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Efficient coupling to plasmonic multipole resonances by using a multipolar incident field David Becerril, Humberto Batiz, Giuseppe Pirruccio, and Cecilia Noguez ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01426 • Publication Date (Web): 26 Feb 2018 Downloaded from http://pubs.acs.org on February 27, 2018

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Efficient coupling to plasmonic multipole resonances by using a multipolar incident field David Becerril, Humberto Batiz, Giuseppe Pirruccio, and Cecilia Noguez∗ Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México D.F. 01000, México E-mail: [email protected] Abstract A method to manipulate the multipolar plasmonic response of a nanostructure in the quasi-static limit is introduced. The theoretical method puts on the same footing geometry, dielectric properties, and incident field and proceeds in two steps: it optimizes the geometry of the nanostructure to maximize the intensity of the scattering cross-section spectrum. This is done by calculating the coupling strengths of the different modes of the system to the external field, which the method naturally provides. Then, it exploits the symmetry of the incident electromagnetic field to enhance or suppress specific orders, which, in turn, tunes the field enhancement. We demonstrate the method by using a plasmonic dimer of nanospheres.

Keywords plasmonics, high multipole, near field enhancement, plasmonic dimer, non-uniform incident field

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ACS Paragon Plus Environment

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Introduction

Since the very beginning of nanoscience, researchers have realized that nanostructured materials possess a very distinct optical response when compared to the bulk ones. Their optical response is dominated by localized surface resonances appearing in the extinction crosssection spectrum at discrete wavelengths and corresponding to the electromagnetic modes of the nanostructure to which the incident light couples. Optical response can be classified in terms of the symmetry of the induced multipole charge distribution inside the object. The relevance of a multipole response was already clear from the solution, found by Mie, to the problem of a metallic sphere illuminated by a plane wave when the long-wave condition is not met, further generalized to the case of aggregates of non-identical spheres. Multipole resonances for many other geometries such as cubes, nanowires, nanorods and dimers have also been studied, 1–5 and have been shown to exist even in the long-wavelength limit. 6–8 Recently, multipole resonances have found application in chiral plasmonics, where circular dichroism and pseudo-chirality has been shown. 9–11 Electric quadrupoles and octupoles moments, and magnetic dipoles also emerge in plasmonic lattices and assemblies of nanoparticles where they give rise to bright and dark modes and are responsible for phenomena such as electromagnetically induced transparency, 12,13 lasing, 14 superradiance 15,16 and non-linear effects. 17 Furthermore, by coupling the external field to the different modes of the system, it is possible to modify the symmetry and spatial distribution of the total electric field intensity in the system, which can be useful to focus light and localize energy to subwavelength hot spots. 18,19 Strong field gradients associated with high order multipoles can be also beneficial for optical tweezing. 20 So far, a lot of attention has been paid to the rational design of resonant nanostructures, which typically is achieved by varying the geometrical parameters and composition of the system. 21,22 For instance, optical tuning obtained by combining materials with different electric permittivities in complex arrays leads to the development of metamaterials and photonic/plasmonic crystals, which allow the engineering of the optical response by tailoring 2


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the photonic energy bands. 23 Metamaterials use non-diffractive arrays of coupled resonant nanoscatterers sustaining magneto-electric resonances 24 to achieve negative refraction, 25–27 superlensing, 28 resonant absorption, 29 cloaking. 30 On the other hand, diffractive arrays sustain hybrids photonic-plasmonic modes arising from the long-range radiative coupling of resonant scatterers and have been widely used to control the light emission of emitters. 31–33 In both cases, the efficiency of the optical resonances is predetermined by the geometrical parameters of the system, such as lattice constant, dimensions and shape of the scatterers. Therefore, after the fabrication no active control on the optical response is possible. Recently it has been proposed and experimentally demonstrated that structuring the incident field leads to an externally tuneable, dramatic change in the optical response of the illuminated object. 33–37 Different methods have been introduced to achieve this, such as wavefront shaping 38–42 and coherent control of absorption and scattering via multiple beam illumination. 43–45 Polarization of the incident light also plays an essential role in selecting the available modes of the nanostructure. 46,47 However, little attention has been paid to the relation between the symmetry of the incident field and the resulting excitation of the resonances of a system. 33,46 The combination of the geometrical design of the nanostructure with a proper symmetry of the illumination field can result in an intriguing additional degree of freedom to tailor the optical response of nanostructures. Here, we propose a general analytical model able to compare and isolate the contributions of the geometry, dielectric response and excitation field in the extinction cross-section. The method quantifies the coupling strength of the incident field to a specific multipole resonance in the quasi-static limit. To present the method, we make use of a dimer of metallic nanospheres placed in proximity to each other, the simplest system used in the generalized multiparticle Mie (GMM) solution and the technique developed in Refs. ( 46,48). However, our method shows how specific plasmonic multipole modes are excited by using a spatially non-uniform incident field. Our method fills the gap between other solutions valid for nanospheres of arbitrary dimension but placed

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at arbitrary distances and in the presence of a spatially uniform incident field. Our method adequately accounts for Coulomb interactions and evanescent fields between the particles which are responsible for the excitation and the mutual coupling of multipoles under uniform and non-uniform incident fields.

2

Theoretical model

We develop a theoretical framework which considers a general multipole electromagnetic field excitation within the quasi-static approximation. The method is a generalization of the spectral representation formalism introduced by Fuchs. 49 Consider a set of N non magnetic polarizable nanospheres of radii ai , described by the same local dielectric function (ω), #» located at positions {Ri } in a host with dielectric constant h . If the nanospheres are sufficiently close to each other near-field coupling due to electromagnetic interactions between them takes place. This coupling results in high order multipole moments induced on each nanosphere, 8 which are assigned a set of indexes (l, m, i), where lm is the multipole order and i the particle. With this convention, the frequency-dependent multipole moments induced on each of the nanospheres, Qlm,i , are: lm,i Qlm,i = −αlm,i (ω)Vtot

= −αlm,i (ω) [Vext + Vind ]

(1) lm,i

,

where αlm,i (ω) are the frequency-dependent multipolarizabilities of the nanospheres, which, by defining the complex spectral variable u(ω) = 1/[1 − (ω)/h ], can be expressed as: 8

αlm,i (ω) =

n0l 2l + 1 a2l+1 , 4π n0l − u(ω) i

(2)

where n0l = l/(2l + 1). To correctly account for the surface scattering damping present in small particles a corrected expression for the dielectric permittivity, (ω), has been em-

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ployed. 7 Notice that αlm,i (ω) are independent of m due to the spherical symmetry of the parlm,i lm,i ticles. Here Vext and Vind are the external and induced potentials felt by the nanospheres, lm,i respectively. The components of Vind represent the potential felt by particle i due to the

induced multipole moments on the rest of the nanospheres that comprise the system. In this formalism, the indexes l and m represent the symmetry and the polarization of the external and induced fields. To simplify notation we use a compact form for the set of indexes µ µ = (l, m, i) and µ0 = (l0 , m0 , j). Using this notation, Vind can be further decomposed as:

0

µ Vind = Aµµ Qµ0 ,

(3)

0

where Aµµ are components of a (LM N ) × (LM N ) matrix, A , that describe the interaction between multipole moments of particles. Here, l = L and m = M are the highest multipole orders considered and N is the total number of nanospheres. Due to the symmetry of the 0

scatterers, Aµµ can be expressed in terms of spherical harmonics and only depend on the geometrical parameters of the system, such as particle radii and separation distances (see Supporting Information). Interestingly, A allows the coupling of high multipole moments even if the external field is spatially uniform when considering the first multipole order. Using Eqs. (2) and (3), Eq. (1) can be recast in matrix form as:

[−u(ω)II + H ]X = F,

(4)

where X and F are vectors whose components are written in terms of the induced multipole moments and the external potential, respectively:

Xµ = q

Fµ

0

= −

Qµ

, la2l+1 i q 0 +1 l0 a2l i0 4π

5

(5)

0

µ Vext .


(6)

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In Eq. (4), I is the unitary matrix and H , is a hermitian matrix given in terms of A , which only depends on the geometrical parameters of the system (see Supporting Information). Note that I and H also have dimension (LM N ) × (LM N ). In Eq. (4) we see that with the help of the spectral variable, the geometrical and dielectric parameters have been separated. The solution to Eq. (4) is given in terms of the frequency-dependent Green’s function, G , associated to H , with components: 0

0 Gµµ

=−

X s

Cµµ,s , u(ω) − ns

(7)

0

in which ns are eigenvalues of H representing the multipole modes of the system. Cµµ,s are components of the real matrix, C s , which describe the coupling strength of the external field through the s-th mode of the system (see Supporting Information). The H of a linear array of nanospheres, is real and symmetric such that ns are real and positive eigenvalues. Furthermore, coupling between multipoles of different index m, i.e. m 6= m0 is not permitted. Therefore, eq. (4) with a given m can be solved separately. Taking this into consideration, for a given allowed value of m, we indicate with C11,s and 0

µ C21,s the coupling strengths of the induced dipole moment (l = 1) by an external field Vext ,

with l0 = 1 and l0 = 2, respectively. While, C12,s and C22,s represent the coupling coefficients of the quadrupole moment (l = 2) induced by an incident field with l0 = 1 and l0 = 2, respectively. Explicitly, l0 = 1 corresponds to a spatially uniform field. In this work, we consider the excitation with l0 = 1 for the case of a dipole excitation and l0 = 2 for a quadrupole excitation. By using Eqs. (4) and (7) the induced multipole moments Qµ can be obtained: Xµ =

X

0

0

Gµµ F µ .

(8)

µ0

We stress that Eq. (6) gives the multipole coefficients of the external potential that needs to be applied to induce the corresponding multipole moment in the nanospheres. The main result of this work is that by introducing a proper external electric field, the optical response 6


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of nanostructures can be modified. Once the induced multipole moments, Qµ , are found, the spatial distribution of the multipole electric near-field intensity is calculated by taking minus the gradient of the electric potential outside of the nanospheres: " E(r) = −∇

# X

gµ (r)Qµ ,

(9)

µ

where the components of the spatial function gµ (r) are given by

glm,i (r) =

4π Ylm (θi , φi ) 2l + 1 |r − Ri |l+1

(10)

with (θi , φi ) being the angles corresponding to the vector (r − Ri ) (see Supporting Infor0

µ mation). For a specific multipole excitation field, Vext , C s , can be obtained by using the

multipole moments, Qµ in Eq. (8) along with their coupling strengths. Because the scattering cross section is proportional to the dipole moment of the system, we use Eq. (8) and the symmetry properties previously mentioned to calculate the dipole moment: l = 1 with m = 0, this is along the z axis in cartesian coordinates: 50 r pz (ω) = −

0 Cl1,s 4πa3 X 0 Fl . 3 u(ω) − ns s

(11)

For a given excitation l0 , its frequency-dependent scattering cross-section can be evaluated as: 50 16π k 4 v Csca (ω) = 3 E02

2 l0 X C 0 1,s l F , s u(ω) − ns

(12)

where v is the volume of the nanosphere and the term between bars is an effective Cartesian dipole moment of the system and E0 is the magnitude of the incident field. From Eq. (12), resonances occur when:

Re[u(ωs )] − ns = 0, and Im[u(ωs )] ' 0 . 7


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At the resonant frequency, the intensity of the scattering cross section depends primarily on 0

the value of two parameters, the coupling strength Cl1,s and the value of Im[u(ωs )]. In principle, all the matrices in the previous equations are infinite-dimensional, L → ∞ and M → ∞. Since the larger l is, the weaker the coupling between multipole moments, numerically, we introduce a cut off at L = lmax . For lmax = 1 we recover the dipole approximation, which is expected to be good only for large particle separations. For lmax = 2 the quadrupole approximation is obtained in which dipole-dipole, dipole-quadrupole and quadrupole-quadrupole moments interactions are taken into account. In general, for a given particle separation an appropriate lmax must be chosen to ensure convergence of the physical properties. 21

3

Results and discussion

To demonstrate our method we use the archetypal system made of two identical spheres of radius a, whose centers are separated by a distance D. We take the axis that joins the centers of the spheres as the z axis. Remember that under this condition there is no coupling between multipole moments with different m index. Moreover, the sign of the multipole moments in the two identical spheres satisfy the relation: 8 Qlm,2 = (−1)l+1 Qlm,1 . Therefore, by exploiting symmetry, the number of multipole moments that need to be evaluated can be considerably lowered. Additionally, the high symmetry of our system allows us to parameterize the solution regarding only one geometrical parameter, i.e., σ = D/2a. In the quasi-static limit, NPs of sizes of 40 nm and smaller can be considered. 51 The separation between NP is also an essential factor since quantum effects may play an important role. The tunneling current may partially quench the excitation of multipolar modes and limit the possible near-field intensity enhancement. However, quantum effects are significant for distances below 0.5 nm. 52 Therefore, the geometric parameter can take values as small as σ = 1.0125 for the largest nanoparticles and shortest distances. One of the advantages of

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this formalism is that it makes explicit that the coupling mechanism of the particles through the excitation of multipoles solely depend on σ. Therefore, it is sufficient to rescale the nanosphere’s diameters and distances by a factor to keep σ, retaining all the results and the physics of the multipolar coupling. Here, we discuss the results for nanospheres with a radius of 12.5 nm and interparticle distances of 0.5 nm and larger., i.e., σ ≥ 1.02. For these values, we see that the influence of the multipolar coupling is still considerable, quantum effects are negligible, and that the quasi-static approximation still holds true. The number of multipole moments i.e., lmax , needed to ensure convergence in the solutions depends on the configuration of the system, or equivalently on the value of σ, with a larger value of lmax needed for a smaller geometric factor. We found that to ensure convergence within the 1% of the solutions for the configurations considered in this work, a cut-off equal to lmax = 150 is needed. In principle, the formalism outlined in the previous section can be used for any material once the dielectric function is known. In this work we present results for silver nanospheres, 53 since these have shown to produce larger field enhancement than gold, for example,. 54 We stress that the extension to 1D and 2D ordered or disordered systems made of an assembly of many particles of different composition and radii is straightforward. In the following section, we discuss the principal result of the model, i.e., the introduction of a proper multipole excitation field in Eqs. (4)-(6). Combining geometry and symmetry of illumination, it is possible to modify the spectrum further, the number of excited modes, and the near field of a nanostructure.

3.1

Excitation of Multipole Modes as a Function of the External Field

We consider two different types of external fields, first the case of a spatially homogeneous external field with a parallel polarization to the axis that joins the centers of the spheres. 0

This is achieved by taking F µ 6= 0 in Eq. 6 with µ0 = (l0 = 1, m0 = 0, i). As a second case, we consider a spatially inhomogeneous external field such that if felt by an isolated 9


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0

sphere induce a quadrupole charge distribution. This is achieved by taking F µ 6= 0 with µ0 = (l0 = 2, m0 = 0, i). 1 with s = 1, ..., 5, as a In Fig. 1(a) we plot the first excitation efficiency coefficients C1,s

function of σ for the external field F 1,0,i . Each curve represents the coupling strength of the P 1 induced dipole moment through each corresponding s−mode. The sum rule, 49 s C1,s = 1, causes a distribution among the available modes of the dimer as a function of σ. For example, for large σ, the excitation efficiency coefficients of the mode with s = 1 dominates, i.e. 1 1 → 0. At large → 1, while coupling strengths associated to s = 2, 3, . . . vanish, C1,s C1,1

interparticle distances, the optical response of the dimer is dominated by the dipole mode of the isolated nanospheres, as expected. Thus, the mode s = 1 is identified with the dipole response of the dimer. As σ decreases, coupling strengths to higher order multipole modes become more important. By accurately selecting σ, we modify the eigenvalues of the system and the coupling strengths and therefore, the resonant frequencies and optical response of the dimer. Thus, it is possible to tune the spectrum of the dimer, i.e., enhance, suppress or shift the resonant peaks by considering only the geometric parameter σ. We label with 1 and 2 the dimers with σ = 1.02 and σ = 1.15, respectively. In Fig. 1(a) these specific geometric parameters are indicated with vertical dashed lines. Once the geometric parameters are fixed, we can calculate the eigenvalues and coupling strengths for 0

both dimers. The most significative values of ns and Cl1,s are listed in Table 1. For dimer 2 1 (σ = 1.15) illuminated by an external field with l0 = 1, we found C1,1 = 0.751 when s = 1 1 and C1,2 = 0.247 when s = 2. Both modes account for the 99.8% of the total coupling

strength to the external field, thus modes with s > 2 become irrelevant at this particular σ. Here, the s = 1 mode, corresponding to the dipole mode of the dimer at large interparticle distances, dominates. For dimer 1 (σ = 1.02) the nanospheres are very close, resulting in 1 larger excitation efficiency coefficients for mode s = 2, with C1,2 = 0.314, followed by s = 3 1 1 with C1,3 = 0.275 and then by s = 1 with C1,1 = 0.242. Also modes with s = 4 and s = 5

have significant coupling and the first five modes account for the 99.6% of the total coupling

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1 for s = 1, ..., 5. b) Scattering cross sections Figure 1: a) Coupling efficiency coefficients C1,s for dimers 1 and 2 made by silver nanospheres with a = 12.5 nm.

with the external field. Once eigenvalues and coupling strengths are calculated, we fix the dielectric function of the material to calculate first the spectral variable u(ω), and then, the resonant frequencies or wavelengths (λs = c/ωs ) from the condition, Re[u(ωs )] = ns , imposed in Eq. (12). The obtained values for both dimers are listed in Table 1, as well as the corresponding values of Im[u(ωs )]. In Fig. 1(b) we plot the scattering cross section spectrum calculated with Eq. (12) for dimers 1 and 2. For dimer 2 (σ = 1.15), the intense peak around λ1 = 370 nm is associated to the dipole mode of the dimer (s = 1). For this configuration, higher order multipole modes are not appreciated because the corresponding coupling strengths are small and located in a range with large values of Im[u(ωs )], compared with s = 1. In general, as Im[u(ωs )] is larger, the energy dissipation increases and the intensity of the resonances 1 1 diminishes. 18 Consequently, even though C1,2 ' (1/3)C1,1 , only a small shoulder associated

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Table 1: Resonant wavelengths (λs ), imaginary part of the spectral function (Im[u(λs )]) l0 with l0 = 1 and evaluated at the resonant wavelength and strongest coupling strengths (C1,s 2) to the modes of the two dimer configurations considered. σ s 1.02 1 2 3 4 5 1.15 1 2 3

λs (nm) 444 375 355 348 344 370 350 342

Im[u(λs )] 0.015 0.037 0.067 0.097 0.113 0.043 0.087 0.117

1 C1,s 0.242 0.314 0.275 0.141 0.024 0.751 0.247 0.001

2 C1,s 0.212 0.124 -0.067 -0.179 -0.094 0.346 -0.328 -0.020

ns 0.130 0.241 0.313 0.361 0.392 0.251 0.361 0.408

with the quadrupole mode is observed around λ = 350 nm. For dimer 1 (σ = 1.02), in Fig. 1(b) we observe 3 distinct peaks and a redshift of all the resonances. The redshift of the dipole mode of the dimer can be understood as a consequence of the external field being parallel to the axis that joins the nanospheres. This field polarizes the particles, such that, the induced local-field is in the same direction as the applied field. Both induced and applied fields are contrary to the restoring force acting on the electron cloud, therefore reducing the resonant frequency. 7 An analogous argument can be used for the redshift of the multipole resonances. As expected, higher order modes beyond the dipole mode have smaller resonant wavelengths, as they require larger energies to be excited. Due P 1 = 1, and because the coupling strengths to the quadrupolar and to the sum rule s C1,s octupolar modes are larger compared to dimer 2, the intensity of the dipole peak reduces 1 1 compared to dimer 1. Even if C1,1 < C1,2/3 , the dipole mode is still more intense because the

corresponding Im[u(ωs )] value is smaller. Incidentally, for certain σ the coupling strengths 1 C1,s , for different s can be made equal.

In Fig. 2 the coupling strengths to the second type of external field with l0 = 2 are 2 shown. Interestingly, we observe that the excitation efficiency coefficients C1,s for all modes

s qualitatively differs from those for an external field with l0 = 1. For instance, all the curves exhibit a maximum and/or a minimum value at a particular σ and eventually all

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excitation efficiency coefficients go to zero for large σ. This is consequence of the sum rule P 2 49 2 with s = 1, ..., 5, as a function of σ for an external Fig. 2(a) displays C1,s s C1,s = 0. field with l0 = 2 and m = 0. This field shows two different amplitude gradients along the direction parallel and perpendicular to the z axis (see Supporting Information). Distinctly, negative values are allowed for some modes, which can be interpreted as an excitation of a 2 given moment opposing the external field. In Table 1 also C1,s with s = 1, ..., 5 are listed.

2 Figure 2: a) Coupling efficiency coefficients C1,s for s = 1, ..., 5 plotted against he dimensionless geometric factor σ. b) Scattering cross sections for dimers 1 and 2 made by silver nanospheres with a = 12.5 nm.

2 2 For dimer 2, the coupling strengths to the quadrupole field, C1,1 and C1,2 are comparable

but of opposite sign. These give rise to an asymmetric structure in the scattering cross section spectrum, shown in Fig. 2(b). Notice that this peak is half as intense when compared with the one shown in Fig. 1(b). This feature arises from the considerable intensity of the quadrupole mode with respect to the dipole one. As in the cases of plane wave illumination

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(l0 = 1), higher multipole modes (s > 1) are not appreciable in the spectrum as a consequence of the small values of the corresponding coefficients for those modes and large Im[u(ω)]. For dimer 1 (σ = 1.02), also shown in Fig. 2(b), the smaller peak found at λ = 370nm, represents the convolution of many high order modes which are spectrally too close to be resolved. Differences in intensity obtained with a l0 = 1 and l0 = 2 external field can be explained by comparing coupling efficiencies. For example, consider dimer 1 (σ = 1.02), with a plane wave external field (l0 = 1), as shown in Fig. 1. For this configuration, coupling strengths to the quadrupole and octupole modes are larger than in the dipolar mode. Therefore, although there is more dissipation at the quadrupole and octupole resonances, the three peaks are still clearly distinguished. Summarizing, for a given geometric factor and dielectric function, the resonant wavelengths and therefore the value of Im[u(ω)] are fixed, and independent of the external field. On the other hand, peak intensities are inversely proportional to Im[u(ω)] and directly comparable to the excitation efficiencies of the mode, whose coupling strength depends on the symmetry of the external field being used.

3.2

Near Field

Once the multipole moments are obtained, it is possible to calculate the electric near-field intensity outside the nanospheres and separately analyze the contribution of each multipole moment by using Eq. (9). Notice that all the multipole moments contribute to the induced field, as explicitly shown in Eq. (3). This means that when a given mode of the dimer s is excited, the spatial distribution of the charge density and electric field are not only determined by the s− multipole moment but also by all the other excited moments that can couple to it. Fig. 3 displays the spatial distribution of the near field intensity normalized by the intensity of the incident field for the dipole, quadrupole and octupole modes of dimer 1 with an external field with l0 = 1. This configuration is equivalent to taking nanosphere’s radius 14


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2

|E| Figure 3: Normalized near field intensity, log10 |E 2 , for dimer 1 illuminated by an external 0| 0 field l = 1, at the (a) dipole (s=1), (b) quadrupole (s=2) and (c) octupole (s=3) resonances at λ1 = 444, λ2 = 375 and λ3 = 355 nm, respectively.

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of 12.5 nm and a separation distance of 0.5 nm. All modes exhibit the maximum of field intensity in the gap of the dimer, which can be enhanced more than four orders of magnitude for modes s = 1 and s = 2. In general, we observe that for all modes, the most significant contribution to the field enhancement and its overall shape are given by the dipole moment, which can be seen in the symmetry of the induced field. Fig. 4 displays the multipole decomposition of the electric near field intensity of mode s = 1 of dimer 1 at λ1 = 444 nm into the first and most significant components with l = 1, l = 2 and l = 3. The near field plots are obtained by selecting the corresponding l in Eq. (3) for a given wavelength and then, calculating the field of each component using Eq. (9). We found that as l increases the intensity of the field associated with each component decreases. This fact follows from the coupling of the external field to high order moments is achieved indirectly via Eq. (3), so the field intensity enhancement associated to them is weak. In Fig. 4 we can also observe the spatial distribution of the near field for each l, which shows a symmetry related to the corresponding component. Higher multipole moments are characterized by pronounced spatial modulations and strong confinements of the electric field intensity around the nanospheres. For example, for l = 1 the two darker spots or nodes are along the azimuth, while for l = 2 there are three dark spots for each nanosphere, and for l = 3 we found four dark spots for each nanosphere. The decomposition of the near field allows us to follow the contributions to the total field. The total enhancement shown in Fig. 3 is obtained by adding these fields as well as contributions from the rest of the components up to lmax . Now, we want to analyze the dimer near field when the spatially structured external field is applied. Fig. 5 displays the spatial distribution of the total electric near-field intensity for the dipole (s = 1), quadrupole (s = 2) and octupole (s = 3) modes of dimer 1 at their corresponding resonant wavelengths: λ1 = 444 nm, λ2 = 375 nm and λ3 = 355 nm respectively. As in the case of plane wave excitation (l0 = 1), for small values of σ the electromagnetic near-field is strongly inhomogeneous. Also for this type of illumination, the

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Figure 4: Dipole l = 1 (a), quadrupole l = 2 (b) and octupole l = 3 (c) contributions to the |E|2 electric near field intensity, log10 |E 2 , for a quadrupole mode s = 1 at λ1 = 444 nm and an 0| 0 external field l = 1.

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enhancement decreases as the order of the mode increases. The effect of illumination with l0 = 2 is to increase the overall field enhancement by 2-3 orders of magnitude concerning the dipole illumination, as one can observe by comparing Fig. 5 and Fig. 3 at all resonant wavelengths. Moreover, it can also be seen that the total area where the near field is enhanced is extended by more than four orders of magnitude. This enhancement is partially due to the spatial inhomogeneity of the incident field. Simultaneously, the intensity enhancement in the regions of the dimer outside the gap decreases more rapidly. Therefore, this kind of illumination might be useful to increase the light-matter interaction in nanoptics experiments or to increase the sensitivity of devices. Notice that even though the intensity of the peaks in the scattering cross-section spectrum in Fig. 2 are smaller (less than half the intensity) than in Fig. 1, field enhancement for the l0 = 2 illumination is still larger even at the quadrupole and octupole resonances. This result suggests that to perform spectroscopic experiments, it is convenient to use the symmetry of the exciting field. Moreover, for experiments, it might prove more useful to tune the emission spectrum to low-intensity high multipole peaks rather than to the more intense dipolar one. 55 In Fig. 6 we show the contributions to the near electric field in the wavelength of the quadrupole resonance at λ2 = 375 nm for an excitation with l0 = 2. In Fig. 6 it can be seen that the most significant contribution comes from the quadrupole moment followed by the dipole moment that shows a similar maximum intensity as the octupole. However, the symmetry of the induced field does not always correspond to the order of the mode that is excited (see Supporting Information). In general, the dominating contribution to the induced field will depend on the chosen external field, the excitation wavelength through eigenmodes and the system’s geometry through the excitation efficiency coefficients.

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2

|E| 0 Figure 5: Normalized electric near field intensity, log10 |E 2 , for an external field l = 2, at 0| the (a) dipole (s=1), (b) quadrupole (s=2) and (c) octupole (s=3) resonances at λ1 = 444, λ2 = 375 and λ3 = 355 nm, respectively.

4

Conclusions

We developed a general analytical framework to describe the optical response of nanostructures in the quasi-static limit that treats on the same footing their geometry, dielectric properties and incident electromagnetic field, being able to isolate the role of each of them precisely. The method is based on the multipole decomposition of both the incident field and the polarizability of the nanostructure. We show that the symmetry of a multipole incident field provides an additional parameter through which it is possible to change the optical response, spatial distribution and near-field enhancement of the illuminated system. Our formalism gives compelling insights on the coupling efficiency of the incident field to 19


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Figure 6: Dipole l = 1 (a), quadrupole l = 2 (b) and octupole l = 3 (c) contributions to the |E|2 normalized near field intensity, log10 |E 2 , for a quadrupole mode s = 2 at λ2 = 375 nm and 0| 0 an external field with l = 2. the optical modes of the nanostructure, facilitating the analysis of the relative interaction between the mentioned variables. To illustrate the method we considered two limiting geometrical configurations of a dimer of metallic nanospheres and two symmetries of the external incident field. We show that for the configuration with large separation, both types of external fields couple mainly to the dipole mode. However, for a small separation, the spatially modulated field strongly couples to more modes than the homogeneous external field and therefore produces less defined peaks in the scattering spectrum, but larger field enhancements which are covering more area.

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Acknowledgement Authors acknowledge the partial support from DGAPA-UNAM projects PAPIIT IN107615 and PAPIIT IA102117, and from CIC-UNAM.

Supporting Information Available Details of the methodology are given as well as the quadrupole external field, and the multipolar contributions to the near-field due to this external field to the dipole resonance. This material is available free of charge via the Internet at http://pubs.acs.org/.

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