Demonstration of resolving power $\lambda/\Delta\lambda > 10,000$ for a space-based x-ray transmission grating spectrometer

We present measurements of the resolving power of a soft x-ray spectrometer consisting of 200 nm-period lightweight, alignment-insensitive critical-angle transmission (CAT) gratings and a lightweight slumped-glass Wolter-I focusing mirror pair. We measure and model contributions from source, mirrors, detector pixel size, and grating period variation to the natural line width spectrum of the Al and Mg K$_{\alpha_1 \alpha_2}$ doublets. Measuring up to 18$^{\rm th}$ diffraction order at characteristic Al-K wavelengths we consistently obtain small broadening due to gratings corresponding to a minimum effective grating resolving power $R_g > 10,000$ with 90\% confidence. Upper limits are often compatible with $R_g = \infty$. Independent fitting of different diffraction orders, as well as ensemble fitting of multiple orders at multiple wavelengths, gives compatible results. Our data leads to uncertainties for the Al-K$_{\alpha}$ doublet line width and line separation parameters 2-3 times smaller than values found in the literature. Data from three different gratings are mutually compatible. This demonstrates that CAT gratings perform in excess of the requirements for the Arcus Explorer mission and are suitable for next-generation space-based x-ray spectrometer designs with resolving power 5-10 times higher than the transmission grating spectrometer on the Chandra X-ray Observatory.