PMCHWT Solver Accelerated by Adaptive Cross Approximation for Efficient Computation of Scattering from Metal Nanoparticles
Abstract
:1. Introduction
2. Materials and Methods
2.1. Generic PMCHWT Formulation for a Metal Nanoparticle
2.2. Triangular–Triangular Cyclic Integral Method for Element Calculation of Impedance Matrix
2.3. The Octree Establishes Grouping
2.4. Overview of the Adaptive Cross Approximation (ACA) Algorithm
Algorithm 1. The detailed process of the ACA algorithm 
Initialization steps:

Next, the k th iteration:

2.5. ACA Algorithm Accelerates Filling Impedance Matrix
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
 Hedayati, M.K.; Faupel, F.; Elbahri, M. Review of Plasmonic Nanocomposite Metamaterial Absorber. Materials 2014, 7, 1221–1248. [Google Scholar] [CrossRef] [PubMed]
 Hutter, E.; Fendler, J.H. Exploitation of localized surface plasmon resonance. Adv. Mater. 2010, 16, 1685–1706. [Google Scholar] [CrossRef]
 Johnson, P.B.; Christy, R.W. Optical constants of the noble metals. Phys. Rev. B 1972, 6, 4370. [Google Scholar] [CrossRef]
 Aizpurua, J.; Hanarp, P.; Sutherland, D.S.; Käll, M.; Bryant, G.W.; García de Abajo, F.J. Optical properties of gold nanorings. Phys. Rev. Lett. 2003, 90, 057401. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Pendry, J.B.; Holden, A.J.; Robbins, D.J.; Stewart, W.J. Magnetism from Conductors and Enhanced Nonlinear Phenomena. IEEE Trans. Microw. Theory Tech. 1999, 47, 2075–2084. [Google Scholar] [CrossRef] [Green Version]
 Cantale, V.; Simeone, F.C.; Gambari, R.; Rampi, M.A. Gold nanoislands on FTO as plasmonic nanostructures for biosensors. Sens. Actuators B Chem. 2011, 152, 206–213. [Google Scholar] [CrossRef]
 Watanabe, M.; Sassa, F.; Hayashi, K. Formation of oriented metal nanostructures by polarized light irradiation for optical sensing. In Proceedings of the 2016 IEEE SENSORS, Orlando, FL, USA, 30 October–3 November 2016. [Google Scholar]
 Choy, W.C.H.; Ren, X.G. PlasmonElectrical Effects on Organic Solar Cells by Incorporation of Metal Nanostructures. IEEE J. Sel. Top. Quantum Electron. 2015, 22, 1–9. [Google Scholar] [CrossRef]
 Liu, Z.Q.; Liu, G.Q.; Huang, S.; Liu, X.S.; Wang, Y.; Liu, M.L.; Gu, G. Enabling Access to the Confined Optical Field to Achieve HighQuality Plasmon Sensing. IEEE Photonics Technol. Lett. 2015, 27, 1212–1215. [Google Scholar] [CrossRef]
 Liu, Z.Q.; Liu, G.Q.; Liu, X.S.; Fu, G.L.; Liu, M.L. Improved Multispectral Antireflection and Sensing of Plasmonic Slits by Silver Mirror. IEEE Photonics Technol. Lett. 2014, 26, 2111–2114. [Google Scholar] [CrossRef]
 Yu, D.M.; Liu, Y.N.; Tian, F.L.; Pan, X.M.; Sheng, X.Q. Accurate thermoplasmonic simulation of metallic nanoparticles. J. Quant. Spectrosc. Radiat. Transf. 2017, 187, 150–160. [Google Scholar] [CrossRef]
 Araújo, M.G.; Taboada, J.M.; Solís, D.M.; Rivero, J.; Landesa, L.; Obelleiro, F. Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers. Opt. Express 2012, 20, 9161–9171. [Google Scholar] [CrossRef] [PubMed]
 Liu, Y.N.; Pan, X.M.; Sheng, X.Q. Skeletonization Accelerated MLFMA Solution of Volume Integral Equation for Plasmonic Structures. IEEE Trans. Antennas Propag. 2018, 66, 1590–1594. [Google Scholar] [CrossRef]
 Taboada, J.M.; Rivero, J.; Obelleiro, F.; Araújo, M.G.; Landesa, L. Methodofmoments formulation for the analysis of plasmonic nanooptical antennas. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 2011, 28, 1341–1348. [Google Scholar] [CrossRef] [PubMed]
 Araújo, M.G.; Taboada, J.M.; Rivero, J.; Solís, D.M.; Obelleiro, F. Solution of largescale plasmonic problems with the multilevel fast multipole algorithm. Opt. Lett. 2012, 37, 416–418. [Google Scholar] [CrossRef] [PubMed]
 Rivero, J.; Taboada, J.M.; Landesa, L.; Obelleiro, F.; Tuñón, I.G. Surface integral equation formulation for the analysis of lefthanded metamaterials. Opt. Express 2010, 18, 15876–15886. [Google Scholar] [CrossRef]
 Araújo, M.G.; Taboada, J.M.; Rivero, J.; Solís, D.M.; Obelleiro, F.; Landesa, L. Electromagnetic Analysis of Metamaterials and Plasmonic Nanostructures with the Method of Moments. IEEE Antennas Propag. Mag. 2012, 54, 81–91. [Google Scholar] [CrossRef]
 Gaffar, M.; Jiao, D. An Explicit and Unconditionally Stable FDTD Method for Electromagnetic Analysis. IEEE Trans. Microw. Theory Tech. 2014, 62, 2538–2550. [Google Scholar] [CrossRef]
 Jin, J.M.; Volakis, J.L. A hybrid finite element method for scattering and radiation by microstrip patch antennas and arrays residing in a cavity. IEEE Trans. Antennas Propag. 1991, 39, 1598–1604. [Google Scholar] [CrossRef]
 Harrington, R.F. Field computation by moment methods; Macmillan: London, UK, 1968; ISBN 0780310144. [Google Scholar]
 Umashanker, K.; Taflove, A.; Rao, S.M. Electromagnetic scattering by arbitrary shaped threedimensional homogeneous lossy dielectric objects. IEEE Trans. Antennas Propag. 1986, 34, 758–766. [Google Scholar] [CrossRef]
 Schaubert, H.; Wilton, D.R.; Glisson, A.W. A tetrahedral modeling method for electromagnetic scattering by arbitrary shaped inhomogeneous dielectric bodies. IEEE Trans. Antennas Propag. 1984, 32, 77–85. [Google Scholar] [CrossRef]
 Kern, A.M.; Martin, O.J.F. Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures. J. Opt. Soc. Am. A 2009, 26, 732–740. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Wu, T.K.; Tsai, L.L. Scattering from arbitrarilyshaped lossy dielectric bodies of revolution. Radio Sci. 1977, 12, 709–718. [Google Scholar] [CrossRef]
 Chang, Y.; Harrington, R.F. A surface formulation for characteristic modes of material bodies. IEEE Trans. Antennas Propag. 1977, 25, 789–795. [Google Scholar] [CrossRef] [Green Version]
 Saad, Y. A flexible innerouter preconditioned GMRES algorithm. SIAM J. Entific Comput. 1993, 14, 461–469. [Google Scholar] [CrossRef]
 Raziman, T.V.; Somerville, W.R.C.; Martin, O.J.F.; Ru, E.C.L. Accuracy of surface integral equation matrix elements in plasmonic calculations. J. Opt. Soc. Am. B 2015, 32, 485–492. [Google Scholar] [CrossRef] [Green Version]
 Bebendorf, M. Approximation of boundary element matrices. Numer. Math. 2000, 86, 565–589. [Google Scholar] [CrossRef]
 Zhao, K.Z.; Vouvakis, M.N.; Lee, J.F. The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems. IEEE Trans. Electromagn. Compat. 2005, 47, 763–773. [Google Scholar] [CrossRef]
 Liu, Z.W.; Tang, D.; Zhang, Z.Y.; Zhang, Y.y.; Wang, X.L.; Jie, S.L. Combination of MLFMA and ACA to accelerate computation of scattering from underground targets. Int. J. Antennas Propag. 2019, 2019, 3456871. [Google Scholar] [CrossRef]
Method  MoM  ACA 

HH polarization calculation time (s)  42.64  28.84 
VV polarization calculation time (s)  42.61  28.86 
Radius (nm)  68.25  456  

Method  MoM  ACA  MoM  ACA 
HH polarization calculation time (s)  41.30  27.90  45.22  30.37 
VV polarization calculation time (s)  41.19  27.47  44.61  30.57 
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Liu, Z.; Xi, L.; Bao, Y.; Cheng, Z. PMCHWT Solver Accelerated by Adaptive Cross Approximation for Efficient Computation of Scattering from Metal Nanoparticles. Micromachines 2022, 13, 1086. https://doi.org/10.3390/mi13071086
Liu Z, Xi L, Bao Y, Cheng Z. PMCHWT Solver Accelerated by Adaptive Cross Approximation for Efficient Computation of Scattering from Metal Nanoparticles. Micromachines. 2022; 13(7):1086. https://doi.org/10.3390/mi13071086
Chicago/Turabian StyleLiu, Zhiwei, Longfeng Xi, Yang Bao, and Ziyue Cheng. 2022. "PMCHWT Solver Accelerated by Adaptive Cross Approximation for Efficient Computation of Scattering from Metal Nanoparticles" Micromachines 13, no. 7: 1086. https://doi.org/10.3390/mi13071086