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.