通过基于自动微分的海森光谱分析实现稳健光学设计和闭式公差分析
Robust optical design and closed-form tolerancing through autodiff-based Hessian spectral analysis
AI总结：
研究通过分析海森矩阵的扰动本征模来洞察性能退化机制，提出用缩放海森矩阵迹作为稳健性指标，经本征模分解可揭示主导敏感方向并给出确定性公差预算，替代蒙特卡洛采样。
摘要：
稳健光学设计需要了解制造和对准公差如何降低系统性能。我们表明，分析海森矩阵的扰动本征模能提供关于性能退化机制（如耦合）的定性见解，这是仅基于雅可比矩阵的经典灵敏度矩阵分析所无法看到的。通过包络定理，我们证明了完全补偿系统的一阶灵敏度与未补偿系统相同；重新聚焦仅通过海森矩阵的舒尔补在二阶起作用。我们提出缩放海森矩阵的迹，即$\Tr(\mathbf{S}\mathbf{H}\mathbf{S})$，作为一个单一的标量稳健性指标。在离轴三镜消像散系统上进行了演示，并扩展到一个二十三参数的表面形状模型，本征模分解揭示了主导灵敏度方向，并产生了确定性的公差预算，取代了昂贵的蒙特卡罗采样。
英文摘要：
Robust optical design demands quantitative knowledge of how manufacturing and alignment tolerances degrade system performance. We show that analysing the perturbation eigenmodes of the Hessian matrix gives qualitative insight about the mechanisms of performance degradation (such as couplings) that is invisible to classical sensitivity-matrix analysis based on the Jacobian alone. Via the envelope theorem, we prove that the first-order sensitivity of the fully compensated system is identical to that of the uncompensated one; refocusing only acts at second order through the Schur complement of the Hessian. We propose the trace of the tolerance-scaled Hessian %,$\Tr(\mathbf{S}\mathbf{H}\mathbf{S})$, as a single scalar robustness metric. Demonstrated on an off-axis three-mirror anastigmat and scaled to a twenty-three-parameter surface-figure model, eigenmode decomposition reveals the dominant sensitivity directions and yields deterministic tolerance budgets that replace costly Monte Carlo sampling.
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