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Shape optimization of optical microscale inclusions

Manaswinee Bezbaruah, Matthias Maier, Winnifried Wollner

TL;DR

The paper tackles the problem of tailoring the macroscopic optical response of plasmonic metamaterials by shaping microscale inclusions. It develops a deformation-based homogenization framework for time-harmonic Maxwell equations, proving well-posedness and regularity of the deformed cell problem and enabling a gradient-based PDE-constrained optimization to match a target permittivity tensor $\varepsilon^{\text{target}}$. An adjoint formulation paired with gradient-based methods (gradient descent or BFGS) is used, with penalty terms ensuring mesh regularity during large deformations. Numerical experiments demonstrate the method’s ability to steer toward epsilon-near-zero configurations and to handle substantial mesh changes, highlighting a practical route for microstructural design in optical metamaterials. The approach lays groundwork for frequency-aware extensions and broader inverse-design of microscale geometries for prescribed macroscopic electromagnetic properties.

Abstract

This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target.

Shape optimization of optical microscale inclusions

TL;DR

The paper tackles the problem of tailoring the macroscopic optical response of plasmonic metamaterials by shaping microscale inclusions. It develops a deformation-based homogenization framework for time-harmonic Maxwell equations, proving well-posedness and regularity of the deformed cell problem and enabling a gradient-based PDE-constrained optimization to match a target permittivity tensor . An adjoint formulation paired with gradient-based methods (gradient descent or BFGS) is used, with penalty terms ensuring mesh regularity during large deformations. Numerical experiments demonstrate the method’s ability to steer toward epsilon-near-zero configurations and to handle substantial mesh changes, highlighting a practical route for microstructural design in optical metamaterials. The approach lays groundwork for frequency-aware extensions and broader inverse-design of microscale geometries for prescribed macroscopic electromagnetic properties.

Abstract

This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target.
Paper Structure (20 sections, 12 theorems, 73 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 20 sections, 12 theorems, 73 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Microscopic model and homogenization theory
  3. Shape optimization approach
  4. Past works
  5. Paper organization
  6. Problem Description
  7. Background: Heterogeneous Maxwell equations
  8. Background: Homogenization results
  9. Deformed cell problem
  10. Well-posedness, regularity and dependence on deformation field
  11. Shape optimization problem and adjoint formulation
  12. Adjoint formulation
  13. Finite element discretization and optimization framework
  14. Numerical illustrations
  15. Influence of the regularization parameters
  16. ...and 5 more sections

Key Result

Theorem 2.1

\newlabelthm:well_posedness0 Let $\mu \in \mathbb{R}_{>0}$ and let $\varepsilon^d, \sigma^d \in L^\infty(\Omega,\mathbb C^{d\times d})$ be bounded, complex and tensor-valued functions such that $\mathrm {Re}\,(\varepsilon^d(\boldsymbol{x}))$ and $\mathrm {Im}\,{(\sigma^d(\boldsymbol{x}))}$ are sym

Figures (6)

  • Figure 1: (a) The unit cell $Y = [0,1]^3$ consisting of 2D graphene inclusions $\Sigma$ with surface conductivity $\sigma$ in an ambient host material with permittivity $\varepsilon$; (b) the plasmonic crystal formed by many scaled and repeated copies of $Y$ in all space directions. \newlabelfig:layered0
  • Figure 1: Epsilon-near-zero testcase: Final geometry obtained for target cases (a), (b) and (c) with increasingly smaller $\varepsilon^{\text{target}}_{xx}$ component. The corresponding deformed (and initial) meshes are shown in (e)-(h). The black region in (a)-(d), as well as the red region in (e)-(h) show the volume surrounded by the interface $\Sigma_h$.
  • Figure 2: A reference unit cell $\hat{Y}$ with a 2D inclusion $\hat{\Sigma}$ that is deformed by a deformation vector field $\boldsymbol{\hat{q}}(\boldsymbol{x})$.
  • Figure 2: Evolution of the optimality, $\|\boldsymbol{\delta c}_h(\boldsymbol{\hat{q}}_h^n)\|/\|\boldsymbol{\delta c}_h(\boldsymbol{\hat{q}}_h^0)\|$, during the BFGS solution process for the three cases (a) $0.5+0.01\textrm{i}$, (b) $0.25+0.005\textrm{i}$, to (c) $0.0$. The thick line for each case is a smoothed Bezier curve (gnuplot builtin) that is overlayed over the actual, oscillatory value.
  • Figure 4: Large deformation testcase: Final geometry obtained for target cases (a), (b) and (c) with increasingly larger $\{\varepsilon^{\text{target}}_{xy}, \varepsilon^{\text{target}}_{yx}\}$ components.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Theorem 2.1: Well-posedness and two-scale convergence amirat2017maier20c
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: Transformation
  • Proof 1
  • Lemma 2.6
  • Proof 2
  • Lemma 2.7
  • Lemma 2.8
  • ...and 22 more