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Topology optimization of freeform large-area metasurfaces

Zin Lin, Victor Liu, Raphaël Pestourie, Steven G. Johnson

arXiv:1902.03179v1physics.optics

TL;DR

Large-area metasurface design calls for handling extensive design spaces and multiple objectives, while earlier topology-optimization methods were limited to small domains. This work combines topology optimization with locally periodic cell simulations and demonstrates large-area two- and three-dimensional lens designs, including multi-angle and multi-wavelength examples.

  • Problem

    Earlier topology-optimization designs for metasurfaces were limited to domains of roughly 10 wavelengths, restricting their use for large-area designs with extensive design spaces and multiple objectives.

    Source: §1 · p. 3, §3 · p. 4

  • Method

    The method combines pixel-wise topology optimization with locally periodic simulations of smaller cells to design large-area metasurfaces.

    Source: §1 · pp. 1–3

  • Results

    Full-wave simulations validate large-area lens examples, including a three-dimensional cylindrical lens with 75% transmission efficiency and multi-frequency and multi-angle designs.

    Source: §3.1 · p. 5, §3.3 · p. 6, §3.2 · p. 6

  • Takeaways & Limitations

    The framework provides an efficient computational approach for inverse design of large-area freeform metasurfaces with multi-angle and multi-wavelength functionality.

    Source: §4 · p. 7

    The locally periodic approximation offers asymptotic advantages for nearly periodic surfaces, while the framework is presented as a pathway toward a wider variety of large aperiodic systems.

    Source: §4 · p. 8

Abstract

from arXiv · show

We demonstrate optimization of optical metasurfaces over $10^5$--$10^6$ degrees of freedom in two and three dimensions, 100--1000+ wavelengths ($\lambda$) in diameter, with 100+ parameters per $\lambda^2$. In particular, we show how topology optimization, with one degree of freedom per high-resolution "pixel," can be extended to large areas with the help of a locally periodic approximation that was previously only used for a few parameters per $\lambda^2$. In this way, we can computationally discover completely unexpected metasurface designs for challenging multi-frequency, multi-angle problems, including designs for fully coupled multi-layer structures with arbitrary per-layer patterns. Unlike typical metasurface designs based on subwavelength unit cells, our approach can discover both sub- and supra-wavelength patterns and can obtain both the near and far fields.

1 Introduction

The paper combines topology optimization with a locally periodic approximation to design large-area metasurfaces with millions of automatically determined degrees of freedom. The approach explores sub- and supra-wavelength patterns and is demonstrated on multi-angle, multi-frequency, and three-dimensional lens designs.

1 Introduction

  • The method extends topology optimization to metasurfaces at least 100 wavelengths across, with millions of automatically determined degrees of freedom.Each high-resolution pixel is a degree of freedom, allowing sub- and supra-wavelength features.Source: §1 · pp. 1–2
  • The method validates two-dimensional multi-frequency and multi-angle lens designs against full-wave simulations and presents three-dimensional monochromatic lenses with about 10^6 degrees of freedom.Its full scattered-field treatment also supports near-field or guided-wave optimization.Source: §1 · p. 2
  • Metasurfaces pattern nanophotonic features across hundreds or thousands of wavelengths, while conventional unit-cell libraries rely on primitive geometries and local periodic modeling.The paper considers both sub- and supra-wavelength features without an effective-medium approximation.Source: §1 · p. 2
  • Topology optimization searches continuous dielectric-permittivity values at spatial points using gradient-based methods and adjoint calculations to handle large design spaces.The passage contrasts this with heuristic searches such as genetic algorithms and particle-swarm methods.Source: §1 · p. 2
  • Combining topology optimization with a locally periodic approximation scales design beyond the roughly 10-wavelength domains of earlier optical topology-optimization methods.Many smaller simulations are combined into one optimization problem for a large surface.Source: §1 · pp. 2–3

2 Formulation

The formulation optimizes electromagnetic fields over pixel-wise dielectric variables, using locally periodic cell simulations and adjoint gradients to handle large metasurface designs. It can target field intensity and impose binary designs, while its implementation solves independent cells in parallel.

2 Formulation

  • The objective can be chosen as a function of electromagnetic fields; for focusing, the formulation maximizes electric-field intensity at a focal point.The far-field intensity objective is obtained by convolving the near field with the free-space Green’s function.Source: §2 · p. 3
  • The design uses a continuous dielectric-permittivity variable, then Heaviside projection filters make the final optimized design binary.The stated examples use a binary filter; geometric and regularization filters or curvature constraints can also be incorporated.Source: §2 · p. 3
  • A locally periodic approximation divides the metasurface into smaller cells simulated independently with periodic boundary conditions.The total near field is approximated by the disjoint intra-cell fields, which are engineered to add constructively at the focus.Source: §2 · pp. 3–4
  • Adjoint calculations provide the objective-function derivative at every pixel for optimization over thousands of degrees of freedom.The adjoint field is found by solving an additional Maxwell equation.Source: §2 · p. 4
  • The framework differs from phase-profile fitting with primitive subwavelength scatterers by optimizing non-intuitive shapes and supra-wavelength cells using their varying field amplitude and phase.Such cells require their field variation to be fully accounted for during optimization.Source: §2 · p. 4

2.1 Numerical Implementation

  • Independent cells are solved with FDFD or RCWA and simulated in parallel with MPI, after which their results are consolidated to compute the global objective and gradients.The structural update uses a gradient-based nonlinear-optimization algorithm; reported gradient discretization error is at most about 0.01%.Source: §2.1 · p. 4

3 Applications

The applications extend the locally periodic approach to multi-objective lens designs, including multi-angle and multi-wavelength 2D lenses and proof-of-concept 3D lenses. Full-wave validation supports the multi-angle and some 3D examples, while high-NA multi-wavelength designs expose a limitation of the locally periodic approximation.

3 Applications

  • For multiple frequencies and incident currents, the design objective can use a maximin formulation recast in differentiable form.This formulation addresses the multi-objective behavior of one metasurface across specified operating conditions.Source: §3 · pp. 4–5

3.1 Metalens Concentrator

  • A 1200λ-diameter, five-layer lens focused at 11 angles with diffraction-limited FWHM of ∼λ and average transmission efficiency T ≈40% in full-wave validation.The design used 240 cells and 3 × 10^5 DoFs; its distinct off-axis output patterns explain why the average efficiency does not violate the cited concentration bound.Source: §3.1 · p. 5

3.2 Multi-wavelength Focusing

  • At three wavelengths, the 200λ1-diameter lens focused near the diffraction limit, with FWHMs of ≈0.71λ1, 0.72λ2, and 0.76λ3 and average transmission efficiency ∼71%.The design used two TiO2 layers and 1.6 × 10^4 DoFs.Source: §3.2 · pp. 5–6
  • Full-wave focusing efficiencies were ∼51%, 50%, and 45% at the three wavelengths, versus LPA predictions of ∼78%, 55%, and 60%, respectively.Rapid surface variation in high-NA multi-wavelength designs produces scattering noise that the conjoined LPA solution does not capture; the focal-intensity objective also does not suppress stray diffraction.Source: §3.2 · p. 6

3.3 Three-dimensional Examples

  • A 3D cylindrical lens with NA=0.71 focused a normally incident beam to a focal line, and whole-lens FDTD showed 75% transmission efficiency.Its λ × λ cells repeat along y but vary along x, with complex geometry within each cell.Source: §3.3 · p. 6
  • The full 3D lens uses 6400 λ × λ cells and 6.4 × 10^5 DoFs; its far-field profile is approximated by combining unit-cell simulations because whole-lens FDTD is impractical with a few CPUs.The optimization used 40 CPUs and took a few minutes.Source: §3.3 · p. 7

4 Conclusion and Outlook

The paper presents an efficient framework for inverse design of large-area freeform metasurfaces and demonstrates it with multi-angle and multi-wavelength metalenses. It identifies aperiodicity as a challenge for locally periodic approximation and points to added constraints or domain decomposition as possible ways to address broader structures.

4 Conclusion and Outlook

  • The framework provides an efficient computational approach to inverse-designing large-area freeform metasurfaces, with multi-angle and multi-wavelength metalenses among its examples.The authors expect large-scale optimization to be useful for problems requiring large design spaces and multiple layers.Source: §4 · p. 7
  • Potential applications include 3D sculpting of far-field intensity, thin-film holography, and far-field design between separated meta-devices.The paper also identifies near-field manipulation of large aperiodic structures as a challenging future problem.Source: §4 · pp. 7–8
  • LPA offers asymptotic advantages for adiabatic, nearly periodic surfaces, while broader aperiodic structures may require added constraints or domain decomposition.Proposed extensions include using the conjoined LPA solution as a preconditioner, mirror boundaries, and overlapping domains.Source: §4 · p. 8
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