Rigorous formulation of space-charge wake function and impedance by solving the three-dimensional Poisson equation

In typical numerical simulations, the space-charge force is calculated by slicing a beam into many longitudinal segments and by solving the two-dimensional Poisson equation in each segment. This method neglects longitudinal leakage of the space-charge force to nearby segments owing to its longitudinal spread over 1/γ. By contrast, the space-charge impedance, which is the Fourier transform of the wake function, is typically calculated directly in the frequency-domain. So long as we follow these approaches, the longitudinal leakage effect of the wake function will remain to be unclear. In the present report, the space-charge wake function is calculated directly in the time domain by solving the three-dimensional Poisson equation for a longitudinally Gaussian beam. We find that the leakage effect is insignificant for a bunch that is considerably longer than the chamber radius so long as the segment length satisfies a certain condition. We present a criterion for how finely a bunch should be sliced so that the two-dimensional slicing approach can provide a good approximation of the three-dimensional exact solution.

Here I n (z) and K n (z) are the modified Bessel functions 3 , j is an imaginary unit, Z 0 is the impedance of free space, L is total chamber length, ω is angular frequency, k = ω/cβ, c is velocity of light, γ = k k/ , and, β and γ are Lorentz-β and γ, respectively. Note that the impedance is purely imaginary.
Accordingly, the space-charge impedance of an ultra-relativistic beam is typically approximated as follows: Thus, the corresponding space-charge wake function is described by the δ-function, which violates the causality condition because the δ-function is an even function. Because the space-charge force spreads out over 1/γ symmetrically in the longitudinal direction, the force excited by the source particle can affect the particles in its front.
To understand more clearly whether this δ-function description of the wake function is a good approximation, especially for a non-relativistic beam, we should calculate the wake function directly in the time domain, instead of calculating the Fourier transformed impedance from the beginning of its derivation.
In the next section, we calculate the wake function by solving the three-dimensional Poisson equation in a cylindrical chamber for a ring-shaped beam with a Gaussian longitudinal distribution.

Three-Dimensional Approach for Determining Space-Charge Transverse Wake Function
Let us start by solving the Poisson equation in cylindrical coordinates for an axisymmetric beam in the rest frame ρ θ ct z ( , , , ) 4 , in a perfectly conductive cylindrical chamber of radius a. The Poisson equation is as follows: where the shape of the beam distribution is assumed to be where Φ is the scalar potential in the rest frame, i 1 = qr b is the dipole moment, and σ z is the longitudinal rms beam size in the lab-frame (ct, ρ, θ, z). The scalar potential Φ and the vector potential A z in the lab-frame are obtained by the following transformations , ρ θ β β γ ρ θ γ β z respectively. The Green function → → ′ G r r ( , ), which satisfies the boundary condition: G = 0 at ρ = a, is given by 5  It is simplified as for ρ < r b , where we use the formula: To obtain the wake function, we need the scalar Φ and the vector A z potentials in the lab-frame for ρ < r b . By substituting Eq. (12) into Eqs (7) and (8), they are calculated as Since the Lorentz force is given by we can define the force ξ ρ  F ( ) in the radial direction felt by the witness particle, which is located at a distance ξ from the source particle, as  Note that the sign of ξ can be positive or negative. Accordingly, the transverse space-charge wake function W T (ξ) is expressed as which violates the causality condition, because the wake function W T (ξ) is the even function of ξ. Hereinafter, we call the first term in Eq. (23) the direct space-charge wake function, and the second term Because the transverse impedance Z T (ω) is defined as for γ  1. It reproduces Eq. (2) for an infinitesimal beam with σ z = 0. In this description, the first and the second terms correspond to the direct and the indirect space-charge impedances, respectively. Surprisingly, Eq. (27) is different from Eq. (1) even for σ z = 0. To identify the causes, we recalculate the impedance in the frequency domain by following the conventional analysis.

Derivation of Impedance in the Frequency Domain
The Maxwell equations can be written as wave equations by assuming that electro-magnetic fields have a time dependency of e jωt . They are described as 2 2 where ρ and → j are the charge density and current density of the beam, respectively. The charge density is expressed as In the cylindrical coordinates (ρ, θ, z) for an axially symmetric structure, the wave equations of the longitudinal component of the electric and magnetic field contain no transverse field component. They are decoupled. For the longitudinal field, there is a source term Because the general solutions of the Maxwell equations are obtained by superposition of the solutions for i m ρ m , we choose i 1 ρ 1 as the source term for calculating the transverse impedance. Let us define the source field specified with superscript S as the solution that satisfies the Maxwell equations with ρ βρ → = j c , (0, 0, ) 1 1 1 and vanishes at ρ → ∞. It is given by Thus, the general solutions of the Maxwell equations for the dipole moment are expressed as inside the vacuum chamber (ρ < a), where A and B are arbitrary coefficients, ε 0 and μ 0 are the dielectric constant and the permeability of vacuum, respectively. The perfectly conductive boundary condition at the chamber wall ρ = a is expressed as Accordingly, the longitudinal electric field and the Lorentz-force are expressed as Finally, the transverse impedance is expressed as is the direct space-charge impedance and the second term

Numerical Examples
Now, we compare the conventional formula (1) with the new formula (51) for the parameters used in ref. 8 (σ z = 0, β = 0.1 or β = 0.9; a = 40 mm, and r b = 10mm). The imaginary parts of the impedances Z T (ω) for different Lorentz-β are shown in Fig. 1. The left and the right panels show the impedances for β = 0.1 and 0.9, respectively. The brown and the black lines show the ω I Z [ ( )] T calculated using the conventional formula (1) and the new formula (51), respectively. The space-charge impedances diminish toward the relativistic γ, as expected. For both non-relativistic (β = 0.1) and relativistic Lorentz-β (β = 0.9) beams, the space-charge impedances obtained using the new formula (51) are damped more rapidly toward high frequencies than those obtained using Eq. (1). The rapid damping of Eq. (51) toward high frequencies indicates that the wake function obtained by the inverse Fourier transform of the new formula (51) should be more deformed relative to the δ-function than that obtained using the conventional formula (1).
Notably, in reference 8 , the numerical simulation results of transverse space-charge impedance were compared with the results obtained using Gluckstern's formula (1), as in Fig. 5 of ref. 8 , and good agreement was found between them. This is because in that study, the same approximate definition of the transverse impedance (see the definition (54)) as that of Gluckstern was used to derive formula (1). We believe that they can reproduce our results (Fig. 1), if they adopt the exact definition of transverse impedance.  Figure 2 shows the indirect space-charge impedance Z T,indirect (ω) (brown) calculated using the new formula (53) and the direct one Z T,direct (ω) (black) calculated using the new formula (52) for different Lorentz-β. The left and the right panels show the impedances Z T,indirect (ω) and Z T,direct (ω) for β = 0.1 and 0.9, respectively. The indirect impedance Z T,indirect (ω) (brown) is damped more rapidly toward high frequencies than the direct one Z T,direct (ω) (black), regardless of the Lorentz-β, which reflects the fact that the indirect effect originates from the image charge on the chamber wall that spreads out by 2a/γ. Thus, the direct space-charge impedance Z T,direct (ω) is dominant in the short range.
Next, we investigate the wake function W T (ξ) by using Eq. (23) in the time domain for the same set of parameters (σ z = 0, β = 0.1 or β = 0.9; a = 40 mm, and r b = 10mm), as in Figs 1 and 2. Figure 3 shows the Lorentz-β dependence of the wake function W T (ξ). The black and brown lines show the wake functions W T (ξ) for β = 0.1 and 0.9, respectively. Owing to the relativistic effect, the absolute value of the wake function W T (ξ) is smaller for β = 0.9 than it is for β = 0.1. Again, the transverse wake functions W T (ξ) violate the conventional causality condition, because they are even functions of ξ.
Here, we separate the indirect and the direct contributions of W T (ξ). The left and the right panels in Fig. 4 show the indirect wake function W T,indirect (ξ) obtained using Eq. (25) and the direct one W T,direct (ξ) obtained using Eq. (24), respectively. The black and the brown lines represent the functions W T,indirect (ξ) and W T,direct (ξ) for β = 0.1, and for β = 0.9, respectively.
Note that the right panel shows the negative values of the direct space-charge wake function W T,direct (ξ). The indirect wake function W T,indirect (ξ) and the direct one W T,direct (ξ) have opposite signs because the former and the latter are generated by the image charge and by a beam, respectively. The indirect space-charge wake function  W T,indirect (ξ) (left) is broader than the direct one (right) because W T,indirect (ξ) is generated by the image charge on the chamber wall, as expected. Because both W T,indirect (ξ) and W T,direct (ξ) become narrower for a larger Lorentz-γ, the space-charge wake functions W T (ξ) approach the δ-functions in an ultra-relativistic beam limit. Figure 5 shows the dependence of the indirect space-charge wake function W T,indirect (ξ) on the chamber radius a with fixed beam offset r b by using Eq. (25). The brown, purple, red and green lines (read using the scale markings on the left axis) represent W T,indirect (ξ) for a = 40 mm, 60 mm, 80 mm, and 100 mm, respectively. The black line (read using the scale markings on the right axis) shows the W T,direct (ξ) determined using Eq. (24) as a reference. The upper and lower panels show the functions W T,indirect (ξ) and W T,direct (ξ) for β = 0.1, and for β = 0.9, respectively. The panels on the right show the normalized (to the same amplitude) versions of the panels on the left. The shape of the indirect space-charge wake function W T,indirect (ξ) becomes broader for β = 0.1 owing to non-relativistic effects. For both β = 0.1 and 0.9, as the chamber radius a increases, W T,indirect (ξ) reduces drastically compared to W T,direct (ξ). In addition, the shape of the indirect space-charge wake functions W T,indirect (ξ) becomes broader as the chamber radius increases. Again, this tendency reflects the fact that the indirect space-charge wake function W T,indirect (ξ) is created by the image charge that spreads out by 2a/γ on the chamber wall.
Finally, the dependences of the indirect wake function W T,indirect (ξ) and the direct one W T,direct (ξ) on the rms bunch length σ z are shown in Fig. 6, which are obtained by using Eqs (24) and (25), respectively, for the case of r b = 10 mm and a = 40 mm, although the wake function W T (ξ), or impedance Z T (ω), is conventionally defined for infinitesimal σ z = 0. The left and right panels in Fig. 6 show the functions W T,indirect (ξ) and W T,direct (ξ) for σ z = 0 m, and for σ z = 1 m, respectively. The dashed (read using the scale marking on the left axis) and the solid (read  using the scale marking on the right axis) lines represent W T,indirect (ξ) and W T,direct (ξ), respectively. The red dashed and purple solid lines show the wake functions for β = 0.1 and the black dashed and green solid lines for β = 0.9, respectively. Notice that the scales of the vertical axes are different in the left and the right panels. Figure 6 clearly shows that the wake function W T (ξ) is more deformed from the δ-function for longer σ z , as expected.
In typical numerical simulations, the space-charge force is obtained by longitudinally slicing a beam into several segments and by solving the two-dimensional Poisson equation. Because we now have the three-dimensionally obtained exact space-charge wake function W T (ξ), we can examine the accuracy of this approximation and identify the conditions under which the approximation is adequately accurate. In the next section, we discuss this issue by solving the two-and the three-dimensional Poisson equations, and by comparing the results.

Comparison of Transverse Space-Charge Wake Functions by Conventional Scheme and Rigorous One
In conventional numerical calculations of space-charge force, the beam is sliced longitudinally into several segments. In each segment, the space-charge force is calculated approximately by solving the two-dimensional Poisson equation, assuming that particles are distributed uniformly in the longitudinal direction in each segment. Let us follow this scheme and approximate the transverse space-charge wake functions theoretically. By comparing the approximated functions with the rigorous ones presented in the section entitled 'Three-Dimensional Approach for Determining Space-Charge Transverse Wake Function' , we can find out how finely we must slice a beam to obtain a good approximation.
Here, we introduce the two-dimensional Poisson equation in the rest frame ρ θ ct z ( , , , ), which is given by where the beam distribution in the i-th segment within the mesh size Δz 2 is given by The scalar potential Φ and the vector potential A z in the lab-frame are given by for two-dimensional space, which satisfies the boundary condition: G two = 0 at ρ = a, is given by (the derivation is given in the appendix) 5,6 ∑ ∑ π ρ π θ θ ρ ρ ρ ρ ρ π ρ π θ θ ρ ρ ρ ρ ρ By using the Green function → → ′ G r r ( , ) two , the scalar potential Φ in the rest frame is calculated as    (77) and (78) and the exact ones (right) obtained using Eqs (24) and (25) for the shorter bunch case of σ z = 10 mm. The purple and red lines show the wake functions for β = 0.1 m and the green and black lines for β = 0.9 m, respectively. For the direct space-charge wake functions W T,direct (ξ) (purple solid and green solid), the agreement is relatively good for relativistic and non-relativistic beams, but agreement of the indirect wake functions W T,indirect (ξ) (red dashed and black dashed) is poor, especially for the non-relativistic beam (red dashed). Figure 8 shows that Eq. (76) cannot approximate the exact wake function for a non-relativistic short bunch, even if we slice the beam into infinite segments because of the non-negligible longitudinal leakage of the indirect space-charge force to nearby segments. In other words, the two-dimensional approximation cannot, in principle, be used to accurately determine the space-charge force for a non-relativistic beam with shorter bunch length, even if the bunch is sliced to infinite segments. This result has an important implication for simulations of space-charge force of an electron bunch at low energy such as just after a DC gun.
It is impractical to slice a bunch into infinite segments. Then, the question is that how finitely must a bunch be sliced to secure good accuracy in case of the two-dimensional approach. To this end, we compare the functions W T,direct (ξ) and W T,indirect (ξ) calculated using Eqs (74) and (75) (approximate results) with those calculated using Eqs (24) and (25) (rigorous results) for various segment sizes 2Δz. Let us consider a long bunch. Figure 9 shows the direct space-charge wake function W T,direct (ξ) (solid) and the indirect one W T,indirect (ξ) (dashed) for a non-relativistic beam with β = 0.1 and σ z = 1 m. The black, red and purple lines show the functions W T,direct (ξ) and W T,indirect (ξ) for 2Δz = σ z /5, for 2Δz = 2σ z /5, and for 2Δz = σ z , respectively. The exact direct wake function W T,direct (ξ) and indirect one W T,indirect (ξ) are represented by the purple solid and the red dashed lines in Fig. 7, respectively. When the segment size 2Δz is equal to 2σ z /5 or smaller, the approximate wake functions W T,direct (ξ) and W T,indirect (ξ) (black and red lines) reproduce the exact ones, as shown in Fig. 7 with good accuracy. By contrast, when the segment size 2Δz is identical to the rms size σ z (purple lines), the functions W T,direct (ξ) and W T,indirect (ξ) deviate significantly from the exact ones. Now, let us fix the segment size to 2Δz = 2σ z /5 and compare the approximate wake functions with the exact ones for different bunch lengths. Figure 10 shows the σ z -dependence of W T,indirect (ξ) according to Eq. (75) and that of W T,direct (ξ) according to Eq. (74) for different Lorentz-β. The upper, middle and lower panels show the wake functions W T,indirect (ξ) and W T,direct (ξ) for σ z = 1 m, for σ z = 0.5 m, and for σ z = 0.1 m, respectively. The left and the right panels in the figure show the wake functions W T,indirect (ξ) and W T,direct (ξ) for β = 0.1, and for β = 0.9, respectively. The dashed and the solid lines show the indirect wake function W T,indirect (ξ) and the direct one W T,direct (ξ), respectively. The black and purple lines show the exact wake functions W T,direct (ξ) and W T,indirect (ξ) determined using Eqs (24) and (25), respectively, while the red and green lines show the approximate ones obtained using Eqs (74) and (75), respectively. In most cases, the approximate wake functions W T,direct (ξ) and W T,indirect (ξ) well reproduce the exact ones by selecting the segment size as 2Δz = 2σ z /5. The approximate result of the indirect space-charge wake function W T,indirect (ξ) with σ z = 0.1 m and β = 0.1 (bottom left), for the non-relativistic and shorter bunch case, shows the most significant deviation from the exact indirect wake functions W T,indirect (ξ). This deviation is reduced as the beam becomes more relativistic, as shown in the result for the beam with β = 0.9 and σ z = 0.1 m (bottom right).  (77) and (78) and the exact (right) results obtained using Eqs (24) and (25) of the indirect wake function W T,indirect (ξ) and the direct wake function W T,direct (ξ) for σ z = 10 mm, where r b = 10 mm and a = 40 mm. The indirect wake function W T,indirect (ξ) (dashed) and the direct wake function (solid) W T,direct (ξ) are read using the scale markings on the left and the right vertical axes, respectively. The purple and red lines show the wake functions for β = 0.1 m and the green and black lines for β = 0.9 m, respectively. Figure 10. The rms bunch length σ z dependence of W T,indirect (ξ) according to Eq. (75) and W T,direct (ξ) according to Eq. (74) for fixed segment size 2Δz = 2σ z /5, where r b = 10 mm and a = 40 mm. The upper, middle and lower panels show the wake functions for σ z = 1 m, for σ z = 0.5 m, and for σ z = 0.1 m, respectively. The left and right panels show the wake functions for β = 0.1, and for β = 0.9, respectively. The dashed (read using the scale markings on the left axis) and solid lines (read using the scale markings on the right axis) show the indirect W T,indirect (ξ) and the direct W T,direct (ξ) space-charge wake functions, respectively. The black and purple lines show the exact wake functions obtained according to Eqs (24) and (25), respectively. The red and green lines show the approximate wake functions obtained using Eqs (74) and (75), respectively.