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Modern x-ray free-electron lasers (XFELs) produce x-ray pulses of exceptional transverse coherence. This is due largely to the process of optical guiding by which the radiation is both refractively guided by the bunched electron beam and gain guided by the preferential amplification of on-axis radiation. These effects may be summarized by an effective index of refraction, which has been used in the past to study the transverse dynamics of the FEL process with significant simplifications and approximations, but never fully self-consistently. We present here a self-consistent method for studying high gain FELs in the linear regime by approximating the FEL equations to second-order in the lateral displacement from the nominal electron beam axis. This is made possible by casting the FEL equations in the language of optical fibers with an appropriately chosen refractive index. We demonstrate that this approach is both fast and highly accurate, indicating that the most important FEL dynamics are inherently second-order. In its full form our method can capture the effects of transverse offsets in both the x-ray beam and the electron beam, making it a versatile tool for studying non-ideal effects in seeded FELs, regenerative amplifier XFELs, and even self-amplified spontaneous emission (SASE) FELs. |
