Mathematical operations and equation solving with reconfigurable metadevices

Dimitrios C. Tzarouchis; Mario Junior Mencagli; Brian Edwards; Nader Engheta

doi:10.1038/s41377-022-00950-1

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Mathematical operations and equation solving with reconfigurable metadevices

Original Articles

- Mathematical operations and equation solving with reconfigurable metadevices
- Light: Science & Applications Vol. 11, Issue 10, Pages: 2365-2377(2022)
- Affiliations：
  
  1.University of Pennsylvania, Department of Electrical and Systems Engineering, Philadelphia, PA, 19104, USA
  2.University of North Carolina at Charlotte, Department of Electrical and Computer Engineering, Charlotte, NC, 28223, USA
- Author bio：
  
  Nader Engheta (engheta@ee.upenn.edu)
- Funds：
- DOI：10.1038/s41377-022-00950-1
  CLC：
- Published：31 October 2022，
  
  Published Online：07 September 2022，
  
  Received：19 November 2021，
  
  Revised：15 July 2022，
  
  Accepted：02 August 2022
- Accepted：
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Tzarouchis, D. C. et al. Mathematical operations and equation solving with reconfigurable metadevices. Light: Science & Applications, 11, 2365-2377 (2022).
DOI：

Tzarouchis, D. C. et al. Mathematical operations and equation solving with reconfigurable metadevices. Light: Science & Applications, 11, 2365-2377 (2022). DOI： 10.1038/s41377-022-00950-1.

摘要

Abstract

Performing analog computations with metastructures is an emerging wave-based paradigm for solving mathematical problems. For such devices

one major challenge is their reconfigurability

especially without the need for a priori mathematical computations or computationally-intensive optimization. Their equation-solving capabilities are applied only to matrices with special spectral (eigenvalue) distribution. Here we report the theory and design of wave-based metastructures using tunable elements capable of solving integral/differential equations in a fully-reconfigurable fashion. We consider two architectures: the Miller architecture

which requires the singular-value decomposition

and an alternative intuitive direct-complex-matrix (DCM) architecture introduced here

which does not require a priori mathematical decomposition. As examples

we demonstrate

using system-level simulation tools

the solutions of integral and differential equations. We then expand the matrix inverting capabilities of both architectures toward evaluating the generalized Moore–Penrose matrix inversion. Therefore

we provide evidence that metadevices can implement generalized matrix inversions and act as the basis for the gradient descent method for solutions to a wide variety of problems. Finally

a general upper bound of the solution convergence time reveals the rich potential that such metadevices can offer for stationary iterative schemes.

关键词

Keywords

references

Trefethen, L. & Bau, D. III. Numerical Linear Algebra (SIAM, Philadelphia, PA, 1997).

H. J. Caulfield, S. Dolev . Why future supercomputing requires optics . Nat. Photonics , 2010 . 4 261 -263 . DOI:10.1038/nphoton.2010.94http://doi.org/10.1038/nphoton.2010.94 .

D. R. Solli, B. Jalali . Analog optical computing . Nat. Photonics , 2015 . 9 704 -706 . DOI:10.1038/nphoton.2015.208http://doi.org/10.1038/nphoton.2015.208 .

Z. Wang, 等 . Resistive switching materials for information processing . Nat. Rev. Mater. , 2020 . 5 173 -195 . DOI:10.1038/s41578-019-0159-3http://doi.org/10.1038/s41578-019-0159-3 .

A. Silva, 等 . Performing mathematical operations with metamaterials . Science , 2014 . 343 160 -163 . DOI:10.1126/science.1242818http://doi.org/10.1126/science.1242818 .

A. Pors, M. G. Nielsen, S. I. Bozhevolnyi . Analog computing using reflective plasmonic metasurfaces . Nano Lett. , 2015 . 15 791 -797 . DOI:10.1021/nl5047297http://doi.org/10.1021/nl5047297 .

G. Wetzstein, 等 . Inference in artificial intelligence with deep optics and photonics . Nature , 2020 . 588 39 -47 . DOI:10.1038/s41586-020-2973-6http://doi.org/10.1038/s41586-020-2973-6 .

N. C. Harris, 等 . Linear programmable nanophotonic processors . Optica , 2018 . 5 1623 -1631 . DOI:10.1038/s41586-020-2973-6http://doi.org/10.1038/s41586-020-2973-6 .

N. M. Estakhri, B. Edwards, N. Engheta . Inverse-designed metastructures that solve equations . Science , 2019 . 363 1333 -1338 . DOI:10.1126/science.aaw2498http://doi.org/10.1126/science.aaw2498 .

M. Le Gallo, 等 . Mixed-precision in-memory computing . Nat. Electronics , 2018 . 1 246 -253 . DOI:10.1038/s41928-018-0054-8http://doi.org/10.1038/s41928-018-0054-8 .

C. Mead . How we created neuromorphic engineering . Nat. Electronics , 2020 . 3 434 -435 . DOI:10.1038/s41928-020-0448-2http://doi.org/10.1038/s41928-020-0448-2 .

M. A. Zidan, 等 . A general memristor-based partial differential equation solver . Nat. Electronics , 2018 . 1 411 -420 . DOI:10.1038/s41928-020-0448-2http://doi.org/10.1038/s41928-020-0448-2 .

Z. Sun, 等 . Solving matrix equations in one step with cross-point resistive arrays . Proc. Natl. Acad. Sci. USA , 2019 . 116 4123 -4128 . DOI:10.1073/pnas.1815682116http://doi.org/10.1073/pnas.1815682116 .

I. M. Vellekoop, A. P. Mosk . Focusing coherent light through opaque strongly scattering media . Opt. Lett. , 2007 . 32 2309 -2311 . DOI:10.1364/OL.32.002309http://doi.org/10.1364/OL.32.002309 .

Y. Zhou, H. Zheng, I. I. Kravchenko, J. Valentine . Flat optics for image differentiation . Nat. Photon. , 2020 . 14 316 -323 . DOI:10.1364/OL.32.002309http://doi.org/10.1364/OL.32.002309 .

M. F. Imani, 等 . Review of metasurface antennas for computational microwave imaging . IEEE Trans. Antennas Propag. , 2020 . 68 1860 -1875 . DOI:10.1109/TAP.2020.2968795http://doi.org/10.1109/TAP.2020.2968795 .

Hougne P. del, G. Lerosey . Leveraging chaos for wave-based analog computation: demonstration with indoor wireless communication signals . Phys. Rev. X , 2018 . 8 041037 DOI:10.1103/PhysRevX.8.041037http://doi.org/10.1103/PhysRevX.8.041037 .

A. Cordaro, 等 . High-index dielectric metasurfaces performing mathematical operations . Nano Lett. , 2019 . 19 8418 -8423 . DOI:10.1021/acs.nanolett.9b02477http://doi.org/10.1021/acs.nanolett.9b02477 .

M. W. Matthès, P. del Hougne, J. de Rosny, G. Lerosey, S. M. Popoff . Optical complex media as universal reconfigurable linear operators . Optica , 2019 . 6 465 -472 . DOI:10.1364/OPTICA.6.000465http://doi.org/10.1364/OPTICA.6.000465 .

J. Feldmann, 等 . Parallel convolutional processing using an integrated photonic tensor core . Nature , 2021 . 589 52 -58 . DOI:10.1038/s41586-020-03070-1http://doi.org/10.1038/s41586-020-03070-1 .

Y. Shen, 等 . Deep learning with coherent nanophotonic circuits . Nat. Photon. , 2017 . 11 441 DOI:10.1038/nphoton.2017.93http://doi.org/10.1038/nphoton.2017.93 .

T. W. Hughes, R. J. England, S. Fan . Reconfigurable photonic circuit for controlled power delivery to laser-driven accelerators on a chip . Phys. Rev. Appl. , 2019 . 11 64014 DOI:10.1103/PhysRevApplied.11.064014http://doi.org/10.1103/PhysRevApplied.11.064014 .

C. Haffner, 等 . All-plasmonic Mach-Zehnder modulator enabling optical high-speed communication at the microscale . Nat. Photon. , 2015 . 9 525 -528 . DOI:10.1038/nphoton.2015.127http://doi.org/10.1038/nphoton.2015.127 .

J. Wu, 等 . Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems . Opt. Express , 2014 . 22 26254 -26264 . DOI:10.1364/OE.22.026254http://doi.org/10.1364/OE.22.026254 .

K. Wu, C. Soci, P. P. Shum, N. I. Zheludev . Computing matrix inversion with optical networks . Opt. Express , 2014 . 22 295 -304 . DOI:10.1364/OE.22.000295http://doi.org/10.1364/OE.22.000295 .

C. Taballione, 等 . 8 × 8 reconfigurable quantum photonic processor based on silicon nitride waveguides . Opt. Express , 2019 . 27 26842 DOI:10.1364/OE.27.026842http://doi.org/10.1364/OE.27.026842 .

W. Bogaerts, 等 . Programmable photonic circuits . Nature , 2020 . 586 207 -216 . DOI:10.1038/s41586-020-2764-0http://doi.org/10.1038/s41586-020-2764-0 .

T. Low, P. Avouris . Graphene plasmonics for terahertz to mid-infrared applications . ACS Nano , 2014 . 8 1086 -1101 . DOI:10.1021/nn406627uhttp://doi.org/10.1021/nn406627u .

M. Morea, K. Zang, T. I. Kamins, M. L. Brongersma, J. S. Harris . Electrically tunable, CMOS-compatible metamaterial based on semiconductor nanopillars . ACS Photon. , 2018 . 5 4702 -4709 . DOI:10.1021/acsphotonics.8b01383http://doi.org/10.1021/acsphotonics.8b01383 .

Y. Fan, 等 . Graphene plasmonics: a platform for 2D optics . Adv. Opt. Mater. , 2019 . 7 1 -14 . DOI:10.1002/adom.201800537http://doi.org/10.1002/adom.201800537 .

M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani . Experimental realization of any discrete unitary operator . Phys. Rev. Lett. , 1994 . 73 58 -61 . DOI:10.1103/PhysRevLett.73.58http://doi.org/10.1103/PhysRevLett.73.58 .

D. A. B. Miller . Self-configuring universal linear optical component [Invited] . Photon. Res. , 2013 . 1 1 -15 . DOI:10.1364/PRJ.1.000001http://doi.org/10.1364/PRJ.1.000001 .

D. A. B. Miller . Sorting out light . Science , 2015 . 347 1423 -1424 . DOI:10.1126/science.aaa6801http://doi.org/10.1126/science.aaa6801 .

W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, I. A. Walmsley . Optimal design for universal multiport interferometers . Optica , 2016 . 3 1460 -1465 . DOI:10.1364/OPTICA.3.001460http://doi.org/10.1364/OPTICA.3.001460 .

F. Zangeneh-Nejad, D. L. Sounas, A. Alù, R. Fleury . Analogue computing with metamaterials . Nat. Rev. Mater. , 2021 . 6 207 -225 . DOI:10.1038/s41578-020-00243-2http://doi.org/10.1038/s41578-020-00243-2 .

D. Psaltis, D. Casasent, M. Carlotto . Iterative color-multiplexed, electro-optical processor . Opt Lett. , 1979 . 4 348 -350 . DOI:10.1364/OL.4.000348http://doi.org/10.1364/OL.4.000348 .

J. Cederquist, S. H. Lee . The use of feedback in optical information processing: a review . Appl. Phys. , 1979 . 18 311 -319 . DOI:10.1007/BF00899683http://doi.org/10.1007/BF00899683 .

R. P. Akins, R. A. Athale, S. H. Lee . Feedback in analog and digital optical image processing: a review . Opt. Eng. , 1980 . 19 193347 DOI:10.1117/12.7972519http://doi.org/10.1117/12.7972519 .

Barrett, R. et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (SIAM, 1994).

J. W. Goodman, M. S. Song . Performance limitations of an analog method for solving simultaneous linear equations . Appl. Opt. , 1982 . 21 502 -506 . DOI:10.1364/AO.21.000502http://doi.org/10.1364/AO.21.000502 .

R. Penrose . A generalized inverse for matrices . Math. Proc. Camb. Philos. Soc. , 1955 . 51 406 -413 . DOI:10.1017/S0305004100030401http://doi.org/10.1017/S0305004100030401 .

L. F. Richardson . The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam . Philos. Trans. Royal Soc. A , 1911 . 210 307 -357 . DOI:10.1098/rsta.1911.0009http://doi.org/10.1098/rsta.1911.0009 .

L. Landweber . An iteration formula for fredholm integral equations of the first kind . Am. J. Math. , 1951 . 73 615 -624 . DOI:10.2307/2372313http://doi.org/10.2307/2372313 .

Shewchuk, Jonathan, R. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Tech. Rep., Carnegie Mellon University (1994).

A. W. Harrow, A. Hassidim, S. Lloyd . Quantum algorithm for linear systems of equations . Phys. Rev. Lett. , 2009 . 103 150502 DOI:10.1103/PhysRevLett.103.150502http://doi.org/10.1103/PhysRevLett.103.150502 .

D. M. Pozar . Microwave engineering , 3rd edn :Hoboken, NJ Wiley , 2005 . DOI:10.1364/OE.25.029233http://doi.org/10.1364/OE.25.029233 .

D. A. B. Miller . Setting up meshes of interferometers - reversed local light interference method . Opt. Express , 2017 . 25 29233 -29248 . DOI:10.1364/OE.25.029233http://doi.org/10.1364/OE.25.029233 .

Cadence. AWR Microwave Office (2020).

Abramowitz, M. & Stegun, I. A. Handbook of mathematical functions with formulas, graphs, and mathematical tables (Washington: U.S. Government Printing Office, 1964).

Kreyszig, E. Advanced Engineering Mathematics (Hoboken: John Wiley & Sons, Inc., 1999), 8th edn.

F. Zangeneh-Nejad, R. Fleury . Topological analog signal processing . Nat. Commun. , 2019 . 10 2058 DOI:10.1038/s41467-019-10086-3http://doi.org/10.1038/s41467-019-10086-3 .

LeVeque, R. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Philadelphia: SIAM, USA, 2007).

S. Jahani, 等 . Controlling evanescent waves using silicon photonic all-dielectric metamaterials for dense integration . Nat. Commun. , 2018 . 9 1893 DOI:10.1038/s41467-018-04276-8http://doi.org/10.1038/s41467-018-04276-8 .

M. B. Mia, 等 . Exceptional coupling in photonic anisotropic metamaterials for extremely low waveguide crosstalk . Optica , 2020 . 7 881 -887 . DOI:10.1364/OPTICA.394987http://doi.org/10.1364/OPTICA.394987 .

T. Song, 等 . Ultracompact photonic circuits without cladding layers . Phys. Rev. X , 2022 . 12 011053 DOI:10.1103/PhysRevX.12.011053http://doi.org/10.1103/PhysRevX.12.011053 .

R. E. Kleinman, P. M. Van den Berg . A modified gradient method for two-dimensional problems in tomography . J. Comput. Appl. Math. , 1992 . 42 17 -35 . DOI:10.1016/0377-0427(92)90160-Yhttp://doi.org/10.1016/0377-0427(92)90160-Y .

J. S. Jensen, O. Sigmund . Topology optimization for nano-photonics . Laser Photon. Rev. , 2011 . 5 308 -321 . DOI:10.1002/lpor.201000014http://doi.org/10.1002/lpor.201000014 .

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