Angel type lobster-eye X-ray optical devices have the unique capability of 4π solid angle focusing with one of the most optimal effective area-to-weight ratios. Since the micropore structures of the micropore optics (MPO) devices reflect X-ray photons, the statistical characteristics of the micropores are critical to the focusing performance. After the MPO devices are thermally curved into sphere profiles, all the millions of micropores parallel to each other on an MPO device initially point towards a common curvature center to focus on X-ray photons. Therefore, the statistical distribution of the directional characteristics of micropores can be described by a virtual sphere surface, which is defined as the X-ray sphere profile with X-ray curvature. The physical sphere profiles of the devices that are formed by thermal bending and can be measured by instruments in visible bands are referred to as optical surface. Under ideal conditions, the change of the optical profile of a device should be consistent with the X-ray profile with the curvature radius difference of half of the device thickness, since both profiles are formed in thermal bending simultaneously. The X-ray curvature can only be measured by the X-ray beamline facility in vacuum, which is related to the X-ray focusing performance of a device. The optical curvature is applied to optics assembly fabrication, i.e., mounting devices onto frame in the air. That means the X-ray performance of the lobster-eye assemblies is closely related to both profiles. In our work, 468 MPO devices applied for wide field telescope (WXT) on the Einstein Probe satellite are tested with the statistical characteristics of X-ray and optical curvature radius are analyzed.

The X-ray curvature of MPO devices can be measured in an X-ray beamline facility, which has a point-like X-ray source far away from MPO chips. A CMOS camera is applied to detect X-ray photons. The centers of the X-ray source CMOS sensor are carefully aligned to form the baseline of the facility. The position and pointing of the MPO devices can be adjusted by a computer-controlled hexapod around the baseline. By aligning the center of the MPO device to the baseline and adjusting the distance between the CMOS and the center of the MPO device, the sharpest focal spot can be observed on the CMOS sensor. The X-ray curvature (XCur) can be measured and characterizes the central X-ray profile of the device. By rotating the MPO devices, the focal spot of a device moves to the CMOS sensor. By fitting the rotation with the shift of the focal spot, a mean X-ray curvature (XCur_scan) of the MPO device is obtained, which characterizes the mean X-ray profile of the whole device. By applying an automatic alignment telescope, the optical curvature can be measured (Cur). All the data from 438 devices are measured and analyzed, including the statistical properties of the difference between X-ray curvature and optical curvature of each chip.

The ideal mean difference among XCur, XCur_scan, and Cur should be zero. However, the XCur and XCur_scan differ from each other by 0.50%, and there are two peak differences between X-ray curvature and optical curvature, i.e., the mean value of XCur is larger than that of Cur by 0.68% and 1.52%, respectively.

Statistical analysis of the curvature radius of the MPO device reveals that during thermal bending, the possibility of parallel microspores pointing to a common center is smaller than expected, because statistically XCur is larger than Cur, and the micropore is not completely columnar but a little tapered. In addition, XCur is statistically larger than XCur_scan, indicating that the central area of the device curves less than the whole device does. For lobster-eye optical devices, the inconsistency among XCur, XCur_scan, and Cur restricts further optimization of the X-ray focusing performance, so further development of the thermal processing is needed.