Analytical formulas for short bunch wakes in a flat dechirper

We develop analytical models of the longitudinal and transverse wakes, on and off axis for a flat, corrugated beam pipe with realistic parameters, and then compare them with numerical calculations, and generally find good agreement. These analytical “first order” formulas approximate the droop at the origin of the longitudinal wake and of the slope of the transverse wakes; they represent an improvement in accuracy over earlier, “zeroth order” formulas. In example calculations for the RadiaBeam/LCLS dechirper using typical parameters, we find a 16% droop in the energy chirp at the bunch tail compared to simpler calculations. With the beam moved to $200\mu \mathrm{m}$ from one jaw in one dechirper section, one can achieve a 3 MV transverse kick differential over a $30\mu \mathrm{m}$ length.

Figure 1

Geometry of a dechirper unit. For a round dechirper the radius of the aperture is designated as $a$; in the flat case, for a vertical dechirper unit, $2a$ represents the gap, and the structure is unbounded in $x$. The blue ellipse represents an electron beam propagating along the $z$ axis.

Figure 2

The function $e^x\mathrm{erfc}(\sqrt{x})$ (blue solid), and the approximation $e^{-\sqrt{x}}$, which differs by less than 1.2% over the range plotted (orange dashes).

Figure 3

Longitudinal wake on axis for the RadiaBeam/LCLS dechirper nominal parameters, comparing the numerical inverse Fourier transform of the general impedance equation, Eq. (10) (blue curve), with the approximation, Eq. (15) (the dashes). The scale factor $s_{0r}=190\mu \mathrm{m}$.

Figure 4

Longitudinal bunch wake for a Gaussian beam on axis, with ${\sigma }_z=10\mu \mathrm{m}$, in a 2-m section of the RadiaBeam/LCLS dechirper. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.

Figure 5

Longitudinal wake off axis by $y=0.5\mathrm{mm}$, comparing the numerical inverse Fourier transform of the general impedance equation, Eq. (10) (blue curve), with the first order approximation, Eq. (15) (the dashes).

Figure 6

Longitudinal bunch wake for a Gaussian beam with ${\sigma }_z=10\mu \mathrm{m}$ offset in the RadiaBeam/LCLS dechirper. The offset is $y=0.5\mathrm{mm}$ and the half-aperture is the nominal $a=0.7\mathrm{mm}$. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.

Figure 7

Quad and dipole wakes on axis for the RadiaBeam/LCLS dechirper nominal parameters, comparing the numerical inverse Fourier transform of the impedance equations, Eqs. (19), (24) (blue curves), with the approximations, Eqs. (22), (23), (27) (the dashes). Note that the curves for $w_{yd}{a}^3$ have been shifted up by 0.02 units to improve visibility.

Figure 8

Quad bunch wake for a Gaussian beam on axis, with ${\sigma }_z=10\mu \mathrm{m}$, in the RadiaBeam/LCLS dechirper. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.

Figure 9

Dipole bunch wake for a Gaussian beam on axis, with ${\sigma }_z=10\mu \mathrm{m}$, in the RadiaBeam/LCLS dechirper. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.

Figure 10

Dipole wake at offset $y=0.5\mathrm{mm}$ for the RadiaBeam/LCLS dechirper, comparing the numerical inverse Fourier transform of the impedance equation, Eq. (29) with $x_0=x$, $y_0=y$ (blue curve), with the approximation, Eq. (34) (the dashes).

Figure 11

Quad wake at offset $y=0.5\mathrm{mm}$ for the RadiaBeam/LCLS dechirper, comparing the numerical inverse Fourier transform of the impedance equation, Eq. (35) (blue curve), with the approximation, Eq. (36) (the dashes).

Figure 12

Dipole bunch wake for a Gaussian beam with ${\sigma }_z=10\mu \mathrm{m}$ offset in the RadiaBeam/LCLS dechirper. The offset is $y=0.5\mathrm{mm}$ and the half-aperture is the nominal $a=0.7\mathrm{mm}$. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.

Figure 13

Vertical quad bunch wake for a Gaussian beam with ${\sigma }_z=10\mu \mathrm{m}$ offset in the RadiaBeam/LCLS dechirper. The offset is $y=0.5\mathrm{mm}$ and the half-aperture is the nominal $a=0.7\mathrm{mm}$. Given are the numerical results of ECHO(2D) (blue), and the analytical zeroth order (red) and 1st order results (green). The bunch shape $\lambda (s)$ with the head to the left is shown in black.