  The surface errors of freeform surfaces reflect the manufacturing
complexities and significantly impact the feasibility of processing designed
optical systems. With multiple degrees of freedom, freeform surfaces pose
challenges in surface tolerance analysis in the field. Nevertheless, current
research has neglected the influence of surface slopes on the directions of ray
propagation. A sudden alteration in the surface slope will lead to a
corresponding abrupt shift in the wavefront, even when the change in surface
sag is minimal. Moreover, within the realm of freeform surface manufacturing,
variation in surface slope across different frequency bands may give rise to
unique surface variation. Within the context of this study, we propose a
tolerance analysis method to analyze surface variation in freeform surfaces
considering surface frequency band slopes based on real ray data. This approach
utilizes real ray data to rapidly evaluate surface variation within a specified
frequency band of surface slopes. Crucially, our proposed method yields the
capability to obtain system surface variation with significant wavefront
aberration, in contrast to previous methodologies. The feasibility and
advantages of this framework are assessed by analyzing a single-mirror system
with a single field and an off-axis two-mirror system. We expect to integrate
the proposed methodology with freeform surface design and manufacturing,
thereby expanding the scope of freeform optics.
