Metasurfaces and Optimal transport
The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 201-224.

This paper provides a theoretical and numerical approach to show existence, uniqueness, and the numerical determination of metalenses refracting radiation with energy patterns. The theoretical part uses ideas from optimal transport and for the numerical solution we study and implement a damped Newton algorithm to solve the semi discrete problem. A detailed analysis is carried out to solve the near field one source refraction problem and extensions to the far field are also mentioned.

Published online:
DOI: 10.5802/smai-jcm.84
Classification: 49N45
Keywords: optimal transport, metasurfaces
Cristian E. Gutiérrez 1; Qingbo Huang 2; Quentin Mérigot 3; Boris Thibert 4

1 Department of Mathematics Temple University Philadelphia, PA 19122
2 Department of Mathematics and Statistics Wright State University Dayton, OH 45435
3 Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France & Institut universitaire de France (IUF)
4 Université Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Metasurfaces and {Optimal} transport},
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     pages = {201--224},
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Cristian E. Gutiérrez; Qingbo Huang; Quentin Mérigot; Boris Thibert. Metasurfaces and Optimal transport. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 201-224. doi : 10.5802/smai-jcm.84. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.84/

[1] The runners-up, Science, Volume 354 (2016) no. 6319, pp. 1518-1523 | DOI

[2] Farhan Abedin; Cristian E Gutiérrez; Giulio Tralli C 1,α estimates for the parallel refractor, Nonlinear Anal., Theory Methods Appl., Volume 142 (2016), pp. 1-25 | DOI | Zbl

[3] Wei Ting Chen; Federico Capasso Will flat optics appear in everyday life anytime soon?, Appl. Phys. Lett., Volume 118 (2021) no. 10, p. 100503 | DOI

[4] Cristian E. Gutiérrez; Qingbo Huang The refractor problem in reshaping light beams, Arch. Ration. Mech. Anal., Volume 193 (2009) no. 2, pp. 423-443 | DOI | MR | Zbl

[5] Cristian E. Gutiérrez; Qingbo Huang The near field refractor, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 4, pp. 655-684 | DOI | Numdam | MR | Zbl

[6] Cristian E. Gutiérrez; Luca Pallucchini Reflection and refraction problems for metasurfaces related to Monge-Ampère equations, Journal Optical Society of America A, Volume 35 (2018) no. 9, pp. 1523-1531 | DOI

[7] Cristian E. Gutiérrez; Luca Pallucchini; Eric Stachura General refraction problems with phase discontinuities on nonflat metasurfaces, Journal Optical Society of America A, Volume 34 (2017) no. 7, pp. 1160-1172 | DOI

[8] Cristian E. Gutiérrez; Ahmad Sabra Aspherical lens design and imaging, SIAM J. Imaging Sci., Volume 9 (2016) no. 1, pp. 386-411 | DOI | MR | Zbl

[9] Cristian E. Gutiérrez; Ahmad Sabra Freeform lens design for scattering data with general radiant fields, Arch. Ration. Mech. Anal., Volume 228 (2018), pp. 341-399 | DOI | MR | Zbl

[10] Cristian E. Gutiérrez; Ahmad Sabra Chromatic aberration in metalenses, Adv. Appl. Math., Volume 124 (2021), pp. 1090-2074 | DOI | MR | Zbl

[11] Jun Kitagawa; Quentin Mérigot; Boris Thibert Convergence of a Newton algorithm for semi-discrete optimal transport, J. Eur. Math. Soc., Volume 21 (2019) no. 9, pp. 2603-2651 | DOI | MR | Zbl

[12] Ren Jie Lin; Vin-Cent Su; Shuming Wang; Mu Ku Chen; Tsung Lin Chung; Yu Han Chen; Hsin Yu Kuo; Jia-Wern Chen; Ji Chen; Yi-Teng Huang et al. Achromatic metalens array for full-colour light-field imaging, Nature nanotechnology, Volume 14 (2019) no. 3, pp. 227-231

[13] Quentin Merigot; Boris Thibert Optimal transport: discretization and algorithms, Handbook of Numerical Analysis, Volume 22, Elsevier, 2021, pp. 133-212

[14] Jocelyn Meyron Initialization procedures for discrete and semi-discrete optimal transport, Comput.-Aided Des., Volume 115 (2019), pp. 13-22 | DOI | MR

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