## Abstract

We applied a multiple-multipole method to calculate the field enhancement of discrete metal nanosphere assemblies due to plasma resonance, thus performing the first full electromagnetic simulation of a variety of nanoparticle assemblies for efficient field focusing, including the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman. Our study captures electromagnetic resonance effects important for optimizing nanoparticle assemblies to achieve maximum electric field focusing. We predict optical frequency electric fields can be enhanced in gold nanoparticle assemblies in aqueous solution by the order of ~450, within a factor of 2 of that achievable in silver nanostructures. We find that both absorption and far-field scattering resonances of nanoparticle assemblies must be carefully interpreted when inferring near-field focusing properties.

© 2008 Optical Society of America

## 1. Introduction

An important problem in optics is to “see” beyond the diffraction limit of light using optical near-fields [1]. Plasma resonance in metal nanostructures provides one means to efficiently couple far-field radiation to nanometer scale volumes, at a so-called “hot-spot”, to produce such effects as surface-enhanced Raman scattering (SERS). SERS is an enhancement of Raman scattering due in large part to enhancement of local electric fields. The Raman shift induced by molecule specific vibrations renders SERS useful in molecular sensing [2]. The Raman scattered optical power due to SERS scales with the fourth power of electric field enhancement,

where *ν _{s}* is the scattered photon frequency.

For this reason, the question of how metal particles should be arranged with respect to each other to produce the strongest field enhancement at the “hot-spots” remains an important one. It is known that linear chains of silver nanoparticles provide a way to concentrate the electromagnetic field in nanoscale regions between the particles [3]. A more efficient field enhancing structure consists of a self-similar chain of particles with decreasing diameters and gaps [4]. Tapered, continuous plasmonic waveguides are a third instance of efficient nanofocusing structures [5]. Moreover, the local optical field can also be controlled using a dielectric core-metal shell nanoparticle [6].

Our study is motivated by recent advances in the fabrication of metal nanostructures. Plasmon resonance in noble metal bispheres was measured by Reinhard, *et al.*, [7] to demonstrate nanometer scale distance calibration. Further, the ability to organize gold nanoparticles into discrete and well-controlled structures using programmable DNA templates [8] provides the possibility for rational design and synthesis of efficient near-field focusing structures for applications such as SERS.

In our work, we have simulated the optical response of discrete assemblies of nanospheres in linear, self-similar and other arrangements using a multiple-multipole technique (also known as the T-matrix method) [9–11]. The mathematical construction of the multiple-multipole method has already been established and simulations on some simple geometries have been performed [12–16]. The most relevant is the recent work by Pellegrini, *et al.*, in which metal bispheres and clusters of small spheres dressing a large sphere have been analyzed. Thus far, there has been no full electromagnetic analysis of the focusing properties of sphere assemblies explicitly designed for efficient field focusing, such as the self-similar structure. The quasi-static approximation or dipole approximation are often made in studies of light scattering by assemblies of spherical nanoparticles; Khlebtsov et al. have conclusively shown that a multipole approach is needed for accurate prediction of far-field light scattering even for simple sub-wavelength bispheres [17]. Our work reveals, for the first time, the complex resonances of various nanoparticle assemblies, particularly the efficient self-similar structure. We provide estimates of electric near-field focusing in these structures without recourse to quasi-static or dipole approximations.

We describe in section 2 the multiple-multipole method, including the recursion relations used for efficient numerical calculation. Simulation results for scattering, absorption and local field enhancement are presented and discussed in section 3 for a variety of structures: nanosphere assemblies arranged in linear chains, self-similar series and in discrete approximations of the truss, cone and tetrahedron. Conclusions drawn from our numerical work are summarized in section 4.

## 2. Numerical technique

#### 2.1 Multiple-multipoles

We solved the electromagnetic scattering problem of aggregates of *N _{s}* small spheres, each characterized by a normalized radius

*x*=

^{j}*ka*=2

^{j}*πm*/

_{medium}a^{j}*λ*, a normalized position

_{vacuum}**R**

^{j}=(

*X*,

^{j}*Y*,

^{j}*Z*) and a relative refractive index ${m}^{j}={m}_{abs}^{j}/{m}_{medium}$ . We use harmonic time dependence exp(-

^{j}*iωt*) for all field quantities. Mie theory accurately describes the scattering of uncoupled spheres, but generalization of Mie theory to multiple-multipoles is necessary to account for successive light scattering amongst nanospheres at sub-wavelength separations.

The multiple-multipole theory, also known as the T-matrix method [9–11] begins with an expansion of the incident, scattered, and internal (to the sphere) electromagnetic field into the well known vector spherical harmonic basis **N**
^{(1)}, **M**
^{(1)}, **N**
^{(3)}, **M**
^{(3)} [9,18–20] about each sphere:

$${\mathbf{E}}_{\mathrm{inc}}^{j}=-\sum _{n=1}^{\infty}\sum _{m=-n}^{n}i{E}_{\mathrm{mn}}\left[{p}_{\mathrm{mn}}^{j}{\mathbf{N}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)+{q}_{\mathrm{mn}}^{j}{\mathbf{M}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)\right]$$

$${\mathbf{E}}_{\mathrm{int}}^{j}=-\sum _{n=1}^{\infty}\sum _{m=-n}^{n}i{E}_{\mathrm{mn}}\left[{d}_{\mathrm{mn}}^{j}{\mathbf{N}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)+{c}_{\mathrm{mn}}^{j}{\mathbf{M}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)\right]$$

$${\mathbf{H}}_{\mathrm{inc}}^{j}=-\frac{1}{\eta}\sum _{n=1}^{\infty}\sum _{m=-n}^{n}{E}_{\mathrm{mn}}\left[{q}_{\mathrm{mn}}^{j}{\mathbf{N}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)+{p}_{\mathrm{mn}}^{j}{\mathbf{M}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)\right]$$

$${\mathbf{H}}_{\mathrm{int}}^{j}=-\frac{1}{{\eta}^{j}}\sum _{n=1}^{\infty}\sum _{m=-n}^{n}{E}_{\mathrm{mn}}\left[{c}_{\mathrm{mn}}^{j}{\mathbf{N}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)+{d}_{\mathrm{mn}}^{j}{\mathbf{M}}_{\mathrm{mn}}^{\left(1\right)}\left(j\right)\right]$$

where
${a}_{mn}^{j},{b}_{mn}^{j},{p}_{mn}^{j},{q}_{mn}^{j},{c}_{mn}^{j}$
and
${d}_{mn}^{j}$
are the expansion coefficients, *η*=*η*
_{0}/*m _{medium}* is the wave impedance of the ambient medium,

*η*=

^{j}*η*/

*m*is the wave impedance in metal sphere

^{j}*j*, and

**N**

^{(1)},

**M**

^{(1)}indicate vector spherical harmonics centered on the origin of sphere

*j*. We assume here that the vector harmonics are defined at each sphere with respect to a common set of direction axes

*x*,

*y*and

*z*. The complete scattered electric field is given by a sum over all

*N*spheres,

_{s}We assume a plane wave polarized in the +x-direction and propagating in the +z-direction, so that the non-trivial incident field coefficients are,

where we take as the spatial coordinates of the origin of sphere *j* with respect to the system origin. The prefactor *E _{mn}*=|

*E*

_{0}|

*i*(2

^{n}*n*+1)(

*n*-

*m*)!/(

*n*+

*m*)!, where |

*E*

_{0}| is the magnitude of the incident wave, is introduced as a normalization constant to maintain numerical stability [9] and to maintain consistency with the single-sphere Mie theory, where

*m*=±1. In the above, the order

*m*and the degree

*n*are integers such that -

*n*≤

*m*≤

*n*, and describes the multipole (i.e. dipole, quadrupole, etc.) nature of the electromagnetic field dressing each sphere. In numerical work, the harmonic degree is truncated to a finite number 1≤

*n*≤

*N*.

The central numerical problem is the efficient calculation of the scattered field coefficients ${a}_{mn}^{j},{b}_{mn}^{j}$ in a self-consistent manner that accounts for successive scattering between all nanospheres. Imposing the usual boundary conditions for electromagnetic fields at the surfaces of the spherical particles gives a set of inhomogeneous linear equations for the coefficients ${a}_{mn}^{j},{b}_{mn}^{j}$ [9,10],

$${b}_{\mathrm{mn}}^{j}={\beta}_{n}^{j}\left({q}_{\mathrm{mn}}^{j}-\sum _{\underset{j\ne l}{j=1}}^{{N}_{s}}\sum _{v=1}^{\infty}\sum _{\mu =-v}^{v}\frac{{E}_{\mu v}}{{E}_{mn}}\left[{a}_{\mu v}^{l}{B}_{\mathrm{mn}}^{\mu v}\left({\mathbf{R}}^{\mathrm{jl}}\right)+{b}_{\mu v}^{l}{A}_{\mathrm{mn}}^{\mu v}\left({\mathbf{R}}^{\mathrm{jl}}\right)\right]\right)$$

where the quantities *α ^{j}_{n}* and

*β*are the single sphere Lorenz-Mie coefficients and ${A}_{mn}^{\mu \nu}$ and ${B}_{mn}^{\mu \nu}$ are the coefficients of the vector harmonic addition theorem [9,10,12] that are functions of the inter-origin vector

^{j}_{n}**R**

^{jl}=

**R**

^{l}-

**R**

^{j}.

The system of Eq. (6) is often re-written in block form in which the matrix **T** represents an interaction operator for the vector harmonic coefficients associated with each sphere,

with the scattering vector
${a}^{j}=[{a}_{mn}^{j}{b}_{mn}^{j}]$
, the incident field vector
${p}^{j}[{a}_{n}^{j}{p}_{mn}^{j}{\beta}_{n}^{j}{q}_{mn}^{j}]$
and **T**
^{ji} being composed of elements of the form
$(E\mu \nu /{E}_{mn}){\alpha}_{n}^{j}{A}_{mn}^{\mu \nu}$
. In the limit of weak interaction between spheres, Mie theory is asymptotically recovered, **a**=(**1-T**+**T**
^{2}-**T**
^{3}+…) **p**. The strong interaction between metal nanospheres in close proximity, illuminated close to plasmon-polariton resonance, prohibits the use of such an expansion. In our work, the linear system of Eq. (7) was solved using a biconjugate gradient method with commercially available software.

#### 2.2 Numerical evaluation of coefficients

The Lorenz-Mie coefficients for sphere *j* depend on the normalized radius *x ^{j}* and relative refractive index

*m*as,

^{j}$${\beta}_{n}^{j}=\frac{{\psi}_{n}^{\prime}\left({x}^{j}\right){\psi}_{n}\left({m}^{j}{x}^{j}\right)-{{m}^{j}\psi}_{n}\left({x}^{j}\right){\psi}_{n}^{\prime}\left({m}^{j}{x}^{j}\right)}{{\xi}_{n}^{\prime}\left({x}^{j}\right){\psi}_{n}\left({m}^{j}{x}^{j}\right)-{{m}^{j}\xi}_{n}\left({x}^{j}\right){\psi}_{n}^{\prime}\left({m}^{j}{x}^{j}\right)}$$

where *ψ _{n}*(

*z*)=

*zj*(

_{n}*z*) and

*ξ*(

_{n}*z*)=

*zh*

^{(1)}

_{n}(

*z*), with

*j*and

_{n}*h*the spherical Bessel and spherical Hankel functions of first kind.

^{(1)}_{n}The vector harmonic addition coefficients are defined implicitly through the vector harmonic addition theorem [9,10,12,21] for decomposing the harmonics **N**
^{(3)}
*(j)*, **M**
^{(3)}
*(j)* about an origin at **R**
^{j} into harmonics **N**
^{(1)}
*(l)*, **M**
^{(1)}
*(l)* about the origin **R**
^{l},

$${\mathbf{N}}_{\mathrm{mn}}^{\left(3\right)}\left(j\right)=\sum _{v=1}^{\infty}\sum _{\mu =-v}^{v}\left[{B}_{\mu v}^{\mathrm{mn}}\left({\mathbf{R}}^{\mathrm{jl}}\right){\mathbf{M}}_{\mu v}^{\left(1\right)}\left(l\right)+{A}_{\mu v}^{\mathrm{mn}}\left({\mathbf{R}}^{\mathrm{jl}}\right){\mathbf{N}}_{\mu v}^{\left(1\right)}\left(l\right)\right].$$

A variety of methods can be used to calculate the tensor coefficients. Efficient numerical evaluation of these addition coefficients is critical. In our work we have combined several techniques briefly described in the following. We have used Mackowski’s strategy of decomposing general translation coefficients into rotations and axial translations,

$${B}_{\mu v}^{\mathrm{mn}}\left({\mathbf{R}}^{\mathrm{jl}}\right)\leftarrow {\left({R}_{\mu m}^{\left(n\right)}\right)}^{-1}{B}_{\mu v}^{\mathrm{mn}}(0,0,\mid {\mathbf{R}}^{\mathrm{jl}}\mid ){R}_{\mu m}^{\left(n\right)}$$

as illustrated in Fig. 1. The rotation matrix
${R}_{\mu m}^{(n)}(\alpha ,\beta ,\gamma )$
is defined by the Euler angles *α*=acos(*X ^{jl}*/|

**R**

^{jl}|)/sin

*β*,

*β*=acos(

*Z*/|

^{jl}**R**

^{jl}|) and

*γ*=0 for rotating the angular momentum z-axis into alignment with the inter-origin vector

**R**

^{jl}=

**R**

^{l}-

**R**

^{j}. The rotation matrix elements are,

with normalization constants

and the rotation operator of quantum mechanics [22]

composed of the Jacobi Polynomial evaluated directly as,

The symmetry relations ${d}_{\mu m}^{(n)}(\beta )={(-1)}^{\mu -m}{d}_{-\mu -m}^{(n)}(\beta )$ and ${d}_{\mu m}^{(n)}(\beta )={(-1)}^{\mu -m}{d}_{m\mu}^{(n)}(\beta )$ were used to further reduce computational time.

The coefficients for positive axial translation of vector harmonics by a distance |**R**
^{jl}| were calculated using the following expressions [12],

where the Gaunt coefficients *a _{p}*=

*a*(

*m*′,

*n*,-

*m*′,

*ν*,

*p*),

*p*=

*n*+

*ν*,

*n*+

*ν*-2,… |

*n*-

*ν*|, were calculated using Bruning and Lo’s recursive technique [11],

where (2*q*-1)!!=(2*q*-1)(2*q*-3)…3·1;(-1)!!≡1. The simplicity of our chosen strategy arises from the fact that axial translations are diagonal in harmonic order *m*=*µ*, while the rotations are diagonal in harmonic degree *η*=*ν*.

Once the scattered wave coefficients
${a}_{mn}^{j},{b}_{mn}^{j}$
are calculated, quantities of direct physical interest are evaluated easily. The electromagnetic fields are given by Eqs. (3) and (4), so that the electric field enhancement, defined as the ratio of |**E**
_{sca}+**E**
_{inc}|/|**E**
_{inc}|, can be evaluated. The time average Poynting vector is also directly evaluated <*S*>=1/2Re{**E**.**H***}. Finally, the normalized far-field scattering, extinction and absorption cross-sections (also known as the efficiencies) were calculated [9,10],

where
$G=\pi {\sum}_{j}^{{N}_{S}}{\left({a}^{j}\right)}^{2}$
is the geometric area of each assembly under broadside illumination. Eqs. (24) and (25) make use of a single set of total scattered field coefficients
${a}_{mn}^{T},{b}_{mn}^{T}$
for an expansion about a sphere (centered at **R**
^{l}) that is chosen as an origin,

$${b}_{\mathrm{mn}}^{T}=\sum _{j=1}^{{N}_{s}}\sum _{v=1}^{\infty}\sum _{\mu =-v}^{v}\left[{A\prime}_{\mathrm{mn}}^{\mu v}\left({\mathbf{R}}^{\mathrm{jl}}\right){b}_{\mu v}^{j}+{B\prime}_{\mathrm{mn}}^{\mu v}\left({\mathbf{R}}^{\mathrm{jl}}\right){a}_{\mu v}^{j}\right].$$

where the
${A}_{mn}^{\text{'}\mu \nu},{B}_{mn}^{\text{'}\mu \nu}$
coefficients are calculated as
${A}_{mn}^{\text{'}\mu \nu},{B}_{mn}^{\text{'}\mu \nu}$
, but with spherical Hankel functions *h _{n}^{(1)}* replaced by spherical Bessel functions

*j*in Eqs. (17) and (18), as appropriate to translation of outward propagating vector harmonics

_{n}**N**

^{(3)},

**M**

^{(3)}[12].

## 3. Results and analysis

#### 3.1 Geometries setup and convergence

The simulations were performed on systems of gold and silver nanoparticles assembled into various geometries and immersed in an aqueous medium (*m _{medium}*=1.3342). The dielectric properties of the silver and gold nanoparticles were assumed to be that of bulk material; measured bulk dielectric values reported by Johnson and Christy [23] were used in all calculations and interpolated as needed. The problem setup is depicted in Fig. 1, where the incident field is propagating in the

*z*-direction and polarized along the

*x*-direction. The structures are always oriented in such a way that the maximum charge polarization, along

*x*, is obtained. The seed structure in the center of our systems, where the “hot-spot” is located, consists of a bisphere whose spheres have 5nm radii and a separation of 0.5nm – close to the limit of what DNA based self-assembly might achieve. Field enhancement is taken as the ratio of the local electric field at the “hot-spot”, the center of the seed bisphere, to the electric field strength in the illuminating field.

While the multiple-multipole theory has been described in the exact case where all harmonic degrees and orders are included, practical computation truncates harmonic degree to 1≤*n*≤*N*. In our work, to achieve convergence in near field enhancement at the level of <5%, we have used *N*=10. This corresponds to a total of 2*N*(*N*+1)=220 vector harmonic expansion components *per sphere*. An example of the convergence in field enhancement is given in Fig. 2 below. The contribution of high degree multipoles is obviously important for the accurate prediction of resonant wavelengths at which field enhancement is maximum, in agreement with Khlebtsov, *et al.*, findings that the dipole approximation is inadequate. The physical explanation is simple, electric fields confined to a characteristic size *d* are decomposed into harmonics extending to up to degree *N*α*a*/*d*, where *a* is particle radius.

#### 3.2 Field enhancement in linear chains

The linear chain geometry simulated here consists of identical gold or silver nanospheres with 5nm radius and interparticle gap of 0.5nm, shown in Fig. 3. Field enhancement is maximum at the middle gap. As the number of spheres *N _{s}* is increased from 2 to 20, the resonant wavelength is red-shifted and the quality factor of the field enhancement resonance increases, as shown in Figs. 4 and 5.

The field enhancement is calculated as the ratio of the electric field magnitude at the focal spot of the structure to the incident plan wave electric field magnitude. The linear particle chain functions as an optical antenna, similar to the continuous metal wire exhibiting a plasma resonance [24], but with an effective resonance determined by the phase velocity of a plasma excitation propagating along a chain of particles. The field enhancement response of gold linear chains is suppressed at free space optical wavelengths shorter than 600nm, due to a reduction in gold dielectric quality factor, *Q*=*d*(*ωε*’)/*dω*/2*ε*” where ε’, ε” are real and imaginary parts of the dielectric constant [25], resulting from the onset of interband absorption at 2eV [23]. The greater material quality factor of silver versus gold is further evidenced by the sharper resonances in field enhancement observed for the silver particle array.

#### 3.3 Field enhancement in self-similar structures with constant number of spheres

The self-similar configuration of metal nanospheres first proposed by Li, Stockman and Bergman [4] was shown to be efficient at focusing optical near fields using quasi-static simulations. The self-similar structure is defined with a central bisphere with sphere radii *a ^{0}* (taken to be 5nm) and interparticle gap

*d*(taken to be 0.5nm). Successive spheres are added to the structure in geometric series with radii

^{0}*a*and nearest neighbour gap

^{j}*d*enlarged by a factor 1/κ, such that

^{j}*a*

^{j+1}=

*a*/

^{j}*κ*and

*d*

^{j+1}=

*d*/

^{j}*κ*. Note that in the limit where κ=1, the linear chain is recovered. We compared the response of arrays with a fixed number of spheres

*N*=4 and 6 and varying geometric factor

_{s}*κ*and length

*L*, illustrated to scale in Fig. 6 and 7. Since the complexity of self-assembly processes increases with number of particles per discrete unit, optimizing field enhancement for fixed particle number

*N*is important.

_{s}The simulated field enhancements for 4-sphere and 6-sphere structures composed of gold and silver particles are plotted in Figs. 8–10. The higher material *Q* of silver compared to gold is apparent in the widths of field enhancement resonances, and also in the clear presence of higher order resonances for silver structures but not gold. Interband absorption in gold above 2eV suppresses field enhancement at *λ*<600nm. The improved efficiency of the geometric series of spheres over the linear chain is due to the increased metallic volume that expels a greater amount of optical energy, and the reduced number of gaps into which optical energy is focused. Importantly, these simulated field enhancement plots reveal that the geometric scale factor *κ* (which implicitly defines structure length for a fixed number of spheres *N _{s}*) that maximizes field enhancement is a non-trivial function of both sphere material and number of spheres. Full electromagnetic simulation without resorting to the quasi-static approximation is required to accurately model the resonances of the plasma wave excitation along a self-similar structure.

#### 3.4 Field enhancement in self-similar structures of fixed length

Simulation of structures of fixed length and varying growth factor *κ* and sphere number *N _{s}*, illustrated in Figs. 12, were also performed. The optimization of structures of fixed linear extent is important for

*in-vivo*applications [26], where the retention or expulsion of particles is dependent on size. The electric field enhancement is plotted in Figs. 13–16 for self-similar structures with length 167.5nm (209.5nm), the length of linear chains of 16 (20) spheres of 5nm radius and 0.5nm interparticle gap.

The field enhancement resonances are red-shifted as the structures approach linear chains (geometric factor *κ*→1). The red-shift can be interpreted as arising from the increase in hybridization of the plasma resonances associated with individual spheres, and hence a reduction in frequency of the lowest order plasma oscillation distributed over the structure [27]. The linear chain does not provide the same field enhancement as self-similar structures with *κ*<1. The precise value of *κ* (which implicitly defines number of spheres *N _{s}* for a fixed length structure) that maximizes field enhancement depends upon both sphere material and physical length of the structure, but the general trend is improved efficiency with smaller

*κ*(fewer spheres

*N*).

_{s}#### 3.5 Near-field distribution in self-similar structures

The spatial distribution of electric field enhancement |**E**
_{sca}+**E**
_{inc}|/|**E**
_{inc}| in the vicinity of the self-similar structures is useful in explaining the role of structure in near-field enhancement and the importance of vector multipole calculation. Field enhancements in the x-z and x-y planes (the x-z plane is the plane of incidence) for self-similar gold structures of 167.5nm length are presented on a logarithmic scale in Figs. 17, 18. The internal field of each sphere is not shown. Each field enhancement plot is taken with incident light at the wavelength that maximizes field enhancement at the focal spot of the structure, summarized in Table 1. These plots reveal the “hot-spots” associated with each structure, with those structures that approach the linear chain having a preponderance of “hot-spots”. Structures with small *κ* are dominated by the large volume spheres, which serve to enhance field in a reduced number of “hot-spots”. The higher dielectric quality factor *Q* of silver versus gold is evident in the increased field enhancement about the silver structures.

The near-field plots also reveal the importance of multiple-multipole calculations. The incident side of the nanospheres shows field enhancement associated with the charge polarization induced by the incident electric field. Although this induced charge distribution is dipolar in nature, the field distribution about spheres is clearly not dipolar due to the strong coupling amongst them. Furthermore, there is a clear distinction in Fig. 17 between the incident side and the shadow side of the nanospheres, a direct manifestation of the retardation of fields. The near-fields of Fig. 17 indicate clearly that as the condition for the quasi-static approximation, *x*≪2π*m _{medium}a*/

*λ*, is better satisfied, by smaller spheres (in the linear chains), field retardation is reduced and the distinction between incident side near-field and shadow side near-field is diminished.

The retardation of fields is also seen in the time average Poynting vector <*S*>=1/2Re{**E**×**H***}. The Poynting vector magnitude is plotted in Fig. 19 for self similar gold structures of 167.5 nm length. Energy is clearly drawn into the structures on the incident side, casting shadow regions for spheres as small as 5nm in radius. It should be noted that although isolated spheres may be well modeled under the quasi-static approximation, the complete structure need not be well modeled using a quasi-static approximation. Shadows in the Poynting vector are indeed visible for each structure in Fig. 19.

The higher order resonances of self-similar structures are complex in nature. The spatial distribution of near-field enhancement in the incident plane for a silver *N _{s}*=8,

*κ*=0.672 selfsimilar structure is plotted in Fig. 20 for the three lowest order resonances in near-field enhancement (taken from Fig. 16). The three near-field resonances appear to be distinguished by different hybridizations of dipolar plasma excitations on the individual spheres. The geometric series structure does not lend itself to a simple quantitative description of resonances expected for a linear chain.

#### 3.6 Far-field properties of self-similar structures

The far-field scattering efficiency *Q _{sca}* of single spheres in the Rayleigh limit can be measured using linear dark-field microscopy techniques [28] and provide indirect information about field enhancement properties [29]. Near-field enhancement and far-field scattering are not simply related in more general metal nanostructures such as nanosphere assemblies. Both the absorption efficiency

*Q*and far-field scattering efficiency are plotted in Figs. 21–24 for selfsimilar structures with a length of 167.5nm.

_{abs}The absorption efficiency and near-field enhancement are resonant at approximately equal wavelengths (refer to Table 1). The co-location of resonances is in agreement with studies of small individual spheres [29]. We also note that linear structures have N_{s}-1 gaps where field enhancement is comparable to that at the central focus (Figs. 17, 18), resulting in significantly increased absorption at the near-field enhancement resonance in comparison to structures with *κ*<1. Strong interband absorption at wavelengths *λ* < 600nm is evident for gold particle assemblies with small *κ*, where absorption is expected to be dominated by the largest spheres.

The resonances in scattering efficiency of self-similar structures in the linear chain limit *κ*→1 closely follow the resonances in field enhancement and absorption efficiency. However, in the limit of small geometric factor *κ*, the scattering efficiency peaks acquire a significant red-shift with respect to the near-field enhancement. The physical origin of this red-shift is unknown, but has been observed in numerical studies of single sphere scattering [29]. The red-shift was found in [29] to increase with sphere size, in agreement with our finding of increased red-shift with decreasing *κ* (and hence larger spheres in the structure). Elastic light scattering data, such as that collected with dark-field microscopy, will therefore need to be carefully interpreted when inferring near-field enhancement properties of self-similar structures with geometric factor *κ*≪1.

#### 3.7 Field enhancement in other geometries

Three other geometries were investigated, as illustrated in Fig. 29: the discrete nanosphere approximations of a cone, pyramid and triangular truss. The near-field enhancement versus wavelength for structures composed of 5nm radius gold nanoparticles with 0.5nm minimum interparticle gaps are illustrated in Fig. 30. As in the case of linear chains, the preponderance of interparticle gaps results in field enhancement inferior to the self-similar structure.

## 4. Conclusions

We have used a multiple-multipole method to characterize the near-field enhancement properties of discrete assemblies of metal nanospheres. We have found that the self-similar geometric series of spheres first proposed by Li, Stockman and Bergman were significantly more efficient than linear chains and other structures (cones, pyramids, triangular trusses). Examination of the spatial distribution of the near-field about self-similar structures demonstrates clearly the need to go beyond dipole approximations and quasi-static approximations to accurately capture the optical properties of these structures. We have found the near-field enhancement of self-similar structures to exhibit complex resonant behaviour. The rational design of such structures – to target a particular operating wavelength for instance – will require full electromagnetic modeling such as that of the multiple-multipole method. We anticipate that recent advances in DNA based self-assembly techniques [7,8] for fabrication of discrete metal nanoparticle structures will lead to the physical realization of the efficient near-field focusing structures analyzed herein. Furthermore, we have found that farfield scattering properties, which have been traditionally used to infer near-field enhancement properties of individual metal nanospheres, will require more careful analysis in the case of self-similar structures with a small geometric factor *κ*.

## Acknowledgment

This work was supported by Le Fonds québécois de la recherche sur la nature et les technologies (FQRNT) and the Canadian Institute for Advanced Research (CIFAR). We are also grateful for the discussions with Prof. Andrew G. Kirk and Prof. Hanadi Sleiman, both from McGill University. We also thank the McGill University Photonic Systems Group for providing computer resources.

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