Modeling and Applications of Nonlinear Metasurfaces by Xiaojun Liu Department of Electrical and Computer Engineering Duke University Date:_______________________ Approved: ___________________________ David R. Smith, Supervisor ___________________________ Steven A. Cummer ___________________________ Qing H. Liu ___________________________ Nan M. Jokerst ___________________________ Willie J. Padilla Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering of Duke University 2018 ABSTRACT Modeling and Applications of Nonlinear Metasurfaces by Xiaojun Liu Department of Electrical and Computer Engineering Duke University Date:_______________________ Approved: ___________________________ David R. Smith, Supervisor ___________________________ Steven A. Cummer ___________________________ Qing H. Liu ___________________________ Nan M. Jokerst ___________________________ Willie J. Padilla An abstract of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Electrical and Computer Engineering of Duke University 2018 Copyright by Xiaojun Liu 2018 Abstract A patterned metasurface can strongly scatter incident light, functioning as an extremely low-profile lens, filter, reflector or other optical devices. Nonlinear metasurfaces‒combine the properties of natural nonlinear medium with novel features such as negative refractive index, magneto-electric coupling‒provide novel nonlinear features not available in nature. Compared to conventional optical components that often extend many wavelengths in size, nonlinear metasurfaces are flexible and extremely compact. Characterization of a nonlinear metasurface is challenging, not only due to its inherent anisotropy, but also because of the rich wave interactions available. This thesis presents an overview of the work by the author in modeling and application of nonlinear metasurfaces. Analytical methods - transfer matrix method and surface homogenization method - for characterizing nonlinear metasurfaces are presented. A generalized transfer matrix method formalism for four wave mixing is derived, and applied to analyze nonlinear interface, film, and metallo-dielectric stack. Various channels of plasmonic and Fabry-perot enhancement are investigated. A homogenized description of nonlinear metasurfaces is presented. The homogenization procedure is based on the nonlinear generalized sheet transition conditions (GSTCs), where an optically thin nonlinear metasurface is modeled as a layer of dipoles radiating at iv fundamental and nonlinear frequencies. By inverting the nonlinear GSTCs, a retrieval procedure is developed to retrieve the nonlinear parameters of the nonlinear metasurface. As an example, we investigate a nonlinear metasurface which presents nonlinear magnetoelectric coupling in near infrared regime. The method is expected to apply to any patterned metasurface whose thickness is much smaller than the wavelengths of operation, with inclusions of arbitrary geometry and material composition, across the electromagnetic spectrum. The second part presents the applications of nonlinear metasurfaces. First, we show that the third-harmonic generation (THG) can be drastically enhanced by the nonlinear metasurfaces – film-coupled nanostripes. The large THG enhancement is experimentally and theoretically demonstrated. With numerical simulations, we present multiple ways to clarify the origin of the THG from the metasurface. Second, the enhanced two-photon photochormism is investigated by integrating spiropyrans with film-coupled nanocubes. This metasurface platform couples almost 100% energy at resonance, and induces isomerization of spiropyrans to merocyanines. Due to the large Purcell enhancement introduced by the film-coupled nanocubes, fluorescence lifetime measurements on the merocyanine form reveal large enhancements on spontaneous emission rate, as well as high quantum efficiency. We show that this metasurface platform is capable of storing information, supports reading and writing with ultra-low power, offering new possibilities in optical data storage. v To my beloved parents and family, for their love, support and encouragement. vi Contents Abstract ......................................................................................................................................... iv List of Tables .................................................................................................................................. x List of Figures ............................................................................................................................... xi Acknowledgements .................................................................................................................... xv 1. Introduction........................................................................................................................... 1 1.1 Metasurfaces ................................................................................................................. 2 1.2 Nonlinearity ................................................................................................................. 3 1.2.1 Nonlinear polarizations .......................................................................................... 4 1.2.2 Nonlinear processes ................................................................................................ 5 1.3 Modelling ...................................................................................................................... 7 2. Modeling of a nonlinear surface, film, and stack ........................................................... 10 2.1 Analytic expression: an interface ............................................................................. 11 2.2 Transfer matrix method: a film ................................................................................ 18 2.2.1 Transfer matrix method ........................................................................................ 19 2.2.2 Nonlinear Polarizations ........................................................................................ 22 2.2.3 Numerical validation ............................................................................................ 27 2.2.4 A thin film in the kreschmann configuration .................................................... 29 2.3 Transfer matrix method: a multilayer stack ........................................................... 32 2.4 Conclusion .................................................................................................................. 35 3. Modeling of nonlinear metasurfaces ............................................................................... 40 vii 3.1 Effective surface description of metasurfaces........................................................ 41 3.1.1 Universal Boundary Conditions ......................................................................... 44 3.1.2 Average fields across a surface............................................................................ 45 3.1.3 Generalized sheet transition condition .............................................................. 48 3.1.4 Nonlinear generalized sheet transition condition ............................................ 50 3.2 Nonlinear surface parameter retrieval ................................................................... 55 3.2.1 Retrieval method ................................................................................................... 55 3.2.2 Connection between surface parameters and bulk parameters ..................... 60 3.3 Applications ............................................................................................................... 66 3.3.1 Nonlinear slab ........................................................................................................ 67 3.3.2 Magnetoelectric nonlinear metasurfaces ........................................................... 71 3.4 Conclusion .................................................................................................................. 81 4. Enhanced nonlinear response from metasurface platforms ......................................... 84 4.1 Third-harmonic generation enhancement ............................................................. 85 4.1.1 Geometry and method .......................................................................................... 86 4.1.2 Result and Discussion ........................................................................................... 90 4.1.3 Conclusions ............................................................................................................ 92 4.2 Clarifying the origin of THG from an isolated film-coupled nanostripe .......... 93 4.2.1 Geometry and method .......................................................................................... 94 4.2.2 Results and discussions ........................................................................................ 96 4.2.3 Generality ............................................................................................................. 103 4.2.4 Conclusion ............................................................................................................ 106 viii 5. Enhanced two photon photochromism in metasurface perfect absorbers ............... 108 5.1 Overview .................................................................................................................