Imaging X-rays by direct focusing offers greater sensitivity and a higher dynamic range compared to techniques based on indirect imaging. The Focusing Optics X-ray Solar Imager (FOXSI) is a sounding rocket payload that uses seven sets of nested Wolter-I figured mirrors to observe the Sun in hard X-rays through direct focusing. Characterizing the performance of these optics is critical to optimize their performance and to understand their resulting data. In this paper, we present a ray-tracing simulation we created and developed to study Wolter-I X-ray mirrors. We validated the accuracy of the ray-tracing simulation by modeling the FOXSI rocket optics. We found satisfactory agreements between the simulation predictions and laboratory data measured on the optics. We used the ray-tracing simulation to characterize a background pattern of singly reflected rays (i.e., ghost rays) generated by photons at certain incident angles reflecting on only one of a two-segment Wolter-I figure and still reaching the focal plane. We used the results of the ray-tracing simulation to understand, and to formulate a set of strategies that can be used to mitigate, the impact of ghost rays on the FOXSI optical modules. These strategies include the optimization of aperture plates placed at the entrance and exit of the smallest Wolter-I mirror used in FOXSI, a honeycomb type collimator, and a wedge absorber placed at the telescope aperture. The ray-tracing simulation proved to be a reliable set of tools to study Wolter-I X-ray optics. It can be used in many applications, including astrophysics, material sciences, and medical imaging.
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