The monolithic Wolter mirror is an effective and promising solution for providing stable X-ray micro- and nano-focusing beams for synchrotron radiation or free-electron laser beamlines. In scientific experiments, nanometer-scale X-ray probes play a crucial role in resolving intricate sample features through scanning. The stability of the beam is a critical factor, especially for scanning nanoprobe applications. Wolter optics approximately satisfies Abbe’s sine condition, making it an advanced optical design for X-ray imaging applications, particularly for imaging off-axis object points. Despite their long history, Wolter mirrors have not been widely used in beamlines due to the extremely high surface figure accuracy required. In recent years, improvements in manufacturing processes have made their gradual application in practical X-ray facilities feasible. Currently, few studies focus on design optimization methods for Wolter mirrors under machining capacity constraints. Wave propagation simulation methods are insufficient for guiding the application of Wolter mirrors in X-ray micro- and nano-probes, as they fail to account for aberration effects and coating material reflectivity. Manufacturing these tube-like mirrors with high precision presents significant challenges in metrology and fabrication. To address these challenges, mirror shapes can be optimized during the design phase, taking metrology and fabrication constraints into account.

The optimization of parameters for Wolter mirrors in X-ray beamlines under machining and experimental constraints is proposed. By incorporating Abbe’s sine condition and considering the relationship between the lengths of ellipsoidal and hyperbolic surfaces, five parameters are selected: working distance, aperture angle, source-to-focus distance, mirror length, and magnification factor. The monolithic and closed Wolter configuration is then determined. The optical performance of the Wolter mirrors, including focused spot size and transmission efficiency, is evaluated through ray-tracing simulations. Working distance and magnification factor are limited by experimental conditions, while practical constraints related to mirror metrology and fabrication, such as mirror length and downstream aperture diameter, also play a role. The highest transmission efficiency is achieved by optimizing the distance between the source and the focus, as well as the aperture angle. The ray-tracing program uses five parameters to establish the Wolter model, as in the design phase. The rotating hyperboloid surface and the rotating ellipsoid surface are integrated into one optical element. The ray-tracing code can track multiple reflections within such a single optical element while adhering to coated material reflectivity, which is useful for investigating alignment errors. The angle and position of each ray obtained through ray tracing are used to calculate the smallest spot size at the focal plane. An ellipsoidal mirror with the same magnification factor as the Wolter mirror is built and the effects of rotational and translational alignment errors on optical performance are compared. By analyzing variations in focused spot size and transmission efficiency, the tolerance limits for alignment errors are determined.

The optimization process for a Wolter mirror with specific constraints such as experimental requirements and fabrication challenges is demonstrated. The designed Wolter mirror achieves the highest transmission efficiency. The results show that when the divergence angle of the light source is large, Wolter mirrors are unsuitable for high-energy X-rays due to their lower transmission efficiency, a disadvantage compared to 1D reflective mirrors. This is primarily due to practical fabrication constraints, such as mirror length and downstream aperture diameter. In the alignment error study, the full width at half maximum (FWHM) of the spot size increases to 120%, and transmission efficiency drops to 80% of the original value as the tolerance limit. The results show that for rotational alignment errors, the tolerance limit of the Wolter mirror is 60 times that of the elliptical mirror with the same magnification factor. The tolerance limit for translational alignment errors in the Wolter mirror is also significantly higher. These results demonstrate that Wolter mirrors have commendable tolerance to fluctuations in beam angle, ensuring stable focused spots. These simulations validate the efficacy and robustness of the designed Wolter mirror and the proposed optimization approach.

The approach that integrates constraints into the optimization framework discussed in this paper enhances the efficiency and reliability of the Wolter mirror design. In addition, the ray-tracing code, based on the XRT program, allows the practical application to beamlines, providing insights into the actual performance of the Wolter optical system and facilitating smoother transitions from concept to implementation. The alignment error tolerance analysis during the design phase assists in designing the control system and developing a compact experimental setup, including both the mirror and sample mounting stages.