In the past year, we have written several blogs on our continuing collaboration with materials scientists from the University of Pittsburgh on fabricating high performing, superomniphobic, nanostructured glass. We are fortunate to continue working with the LAMP team this year on related materials science projects.

In this post, we discuss our article Applying Bayesian Optimization to Understand Tradeoffs for Antireflective Optical Designs, by Sajad Haghanifar, Michael McCourt, Bolong Cheng, and Paul Leu, which appeared at the NeurIPS 2019 Machine Learning and the Physical Science workshop. We adapt Bayesian optimization to a different aspect of materials science research, numerical simulation.

For this paper, we study the antireflective characteristic and its relationship with the various design parameters of nanostructures through electrodynamic simulation. One of the challenges is to efficiently search for the Pareto optimal set of design parameters that jointly minimizes light reflection at multiple angles of incidence. The figure below shows the two types of nanostructures (nanowires and nanocones) and the various design parameters that we are simulating.

Figure 1: Nanowires (left) and nanocones (right) with their respective design parameters.

Two different reflective properties contribute to the comprehensive reflective performance of a material: the reflection of light projected onto the glass perpendicularly and the reflection of light at some obtuse angle. More discussion is provided in this video, produced by the University of Pittsburgh. These metrics may be correlated for part of the fabrication space, but they compete with each other near optimal performance.

Because we want both metrics to be extremely low, this problem can be naturally formulated as a multiobjective optimization. One particularly effective strategy of adapting Bayesian optimization to solve multiobjective problems is the epsilon constraint method, namely, we are solving the constrained optimization problem

\begin{align}
\min \;&\Rnormal(x), \\
\mbox{subject to }&\Runpolarized(x) < \hat{R}.
\end{align}

where \(\Rnormal\), \(\Runpolarized\) are the reflection of the glass at 0° and 60° angles of incidence respectively. \(\hat{R}\) is a threshold parameter that is adaptively changed as we observe more samples in order to search for different sections of the Pareto efficient frontier.  (Note: We used SigOpt’s Multimetric feature for this project, which implements the epsilon constraint method as part of its workflow.)

We compared our method against NSGA-II, a popular genetic algorithm for multiobjective optimization, on optimizing nanocone structures with 2 x 2 unit cell. We show that our method vastly outperforms NSGA-II (using this native implementation) for a fixed function evaluation budget of 500.

Figure 2: (Left) Sample 2×2 nanocone layout, with the unit cell labeled in red. (Right) Multiobjective BO compared to NSGA-II on optimizing 2×2 nanocone structures. This problem has 13 design parameters. NSGA-II uses a population size of 25 (tuned to perform optimally) and ran for 20 generations.

Additionally, we confirm our conjecture that we can drastically improve the reflection performance of the glass by searching in a larger design space. We confirm that with experiments on the uniform case (1×1 periodic domain shown in Figure 1 rather than the 2×2 domain shown in Figure 2). In Figure 3, we observe that the Pareto optimal designs of nanocones completely dominate those of nanowires. The SigOpt experiments that we ran can be viewed online: nanowires and nanocones.

Figure 3. (Left) All simulation results for uniform nanowire and uniform nancone settings. (Right) Comparison of the Pareto frontier of the two settings.

While there are many fabrication techniques presented in the research literature for creating nanostructures, the precision of the current state-of-the-art cannot consistently fabricate nanostructures at the resolution suggested by the BO algorithm and/or is not scalable to fabricating glass of surface area larger than 1 cm². Nonetheless, we were able to fabricate an approximation, a SEM image of which is provided in the figure below.

Figure 4. The fabrication of one of the Pareto efficient results for the uniform nanocone setting; greater zoom provided on the right.

These results provide valuable insight into where in the vast design parameter space we should strive for when fabricating the glass. As we continue to work with our collaborators, we look forward to investigating new topics such as incorporating fabrication feasibility into the optimization process and building a joint statistical model of fabrication and simulation results.

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