# Theory and optical design of x-ray echo spectrometers

## TL;DR

This paper advances the theory and optical design of x-ray echo spectrometers, highlighting their broadband capabilities, potential for higher resolution, and practical design considerations for improved inelastic x-ray scattering measurements.

## Contribution

It provides a detailed theoretical framework and optical design principles for x-ray echo spectrometers, enhancing their feasibility and performance assessment.

## Key findings

- Optical designs for 1-meV and 0.1-meV resolution spectrometers.
- Analysis of spectral resolution and refocusing conditions.
- Assessment of tolerances and optical components impact.

## Abstract

X-ray echo spectroscopy, a space-domain counterpart of neutron spin echo, is a recently proposed inelastic x-ray scattering (IXS) technique. X-ray echo spectroscopy relies on imaging IXS spectra, and does not require x-ray monochromatization. Due to this, the echo-type IXS spectrometers are broadband, and thus have a potential to simultaneously provide dramatically increased signal strength, reduced measurement times, and higher resolution compared to the traditional narrow-band scanning-type IXS spectrometers. The theory of x-ray echo spectrometers presented in [1] is developed here further with a focus on questions of practical importance, which could facilitate optical design and assessment of the feasibility and performance of the echo spectrometers. Among others, the following questions are addressed: spectral resolution, refocusing condition, echo spectrometer tolerances, refocusing condition adjustment, effective beam size on the sample, spectral window of imaging and scanning range, impact of the secondary source size on the spectral resolution, angular dispersive optics, focusing and collimating optics, and detector's spatial resolution. Examples of optical designs and characteristics of echo spectrometers with 1-meV and 0.1-meV resolutions are presented.