Analytical treatment of the wakefields driven by transversely shaped beams in a planar slow-wave structure

The suppression of transverse wakefield effects using transversely elliptical drive beams in a planar structure is studied with a simple analytical model that unveils the geometric nature of this phenomenon. By analyzing the suggested model we derive scaling laws for the amplitude of the longitudinal and transverse wake potentials as a function of the Gaussian beam ellipticity—${\sigma }_x/a$. We explicitly show that in a wakefield accelerator application it is beneficial to use highly elliptical beams for mitigating transverse forces while maintaining the accelerating field. We consider two scaling strategies: (1) aperture scaling, where we keep a constant charge to have the same accelerating gradient as in a cylindrical structure and (2) charge scaling, where aperture is the same as in the cylindrical structure and charge is increased to match the gradient.

Figure 1

Initial plane $\psi$ and schematics of the conformal mapping of the $\omega$ plane.

Figure 2

Field lines of the transverse wake potential that is given by Eq. (14) for the source (red circle) located in the center of the wakefield structure (left panel) and displaced from the center (right panel) $y_0=0.2a$.

Figure 3

$x$-dependence for the longitudinal wake potential (left panel) and $y$-component of the transverse wake potential (right panel) for a point particle. Both plots are normalized to be unity at the point $x=0$. The red shaded area on the right panel shows the focusing region of $w_y$.

Figure 4

$x$-dependence for the longitudinal wake potential (left panel) and $y$ part of the transverse wake potential (right panel) for two point particle separated by a distance equal to the distance to the location of the minimum of the $w_y$. Dashed lines are normalized wake potentials of each particle separately. Solid lines are the resulting wake potentials.

Figure 5

Normalized maximum of the longitudinal wake potential (red) and maximum of the $y$-component of the transverse wake potential (blue) for the Gaussian distribution as a functions of the ${\sigma }_x/a$. Dashed lines are asymptote given by Eqs. (20) and (21).

Figure 6

Normalized maximum of the longitudinal wake potential (red) and $x$-component of the transverse wake potential at $x={\sigma }_x$ (blue) for the Gaussian distribution as a functions of the ${\sigma }_x/a$. Dashed line is the asymptote given by Eq. (24) and multiplied by the same normalization coefficient as blue $x$-component of the transverse wake potential at $x={\sigma }_x$.