
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized axial magnetic field frequency  0 k wc in the presence of an ion-channel, with i kwc  0.2 , for group I
Figure 3. The normalized gain
Gs
1
3
orbits. The parameters are k w  3.14 cm , nb  8  10 cm , and
12
Bw  1kG .
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Figure 4. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized axial magnetic field frequency  0 k wc in the presence of an ion-channel, with i
1
3
orbits. The parameters are k w  3.14 cm , nb  8  10 cm , and
Fig. 3 shows the normalized gain
Gs
12
kwc  0.2 , for group II
Bw  1kG .
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function
 0 k w c in the presence of an ion-channel when it is held constant
with i kwc  0.2 , for group I orbits. Fig. 3 shows that  is less than 1 and the gain decreases with considering the
of the normalized axial magnetic field frequency
self-fields. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of
the normalized axial magnetic field frequency
 0 k w c in the presence of an ion-channel when it is held constant with
i kwc  0.2 , for group II orbits, are shown in Fig. 4.
Figure 5. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized ion-channel frequency i k wc in the presence of an axial magnetic field, with
1
3
orbits. The parameters are k w  3.14 cm , nb  8  10 cm , and
12
0 kwc  0.2 , for group I
Bw  1kG .
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Figure 6. The normalized gain
Gs
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function of the
normalized ion-channel frequency  i k w c in the presence of an axial magnetic field, with
1
3
orbits. The parameters are k w  3.14 cm , nb  8  10 cm , and
Fig. 5 shows the normalized gain
Gs
ratio
0 kwc  0.2 , for group II
Bw  1kG .
G0  (solid lines) and the gain ratio  Gs Gs 0  (dotted lines) as a function
of the normalized ion-channel frequency
with
12
i k w c
in the presence of an axial magnetic field when it is held constant
0 kwc  0.2 , for group I orbits. For group II orbits, the normalized gain Gs G0  (solid lines) and the gain
 Gs Gs 0  (dotted lines) as a function of the normalized ion-channel frequency  i k w c in the presence of
an axial magnetic field when it is held constant with
0 kwc  0.2 , are shown in Fig. 6. It should be noted that the
foregoing result has been shown that the Budker condition eliminates the group II orbit in a FEL with ion-channel
guiding. While, when both the ion-channel and the axial magnetic field are present, there are two groups of electron
trajectories, because electron beam will be guided by the axial magnetic field.
Conclusion
In this work, the effects of self-fields on gain in a helical wiggler free electron laser with axial magnetic field and ionchannel guiding is investigated. The self-electric field is derived from Poisson,s equation and the self-magnetic field is
obtained from Ampere,s law by a self-consistent method. A detailed analysis of electron interaction with radiation field
in the presence of self-fields is presented.
It should be noted that the wiggler-induced self-magnetic field decreases the effective wiggler magnetic field for group I
orbits, whereas for group II orbits it increases the effective wiggler magnetic field. Therefore, we expect that in the
presence of self-fields the gain decreases for group I orbits and increases for group II orbits.
The results of the present one-dimensional analysis are valid for v w v||  1 . When v w is large the radial variation of
the wiggler field can no longer be neglected and the present results are not valid. The radial variation of the wiggler field
is also negligible for thin beams for which the beam radius is much smaller than the wiggler wavelength.
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