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Polarizabilities of Complex Individual Dielectric Or Plasmonic Nanostructures Polarizabilities of complex individual dielectric or plasmonic nanostructures Adelin Patoux, Clément Majorel, Peter Wiecha, Aurelien Cuche, Otto Muskens, Christian Girard, Arnaud Arbouet To cite this version: Adelin Patoux, Clément Majorel, Peter Wiecha, Aurelien Cuche, Otto Muskens, et al.. Polarizabilities of complex individual dielectric or plasmonic nanostructures. Physical Review B, American Physical Society, 2020, 101 (23), 10.1103/PhysRevB.101.235418. hal-02991773 HAL Id: hal-02991773 https://hal.archives-ouvertes.fr/hal-02991773 Submitted on 16 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW B 101, 235418 (2020) Polarizabilities of complex individual dielectric or plasmonic nanostructures Adelin Patoux ,1,2,3 Clément Majorel,1 Peter R. Wiecha ,1,4,* Aurélien Cuche,1 Otto L. Muskens ,4 Christian Girard,1 and Arnaud Arbouet1,† 1CEMES, Université de Toulouse, CNRS, Toulouse, France 2LAAS, Université de Toulouse, CNRS, Toulouse, France 3AIRBUS DEFENCE AND SPACE SAS, Toulouse, France 4Physics and Astronomy, Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, United Kingdom (Received 9 December 2019; accepted 25 March 2020; published 8 June 2020) When the sizes of photonic nanoparticles are much smaller than the excitation wavelength, their optical response can be efficiently described with a series of polarizability tensors. Here, we propose a universal method to extract the different components of the response tensors associated with small plasmonic or dielectric particles. We demonstrate that the optical response can be faithfully approximated, as long as the effective dipole is not induced by retardation effects, hence do not depend on the phase of the illumination. We show that the conventional approximation breaks down for a phase-driven dipolar response, such as optical magnetic resonances in dielectric nanostructures. To describe such retardation induced dipole resonances in intermediate-size dielectric nanostructures, we introduce “pseudopolarizabilities” including first-order phase effects, which we demonstrate at the example of magnetic dipole resonances in dielectric spheres and ellipsoids. Our method paves the way for fast simulations of large and inhomogeneous metasurfaces. DOI: 10.1103/PhysRevB.101.235418 I. INTRODUCTION where ε1 (respectively ε2) is the dielectric constant of the medium (respectively the nanosphere). From relation (1), we In a multitude of topical areas in contemporary physics can extract the extinction spectrum via the imaginary part of and chemistry, the concept of the polarizability has proven α(ω ). Consequently, the extinction spectra of a sample con- to be extremely useful. In particular, in the physics of gases 0 taining a large number N of such noninteracting nanoparticles and surfaces, the dynamic polarizability tensor of molecules α (ω ) is given by [15,16] appears explicitly in the description, for example, of the Van i 0 der Waals dispersion energy, or in the description of the 8π 2 N Raman scattering process [1–6]. During the 1970s, A. D. I (λ ) = Im(α (ω )) , (2) ext 0 n λ i 0 Buckingham wrote a founding paper on this subject in which 1 0 i an exhaustive list of linear polarizabilities is proposed [7]. where λ represents the incident wavelength, n the refractive Although this work was restricted to atomic and molecular 0 1 index of the environment, and “Im” the imaginary part. systems, it represents a valuable stand of the various possible The sphere represents the highest symmetry, belonging to contributions as well as their ranking, in terms of electric and the isotropic symmetry group. As stated above, in this case, all magnetic multipolar polarizabilities. the diagonal elements of the polarizability are identical, and The theoretical study of the linear optical response of the system displays a scalar response defined by α = αδ small metallic or dielectric particles has also been extensively ij ij [see Eqs. (1) and (2)]. When transforming the sphere into investigated in the last decades. In particular, in the context an ellipsoid of symmetry group D∞ , the polarizability must of plasmonics, the concept of polarizability is often applied h be defined with two independent components [17], and for to the description of plasmon spectra of sub-wavelength sized even lower symmetry, all components α of the polarizabil- noble metal particles [8–14]. In many situations, single metal ij ity tensor must be calculated. This situation corresponds to particles can be schematized by a sphere of radius a,in high anisotropy induced by a complex shape of particles (or which case their optical response can be described by a nanocavities). Note that other kinds of anisotropy can come scalar, frequency-dependent polarizability α(ω ). Then, the 0 from the intrinsic anisotropy of the dielectric constant of the polarizability tensor is diagonal and all tensor elements are particle but also from the surface of another object [18]. In the identical. In cgs units, it reads [15] latter case, the concept of effective polarizability is generally ε (ω ) − ε introduced, and the final symmetry of the particle is dressed by α (ω ) = a3 2 0 1 , (1) ij 0 the symmetry of the surface (i.e., D∞h, for a perfectly planar ε2(ω0 ) + 2ε1 surface). As illustrated by these examples, the design of nanos- tructure polarizabilities starts with the conception of a refer- *[email protected] ence geometry by intuitive considerations. Such an approach, †[email protected] however, is limited to rather simple problems. In case of 2469-9950/2020/101(23)/235418(13) 235418-1 ©2020 American Physical Society ADELIN PATOUX et al. PHYSICAL REVIEW B 101, 235418 (2020) and H refer to electric and magnetic fields) to describe the incident and total electromagnetic fields, we can define a unique (6 × 6) dyadic tensor K(r, r ,ω0 ) operating in the volume V of the source zone and establishing the link between F0(r,ω0 ) and F(r,ω0 ): F(r,ω0 ) = K(r, r ,ω0 ) · F0(r ,ω0 )dr . (6) V Actually, the (6 × 6) superpropagator K(r, r ,ω0 )iscom- posed of four mixed (3 × 3) dyadic tensors: EE EH K (r, r ,ω0 ) K (r, r ,ω0 ) FIG. 1. Sketch used to implement the concept of generalized K(r, r,ω ) = (7) 0 HE , ,ω HH , ,ω electromagnetic propagator. (a) Transparent reference medium with K (r r 0 ) K (r r 0 ) ε (ω ) = n2 and μ = 1, and (b) material system of arbitrary shape, 1 0 1 1 EE , ,ω also called the source zone, embedded in the reference medium in which the first one, K (r r 0 ) that describes the [permittivity εs(ω0 ) and permeability μs(ω0 )]. electric-electric field couplings was introduced in the early beginning of near-field optics [21]. The three other contribu- EH HE HH tions, i.e., K (r, r ,ω0 ), K (r, r ,ω0 ), and K (r, r ,ω0 ), complex structures or complicated phenomena, the intuitive account for coupling with the magnetic field. All these method often fails, as unexpected effects such as polarization propagators are related to the corresponding mixed field conversion occur in the polarizability tensors. In this work, susceptibilities SEE, SEH, SHE, and SHH [9,22], associated we propose a numerical method to extract the polarizability with the source zone: tensors for complex shaped metallic and dielectric nanos- EE EE tructures through a volume discretization technique, which K (r, r ,ω0 ) = δ(r − r )I + χe(ω0 ) · S (r, r ,ω0 ), uses the concept of a generalized propagator. Furthermore, EH , ,ω = χ ω · EH , ,ω , in order to faithfully describe also magnetic optical effects K (r r 0 ) m( 0 ) S (r r 0 ) in dielectric nanostructures, where the conventional dipolar HE HE K (r, r ,ω0 ) = χe(ω0 ) · S (r, r ,ω0 ), polarizability approximation fails, we introduce “pseudopo- HH HH larizabilities” that include phase-induced magnetic dipole K (r, r ,ω0 ) = δ(r − r )I + χm(ω0 ) · S (r, r ,ω0 ). (8) resonances, similar to some homogenization approaches for metamaterials [19,20], but at the level of a single, isolated As explained in Refs. [21,22], these dyadic tensors can be structure. Our pseudopolarizabilities might then be used to numerically computed by performing a volume discretization construct aperiodic or random meta-surface-like assemblies of the source zone together with a Dyson sequence procedure without periodicity. [21] or other numerical inversion techniques, to extract the various field susceptibilities in the source zone. II. A GENERALIZED ELECTROMAGNETIC PROPAGATOR FOR ARBITRARY SHAPED III. EXTRACTION OF POLARIZABILITIES PARTICLES OR CAVITIES OF SMALL NANOSTRUCTURES The concept of the generalized electric field propagator The volume discretization of the source region leads to a previously described in Ref. [21] can be easily extended to mesh of N identical elementary volumes v. Such a proce- the general case of metasystems displaying both an electric dure converts integrals over the source volume V into discrete and a magnetic linear response. In this case, the source summations. In Ref. [16] we have gathered the expressions zone as depicted in Fig. 1 is characterized by the following of the discretization volume elements v for both cubic susceptibility tensor, where I is the identity tensor: and hexagonal compact discretization grids together with the corresponding Green’s function renormalization terms. The χ ω e( 0 ) I 0 electric polarization at the ith cell in the source region can χ(ω0 ) = , (3) 0 χm(ω0 )I be written as follows: χ ω χ ω 2 where e( 0 ) and m( 0 ) are related to the permittivity P(ri,ω0 ) = v χe(ω0 ) εs(ω0 ), respectively, the permeability μs(ω0 ) of the source N zone: EE × (K (ri, r j,ω0 ) · E0(r j,ω0 ) ε (ω ) − ε (ω ) χ ω = s 0 1 0 j=1 e( 0 ) π (4) 4 EH + K (ri, r j,ω0 ) · H0(r j,ω0 )).
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