In this Review, the theory and applications of optical micro-and nano-resonators are presented from the underlying concept of their natural resonances, the so-called quasi-normal modes (QNMs). QNMs are the basic constituents governing the response of resonators. Characterized by complex frequencies, QNMs are initially loaded by a driving field and then decay exponentially in time due to power leakage or absorption. Here, the use of QNM-expansion formalisms to model these basic effects is explored. Such modal expansions that operate at complex frequencies distinguish from the current user habits in electromagnetic modeling, which rely on classical Maxwell's equation solvers operating at real frequencies or in the time domain; they also bring much deeper physical insight into the analysis. An extensive overview of the historical background on QNMs in electromagnetism and a detailed discussion of recent relevant theoretical and numerical advances are therefore presented. Additionally, a concise description of the role of QNMs on a number of examples involving electromagnetic resonant fields and matter, including the interaction between quantum emitters and resonators (Purcell effect, weak and strong coupling, superradiance, . . . ), Fano interferences, the perturbation of resonance modes, and light transport and localization in disordered media is provided. (3 of 38)www.advancedsciencenews.com www.lpr-journal.org Figure 2. Examples of applications whose analysis benefit from QNM-expansion approaches. a) Nonlocal plasmonics. QNMs can be used to predict the nonlocal response of free electron gas on the Purcell factor of an emitter placed in the near-field of a gold nanorod. Predictions from two different nonlocal models-the hydrodynamic Drude model (HDM) and the generalized nonlocal optical response (GNOR) Model-are shown and compared to classical predictions obtained with a Drude model. The inset shows the electric field intensity of the nonlocal GNOR QNM. Adapted with permission. [53] Copyright 2017, Optical Society of America. b) QNM expansion of the scattering matrix. (Upper panel) Absorption cross section of a multi-layered metallic-dielectric sphere in air, demonstrating the good agreement between the results obtained with Mie's scattering theory (dashed black curve) and a QNM-expansion formalism for the scattering matrix (solid red curve) computed with the QNM eigenfrequencies shown in the Lower panel. Reproduced with permission. [40] Copyright 2017, American Physical Society. c) Quantum hybrids. Supperradiant and subradiant decay rates m and energies m of a quantum hybrid formed by 100 molecules that are randomly distributed and oriented around a silver nanorod (diameter 30 nm, length 100 nm), predicted with the QNM formalism. γ 0 denotes the decay rate of every individual molecule in the vacuum. The left inset shows the superradiant state of the hybrid, with a large cooperativity involving more than one half of the molecules. Reproduced with permission. [28] Copyright 2017, American Physical Society. d) Quant...

The specificity of modal-expansion formalisms is their capabilities to model the physical properties in the natural resonance-state basis of the system in question, leading to a transparent interpretation of the numerical results. In electromagnetism, modal-expansion formalisms are routinely used for optical waveguides. In contrast, they are much less mature for analyzing open non-Hermitian systems, such as micro and nanoresonators. Here, by accounting for material dispersion with auxiliary fields, we considerably extend the capabilities of these formalisms, in terms of computational effectiveness, number of states handled and range of validity. We implement an efficient finite element solver to compute the resonance states, and derive new closed-form expressions of the modal excitation coefficients for reconstructing the scattered fields. Together, these two achievements allow us to perform rigorous modal analysis of complicated plasmonic resonators, being not limited to a few resonance states, with straightforward physical interpretations and remarkable computation speeds. We particularly show that, when the number of states retained in the expansion increases, convergence towards accurate predictions is achieved, offering a solid theoretical foundation for analyzing important issues, e.g. Fano interference, quenching, coupling with the continuum, which are critical in nanophotonic research.

The classical treatment of plasmonics is insufficient at the nanometer-scale due to quantum mechanical surface phenomena. Here, an extension of the classical paradigm is reported which rigorously remedies this deficiency through the incorporation of first-principles surface response functions-the Feibelman d parameters-in general geometries. Several analytical results for the leading-order plasmonic quantum corrections are obtained in a first-principles setting; particularly, a clear separation of the roles of shape, scale, and material is established. The utility of the formalism is illustrated by the derivation of a modified sum rule for complementary structures, a rigorous reformulation of Kreibig's phenomenological damping prescription, and an account of the small-scale resonance shifting of simple and noble metal nanostructures. DOI: 10.1103/PhysRevLett.118.157402 Classical treatments of plasmonics require specification of just two elements: geometry, involving shape and scale, and dielectric environment, supplied through local bulk dielectric functions. In the deep subwavelength regime, i.e., in the nonretarded limit, even the element of scale is rendered superfluous by scale-invariant governing equations. As the geometric scale is reduced further, below 10-20 nm in metals, toward the intrinsic quantum mechanical length scales of the plasmon-supporting electron gas, the classical approach inevitably deteriorates, as established by numerous experiments [1][2][3][4][5][6][7][8][9][10]. The main shortcomings of the classical approach can be divided into three categories [11], resulting from the neglect of (i) spill-out of the conduction electron's wave function beyond the material boundaries [12], (ii) nonlocality, i.e., the momentum dependence of the bulk response functions [13], and (iii) incomplete accounting of internal electron dynamics, especially surface-enabled plasmon damping by electron-hole pair creation [10,14]. In the subnanometer domain, additional shortcomings are expected to materialize, e.g., due to size quantization [15,16] and the breakdown of jellium treatments [17,18]. Jointly, these shortcomings and their impact on plasmonic observables (modal spectrum, field enhancements, local density of states, etc.) motivate and define the field of quantum nanoplasmonics [12,13,17].While time-dependent density-functional theory (TDDFT) [19], in principle, can bridge the gap between classical and quantum nanoplasmonics, its explicit application is, in practice, limited to few-atom clusters and systems of high spatial symmetry due to computational constraints. A sizable fraction of nanoplasmonic structures of interest [20][21][22], thus, fall in a region which is simultaneously inaccessible to conventional, explicit TDDFT and beyond the validity of classical plasmonics, roughly spanning characteristic geometric scales L ∼ 2-20 nm. In this Letter, we provide a simple and general answer to the central question raised by this dichotomy: namely, what are the leading-order nonclassical corrections to cla...

Quantum effects of plasmonic phenomena have been explored through ab initio studies, but only for exceedingly small metallic nanostructures, leaving most experimentally relevant structures too large to handle. We propose instead an effective description with the computationally appealing features of classical electrodynamics, while quantum properties are described accurately through an infinitely thin layer of dipoles oriented normally to the metal surface. The nonlocal polarizability of the dipole layer-the only introduced parameter-is mapped from the free-electron distribution near the metal surface as obtained with 1D quantum calculations, such as time-dependent density-functional theory (TDDFT), and is determined once and for all. The model can be applied in two and three dimensions to any system size that is tractable within classical electrodynamics, while capturing quantum plasmonic aspects of nonlocal response and a finite work function with TDDFT-level accuracy. Applying the theory to dimers, we find quantum corrections to the hybridization even in mesoscopic dimers, as long as the gap itself is subnanometric.

Local, bulk response functions, e.g. permittivity, and the macroscopic Maxwell equations completely specify the classical electromagnetic problem, which features only wavelength λ and geometric scales. The above neglect of intrinsic electronic length scales L e leads to an eventual breakdown in the nanoscopic limit. Here, we present a general theoretical and experimental framework for treating nanoscale electromagnetic phenomena. The framework features surface-response functions-known as the Feibelman d-parameters-which reintroduce the missing electronic length scales. As a part of our framework, we establish an experimental procedure to measure these complex, dispersive surface response functions, enabled by quasi-normal-mode perturbation theory and observations of pronounced nonclassical effects-spectral shifts in excess of 30% and the breakdown of Kreibig-like broadening-in a quintessential multiscale architecture: film-coupled nanoresonators, with feature-sizes comparable to both L e and λ.

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 rises 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 twodimensional plasmonic crystal, a two-dimensional metal grating, and a three-dimensional nanopatch antenna on a metal substrate, in the perspective to elaborate standards for the computation of resonance modes.

Microcavities and nanoresonators are characterized by their quality factors Q and mode volumes V. While Q is unambiguously defined, there are still questions on V and in particular on its complexvalued character, whose imaginary part is linked to the non-Hermitian nature of open systems. Helped by cavity perturbation theory and near field experimental data, we clarify the physics captured by the imaginary part of V and show how a mapping of the spatial distribution of both the real and imaginary parts can be directly inferred from perturbation measurements. This result shows that the mathematically abstract complex mode volume in fact is directly observable.This supplementary Material provides complementary results and discussions to the main text, starting with a Section detailing discussing the reliability of the Δ -measurements, followed by a formal comparison of the classical perturbation formula of Eqs. (1) and (2), a study of the accuracy of Eq. (2) for predicting resonance shifts, and by an analytical study of the domain of validity of Eq. (2) that leads to upper bounds for the maximum perturber strength.

We propose a new interpretation for light confinement in picocavities formed by ultrasmall metallic protuberances inside the gap of metal−insulator−metal nanoresonators. We demonstrate that the protuberances support dark resonances with mode volumes comparable to their geometric volumes and that their brightness can be enhanced by several orders of magnitude when they are strongly coupled with the modes of nanoresonators with nanometric dielectric spacers. With a simple and accurate closed-form expression, we clarify the role of gap plasmons in this coupling. Based on this understanding, we propose a general strategy, exploiting strong coupling to design extremely localized modes with cubic nanometer volumes and so-far unreached brightness.