Quasinormal mode solvers for resonators with dispersive materials

Efficient method for modeling large-scale arrays of optical nanoresonators based on the coupling theory of quasinormal mode

We propose an efficient method for calculating the electromagnetic field of a large-scale array of optical nanoresonators based on the coupling theory of quasinormal mode (QNM). In this method, two approaches of the scattered-field reconstruction and stationary-phase-principle calculated plane-wave expansion are developed to obtain the regularized QNM (RQNM) in different regions. This accurate and efficient calculation of RQNM resolves the far-field divergence issue of QNMs in the QNM-coupling theory, thus enabling a rapid computation of the electromagnetic field of a large-scale array of optical nanoresonators, which is a challenging task for full-wave numerical methods. Using this method, we consider the numerical example of the radiation problem of a single point source in a large-scale periodic array of optical nanoantennas. In comparison to full-wave numerical methods, this method significantly reduces the computation time by 1∼2 orders of magnitude while maintaining accuracy. The high computational efficiency and physical intuitiveness of the method enables to clarify the impact of array size (exceeding 50 × 50 wavelengths), period and field-coupling range (far beyond the tight-binding approximation) on the optical response. The proposed method and results can provide an efficient tool and guidance for the design of large-scale arrays of optical nanoresonators.

Manipulating the quasi-normal modes of radially symmetric resonators

The frequency response of a resonator is governed by the locations of its quasi-normal modes in the complex frequency plane. The real part of the quasi-normal mode determines the resonance frequency and the imaginary part determines the width of the resonance. For applications such as energy harvesting and sensing, the ability to manipulate the frequency, linewidth and multipolar nature of resonances is key. Here, we derive two methods for simultaneously controlling the resonance frequency, linewidth and multipolar nature of the resonances of radially symmetric structures. Firstly, we formulate an eigenvalue problem for a global shift in the permittivity of the structure to place a resonance at a particular complex frequency. Next, we employ quasi-normal mode perturbation theory to design radially graded structures with resonances at desired frequencies.

Dispersive perfectly matched layers and high-order absorbing boundary conditions for electromagnetic quasinormal modes

Resonances, also known as quasinormal modes (QNMs) in the non-Hermitian case, play a ubiquitous role in all domains of physics ruled by wave phenomena, notably in continuum mechanics, acoustics, electrodynamics, and quantum theory. The non-Hermiticity arises from the system losses, whether they are material (Joule losses in electromagnetism) or linked to the openness of the problem (radiation losses). In this paper, we focus on the latter delicate matter when considering bounded computational domains mandatory when using, e.g., finite elements. We address the important question of whether dispersive perfectly matched layer (PML) and high-order absorbing boundary conditions offer advantages in QNM computation and modal expansion of the optical responses compared with nondispersive PMLs.

Tomographic Reconstruction of Quasistatic Surface Polariton Fields

We theoretically investigate the tomographic reconstruction of the three-dimensional photonic environment of nanoparticles. As input for our reconstruction we use electron energy loss spectroscopy (EELS) maps for different rotation angles. We perform the tomographic reconstruction of surface polariton fields for smooth and rough nanorods and compare the reconstructed and simulated photonic local density of states, which are shown to be in very good agreement. Using these results, we critically examine the potential of our tomography scheme and discuss limitations and directions for future developments.

Quasinormal Mode Expansion Theory for Mesoscale Plasmonic Nanoresonators: An Analytical Treatment of Nonclassical Electromagnetic Boundary Condition

Nonclassical quantum effects will significantly affect the optical response of plasmonic nanoresonators with mesoscale feature sizes between about 2 and 20 nm, and can be fully described by the nonclassical electromagnetic boundary condition (NEBC) expressed with the surface-response Feibelman d parameters. In this Letter, a quasinormal mode (QNM) expansion theory under the NEBC is proposed. By adopting the easily solved classical QNMs under the classical electromagnetic boundary condition as a complete set of basis functions, rigorous expansions of the nonclassical source-free QNMs and source-excited electromagnetic field under the nonperturbative NEBC are provided. With the obtained nonclassical QNMs as basis functions, expansions of the nonclassical source-excited field and Green's function tensor are further obtained. These expansions have a fully analytical dependence on the NEBC and classical QNMs, thus transparently unveiling their impact on the nonclassical QNMs and source-excited electromagnetic field. For instance, a new expression of mode volume is proposed for analyzing the nonclassically corrected Purcell factor. The proposed theory is physically intuitive and computationally efficient which is enabled by the dominance of a small set of classical QNMs, thus providing an efficient tool for understanding and designing mesoscale plasmonic nanoresonators.

Body-of-revolution finite-difference time-domain modeling of hybrid-plasmonic ring resonators

Development of a computational technique for the analysis of quasi-normal modes in hybrid-plasmonic resonators is the main goal of this research. Because of the significant computational costs of this analysis, one has to take various symmetries of these resonators into account. In this research, we consider cylindrical symmetry of hybrid-plasmonic ring resonators and implement a body-of-revolution finite-difference time-domain (BOR-FDTD) technique to analyze these resonators. We extend the BOR-FDTD method by proposing two different sets of auxiliary fields to implement multi-term Drude-Lorentz and multi-term Lorentz models in BOR-FDTD. Moreover, we utilize the filter-diagonalization method to accurately compute the complex resonant frequencies of the resonators. This approach improves numerical accuracy and computational time compared to the Fourier transform method used in previous BOR-FDTD methods. Our numerical analysis is verified by a 2D axisymmetric solver in COMSOL Multiphysics.

Spontaneous emission enhancement by rotationally-symmetric optical nanoantennas: impact of radially and axially propagating surface plasmon polaritons

The excitation and radiation properties of rotationally-symmetric optical nanoantennas are independent of the azimuth angle, which enables great convenience and superior performances in practical applications. However, for rotationally-symmetric nanoantennas, the physical mechanisms behind their resonance properties remain to be clarified. In this paper, firstly, for a simple single-nanocylinder-on-mirror antenna (S-antenna), we establish a first-principles-based semianalytical model of surface plasmon polariton (SPP) by considering an intuitive multiple-scattering process of the radially-propagating gap surface plasmon (RGSP) in the nanogap and the axially-propagating surface plasmon (ASP) on the nanocylinder. The model can comprehensively reproduce all the radiation properties of the S-antenna such as the total and radiative emission rates, SPP excitation rates, and far-field radiation pattern. The model indicates that when the antenna radius is small (respectively, large), the enhancement of spontaneous emission mainly results from the resonance of ASP (respectively, RGSP). To show the wide applicability of the SPP model along with its unveiled decisive role of the RGSP and ASP in the spontaneous emission enhancement for other rotationally-symmetric nanoantennas of cylindrical shapes, we extend the SPP model to a more complex ring-nanocylinder-on-mirror antenna (R-antenna) that supports two ASPs. Moreover, to provide an explicit explanation of the resonance properties of the R-antenna, we further establish a semianalytical model for the resonant modes (called quasinormal modes, QNMs) supported by the R-antenna based on the SPP model, which quantitatively reveals the role of the RGSP and ASP in forming the antenna resonant modes and the resultant enhancement of spontaneous emission.

Normalization, orthogonality, and completeness of quasinormal modes of open systems: the case of electromagnetism [Invited]

The scattering of electromagnetic waves by resonant systems is determined by the excitation of the quasinormal modes (QNMs), i.e. the eigenmodes, of the system. This Review addresses three fundamental concepts in relation to the representation of the scattered field as a superposition of the excited QNMs: normalization, orthogonality, and completeness. Orthogonality and normalization enable a straightforward assessment of the QNM excitation strength for any incident wave. Completeness guarantees that the scattered field can be faithfully expanded into the complete QNM basis. These concepts are not trivial for non-conservative (non-Hermitian) systems and have driven many theoretical developments since initial studies in the 70's. Yet, they are not easy to grasp from the extensive and scattered literature, especially for newcomers in the field. After recalling fundamental results obtained in initial studies on the completeness of the QNM basis for simple resonant systems, we review recent achievements and the debate on the normalization, clarify under which circumstances the QNM basis is complete, and highlight the concept of QNM regularization with complex coordinate transforms.

Nanophotonic Chiral Sensing: How Does It Actually Work?

Nanophotonic chiral sensing has recently attracted a lot of attention. The idea is to exploit the strong light-matter interaction in nanophotonic resonators to determine the concentration of chiral molecules at ultralow thresholds, which is highly attractive for numerous applications in life science and chemistry. However, a thorough understanding of the underlying interactions is still missing. The theoretical description relies on either simple approximations or on purely numerical approaches. We close this gap and present a general theory of chiral light-matter interactions in arbitrary resonators. Our theory describes the chiral interaction as a perturbation of the resonator modes, also known as resonant states or quasi-normal modes. We observe two dominant contributions: A chirality-induced resonance shift and changes in the modes' excitation and emission efficiencies. Our theory brings deep insights for tailoring and enhancing chiral light-matter interactions. Furthermore, it allows us to predict spectra much more efficiently in comparison to conventional approaches. This is particularly true, as chiral interactions are inherently weak and therefore perturbation theory fits extremely well for this problem.

Efficient method for the calculation of the optical force of multiple nanoparticles based on the coupling theory of quasinormal modes

An efficient method is proposed for the calculation of the optical force of multiple nanoparticles. In this method, the optical force is calculated by integrating the Maxwell stress tensor (MST) over a closed surface encompassing the nanoparticle. The electromagnetic (EM) field required for evaluating the MST is computed with the coupling theory of quasinormal modes (QNMs), in which the EM field is expanded onto a small set of QNMs of each nanoparticle. Once these dominant modes, which are eigensolutions of source-free Maxwell equations with complex eigenfrequencies, are known, any variation of the interparticle distance, illumination polarization, or wavelength can be treated analytically. Comparisons with the full-wave numerical method demonstrate the accuracy and efficiency of the formalism. With the formalism, force maps are calculated at remarkable computation speed, providing a promising simulation tool for applications such as plasmon tweezer and photoinduced force microscopy.

Efficient method for the calculation of the optical force of a single nanoparticle based on the quasinormal mode expansion

An efficient method for the calculation of the optical force of a single nanoparticle is proposed based on the expansion of quasinormal modes (QNMs), which are eigensolutions of source-free Maxwell's equations with complex eigenfrequencies. In this method, the optical force is calculated by integrating the Maxwell stress tensor (MST) over a closed surface encompassing the nanoparticle. The electromagnetic (EM) field required for evaluating the MST is computed by a rigorous modal analysis, in which the EM field is expanded onto a small set of QNMs. Once the QNMs of the nanoparticle are solved, their excitation coefficients are obtained analytically. This means that additional full-wave computations are not required if the nanoparticle's location and the wavelength or distribution of the excitation field vary. Comparisons with full-wave numerical calculations of optical force evidence the high efficiency and accuracy of our formalism.

Combining density functional theory with macroscopic QED for quantum light-matter interactions in 2D materials

A quantitative and predictive theory of quantum light-matter interactions in ultra thin materials involves several fundamental challenges. Any realistic model must simultaneously account for the ultra-confined plasmonic modes and their quantization in the presence of losses, while describing the electronic states from first principles. Herein we develop such a framework by combining density functional theory (DFT) with macroscopic quantum electrodynamics, which we use to show Purcell enhancements reaching 10⁷ for intersubband transitions in few-layer transition metal dichalcogenides sandwiched between graphene and a perfect conductor. The general validity of our methodology allows us to put several common approximation paradigms to quantitative test, namely the dipole-approximation, the use of 1D quantum well model wave functions, and the Fermi's Golden rule. The analysis shows that the choice of wave functions is of particular importance. Our work lays the foundation for practical ab initio-based quantum treatments of light-matter interactions in realistic nanostructured materials.

Quasinormal modes expansions for nanoresonators made of absorbing dielectric materials: study of the role of static modes

The interaction of light with photonic resonators is determined by the eigenmodes of the system. Modal theories based on quasinormal modes provide a natural tool to calculate and understand light scattering by nanoresonators. We show that, in the case of resonators made of absorbing dielectric materials, eigenmodes with zero eigenfrequency (static modes) play a key role in the modal formalism. The excitation of static modes builds a non-resonant contribution to the modal expansion of the scattered field. This non-resonant term plays a crucial physical role since it largely contributes to the off-resonance signal to which resonances are added in amplitude, possibly leading to interference phenomena and Fano resonances. By considering light scattering by a silicon nanosphere, we quantify the impact of static modes. This study shows that the importance of static modes is not just formal. Static modes are of prime importance in an expansion truncated to only a few modes.

On the pole expansion of electromagnetic fields

In several publications, it has been shown how to calculate the near- or far-field properties for a given source or incident field using the resonant states, also known as quasi-normal modes. As previously noted, this pole expansion is not unique, and there exist many equivalent formulations with dispersive expansion coefficients. Here, we approach the pole expansion of the electromagnetic fields using the Mittag-Leffler theorem and obtain another set of formulations with constant weight factors for each pole. We compare the performance and applicability of these formulations using analytical and numerical examples. It turns out that the accuracy of all approaches is rather comparable with a slightly better global convergence of the approach based on a formulation with dispersive expansion coefficients. However, other expansions can be superior locally and are typically faster. Our work will help with selecting appropriate formulations for an efficient description of the electromagnetic response in terms of the resonant states.

Continuous family of exact Dispersive Quasi-Normal Modal (DQNM) expansions for dispersive photonic structures

In photonics, Dispersive Quasi-Normal Modes (DQNMs) refer to optical resonant modes, solutions of spectral problems associated with Maxwell's equations for open photonic structures involving dispersive media. Since these DQNMs are the constituents determining optical responses, studying DQNM expansion formalisms is the key to model the physical properties of a considered system. In this paper, we emphasize the non-uniqueness of the expansions related to the over-completeness of the set of modes and discuss a family of DQNM expansions depending on continuous parameters that can be freely chosen. These expansions can be applied to dispersive, anisotropic, and even non-reciprocal materials. As an example, we particularly demonstrate the modal analysis on a 2-D scattering model where the permittivity of a silicon object is drawn directly from actual measurement data.

On the applicability of quantum-optical concepts in strong-coupling nanophotonics

Rooted in quantum optics and benefiting from its well-established foundations, strong coupling in nanophotonics has experienced increasing popularity in recent years. With nanophotonics being an experiment-driven field, the absence of appropriate theoretical methods to describe ground-breaking advances has often emerged as an important issue. To address this problem, the temptation to directly transfer and extend concepts already available from quantum optics is strong, even if a rigorous justification is not always available. In this review we discuss situations where, in our view, this strategy has indeed overstepped its bounds. We focus on exciton-plasmon interactions, and particularly on the idea of calculating the number of excitons involved in the coupling. We analyse how, starting from an unfounded interpretation of the term N/V that appears in theoretical descriptions at different levels of complexity, one might be tempted to make independent assumptions for what the number N and the volume V are, and attempt to calculate them separately. Such an approach can lead to different, often contradictory results, depending on the initial assumptions (e.g. through different treatments of V as the-ambiguous in plasmonics-mode volume). We argue that the source of such contradictions is the question itself-How many excitons are coupled?, which disregards the true nature of the coupled components of the system, has no meaning and often not even any practical importance. If one is interested in validating the quantum nature of the system-which appears to be the motivation driving the pursuit of strong coupling with small N-one could instead focus on quantities such as the photon emission rate or the second-order correlation function. While many of the issues discussed here may appear straightforward to specialists, our target audience is predominantly newcomers to the field, either students or scientists specialised in different disciplines. We have thus tried to minimise the occurrence of proofs and overly-technical details, and instead provide a qualitative discussion of analyses that should be avoided, hoping to facilitate further growth of this promising area.

Modal analysis of photonic and plasmonic resonators

Determining the electromagnetic field response of photonic and plasmonic resonators is a formidable task in general. Field expansions in terms of quasi-normal modes (QNMs) are often used, since only a few of these modes are typically required for an accurate field description. We show that by exploiting the structure of Maxwell's equations, conjugate-symmetric frequency-domain field expansions can be efficiently computed via a Lanczos-type algorithm. Dominant QNMs can be identified a posteriori with error control and without a priori mode selection. Discrete QNM approximations of resonating nanostructures are presented and the spontaneous decay rate of a quantum emitter is also considered.

Nonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materials

Any optical structure possesses resonance modes, and its response to an excitation can be decomposed onto the quasinormal and numerical modes of a discretized Maxwell operator. In this paper, we consider a dielectric permittivity that is an N-pole Lorentz function of the frequency. Even for discretized operators, the literature proposes different formulas for the coefficients of the quasinormal-mode expansion, and this comes as a surprise. We propose a general formalism, based on auxiliary fields, which explains why and evidences that there is, in fact, an infinity of mathematically sound possible expansion coefficients. The nonuniqueness is due to a choice of the linearization of Maxwell's equations with respect to frequency and of the choice of the form of the source term. Numerical results validate the different formulas and compare their accuracy.

Discontinuities in photonic waveguides: rigorous Maxwell-based 3D modeling with the finite element method

In this paper, a general methodology to study rigorous discontinuities in open waveguides is presented. It relies on a full vector description given by Maxwell's equations in the framework of the finite element method. The discontinuities are not necessarily small perturbations of the initial waveguide and can be very general, such as plasmonic inclusions of arbitrary shapes. The leaky modes of the invariant structure are first computed and then injected as incident fields in the full structure with obstacles using a scattered field approach. The resulting scattered field is finally projected on the modes of the invariant structure making use of their bi-orthogonality. The energy balance is discussed. Finally, the modes of open waveguides periodically structured along the propagation direction are computed. The relevant complex propagation constants are compared to the transmission obtained for a finite number of identical cells. The relevance and complementarity of the two approaches are highlighted on a numerical example encountered in infrared sensing. Open source models allowing us to retrieve most of the results of this paper are provided.

Quasinormal-mode analysis of grating spectra at fixed incidence angles

Grating spectra exhibit sharp variations of the scattered light, known as grating anomalies. The latter are due to resonances that have fascinated specialists of optics and physics for decades and are today used in many applications. We present a comprehensive theory of grating anomalies and develop a formalism to expand the field scattered by metallic or dielectric gratings into the basis of its natural resonances, thereby enabling the possibility to reconstruct grating spectra measured for fixed illumination angles as a sum over every individual resonance contribution with closed-form expressions. This gives physical insights into the spectral properties and direct access to the resonances to engineer the spectral response of gratings and their sensitivity to tiny perturbations.