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MIQCP and MISOCP-Based Solution Methods for the Multi-Layer Thin Films Problem

Deniz Tuncer, Burak Kocuk

TL;DR

This study tackles designing multi-layer thin-film coatings to maximize average reflectance across wavelengths by formulating an exact nonconvex MIQCP and a convex MISOCP relaxation. The approach leverages transfer matrices and a bilinearization strategy to model layer effects, and introduces bound tightening and SOC-based SOC-outer approximations to enable tractable convex relaxations. Key findings show near-100% reflectance in the visible and about 95% across a broad spectrum with a practical number of layers, with the MISOCP offering significant speedups at a modest cost to optimality. The work advances robust, multi-wavelength reflector design and highlights avenues for improving broad-spectrum performance and relaxation tightness for scalability.

Abstract

The Multi-Layer Thin Films Problem is a materials science problem that aims to enhance the reflectance of a metallic substrate by designing multi-layer coatings composed of different dielectric materials and thicknesses. While previous studies on the problem mostly rely on heuristic approaches and are designed for single wavelength applications, this work addresses the problem using global optimization techniques for multiple wavelengths. We develop an exact nonconvex mixed-integer quadratically constrained programming (MIQCP) model to solve this problem. We also develop a mixed-integer second-order cone programming relaxation that has computational advantage over the MIQCP model. Our numerical experiments yield solutions that have average reflectance of 99% over the visible spectrum (380-770 nm) and 95% over the broad spectrum (300-3000 nm).

MIQCP and MISOCP-Based Solution Methods for the Multi-Layer Thin Films Problem

TL;DR

This study tackles designing multi-layer thin-film coatings to maximize average reflectance across wavelengths by formulating an exact nonconvex MIQCP and a convex MISOCP relaxation. The approach leverages transfer matrices and a bilinearization strategy to model layer effects, and introduces bound tightening and SOC-based SOC-outer approximations to enable tractable convex relaxations. Key findings show near-100% reflectance in the visible and about 95% across a broad spectrum with a practical number of layers, with the MISOCP offering significant speedups at a modest cost to optimality. The work advances robust, multi-wavelength reflector design and highlights avenues for improving broad-spectrum performance and relaxation tightness for scalability.

Abstract

The Multi-Layer Thin Films Problem is a materials science problem that aims to enhance the reflectance of a metallic substrate by designing multi-layer coatings composed of different dielectric materials and thicknesses. While previous studies on the problem mostly rely on heuristic approaches and are designed for single wavelength applications, this work addresses the problem using global optimization techniques for multiple wavelengths. We develop an exact nonconvex mixed-integer quadratically constrained programming (MIQCP) model to solve this problem. We also develop a mixed-integer second-order cone programming relaxation that has computational advantage over the MIQCP model. Our numerical experiments yield solutions that have average reflectance of 99% over the visible spectrum (380-770 nm) and 95% over the broad spectrum (300-3000 nm).
Paper Structure (13 sections, 17 equations, 4 figures, 10 tables, 1 algorithm)

This paper contains 13 sections, 17 equations, 4 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimization Formulation
  3. Generic Optimization Formulation
  4. Discrete Material Thicknesses
  5. Solution Method
  6. Bound Tightening
  7. Relaxation by Overapproximation of $D_s^\lambda (w^\lambda)$
  8. Computational Experiments
  9. Computational Setting
  10. Experiments for Single Wavelength
  11. Experiments for the Visible Spectrum
  12. Experiments for the Broad Spectrum
  13. Conclusion

Figures (4)

  • Figure 1: Illustration of two layers of coating on top of a substrate.
  • Figure 5: Reflectance profiles of uncoated substrates.
  • Figure 6: Reflectance profiles with $N=14$ layers optimized for the visible spectrum for the MIQCP-based method.
  • Figure 7: Reflectance profiles with $N=20$ layers optimized for the broad spectrum.