Shape Deformation of Nanoresonator: A Quasinormal-Mode Perturbation Theory

Wei Yan, Philippe Lalanne, and Min Qiu

Phys. Rev. Lett. 125, 013901 – Published 1 July 2020

Abstract

When material parameters are fixed, optical responses of nanoresonators are dictated by their shapes and dimensions. Therefore, both designing nanoresonators and understanding their underlying physics would benefit from a theory that predicts the evolutions of resonance modes of open systems—the so-called quasinormal modes (QNMs)—as the nanoresonator shape changes. QNM perturbation theories (PTs) are one ideal choice. However, existing theories developed for tiny material changes are unable to provide accurate perturbation corrections for shape deformations. By introducing a novel extrapolation technique, we develop a rigorous QNM PT that faithfully represents the electromagnetic fields in perturbed domain. Numerical tests performed on the eigenfrequencies, eigenmodes, and optical responses of deformed nanoresonators evidence the predictive force of the present PT, even for large deformations. This opens new avenues for inverse design, as we exemplify by designing super-cavity modes and exceptional points with remarkable ease and physical insight.

Received 9 September 2019
Revised 25 April 2020
Accepted 1 June 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.013901

Physics Subject Headings (PhySH)

Classical optics Light propagation, transmission & absorption Photonics Plasmonics

Optical microcavities

Perturbative methods

Atomic, Molecular & Optical

Authors & Affiliations

Wei Yan^1,2,*, Philippe Lalanne ^3,†, and Min Qiu^1,2,‡

¹Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang Province, School of Engineering, Westlake University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
²Institute of Advanced Technology, Westlake Institute for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China
³Laboratoire Photonique, Numérique et Nanosciences (LP2N), IOGS-University of Bordeaux-CNRS, 33400 Talence cedex, France

^*wyanzju@gmail.com
^†philippe.lalanne@institutoptique.fr
^‡qiumin@westlake.edu.cn

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Vol. 125, Iss. 1 — 3 July 2020

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Images

Figure 1
Overview of the PT framework and notations. (a) A body (nanoresonator) is deformed with boundary changing from $\partial V_{res}$ to $\partial V_{res}^{'}$ . (b) Geometrical deformation is parametrized by $h (r_{\partial V_{res}}) \hat{n}$ . (c) The perturbed body is modeled as the unperturbed body dressed (augmented) by a surface-polarization distribution, ${\tilde{P}}_{Geom}$ given by Eq. (1b). In (b)–(c), ${\tilde{E}}^{\pm δ [h]} \equiv \tilde{E} (r_{\partial V_{res}} \pm δ [h])$ with $δ [h] \equiv 0^{-}$ for $h > 0$ and otherwise $δ [h] \equiv 0^{+}$ .
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Figure 2
Validation of the perturbation theory for a silicon rod (radius $a = 200 nm$ , height $d = 400 nm$ , and permittivity $ϵ_{si} = 12.96$ ) in air. (a) Field distribution of a (1,1,1) Mie’s mode on the rod surface. The QNM frequency is ${\tilde{ω}}_{u} = 0.74 - 0.034 i eV$ . (Top) Real part of perpendicular normalized displacement field; (bottom) amplitude of real part of (vectorial) parallel normalized electric fields. The arrows specify the field direction. (b) Modal eigenfrequency shifts, $Δ \tilde{ω} \equiv Δ \tilde{Ω} - i Δ \tilde{Γ} / 2$ , as the nanorod radius varies either uniformly (left, $h = b$ ) or sinusoidally [right, $h = b \sin (2 π z / d)$ ]. The (left) linear and (right) quadratic dependencies of $Δ \tilde{ω}$ on deformation parameter $b$ are accurately predicted using the first-order and second-order PTs, respectively.
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Figure 3
Test for large deformations of a silver sphere (50-nm radius) in air. Two deformations, into spheroids or cuboids, are considered. Silver is modeled by a Drude permittivity $ϵ_{Ag} = 1 - ω_{p}^{2} / (ω^{2} + i ω γ)$ with $ℏ ω_{p} = 9 eV$ and $ℏ γ = 0.021 eV$ . (a) (Left) Eigenfrequencies ${\tilde{ω}}_{u} \equiv Ω_{u} - i Γ_{u} / 2$ of dipole (blue) and quadrupole (red) QNMs. Degeneracy factors are given in parenthesis. (Right) Amplitudes of perpendicular normalized electric displacement fields for dipole, quadrupole and hexapole QNMs with azimuthal order $m = 0$ . (b) Eigenfrequencies ${\tilde{ω}}_{p} \equiv Ω_{p} - i Γ_{p} / 2$ of perturbed dipole QNMs for spheroids and cuboids as aspect ratios $b / a$ vary (for small deformations, $b / a ≃ 1$ ). (c) Extinction-cross-section spectra of cuboids for several values of $b / a$ . $2 \times 15$ QNMs are used in (b),(c); additional static QNMs at zero frequency are taken into account in (c) [9].
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Figure 4
Application of the PT for designing exceptional point (EP) and high- $Q$ super-cavity mode (SCM). (a) A Si dumbbell-shaped nanoresonator sits on an Au substrate coated with a 50-nm-thick ${SiO}_{2}$ layer. The dumbbell consists of three equal-high cylinders (total height $h = 500 nm$ ) with diameters $D_{1}$ , $D_{2}$ , and $D_{1}$ (from top to bottom). The Au permittivity is approximated by the Lorentz-Drude model $ϵ_{Au} (ω) = ϵ_{\infty} - ω_{p, 1}^{2} / (ω^{2} + i γ_{1} ω) - ω_{p, 2}^{2} / (ω^{2} - ω_{0}^{2} + i γ_{2} ω)$ with $ϵ_{\infty} = 6$ , $ℏ ω_{p, 1} = 8.67 eV$ , $ℏ γ_{1} = 0.1 eV$ , $ℏ ω_{p, 2} = 3.65 eV$ , $ℏ γ_{2} = 2.15 eV$ , $ℏ ω_{0} = 7.38 eV$ . (b) Field distributions of azimuthal-component electric fields, $Re ({\tilde{E}}_{u; ϕ})$ , of two modes $M_{1}$ and $M_{2}$ with azimuthal order $m = 0$ obtained for $D_{1} = D_{2} = 1.2 h$ . (c) Eigenfrequencies calculated with Eqs. (2) for perturbed QNMs resulting from $M_{1} - M_{2}$ coupling as the parameter space ( $D_{1} - D_{2}$ ) is spanned. (d) Eigenfrequencies for $D_{2} = 1.2558 h$ . The EP is labeled. (e) Resonance frequencies and quality factors for $D_{1} = D_{2}$ . The SCM with $Q_{p} ≃ 240$ is labeled in the lower panel. (f) SCM-field distribution $Re ({\tilde{E}}_{p; ϕ})$ .
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