Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
Theoretical approach of the photoinjector exit aperture
in#uence on the wake "eld driven by an electron beam
accelerated in an RF gun of free-electron laser `ELSAa
Wa'el Salah *, J.-M. Dolique
Amman College for Engineering Technology, Balq+a Applied University, P.O. Box 15008, Amman 11134, Jordan
Commissariat a% l +energie atomique } PTN, BP 12, 91680 Bruye% re-le-chaL tel, France
Received 20 April 1999; received in revised form 28 October 1999; accepted 20 November 1999
Abstract
The wake "eld generated in the cylindrical cavity of an RF photoinjector, by a strongly accelerated electron beam, has
been analytically calculated (Salah, Dolique, Nucl. Instr. and Meth. A 431 (1999) 27) under the assumption that the
perturbation of the "eld map by the exit hole is negligible as long as the ratio: exit hole radius/cavity radius is lower than
approximately 1/3. Shown experimentally in the di!erent context of a long accelerating structure formed by a sequence of
bored pill-box cavity (Figuera et al., Phys. Rev. Lett. 60 (1988) 2144; Kim et al., J. Appl. Phys. 68 (1990) 4942), this
often-quoted result must be checked for the wake "eld map excited in a photo injector cavity. Further, in the latter case,
the empirical rule in question can be broken more easily because, due to causality, the cavity radius to be considered is
not the physical radius but that of the part of the anode wall around the exit hole reached by the beam electromagnetic
in#uence. We present an analytical treatment of the wake "eld driven in a photoinjector by the accelerated electron beam
which takes this hole e!ect into account, whatever the hole radius may be. 2000 Elsevier Science B.V. All rights
reserved.
Keywords: **ELSA'' photoinjector; Wake "eld; Perturbation due to the exit aperture
1. Introduction
For some years, the RF photoinjector has been
known as a source of low-emittance and highbrightness electron beam. This quality is an
essential parameter of the free-electron laser [1,2].
Emittance growth is due to the electromagnetic
reaction of the acceleration structure walls with the
beam-generated "eld } that is, the so-called wake
"eld which generally results in diminished perfor* Correspondence address. P.O. Box 2059, Russaifeh 13710,
Jordan. Fax: 962-6-4894-292.
mance of the free-electron laser [3]. Wake "elds are
usually considered for ultrarelativistic coasting
beam [4}10]. The electromagnetic response of
a discontinuous conducting wall to an exciting
charged particle is experienced only by the particles
located downstream, in the wake. Furthermore, the
self-"eld, or space-charged "eld, is negligible so that
the only forces acting on a beam particle are the
wake "eld and a possible focusing force. At the end,
the beam is assumed to be coming from !R and
going to #R.
For a photoinjector, the situation is very di!erent. The beam is strongly accelerated from thermal
0168-9002/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 3 1 0 - 8
310
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
to transrelativistic velocity. If intense, its self-"eld
must be taken into account and added to the cavity
response [11]. In addition, an electron beam appears at the cathode surface at t"0, the time when
laser illumination begins, so that causality governs
both synchronous and radiation "elds.
The photoinjector considered is the `ELSAa facility [11] (CEA, Bruyère-le-Cha( tel) schematised in
Fig. 1 which works at 144 MHz.
The wake "eld thus de"ned has been analytically
calculated for a pill-box photoinjector model [11].
Radially, this modelling is relevant because the
only parts of the photoinjector RF cavity walls that
the beam "eld has time to reach, during beam
acceleration, are located not far from the axis on
both cathode and anode. At the anode, however,
the question of the exit hole in#uence arises. This
has been neglected in Ref. [11] by applying an
empirical rule [12,13] according to which the hole
in#uence is negligible as long as r /R(1/3, where
r and R are the hole and cavity radii, respectively
(Fig. 1). Though often quoted, this rule is based on
a single experimental study worked out on an ultrarelativistic coasting beam crossing a set of bored
cavities. Its validity for the low to transrelativistic
energy beam accelerated in a photoinjector is questionable.
The aim of this paper is to investigate theoretically the aperture e!ects on the above-de"ned
wake "eld.
2. Dynamics of electron beam
The velocity and the acceleration of electrons are
obtained by using the fundamental relation of dynamics under relativistic form [11]:
dp
F"
dt
(1)
where p"Cm* with ! being the relativistic mass
factor.
During the beam dynamics, the following hypothesis is supposed:
1. The amplitude E of the accelerating "eld of
the RF cavity which works at 144 MHz may be
considered as constant for beam pulse length q
of the order of 100 ps [14].
2. The current density remains constant during the
photoemission [14].
3. Paraxial approximation is used for the beam
dynamics; the electrons displaced parallel to the
axis of symmetry.
Under these conditions, the velocity b(z, t) and
the acceleration c(z, t) will be parallel to E and
independent of time:
b(z, t)"b(z)u
X
c(z)"c(z)u
X
with
(2)
((1#Hz(t))!1
b(z)"
1#Hz(t)
(3)
c(z)"1#Hz
(4)
1
z(t)"
(1#(Hc(t!t ))!1
X
H
Fig. 1. ELSA photoinjector (144 MHz cavity).
mc
H\"
eE
(5)
(6)
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
311
where e and m are the charge and mass of the
electron, respectively, c is the velocity of light, z(t) is
the longitudinal coordinate of the electron at time t,
and t is the time at which the element z of the beam
X
leaves the photocathode.
3. Expansion of the zone of beam electromagnetic
in6uence
Before trying to calculate the map (E, B)(X, t) of
the beam-generated, time-dependent electromagnetic "eld, in the presence of the cavity conducting
walls, one has to know which parts of these cavity
walls are able to play a role in the "eld map building. Owing to causality, three phases have to be
distinguished. In the "rst, t(g/c (where t"0 corresponds to the beginning of the photoemission)
the beam-generated electromagnetic "eld has not
yet reached the exit wall (anode), the situation is
schematised in Fig. 2. In the second, g/c(t(t
E
(where t is the time at which the beam head
E
reaches the photoinjector exit) the beam-generated
electromagnetic "eld has reached the exit wall, but
Fig. 3. Second phase: g/c(t(t (200(t(250 ps for the
E
above parameter).
the beam head is still above the exit aperture
(Fig. 3). In the third, t (t(t (where the beam
E
OE
t is the time at which the beam back reaches the
OE
photoinjector exit) the beam penetrates the exit
aperture. This phase will be studied later.
For the photoinjector of `ELSAa facility,
E "30 MV/m, q"30 ps, g"6 cm, these three
phases correspond to: t(200 ps, 200(t(250 ps
and 250(¹(280 ps, respectively.
3.1. First phase
The situation is schematised in Fig. 2. The
(E, B) (X, t) "eld map is the same as that already
calculated in this phase for the closed cavity [11].
3.2. Second phase
Fig. 2. First phase: t(g/c, where t"0 corresponds to the
beginning of photoemission (t(200 ps for the above parameters).
In this phase, we have to solve Maxwell's equations in the unbounded domain of Fig. 3. There is
no rigorous analytical solution which would take
into account the boundary conditions of the "rst
step, as well as
U(r )r)R, z"g, t)"0
(7)
312
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
axial current pro"le - (z, t) is uniform, with sti!
front and back, and time length q.
According to symmetries, the only non-vanishing components, in cylindrical coordinates, are
E , E , and B . For B the following condition must
X P
F
F
be satis"ed:
1 *
(rB )
F
r *r
A (r )r)R, z"g, t)"0
P and
(8)
where U and A are the scalar and vector potentials,
respectively.
However, a good approximation is obtained by
assigning these conditions to the general solution in
the domain of Fig. 4. This general solution is the
sum of synchronous and radiation "elds.
3.2.1. Synchronous xeld
This is the particular solution of Maxwell's equations which corresponds to the beam right-hand
sides o(r, z, t) and J(r, z, t) in Lorentz gauge:
1 *U
e. A#
"0
c *t
(13)
"0
P0
which imposes A (R)"0.
X
The solution of Eq. (11) can be written as a development of Dini series of order zero:
Fig. 4. The unbounded domain.
*A
X (r )r)R, z"g, t)"0
*z r
A (r, z, t) " a (z, t) J j
X
L
L R
L
2
r
a (z, t) "
dr.
A (r, z, t) r J j
L
J ( j )R
X
L R
L
(14)
Substituting Eq. (14) into Eq. (11) and using
Fourier's transformations and Green's function
techniques leads to
r
a
J j
J j
4k Ic L R L R
A (r, z, t) " X
paR
j J ( j )
L L
L
cos (hz)
;
h(h#( jL /R)
(9)
o
䊐U(r, z, t)"
(10)
e
䊐A(r, z, t)"k j
(11)
where 䊐 is the Alembertien operator, o(r, z, t) and
j(r, z, t) are the current and charge density, respectively:
I- (z, t)
o(r, z, t)"
[1!H(r!a)]
pab(z)c
j(r, z, t)"b(z)co(r, z, t)u
(12)
X
describe a radially uniform beam of radius a and
velocity v(z)"b(z)c, carrying a total current I, is
assumed to be radially uniform with radius a. Its
;
R
j sin c h# L (t!t)
R
;[sin(hz (t))!sin(hz (t))] dt dh
D
O
(15)
where I is the total current carried by the beam, a is
the beam radius, j is the nth zero of Bessel function
L
J , and z (t) and z (t) are the coordinates of the
O
D
beam's back and head, respectively:
1
z (t)" ((1#(Hct)!1)
D
H
1
z (t)" ((1#(Hc(t!q))!1).
O
H
(16)
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
By the gauge of Lorentz (Eq. (9)) we get the scalar
potential:
j
a
LR
r
4I
J j
J
U(r, z, t) "
L R j J (j )
ne aR
L L
L
;
sin (hz)
h#( j /R)
L
;
R
j (1!cos c h# L (t!t)
R
;[sin(hz (t))!sin(hz (t))] dt dh.
D
O
(17)
3.2.2. Radiation xeld
This is a general solution of homogeneous Maxwell's equation. Taking the boundary conditions
into account, this general solution can be written in
the form
r
A (r, z, t) "A J j
X
LR
L
cos(hz)C (h)
L
j ;exp ic h# L t dh
R
(18)
where A "k IR and C (h) is an arbitrary di
L
mensionless function:
r
U(r, z, t) "U R J j
LR
L
sin(hz)CU (h)
L
j ;exp ic h# L t dh
R
(19)
where U "1/e c and CU (h) is an arbitrary dimen
L
sionless function.
By the gauge of Lorentz (Eq. (9)) we get
A
i
C (h)"
R L
c
j h# L
U CU (h)
L
R
where i"(!1.
(20)
313
3.2.3. Calculation of C (h) and CU (h)
L
L
These functions are expanded in Laurent series:
C (h)" C
(h R)\K
(21)
L
LK
K
CU (h)" CU (h R)\K.
(22)
L
LK
K
To calculate C
(h) and CU (h), the above
LK
LK
boundary condition (Eq. (7)) is written for
U(r, z, t)"U(r, z, t) #U(r, z, t) and integrated
on r in its validity domain: r )r)R. One is led to
an in"nite system of linear algebraic equations:
K (t) C
"¸(t)
LK
LK
LK
where
(23)
r
K (t)"RJ j LK
L R
h sin(hg)
(hR)K (h#( jL /R)
j ;exp ic h# L
t dh
R
(24)
r
r
J j
J j 4ic L R L R
¸(t)"
j J (j )
paR
L L
L
sin (hz)
;
h#( j /R)
L
;
R
j cos c h# L (t!t)
R
;[sin (hz (t))!sin (hz (t))] dt dh.
D
O
(25)
A good approximate solution is found by truncating the system to a "nite size: n)N, m)M, with
N&100 and M)10. The 2N(M#1) unknowns
are then calculated by numerically solving the system of 2N(M#1) equations:
K (t ) C
"¸(t ), i3[1, 2,2, N M] (26)
LK G LK
G
LK
C (h) and CU (h) being calculated, U(r, z, t) and
L
L
A(r, z, t) are deduced, from which the "elds result.
314
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
4. Chosen parameters
The photoinjector of `ELSAa facility (CEA,
PTN, Bruyeres-le-Chatel) is taken as an example. It
has a wide range of possible working parameters.
The chosen parameter set for some sample wake
"eld maps represented here under is: I"100 Am,
pa"1 cm (a is the radius of the beam)
E "30 MV/m, r "2 cm (r is the exit aperture
radius) and q 30 ps.
5. Results
The electric and magnetic "elds can be obtained
from the rebuilt potentials. Reduced coordinates
and quantities will be used, based upon the characteristic length H\"mc/eE , where e and m are
the electron charge and mass, respectively, and
E the RF electric "eld amplitude on the photo
injector cathode.
According to symmetries, the only non-vanishing components, in cylindrical coordinates, are
E , E , and B .
X P
F
There are two cases where the "eld maps obtained by the method described in this paper must
be identical to those previously obtained for
a closed cavity where analytical expressions of
the (E, B)(x, t) map have been obtained as development on the complete base cavity normal
modes [11]:
Fig. 5. E on the beam axis, as given by the closed- or openX
cavity modelling respectively, for t"t /4.
E
(a) in the above `"rst phasea (t(200 ps);
(b) when the aperture radius r P0 vanishes.
These two tests have been successfully carried
out, as shown in Figs. 5}7 given as examples.
Fig. 8 shows E on the beam axis, as given by the
P
closed-or open-cavity modelling, respectively, for
t"t "250 ps (t is the time at which the beam
E
E
head reaches the photoinjector exit z"g),
E "30 MV/m, r "2 cm (r is the exit aperture
radius) and q"30 ps. It emphasizes the strong in#uence of the exit aperture.
Fig. 9 shows B on the beam axis, as given by
F
the closed cavity modelling t"t "250 ps,
E
E "30 MV/m, r "2 cm (r is the exit aperture
radius) and q"30 ps.
Fig. 6. B on the beam axis, as given by the closed- or openF
cavity modelling, respectively, for t"t /4.
E
6. Conclusion
The wake "eld generated in the cylindrical cavity
of an RF photoinjector, by a strongly accelerated
electron beam, has been analytically calculated
[11] under the assumption that the perturbation of
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
Fig. 7. E on the beam axis, as given by the closed- or openP
cavity modelling, respectively, for t"t and r "0 cm.
E
315
Fig. 9. B on the beam axis, as given by the closed- cavity
F
modelling for t"t "250 ps and r "2 cm.
E
for the wake "eld map excited in a photoinjector
cavity. Further, in the latter case, the empirical rule
in question can be broken more easily because, due
to causality, the cavity radius to be considered is
not the physical radius but that of the part of the
anode wall around the exit hole reached by the
beam electromagnetic in#uence. We present an
analytical treatment of the wake "eld driven in
a photoinjector by the accelerated electron beam
which takes this hole e!ect into account, whatever
the hole radius may be.
Acknowledgements
Fig. 8. E on the beam axis, as given by the closed- or openP
cavity modelling, respectively, for t"t "250 ps and
E
r "2 cm.
the "eld map by the exit hole is negligible as long as
the ratio: exit hole radius/cavity radius is lower
than approximately 1/3. Shown experimentally in
the di!erent context of a long accelerating structure
formed by a sequence of bored pill-box cavity
[12,13], this often-quoted result must be checked
We wish to thank Prs. J.-P. Girardeau-Montaut
of Claude Bernard university, Lyon I. France, and
J.A. Riche of CERN, Geneva for helpful discussion
of these results.
References
[1]
[2]
[3]
[4]
[5]
Ph. Guimbal et al., Nucl. Instr. and Meth. A 341 (1994) 43.
J.-M. Madey et al., Phys. Rev. Lett. 38 (1977) 892.
S. Joly et al., Nucl. Instr. and Meth. A 341 (1994) 386.
Zohreh Parsa, Nucl. Instr. and Meth. A 318 (1994) 259.
K. Nakajima et al., Phys. Scr. T 52 (1994) 61.
316
[6]
[7]
[8]
[9]
[10]
W. Salah, J.-M. Dolique / Nuclear Instruments and Methods in Physics Research A 447 (2000) 309}316
Y. Nishida et al., Phys. Scr. T 52 (1994) 65.
P.K. Shukla, Phys. Scr. T 52 (1994) 73.
A. Ogata et al., Phys. Scr. T 52 (1994) 69.
D.A. Johnson et al., Phys. Scr. T 52 (1994) 77.
R.K. Cooper, Part. Accel. 12 (1982) 1.
[11] W. Salah, J.-M. Dolique, Nucl. Instr. and Meth. A 431
(1999) 27.
[12] H. Figuera et al., Phys. Rev. Lett. 60 (1988) 2144.
[13] Kim et al., J. Appl. Phys. 68 (1990) 4942.
[14] R. Deicas et al., Nucl. Instr. and Meth. A 318 (1992) 121.