Efficient field calculation of 3D bunches on general trajectories

The program TRAFIC4 calculates the effects of space charge forces and coherent synchrotron radiation on bunches moving on curved trajectories. It calculates the fields acting on the particles, as they travel along the beamline, from first principles. The bunch is modeled by small overlapping 1d, 2d or 3d continuous Gaussian sub-bunches. Previously, a 1d or 2d integration of the retarded source distribution was used to calculate the fields of 1d and 2d sub-bunches. A new approximation is proposed to avoid the need for multi-dimensional integration: a 2d or 3d sub-bunch can be interpreted as the convolution of a 1d sub-bunch with a transverse density function. In the same way, the field can be obtained by a convolution of the 1d-field with the transverse density function. The near field of a 1d sub-bunch can be split into a singular part which is dominated by local effects and a residual part, which depends essentially on long-range interactions. As the singular part can be described by analytical functions, the convolution can be performed efficiently. The residual part depends weakly on the transverse offset. Therefore, only one or few sampling points are needed for the convolution, which significantly reduces the numerical effort.