Computation of Wakefields in Arbitrarily Cross-Shaped Waveguides for Particle Accelerator Applications M. Jim´enez Universidad Polit´ecnica de Cartagena - Campus Muralla del Mar-Antigones, 30202 Cartagena (Murcia), Spain
S. Marini Dpto de F´ısica, Ing. de Sistemas y Teor´ıa de la Se˜ nal - Universidad de Alicante, 03690 San Vicente del Raspeig, Alicante, Spain
B. Gimeno∗ Dpto de F´ısica Aplicada y Electromagnetismo - ICMUV, Universidad de Valencia c/ Dr. Moliner, 50, 46100 Burjasot (Valencia), Spain
´ A. Alvarez, F. Quesada Universidad Polit´ecnica de Cartagena - Campus Muralla del Mar-Antigones, 30202 Cartagena (Murcia), Spain
V. E. Boria, P. Soto, S. Cogollos Departamento de Comunicaciones - ITEAM, Universidad Polit´ecnica de Valencia - Camino de Vera s/n, 46022 Valencia, Spain
D. Raboso European Space Agency (ESA), European Space Research and Technology Center (ESTEC), 2200 AG Noordwijk, The Netherlands (Dated: June 30, 2011) We propose a technique for the calculation of the electromagnetic fields radiated by charges in motion within homogeneous cross-section, arbitrarily-shaped waveguides. The fields are found out from the waveguide geometry and the equation of motion of the charges, by means of the classical Green’s function method. The electric and magnetic 3D-dyadic Green’s functions of such waveguides are formulated in modal expansion and numerically solved for non-standard waveguides. The convolution of the Green’s function and the charges in motion gives the electromagnetic field. This technique allows to study cases like a charge in constant velocity travelling through a uniform waveguide. Furthermore, we have derived a formulation for the computation of the wakefields of arbitrary cross-section uniform waveguides from the resulting field expressions. Examples of charged particles moving in the axial direction of such waveguides are included. I.
INTRODUCTION
Actual applications of accelerators and storage rings introduce high restrictive design constraints on beam intensity and emittance.1–3 In order to achieve optimum performance, an accurate understanding of the involved physics is required. In this sense, the evaluation of the fields radiated by a charged particle moving linearly at constant velocity within a beampipe is particularly important, since it may influence the motion of trailing particles.4–6 The electromagnetic field created by a charge is scattered on the metallic walls of the waveguide and acts back on trailing charges, thus inducing energy loss, beam instabilities and some secondary effects like the heating of sensitive components.7 The interaction with the structure can be described by impedances in the frequency-domain, or equivalently by wakefields in the time-domain.8,9 These parameters have to be taken into account during the design of an accelerator, as they restrict the choice of materials and shape of components.10 The wakefields critically depend on the geometry of the structure.11 In this sense, the absence of analytical solutions for predicting the radiation within many geometries
demands the development of numerical techniques for the analysis of arbitrary waveguides.12–14 The radiation of particles within waveguides has deserved the attention of many researchers in different fields of the electromagnetic theory.15–18 Usually, particle-in-cell codes are used to compute wakefields. Here, we present a full-wave modal method for the analysis of the electromagnetic radiation of charges in motion within uniform waveguides with arbitrary cross-section. The method is based on the dyadic three-dimensional electric and magnetic Green’s functions formulation in the frequency-domain. The radiated fields are obtained from the convolution of the Green’s functions with the current distribution of the bunch. Then, the fields are expressed in the time-domain by means of the Fourier’s Transform technique, from which the wakefields are finally derived. For the calculation of the modal expansion of an arbitrarily-shaped uniform waveguide, the two-dimensional Boundary IntegralResonant Mode Expansion (BI-RME) method is used. The BI-RME method19,20 has been revealed as a robust and high computational efficient modal technique for the characterization of arbitrary waveguides.
2 II. A.
THEORY
Basic formulation
The waveguide analyzed in this work is an arbitrary cross-section homogeneous waveguide, uniform along the direction of propagation, which coincides with the z axis (see Fig. 1). Losses are not considered, and the guide is filled with vacuum, ε0 being the electric permittivity of free space and µ0 the magnetic permeability of free √ space; speed of ligth is given by c = 1/ µ0 ε0 . In this context, the vector position is divided in its transverse and axial components, r = rt + z b z. For this uniform cross-section waveguide region, it is well known that the solutions of Maxwell equations in the frequency-domain can be expressed in terms of TE and TM modes.19,21–23 The transverse electric and magnetic fields can be decomposed into an infinite set of waveguide modes: X Et (r) = Vm (z) em (rt ) (1a) m
Ht (r) =
X
Im (z) hm (rt )
(1b)
Y
X
Z µ 0 ε0 CS
FIG. 1: Schematic of an arbitrary charge distribution moving inside a uniform arbitrarily-shaped cross-section waveguide region.
density J (r, t) = ρ(r, t) v(r, t), where v is the velocity vector. The first step in the formulation relies on the evaluation of the Fourier transform of the time-domain current density,
m
where Vm and Im are the voltage and current modal amplitudes, and em and hm are the electric and magnetic normalized vector modal functions, respectively. On the other hand, the axial components are expressed in terms of the scalar potentials Φm : 1 X TM I (z) kt2m ΦTmM (rt ) Ez (r) = iωε0 m m 1 X TE Hz (r) = V (z) kt2m ΦTmE (rt ) iωµ0 m m
(2a)
TE Zm
+∞ −∞
J (r, t) e−iωt dt
(4)
(2b)
√ i being the imaginary unit i = −1; ktm are the modal transverse wavenumbers, and ω is the angular frequency; a time-harmonic dependence eiωt is assumed and omitted throughout this document. The normalized vector modal functions are obtained as eTmM = −∇t ΦTmM , hTmE = −∇t ΦTmE and hm = b z × em . Any electromagnetic field in the frequency-domain existing inside a waveguide can be expressed as a superposition of these modal vectors, which obviously satisfy the boundary conditions on the walls of the waveguide. The modal characteristic impedances are given by ω µ0 kz m TM = , Zm = kz m ω ε0
J(r) =
Z
J(r) being the frequency-domain current density. The next step consists on the evaluation of the frequencydomain electric and magnetic fields radiated by such harmonic currents, which will be performed by means of the following volume integrals:
E(r) = H(r) =
(3)
kzm being the propagation factor in the axial direction. The dispersion relationship satisfied by these modes is √ k 2 = kt2m + kz2m , where k = ω µ0 ε0 is the free space wavenumber. In our problem we have a time-varying arbitrarilyshaped charged distribution radiating inside the waveguide region, as depicted in Fig. 1, which is described by its volumetric charge density ρ(r, t) as well as its current
Z
Z
(5a)
V
Ge (r, r′ ) · J(r′ ) dV ′ Gm (r, r′ ) · J(r′ ) dV ′
(5b)
V
where Ge (r, r′ ) and Gm (r, r′ ) are the frequency-domain three-dimensional dyadic electric and magnetic Green’s functions of the infinite waveguide region, respectively. In the spectral domain, the dyadic electric and magnetic Green’s functions are expressed in terms of the normalized electric and magnetic TE and TM vector modal
3 functions, as follows22,24–26 ′ 1 X TM TM Ge (r, r′ ) = − Z em (rt ) eTmM (r′t )e−ikzm |z−z | 2 m m ′ 1 X TE TE Zm em (rt ) eTmE (r′t )e−ikzm |z−z | − 2 m ′ u(z, z ′ ) X 2 T M − ktm Φm (rt ) b z eTmM (r′t ) e−ikzm |z−z | i 2 ω ε0 m ′ u(z, z ′ ) X 2 T M + z ΦTmM (r′t ) e−ikzm |z−z | k e (rt ) b i 2 ω ε 0 m tm m −
′ 1 X kt4m T M Φm (rt ) ΦTmM (r′t ) b zb z e−ikzm |z−z | TM 2 ω 2 ε20 m Zm
δ(r − r′ ) b zb z (6a) i ω ε0 ′ u(z, z ′ ) X T M Gm (r, r′ ) = − hm (rt ) eTmM (r′t )e−ikzm |z−z | 2 m ′ u(z, z ′ ) X T E hm (rt ) eTmE (r′t )e−ikzm |z−z | − 2 m ′ 1 X TE 2 TE − Zm ktm Φm (rt ) b z eTmE (r′t ) e−ikzm |z−z | i 2 ωµ0 m −
+
′ 1 X kt2m T M hm (rt ) b z ΦTmE (r′t ) e−ikzm |z−z | T M i 2 ωµ0 m Zm (6b)
where the sign function u(z, z ′ ) is given by −1 , z < z ′ ′ 0 , z = z′ u(z, z ) = +1 , z > z ′
(7)
and δ(r − r′ ) stands for the Dirac delta function. Finally, we derive the time-domain electric and magnetic fields using the standard inverse Fourier transform,
E(r, t) = H(r, t) =
Z
+∞
−∞ Z +∞
E(r) eiωt dω
(8a)
H(r) eiωt dω
(8b)
−∞
The present formulation is completely general, and now it will be applied to analyze several problems of radiation of charged particles. In order to compute the electromagnetic fields inside a hollow arbitrarily-shaped waveguide with uniform cross-section, the BI-RME19,20,27 method has been used. This numerical technique efficiently characterizes such arbitrarily-shaped waveguides thanks to the solution of first kind Fredholm integral equations resulting into eigenvalue problems. The main advantage of this method is the use of an exact kernel defined by means of dyadic Green’s functions expressed in terms of fast convergent series. The terms of these series are rational functions of the frequency, which can be truncated without
a significant loss of accuracy. The classical implementation of the method20 always consider an arbitrary profile CS, uniform in the longitudinal direction (z-axis), composed of smaller straight arcs and completely enclosed within a rectangular waveguide. Recently this method has been revisited and extended in order to cope with arbitrary profiles defined by the combination of linear, circular and/or elliptical segments .27 B. Study of the fields radiated by a charged particle uniformly moving in the axial direction
A particle accelerator structure basically consists on a waveguide circuit alternating accelerating cavities and thru-sections. The particles, packed in bunches, are injected into the structure and accelerated up to relativistic velocities by the action of the accelerating cavities which contain RF high-power electromagnetic fields. In the present section, the electromagnetic fields radiated by such particles within the beampipe of a particle accelerator or within a generic homogeneous waveguide are discussed. Since the particles velocity is unchanged between successive accelerating cavities, they can be reduced to the study of constant velocity particles travelling along an infinite homogeneous waveguide, as depicted in Fig. 2. In this figure, there is a source particle carrying charge q uniformly moving in the z direction with constant velocity v; we assume that v ≥ 0. The charge and current densities of this charge are represented in the time-domain by ρ(r′ , t) = q δ(x′ − x0 ) δ(y ′ − y0 ) δ(z ′ − v t) J~ (r′ , t) = ρ(r′ , t) v b z
(9a) (9b)
respectively. Thus, the cartesian coordinates (x0 , y0 ) define the transverse position of the particle. Following the presented technique, now we have to calculate the frequency-domain current density applying (4) to (9), easily obtaining ′
z J(r′ ) = q δ(x′ − x0 ) δ(y ′ − y0 ) e−iω z /v b
(10)
Secondly, the frequency-domain electric and magnetic fields can be analytically derived by inserting (10) in (5), thus obtaining q X 2 TM e−i ω z/v Et (r) = ktm em (rt )ΦTmM (r′t ) 2 v ε0 m ω − kz2m v (11a) −i ω z/v X iq e Ez (r) = − k 4 ΦT M (rt )ΦTmM (r′t ) 2 ω ε 0 m tm m ω − kz2m v
(11b)
Ht (r) = q
X m
Hz (r) = 0
e−i ω z/v (11c) kt2m hTmM (rt )ΦTmM (r′t ) 2 ω 2 − k zm v (11d)
4 After applying the inverse Fourier transformation to these expressions (8), we finally obtain the time-domain fields radiated by the charged particle, qγ X E~t (r, t) = kt eT M (rt )ΦTmM (r′t )e−ktm γ|vt−z| 2 ε0 m m m (12a) z q · u t− Ez (r, t) = − 2ε0 v X kt2m ΦTmM (rt )ΦTmM (r′t )e−ktm γ|vt−z| (12b) m
X ~ t (r, t) = q v γ H ktm hTmM (rt )ΦTmM (r′t )e−ktm γ|vt−z| 2 m (12c) Hz (r, t) = 0
v→c−
q zX δ t− kt eT M (rt )ΦTmM (r′t ) 2 ε0 v m m m (13a)
lim− Ez (r, t) = 0
v→c
~ t (r, t) = lim− H
v→c
a FIG. 3: Electric field lines of a charged particle travelling along a rectangular waveguide of cross-section (a, b) at constant velocity. Results on the transverse plane for two different velocities: the red lines stand for β = 0.9 and the green lines for β = 0.7.
(12d)
p where γ ≡ 1/ 1 − β 2 is the relativistic factor, β ≡ v/c being the velocity in terms of the speed of light in vacuum. Note that only the TM modes are being excited. It should be observed that the magnetic field is zero for a static charge (v = 0). It is also worth to analyse these expressions in the case that the particle velocity approaches to the speed of light limit (ultrarelativistic case). Then, the field power concentrates on a cross-plane moving together with the charge. The transverse components of the fields turn into a Dirac-delta, meanwhile the longitudinal field vanishes: lim E~t (r, t) =
b
(13b)
qc zX δ t− kt hT M (rt )ΦTmM (r′t ) 2 v m m m (13c)
lim Hz (r, t) = 0
(13d)
v→c−
This effect is observed in Figs. 3 and 4. In this example, the electric field lines of a point-charge moving at constant velocity within a rectangular waveguide are shown for two different particle velocity.
z q
v
z’ = vt FIG. 2: A charge q travels through an homogeneous waveguide at constant velocity v.
y z
v
FIG. 4: Electric field lines of a charged particle travelling along a rectangular waveguide of cross-section (a, b) at constant velocity. Results on the YZ-plane for two different velocities: the red lines stand for β = 0.9 and the green lines for β = 0.7.
C.
Study of the wakefields
The electromagnetic fields created by the particles in the previous section induce surface charges and currents in the walls of the beampipe, which act back on particles and beams travelling behind. As the particle velocity increases, the fields compress in the transverse plane and the effect of the surface charges becomes significant. The trajectory and the velocity of travelling particles are modified by the presence of such surface charges, thus resulting in bunch instabilities. It is a convention for relativistic electron beams to know this scattered radiation as wakefields, although it also propagates in front of the source charge for v < c. The wakefield effect is analysed in the frame of the actual work by means of the definition of a δ-function wake potential. This function characterizes the net impulse delivered from a unit-strength source charge to a trailing charge along an homogeneous waveguide section of length L. Both charges travel at the same velocity v along the same or parallel trajectories, spaced in the axial direction by a distance s (s can be greater or smaller than L). The δ-function wake potential has been defined as in section 11.3 of5 , here adapted to particles with a velocity below the limit c and travelling within a
5 lossless waveguide: Z
L
z+s dz + E~t r, t = v 0 Z v µ0 L ~ t r, t = z + s dz + zb × H q v 0 (14a) Z L z+s 1 dz (14b) Ez r, t = wz (r, r′ , s) = − q 0 v
wt (r, r′ , s) =
1 q
where wz and wt are the longitudinal and transverse δfunction wake potentials; r′ and r stand for the initial position of the leading and the trailing charges, respectively. Applying these definitions to the field expressions (12), one obtains: L X kt eT M (rt )ΦTmM (r′t )e−ktm γs 2 γ ε0 m m m (15a) X L wz (r, r′ , s) = k 2 ΦT M (rt )ΦTmM (r′t )e−ktm γs zˆ 2 ε 0 m tm m (15b)
wt (r, r′ , s) =
TABLE I: Waveguides used in section III and cut-off frequencies for the first two modes. Dimensions are expressed in mm and cut-off frecuencies in GHz. WG A B C D E
Figure 6(a) 6(b) 6(b) 6(c) 6(d)
Dimensions a1 = 44, b1 = 36, r1 = 22 a2 = 39, b2 = 34, r2 = 3 a2 = 44, b2 = 36, r2 = 15.3 a3 = 46, b3 = 32, h = 34 r4 = 22
fc1 3.84 3.86 3.80 3.82 3.99
fc2 4.49 4.43 4.56 5.22 5.21
dimensions of the guides were tuned to obtain similar cut-off frequencies for the two first propagative modes. In Fig. 6 there is a picture of the different geometries analysed, whose dimensions and cut-off frequencies are related in Table I. Note that in most of the geometries the cut-off frequencies cannot be tuned independently, so the achieved results are not so close for the second mode. y
b1
Note that last expressions are zero at the physical limit v → c− for any separation between charges s 6= 0. It means that no wakefield is present in a lossless homogeneous waveguide for charges travelling at the speed of light. Wakefields appear as consequence of wall discontinuities, finite conductivity of the material and a velocity of charges less than the ultrarelativistic limit.
y
r1
r1
x
b2
r2 x
a1
a2
(a)
(b)
y y
III. A COMPARATIVE STUDY OF THE WAKE POTENTIALS FOR DIFFERENT WAVEGUIDE GEOMETRIES
h
b3
x
r4
x
a3 (c)
(d)
FIG. 6: Cross-section of the waveguides studied in section III: (a) LHC beampipe. (b) Rounded-corner rectangular waveguide. (c) LHC beampipe modified with elliptical side walls. (d) Circular waveguide. FIG. 5: Electric field radiated by a charged point moving at constant velocity inside a uniform elliptic waveguide.
Next, the presented formulation will be applied to the analysis of wakefields within structures of arbitrary geometry, thus proving the capabilities of this method. An homogeneous waveguide with a cross-section similar to the beampipe used in the CERN LHC Laboratory28 has been modelled by means of a BI-RME based tool (see Fig. 5). The wake potential in this structure is compared with a total of four waveguides, which have different crosssections. In order to have comparable structures, the
Next, the wakefield of a point charge is analysed for the proposed waveguides. The potentials have been computed using 1800 TM-modal vectors, previously obtained by a BI-RME simulation tool. The source charge moves along the z-direction at the center of the waveguides. In Fig. 7-9, the wake potential is evaluated on the cross axes for several velocities. Only the positive semi-axes are considered because of symmetry. Despite results are similar for all the geometries, some conclusions can be extracted from these plots. The lengthwise force induced on a trailing charge is lighter in rounded-corner rectangular waveguides, whereas the response is worse in a circular
6
1800 A B C D E
Wz / L (V pC−1 m−1)
1600 1400
posed. A comparative example is included to show the capabilities of the presented method. The required anal0
−1
Wx / L (V nC−1 m−1)
waveguide; the same happens for wx , but it is not the case for wy . It is also worthwhile to mention that the wakefields produced in the waveguides A and D are very close. Finally, the dependence of the wake potential on the velocity of the particles is shown in Fig. 10, where the vanishing of wz as the velocity increases is evident.
−2
−3
A B
−4
C D E
−5
1200 1000
−6
0
2
4
6
8
800
10
12
14
16
18
20
18
20
x (mm)
600
(a)
400 200 0
0
2
4
6
8
10
12
14
16
y (mm)
0 A −0.005
Wx / L (V pC−1 m−1)
(a)
14 A B C D E
Wz / L (V pC−1 m−1)
12
10
B C
−0.01
D E
−0.015
−0.02
−0.025
8 −0.03
0
2
4
6
8
10
12
14
16
x (mm)
6
4
(b) 2
0
0
2
4
6
8
10
12
14
16
y (mm)
(b)
FIG. 7: wz /L on the positive y-semiaxis for different particle velocities. The source charge is located at the center of the waveguide; s = 10 µm. (a) β = 1 − 10−5 (b) β = 1 − 10−7
IV.
CONCLUSIONS
The calculation of the wakefields is a primary task in the evaluation of the suitability of a beampipe. In this sense, the use of δ-function wake potentials is a good strategy for predicting the response of a waveguide to real bunches of particles. We have presented a method for the calculation of the wake potential for point charges in uniform waveguides with arbitrarily shaped cross-section. An expression for the wake potentials as a function of the vector modal functions of the waveguide has been pro-
FIG. 8: wx /L on the positive x-semiaxis for different particle velocities. The source charge is located at the center of the waveguide; s = 10 µm. (a) β = 1 − 10−5 (b) β = 1 − 10−7
ysis of arbitrary waveguides for the example are accomplished by a BI-RME based tool. The flexibility and efficiency of the BI-RME method makes possible the accurate computation of the modes of arbitrary waveguides.
Acknowledgments
The authors would like to thank ESA/ESTEC for having cofunded this research activity through the Network Partnering Initiative program and through the project “Multipactor Analysis in Planar Transmission Lines” (Contract no 20841/08/NL/GLC). We also are grateful to the Spanish goverment and the local Council of Murcia for their support through the projects CICYT Ref. TEC2010-21520-C04-04 and SENECA Ref. 08833/PI/08, respectively.
7
6
Wy / L (V nC−1 m−1)
5
4
3 A B
2
C D
1
E 0
0
2
4
6
8
10
12
14
16
y (mm)
(a)
0.035
Wy / L (V pC−1 m−1)
0.03
0.025
0.02 A 0.015 B C
0.01
D E
0.005
0
0
2
4
6
8
10
12
14
16
y (mm)
(b)
FIG. 9: wy /L on the positive y-semiaxis for different particle velocities. The source charge is located at the center of the waveguide; s = 10 µm. (a) β = 1 − 10−5 (b) β = 1 − 10−7
10 log(Wz / L) (V mC−1 m−1)
40 20
0 −20
−40 A −60
B C
−80
D −100
−120 0.5
E
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
ln(γ)
FIG. 10: Dependence of wz on the velocity. The leading charge travels along the center of the waveguide; s = 1 mm. A logarithmic scale has been chosen to emphasize differences at high velocities.
8
∗ 1
2
3
4
5
6
7
8 9
10 11
12
13
14
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