Mathematical analysis of plasmonics resonances for nanoparticles and applications Matias Ruiz To cite this version: Matias Ruiz. Mathematical analysis of plasmonics resonances for nanoparticles and applications. Plasmas. Université Paris sciences et lettres, 2017. English. NNT : 2017PSLEE054. tel-01827839 HAL Id: tel-01827839 https://tel.archives-ouvertes.fr/tel-01827839 Submitted on 2 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences Lettres – PSL Research University préparée à l’École Normale Supérieure Analyse Mathématique de École doctorale n°386 Résonances Plasmoniques Spécialité: Mathématiques pour des Nanoparticules Appliquées et Applications Soutenue le 27.06.2017 Mathematical Analysis of Plasmonic Resonances for Nanoparticles and Applications Composition du Jury : par Matias Ruiz M. Habib Ammari ETH Zurich Directeur de thèse M. Oscar Bruno California Institute of Technology Rapporteur M. Gilles Lebeau Université de Nice - Sophia Antipolis Rapporteur Mme. Liliana Borcea University of Michigan Examinatrice M. Josselin Garnier École Polytechnique Examinateur M. Bertrand Maury Université Paris Sud Examinateur M. Gabriel Peyré École Normale Supérieure Président du jury iii Acknowledgements I would like to start by thanking the most important person in the achieve- ment of this Ph.D. Habib, I couldn’t have asked for a better advisor. I have learnt so much from you. You are not only an amazing scientist but also a great leader and person. I truly hope that our friendship and scientific collaboration will last long years. There is someone else who has been of key importance in the success of these three years of hard work. Hai Zhang, your enormous talent has and will be an inspiration for me. I thank you for our fruitful collaboration and for, more than once, setting me in the right path of mathematical discovery. I hope this is only the beginning of many works together. I would also like to greatly thank the members of my jury: Oscar Bruno and Gilles Lebeau, who accepted the time consuming mission of being my "rapporteurs" and Lilianna Borcea, Josselin Garnier, Bertrand Maury and Gabriel Peyré, who kindly agreed to be my "examinateurs". It is a great honor to have such world class researchers judging my work. Research is done better when working in groups. I would first like to thank Sanghyeon Yu, with whom I have already cosigned 4 research papers. You are an amazing mathematician and a very good friend. I hope our collaboration doesn’t stop with the end of this work. Pierre Millien and Francisco Romero, it has also been a privilege to work with you. I have really enjoyed the time we spent together, trying to solve a math problem, discussing about science or just visiting a city during a conference. I would also like to thank Mihai Putinar and Wei Wu, with whom I have cosigned one paper. Habib’s team is a big family where I have encountered wonderful people. Wenglong and Timothée, it has been a truly pleasure to have started and finished this experience along your side. I would also like to thank the other members (or former members) of Habib’s team that I had the fortune to met. Abdul, Brian, David, Alexander, Giovanni, Guanghui and Tomas. I thank you for all the moments we shared. The DMA is full of good people, starting by Bénédicte, Zaïna and Lara who always had the best disposition to help me with all the administrative issues during these three years. You were always there when I needed your help and for that, a thousand thanks. I had the chance to have my desk in the best office of the department, the "bureau de l’ambiance". Thank you Maxence, Jessica and Jeremy for making every day so pleasant and for considerably improving my level on baby foot. Of course, I cannot talk about the "bureau de l’ambiance" with- out mentioning its topological dual, the "bureau des plaisirs". Thank you Guillaume, Yichao and Wei for all the good moments. I also thank Charles, Jaime, Thibault and Rodolfo for your sympathy. When you are away from home, your friends become your family. I would like to thank all my friends who have been to my side in this process of expatriation. In particular, Jorge Gonazález, Sofia Olivares and Felipe Soto who arrived with me in France six and a half years ago. Thank you also Ignacio Atal, Francisco Vial (jedi master), Tomas Yany, Michel Lemaire and Juan Pablo de la Cruz. Special greetings to my good friend and bestman at my wedding, Arsalan Shoysoronov. I am proud to have you as a friend. iv I also happen to have real family in France. Thank you Carla and Lolo for feeding me and sharing so many good moments. Nada de esto sería posible sin el apoyo de mi familia. Papá (Cesar), Mamá (Olga), ustedes lucharon incansablemente para que yo y mis hermanos tuviéramos siempre la mejor educación. Este trabajo es de ustedes. No puedo dejar de agradecer a dos personas cuyo apoyo ha sido funda- mental. Carmen Luz Palacios y Fernando Salinas, les estaré eternamente agradecido. Este trabajo también se lo dedico a ustedes. And last but not least, I would like to thank Despi, my lovely and beau- tiful wife. You just make my life so happy. Thank you for having my back no matter what and for being my partner in life and crime. All of this would be meaningless if I couldn’t share it with you. v Contents Introduction 1 I Mathematical Analysis of Plasmonic Resonances 7 1 The Quasi-Static Limit 9 1.1 Introduction . 10 1.2 Preliminaries . 10 1.2.1 Layer potentials for the Laplace equation . 10 1.3 Layer potential formulation for the scattering problem . 12 1.3.1 Contracted generalized polarization tensors . 13 1.4 Plasmonic resonances for the anisotropic problem . 15 1.5 A Maxwell-Garnett theory for plasmonic nanoparticles . 19 1.6 Concluding remarks . 23 2 The Helmholtz Equation 25 2.1 Introduction . 26 2.2 Preliminaries . 26 2.2.1 Layer potentials for the Helmholtz equation . 26 2.3 Layer potential formulation for the scattering problem . 28 2.3.1 Problem formulation and some basic results . 28 2.3.2 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . 32 2.4 Scattering and absorption enhancements . 37 2.4.1 Far-field expansion . 37 2.4.2 Energy flow . 38 2.4.3 Extinction, absorption, and scattering cross-sections and the optical theorem . 39 2.4.4 The quasi-static limit . 41 2.4.5 An upper bound for the averaged extinction cross-section 43 Bound for ellipses . 46 Bound for ellipsoids . 48 2.5 Link with the scattering coefficients . 48 2.5.1 The notion of scattering coefficients . 49 2.5.2 The leading-order term in the expansion of the scat- tering amplitude . 50 2.6 Concluding remarks . 53 3 The Full Maxwell Equations 55 3.1 Introduction . 56 3.2 Preliminaries . 57 3.3 Layer potential formulation for the scattering problem . 60 3.3.1 First-order correction to plasmonic resonances and field behavior at the plasmonic resonances . 70 vi 3.4 The extinction cross-section at the quasi-static limit . 76 3.5 Explicit computations for a spherical nanoparticle . 77 3.5.1 Vector spherical harmonics . 77 3.5.2 Explicit representations of boundary integral operators 79 3.5.3 Asymptotic behavior of the spectrum of (r) . 81 WB 3.5.4 Extinction cross-section . 83 3.6 Explicit computations for a spherical shell . 86 3.6.1 Explicit representation of boundary integral operators 87 3.6.2 Asymptotic behavior of the spectrum of sh(r ) . 90 WB s 3.7 Concluding remarks . 92 II Applications 95 4 Heat Generation with Plasmonic Nanoparticles 97 4.1 Introduction . 98 4.2 Setting of the problem and the main results . 98 4.3 Heat generation . 101 4.3.1 Small volume expansion of the inner field . 102 Rescaling . 102 Proof of Theorem 4.2.1 . 103 4.3.2 Small volume expansion of the temperature . 106 Re-scaling of the equations . 107 Proof of Theorem 4.2.2 . 107 4.3.3 Temperature elevation at the plasmonic resonance . 109 4.3.4 Temperature elevation for two close-to-touching particles110 4.4 Numerical results . 111 4.4.1 Single-particle simulation . 112 Single-particle Helmholtz resonance . 112 Single-particle surface heat generation . 113 4.4.2 Two particles simulation . 115 Two particles Helmholtz resonance . 115 Two particles surface heat generation . 115 4.5 Concluding remarks . 116 5 Plasmonic Metasurfaces 119 5.1 Introduction . 120 5.2 Setting of the problem . 121 5.3 One-dimensional periodic Green function . 122 5.4 Boundary layer corrector and effective impedance . 124 5.5 Numerical illustrations . 128 5.5.1 Setup and methods . 128 5.5.2 Results and discussion . 129 5.6 Concluding remarks . 129 6 Shape Recovery of Algebraic Domains 133 6.1 Introduction . 134 6.2 Plasmonic resonance for algebraic domains . 135 6.2.1 Algebraic domains of class . 135 Q 6.2.2 Explicit computation of the Neumann-Poincaré operator135 vii 6.2.3 Asymptotic behavior of the Neumann-Poincaré opera- tor .