Wide field-of-view freeform off-axis four-mirror reflective system design for geosynchronous belt observation

To address sub-meter space debris monitoring demands in the geosynchronous belt, this paper proposes a wide-field unobscured off-axis freeform four-mirror reflective optical system. A parametric mathematical model derived from the detection signal-to-noise ratio mechanism is established to correlate the threshold entrance pupil diameter, observation phase angle, focal length, and detectable debris radius, supporting rational matching and selection of core specifications. A joint merit function is constructed by combining an independent obscuration penalty function and low-order distortion-free aberration terms derived under the pupil-field four-dimensional ray propagation framework, enabling rapid initial configuration screening and efficient system design. The system features a 400-700 nm band, a 300 mm aperture, a 620 mm focal length, and a 15^∘×12^∘ FoV, achieving <5% distortion and >95% encircled energy in 20µm×20µm binned four-pixel areas. Modeling confirms its capability to detect 0.5 m radius debris in dawn-dusk sun-synchronous orbits. This work meets the detection requirements and provides a reference for similar developments.