The resonator is one of the main building blocks of a plethora of photonic and microwave devices from nanolasers to compact biosensors and magnetic resonance scanners. The symmetry of the resonators is tightly related to their mode structure and multipole content which determines the linear and non-linear response of the resonator. Here, we develop the algorithm for the classification of eigenmodes in resonators of the simplest shapes depending on their symmetry group. For each type of mode, we find its multipole content. As an illustrative example, we apply the developed formalism to the analysis of dielectric triangular prism and demonstrate the formation of high-Q resonances originated due to suppression of the scattering through the main multipole channel. The developed approach one to engineer, predict, and explain scattering phenomena and optical properties of resonators and meta-atoms basing only on their symmetry without the need for numerical simulations and it can be used for the design of new photonic and microwave devices.
We propose the generalization of the Fourier modal method aimed at calculating localized eigenmodes of integrated optical resonators. The method is based on constructing the analytic continuation of the structure's scattering matrix and calculating its poles. The method allows one to calculate the complex frequency of the localized mode and the corresponding field distribution. We use the proposed method to calculate the eigenmodes of rectangular dielectric block located on metal surface. We show that excitation of these modes by surface plasmon-polariton (SPP) results in resonant features in …
We experimentally show an all-optical multipolar decomposition of the lowest-order Eigenmodes of a single gold nanoprism using azimuthally and radially polarized cylindrical vector beams. By scanning the particle through these tailored field distributions, the multipolar character of the Eigenmodes gets encoded into 2D-scanning intensity maps even for higher-order contributions to the Eigenmode that are too weak to be discerned in the direct far-field scattering response. This method enables a detailed optical mode analysis of individual nanoparticles.
The polarization exchange effect in a twisted-nematic Fabry-Pérot resonator is experimentally confirmed in the regimes of both uniform and electric-field-deformed twisted structures. The polarization of output light in the transmission peaks is shown to be linear rather than elliptical. The polarization deflection from the nematic director grows from $0^\circ$ to $90^\circ$ angle and exchanges the longitudinal and transverse directions. Untwisting of a nematic by a voltage leads to the rotation of the polarization plane of light passing through the resonator. The polarization exchange effect a…
Terahertz time domain spectroscopy and rigorous simulations are used to probe the coupling between a dark and a bright plasmonic eigenmode in a metamaterial with broken symmetry. The metamaterial consists of two closely spaced split ring resonators that have their gaps in non-identical positions within the ring. For normal incidence and a fixed polarization both lowest order eigenmodes of the split ring resonators can be excited; although one of them has to be regarded as dark since coupling is prohibited because of symmetry constraints. Emphasis in this work is put on a systematic evaluation …
Codes were written to simulate the propagation of monochromatic light through a bare optical resonator, using a computational Fourier method to solve the Huygens-Fresnel integral. This was used, in the Fox-Li method, to find the lowest-loss eigenmodes of arbitrary cavity designs. An implicit shift `hopping' method was employed to allow a series of increasingly higher-loss eigenmodes to be found, limited in number by computational time. Codes were confirmed in their accuracy against the literature, and were used to investigate a number of different cavity configurations. In addition to confirmi…
We use a spatially resolved cavity ring-down technique to show that the 2D eigenmode of an unstable optical cavity has a fractal pattern, i.e. it looks the same at different length scales. In agreement with theory, we find that this pattern has the maximum conceivable roughness, i.e., its fractal dimension is 3.01 plus\minus 0.04. This insight in the nature of unstable cavity eigenmodes may lead to better understanding of wave dynamics in open systems, for both light and matter waves.
Coupled-resonator optical waveguides (CROWs), which play a significant role in modern photonics, achieve waveguiding through near-field coupling between tightly localized resonators. The coupling factor, a critical parameter in CROW theory, determines the coupling strength between two resonators and the waveguiding dispersion of a CROW. However, the original CROW theory proposed by Yariv et al. only demonstrated one value of coupling factor for a multipole resonance mode. Here, by imaging the tight-binding Bloch waves on a CROW consisting of designer-surface-plasmon resonators in the microwave…
We present a bi-orthogonal approach for modeling the response of localized electromagnetic resonators using quasinormal modes, which represent the natural, dissipative eigenmodes of the system with complex frequencies. For many problems of interest in optics and nanophotonics, the quasinormal modes constitute a powerful modeling tool, and the bi-orthogonal approach provides a coherent, precise, and accessible derivation of the associated theory, enabling an illustrative connection between different modeling approaches that exist in the literature.
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) approac…
The eigenmodes of unstable canonical optical resonators possess fractal structure in their transverse intensity cross-sections [Karman et al., Nature 402, 138 (1999)]. In one particular plane, the magnified self-conjugate plane, this structure can be explained in terms of a combination of imaging and diffraction [Courtial and Padgett, PRL 85, 5320 (2000)]. Here we show that this combination of imaging and diffraction simultaneously occurs in the longitudinal direction, resulting in three-dimensional self-similar fractal structure around the centre of the magnified self-conjugate plane.
A novel zero-order resonator has been designed by an optical surface transformation (OST) method. The resonator proposed here has many novel features. Firstly, the mode volume can be very small (e.g. in the subwavelength scale). Secondly, the resonator is open (no reflecting walls are utilized) and resonant effects can be found in a continuous spectrum (i.e. a continuum of eigenmodes). Thirdly, we only need one homogenous medium to realize the proposed resonator. The shape of the resonator can be a ring structure of arbitrary shape. In addition to the natural applications (e.g. optical storage…
Coupled-resonator optical waveguides (CROWs) are known to have interesting and useful dispersion properties. Here, we study the transport in these waveguides in the general case where each resonator is open and asymmetric, i.e., is leaky and possesses no mirror-reflection symmetry. Each individual resonator then exhibits asymmetric backscattering between clockwise and counterclockwise propagating waves, which in combination with the losses induces non-orthogonal eigenmodes. In a chain of such resonators, the coupling between the resonators induces an additional source of non-hermiticity, and a…
We generalize normal mode expansion of Green's tensor $\bar{\bar{G}}(\bf{r},\bf{r}')$ to lossy resonators in open systems, resolving a longstanding open challenge. We obtain a simple yet robust formulation, whereby radiation of energy to infinity is captured by a complete, discrete set of modes, rather than a continuum. This enables rapid simulations by providing the spatial variation of $\bar{\bar{G}}(\bf{r},\bf{r}')$ over both $\bf{r}$ and $\bf{r}'$ in one simulation. Few eigenmodes are often necessary for nanostructures, facilitating both analytic calculations and unified insight into compu…
The theoretical work of V.B. Braginsky predicted that radiation pressure can couple the mechanical, mirror-eigenmodes of a Fabry-Perot resonator to it's optical modes, leading to a parametric oscillation instability. This regime is characterized by regenerative mechanical oscillation of the mechanical mirror eigenmodes. We have recently observed the excitation of mechanical modes in an ultra-high-Q optical microcavity. Here, we present a detailed experimental analysis of this effect and demonstrate that radiation pressure is the excitation mechanism of the observed mechanical oscillations.
We propose a novel scheme for enhancing the quality factor of coupled resonators optical waveguides (CROWs) when operating near a degenerate band edge (DBE). A DBE is a four-mode exceptional point of degeneracy (EPD) occurring when four Bloch eigenmodes coalesce providing a resonance condition with a giant enhancement in fields. We report an unprecedented scaling law of quality factor of CROWs when operating at the DBE, even in the presence of losses and structural perturbations. Remarkably, the Q factor of the proposed CROW can be engineered to exceed that of a single ring resonator having a …
We develop a theoretical formalism for collectively responding point scatterers where the radiating electromagnetic fields from each emitter are considered in the electric dipole, magnetic dipole, and electric quadrupole approximation. The contributions of the electric quadrupole moment to electromagnetically-mediated interactions between the scatterers are derived in detail for a system where each scatterer represents a linear $RLC$ circuit resonator, representing common metamaterial resonators in radiofrequency, microwave, and optical regimes. The resulting theory includes a closed set of eq…
Interference phenomena render tailoring propagation of electromagnetic waves by controlling phases of several scattering channels. Huygens element, being a representative example of this approach, allows enhancement of the scattering from an object in a forward direction, while the reflection is suppressed. However, a typical resonant realization of Huygens element employs constructive interference between electric and magnetic dipolar resonances that makes it relatively narrowband. Here we develop the concept of a broadband resonant Huygens element, based on a circular array of vertically ali…
Ultra-narrow optical spectral features resulting from highly dispersive light-matter interactions are essential for a broad range of applications such as spectroscopy, slow-light, and high-precision sensing. Features approaching sub-MHz, or equivalently, Q-factors approaching ~1 billion and beyond, are challenging to obtain in solid-state systems, ultimately limited by loss. We present a novel approach to achieve tunable sub-MHz spectral features, at room temperature, without resonators. We exploit gain-enhanced polarization pulling in a twisted birefringent medium where polarization eigenmode…
Real-space imaging reveals symmetry-selected nonlinear energy routing in a mechanical resonator
Tokyo University of Agriculture and Technology（东京农工大学）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）；Xi'an Jiaotong University（西安交通大学）；University of Tokyo（东京大学）
We design a three-way silicon optical waveguide with the Bloch dispersion relation supporting a stationary inflection point (SIP). The SIP is a third order exceptional point of degeneracy (EPD) where three Bloch modes coalesce forming the frozen mode with greatly enhanced amplitude. The proposed design consists of a coupled resonators optical waveguide (CROW) coupled to a parallel straight waveguide. At any given frequency, this structure supports three pairs of reciprocal Bloch eigenmodes, propagating and/or evanescent. In addition to full-wave simulations, we also employ a so-called ''hybrid…
Creation of correlated photon pairs is one of the key topics in contemporary quantum optics. Here, we theoretically describe the generation of photon pairs in the process of spontaneous parametric down-conversion in a resonant spherical nanoparticle made of a dielectric material with bulk $χ^{(2)}$ nonlinearity. We pick the nanoparticle size that satisfies the condition of resonant eigenmodes described by Mie theory. We reveal that highly directional photon-pair generation can be observed utilising the nonlinear Kerker-type effect, and that this regime provides useful polarisation correlations…
In modern nanophotonics, multipolar interference plays an indispensable role to realize novel optical devices represented by metasurfaces with unprecedented functionalities. Not only to engineer sub-wavelength structures that constitute such devices but also to realize and interpret unnatural phenomena in nanophotonics, a program that efficiently carries out multipole expansion is highly demanded. MENP is a MATLAB program for computation of multipole contributions to light scattering from current density distributions induced in nanophotonic resonators. The main purpose of MENP is to carry out…
The nonlocal order parameter equation for nonlinear optical resonator is approximated in the limit of high Fresnel number by the Swift- Hohenberg equation. The exact soliton and periodic solutions of the Swift- Hohenberg equation for optical resonator with thin slice gain element are obtained. It is shown that hyperbolic "flat" secant, "chirped" secant and Jacobi elliptic functions solutions in one transverse dimension are nonlinear eigenmodes for plane - parallel, confocal and quasiconfocal geometries.
We discuss the statistical properties of lifetimes of electromagnetic eigenmodes in dielectric microresonators with fully chaotic ray dynamics. Using the example of a resonator of stadium geometry, we find that a recently proposed random-matrix model very well describes the lifetime statistics of long-lived resonances, provided that two effective parameters are appropriately renormalized. This renormalization is linked to the formation of anomalously short-lived resonances, a mechanism also known from the fractal Weyl law and the resonance trapping phenomenon.
The Purcell factor quantifies the change of the radiative decay of a dipole in an electromagnetic environment relative to free space. Designing this factor is at the heart of photonics technology, striving to develop ever smaller or less lossy optical resonators. The Purcell factor can be expressed using the electromagnetic eigenmodes of the resonators, introducing the notion of a mode volume for each mode. This approach allows to use an analytic treatment, consisting only of sums over eigenmode resonances, a so-called spectral representation. We show in the present work that the expressions f…
We developed a fast numerical algorithm for solving the three dimensional vectorial Helmholtz equation that arises in electromagnetic scattering problems. The algorithm is based on electric field integral equations and is essentially a boundary element method. Nystrom's quadrature rule with a triangular grid is employed to linearize the integral equations, which are then solved by using a right-preconditioned iterative method. We apply the fast multipole technique to accelerate the matrix-vector multiplications in the iterations. We demonstrate the broad applications and accuracy of this metho…
Freestanding Silicon nitride (SiN) devices are central to the field of nanomechanical resonators and for other technology applications such as transmission electron imaging and nanopore bioassays. The nanofabrication techniques used for fabricating these devices often lack flexibility. While photolithography requires printing of an expensive photomask for each new design iteration, electron-beam lithography is extremely slow and commands high equipment cost. Here we demonstrate maskless rapid prototyping of freestanding SiN nanomechanical resonators fabricated by femtosecond laser ablation of …
We demonstrate for the first time the occurrence of a sixth order exceptional point of degeneracy (EPD) in a realistic multimode optical photonic structure by using a modified periodic coupled resonators optical waveguide (CROW), at the optical wavelength ${λ_e} = 1550 nm$. The 6th order EPD is obtained in a CROW without the need of loss or gain and such an EPD corresponds to a very special band edge of the periodic photonic structure where six eigenmode coalesce, so we refer to it as the 6th order degenerate band edge (6DBE). Moreover, we report a new scaling law of the quality factor Q of an…
We study the spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator observed experimentally by Xu and Coen in Opt.~Lett.~{\bf 39}, 3492 (2014). We perform a detailed stability analysis of the Lugiato-Lefever model for the optical Kerr resonators and analyze the temporal bifurcation structure of stationary symmetric and the emerging asymmetric states as a function of the pump power. For intermediate pump powers a pitchfork loop is responsible for the destabilization of symmetric states towards stationary asymmetric ones while at large pump power…
We develop a general formalism to describe the propagation of a near-resonant electromagnetic field in a medium composed of magnetodielectric resonators. As the size and the spatial separation of nanofabricated resonators in a metamaterial array is frequently less than the wavelength, we describe them as discrete scatterers, supporting a single mode of current oscillation represented by a single dynamic variable. We derive a Lagrangian and Hamiltonian formalism for the coupled electromagnetic fields and oscillating currents in the length gauge, obtained by the Power-Zienau-Woolley transformati…
The high-Q {\it subwavelength} resonances in an isolated dielectric disk modes can be achieved by avoided crossing (anticrossing) of the nonorthogonal TE resonances under variation of the aspect ratio as it was reported by Rybin {\it et al} [Phys. Rev. Lett. 119, 243901 (2017)]. Traversing over two parameters, the aspect ratio and the distance between two disks, enhances the Q factor by several times compared to the case of the former case of one-parametric avoided crossing in the isolated disk. Therefore successive two-parametric avoided crossing gives multiplicative gain in the Q-factor as i…
Optical resonances in 1D photonic crystal microcavities are investigated numerically using finite-element light scattering and eigenmode solvers. The results are validated by comparison to experimental and theoretical findings from the literature. The influence of nanometer-scale geometry variations on the resonator performance is studied. Limiting factors to ultra-high Q-factor performance are identified.
Systematic Design and Demonstration of Multipole, Coupled-Cavity Integrated Photonic Bandpass Filters with High FSR, High-Q Single-Mode Microresonators in Low-Loss Silicon Nitride Platform
Georgia Institute of Technology（佐治亚理工学院）；OEwaves Inc.（OEwaves公司）
Quasi-Bound States in the Continuum-Induced Second Harmonic Generation Enhancement in High-$Q$ Triple Dielectric Nanoresonators
Institute for Advanced Study, Nanchang University（南昌大学高等研究院）；School of Education, Nanchang Institute of Science and Technology（南昌理工学院教育学院）；School of Optoelectronic Engineering, Guangdong Polytechnic Normal University（广东技术师范大学光电工程学院）；School of Information Engineering, Nanchang University（南昌大学信息工程学院）；School of Physics and Electronic Science, East China Normal University（华东师范大学物理与电子科学学院）
Research on photonics and metamaterials constantly challenges our intuitive understanding of the behaviour of light. In recent years we have seen negative refraction, focusing of light by a flat slab, a ``perfect'' prism, and an ``invisibility cloak'' [1-6]. It is generally understood that the cause of this unusual behaviour is the strong (anomalous) dispersion, i.e., dependence of the material properties on the frequency of light. Dispersion can be either due to a natural microscopic resonance of the material as with surface plasmons-polaritons, or due to an effective resonance (band-gap) of …
Electromagnetic scattering on subwavelength structures keeps attracting attention owing to abroad range of possible applications, where this phenomenon is in use. Fundamental limits of scattering cross-section, being well understood in spherical geometries, are overlooked in cases of low-symmetry resonators. Here, we revise the notion of superscattering and link this property with symmetry groups of the scattering potential. We demonstrate pathways to spectrally overlap several eigenmodes of a resonator in a way they interfere constructively and enhance the scattering cross-section. As a parti…
