Time dependence of Bragg forward scattering and self-seeding of hard x-ray free-electron lasers

TL;DR

This paper develops an analytic framework for understanding the time-dependent behavior of Bragg scattering in x-ray free-electron lasers, linking temporal delays to spatial shifts, and applies it to improve seed coherence in FELs.

Contribution

It introduces a coupled wave dynamical theory for x-ray diffraction that analytically describes the spatiotemporal response of Bragg scattering to short pulses.

Findings

Derived analytic expressions for x-ray pulse profiles after Bragg scattering.

Established a relationship between time delay and transverse spatial shift of the wave.

Provided an analytic model for forward scattering in monochromatically seeded FELs.

Abstract

Free-electron lasers (FELs) can now generate temporally short, high power x-ray pulses of unprecedented brightness, even though their longitudinal coherence is relatively poor. The longitudinal coherence can be potentially improved by employing narrow bandwidth x-ray crystal optics, in which case one must also understand how the crystal affects the field profile in time and space. We frame the dynamical theory of x-ray diffraction as a set of coupled waves in order to derive analytic expressions for the spatiotemporal response of Bragg scattering from temporally short incident pulses. We compute the profiles of both the reflected and forward scattered x-ray pulses, showing that the time delay of the wave $\tau$ is linked to its transverse spatial shift $\Delta x$ through the simple relationship $\Delta x = c\tau \cot\theta$, where $\theta$ is the grazing angle of incidence to the diffracting planes. Finally, we apply our findings to obtain an analytic description of Bragg forward scattering relevant to monochromatically seed hard x-ray FELs.