Quasimodal expansion of electromagnetic fields in open two-dimensional structures

Abstract

A quasimodal expansion method (QMEM) is developed to model and understand the scattering properties of arbitrary shaped two-dimensional (2-D) open structures. In contrast with the bounded case which have only discrete spectrum (real in the lossless media case), open resonators show a continuous spectrum composed of radiation modes and may also be characterized by resonances associated to complex eigenvalues (quasimodes). The use of a complex change of coordinates to build Perfectly Matched Layers (PMLs) allows the numerical computation of those quasimodes and of approximate radiation modes. Unfortunately, the transformed operator at stake is no longer self-adjoint, and classical modal expansion fails. To cope with this issue, we consider an adjoint eigenvalue problem which eigenvectors are bi-orthogonal to the eigenvectors of the initial problem. The scattered field is expanded on this complete set of modes leading to a reduced order model of the initial problem. The different contributions of the eigenmodes to the scattered field unambiguously appears through the modal coefficients, allowing us to analyze how a given mode is excited when changing incidence parameters. This gives new physical insights to the spectral properties of different open structures such as nanoparticles and diffraction gratings. Moreover, the QMEM proves to be extremely efficient for the computation of Local Density Of States (LDOS).

General theory of spontaneous emission near exceptional points

“…Note that this relation holds also with Bloch-periodic boundary conditions (and the surface-term correction found in [86] does not appear in our formulation). Although the matrix A is no longer symmetric in the k-periodic problem, it satisfies the relation A(k) T = A(−k), and since we are essentially relating k-right eigenvectors to (−k)-left eigenvectors, the relation above remains unchanged.…”

Purcell factor of spherical Mie resonators

“…The normal modes of the system are solutions of Maxwell equations in the absence of sources, with outgoing radiation conditions, and with a certain normalization [26,[28][29][30]. Due to the outgoing radiation conditions, the Hamiltonian of the system is non-hermitian and the normal modes of the system have complex eigenfrequencies ω µ [26,31,32]. Let us notice that a normalization condition of the kind V |E µ (r, ω µ )| 2 dV cannot be applied to normal modes with resonant complex frequencies ω µ because E µ (|r| → ∞, ω µ ) → ∞ [20,[26][27][28][29]33].…”

Ultracompact resonator with high quality-factor based on a hybrid grating structure

“…In this picture, a resonator is viewed as a passive open system with only out-going emission from the resonator, which determines the boundary conditions. Its resonating modes are referred to as quasi modes [25], since they decay with time. The field evolution during one round trip in the cavity is expressed by a matrix U = R bot R top where R top and R bot are the reflectivity matrices from the top and the bottom sections of the structure, respectively, as shown in Fig.…”

“…the cap layer, each Bloch mode corresponds to a single spatial harmonic (or diffraction order) [16]. For Q-factor estimation, a method based on the quasi-normal mode (QNM) [25] picture is employed among several approaches [26], due to its accuracy and numerical efficiency for both 2D and 3D simulations. In this picture, a resonator is viewed as a passive open system with only out-going emission from the resonator, which determines the boundary conditions.…”