Abstract
We present a power-fit formula, obtained from a variational analysis using three-dimensional free-electron laser theory, for the gain length of a high-gain free-electron laser's fundamental mode in the presence of diffraction, uncorrelated energy spread, and longitudinal space-charge effects. The approach is inspired by the work of Xie [Nucl. Instrum. Methods Phys. Res., Sect. A 445, 59 (2000)], and provides a useful shortcut for calculating the gain length of the fundamental Gaussian mode of a free-electron laser having strong space-charge effects in the 3D regime. The results derived from analytic theory are in good agreement with detailed numerical particle simulations that also include higher-order space-charge effects, supporting the assumptions made in the theoretical treatment and the variational solutions obtained in the single-mode limit.
Highlights
High-gain free-electron lasers (FELs) operate on the principle that tunable, narrow bandwidth light pulses can be emitted and amplified many orders of magnitude by the strong beam-radiation instability affecting a relativistic electron beam (e-beam) traversing a periodic undulator
The distance along the undulator it takes for the power of the emitted light to increase by a factor of e during the exponential growth regime is known as the power gain length
We focus on the three of highest relevance to optical regime FELs: the diffraction parameter, d 1⁄4 L1D=2k2x, which quantifies the extent to which transverse effects contribute to the gain for a FEL with wavelength 1⁄4 2=k; the scaled energy spread parameter 1⁄4 2ku L1D, which captures the contribution of the rms uncorrelated e-beam energy spread ; and "p 1⁄4 2pL1D, the space-charge parameter, which is scaled to be twice the plasma phase advance over a one-dimensional gain length
Summary
High-gain free-electron lasers (FELs) operate on the principle that tunable, narrow bandwidth light pulses can be emitted and amplified many orders of magnitude by the strong beam-radiation instability affecting a relativistic electron beam (e-beam) traversing a periodic undulator. There has been increased interest in obtaining high average power using high-gain amplifiers operating at $100 MeV e-beam energies and Ie $ 1 kA currents where longitudinal space charge will strongly affect the FEL performance [10,11]. The investigation of longitudinal space-charge waves in high-brightness beams is still currently of great theoretical interest [12,13], following past emphases on understanding transverse beam-plasma oscillations [14] With all this in mind, we note that there is currently no handy formulation for quickly predicting important FEL characteristics when space charge has significant influence, despite the fact that there has been strong historical interest and recent experimental investigations into FELs that operate under these conditions. We emphasize that the results obtained are in general limited to FELs that operate from the optical to the far IR, or otherwise satisfy the stated parametric constraints
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