Quasinormal mode theory for nanoscale electromagnetism informed by quantum surface response

Qiang Zhou, Pu Zhang, and Xue-Wen Chen

Phys. Rev. B 105, 125419 – Published 29 March 2022

Abstract

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 $d$ 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 semianalytical 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 plane-wave excitations in the nanostructures.

Received 28 November 2021
Revised 11 March 2022
Accepted 16 March 2022

DOI:https://doi.org/10.1103/PhysRevB.105.125419

Physics Subject Headings (PhySH)

Nanophotonics Plasmonics Quantum description of light-matter interaction

Atomic, Molecular & Optical

Authors & Affiliations

Qiang Zhou, Pu Zhang ^*, and Xue-Wen Chen ^†

School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong Univeristy of Science and Technology, Luoyu Road 1037, Wuhan, 430074, People's Republic of China

^*Corresponding author: puzhang0702@hust.edu.cn
^†Corresponding author: xuewen_chen@hust.edu.cn

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Issue

Vol. 105, Iss. 12 — 15 March 2022

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Images

Figure 1
(a) The quantum surface responses (QSRs) of a metallic nanostructure in terms of effective surface current $K$ and polarization $Π$ . The zoomed-in view illustrates the QSRs characterized by Feibelman's $d$ parameters at the metal-dielectric surface. (b) Principal QNM eigenfrequencies of a sodium ( $r_{s} = 4, ℏ γ = 0.1$ eV) nanosphere in vacuum obtained numerically by solving the generalized linear eigenvalue problem Eq. ((5), (6)) and analytically by locating the poles of the generalized Mie coefficients [32]. $d_{⊥}$ is taken from Fig. 2 (a).
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Figure 2
The QSR informed QNMs of sodium (left panel) and gold (right panel) nanosphere dimers. The sphere diameter is 60 nm. (a), (b) The Feibelman's $d_{⊥}$ parameters for the metal-air interfaces calculated with high-level theories (TDDFT for sodium [27] and specular reflection model for gold [31]) and the multiple Lorentzian fittings. The fitting for sodium is according to Ref. [32]. (c)–(h) The first 12 QNM eigenfrequencies for various gap sizes calculated with the QSRs (blue squares) and under LRA (green triangles). (i)–(m) The ratios between the surface ( $N_{s}$ ) and the complete ( $N$ ) QNM normalization factors of the first 12 QNMs for various gap sizes.
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Figure 3
The QNM analysis of emission enhancement for a dipolar emitter at the center of a Na nanosphere dimer (60 nm diameter and 5 nm gap). (a) The QNM spectrum calculated with the QSRs (gray squares) and under LRA (green triangles). The gray scale is weighted by the effective mode volumes ( $V_{eff}$ ) evaluated at the dipole position. The red asterisks denote the poles of $d_{⊥} (\tilde{ω})$ . (b) The QNM construction of the Purcell factor spectra. The thin blue traces denote the individual contributions of the QSR informed QNMs. The construction of the total spectrum by $\sim 300$ QNMs is plotted with the red trace and benchmarked with the QSR-informed full numerical simulation denoted by black circles. The green dashed and orange dash-dotted traces depict the spectra calculated under LRA and constructed with the QNMs normalized by only $N_{v}$ , respectively.
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Figure 4
The QNM analysis for a gold nanoparticle on mirror (NPoM) under the illumination of a plane wave. (a) The schematic of the NPoM structure. (b) The extinction cross section spectra for various incidence angles are calculated with the QSRs (red) and under LRA (green). The blue and magenta dashed lines denote the respective contributions from the dominant QNMs $M_{0, 1}$ . (c) For the $θ = 15^{o}$ incidence at $λ = 705$ nm, $Re {E_{z}}$ is constructed with $Re {{\tilde{E}}_{z}}$ of $M_{0, 1}$ on the cut plane at gap center; cf. the red line segment in (a). (d) The QNM construction of light intensity enhancement at $(x, y) = (8, 0)$ nm on the cut plane; cf. the gray asterisks in (a) and (c). The spectra constructed by eight QNMs, obtained by the QSR-informed and classical LRA full numerical simulations are plotted with the red trace, black circles, and green dashed trace, respectively. The thin blue, magenta, and orange traces denote the respective contributions from $M_{0, 1}$ and the cross coupling. The inset shows the intensity enhancement along the line $y = 0$ nm with the QSRs (red, $λ = 705$ nm) or under LRA (green, $λ = 723$ nm).
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