Geometry-constrained bilevel optimization of advanced Kirkpatrick-Baez optics: closed-form feasibility and multi-objective design.

Grazing-incidence micro-nano focusing must balance focal size, aberration tolerance, and mirror manufacturability under tight mechanical and total-reflection constraints. Advanced Kirkpatrick-Baez (AKB) optics reduce off-axis aberrations through a four-reflection Wolter-type geometry, but their design space is high-dimensional and strongly constrained. We present a geometry-driven bilevel optimization framework for AKB systems. An analytical inner stage derives admissible eccentricity bounds under the adopted grazing-incidence approximation and uses monotonic relations to reduce the effective search space for magnification and numerical aperture. An outer stage then applies NSGA-II/III to optimize the remaining layout variables under manufacturability and tolerance constraints. In the ablation study, the reduced formulation shows improved convergence behavior and empirical feasible-solution discovery relative to direct full-space search. The framework produces well-sampled Pareto fronts within minutes and clarifies key parameter couplings, including working distance, mirror spacing, and grazing angle. Two design studies are presented: a total-reflection AKB system with an approximately 80 nm calculated focus at a 200 mm working distance, and a theoretically proposed mechanically compatible multilayer configuration predicted to achieve a sub-7 nm focus at 50 mm. Alignment sensitivity is further evaluated by wave-optical simulation.