Design of a High-magnification Compact Off-axis Three-mirror Afocal Optical System

High-magnification beam expansion optical systems are critical components in space-based active optical detection and long-range laser transmission， imposing stringent requirements on beam collimation performance， energy utilization efficiency， and structural compactness. Increasing the beam expansion ratio can effectively suppress laser beam divergence and improve far-field spot quality， but it typically leads to a significant increase in system volume and a sharp rise in the difficulty of aberration control. Traditional transmissive beam expansion systems are limited by their clear aperture and face challenges such as thermal effects and material absorption under high-power laser conditions； coaxial reflective systems inevitably suffer from central obscuration and energy loss. Although off-axis reflective systems can avoid these defects， existing designs generally exhibit limitations in beam expansion ratio， design complexity， and excessive overall volume. To address these issues， this paper proposes a design method for a high-magnification compact off-axis three-mirror afocal optical system， aiming to achieve a high beam expansion ratio while maintaining system compactness and reducing the structural design difficulty of off-axis systems.The proposed design method consists of three stages：　coaxial initial structure construction， direct off-axis processing based on the focus transfer principle， and freeform surface-based aberration optimization. First， the initial structure of the coaxial three-mirror afocal system is derived based on Gaussian optics and primary aberration theory. By appropriately selecting the conic constants of the primary， secondary， and tertiary mirrors， spherical aberration， coma， astigmatism， and distortion are simultaneously eliminated under zero field conditions， resulting in a coaxial initial structure with highly balanced aberrations. Second， a direct off-axis method based on the focus transfer principle is introduced. By aligning the virtual focus of the secondary mirror with the image-side focus of the primary mirror， and the real focus of the secondary mirror with the object-side focus of the tertiary mirror， while presetting the tilt angles of the secondary and tertiary mirrors， the decenter amounts and mirror spacings of each reflector can be analytically determined， thereby directly constructing an unobscured “ring-shaped” off-axis three-mirror afocal layout， avoiding potential issues of aberration accumulation and beam expansion ratio drift that may occur in progressive off-axis processes. Finally， to expand the system field of view and correct non-rotationally symmetric aberrations introduced by decenter and tilt， freeform surfaces characterized by XY polynomials are used for optimization. By establishing a correspondence between XY polynomial terms and Seidel aberrations， and gradually introducing selected polynomial terms based on quantitative analysis of dominant residual aberrations at different optimization stages， the controllability and interpretability of the optimization process are enhanced.Based on the above method， a high-magnification compact off-axis three-mirror afocal optical system was successfully designed. The system has an effective clear aperture of 500 mm， a beam expansion ratio of 20×， and a full field of view of 1 mrad. The overall structure is constrained within an envelope circle of 900 mm in diameter， with a system length to entrance pupil diameter ratio of 1.78， demonstrating high structural compactness. Image quality evaluation results show that the RMS wavefront error across the full field of view is better than λ/14 （λ=632.8 nm）， and the Strehl ratio is consistently above 0.8， indicating excellent beam transmission performance. The freeform surface sag distribution is smooth and continuous without significant abrupt changes， meeting machining process requirements. To further verify the engineering feasibility of the system， systematic tolerance analysis was conducted. Monte Carlo methods were used for statistical simulation of freeform surface coefficients， curvature radii， and alignment errors， and sensitivity analysis identified the top ten freeform surface coefficients with the most significant impact on system performance， defining their tolerance control ranges. Results from 2 000 random simulations show that over 98% of the samples meet the design requirement of RMS wavefront error better than λ/14， fully validating the stability and robustness of the system under manufacturing and alignment deviations.In summary， this paper proposes an off-axis three-mirror afocal optical system design method suitable for high-magnification beam expansion requirements. The designed system effectively balances beam expansion ratio and structural compactness while maintaining unobscured characteristics and high energy utilization efficiency. This method significantly reduces the optimization complexity in the transition from coaxial to off-axis structures， improves design efficiency， and provides a feasible and promotable technical approach for the design of high-magnification beam expansion systems in the field of space laser transmission and active optical detection.