Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations Maxim Elizarov, Andrea Fratalocchi PRIMALIGHT, Faculty of Electrical Engineering, Applied Mathematics and Computational Science King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia Introduction Theory Simulation Setup Results Conclusion One of important steps for achieving more complex computational In this work, we propose an inverse design framework for developing devices based on photonics is to control light at the nanoscale. For materials with desired distribution of refractive index. We apply this we need to engineer the materials with tailored refractive index inverse design to transformation optics and conformally map desired distribution (significantly high and significantly low refractive index distribution of a virtual domain to the physical magnitudes). Inverse design is a powerful method for making domain of a material. We show that it is possible to create this photonic devices such as photonic crystals [1], and metasurfaces [2], distribution by geometrical deformation of a reflective substrate. as well as transformation optics approach [3][4][5]. Particularly, we discuss design of the fundamental optical nanocomponents, including nanoresonators with ultra-high (n ≈ 100) and near-zero (n ≈ 0.02) nondispersive refractive index. References [1] P. I. Borel et al. , “Topology optimization and fabrication of photonic crystal structures,” Opt. Express , vol. 12, no. 9, p. 1996, May 2004, doi: 10.1364/OPEX.12.001996. [2] Z. Liu et al ., “Generative Model for the Inverse Design of Metasurfaces,” Nano Lett. , vol. 18, no. 10, pp. 6570–6576, Oct. 2018, doi: 10.1021/acs.nanolett.8b03171. [3] P. Mao et al. , “Broadband single molecule SERS detection designed by warped optical spaces,” Nat. Commun. , vol. 9, no. 1, p. 5428, Dec. 2018, doi: 10.1038/s41467-018-07869-5. [4] H. Galinski et al. , “Scalable, ultra-resistant structural colors based on network metamaterials,” Light Sci. Appl. , vol. 6, no. 5, pp. e16233–e16233, May 2017, doi: 10.1038/lsa.2016.233. [5] Y. Tian et al. , “Enhanced Solar-to-Hydrogen Generation with Broadband Epsilon-Near-Zero Nanostructured Photocatalysts,” Adv. Mater. , vol. 29, no. 27, p. 1701165, Jul. 2017, doi: 10.1002/adma.201701165.
Inverse design of materials with arbitrary distribution of refractive <latexit sha1_base64="dZvWMb6uQ4Sz/lqdrKARXTsNXa8=">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</latexit> <latexit sha1_base64="HRAmyuTarK2f/Qo2ybUVBIU38Pc=">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</latexit> <latexit sha1_base64="nAtb42up5KcZHETrA8g3ehQy2MU=">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</latexit> <latexit sha1_base64="4J0MUWKafXbFPHFHEtN2O8Qaero=">ACXicbVFdS8MwFE2r82NOnfrgy/BIQjKaIegPgiCLz4qOBXWbdxm6RaWpiW5FUbpn/RNX/wrprOITi8kOZxzDzc5CVMpDHrem+MuLdWVtfW6xuNza3t5s7uo0kyzXiXJTLRzyEYLoXiXRQo+XOqOcSh5E/h9KbUn164NiJRDzhLeT+GsRKRYICWGjYxiDSwPIgBJ2FE1aBT5MxuFZ2CRgFy0KFBmMiRmcX2sLQRfGtUrSGk0BKCFAEXPzCy8ar7xhs+W1vXnRv8CvQItUdTdsvgajhGUxV8gkGNPzvRT7eTmeSV7Ug8zwFNgUxrxnoQI7t5/P0ynokWVGNEq0XQrpnP3pyCE25f1sZ5mCWdRK8j+tl2F0c+FSjPkin0NijIbR0LqOlIaM5QziwApoW9K2UTsLmi/ZC6DcFfPJf8Nhp+2fty/uz1vVpFcaOSCH5Jj45Jxck1tyR7qEkXeHOtO3flwa27D3fpqdZ3Ks0d+lbv/CSRTt/M=</latexit> index by geometrical deformations Maxim Elizarov, Andrea Fratalocchi Introduction Theory Simulation Setup Results Conclusion Theory of inverse designed materials via geometrical deformations The starting configuration is a universal basic structure in a virtual domain In the transformed space 𝒔 " , Maxwell equations remains identically the same with composed by a reflective substrate with a semi-infinite material defined on the introduction of a new material with dielectric permittivity 𝜻′ 𝒔′ and magnetic top possessing an inhomogeneous, nondispersive, and anisotropic permeability 𝝂′(𝒔′) expressed by transformation optics formulas via Jacobian matrix ∇Ω . This material generates a transformed refractive index 𝒐′ 𝒔′ = refractive index 𝒐 𝒔 = 𝜻 𝒔 𝝂(𝒔), with 𝜻 and 𝝂 dielectric permittivity 𝜻′ 𝒔 " 𝝂′(𝒔′) that incorporates the effects of the geometrical deformation (2), and magnetic permeability tensors, respectively, and 𝒔 𝒚, 𝒛, 𝒜 position coordinates. providing an identical evolution to light when propagating in the spaces 𝒔 or 𝒔 " . For materials with purely dielectric or purely magnetic response, the The main idea of this work is to inverse design the deformation (2) so that the propagation of light arises from a single vectorial wave equation defined in transformed refractive index is that of a vacuum, with 𝒐′ 𝒔′ = 𝟐 . The deformation the positive semi-infinite space for 𝑧 > 0: is found by solving (3) in the scalar limit with 𝜻 and 𝝂 described by a single arbitrary ( scalar component for a desired refractive index distribution. n 2 ∂ 2 ψ = E , for µ = 1 , (𝟐) ∂ t 2 + r ⇥ r ⇥ ψ = 0 c 2 = H , for ✏ = 1 , We wish to study the realization of the material present in Eq(1) and characterized by the refractive index 𝒐(𝒔) , by using a geometrical deformation of coordinates 𝒔 " = Ω(𝒔) with: x 0 x 0 ( x, y, z ) 2 � � d Ω ( u ) r 0 = (𝟒) = = Ω ( r ) , (𝟑) n 2 ( r ) = � � y 0 y 0 ( x, y, z ) � � du z 0 z 0 ( x, y, z ) � �
Inverse design of materials with arbitrary distribution of refractive <latexit sha1_base64="iaXC1zaxBmvXWJLY4uRjiVvkgIk=">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</latexit> index by geometrical deformations Maxim Elizarov, Andrea Fratalocchi Introduction Theory Simulation Setup Results Conclusion Pseudospectral technique Statistical learning approach To inverse design Eq(3), we have used a pseudospectral technique based We solve Eq(4) by statistical learning via nonlinear regression. Given an analytic on suitably defined rational Chebyshev polynomials in the complex distribution of refractive index 𝒐 𝒔 that the transformation Ω(𝑣) is expected to domain. Our approach expands the unknown deformation Ω(𝑣) as generate in the space 𝒔 , we create a learning dataset composed of a discrete number follows: 𝑜 = 0,1, … of refractive index values 𝒐 𝒔 𝒐 calculated on a series of points 𝒔 𝒐 . Each dataset element 𝒐 𝒔 𝒐 and 𝒔 𝒐 , when substituted in (4), originates a nonlinear equation for the regression coefficients 𝛾 < . These are then calculated via nonlinear 2 � ◆� ∞ 2 ia ✓ u − ia n 2 ⇣ r ⌘ � � X (𝟓) least-square, by solving the regression equations for all the elements of the dataset = � 1 + β m +1 ( m + 1) · U m � � ( u + ia ) 2 · u + ia a � � with available large-scale optimization libraries. Once the coefficients 𝛾 < are � m =0 evaluated, the deformation of coordinates Ω(𝑣) can be in any point of the space. with 𝑉 𝑛 the second kind of Chebyshev polynomial of order m , a an arbitrary spatial scaling constant, and 𝛾 < coefficients with 𝛾 ? = C ∑ <AB 𝛾 < . Loss function Number of modes
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