Information and Signal Theory Department, Technical University of Cartagena, Cartagena, Spain
Corresponding author: F. D. Quesada Pereira, Information and Signal Theory Department, Technical University of Cartagena, Campus de la Muralla del Mar, Cuartel de Antigones, C. P. ES-30202, Cartagena, Spain. (firstname.lastname@example.org)
Corresponding author: F. D. Quesada Pereira, Information and Signal Theory Department, Technical University of Cartagena, Campus de la Muralla del Mar, Cuartel de Antigones, C. P. ES-30202, Cartagena, Spain. (email@example.com)
 A technique for the accurate computation of the time domain electromagnetic fields radiated by a charged distribution traveling along an arbitrarily shaped waveguide region is presented. Based on the transformation (by means of the standard Fourier analysis) of the time-varying current density of the analyzed problem to the frequency domain, the resulting equivalent current is further convolved with the dyadic electric and magnetic Green's functions. Moreover, we show that only the evaluation of the transverse magnetic modes of the structure is required for the calculation of fields radiated by particles traveling in the axial direction. Finally, frequency domain electric and magnetic fields are transformed back to the time domain, just obtaining the total fields radiated by the charged distribution. Furthermore, we present a method for the computation of the wakefields of arbitrary cross-section uniform waveguides from the resulting field expressions. Several examples of charged particles moving in the axial direction of such waveguides are included.
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 From radiotherapy to nuclear physics research, particle accelerators constitute a powerful device under continuous development. Actual applications of accelerators and storage rings introduce high restrictive design constraints on beam intensity and emittance [Salah and Dolique, 1999; Salah, 2004; Gai et al., 1997]. 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 [Panofsky and Wenzel, 1956; Wangler, 2008; Zotter and Kheifets, 1998]. In a particle accelerator, the beampipe is excited by an electromagnetic field during the accelerating stage to store and accelerate beams of charged particles. The particles are usually packed into bunches and launched at the appropriate time to take advantage of the accelerating phases, in order to achieve relativistic velocities [Bane et al., 1985]. After the accelerating circuit, the velocity of a beam should remain constant; but the electromagnetic radiation of moving charges may influence the motion of other particles and bunches. 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 [Figueroa et al., 1988]. The interaction with the structure can be described by impedances in the frequency domain, or equivalently by wakefields in the time domain [Yokoya, 1993a; Stupakov et al., 2007]. 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 [Burov and Danilov, 1999].
 The wakefields critically depend on the geometry of the structure [Danilov, 2000]. The radiation of particles within waveguides has deserved the attention of many researchers in different fields of the electromagnetic theory [Rosing and Gai, 1990; Ng, 1990; Xiao et al., 2001; Hess et al., 2007]. The solution of wakefields for rectangular and circular waveguides is well known [Wangler, 2008; Zotter and Kheifets, 1998; Gluckstern et al., 1993]; moreover, in Gluckstern et al.  a formula is given for the specific case of charges traveling on the symmetry axes within an elliptical guide. This formula has been extensively followed to estimate the wakefields in beampipes similar to the elliptical geometry [Rumolo et al., 2001; Zimmermann, 1997]. The absence of analytical expressions for predicting the radiation within many geometries demands the development of numerical techniques for the analysis of arbitrary waveguides [Kim et al., 1990; Jing et al., 2003; Lutman et al., 2008; Zagorodnov, 2006]. Usually, particle-in-cell codes are used to compute wakefields. In this article, we present an alternative full-wave modal method for the analysis of the electromagnetic radiation of charges in motion within uniform waveguides with arbitrary cross-section. The method here presented 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. The effect of the velocity on beams has been traditionally omitted in the accelerators publications, where the ultrarelativistic approach is assumed [Gluckstern et al., 1993; Yokoya, 1993b; Kim et al., 1990; Bane et al., 1985; Rosing and Gai, 1990; Iriso-Ariz et al., 2003; Lutman et al., 2008; Palumbo et al., 1984]. In this sense, the present work represents a contribution since different velocities for the particles can be used in the proposed formulation. The presented formulation is completely analytic except in the calculation of the modes of the waveguide, which must be numerically computed. This fact reduces significantly the associated error of the method to the accuracy of the numerical technique used for the modal analysis of the structure. Moreover, the proposed formulation can treat waveguides of arbitrary shapes, what makes it a suitable method when non-canonical geometries are considered. In this sense, this method gives a practical contribution, since many modern accelerating structures are based on non-canonical waveguides.
 The paper has been organized in three sections. In the next one, the theoretical formulation of the problem is detailed. This section is divided in three blocks, corresponding to the dyadic electric and magnetic Green's functions, the fields created by moving charged particles, and the resulting wakefields. Afterwards, some examples of charges moving lengthwise within arbitrary waveguides are tackled by the presented derivation. Finally, conclusions are outlined.
2.1. Basic Formulation
 The type of geometry analyzed in this paper consists of an arbitrary cross-section homogeneous waveguide that is uniform along the direction of propagation, which coincides with thez axis (see Figure 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 light is given by . In this context, the vector position is divided in its transverse and axial components, . 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 [Conciauro et al., 2000; Collin, 1991; Felsen and Marcuvitz, 1994; Jackson, 1999]. The transverse electric and magnetic fields can be decomposed into an infinite set of waveguide modes:
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. The normalization condition satisfied by these functions is given by
where CSis the waveguide cross-section, andδm,n is the Kronecker delta function. On the other hand, the axial components are expressed in terms of the scalar potentials Φm [Felsen and Marcuvitz, 1994]:
i being the imaginary unit ; are the modal transverse wave numbers, and the frequency is . A time-harmonic dependence eiωt is assumed and omitted throughout this paper. The modal characteristic impedances are given by and being the propagation factor in the axial direction. The dispersion relationship satisfied by these modes is where is the free space wave number. An exponential factor of the form is assumed inside the modal amplitudes V(z) and I(z), for waves traveling in the positive z-direction.
 In our problem we have a time-varying arbitrarily shaped charged distribution radiating inside a waveguide region, as depicted inFigure 1, which is described by its volumetric charge density ρ(r, t) as well as its current density , where v is the velocity vector; note that both densities are also related through the continuity equation [Jackson, 1999]. In our formulation, the first step relies on the evaluation of the Fourier transform of the time domain current density,
J being the frequency domain current density. The next step consists on the evaluation of the frequency domain electric and magnetic fields radiated by such harmonic currents, which will be performed by means of the following volume integrals:
where and are the frequency domain three-dimensional dyadic electric and magnetic Green's functions of the infinite waveguide region, respectively. The evaluation of Green's functions is a classical problem of electromagnetics, which has been extensively treated in the technical literature [Collin, 1991; Felsen and Marcuvitz, 1994; Tai, 1993; Hanson and Yakovlev, 2002]. The dyadic Green's functions of an infinite uniform cross-section waveguide represents the electric and magnetic fields radiated by a time-harmonic unit impulse current source placed at an arbitrary location given by the vector positionr′. The source element can be oriented along any direction, thus allowing to solve three-dimensional current problems. 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 functions, as follows [Felsen and Marcuvitz, 1994; Wang, 1978; Rahmat-Samii, 1975; Deshpande, 1997]
where the sign function u(z − z′) and the Dirac delta function δ(r − r′) have been introduced.
 Finally, we must derive the time domain electric and magnetic fields using the standard inverse Fourier transform,
The present formulation is completely general, and now it will be applied to analyze several problems of radiation of charged particles. The presence of arbitrarily shaped cross-section waveguides will be considered in this work.
2.2. Study of the Fields Radiated by a Charged Particle Uniformly Moving in the Axial Direction
 A particle accelerator structure basically consists of a waveguide circuit alternating accelerating cavities and through-sections. The particles, packed in bunches, are injected into the structure and accelerated up to ultrarelativistic 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 traveling along an infinite homogeneous waveguide. There is a source particle carrying chargeq uniformly moving in the z direction with constant velocity v; we assume that v ≥ 0. The charge and current densities of this particle are represented in the time domain by
respectively. Thus, the Cartesian coordinates r0 = (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 (8), easily obtaining
Secondly, the frequency domain electric and magnetic fields can be analytically derived by inserting (9) into (5), thus obtaining
After applying the inverse Fourier transformation (7) to these expressions, we finally obtain the time domain fields radiated by the charged particle (see Appendix A for a deeper explanation of the inverse Fourier transformation),
where 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). Due to the analytical nature of the method, it achieves good accuracy as compared to numerical techniques based on differential equations, such as finite differences or finite elements. This is because these techniques require the discretization of the whole volume, and the accuracy strongly depends on how fine the structure is discretized.
 It is worth to analyze these expressions in the case that the particle velocity approaches 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, whereas the longitudinal field vanishes (seeAppendix B for details):
2.3. 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 traveling behind. The trajectory and the velocity of traveling 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 these space forces as wakefields, although they also propagate in front of the source charge for v < c. The wakefield effect is analyzed 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 lengthL. 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 ofWangler , here adapted to particles with a velocity below the limit c and traveling within a lossless waveguide:
where wz and wt are the longitudinal and transverse δ-function 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 (11), one obtains (see Appendix C):
Note that last expressions are zero at the physical limit v → c− for any separation between charges s ≠ 0. It means that no wakefield is present in a lossless homogeneous waveguide for charges traveling 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. The integrated effect over a finite distribution of charged particles is described by the wake potential. The wake potential of a complete bunch on a single charge can be determined by the convolution of the wake δ-function with the charge distribution of the bunch.
 The convergence of an expression in infinite series like (14) is dominated by the exponential part; the larger the exponent is, the faster the series converges to a realistic solution. This effect is shown in Figure 2, where the wake potential in a waveguide example is approached by different number of terms in the summation expression. The convergence of the series (14) is also guaranteed in the limit s approaches zero, as is demonstrated in Álvarez Melcón and Mosig . This limit represents the waveguide cross-plane that contains the source charge. In this case, the exponent in the series vanishes and the number of modes required for convergence is extremely high. Thus, it is more efficient to tackle thes= 0 problem by means of a numerical approaching method, such as to estimate the value of fields and wake-potentials from the solution of a series of decreasing distance points, or by using extrapolation techniques as inÁlvarez Melcón and Mosig .
3. Numerical Results
 Next, the presented formulation will be applied to the analysis of wakefields within structures of arbitrary geometry, thus proving the capabilities of this method. In a first example, a cross-shaped waveguide has been chosen for discussing the effect of rounding the corners of the inner waveguide walls. In the second example, several geometries are compared to a waveguide similar to the CERN Large Hadron Collider (LHC) beampipe.
3.1. Rounded-Corner Cross-Shaped Waveguides
 It is well known that the electromagnetic fields within a waveguide are particularly strong near convex wedges. The power focusing in these areas gives rise to undesirable consequences, like the heating of walls and the attraction of free particles. The accumulation of charged particles in a small region of the beampipe will produce changes in the resistivity of the component and multipactoring effects, both causing losses and defocusing in the beam. For this reason, sharp wedges are avoided in the design of accelerating structures as far as possible.
 With the aim of studying the influence of corners in the geometry of a beampipe, an homogeneous cross-shaped waveguide with rounded corners (seeFigure 3) has been analyzed. Three different radii of curvature r for the corners of the waveguide are considered for comparison. The size of the arms of the cross has been selected to a = 3 mm. The wakefields for two different trajectories of a source particle are compared.
 For the calculation of the modal expansion of the waveguide, the two-dimensional Boundary Integral-Resonant Mode Expansion (BI-RME) method was used. The BI-RME method [Conciauro et al., 1984, 2000] has been revealed as a robust and high computational efficient modal technique for the characterization of arbitrary waveguides, whose accuracy is discussed in Conciauro et al. , Cogollos et al.  and Bozzi et al. . The accuracy of the results shown in this example and in the next one is only dependent on the number of modes used in the summation expressions (11) and (14), and on the accuracy of the numerical computation of such modes. The BI-RME method here used allows to compute a large number of modes with high precision on a standard PC [Conciauro et al., 1984; Cogollos et al., 2003; Bozzi et al., 2001]. A total of 450 electric TM-modal vectors were generated by a BI-RME based tool [Cogollos et al., 2003] and used to evaluate the δ-function wake potential at any point.
Figures 4 and 5show the wake-potential distribution in theXY-plane ats = 5 μm behind the source. In Figure 4 the source particle is on the center of the waveguide and the normalized velocity is β = 1 − 10−7; on the other hand, in Figure 5 the source is within one of the arms of the waveguide, and the normalized velocity is β = 1 − 10−6. The CPU time used for the computation of a unique point is 42 seconds in a standard PC at 3.2 GHz.
 Cuts along the y-axis of preceding pictures are found inFigures 6 and 7, where wx has been omitted (this component is null on the y-axis because of the symmetry of the problem). The maximum value forwz varies with the radius of curvature, but obviously coincides with the source location, as can be seen in Figures 6b and 7b. The behavior of the transverse wake potential is different. From Figure 6a it is observed that, when the source is at the center, the maximum magnitude for wy lays at a point within the interval ±(a/2, a/2 + r); this interval coincides with the narrowing of the waveguide at the beginning of one arm of the cross. On the other hand, when the source is shifted from the center (Figure 7a), the transverse wake potential has an offset related to the deviation of the leading charge.
 The different scales for the wake potential intensity shown in both trajectories can be explained by the difference in velocity. Note that the electric field distribution is contracted into a disk perpendicular to the direction of motion with a narrow angular spread of the order of 1/γ, as reported in Wangler . In Figure 8, the wake potential is plotted for a testing and a trial charge depending on the velocity. It is shown that the intensity of the longitudinal wake potential decreases with the velocity as a consequence of the electric field contraction.
3.2. A Comparative Study of the Wake Potentials for Different Waveguide Geometries
 A homogeneous waveguide with a cross-section similar to the beampipe used in the CERN LHC Laboratory [Iriso-Ariz et al., 2003] has been modeled by means of a BI-RME based tool. The wake potential in this structure is compared for several waveguides, which have different cross-sections. A total of four waveguide geometries are considered. In order to obtain comparable structures, the dimensions of the guides are such that the cut-off frequencies of the two first propagating modes are close enough to that of the original beampipe. InFigure 9there is a picture of the different geometries analyzed, whose dimensions and cut-off frequencies are related inTable 1. 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.
Table 1. Waveguides Used in Section 3.2and Cut-Off Frequencies for the First Two Modesa
Dimensions are expressed in mm and cut-off frequencies in GHz.
 Next, the wakefield of a point charge is analyzed 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 thez-direction at the center of the waveguides. InFigures 10, 11, and 12, 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 smaller in rounded-corner rectangular waveguides, whereas the response is worse in a circular waveguide; the same happens forwx, but it is not the case for wy. The symmetry of the circular waveguide E produces equal forces in magnitude along the transverse axes here considered. In the A waveguide, two straight cuts are introduced across the y-axis. These two cuts have the effect of increasingwy with respect to the wx. 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 Figure 13.
 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 proposed. The method consists of deriving the electromagnetic fields radiated by a point charge through the three-dimensional dyadic Green's functions of the waveguide; the wake potential is further extracted from the time domain expression of the fields. A few examples are included in the article to show the capabilities of the presented method. Modal analysis of the geometries in the examples are accomplished by a BI-RME based tool. The flexibility and efficiency of the BI-RME method makes possible an accurate computation of the modes of arbitrary waveguides. In the examples, the influence of the trajectory and the velocity of charges on the wakefields is compared for different geometries, emphasizing the rounded-corner effect of the waveguides used in particle accelerator structures.
Appendix A:: Frequency to Time Domain Transformation of Field Expressions
 The calculation of the time domain field expressions (11) is outlined in the present appendix. For this purpose, the inverse Fourier transform (7) will be applied to the frequency domain field equations (10). The transformation of the electric field transverse component is tackled next. From (10a) one can write:
In order to solve this equation, the following Fourier pair has to be used:
The application of this last relation and the shifting property of Fourier transform into (A2) leads to the equation (11a). The procedure followed for the calculation of the transverse component of the magnetic field is completely similar to the previous one.
 On the other hand, the next integral must be solved for the calculation of the longitudinal field (10b):
Note that this integral is like (A2) but multiplied by the factor . In order to solve it, the integration property of Fourier transform has to be applied to the Fourier pair (A3):
Finally, the shifting term z/v is applied, thus obtaining the field expression (11b).
Appendix B:: Ultrarelativistic Case Fields Derivation
 The derivation of the expression (12) for the time domain fields of the ultrarelativistic case is shown in this appendix. The analysis starts on the transverse component of the electric field. The ultrarelativistic limit of (11a) is:
Knowing the previous expression constitutes an indeterminate form except for zero value of the exponent (ct = z). L'Hopital's rule applies when the exponent is different from zero, obtaining:
On the other hand, this limit diverges when the exponent is zero:
These two results can be combined by means of the Dirac-delta function, thus leading to the expression
which is equivalent to (12a). A similar analysis can be followed for the transverse magnetic field (12c). For the analysis of the longitudinal electric field, the ultrarelativistic limit is applied to (11b):
The resulting limit can be analyzed for zero and different from zero values of the exponent:
This time, the limit is zero in both cases, and hence the null expression for the longitudinal field (12b), as predicted by Lorentz.
Appendix C:: Wake Potentials Derivation
 The derivation of the δ-function wake potential(14) from the definition given in (13) is detailed in the present appendix. First of all, the time domain field expression (11) is used in (13):
where the following integrals have been introduced, obtaining:
On the other hand, knowing that and v2γμ0/2 = γ/(2ε0) − 1/(2γε0), the transverse component (C1a) can be simplified as
Finally, assuming positive distance s, the previous expressions yield the equation (13).
 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 20841/08/NL/GLC). We also are grateful to the Spanish government 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.