Longitudinal impedance and wake from XFEL undulators. Impact on current-enhanced SASE schemes

TL;DR

This paper derives a comprehensive model for the longitudinal impedance and wake functions in XFEL undulators, considering finite bunch size and arbitrary parameters, and assesses their impact on current-enhanced SASE schemes like ESASE.

Contribution

It provides a new analytical expression for impedance and wake functions in undulators with finite transverse size, extending previous models that assumed line density distributions.

Findings

Impedance-induced energy spread can significantly degrade FEL performance.

The derived expressions are applicable to practical LCLS configurations.

Contrasts with prior literature by including undulator effects in wake calculations.

Abstract

In this article we derive longitudinal impedance and wake function for an undulator setup with arbitrary undulator parameter, taking into account a finite transverse size of the electron bunch. Earlier studies considered a line density-distribution of electrons instead. We focus our attention on the long-wavelength asymptote (compared with resonance wavelength), at large distance of the electron bunch from the undulator entrance compared to the overtaking length, and for large vacuum-chamber size compared to the typical transverse size of the field. These restrictions define a parameter region of interest for practical applications. We calculate a closed expression for impedance and wake function that may be evaluated numerically in the most general case. Such expression allows us to derive an analytical solution for a Gaussian transverse and longitudinal bunch shape. Finally, we study the feasibility of current-enhanced SASE schemes (ESASE) recently proposed for LCLS, that fall well-within our approximations. Numerical estimations presented in this paper indicate that impedance-induced energy spread is sufficient to seriously degrade the FEL performance. Our conclusion is in contrast with results in literature, where wake calculations for the LCLS case are given in free-space, as if the presence of the undulator were negligible.