Analytic and Quasi-Analytic Solutions of
Wake-Fields
Samer Banna, David Schieber and Levi Schachter
Department of Electrical Engineering, Technion—HT, Haifa 32000, ISRAEL
Abstract. The analysis of wake-fields generated by moving charges in the vicinity of dielectric and
metallic bodies is essential for the design of the next generation of optical acceleration structures.
Recently, we have investigated analytically the wake-field associated with three different geometries. I. In the case of an electron bunch moving parallel to a dielectric cylinder, it was shown that
for the relativistic regime (y >> 1) the circular harmonic of order zero contributes a decelerating
force inversely proportional to y, whereas the circular harmonics of nonzero order contribute a yindependent force. II. For a line charge moving in the vicinity of a dielectric cylinder, it was shown
that the emitted energy increases logarithmically with the kinetic energy (7— 1) of the line charge
and its transverse kick is inversely proportional to the kinetic energy for the ultra-relativistic case.
III. When an electron-bunch traverses a geometric discontinuity a time-domain method was developed for the evaluation of the wake-field employing the independence of the exponential functions,
that control the temporal behavior of the field.
INTRODUCTION
The analysis of wake-fields generated by moving charges in the vicinity of dielectric
and metallic bodies is essential for the design of the next generation of optical acceleration structures. On the one hand, such an analysis contributes to the evaluation of the
deceleration force facilitating an appropriate design to minimize this force and ensure
efficient acceleration process. On the other hand, it enables to determine the impact of
beam characteristics and geometrical parameters of the structure on the wake-field, as
the latter may be responsible to the increase of the beam emittance and/or excitation of
high order modes. Moreover, the bunch's stability depends mainly on its interaction with
the wake-fields generated by particles moving ahead of it [1].
Throughout the years, many studies have focused on the analysis of electromagnetic
wake-field generated by a relativistic bunch of particles in different waveguide structures
[2-9]. However, due to the complexity of the problem there are only a few analytic
or quasi-analytic solutions. Such solutions contribute to have better understanding of
the wake-fields properties. Recently, Schachter and Schieber [7] have investigated the
deceleration force acting on a point-charge moving in a vacuum tunnel bored in a
dielectric or metallic medium. A similar investigation has been done [8] in the case
of a point-charge moving in free space above a dielectric half-space. In addition, they
have examined [9] the maximum gradient that acts on a line charge moving above an
open periodic structure of arbitrary geometry.
We present here a quasi-analytic expressions for the wake-field in the frequency and
time-domain in three different configurations where metallic or dielectric structures
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
516
are involved. In the first case, the wake-field of an electron bunch moving parallel
to a dielectric cylinder is considered [10]. In the second, the radiation characteristics
generated by a relativistic line charge moving in the vicinity of a dielectric cylinder is
determined [11]. A time-domain solution for the wake-field of an electron bunch moving
in a cylindrical waveguide consisting of geometrical discontinuity is obtained in the third
case [12]. The analysis in all three cases relies on two assumptions: firstly, an "external"
force is acting on the moving charge keeping its velocity constant or equivalently the
energy transferred to radiation is negligible comparing to the initial kinetic energy of the
bunch. Secondly, the transverse motion of the moving charge is neglected.
The topics are organized as follows: in the next section a general approach to obtain an
analytic or quasi-analytic solution for the wake-field of a moving charge in the vicinity
of dielectric or metallic bodies, is presented. This will be followed by introducing the
wake-field calculations in three problems as mentioned above. In the last section, we
summarize the main analytic results corresponding to the different problems.
APPROACH TO SOLUTION
An electron bunch moving in a free space generates a current density / parallel to the
direction of its motion. As this current density is confined to one direction and no additional currents are excited, it suffices to consider only one component of the magnetic
vector potential A^p\ superscript p indicating this is to be the primary field. This potential satisfies the non-homogeneous wave equation. The next step in the solution is
to determine the secondary field (superscript s) due to the presence of an obstacle, e.g.
a dielectric cylinder or an azimuthally symmetric discontinuity in a cylindrical waveguide. The secondary field satisfies the homogeneous wave equation. Both secondary and
primary fields together must satisfy the boundary conditions. Based on the constraints
imposed by the boundary conditions the explicit expression for the secondary wake-field
is achieved, and accordingly, one may evaluate the total energy emitted by the moving
bunch. In the following sections some components of the primary field in each configuration are presented followed by the secondary field and brief analysis of its impact on
the bunch.
A POINT CHARGE MOVING PARALLEL TO A DIELECTRIC
CYLINDER
Consider a dielectric (er) cylinder of radius R. The axis of a cylindrical coordinate system
(r, 0, z) coincides with that of the cylinder. Parallel to this axis, at a radius r = h > R and
at an angle 0 = 00, a point charge is moving at a velocity v0 - Figure 1. At the surface
of the cylinder, the longitudinal component of the primary electric field is given by
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FIGURE 1. Point charge moving parallel to a dielectric cylinder.
(1)
wherein r = M/cy]8, ]8 = v0/c, 7= [1 -jS2]'1/2, av = -(q^/4n2)Kv(rh), and
/ v (£), ^TV(^) a16 modified Bessel functions of order v, of the first and second type
respectively.
The presence of the cylinder alters the field distribution in the whole space. This
change is represented by the so called secondary field whose longitudinal components
read
BvJv(Ar)
r<R
DvJv(Ar)
r<R
(2)
(3)
wherein A = |co| >/er — l/j32/c; / v (<^) is the Bessel function of the first kind and order
v. Formulation of the boundary conditions at r = R requires, in addition to Eqs. (l)-(3),
also the evaluation of the azimuthal components of the electromagnetic field that may
be derived from the longitudinal components. Continuity of the tangential components
imposes four conditions required in order to determine the four unknown amplitudes
A v , # v , Cv and Dv. Being interested only in the longitudinal reaction force acting on the
point charge, we shall state here only the explicit expression of the scattered amplitude
of E^\ namely,
vy 2
2
JvKvIv
j(oav
r yy /
1
+
["0 V
2
r°lvJv + £r
-[
°
W
°ir°
i
tfvrrIvJv\ lKv/v+ rfj$KvJv
r
°
° If
Jfr ^ H
(4)
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where J v (£) stands for the derivative with respect to the argument £ of the Bessel
o
function of the first kind and order v; K v (<^) stands for the derivative of the modified
Bessel function of the second kind, order v also with respect to % and Q = G)R/c,
2
, /v =
Now, we can proceed one step further and calculate the electric-field along the path
of the charged particle <?(T = t — z/v0) = E^\r = h: (j) = 0 0 ,z,f). For this purpose, we
split the expression into two parts: a) that comprising the circular harmonic of order zero
(v = 0); b) all the remaining harmonics (v ^ 0).
In the limiting case 7^> 1, and for very large radius of curvature for the cylinder i.e.
R^>h — R and h^>h — Rwe may approximate the decelerating force contributed by the
circular harmonic of zero order to read
where A = q/2nh denotes the charge per-unit length. This expression clearly indicates
that the decelerating force is inversely proportional to the particle's momentum and to
the distance of the charge from the cylinder surface (h — R). This runs contrary to the
result occurring for the case of a point charge moving inside a symmetric tunnel bored in
a dielectric material, in which case the decelerating force for y^ 1, is y-independent [7].
However this is almost exactly the expression for the decelerating field acting on a line
charge (A) moving at a hight A from a dielectric half-space [10] - this would correspond
to a distance h — R in the case investigated here.
Regarding the same assumptions mentioned above, the decelerating electric field for
the nonzero circular harmonics reads
This expression is identical with the reaction field [8] on a point charge moving at a
height A = (h — R) above a dielectric half-space when y ^> 1 which explicitly reads
x2.
(7)
The last two expressions clearly imply a y-independent decelerating field for very high
y. Moreover, the high-order (v > 0) wake behind the particle is independent of the
dielectric coefficient.
A LINE CHARGE MOVING IN THE VICINITY OF A
DIELECTRIC CYLINDER
Consider a line charge carrying the charge per unit length A [C/m] and moving in free
space (CQ, jU0) at a constant velocity v0 above an infinite dielectric cylinder of radius R
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and relative dielectric coefficient er. The axis of a cylindrical coordinate system (r, 0,z)
coincides with that of the cylinder. The line charge is located at a distance y = h from
the cylinder's axis and moves along the .x-axis, as illustrated in Figure 2. The primary
magnetic field, at the dielectric cylinder surface i.e. at r = R, in a cylindrical coordinate
system, reads
co
co .
co
-j — ACOS0 + — /csm<p± — — 1h
(8)
47T
Accordingly, the secondary magnetic field in cylindrical coordinate has the following
form:
r>R
(9)
r<R,
where H^(Q is the nih order Hankel function of second kind.
Applying the orthogonality of the azimuthal harmonics, the boundary conditions at
r = R entail a set of two algebraic linear equations that in a matrix form read
(10)
-Hn wherein H® (£)
——Vn
\ 7*lo
- e Vo7
is the derivative of
and
= — e vo^ , with Un and Vn are given by the following terms
4n
(H)
With the explicit expression for the secondary electromagnetic field, it is possible to
evaluate the total energy emitted by the line charge due to the wake. Using the following
normalized quantities,
= — h, An = ——,— ,
c
A/47F
energy is given by the following set of equation
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——
2 ,—— , the normalized emitted
A /7re0
FIGURE 2.
Line charge moving in the vicinity of a dielectric cylinder
tff) (of) y,
/„
(12)
Of special interest is also the spectrum of the radiated power or the longitudinal
impedance denned as
1
(13)
Using the normalized quantities notation the two are related
cm
ii
(14)
where Z,, is the normalized longitudinal impedance, defined by Z,, = ZJ2r\Qh. Representative curves of Z,, as a function of Q are presented in Figure 3. Simulations reveal
the following properties of Z,,:
• Main peak of the spectrum (Z k) increases with er while the frequency of the main
peak (&peak) decreases.
• Main peak location shifts towards higher frequencies when the line charge gets further
away from the cylinder surface (Qpeak ^ \/k/R) but at the same time its intensity is
significantly reduced (Z k ^ R/h).
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1.0
2.0
er=2
yfi= 1000
0.8
1.5
= 0.9
0.6
1.0
0.4
0.5
0.2
0.0
25
50
75
100
0.0
0
25
50
75
100
n
FIGURE 3.
Representative curves of Z,, versus Q.
• Main peak of the longitudinal impedance increases with y/3 for a given values of er
and h/R. The increase is proportional to (y/3)1/3.
• Width of the spectrum of the main peak increases as y/3 increases.
• Spectrum of the emitted energy has many secondary peaks; their number (and frequency) increases with the increase of y/3. For the ultra-relativistic case the contribution of these peaks is dominant.
• For high frequencies i.e. (QR/h » 1), the longitudinal impedance is inversely proportional to Q.
In conclusion, the frequency where the spectrum reaches its peak value increases as each
one of the parameters (y/3 , er and h/R) increases. By changing one of these parameters,
it is possible to design a system that will radiate energy whose spectrum will peak at a
desired wavelength.
As the spectrum of the emitted energy was analyzed it is possible to proceed one step
further and investigate the total emitted energy. For this purpose different simulations
were performed and the main conclusions were:
• The emitted energy for intermediate and ultra-relativistic energies may be approximated by
W~aln[b(y-l)]
(15)
wherein a and b are constants for given values of er and R/h. These constants have
different values for intermediate as well as for ultra-relativistic energies. For example,
in the ultra-relativistic regime (y > 1000), for er = 2 and R/h = 0.1: a = 1.5 and
b = 4x 103, while for intermediate energies (2 < y < 250): a = 1 and b = 0.05 for
the same values of er and R/h.
The emitted energy decreases rapidly as h/R is increased. This clearly as the line
charge is closer to the dielectric cylinder, it experiences a larger secondary field, and as
a result, a stronger deceleration force acts on the line charge. For intermediate energies
(5 < y < 100) the dependence of the emitted energy on the geometric ratio h/R (for
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given values of 7 and er) may be, roughly approximated by
- -h
W~a-e~ R
(16)
wherein a and b are constants (for given 7 and er). Specifically, for er = 4 and 5 < 7 <
100 the values of these constants vary between 5 < a < 15 and 3x 10~4 < b < 0.3.
To quote an example, for 7= 5 and er = 4, a=5 and £=0.3. For the ultra-relativistic
regime, the emitted energy decreases linearly as a function of h/R.
• The emitted energy is almost independent on er for high values of 7/3.
In addition to the longitudinal effect which naturally is translated into deceleration
force, there is also the transverse "kick" (APy) from the structure to the charge. For ultrarelativistic energies simulations indicate that the transverse kick is almost independent
of the values of R/h and er while for intermediate energies, it increases as the line
charge approaches the dielectric cylinder. This fact may be understood in terms of larger
attraction force acting on the line charge by its image in the cylinder. Secondly, for ultrarelativistic energies the transverse kick is inversely proportional to 7. The behavior of
APy in this case may be approximated by
A2AZ7]0 3
AP
~
——^x-.
V
y
2n
7
(17)
Dielectrics posses an important feature that can be used in accelerator structure
namely the frequency dependence of the dielectric coefficient. Accordingly, one may
design an accelerator whose operating frequency will be significantly smaller than the
first main peak of the wake generated by the bunch. If the system is further designed
such that at a frequency Qc the dielectric coefficient drops from er to unity, then the
contribution of frequencies greater than Qc to the decelerating force, is virtually zero.
Consequently, a significant fraction of the energy in the wake may be suppressed. A
clear picture is revealed when examining the spectrum of the radiated energy represented
here by the longitudinal impedance Z,,. Consider a system driven by a 1.06jUm laser; if
the dielectric material is frequency independent then the first main peak of the wake's
spectrum occurs at lOOnm. Choosing a material that at a certain wavelength, say at
200nm, its dielectric coefficient drops to unity, the energy of the wake could be reduced
significantly (about 50 percent smaller than the case of a frequency independent er). For
the ultra-relativistic regime 7/3 = 2000, £r = 4, and R/h = 0.5_the emitted energy as a
function of Qc may be roughly approximated by W ~ 0.6
A BUNCH TRAVERSING A GEOMETRIC DISCONTINUITY
Consider a point charge (q) moving at a constant velocity v0 along the axis of a
cylindrical waveguide consisting of two uniform cylindrical waveguides with radii Rl
and R2 respectively, as illustrated in Figure 4. The location of the discontinuity (due
to the radii mismatching) is chosen as the reference point for the z-axis (z = 0) of a
cylindrical coordinate system.
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2R2
/?1
FIGURE 4. Point charge approaching a geometric discontinuity.
The primary field (superscript p) may be derived from the magnetic vector potential
in the time-domain is given by [13]
r<0, z<0
(18)
r>0, z>0
I av2J(,{pv—)e
V=l
^2
otherwise,
J\ •
Knowing the explicit form of the primary solution as presented in (18) is critical for
the overall solution since the boundary conditions have to be satisfied at any time and
consequently, the primary field determines the temporal behavior of all secondary field
components. It is important to point out that the primary solution is not defined at t = 0,
therefore from the perspective of the source, our solution reflects two time intervals:
the first, —oo < t < 0, when the particle is in the left hand side of the discontinuity,
whereas during the second time interval, 0 < t < oo, the particle is in the right hand side.
Moreover, it is important to emphasize that the solution introduced here describes only
the evanescent field contribution in the region of the discontinuity.
Let us focus now on the solution during the time interval when the particle is along the
left hand side of the discontinuity (t < 0). As already indicated, the time-dependence on
both sides of the discontinuity is dictated by the source and it has the form: exp(Q5 lt).
At this point the only quantity that needs to be determined is the longitudinal variation of
the secondary field since the transverse variation is identical to that of the primary field.
In the left hand side (z < 0) the longitudinal variation of the secondary field is dictated
by: (i) the transverse boundary condition [/0(/V/^i)L (u) the time variation imposed by
the primary field [exp(Q5 x f)], (Hi) the wave equation and (iv) the convergence condition
at z —»• —oo. Consequently, the field in the left hand side is, expf Q 51vz J. Similarly, in
the right hand side (z > 0) the transverse variation is dictated by a different geometric
parameter [J0(pvr/R2)] which in conjunction with the time variation, the wave equation
and the condition for convergence at z —> oo entail the following longitudinal behavior:
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exp — z\ (^M + (^R )
. Consequently, the secondary magnetic vector potential
reads,
Z<0
z>0.
(19)
The two sets of amplitudes, T5jV and p5, may now be determined by imposing the
boundary conditions at z = 0 for the time interval — °° < t < 0. Explicitly, applying
the orthogonality properties of the Bessel function and, most importantly, bearing in
mind that the functions describing the time-dependence, i.e. exp(Q 5l r), are linearly
independent functions for any value of s. It is important to emphasize that no matrix
inversion was necessary for establishing these amplitudes as it is the case for solutions of
similar problems in the frequency-domain. This fact reduces dramatically the complexity
of the numerical calculation.
A similar approach may be employed for the second time interval when the particle
is in the right hand side of the discontinuity (t > 0). The primary field dictates the timedependence, exp(—Qv 2r) thus
z<0
z>0.
(20)
As clearly revealed by the equations for the secondary magnetic vector potential, the
field is evanescent therefore it does not contribute directly to radiation. However, the
latter is a result of the acceleration or deceleration as the charge approaches or leaves
the discontinuity. An indication regarding the energy exchange is possible based on the
secondary longitudinal electric field since it enables us to determine the energy required
to maintain the charge in uniform motion namely, W = /Too & I dv/zEz(sec). This quantity
converges only if the electron bunch has a finite width (Rb) and finite length (Az). A
simple manifestation of the analytic method presented above is the numerical evaluation
of the energy (W) required to maintain the relativistic bunch at a constant velocity. This
energy may be approximated by
2n£QR}
(-;) (i
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This approximation is valid for the following set of parameters (I.) 0 < Az < 1, (II.)
0.1 < Rb/Ri < R2/R\ < 1an^ (HI-) 7> 10- The last approximation is convenient since
it illustrates the divergence of the energy for the case where Rb = Q (i.e. a point-charge)
and vanishes when the waveguide is uniform (i.e. Rl = R2). It indicates that the energy
is inversely proportional to the radial width and is linearly dependent on the ratio R2/R\ •
Another fact that may be revealed from this expression is that the energy vanishes as the
bunch moves at ultra relativistic velocity i.e. j^z/Rl ^> 1 indicating that as the bunch
moves faster it almost does not experience the presence of the discontinuity.
CONCLUSIONS
We have established several analytic and quasi-analytic solutions of the wake-fields in
three different geometries. In the first geometry the wake-field of an electron bunch moving parallel to a dielectric cylinder was considered. It was shown that for a relativistic
bunch (7^> 1) the circular harmonic of order zero contributes a decelerating force inversely proportional to y, whereas the circular harmonics of nonzero order contribute
a /-independent force. The second geometry considered was a line charge moving in
the vicinity of a dielectric cylinder. Among the important results demonstrated were the
logarithmic increase in the emitted energy as a function of the kinetic energy (7— 1) of
the line charge as well as the dependence of the transverse kick on this quantity. Finally,
a time domain solution of the wake-field of an electron-bunch approaching a geometric
discontinuity in a metallic waveguide was established.
ACKNOWLEDGMENTS
This study was supported by the United-States Department of Energy and by Israel
Science Foundation.
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