Inverse design of materials with arbitrary distribution of - - PowerPoint PPT Presentation

Maxim Elizarov, Andrea Fratalocchi

Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations

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

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.

One of important steps for achieving more complex computational devices based on photonics is to control light at the nanoscale. For this we need to engineer the materials with tailored refractive index distribution (significantly high and significantly low magnitudes). Inverse design is a powerful method for making photonic devices such as photonic crystals [1], and metasurfaces [2], as well as transformation optics approach [3][4][5]. In this work, we propose an inverse design framework for developing materials with desired distribution of refractive index. We apply inverse design to transformation optics and conformally map desired refractive index distribution of a virtual domain to the physical domain of a material. We show that it is possible to create this distribution by geometrical deformation of a reflective substrate. Particularly, we discuss design

• f

the fundamental

• ptical

nanocomponents, including nanoresonators with ultra-high (n ≈ 100) and near-zero (n ≈ 0.02) nondispersive refractive index.

Theory of inverse designed materials via geometrical deformations Maxim Elizarov, Andrea Fratalocchi

Introduction Theory Simulation Setup Results Conclusion

In the transformed space 𝒔", Maxwell equations remains identically the same with the introduction of a new material with dielectric permittivity 𝜻′ 𝒔′ and magnetic permeability 𝝂′(𝒔′) expressed by transformation optics formulas via Jacobian matrix ∇Ω. This material generates a transformed refractive index 𝒐′ 𝒔′ = 𝜻′ 𝒔" 𝝂′(𝒔′) that incorporates the effects of the geometrical deformation (2), providing an identical evolution to light when propagating in the spaces 𝒔 or 𝒔". The main idea of this work is to inverse design the deformation (2) so that the transformed refractive index is that of a vacuum, with 𝒐′ 𝒔′ = 𝟐. The deformation 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. The starting configuration is a universal basic structure in a virtual domain composed by a reflective substrate with a semi-infinite material defined on top possessing an inhomogeneous, nondispersive, and anisotropic refractive index 𝒐 𝒔 = 𝜻 𝒔 𝝂(𝒔), with 𝜻 and 𝝂 dielectric permittivity and magnetic permeability tensors, respectively, and 𝒔 𝒚, 𝒛, 𝒜 position coordinates. For materials with purely dielectric or purely magnetic response, the propagation of light arises from a single vectorial wave equation defined in the positive semi-infinite space for 𝑧 > 0: 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: n2 c2 ∂2ψ ∂t2 + r ⇥ r ⇥ ψ = 0

( = E, for µ = 1, = H, for ✏ = 1,

(𝟐) r0 =   x0 y0 z0   =   x0(x, y, z) y0(x, y, z) z0(x, y, z)   = Ω(r),

(𝟑) n2(r) =

• dΩ(u)

du

• 2

(𝟒)

Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations

Maxim Elizarov, Andrea Fratalocchi

Introduction Theory Simulation Setup Results Conclusion

Statistical learning approach

We solve Eq(4) by statistical learning via nonlinear regression. Given an analytic distribution of refractive index 𝒐 𝒔 that the transformation Ω(𝑣) is expected to generate in the space 𝒔, we create a learning dataset composed of a discrete number 𝑜 = 0,1, … of refractive index values 𝒐 𝒔𝒐 calculated on a series of points 𝒔𝒐. Each dataset element 𝒐 𝒔𝒐 and 𝒔𝒐, when substituted in (4), originates a nonlinear equation for the regression coefficients 𝛾<. These are then calculated via nonlinear least-square, by solving the regression equations for all the elements of the dataset with available large-scale optimization libraries. Once the coefficients 𝛾< are evaluated, the deformation of coordinates Ω(𝑣) can be in any point of the space.

Pseudospectral technique

To inverse design Eq(3), we have used a pseudospectral technique based

• n suitably defined rational Chebyshev polynomials in the complex
• domain. Our approach expands the unknown deformation Ω(𝑣) as

follows: with 𝑉𝑛 the second kind of Chebyshev polynomial of order m, a an arbitrary spatial scaling constant, and 𝛾< coefficients with 𝛾? = ∑<AB

C

𝛾<. (𝟓) n2 ⇣r a ⌘ =

• 1 +

2ia (u + ia)2 ·

∞

X

m=0

βm+1(m + 1) · Um ✓u − ia u + ia ◆

• 2

Number of modes Loss function

Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations

Maxim Elizarov, Andrea Fratalocchi

Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations

Introduction Theory Simulation Setup Results Conclusion

Simulation results

The hole family of lower refractive index distributions is simpler to compute using initial high index calculations results. a,b,c,– initial mapping with refrac-tive index amplitude

• f

100; e,f– refractive index distribution with amplitude 20; g,h– refractive index distribution with amplitude 70

• 6
• 4
• 2

2 4 6 20 80 40 60 100 x/L target n predicted n refractive index n(x,y=0) b a

• 15
• 10
• 5

5 10 15

• 20
• 15
• 10
• 5

5 x/L y/L

100 80 60 40 20 x/L 0.30 0.25 0.20 0.15 0.10 0.05 0.00 y/L

• 1.0
• 0.5

0.0 0.5 1.0 n, refractive index 1 2

• 2
• 1

5 10 15 20 n(x,y’=0) x/L 10 20 30 40 50 60 70 n(x,y’=0) x/L 1 2

• 2
• 1

c f g h e

• 15
• 10
• 5

5 10 15

• 20
• 15
• 10
• 5

5 x/L y’/L

• 15
• 10
• 5

5 10 15

• 20
• 15
• 10
• 5

5 x/L y’/L

Epsilon-near-zero material: index distribution on the surface and corresponding mapping

• 15
• 10
• 5

5 10 15

• 20
• 15
• 10
• 5

5 x/L y/L

• 6
• 4
• 2

2 4 6 20 50 30 40 60 10 x/L refractive index n(x,y=0) target n predicted n 0.3 0.2 0.1 0.4 0.0 y/L

Separated Gaussians distributions with amplitudes of 30 and 60; d refractive index on a surface, e corresponding deformation of reflective substrate, f equivalent refractive index colormap

Maxim Elizarov, Andrea Fratalocchi

Inverse design of materials with arbitrary distribution of refractive index by geometrical deformations

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

• 1. We developed a new inverse design framework for engineering materials with

arbitrary refractive index distribution

• 2. We demonstrated that refractive index of material can be altered by local

deformations of a reflective substrate

• 3. One simulation of high or low index material can be used to calculate similar

distributions and corresponding deformations of lower amplitudes Contacts Email: maxim.elizarov@kaust.edu.sa, andrea.fratalocchi@kaust.edu.sa