Manufacturable blazed metasurface gratings designed by 3D topology optimization model

Abstract

We present the generalization of our FEM-based topology optimization framework to 3D blazed metasurfaces operating in reflection over the visible and near-infrared range [400-1,500]nm. The design region is described through a density-based SIMP interpolation and optimized using the adjoint method, enabling the treatment of several tens of thousands degrees of freedom. A first approach directly applies topology optimization to the 3D Finite Element mesh (mesh-based), yielding a freeform structure that achieves an average diffraction efficiency of 62% in order -1 over two octaves under the targeted incidence. However, such patterns remain difficult to manufacture. We therefore introduce a pillar-based parameterization, embedding fabrication constraints within the optimization loop. The resulting binary metasurface, compatible with e-beam lithography and Reactive Ion Etching techniques, achieves an average efficiency of 57% over the same spectral band in s-polarization, with low polarization dependence. This work demonstrates that large-scale 3D topology optimization can bridge the gap between broadband optical performance and realistic nanofabrication constraints for blazed metasurfaces.

Figures (10)

• Figure 1: The blazing effect of a sawtooth grating in reflection, with a white light incident radiation. The diffraction phenomenon is illustrated with 4 diffraction orders: $-2$, $-1$, 0 and 1. The grating is blazed, i.e. most part of the diffracted light is on a specific order (here the $-1$st).
• Figure 2: (a) The reference sawtooth blazed grating. It is mono-periodic or 1D with a period denoted $d_y$ and the so-called blaze angle $\alpha_b$. (b) Pillar-shaped bi-periodic (or 2D) metasurface grating. One period of the pattern is highlighted with a darker color; its surface is $d_x \times d_y$.
• Figure 3: Description of the numerical domain for a bi-periodic grating and definition of the incident electric field. In every domain, the permittivity tensor is defined. The region of interest (where the metasurface lies) is the design region $\Omega_d$ (surrounded by the orange cuboid). The material where the incident radiation travels is the superstrate $\mathcal{S}^+$ (in transparent gray). The reflecting metallic layer is the substrate denoted $\mathcal{S}^-$ (in gray). The surface on top of $\mathcal{S^+}$ is the interface where the diffracted field is analyzed, called $\Gamma^+$ (in red). The periodicity of the pattern enables to restrain the domain to a sole $d_x\times d_y$ period, with absorbing Perfectly Matched Layers up and down ($\text{PML}^{\pm}$, in violet). The incident electric field is perpendicular to the direction of propagation given by the vector $\mathbf{k}^+_{\downarrow}$. It is defined by the polar angle $\theta_i$, the azimuthal angle $\varphi_i$ and the polarization angle $\psi_i$.
• Figure 4: (a) Mesh-based density disctribution. Every tetrahedron $\mathcal{T}_i \in {\mathfrak{T}\,{}}_{\,\Omega_d}$ has got a density $\rho_i$ that is optimized. It can be seen as a freeform optimization (a tetrahedron is very small compared to the pattern size). (b) Pillar-based density distribution. Same principle but the density is defined by cuboids that have a manufacturable size.
• Figure 5: (a) 3D view of the optimal 3D pattern on a single period for a 600 nm-thick design region with SiO$_2$ as chosen dielectric. (b) Same pattern viewed on the plane $x=d_x$ (similar to the point of view of 2D gratings). (c) Spectral response of the grating on the wavelength range of interest (blue line) as compared to the response for the targeted incidence of the conical optimal design (gray line).