At the point of saturation in a high-gain free-electron laser (FEL) the light is fully transversely coherent. The number and evolution of the transverse modes is important for the effective tune-up and subsequent operation of FELs based on the photon beam characterization and in designing multi-module devices that rely on relatively stable saturation distances in each module. In the latter, this is particularly critical since each section will seed another module. Overall, in a single- or multi-module device, experimental users will desire stability in power and in photon beam quality. Using a numerical simulation code, the evolution of the transverse modes in the high-gain free-electron laser (FEL) is examined and is discussed. In addition, the transverse modes in the first few higher nonlinear harmonics are investigated.
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