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Adjoint-Based Gradient Evaluation for Metasurface Inverse Design via Affine Geometric Transformations

Vincenzo Mottola, Luisa Faella, Carlo Forestiere, Antonello Tamburrino

Abstract

The sharp increasing in fabrication capabilities of nanomaterials, and complex structures such as meta-surfaces and metalens, has opened to the possibility of employing them for accurately control the electromagnetic field, beyond the possibility ensured by traditional devices. The demand for large scale structures and more complex functionalities from meta-surfaces lead to the research for advanced techniques of inverse design, able to conjugate the ability to produce effective designs and limited computational cost. Among the various approaches for inverse design of large meta-surfaces, the ones based on the adjoint variable method are appealing since able to ensure a minimal computational cost for the gradient computation of the cost function. In this work, a systematic methodology for the application of the adjoint variable method for large meta-surface design is presented. The method is based on: (i) a parametrization of the relevant geometric parameters of the meta-atoms, (ii) the fast computation of the gradient with respect such parameters, allowing for the implementation of general affine transformations during the optimization process. The main findings are first theoretically justified and a numerical validation is provided to show the effectiveness of the proposed approach.

Adjoint-Based Gradient Evaluation for Metasurface Inverse Design via Affine Geometric Transformations

Abstract

The sharp increasing in fabrication capabilities of nanomaterials, and complex structures such as meta-surfaces and metalens, has opened to the possibility of employing them for accurately control the electromagnetic field, beyond the possibility ensured by traditional devices. The demand for large scale structures and more complex functionalities from meta-surfaces lead to the research for advanced techniques of inverse design, able to conjugate the ability to produce effective designs and limited computational cost. Among the various approaches for inverse design of large meta-surfaces, the ones based on the adjoint variable method are appealing since able to ensure a minimal computational cost for the gradient computation of the cost function. In this work, a systematic methodology for the application of the adjoint variable method for large meta-surface design is presented. The method is based on: (i) a parametrization of the relevant geometric parameters of the meta-atoms, (ii) the fast computation of the gradient with respect such parameters, allowing for the implementation of general affine transformations during the optimization process. The main findings are first theoretically justified and a numerical validation is provided to show the effectiveness of the proposed approach.
Paper Structure (23 sections, 59 equations, 11 figures)

This paper contains 23 sections, 59 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Key results overview
  3. Fast gradient computation
  4. The adjoint problem
  5. The gradient of the similarity function
  6. Rotations
  7. Anisotropic expansion
  8. Translation
  9. Evaluation of the function $f$
  10. The adjoint source
  11. Magnitude of the scalar product
  12. Norm of the difference
  13. Angle between the target and forward field
  14. Squared difference of the squared norms
  15. Angle between the square of the norms
  16. ...and 8 more sections

Figures (11)

  • Figure 1: The meta-lens considered in the TEz propagation framework, along with the exit plane $S$ on which the target field is assigned.
  • Figure 2: A generic affine transformation is applied to a circular meta-atom.
  • Figure 3: Left: initial design, corresponding to the vector parameter $\mathbf{p}$. Center: updated design, corresponding to the vector parameter $\mathbf{p}'$. Right: regions $\chi_a$ and $\chi_b$ arising from the variation in the design parameters.
  • Figure 4: Geometry of the scatterers considered in the simulations.
  • Figure 5: Comparison between the gradient with respect to a rotation in the meta-atom, computed via adjoint method and a finite difference scheme. The gradient is computed for different values of the rotation angle.
  • ...and 6 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3