Robustness to deposition error is conventionally treated as a post-hoc Monte-Carlo check in multilayer anti-reflection (AR) coating design: a nominally optimal stack is found first, and its manufacturing tolerance is assessed afterwards. We move robustness directly into the design objective. Instead of minimizing the mean reflectance, we minimize the conditional value-at-risk (CVaR) of the band-mean reflectance under the distribution of deposition errors, so that the optimizer is driven by the worst tail of likely fabrication outcomes rather than by an idealized nominal spectrum. This yield-aware formulation is combined with an Optimal-Transport Conditional Flow Matching (OT-CFM) generator operating under a hard manufacturability constraint enforced by a smooth box reparameterization, and with a differentiable transfer-matrix forward model validated against the analytic quarter-wave result to machine precision. On a six-layer MgF<InlineEquation ID="IEq1"><EquationSource Format="TEX">\(_2\)</EquationSource></InlineEquation>/SiO<InlineEquation ID="IEq2"><EquationSource Format="TEX">\(_2\)</EquationSource></InlineEquation> stack evaluated on <InlineEquation ID="IEq3"><EquationSource Format="TEX">\(10^5\)</EquationSource></InlineEquation> held-out deposition realizations applied identically to every design, the CVaR-robust design lowers the 99th-percentile band-mean reflectance from 1.825 0.0257 pp (95% paired-bootstrap confidence interval <InlineEquation ID="IEq4"><EquationSource Format="TEX">\([+0.0230,+0.0274]\)</EquationSource></InlineEquation> pp; <InlineEquation ID="IEq5"><EquationSource Format="TEX">\(+1.41\%\)</EquationSource></InlineEquation> relative, CI <InlineEquation ID="IEq6"><EquationSource Format="TEX">\([+1.26,+1.50]\%\)</EquationSource></InlineEquation>), at the cost of a 0.011 pp increase in nominal mean reflectance. Because both designs are scored on the same realizations, this is a paired comparison in which the shared sampling noise cancels; the improvement is small in absolute terms, and we report it together with the sampling uncertainty that establishes it. Repeating the entire optimize-then-evaluate workflow under five independent random seeds reproduces the effect in every run (<InlineEquation ID="IEq7"><EquationSource Format="TEX">\(+1.22 \pm 0.10\%\)</EquationSource></InlineEquation>, mean ± s.e.m.). A deposition-error sweep shows the advantage is absent at <InlineEquation ID="IEq8"><EquationSource Format="TEX">\(\sigma = 4 nm\)</EquationSource></InlineEquation>, opens above that break-even point, and then plateaus: the absolute tail reduction increases monotonically across the sweep, whereas the relative gain rises to <InlineEquation ID="IEq9"><EquationSource Format="TEX">\(+2.34\%\)</EquationSource></InlineEquation> at <InlineEquation ID="IEq10"><EquationSource Format="TEX">\(\sigma = 12 nm\)</EquationSource></InlineEquation> and settles at <InlineEquation ID="IEq11"><EquationSource Format="TEX">\(+2.19\%\)</EquationSource></InlineEquation> at 16 nm. A material-choice ablation, restricted to the three six-layer recipes tested here on fused silica over 430 nm to 780 nm, documents a negative result: within this design space, introducing high-index TiO<InlineEquation ID="IEq12"><EquationSource Format="TEX">\(_2\)</EquationSource></InlineEquation> degrades AR performance.

Yield-aware generative inverse design of anti-reflection coatings via optimal-transport flow matching with a conditional-value-at-risk objective

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2026年8月5日

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Erfan Ghapanvari

摘要

Robustness to deposition error is conventionally treated as a post-hoc Monte-Carlo check in multilayer anti-reflection (AR) coating design: a nominally optimal stack is found first, and its manufacturing tolerance is assessed afterwards. We move robustness directly into the design objective. Instead of minimizing the mean reflectance, we minimize the conditional value-at-risk (CVaR) of the band-mean reflectance under the distribution of deposition errors, so that the optimizer is driven by the worst tail of likely fabrication outcomes rather than by an idealized nominal spectrum. This yield-aware formulation is combined with an Optimal-Transport Conditional Flow Matching (OT-CFM) generator operating under a hard manufacturability constraint enforced by a smooth box reparameterization, and with a differentiable transfer-matrix forward model validated against the analytic quarter-wave result to machine precision. On a six-layer MgF\(_2\)/SiO\(_2\) stack evaluated on \(10^5\) held-out deposition realizations applied identically to every design, the CVaR-robust design lowers the 99th-percentile band-mean reflectance from 1.825 0.0257 pp (95% paired-bootstrap confidence interval \([+0.0230,+0.0274]\) pp; \(+1.41\%\) relative, CI \([+1.26,+1.50]\%\)), at the cost of a 0.011 pp increase in nominal mean reflectance. Because both designs are scored on the same realizations, this is a paired comparison in which the shared sampling noise cancels; the improvement is small in absolute terms, and we report it together with the sampling uncertainty that establishes it. Repeating the entire optimize-then-evaluate workflow under five independent random seeds reproduces the effect in every run (\(+1.22 \pm 0.10\%\), mean ± s.e.m.). A deposition-error sweep shows the advantage is absent at \(\sigma = 4 nm\), opens above that break-even point, and then plateaus: the absolute tail reduction increases monotonically across the sweep, whereas the relative gain rises to \(+2.34\%\) at \(\sigma = 12 nm\) and settles at \(+2.19\%\) at 16 nm. A material-choice ablation, restricted to the three six-layer recipes tested here on fused silica over 430 nm to 780 nm, documents a negative result: within this design space, introducing high-index TiO\(_2\) degrades AR performance.