Active Support Correction and Variable Triangular Layout Optimization for an Annular Thin Mirror in Optical Remote-Sensing Systems

Surface-figure stability is essential to high-resolution optical remote-sensing systems that employ lightweight annular primary mirrors. This study proposes a variable triangular active support layout for a 660 mm annular thin mirror with a 100 mm central aperture. The layout is parameterized within a one-sixth annular sector and constructed over the full aperture by six-fold rotational replication. Finite-element influence functions, Annular Zernike modal fitting, and Kriging surrogate modeling are integrated to assess modal controllability while reducing the cost of repeated finite-element analyses. The objective function accounts for surface-figure maintenance under gravity, correction residuals for representative low-order Annular Zernike modes, geometric constraints, and the actuator-force limit. Optimization is performed using the covariance matrix adaptation evolution strategy (CMA-ES), followed by local refinement. For a 36-point support configuration, the optimized variable triangular layout satisfies the 12.66 nm RMS residual requirement under axial and radial gravity and for all representative Z4–Z11 target surfaces. Among the annular, triangular, square, hexagonal, Fibonacci, and proposed layouts, the proposed layout yields the lowest composite objective-function value, a mean corrected RMS residual of 2.15 nm, and an effective response-matrix rank of 35. Monte Carlo simulations with independent ±1 mm support-position perturbations further demonstrate its robustness to installation errors.