Models of Soft-Edge Elements

Similarly to other beam dynamics codes, most beamline elements are modeled by using an ideal, hard-edge approximation for the potentials and fields. In particular, this model assumes that the fields are independent of the longitudinal path-length coordinate \(s\). However, ImpactX also supports several additional soft-edge element models. The models include:

  • RFCavity - soft-edge model of an RF cavity using on-axis field data \((z,E_z(z))\)

  • SoftSolenoid - soft-edge model of a solenoid using on-axis field data \((z,B_z(z))\)

  • SoftQuadrupole - soft-edge model of a solenoid using on-axis field gradient data \((z,\partial B_y(z)/\partial x)\)

For these elements, the user may specify the on-axis field or field gradient in one of two forms:

  • using tabulated on-axis data for the field or gradient (depending on the element type)

  • using a set of precomputed Fourier coefficients obtained from the on-axis data

The vector potential off-axis is then determined from Maxwell’s equations. In either of the above cases, the field is represented internally using a set of Fourier coefficients. If \(g(z)\) denotes the on-axis field profile, the Fourier coefficients are defined such that:

\[\begin{split}\begin{align} g(z) &= \frac{c_0}{2} + \sum_{j=1}^{\rm ncoef}c_j\cos\left(\frac{2\pi j(z-z_{\rm mid})}{L}\right)+\sum_{j=1}^{\rm ncoef}s_j\sin\left(\frac{2\pi j(z-z_{\rm mid})}{L}\right) \\ z_{\rm mid}&=(z_{\rm max}+z_{\rm min})/2,\quad\quad L=z_{\rm max}-z_{\rm min}. \end{align}\end{split}\]

An example illustrating the construction of the Fourier coefficients from on-axis data appears here.

The internal representation of the field is scaled based on the user-specified inputs. In particular, the longitudinal coordinate is scaled to coincide with the element length. Likewise, the field strength is scaled based on input parameters (called escale, bscale, or gscale for the three element types above).

Examples demonstrating the use of soft-edge elements include: