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.
Figure 1. Schematic of an arbitrary charge distribution moving inside a uniform arbitrarily shaped cross-section waveguide region.
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 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 .
Figure 2. Influence of the number of modes used in the solution of the δ-function wake potential(14) for the waveguide geometry represented in Figure 3. Comparison between several charge velocities for a fixed distance s = 5 mm (the wake function has been normalized to make the comparison possible). Note that the number of modes necessary for the convergence of the solution reduces when the velocity increases; similar effect would produce an increment in the distance s.
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