Compared with refractive systems, reflective optical systems have the advantages of no chromatic aberration, high thermal stability, and wide working spectral range. However, coaxial reflective systems suffer from low light utilization due to central obscuration, making it difficult to achieve a large field of view. Off-axis reflective systems can solve this problem, and as a result, the research focus has gradually shifted from coaxial to off-axis designs. The off-axis three-mirror system has a limited number of mirrors, corresponding to fewer optimization parameters, and often requires the introduction of eccentricity and tilt to improve image quality during later stages of optimization, which complicates subsequent detection and assembly. Therefore, we use an off-axis four-mirror system to increase the number of optimization parameters. The primary mirror, tertiary mirror, secondary mirror, and quaternary mirror are machined in pairs on two common substrates, avoiding the need for eccentricity and tilt. This reduces the degree of freedom for installation and alignment from 20 to 10.

First, the ray heights, refraction invariants, and incident/refraction angles of the chief and marginal rays on each paraxial surface are derived and substituted into the Seidel aberration expressions to establish the relationship between aberration coefficients and system parameters. Then, the differential evolution algorithm is used to solve the resulting implicit equations, producing multiple well-corrected and compact coaxial four-mirror system configurations. To enable the shared-substrate design, the system is constrained such that d₁=-d₂=d₃, r₁=r₃, r₂=r₄, k₁=k₃, and k₂=k₄. A configuration with good imaging performance and compact structure is selected as the initial design. The field-of-view bias method is then applied to eliminate obscuration, and the field of view and aperture are gradually increased until the design requirements are met. Finally, a tolerance analysis is conducted, and the results show that the tolerance distribution is reasonable and the design meets the required technical specifications.

By substituting the ray tracing results into the Seidel aberration formula, relationships between aberration coefficients and system structural parameters are established. The differential evolution algorithm yields an initial structure with excellent multi-component image quality (Table 2). The initial layout and its corresponding performance are shown in Fig. 4. Building on this structure, the field-of-view bias method is used to eliminate obscuration, and both the field of view and aperture are gradually expanded (Fig. 6). When aspherical surfaces alone are insufficient for aberration correction, the primary mirror is designed as a freeform surface to increase the degrees of freedom for optimization. The final system parameters are listed in Tables 3 and 4, and the complete optical payout is presented in Fig. 7. Mirrors M1 and M3 share a single substrate, while M2 and M4 share another, requiring only two structural supports. No eccentricity or tilt is introduced during optimization, ensuring that the mirrors remain in quasi-coaxial alignment and simplifying assembly. The system’s imaging performance is shown in Fig. 8. The MTF curve is smooth and exceeds 0.45 at 50 lp/mm, indicating excellent imaging quality. The tolerance analysis results (Table 8) confirm that the surface tolerance distribution is reasonable and meets the required technical specifications.

In this paper, we propose a wide field-of-view off-axis four-mirror optical system based on two integrated mirror substrates. Initial configurations are derived using Seidel aberration theory and optimized globally through a differential evolution algorithm. A compact, well-corrected coaxial structure is selected and converted into an off-axis system using field-of-view biasing. Freeform surfaces are introduced to address aberrations caused by the expanded field and aperture. M1 and M3, as well as M2 and M4, are fabricated on the same substrates, eliminating the need for decenter or tilt adjustments and significantly reducing manufacturing and alignment complexity. The final system achieves a focal length of 1000 mm, an F-number of 10.5, and a field of view of 23°×0.6°. Both the optical performance and tolerance analysis demonstrate the system’s excellent imaging capabilities and practical feasibility.