Quasi-normal Mode Theory Research Articles

Quasinormal modes and coupled mode theory of 1D metal‐dielectric photonic crystals

AbstractMetal‐dielectric photonic crystals (MDPCs) are a class of photonic structure formed through the coupling of metallic microcavities. Interaction of the cavities results in hybridization of the underlying microcavity states to form photonic bands defined by the crystal periodicity, metal‐to‐dielectric ratio, and number of unit cells. In this study, we provide analytical solutions for the resonant states of discrete 1D MDPCs with an arbitrary number of cavities using quasinormal mode (QNM) theory. Our results show that the QNM solutions closely match the predictions of coupled mode theory (CMT) in the tight‐binding regime and diverge as the metal thickness vanishes. We apply the QNM results to the CMT model to find analytical expressions for the inter‐cavity coupling coefficients. Additionally, we analyze the quality factor, transmission, and absorption of both ideal and lossy MDPCs, revealing important trends in bulk optical properties.

Origination of the chiroptical effect in plasmonic nano-structures in the view of quasi-normal mode theory.

General chiroptical effects describe all of the interaction differences between light carrying opposite spins and chiral matters, such as circular dichroism, optical activity, and chiral Raman optical activity, and have been proven to hold great promise for extensive applications in physics, chemistry, and biology. However, the underlying physical mechanism is usually explained intangibly by the twisted currents in chiral geometry, where the cross coupling between the electric and magnetic dipoles breaks the degeneracy of the helicity eigenmodes. In this Letter, we construct a clear sight on the origination of the chiroptical effect in the view of the eigenstates of a non-Hermitian system, i.e., quasi-normal modes (QNMs). The intrinsic chiroptical effect comes from the chiral QNMs, which have distinct excitation and emission differences in both phase and intensity for lights carrying opposite spins, while the extrinsic chiroptical effect coming from the achiral QNMs requires specific illumination and observation conditions, where the low symmetrical QNM can generate chiroptical effects in both absorption and scattering, but the highly symmetrical QNMs can only generate chiroptical effects in scattering through the coherent superposition of several QNMs. Our findings offer an in-depth understanding of the chiroptical effect and have the potential to bring broad inspiration to the design and applications of chiroptical effects.

Certifying Multimode Light-Matter Interaction in Lossy Resonators.

Quantum models based on few-mode master equations have been a central tool in the study of resonator quantum electrodynamics, extending the seminal single-mode Jaynes-Cummings model to include loss and multiple modes. Despite their broad application range, previous approaches within this framework have either relied on a Markov approximation or a fitting procedure. By combining ideas from pseudomode and quasinormal mode theory, we develop a certification criterion for multi-mode effects in lossy resonators. It is based on a witness observable, and neither requires a fitting procedure nor a Markov approximation. Using the resulting criterion, we demonstrate that such multi-mode effects are important for understanding previous experiments in x-ray cavity QED with Mössbauer nuclei and that they allow one to tune the nuclear ensemble properties.

Hybrid epsilon-near-zero modes of photonic gap antennas

We demonstrate that in photonic gap antennas composed of an epsilon-near-zero (ENZ) layer embedded within a high-index dielectric, hybrid modes emerge from the strong coupling between the ENZ thin film and the photonic modes of the dielectric antenna. These hybrid modes show giant electric field enhancements, large enhancements of the far-field spontaneous emission rate, and a unidirectional radiation response. We analyze both parent and hybrid modes using quasinormal mode theory and find that the hybridization can be well understood using a coupled oscillator model. Under plane-wave illumination, hybrid ENZ antennas can concentrate light and achieve local electric field amplitudes more than two orders of magnitude greater than that of the incident wave. This places them on par with the best plasmonic antennas. In addition, the far-field spontaneous emission rate of a dipole embedded at the antenna hotspot can be increased by more than three orders of magnitude as compared to that in free space, with nearly perfect unidirectional emission.

Quasinormal mode theory for nanoscale electromagnetism informed by quantum surface response

We report a self-consistent quasinormal mode theory for nanometer scale electromagnetism where the possible nonlocal and quantum effects are treated through quantum surface responses. With Feibelman's frequency-dependent \textit{d} parameters to describe the quantum surface responses, we formulate the source-free Maxwell's equations into a generalized linear eigenvalue problem to define the quasinormal modes. We then construct an orthonormal relation for the modes and consequently unlock the powerful toolbox of modal analysis. The orthonormal relation is validated by the reconstruction of the full numerical results through modal contributions. Significant changes in the landscape of the modes are observed due to the incorporation of the quantum surface responses for a number of nanostructures. Our semi-analytical modal analysis enables transparent physical interpretation of the spontaneous emission enhancement of a dipolar emitter as well as the near-field and far-field responses of planewave excitations in the nanostructures.

Quasinormal Mode Theory of Chiral Power Flow from Linearly Polarized Dipole Emitters Coupled to Index-Modulated Microring Resonators Close to an Exceptional Point

Chiral emission can be achieved from a circularly polarized dipole emitter in a nanophotonic structure that possess special polarization properties such as a polarization singularity, namely with right or left circularly polarization (C-points). Recently, Chen et al. [Nature Physics 16, 571 (2020)] demonstrated the surprising result of chiral radiation from a linearly-polarized (LP) dipole emitter, and argued that this effect is caused by a decoupling with the underlying eigenmodes of a non-Hermitian system, working at an exceptional point (EP). Here we present a quasinormal mode (QNM) approach to model a similar index-modulated ring resonator working near an EP and show the same unusual chiral power flow properties from LP emitters, in direct agreement with the experimental results. We explain these results quantitatively without invoking the interpretation of a missing dimension (the Jordan vector) and a decoupling from the cavity eigenmodes, since the correct eigenmodes are the QNMs which explain the chiral emission using only two cavity modes. By coupling a LP emitter with the dominant two QNMs of the ring resonator, we show how the chiral emission depend on the position and orientation of the emitter, which is also verified by the excellent agreement with respect to the power flow between the QNM theory and full numerical dipole solutions. We also show how a normal mode solution will fail to capture the correct chirality since it does not take into account the essential QNM phase. Moreover, we demonstrate how one can achieve frequency-dependent chiral emission, and replace lossy materials with gain materials in the index modulation to reverse the chirality.

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.

Quasi-normal mode theory of the scattering matrix, enforcing fundamental constraints for truncated expansions

We develop a quasi-normal mode theory (QNMT) to calculate a system's scattering $S$ matrix, simultaneously satisfying both energy conservation and reciprocity even for a small truncated set of resonances. It is a practical reduced-order (few-parameter) model based on the resonant frequencies and constant mode-to-port coupling coefficients, easily computed from an eigensolver without the need for QNM normalization. Furthermore, we show how low-$Q$ modes can be separated into an effective slowly varying background response $C$, giving an additional approximate formula for $S$, which is useful to describe general Fano-resonant phenomena. We demonstrate our formulation for both normal and fixed-angle oblique plane-wave incidence on various electromagnetic metasurfaces.

Bright Optical Eigenmode of 1 nm^{3} Mode Volume.

We report on the discovery and rationale to devise bright single optical eigenmodes that feature quantum-optical mode volumes of about 1 nm^{3}. Our findings rely on the development and application of a quasinormal mode theory that self-consistently treats fields and electron nonlocality, spill-out, and Landau damping around atomistic protrusions on a metallic nanoantenna. By outpacing Landau damping with radiation via properly designed antenna modes, the extremely localized modes become bright with radiation efficiencies reaching 30% and could provide up to 4×10^{7} times intensity enhancement.

Hot Electron Generation through Near-Field Excitation of Plasmonic Nanoresonators

We theoretically study hot electron generation through the emission of a dipole source coupled to a nanoresonator on a metal surface. In our hybrid approach, we solve the time-harmonic Maxwell's equations numerically and apply a quantum model to predict the efficiency of hot electron generation. Strongly confined electromagnetic fields and the strong enhancement of hot electron generation at the metal surface are predicted and are further interpreted with the theory of quasinormal modes. In the investigated nanoresonator setup, both the emitting source and the acceptor resonator are localized in the same volume, and this configuration looks promising to achieve high efficiencies of hot electron generation. By comparing with the efficiency calculated in the absence of the plasmonic nanoresonator, that is, the dipole source is located near a flat, unstructured metal surface, we show that the effective excitation of the modes of the nanoresonator boosts the generation efficiency of energetic charge carriers. The proposed scheme can be used in tip-based spectroscopies and other optoelectronic applications.

Temporal switching to extend the bandwidth of thin absorbers

Wave absorption in time-invariant, passive thin films is fundamentally limited by a trade-off between bandwidth and overall thickness. In this work, we investigate the use of temporal switching to reduce signal reflections from a thin grounded slab over broader bandwidths. We extend quasi-normal mode theory to time switching, developing an ab initio formalism that can model a broad class of time-switched structures. Our formalism provides optimal switching strategies to maximize the bandwidth over which minimal reflection is achieved, showing promising prospects for time-switched nanophotonic and metamaterial systems to overcome the limits of time-invariant, passive structures.

Quasinormal-mode theory of elastic Purcell factors and Fano resonances of optomechanical beams

We introduce a quasinormal mode theory of mechanical open-cavity modes for optomechanical resonators, and demonstrate the importance of using a generalized (complex) effective mode volume and the phase of the quasinormal mode. We first generalize and fix the normal mode theories of the elastic Purcell factor, and then show a striking example of coupled quasinormal modes yielding a pronounced Fano resonance. Our theory is exemplified and confirmed by full three-dimensional calculations on optomechanical beams, but the general findings apply to a wide range of mechanical cavity modes. This quasinormal mechanical mode formalism, when also coupled with a quasinormal theory of optical cavities, offers a unified framework for describing a wide range of optomechanical structures where dissipation is an inherent part of the resonator modes.

Degenerate quasi-normal mode theory for near-field radiation between plasmonic structures.

Near-field radiation can exceed the blackbody radiation limit due to the contributions from evanescent waves. One promising approach to further enhance near-field radiation beyond existing bulk materials is to utilize metamaterials or metasurfaces made from subwavelength plasmonic structures. In this work, we investigate the near-field thermal radiation between complex plasmonic structures with higher-order symmetry and degeneracy, which is crucial for understanding the radiative heat exchange between metamaterials or metasurfaces at extremely small gaps. We demonstrate that the introduction of degeneracy can drastically boost near-field thermal radiation between plasmonic structures. The enhancement of near-field thermal radiation originates from the emergence of degenerate resonance modes and the secondary emission of thermal photons due to the nonzero coupling between the degenerate modes. Our study provides new pathways for designing high-intensity near-field thermal emitters and absorbers for thermophotovoltaics, thermal management, and infrared spectroscopy.

Coupling theory of quasinormal modes for lossy and dispersive plasmonic nanoresonators

A quasinormal mode (QNM) coupling theory is proposed for a system of coupled resonators. The theory is generally applicable to systems with energy loss and arbitrary frequency dispersion of permittivity such as plasmonic nanoresonators. The QNM coupling coefficients in the theory are obtained with first-principle calculations without fitting the full-wave numerical results or experimental data. The theory is built up based on the Green's-function formalism, which shows that the coupling between resonators originates from the mutual excitation of scattered field of each resonator. With the theory, the source-excited scattered field or the source-free QNMs of the coupled system can be obtained as an expansion in terms of QNMs of each individual resonator in the system. The theory provides an intuitive physical picture and also shows superior computational efficiency compared with the full-wave numerical method if the scattered field of each resonator in the system can be expanded with a small set of QNMs. The theory is tested against full-wave numerical method and is used to design metallic nanoantennas with a broadband enhancement of the spontaneous emission rate.

Quasinormal-Mode Non-Hermitian Modeling and Design in Nonlinear Nano-Optics

Based on quasinormal-mode theory, we propose a novel approach enabling a deep analytical insight into the multi-parameter design and optimization of nonlinear photonic structures at subwavelength scale. A key distinction of our method from previous formulations relying on multipolar Mie-scattering expansions is that it directly exploits the natural resonant modes of the nanostructures, which provide the field enhancement to achieve significant nonlinear efficiency. Thanks to closed-form expression for the nonlinear overlap integral between the interacting modes, we illustrate the potential of our method with a two-order-of-magnitude boost of second harmonic generation in a semiconductor nanostructure, by engineering both the sign of $\chi^{(2)}$ at subwavelength scale and the structure of the pump beam.

Scale Law of Far-Field Thermal Radiation from Plasmonic Metasurfaces.

Although metamaterials or metasurfaces consisting of patterned subwavelength structures have been widely employed for thermal emission control, the collective behavior of the emitter array in a metasurface still remains unclear. Here, based on the quasinormal mode theory, we derive a new scale law to elucidate the far-field thermal emission from a metasurface composed of densely packed plasmonic nanoemitters. The tight binding method is used to approximate the collective resonant mode of the emitter array. Because of in-phase near-field interaction, the thermal radiation from a single emitter in a metasurface is suppressed by its adjacent emitters. We find that the overall far-field thermal radiation from a metasurface can be either positively or negatively correlated with the packing density of the emitters, depending on the mode properties of the single emitter. This new scale law thus serves as a general guideline for designing metasurfaces with desired thermal emission properties.

Quasi-normal mode expansion as a tool for the design of nanophotonic devices

Many nanophotonic devices rely on the excitation of photonic resonances to enhance light-matter interaction. The understanding of the resonances is therefore of a key importance to facilitate the design of such devices. These resonances may be analyzed by use of the quasi-normal mode (QNM) theory. Here, we illustrate how QNM analysis may help study and design resonant nanophotonic devices. We will in particular use the QNM expansion of far-field quantities based on Riesz projection to design optical antennas.

Effective field theory of black hole quasinormal modes in scalar-tensor theories

The final ringdown phase in a coalescence process is a valuable laboratory to test General Relativity and potentially constrain additional degrees of freedom in the gravitational sector. We introduce here an effective description for perturbations around spherically symmetric spacetimes in the context of scalar-tensor theories, which we apply to study quasi-normal modes for black holes with scalar hair. We derive the equations of motion governing the dynamics of both the polar and the axial modes in terms of the coefficients of the effective theory. Assuming the deviation of the background from Schwarzschild is small, we use the WKB method to introduce the notion of “light ring expansion”. This approximation is analogous to the slow-roll expansion used for inflation, and it allows us to express the quasinormal mode spectrum in terms of a small number of parameters. This work is a first step in describing, in a model independent way, how the scalar hair can affect the ringdown stage and leave signatures on the emitted gravitational wave signal. Potential signatures include the shifting of the quasi-normal spectrum, the breaking of isospectrality between polar and axial modes, and the existence of scalar radiation.

Nonlocal quasinormal modes for arbitrarily shaped three-dimensional plasmonic resonators

Nonlocal effects have been shown to be responsible for a variety of non-trivial optical effects in small-size plasmonic nanoparticles, beyond classical electrodynamics. However, it is not clear whether optical mode descriptions can be applied to such extreme confinement regimes. Here, we present a powerful and intuitive quasinormal mode description of the nonlocal optical response for three-dimensional plasmonic nanoresonators. The nonlocal hydrodynamical model and a generalized nonlocal optical response model for plasmonic nanoresonators are used to construct an intuitive modal theory and to compare to the local Drude model response theory. Using the example of a gold nanorod, we show how an efficient quasinormal mode picture is able to accurately capture the blueshift of the resonances, the higher damping rates in plasmonic nanoresonators, and the modified spatial profile of the plasmon quasinormal modes, even at the single mode level. We exemplify the use of this theory by calculating the Purcell factors of single quantum emitters, the electron energy-loss spectroscopy spatial maps, as well as the Mollow triplet spectra of field-driven quantum dots with and without nonlocal effects for different size nanoresonators. Our nonlocal quasinormal mode theory offers a reliable and efficient technique to study both classical and quantum optical problems in nanoplasmonics.

Perfect Thermal Emission by Nanoscale Transmission Line Resonators.

Thermal radiation with a narrow-band emission spectrum is of great importance in a variety of applications such as infrared sensing, thermophotovoltaics, radiation cooling, and thermal circuits. Although resonant nanophotonic structures such as metamaterials and nanocavities have been demonstrated to achieve the narrow-band thermal emission, maximizing their radiation power toward perfect emission still remains challenging. Here, based on the recently developed quasi-normal mode theory, we prove that thermal emission from a nanoscale transmission line resonator can always be maximized by tuning the waveguiding loss of the resonator or bending the structure. By use of nanoscale transmission line resonators as basic building blocks, we experimentally demonstrate a new type of macroscopic perfect and tunable thermal emitters. Our experimental demonstration in conjunction with the general theoretical framework from the quasi-normal mode theory lays the foundation for designing tunable narrow-band thermal emitters with applications in thermal infrared light sources, thermal management, and infrared sensing and imaging.