# Topology optimization of blazed metasurface gratings for high-efficiency spectrographs

Simon Ans, Frédéric Zamkotsian, Guillaume Demésy, Quentin Tanguy, Andrei

Mursa, Roland Salut, Nicolas Passilly

► To cite this version:

Simon Ans, Frédéric Zamkotsian, Guillaume Demésy, Quentin Tanguy, Andrei Mursa, et al. Topology optimization of blazed metasurface gratings for high-efficiency spectrographs. 2025 International Conference on Optical MEMS and Nanophotonics (OMN), Jul 2025, Chiangmai, Thailand. pp.1-2, 10.1109/OMN65869.2025.11125979. hal-05465056

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# Topology optimization of blazed metasurface gratings for high-efficiency spectrographs

Simon Ans¹,², Frédéric Zamkotsian¹, Guillaume Demésy², Quentin Tanguy³, Andrei Mursa³, Roland Salut³, and Nicolas Passilly³

¹Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France

²Aix Marseille Univ, CNRS, Centrale Med, Institut Fresnel, Marseille, France

³FEMTO-ST, CNRS, UFC, Besançon, France

## Abstract

A topology optimized metasurface grating is presented, designed using a 3D, in-house developed Finite Element model. It is manufacturable and exhibits nearly 60% of average diffraction efficiency on the –1st order between 400 and 1,500 nm.

**Keywords:** blazed gratings, metasurface, broadband wavelength, topology optimization, Maxwell's equations, Finite Element Method, e-beam lithography.

## 1 Introduction

For Earth and Universe Observation, a new generation of spectro-imagers is being developed [1]. To efficiently disperse the incident light on a given diffraction order (in reflection for this work) for a wavelength range wider than that of a grating with a sawtooth profile, a metasurface configuration is proposed. We have generalized our 2D conical code [2] to the 3D case using two approaches.

The first approach is based on the Finite Element mesh generated by the resolution of the Maxwell Equations for this problem. In the conical case of our previous work, an average efficiency on the [400–1,500] nm wavelength range of 79% has been achieved. This is a valuable proof of concept, but the goal now is to obtain a manufacturable optimized grating. The second approach, described in this paper, is based on manufacturable pillars of dielectric to reach this objective.

## 2 Theoretical and Numerical Models

First, a natural development of Ref. [2] is the topology optimization of the distribution of the relative permittivity $ε_r$ into the design region based on the Finite Element mesh generated by the resolution of Maxwell's Equations; $ε_r$ is considered constant by part in each tetrahedron. Second, a pillar-based permittivity distribution is presented for a fabrication purpose; $ε_r$ is considered constant by part inside manufacturable cuboids.

The function of merit in the optimization process is the diffraction efficiency of the grating in reflection on the order –1, for the incidence of a plane wave. This quantity is computed with an open-source code that solves Maxwell's equations in the frequency domain [3]. Then, the Jacobian of this merit function is calculated using a so-called adjoint method. The latter Jacobian is applied
in a gradient descent to optimize the permittivity distribution that maximizes the objective. Finally, a binarization and a connectedness filter are added to obtain a readable shape. The theory and the code are detailed in Ref. [2] for the conical incidence, namely for a mono-periodic 2D grating with a 3D incident plane wave. However, the Partial Differential Equation (PDE) remains the same in 3D, namely for a bi-periodic pattern. The weak formulation is even simpler to write, however more costly to solve.

In addition, the resolution is performed using a Finite Element mesh generated by the open source software Gmsh [4] and its associated solver GetDP [5]. Every step of the optimization necessitates three resolutions: one of the direct problem and two of the adjoint problem, in order to compute the Jacobian of the merit function. Gradient descent is carried out using the globally convergent method of moving asymptotes (GCMMA) [6] implemented in the NLopt package [7]. The whole process is monitored by an automated, in-house developed code.

## 3 Pillar-Based Bi-periodic Pattern

The mesh-based algorithm is an interesting proof of concept for the topology optimization process, providing very high efficiency gratings. Nevertheless, a pillar bi-periodic pattern is hereby presented, for it can be manufactured. By "bi-periodic", we mean a first period which is the grating pitch (along the (Oy) axis) and a second one which is the reproduction of the pattern along the grating lines (the (Ox) axis).

The proposed metasurface is showed in Fig. 1a-b. A single period is exhibited in Fig. 1a. A more global scope is illustrated in Fig. 1b, with 2 pitches along the (Oy) axis reproduced 10 times along the grating lines, (Ox) axis (the elementary period is surrounded by the orange rectangle). This

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Single period of the optimized metasurface, with dimensions. The dielectric in blue is Si₃N₄, and the etch stop layer (in orange) is Al₂O₃. (b) 2 periods of the grating reproduced 10 times along the (Ox) axis (grating lines), view from above. The elementary period of Fig. (a) is surrounded by the orange rectangle. (c) Spectral responses of different optimized gratings, on the diffraction order -1, with the targeted incidence: mesh-based, 300 nm-thick, Si₃N₄ layer (green cut view of the pattern, associated with the dotted green line); metasurface of Fig. (a)-(b) (blue solid line); non-optimized metasurface manufactured and presented in Ref. [8] (SEM view from above, associated with the dotted red line).</figcaption></figure>
metasurface is made of 200 nm-thick elementary pillars of Si₃N₄ (in blue) or air with 82.5 nm-long edges, deposited on a 50 nm-thick etch stop layer of alumina (Al₂O₃, in orange). This pattern is deposited on a reflective silver layer. Such dimensions have already been etched with adjusted e-beam lithography [8].

Fig. 1c shows the spectral response of the proposed pattern on the [400-1,500] nm wavelength range (blue graph), with an oblique incident plane wave which wavevector is neither coplanar with the (Oxz) plane nor the (Oyz) plane. It has an average diffraction efficiency of 58%. Compared to the efficiency of a classical sawtooth profile of 46%, a 26% improvement in relative terms is achieved. This optimal pillar pattern is moreover compared to the best spectral response obtained with a topology optimized, mesh-based, mono-periodic structure with a 300 nm-thick Si₃N₄ pattern (green cut view of the pattern, associated with the dotted green line). The latter reaches an average diffraction efficiency of 65%. Finally, the theoretical response of the manufactured structure from Ref. [8] is displayed (SEM view from above, associated with the dotted red line), showing an average efficiency of 42%.

This analysis highlights the performance threshold given by the mesh-based optimization. The average efficiency reached by the green mono-periodic pattern (a periodic pitch propagated along the (Ox) axis) is a priori impossible to exceed. Reaching 58% instead of 65% with such a heavy constraint (pillars vs freeform) is an excellent result for a manufacturable device. The expected performance of the metasurface currently fabricated (in red) is displayed to confirm this statement. The planned improvement in manufacturable blazed metasurfaces is impressive. Furthermore, the response on the two octaves is flatter for the metasurface design, while the typical response for a sawtooth design has a high maximum but decreases strongly in the infrared region.

This optimized configuration is yet to be adapted to other manufacturability constraints. On the one hand, the large gap without dielectric (clearly visible in Fig. 1b) is an issue for the etching. On the other hand, the deposition of Si₃N₄ on Al₂O₃ is not optimal due to an oxidation risk on the silver layer. Moreover, the refractive indices of the deposited dielectrics might be different from the litterature. New optimizations would then adjust the configuration to these indices in order to ensure the best blazing effect with the manufactured samples.

## 4 Conclusion

The topology optimization of blazed metasurface gratings in broadband (two octaves, between 400 and 1,500 nm) has led to a pillar-based design that can be manufactured. This structure exhibits a 58% of average diffraction efficiency on the order -1, which is 26% higher than the sawtooth design in relative terms.

The next step of this project is to manufacture this structure using adjusted e-beam lithography. This fabrication process includes some ongoing adjustments. The sample's feedback will then be analyzed and compared to the models, using far- and near-field characterization techniques.

**Acknowledgements.** This project has been partially funded by CNES and Thales Alenia Space with a PhD grant. Part of it has also been supported by the French RENATECH network ans its FEMTO-ST technology facility (MIMENTO). This same part has been partially funded by the CNRS PEPS-INSIS program through the project RESO-BATMAN.
References

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[8] S. Ans, G. Demésy, F. Zamkotsian, et al. Nanostructured blazed gratings for high performance spectrographs. In *Proc. SPIE 12990, SPIE Photonics Europe – Metamaterials XIV*, 2024.