Quasinormal mode solvers for resonators with dispersive materials.

Optical resonators are widely used in modern photonics. Their spectral response and temporal dynamics are fundamentally driven by their natural resonances, the so-called quasinormal modes (QNMs), with complex frequencies. For optical resonators made of dispersive materials, the QNM computation requires solving a nonlinear eigenvalue problem. This raises a difficulty that is only scarcely documented in the literature. We review our recent efforts for implementing efficient and accurate QNM solvers for computing and normalizing the QNMs of micro- and nanoresonators made of highly dispersive materials. We benchmark several methods for three geometries, a two-dimensional plasmonic crystal, a two-dimensional metal grating, and a three-dimensional nanopatch antenna on a metal substrate, with the perspective to elaborate standards for the computation of resonance modes.

[1]  Richard M. More,et al.  Properties of Resonance Wave Functions , 1973 .

[2]  David A. Powell,et al.  Interference between the modes of an all-dielectric meta-atom , 2016, 1610.04980.

[3]  Tuomas Vallius,et al.  Reformulation of the Fourier modal method with adaptive spatial resolution: application to multilevel profiles. , 2002, Optics express.

[4]  David A. Powell,et al.  Resonant dynamics of arbitrarily shaped meta-atoms , 2014, 1405.3759.

[5]  David R. Smith,et al.  Controlled-reflectance surfaces with film-coupled colloidal nanoantennas , 2012, Nature.

[6]  D. Whittaker,et al.  Scattering-matrix treatment of patterned multilayer photonic structures , 1999 .

[7]  Quenching, Plasmonic, and Radiative Decays in Nanogap Emitting Devices , 2015, 1510.06693.

[8]  Philippe Lalanne,et al.  Modal Analysis of the Ultrafast Dynamics of Optical Nanoresonators , 2017 .

[9]  Benjamin Vial,et al.  Quasimodal expansion of electromagnetic fields in open two-dimensional structures , 2013, 1311.3244.

[10]  Harald Giessen,et al.  Matched coordinates and adaptive spatial resolution in the Fourier modal method. , 2009, Optics express.

[11]  Rob Remis,et al.  A Krylov Stability-Corrected Coordinate-Stretching Method to Simulate Wave Propagation in Unbounded Domains , 2012, SIAM J. Sci. Comput..

[12]  S Hughes,et al.  Generalized effective mode volume for leaky optical cavities. , 2012, Optics letters.

[13]  H. Giessen,et al.  Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates. , 2011, Journal of the Optical Society of America. A, Optics, image science, and vision.

[14]  M Garcia-Vergara,et al.  Extracting an accurate model for permittivity from experimental data: hunting complex poles from the real line. , 2016, Optics letters.

[15]  VLADIMIR DRUSKIN,et al.  Near-Optimal Perfectly Matched Layers for Indefinite Helmholtz Problems , 2015, SIAM Rev..

[16]  P. Lalanne,et al.  Bragg waveguide grating as a 1d photonic band gap structure: COST 268 modelling task , 2002 .

[17]  J. N. Lyness,et al.  A Numerical Method for Locating the Zeros of an Analytic Function , 1967 .

[18]  Marc Van Barel,et al.  Computing the Zeros of Analytic Functions , 2000 .

[19]  H Giessen,et al.  From Dark to Bright: First-Order Perturbation Theory with Analytical Mode Normalization for Plasmonic Nanoantenna Arrays Applied to Refractive Index Sensing. , 2016, Physical review letters.

[20]  T. Ishihara,et al.  Quasiguided modes and optical properties of photonic crystal slabs , 2002 .

[21]  S. Burger,et al.  Benchmarking five numerical simulation techniques for computing resonance wavelengths and quality factors in photonic crystal membrane line defect cavities. , 2017, Optics express.

[22]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[23]  Philippe Lalanne,et al.  Light Interaction with Photonic and Plasmonic Resonances , 2017, Laser & Photonics Reviews.

[24]  Lucas Chesnel,et al.  T-Coercivity for the Maxwell Problem with Sign-Changing Coefficients , 2014 .

[25]  Lifeng Li.  Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings , 1996 .

[26]  Philippe Lalanne,et al.  Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization , 1998 .

[27]  Benfeng Bai,et al.  Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries , 2005 .

[28]  J. Mørk,et al.  Three-dimensional integral equation approach to light scattering, extinction cross sections, local density of states, and quasi-normal modes , 2013, 1305.5263.

[29]  Frank Schmidt,et al.  Adaptive finite element method for simulation of optical nano structures , 2007, 0711.2149.

[30]  Frédéric Zolla,et al.  Photonics in highly dispersive media: the exact modal expansion. , 2018, Optics letters.

[31]  T. Gaylord,et al.  Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings , 1995 .

[32]  J. P. Webb,et al.  Hierarchal Scalar and vector Tetrahedra , 1992, Digest of the Fifth Biennial IEEE Conference on Electromagnetic Field Computation.

[33]  Lifeng Li,et al.  Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors , 2003 .

[34]  P Lalanne,et al.  Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure. , 2013, Optics express.

[35]  Philippe Lalanne,et al.  Aperiodic-Fourier modal method for analysis of body-of-revolution photonic structures. , 2014, Journal of the Optical Society of America. A, Optics, image science, and vision.

[36]  Qing Huo Liu,et al.  Three-dimensional dispersive metallic photonic crystals with a bandgap and a high cutoff frequency. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[37]  Christophe Geuzaine,et al.  Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures , 2018, Comput. Phys. Commun..

[38]  Jean-Luc Pelouard,et al.  Nearly perfect Fano transmission resonances through nanoslits drilled in a metallic membrane. , 2010, Physical review letters.

[39]  Olivier J. F. Martin,et al.  Mode analysis of second-harmonic generation in plasmonic nanostructures , 2016 .

[40]  Lei Wei,et al.  A Lanczos model-order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media , 2016, J. Comput. Phys..

[41]  Patrick Ciarlet,et al.  Mesh requirements for the finite element approximation of problems with sign-changing coefficients , 2018, Numerische Mathematik.

[42]  Lifeng Li,et al.  Use of Fourier series in the analysis of discontinuous periodic structures , 1996 .

[43]  Philippe Lalanne,et al.  Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[44]  Jean-Michel Gérard,et al.  Solid-State Cavity-Quantum Electrodynamics with Self-Assembled Quantum Dots , 2003 .

[45]  Caroline Fossati,et al.  The finite element method as applied to the diffraction by an anisotropic grating. , 2007, Optics express.

[46]  J. E. Román,et al.  Eigenmode computations of frequency-dispersive photonic open structures: A non-linear eigenvalue problem , 2018 .

[47]  Yu Luo,et al.  Description of van der Waals interactions using transformation optics. , 2013, Physical review letters.

[48]  P. Lalanne,et al.  Structural Slow Waves: Parallels between Photonic Crystals and Plasmonic Waveguides , 2018, ACS Photonics.

[49]  Gérard Granet,et al.  Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution , 1999 .

[50]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

[51]  F. Zolla,et al.  Resonant metamaterial absorbers for infrared spectral filtering: quasimodal analysis, design, fabrication, and characterization , 2014, 1402.0387.

[52]  L. M. Delvest,et al.  A Numerical Method for Locating the Zeros of an Analytic Function , 2010 .

[53]  R. T. Hill,et al.  Probing the Ultimate Limits of Plasmonic Enhancement , 2012, Science.

[54]  Brahim Guizal,et al.  Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization , 1996 .

[55]  Philippe Lalanne,et al.  Rigorous modal analysis of plasmonic nanoresonators , 2017, 1711.05011.

[56]  Stefano Selleri,et al.  Modelling leaky photonic wires: A mode solver comparison , 2007 .

[57]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[58]  Sven Burger,et al.  A Riesz-projection-based method for nonlinear eigenvalue problems , 2018, J. Comput. Phys..

[59]  ANG,et al.  Benchmarking five numerical simulation techniques for computing resonance wavelengths and quality factors in photonic crystal membrane line defect cavities , 2018 .

[60]  Lucas Chesnel,et al.  Two-dimensional Maxwell's equations with sign-changing coefficients , 2014 .

[61]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[62]  David R. Smith,et al.  Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas , 2014, Nature Photonics.

[63]  P Lalanne,et al.  Theoretical and computational concepts for periodic optical waveguides. , 2007, Optics express.

[64]  Vicente Hernández,et al.  SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems , 2005, TOMS.

[65]  R. More.  THEORY OF DECAYING STATES. , 1971 .

[66]  Silvania F. Pereira,et al.  Numerical analysis of a slit-groove diffraction problem , 2007 .

[67]  S. Burger,et al.  Simulations of high-Q optical nanocavities with a gradual 1D bandgap. , 2013, Optics express.

[68]  Young,et al.  Completeness and time-independent perturbation of the quasinormal modes of an absorptive and leaky cavity. , 1994, Physical Review A. Atomic, Molecular, and Optical Physics.

[69]  L. L. Doskolovich,et al.  Numerical Methods for Calculating Poles of the Scattering Matrix With Applications in Grating Theory , 2012, Journal of Lightwave Technology.

[70]  Paul Urbach,et al.  Efficient computation of the spontaneous decay rate of arbitrarily shaped 3D nanosized resonators: a Krylov model-order reduction approach , 2016 .

[71]  T. Weiss,et al.  How to calculate the pole expansion of the optical scattering matrix from the resonant states , 2018, Physical Review B.

[72]  Lifeng Li,et al.  New formulation of the Fourier modal method for crossed surface-relief gratings , 1997 .

[73]  P Lalanne,et al.  Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators. , 2013, Physical review letters.

[74]  Philippe Lalanne,et al.  Stable and efficient bloch-mode computational method for one-dimensional grating waveguides. , 2002, Journal of the Optical Society of America. A, Optics, image science, and vision.

[75]  Christophe Geuzaine,et al.  A general environment for the treatment of discrete problems and its application to the finite element method , 1998 .

[76]  Thomas F. Krauss,et al.  Photonics: The Vs and Qs of optical microcavities , 1997, Nature.

[77]  S. Burger,et al.  Advanced finite element method for nano-resonators , 2006, SPIE OPTO.

[78]  김영훈.  2000 , 2001, The Winning Cars of the Indianapolis 500.

[79]  Oliver Benson,et al.  Riesz-projection-based theory of light-matter interaction in dispersive nanoresonators , 2018, Physical Review A.

[80]  Surface and bulk modes for periodic structures of negative index materials , 2006 .

[81]  P. Lalanne,et al.  Highly improved convergence of the coupled-wave method for TM polarization and conical mountings , 1996, Diffractive Optics and Micro-Optics.

[82]  J P Hugonin,et al.  Use of grating theories in integrated optics. , 2001, Journal of the Optical Society of America. A, Optics, image science, and vision.

[83]  J. Baumberg,et al.  Implementation of the Natural Mode Analysis for Nanotopologies Using a Volumetric Method of Moments (V-MoM) Algorithm , 2014, IEEE Photonics Journal.

[84]  H. Giessen,et al.  Analytical normalization of resonant states in photonic crystal slabs and periodic arrays of nanoantennas at oblique incidence , 2017 .

[85]  M. Gurioli,et al.  Mapping complex mode volumes with cavity perturbation theory , 2018, Optica.

[86]  Effective mode volumes and Purcell factors for leaky optical cavities , 2011, 1107.4601.

[87]  R. Ge,et al.  Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach. , 2014, Optics letters.

[88]  E. Muljarov,et al.  Exact mode volume and Purcell factor of open optical systems , 2014, 1409.6877.

[89]  Guoquan Zhang,et al.  Fully vectorial modeling of cylindrical microresonators with aperiodic Fourier modal method. , 2014, Journal of the Optical Society of America. A, Optics, image science, and vision.

[90]  Boris Gralak,et al.  Calculation and analysis of the complex band structure of dispersive and dissipative two-dimensional photonic crystals , 2015, 1512.01508.

[91]  Aaswath Raman,et al.  Photonic band structure of dispersive metamaterials formulated as a Hermitian eigenvalue problem. , 2010, Physical review letters.

[92]  Richart E. Slusher,et al.  Optical Processes in Microcavities , 1994 .