Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect

doi:10.1364/JOSAA.27.002411

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Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect

Huai-Yi Xie, Ming-Yaw Ng, and Yia-Chung Chang »View Author Affiliations

JOSA A, Vol. 27, Issue 11, pp. 2411-2422 (2010)
http://dx.doi.org/10.1364/JOSAA.27.002411

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▸ Article Outline
– Abstract
1. INTRODUCTION
2. MATHEMATICAL DESCRIPTIONS OF A PARTICLE OF ARBITRARY SHAPE
3. MATHEMATICAL FORMULATION FOR LIGHT SCATTERING
4. SUMMARY AND CONCLUSION
– References and links

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Abstract

We derive analytical solutions for the scattering of electromagnetic waves by a nanoparticle with nearly spherical shape and nonlocal dielectric function by using an extended Mie scattering theory with additional boundary conditions. A perturbation method is used to treat the correction due to deviation from the spherical shape. A surface characteristic function is introduced to describe the non-spherical surface profile of the nanoparticle, and it plays an important role in our analytical formulation. Complex surface plasmon modes are obtained. It is found that not only the transverse but also the longitudinal surface plasmon modes of the nanoparticle are excited due to the nonlocal effect. Our analytical formulation provides an alternative method for investigating the optical behaviors of the surface plasmon of nanoparticles with nearly spherical shape and nonlocal effect.

1. INTRODUCTION

Metallic nanoparticles with various sizes and shapes have been studied extensively by scientists and engineers both theoretically and experimentally [1

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

, 2

Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002). [CrossRef] [PubMed]

, 3

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]

, 4

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]

, 5

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]

, 6

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005). [CrossRef] [PubMed]

, 7

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006). [CrossRef] [PubMed]

, 8

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003). [CrossRef]

, 9

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]

]. The surface plasmon resonance of the nanoparticles leads to interesting optical properties in near and far fields such as highly local-field confinement, dramatic local-field enhancement [10

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

], large scattering cross section, and high sensitivity to the surrounding medium [8

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003). [CrossRef]

, 9

, 11

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008). [CrossRef] [PubMed]

]. For a spherical nanoparticle, the resonance properties can be understood by a dipole model, and the higher-order terms are negligible [1

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

]. However, the contributions of higher-order terms become significant for a non-spherical nanoparticle due to the reduced symmetry. Higher-order surface plasmon resonance of metallic nanoparticles exhibits not only high field localization and field enhancement in the vicinity of metallic nanoparticles, but also high sensitivity to the variation of the surrounding medium as compared with the dipole term. A metallic nanoparticle of low symmetry shape exhibits higher-order resonance spectra, and the enhanced local field in the vicinity of the nanoparticle can be achieved around

10^{3} - 10^{4}

times of the incident field intensity [5

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]

, 12

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef] [PubMed]

Various computational methods based on solving Maxwell’s equations have been used for modeling the complex plasmon resonance of irregular-shaped metallic nanoparticles [5

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]

, 9

, 12

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef] [PubMed]

]. However, the optical property of each plasmon mode of the nanoparticle cannot be analyzed easily by these numerical methods as compared with an analytical method. Mie developed mathematical formulations to the scattering problems of a spherical sphere in 1908 [13

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908). [CrossRef]

]. Furthermore, Asano and Yamamoto generalized Mie solutions to a spheroid particle [14

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]

]. By considering electromagnetic field scattering of a particle of arbitrary shape, Erma derived generalized mathematical formulations by using a perturbation method [15

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]

]. The non-spherical surface of the particle is described by a surface characteristic function f. According to the analytical forms of the field coefficients, the higher-order perturbed terms are linear combinations of the lower-order terms and are related to the surface characteristic function. In other words, the higher-order plasmon modes can be controlled by considering an appropriate surface characteristic function of the particle. For large particles, only local dielectric functions are considered in the formulation, and the relation between the displacement

\overset{⃑}{D} (\overset{⃑}{r}, ω)

and the electric field

\overset{⃑}{E} (\overset{⃑}{r}, ω)

is described as

\overset{⃑}{D} (\overset{⃑}{r}, ω) = ε (\overset{⃑}{r}, ω) \overset{⃑}{E} (\overset{⃑}{r}, ω)

. The local dielectric function

ε (\overset{⃑}{r}, ω)

is usually described by the Drude model. However, when the size of particles is smaller than 10 nm (in radius), the quantum effects of the particle become non-negligible [16

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]

]. The earlier literature describes the quantum effects of the particle by using density functional theory [17

W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986). [CrossRef]

]. In that work, the particle size is limited to 1 nm because of the limitation in computation resources. To deal with larger nanoparticles, we use a different way to describe the nonlocal effects of the particle [16

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]

The particle with nonlocal optical response is described by

\overset{⃑}{D} (\overset{⃑}{r}, ω) = \int ε (\overset{⃑}{r}, {\overset{⃑}{r}}^{'}, ω) \overset{⃑}{E} ({\overset{⃑}{r}}^{'}, ω) d^{3} {\overset{⃑}{r}}^{'},

(1)

where

ε (\overset{⃑}{r}, {\overset{⃑}{r}}^{'}, ω)

is a nonlocal dielectric function of the particle. For a continuous medium with an infinite size, the translational invariance leads to a dielectric function of the form

ε (\overset{⃑}{r}, {\overset{⃑}{r}}^{'}, ω) = ε (\overset{⃑}{r} - {\overset{⃑}{r}}^{'}, ω)

. For this case, it is convenient to rewrite Eq. (1) in the momentum space as in the following form:

\overset{⃑}{D} (\overset{⃑}{k}, ω) = ε (\overset{⃑}{k}, ω) \overset{⃑}{E} (\overset{⃑}{k}, ω) .

(2)

From Maxwell’s equations, the electric field is discontinuous on the surface of the particle, and the electric field varies rapidly at this surface. Hence, the electric field has large

\overset{⃑}{k}

components, and the nonlocal optical effects of the particle play an important role at the surface of the particle [18

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

]. Since a smaller particle has a larger surface-to-volume ratio, surface effects must dominate, and nonlocal optical effects of the particle become apparent. In a local approximation, the dielectric function takes the form

ε (\overset{⃑}{r} - {\overset{⃑}{r}}^{'}, ω) = ε (\overset{⃑}{r}, ω) δ (\overset{⃑}{r} - {\overset{⃑}{r}}^{'})

. Namely, the Fourier transform of

ε (\overset{⃑}{r} - {\overset{⃑}{r}}^{'}, ω)

becomes independent of

\overset{⃑}{k}

. Rather than using the local approximation, we assume that the dielectric function of the particle depends not only on the frequency ω but also on the momentum

\overset{⃑}{k}

for

\overset{⃑}{r}

within the particle. This is again an approximation since

\overset{⃑}{k}

is no longer a good quantum number for a particle, where the translational invariance is not valid. However, such an approximation allows us to treat the nonlocal effect analytically. Two simple models such as the hydrodynamics model [19

G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979). [CrossRef]

] and the Lindhard model [20

J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).

] can be used to calculate the

\overset{⃑}{k}

-dependent dielectric function. On the other hand, one can obtain a correct nonlocal dielectric function of the particle by including all relevant electronic excitations [21

Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009). [CrossRef] [PubMed]

, 22

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002). [CrossRef]

], which however would not allow analytical formulations.

In order to deal with the scattering problems of spherical particles with nonlocal effect, Fuchs and Claro used the method of semi-classical infinite barrier (SCIB) to import the dependence of the momentum vector

\overset{⃑}{k}

for the dielectric function of the particle with the spherical [23

R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987). [CrossRef]

] or spherical shell [24

R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988). [CrossRef]

] boundary. Fictitious external charges at the surface of the particle were introduced, and thus the continuity to the normal components of the displacement is broken at the boundary of the particle. Furthermore, by using the Fourier transform method, the dependence of the momentum

\overset{⃑}{k}

can be included. Moreover, Leung and co-workers discussed the interaction between a molecule and a sphere [16

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]

] or a spherical shell [25

R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006). [CrossRef] [CrossRef]

, 26

R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006). [CrossRef] [CrossRef]

] with both hydrodynamics and Lindhard nonlocal functions. Xie et al. also investigated the energy transfer between a donor and an acceptor near a sphere or a spherical shell with the hydrodynamic nonlocal function [27

H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009). [CrossRef]

]. However, the method of SCIB is limited to electrostatics. When a high frequency incident wave is considered, the retardation effects must be included, and exact electrodynamics methods need to be used. Ruppin developed mathematical formulations for dealing with spherical particles with nonlocal optical response [28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

]. These formulations include not only transverse waves but also longitudinal waves inside the particle [18

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

]. However, the original Mie scattering theory only includes transverse waves inside the particle within local approximation. Moreover, the additional boundary condition (ABC) is needed [18

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

] once the

\overset{⃑}{k}

dependence is introduced for the dielectric function of the particle. This method for solving the scattering problems with nonlocal optical response is called “extended Mie scattering theory” [28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

]. Yannopapas used this theory to handle a two-dimensional array of spheres with nonlocal optical response by using both hydrodynamics and Lindhard nonlocal functions [29

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008). [CrossRef]

]. Schoonover et al. discussed some general approximate techniques for the calculation of fields scattered from nonlocal media [30

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005). [CrossRef]

]. Furthermore, the stratified sphere with nonlocal optical response has also been investigated by Moroz [31

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005). [CrossRef]

]. Moreover, the shapes of the particles such as triangles with nonlocal optical response under the hydrodynamics model have been reported by numerical methods such as the finite difference method [32

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009). [CrossRef] [PubMed]

In this work, we derive analytical solutions to a nearly spherical particle with nonlocal optical response. The extended Mie scattering theory [28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

] and the perturbation method [15

] are both used to calculate the complex surface plasmon modes. In local optics, the transverse surface plasmon modes are excited. However, not only transverse but also longitudinal surface plasmon modes will be excited due to the nonlocal effect. The paper is organized as follows: In Section 2, the mathematical descriptions of a particle of arbitrary shape are presented. In Section 3, the mathematical formulations for light scattering from a nearly spherical particle including both local and nonlocal dielectric functions are constructed, and analytical forms of complex surface plasmon modes are derived. Finally in Section 4, we give a summary and conclusion.

2. MATHEMATICAL DESCRIPTIONS OF A PARTICLE OF ARBITRARY SHAPE

The geometry of a particle of arbitrary shape is described in Fig. 1 . The surface of the particle (solid line in Fig. 1) is described by the following equation [15

r_{s} = a (1 + η f (θ, φ)),

(3)

where a is the radius of a sphere (dashed line in Fig. 1) that most closely resembles the particle, η is a small parameter for a nearly spherical particle, and

f (θ, φ)

is a surface characteristic function. In order to apply the perturbation method, the following mathematical constraint must be imposed:

| η f (θ, φ) | ⪡ 1.

(4)

According to Eq. (4), complex surface plasmon modes can be represented in analytical forms by using perturbation method. In the following context, we will first follow Erma’s work [15

] in the case of a homogeneous particle with a local dielectric function

ε (ω)

. Next we will generalize the situation to include nonlocal optical effects with the Fourier transform of the nonlocal dielectric function described by

ε (\overset{⃑}{k}, ω)

3. MATHEMATICAL FORMULATION FOR LIGHT SCATTERING

3A. Case (i): Local Optical Response

In this section, we establish the mathematical formulations for light scattering from a nearly spherical particle with (i) local and (ii) nonlocal optical responses by using the perturbation method.

This case has been established in the literature [15

], and we simply follow Erma’s work and add brief discussions. The physical model is shown in Fig. 2a . The whole space

(R^{3})

is divided into two regions. One is inside the particle (region 2) with a local dielectric function

ε_{2}

, and the other is outside the particle (region 1) with a local dielectric function

ε_{1}

. The boundary surface of the particle is denoted by S. The incident electric field is denoted

{\overset{⃑}{E}}_{i}

. The scattering (in region 1) and transverse (in region 2) fields are denoted by

{\overset{⃑}{E}}_{s}

and

{\overset{⃑}{E}}_{t}

, respectively. First, we write down two transverse vector spherical functions

\overset{⃑}{M}

and

\overset{⃑}{N}

that satisfy the vector Helmholtz equations (time dependence

\sim e^{- i ω t}

{\overset{⃑}{M}}_{n m} = z_{n} (ρ) {\overset{⃑}{m}}_{n m} (θ, φ),

{\overset{⃑}{N}}_{n m} = \frac{z_{n} (ρ)}{ρ} n (n + 1) o_{n m} (θ, φ) {\hat{e}}_{r} + \frac{1}{ρ} \frac{d [ρ z_{n} (ρ)]}{d ρ} {\overset{⃑}{n}}_{n m} (θ, φ),

(5)

where the three vector functions [dependent only on

(θ, φ)

] are defined via

{\overset{⃑}{m}}_{n m} (θ, φ) = e^{i m φ} [\frac{i m}{sin θ} P_{n}^{m} (cos θ) {\hat{e}}_{θ} - \frac{d P_{n}^{m} (cos θ)}{d θ} {\hat{e}}_{φ}],

{\overset{⃑}{n}}_{n m} (θ, φ) = e^{i m φ} [\frac{d P_{n}^{m} (cos θ)}{d θ} {\hat{e}}_{θ} + \frac{i m}{sin θ} P_{n}^{m} (cos θ) {\hat{e}}_{φ}],

o_{n m} (θ, φ) = P_{n}^{m} (cos θ) e^{i m φ} .

(6)

Here,

ρ_{i} = k_{i} r

(with

i = 1, 2

), where

k_{i}

is the wave vector in the region i.

z_{n}

is an appropriate spherical Bessel function, and

P_{n}^{m}

is an associated Legendre function. Unlike Erma’s work [15

], we choose a function with an exponential form

(\sim e^{i m φ})

in φ dependence. This has an advantage for obtaining more compact analytical forms. The orthogonal properties of the three vector functions

{\overset{⃑}{m}}_{n m}

{\overset{⃑}{n}}_{n m}

, and

o_{n m}

are

\int d Ω [\begin{array}{l} {\overset{⃑}{m}}_{n m} (θ, φ) \cdot {\overset{⃑}{m}}_{n^{'} m^{'}}^{*} (θ, φ) \\ {\overset{⃑}{n}}_{n m} (θ, φ) \cdot {\overset{⃑}{n}}_{n^{'} m^{'}}^{*} (θ, φ) \\ o_{n m} (θ, φ) o_{n^{'} m^{'}}^{*} (θ, φ) n (n + 1) \end{array}] = C_{n m} δ_{n n^{'}} δ_{m m^{'}},

(7)

where Ω is a solid angle, the notation

^{*}

denotes complex conjugate, and the normalized constant

C_{n m}

4 π {[n (n + 1)] / [2 n + 1]} [(n + m)! / (n - m)!]

Because both the scattering and transverse fields (

{\overset{⃑}{E}}_{s}

and

{\overset{⃑}{E}}_{t}

) are the solutions of the vector Helmholtz equations, the forms of these two fields can be written as the linear combinations of two vector spherical functions

\overset{⃑}{M}

and

\overset{⃑}{N}

{\overset{⃑}{E}}_{s} = \sum_{n, m} (a_{n m} {\overset{⃑}{M}}_{n m}^{s} + b_{n m} {\overset{⃑}{N}}_{n m}^{s}),

{\overset{⃑}{E}}_{t} = \sum_{n, m} (c_{n m} {\overset{⃑}{M}}_{n m}^{t} + d_{n m} {\overset{⃑}{N}}_{n m}^{t}),

(8)

and the corresponding magnetic fields are

{\overset{⃑}{H}}_{s} = \sum_{n, m} χ_{1} (a_{n m} {\overset{⃑}{N}}_{n m}^{s} + b_{n m} {\overset{⃑}{M}}_{n m}^{s}),

{\overset{⃑}{H}}_{t} = \sum_{n, m} χ_{2} (c_{n m} {\overset{⃑}{N}}_{n m}^{t} + d_{n m} {\overset{⃑}{M}}_{n m}^{t}),

(9)

where

χ_{i} \equiv c k_{i} {(i ω)}^{- 1}

i = 1, 2

, and the two superscripts s and t are associated with

z_{n} (ρ) = h_{n}^{(1)} (ρ)

and

z_{n} (ρ) = j_{n} (ρ)

in Eq. (5), respectively.

In region 1, the total fields are

{\overset{⃑}{E}}_{1} = {\overset{⃑}{E}}_{i} + {\overset{⃑}{E}}_{s}

and

{\overset{⃑}{H}}_{1} = {\overset{⃑}{H}}_{i} + {\overset{⃑}{H}}_{s}

. In region 2, the total fields are

{\overset{⃑}{E}}_{2} = {\overset{⃑}{E}}_{t}

and

{\overset{⃑}{H}}_{2} = {\overset{⃑}{H}}_{t}

. In order to obtain these four coefficients

(a_{n m}, b_{n m}, c_{n m}, d_{n m})

, we use the appropriate boundary conditions on the surface of the particle:

\hat{n} \times {\overset{⃑}{E}}_{1} = \hat{n} \times {\overset{⃑}{E}}_{2} ∣_{\overset{⃑}{r} ∊ S},

\hat{n} \times {\overset{⃑}{H}}_{1} = \hat{n} \times {\overset{⃑}{H}}_{2} ∣_{\overset{⃑}{r} ∊ S},

(10)

where

\hat{n}

is the unit vector perpendicular to the surface of the particle [15

]. Two boundary conditions for both electric and magnetic field identities are obtained:

the electric field $\overset{⃑}{E}$ :

$ρ_{1 s} {\overset{⃑}{E}}_{i t} (ρ_{1 s}) + \sum_{n, m} [a_{n m} ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}) {\overset{⃑}{m}}_{n m} + b_{n m} \frac{d (ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}))}{d ρ_{1 s}} {\overset{⃑}{n}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times {E_{i r} (ρ_{1 s}) + \sum_{n, m} [b_{n m} \frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} n (n + 1) o_{n m}]} = κ \sum_{n, m} {c_{n m} ρ_{2 s} j_{n} (ρ_{2 s}) {\overset{⃑}{m}}_{n m} + d_{n m} \frac{d (ρ_{2 s} j_{n} (ρ_{2 s}))}{d ρ_{2 s}} {\overset{⃑}{n}}_{n m}} + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{n, m} [d_{n m} \frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} n (n + 1) o_{n m}],$

(11)
the magnetic field $\overset{⃑}{H}$ :

$ρ_{1 s} {\overset{⃑}{H}}_{i t} (ρ_{1 s}) + \sum_{n, m} χ_{1} [a_{n m} \frac{d (ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}))}{d ρ_{1 s}} {\overset{⃑}{n}}_{n m} + b_{n m} ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}) {\overset{⃑}{m}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times {H_{i r} (ρ_{1 s}) + \sum_{n, m} [χ_{1} a_{n m} \frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} n (n + 1) o_{n m}]} = κ \sum_{n, m} χ_{2} {c_{n m} \frac{d (ρ_{2 s} j_{n} (ρ_{2 s}))}{d ρ_{2 s}} {\overset{⃑}{n}}_{n m} + d_{n m} ρ_{2 s} j_{n} (ρ_{2 s}) {\overset{⃑}{m}}_{n m}} + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{n, m} [χ_{2} c_{n m} \frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} n (n + 1) o_{n m}],$

(12)
where $ρ_{10} = k_{1} a$ , $ρ_{1 s} = k_{1} r_{s}$ , $ρ_{2 s} = k_{2} r_{s}$ , and $κ = k_{1} / k_{2}$ . $(E_{i r}, H_{i r})$ and $({\overset{⃑}{E}}_{i t}, {\overset{⃑}{H}}_{i t})$ denote the solutions to the radial and tangential electric/magnetic fields, respectively for the incident wave [15
V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]
].

Finally, we will solve the generalized forms of the four coefficients

(a_{n m}, b_{n m}, c_{n m}, d_{n m})

. We expand these four coefficients in a power series of the small parameter η given by

(a_{n m}, b_{n m}, c_{n m}, d_{n m}) = \sum_{j = 0}^{\infty} (a_{n m}^{j}, b_{n m}^{j}, c_{n m}^{j}, d_{n m}^{j}) η^{j},

(13)

where the superscript j denotes the

j th

-order coefficients in the expansion of η. The zero-order coefficients

(a_{n m}^{0}, b_{n m}^{0}, c_{n m}^{0}, d_{n m}^{0})

represent the solutions of the unperturbed system, and these coefficients can be found by the original Mie scattering theory. Moreover, we expand some functions in Eqs. (11, 12) in Taylor series at

r = a

ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} α_{n}^{j} ρ_{10}^{j} f^{j}}{j!}, α_{n}^{j} \equiv {\frac{d^{j} [ρ_{10} h_{n}^{(1)} (ρ)]}{d ρ^{j}} |}_{ρ_{10}},

ρ_{2 s} j_{n} (ρ_{2 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} β_{n}^{j} ρ_{20}^{j} f^{j}}{j!}, β_{n}^{j} \equiv {\frac{d^{j} [ρ j_{n} (ρ)]}{d ρ^{j}} |}_{ρ_{20}},

\frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} = \sum_{j = 0}^{\infty} \frac{η^{j} γ_{n}^{j} ρ_{10}^{j} f^{j}}{j!}, γ_{n}^{j} \equiv {\frac{d^{j}}{d ρ^{j}} [\frac{h_{n}^{(1)} (ρ)}{ρ}] |}_{ρ_{10}},

\frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} = \sum_{j = 0}^{\infty} \frac{η^{j} δ_{n}^{j} ρ_{20}^{j} f^{j}}{j!}, δ_{n}^{j} \equiv {\frac{d^{j}}{d ρ^{j}} [\frac{j_{n} (ρ)}{ρ}] |}_{ρ_{20}},

(14)

where

ρ_{20} = k_{2} a

ρ_{1 s} {\overset{⃑}{E}}_{i t} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(ρ {\overset{⃑}{E}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!},

{(ρ {\overset{⃑}{E}}_{i t})}^{(j)} \equiv {\frac{\partial^{j} (ρ {\overset{⃑}{E}}_{i t} (ρ))}{\partial ρ^{j}} |}_{ρ_{10}},

E_{i r} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(E_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{j!},

{(E_{i r})}^{(j)} \equiv \frac{\partial^{j}}{\partial ρ^{j}} E_{i r} (ρ) ∣_{ρ_{10}},

ρ_{1 s} {\overset{⃑}{H}}_{i t} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(ρ {\overset{⃑}{H}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!}, {(ρ {\overset{⃑}{H}}_{i t})}^{(j)} \equiv {\frac{\partial^{j} (ρ {\overset{⃑}{H}}_{i t} (ρ))}{\partial ρ^{j}} |}_{ρ_{10}},

H_{i r} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(H_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{j!}, {(H_{i r})}^{(j)} \equiv \frac{\partial^{j}}{\partial ρ^{j}} H_{i r} (ρ) ∣_{ρ_{10}} .

(15)

Substituting Eqs. (13, 14, 15) into Eqs. (11, 12) leads to the following equations for the four coefficients

(a_{n m}^{j}, b_{n m}^{j}, c_{n m}^{j}, d_{n m}^{j})

(\begin{matrix} α_{n}^{0} & 0 & - κ β_{n}^{0} & 0 \\ χ_{1} α_{n}^{1} & 0 & - κ χ_{2} β_{n}^{1} & 0 \\ 0 & α_{n}^{1} & 0 & - κ β_{n}^{1} \\ 0 & χ_{1} α_{n}^{0} & 0 & - κ χ_{2} β_{n}^{0} \end{matrix}) (\begin{matrix} a_{n m}^{j} \\ b_{n m}^{j} \\ c_{n m}^{j} \\ d_{n m}^{j} \end{matrix}) = \frac{1}{C_{n m}} \int d Ω (\begin{matrix} {\overset{⃑}{A}}_{j} \cdot {\overset{⃑}{m}}_{n m}^{*} \\ {\overset{⃑}{B}}_{j} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{A}}_{j} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{B}}_{j} \cdot {\overset{⃑}{m}}_{n m}^{*} \end{matrix}),

(16)

where

({\overset{⃑}{A}}_{j}, {\overset{⃑}{B}}_{j})

are given by

{\overset{⃑}{A}}_{j} = κ \sum_{q = 0}^{j - 1} \sum_{n, m} [\frac{c_{n m}^{q} β_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m} + \frac{d_{n m}^{q} β_{n}^{j - q + 1} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m}] + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{j - 1} \sum_{n, m} [\frac{d_{n m}^{q} δ_{n}^{j - 1 - q} ρ_{20}^{j - 1 - q} f^{j - 1 - q}}{(j - 1 - q)!} n (n + 1) o_{n m}] - \sum_{q = 0}^{j - 1} \sum_{n, m} [\frac{a_{n m}^{q} α_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m} + \frac{b_{n m}^{q} α_{n}^{j - q + 1} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m}] - \frac{{(ρ {\overset{⃑}{E}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!} - (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(E_{i r})}^{(j - 1)} ρ_{10}^{j} f^{j - 1}}{(j - 1)!} + \sum_{q = 0}^{j - 1} \sum_{n, m} \frac{b_{n m}^{q} γ_{n}^{j - 1 - q} ρ_{10}^{j - q} f^{j - 1 - q}}{(j - 1 - q)!} n (n + 1) o_{n m}],

(17)

{\overset{⃑}{B}}_{j} = κ \sum_{q = 0}^{j - 1} \sum_{n, m} χ_{2} [\frac{c_{n m}^{q} β_{n}^{j - q + 1} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m} + \frac{d_{n m}^{q} β_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m}] + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{j - 1} \sum_{n, m} χ_{2} \frac{c_{n m}^{q} δ_{n}^{j - 1 - q} ρ_{20}^{j - 1 - q} f^{j - 1 - q}}{(j - 1 - q)!} n (n + 1) o_{n m} - \frac{{(ρ {\overset{⃑}{H}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!} - (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(H_{i r})}^{(j - 1)} ρ_{10}^{j} f^{j - 1}}{(j - 1)!} + χ_{1} \sum_{q = 0}^{j - 1} \sum_{n, m} \frac{a_{n m}^{q} γ_{n}^{j - 1 - q} ρ_{10}^{j - q} f^{j - 1 - q}}{(j - 1 - q)!} n (n + 1) o_{n m}] - \sum_{q = 0}^{j - 1} \sum_{n, m} χ_{1} [\frac{a_{n m}^{q} α_{n}^{j - q + 1} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m} + \frac{b_{n m}^{q} α_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m}] .

(18)

Using Eq. (15), we obtain these four coefficients

(a_{n m}^{j}, b_{n m}^{j}, c_{n m}^{j}, d_{n m}^{j})

(\begin{matrix} a_{n m}^{j} \\ b_{n m}^{j} \\ c_{n m}^{j} \\ d_{n m}^{j} \end{matrix}) = \frac{1}{C_{n m}} \int d Ω {(\begin{matrix} α_{n}^{0} & 0 & - κ β_{n}^{0} & 0 \\ χ_{1} α_{n}^{1} & 0 & - κ χ_{2} β_{n}^{1} & 0 \\ 0 & α_{n}^{1} & 0 & - κ β_{n}^{1} \\ 0 & χ_{1} α_{n}^{0} & 0 & - κ χ_{2} β_{n}^{0} \end{matrix})}^{- 1} (\begin{matrix} {\overset{⃑}{A}}_{j} \cdot {\overset{⃑}{m}}_{n m}^{*} \\ {\overset{⃑}{B}}_{j} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{A}}_{j} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{B}}_{j} \cdot {\overset{⃑}{m}}_{n m}^{*} \end{matrix}),

(19)

where

j ∊ N

. Hence, the general solutions to the coefficients for each order of η are solved, and we obtain the four coefficients

(a_{n m}, b_{n m}, c_{n m}, d_{n m})

by using Eq. (13). In the following case, we will discuss the scattering from a nearly spherical particle with nonlocal optical response.

3B. Case (ii): Nonlocal Optical Response

In this case, we need to consider the longitudinal surface plasmon modes [18

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

, 28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

, 31

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005). [CrossRef]

]. Unlike the original Mie scattering theory, it allows not only transverse waves but also longitudinal waves to propagate inside a particle. The transverse and longitudinal waves inside the particle satisfy different dispersion relations [28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

ε_{t} (k_{t}, ω) = \frac{c^{2}}{ω^{2}} k_{t}^{2},

(20)

ε_{ℓ} (k_{ℓ}, ω) = 0,

(21)

where

ε_{t} (k_{t}, ω)

and

ε_{ℓ} (k_{ℓ}, ω)

denote the transverse and longitudinal dielectric functions of the particle, respectively.

{\overset{⃑}{k}}_{t}

and

{\overset{⃑}{k}}_{ℓ}

are complex wave-vector roots to Eqs. (20, 21) inside the particle. Furthermore, we introduce the longitudinal vector spherical function

{\overset{⃑}{L}}_{n m}

that satisfies

\nabla \times {\overset{⃑}{L}}_{n m} = 0

and has the following form:

{\overset{⃑}{L}}_{n m} = k \frac{d z_{n} (ρ)}{d ρ} o_{n m} (θ, φ) {\hat{e}}_{r} + k \frac{z_{n} (ρ)}{ρ} {\overset{⃑}{n}}_{n m} (θ, φ),

(22)

and the other functions have been defined for transverse vector spherical functions.

The physical model is shown in Fig. 2b. When we consider both longitudinal and transverse surface plasmon modes, the interior electric field

{\overset{⃑}{E}}_{2}

is the sum of transverse

({\overset{⃑}{E}}_{t})

and longitudinal

({\overset{⃑}{E}}_{ℓ})

electric fields:

{\overset{⃑}{E}}_{2} = {\overset{⃑}{E}}_{t} + {\overset{⃑}{E}}_{ℓ} = \sum_{n, m} (c_{n m} {\overset{⃑}{M}}_{n m}^{t} + d_{n m} {\overset{⃑}{N}}_{n m}^{t} + e_{n m} {\overset{⃑}{L}}_{n m}^{ℓ}),

(23)

where the superscript ℓ denotes the longitudinal modes, which is associated with the wave vector

{\overset{⃑}{k}}_{ℓ}

(not

{\overset{⃑}{k}}_{2}

). The electric field outside the particle is

{\overset{⃑}{E}}_{1} = {\overset{⃑}{E}}_{i} + {\overset{⃑}{E}}_{s}

, where

{\overset{⃑}{E}}_{s}

is given by Eq. (8). Since the longitudinal vector spherical function satisfies

\nabla \times \overset{⃑}{L} = 0

, the magnetic fields inside and outside the particle are still treated the same way as in case (i). In order to obtain the five coefficients

(a_{n m}, b_{n m}, c_{n m}, d_{n m}, e_{n m})

, we use the boundary conditions in Eq. (10) and the ABC [28

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

\hat{n} \cdot \frac{\partial}{\partial t} {\overset{⃑}{E}}_{1} ∣_{\overset{⃑}{r} ∊ S} = \hat{n} \cdot \frac{\partial}{\partial t} {\overset{⃑}{E}}_{2} ∣_{\overset{⃑}{r} ∊ S} .

(24)

Namely, the normal component of the displacement current is continuous at the boundary S of the particle. Note that the ABC exists only when the nonlocal effect is introduced. Hence we get three boundary conditions:

the electric field $\overset{⃑}{E}$ :

$ρ_{1 s} {\overset{⃑}{E}}_{i t} (ρ_{1 s}) + \sum_{n, m} [a_{n m} ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}) {\overset{⃑}{m}}_{n m} + b_{n m} {\frac{d (ρ h_{n}^{(1)} (ρ))}{d ρ} |}_{ρ_{1 s}} {\overset{⃑}{n}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times {E_{i r} (ρ_{1 s}) + \sum_{n, m} [b_{n m} \frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} n (n + 1) o_{n m}]} = κ \sum_{n, m} {c_{n m} ρ_{2 s} j_{n} (ρ_{2 s}) {\overset{⃑}{m}}_{n m} + [d_{n m} {\frac{d (ρ j_{n} (ρ))}{d ρ} |}_{ρ_{2 s}} + e_{n m} ρ_{2 s} j_{n} (ρ_{3 s})] {\overset{⃑}{n}}_{n m}} + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times \sum_{n, m} [d_{n m} \frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} n (n + 1) + e_{n m} {\frac{d (j_{n} (ρ))}{d ρ} |}_{ρ_{3 s}} k_{ℓ}] o_{n m},$

(25)
the magnetic field $\overset{⃑}{H}$ :

$ρ_{1 s} {\overset{⃑}{H}}_{i t} (ρ_{1 s}) + \sum_{n, m} χ_{1} [a_{n m} {\frac{d (ρ h_{n}^{(1)} (ρ))}{d ρ} |}_{ρ_{1 s}} {\overset{⃑}{n}}_{n m} + b_{n m} ρ_{1 s} h_{n}^{(1)} (ρ_{1 s}) {\overset{⃑}{m}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times {H_{i r} (ρ_{1 s}) + \sum_{n, m} [χ_{1} a_{n m} \frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} n (n + 1) o_{n m}]} = κ \sum_{n, m} χ_{2} {c_{n m} {\frac{d (ρ j_{n} (ρ))}{d ρ} |}_{ρ_{2 s}} {\overset{⃑}{n}}_{n m} + d_{n m} ρ_{2 s} j_{n} (ρ_{2 s}) {\overset{⃑}{m}}_{n m}} + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{n, m} [χ_{2} c_{n m} \frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} n (n + 1) o_{n m}],$

(26)
from ABC:

$ρ_{1 s} {E_{i r} (ρ_{1 s}) + \sum_{n, m} [b_{n m} \frac{h_{n}^{(1)} (ρ_{1 s})}{ρ_{1 s}} n (n + 1) o_{n m}]} - η \frac{\partial f}{\partial θ} ρ_{10} {E_{i θ} + \sum_{n, m} [a_{n m} h_{n}^{(1)} (ρ_{1 s}) u_{n m} + b_{n m} {\frac{d (ρ h_{n}^{(1)} (ρ))}{ρ d ρ} |}_{ρ_{1 s}} t_{n m}]} - \frac{η}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} {E_{i φ} - \sum_{n, m} [a_{n m} h_{n}^{(1)} (ρ_{1 s}) t_{n m} - b_{n m} {\frac{d (ρ h_{n}^{(1)} (ρ))}{ρ d ρ} |}_{ρ_{1 s}} u_{n m}]} = \sum_{n, m} {ρ_{1 s} [d_{n m} \frac{j_{n} (ρ_{2 s})}{ρ_{2 s}} n (n + 1) + e_{n m} {\frac{d (j_{n} (ρ))}{d ρ} |}_{ρ_{3 s}} k_{ℓ}] o_{n m} - η \frac{\partial f}{\partial θ} ρ_{10} [c_{n m} j_{n} (ρ_{2 s}) u_{n m} + (d_{n m} {\frac{d (ρ j_{n} (ρ))}{ρ d ρ} |}_{ρ_{2 s}} + e_{n m} \frac{j_{n} (ρ_{3 s})}{ρ_{3 s}} k_{ℓ}) t_{n m}] - \frac{η}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} [- c_{n m} j_{n} (ρ_{2 s}) t_{n m} + (d_{n m} {\frac{d (ρ j_{n} (ρ))}{ρ d ρ} |}_{ρ_{2 s}} + e_{n m} \frac{j_{n} (ρ_{3 s})}{ρ_{3 s}} k_{ℓ}) u_{n m}]},$

(27)
where $ρ_{1 s} = k_{1} r_{s}$ , $ρ_{2 s} = k_{2} r_{s}$ , $ρ_{3 s} = k_{ℓ} r_{s}$ [ $k_{1}$ is the magnitude of the incident wave vector, $k_{2}$ is the complex root to Eq. (20), and $k_{ℓ}$ is the complex root to Eq. (21)], $E_{i (θ, φ)}$ denotes the components of electric field in the ${\hat{e}}_{(θ, φ)}$ direction for the incident wave,

$t_{n m} = \frac{d P_{n}^{m} (cos θ)}{d θ} e^{i m φ} and u_{n m} = \frac{i m P_{n}^{m} (cos θ)}{\sin θ} e^{i m φ} .$

Using the same methods as in case (i), we also write the five coefficients $(a_{n m}, b_{n m}, c_{n m}, d_{n m}, e_{n m})$ in power series of η:

$(a_{n m}, b_{n m}, c_{n m}, d_{n m}, e_{n m}) = \sum_{j = 0}^{\infty} (a_{n m}^{j}, b_{n m}^{j}, c_{n m}^{j}, d_{n m}^{j}, e_{n m}^{j}) η^{j},$

(28)
where the zeroth-order coefficients $(a_{n m}^{0}, b_{n m}^{0}, c_{n m}^{0}, d_{n m}^{0}, e_{n m}^{0})$ represent the solutions for a sphere with optical nonlocal response, and we can get these coefficients by following Ruppin’s work [28
R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]
]. We expand some functions in Eqs. (25, 26, 27) that have not been presented in Eqs. (14, 15) as the Taylor series at $r = a$ :

$j_{n} (ρ_{3 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} Δ_{n}^{j} ρ_{30}^{j} f^{j}}{j!}, Δ_{n}^{j} \equiv \frac{d^{j}}{d ρ^{j}} [j_{n} (ρ)] ∣_{ρ_{30}},$

$ρ_{3 s} j_{n} (ρ_{3 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} ξ_{n}^{j} ρ_{30}^{j} f^{j}}{j!}, ξ_{n}^{j} \equiv \frac{d^{j}}{d ρ^{j}} [ρ_{30} j_{n} (ρ)] ∣_{ρ_{30}},$

$h_{n}^{(1)} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} ζ_{n}^{j} ρ_{10}^{j} f^{j}}{j!}, ζ_{n}^{j} \equiv \frac{d^{j}}{d ρ^{j}} [h_{1}^{(1)} (ρ)] ∣_{ρ_{10}},$

$j_{n} (ρ_{2 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} Ω_{n}^{j} ρ_{20}^{j} f^{j}}{j!}, Ω_{n}^{j} \equiv \frac{d^{j}}{d ρ^{j}} [j_{n} (ρ)] ∣_{ρ_{20}},$

$\frac{j_{n} (ρ_{3 s})}{ρ_{3 s}} = \sum_{j = 0}^{\infty} \frac{η^{j} \nabla_{n}^{j} ρ_{30}^{j} f^{j}}{j!}, \nabla_{n}^{j} \equiv {\frac{d^{j}}{d ρ^{j}} [\frac{j_{n} (ρ)}{ρ}] |}_{ρ_{30}},$

(29)
where $ρ_{30} = k_{ℓ} a$ ,

$ρ_{1 s} E_{i r} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(ρ E_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{p!},$

${(ρ E_{i r})}^{(j)} \equiv \frac{\partial^{j}}{\partial ρ^{j}} (ρ_{10} E_{i r} (ρ)) ∣_{ρ_{10}},$

$E_{i θ} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(E_{i θ})}^{(j)} ρ_{10}^{j} f^{j}}{p!}, {(E_{i θ})}^{(j)} \equiv \frac{\partial^{j}}{\partial ρ^{j}} E_{i θ} (ρ) ∣_{ρ_{10}},$

$E_{i φ} (ρ_{1 s}) = \sum_{j = 0}^{\infty} \frac{η^{j} {(E_{i φ})}^{(j)} ρ_{10}^{j} f^{j}}{j!}, {(E_{i φ})}^{(j)} \equiv \frac{\partial^{j}}{\partial ρ^{j}} E_{i φ} (ρ) ∣_{ρ_{10}},$

(30)
Next we substitute Eqs. (14, 15, 28, 29, 30) into Eqs. (25, 26, 27) and get the following identities:

$\sum_{j = 0}^{\infty} \frac{η^{j} {(ρ {\overset{⃑}{E}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} [\frac{a_{n m}^{q} α_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m} + \frac{b_{n m}^{q} α_{n}^{j - q + 1} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{j = 0}^{\infty} {η^{j} [\frac{{(E_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + \sum_{q = 0}^{j} \sum_{n, m} \frac{b_{n m}^{q} γ_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) o_{n m}]} = κ \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} {\frac{c_{n m}^{q} β_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m} + [\frac{d_{n m}^{q} β_{n}^{j - q + 1} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} + \frac{k_{2}}{k_{ℓ}} \frac{e_{n m}^{q} ξ_{n}^{j - q} ρ_{30}^{j - q} f^{j - q}}{(j - q)!}] {\overset{⃑}{n}}_{n m}} + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} [\frac{d_{n m}^{q} δ_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) + \frac{e_{n m}^{q} Δ_{n}^{j - q + 1} ρ_{30}^{j - q} f^{j - q}}{(j - q)!}] o_{n m},$

(31)

$\sum_{j = 0}^{\infty} \frac{η^{j} {(ρ {\overset{⃑}{H}}_{i t})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} χ_{1} η^{j} [\frac{a_{n m}^{q} α_{n}^{j - q + 1} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m} + \frac{b_{n m}^{q} α_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \times \sum_{j = 0}^{\infty} {η^{j} [\frac{{(H_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + χ_{1} \sum_{q = 0}^{j} \sum_{n, m} \frac{a_{n m}^{q} γ_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) o_{n m}]} = κ \sum_{p = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} χ_{2} η^{j} [\frac{c_{n m}^{q} β_{n}^{j - q + 1} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{n}}_{n m} + \frac{d_{n m}^{q} β_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} {\overset{⃑}{m}}_{n m}] + η ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} χ_{2} η^{j} \frac{c_{n m}^{q} δ_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) o_{n m},$

(32)

$\sum_{j = 0}^{\infty} \frac{η^{j} {(ρ E_{i r})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} \frac{b_{n m}^{q} ζ_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) o_{n m} - η \frac{\partial f}{\partial θ} ρ_{10} [\sum_{j = 0}^{\infty} \frac{η^{j} {(E_{i θ})}^{(j)} ρ_{10}^{j} f^{j}}{j!} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} \frac{a_{n m}^{q} ζ_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} u_{n m} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} \frac{b_{n m}^{q} (γ_{n}^{j - q} + ζ_{n}^{j - q + 1}) ρ_{10}^{j - q} f^{j - q}}{(j - q)!} t_{n m}] - \frac{η}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} [\sum_{j = 0}^{\infty} \frac{η^{j} {(E_{i φ})}^{(j)} ρ_{10}^{j} f^{j}}{j!} - \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} \frac{a_{n m}^{q} ζ_{n}^{j - q} ρ_{10}^{j - q} f^{j - q}}{(j - q)!} t_{n m} + \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} \frac{b_{n m}^{q} (γ_{n}^{j - q} + ζ_{n}^{j - q + 1}) ρ_{10}^{j - q} f^{j - q}}{(j - q)!} u_{n m}] = \sum_{j = 0}^{\infty} \sum_{q = 0}^{j} \sum_{n, m} η^{j} {\frac{k_{1}}{k_{2}} \frac{d_{n m}^{j} Ω_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} n (n + 1) o_{n m} + k_{1} \frac{e_{n m}^{q} (ξ_{n}^{j - q + 1} + Δ_{n}^{j - q}) ρ_{30}^{j - q} f^{j - q}}{(j - q)!} o_{n m} - η \frac{\partial f}{\partial θ} ρ_{10} [\frac{c_{n m}^{q} Ω_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} u_{n m} + \frac{d_{n m}^{q} (δ_{n}^{j - q} + Ω_{n}^{j - q + 1}) ρ_{20}^{j - q} f^{j - q}}{(j - q)!} t_{m n} + \frac{e_{n m}^{q} \nabla_{n}^{j - q} ρ_{30}^{j - q} f^{j - q}}{(j - q)!} k_{ℓ} t_{n m}] - \frac{η}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} [\frac{- c_{n m}^{q} Ω_{n}^{j - q} ρ_{20}^{j - q} f^{j - q}}{(j - q)!} t_{n m} + \frac{d_{n m}^{q} (δ_{n}^{j - q} + Ω_{n}^{j - q + 1}) ρ_{20}^{j - q} f^{j - q}}{(j - q)!} u_{n m} + \frac{e_{n m}^{q} \nabla_{n}^{j - q} ρ_{30}^{j - q} f^{j - q}}{(j - q)!} k_{ℓ} u_{n m}]} .$

(33)
Since each order of the small parameter η in Eqs. (31, 32, 33) can form a set of bases, that is, ${1, η, η^{2}, η^{3}, \dots}$ , the coefficients for any given order of η must be equal on both sides of these equations. Hence we write the identities for order $η^{p}$ (the superscript p is arbitrary) as

$\frac{{(ρ {\overset{⃑}{E}}_{i t})}^{(p)} ρ_{10}^{p} f^{p}}{p!} + \sum_{q = 0}^{p} \sum_{n, m} [\frac{a_{n m}^{q} α_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m} + \frac{b_{n m}^{q} α_{n}^{p - q + 1} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m}] + (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(E_{i r})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} γ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m}] = κ \sum_{q = 0}^{p} \sum_{n, m} {\frac{c_{n m}^{q} β_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m} + [\frac{d_{n m}^{q} β_{n}^{p - q + 1} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} + \frac{k_{2}}{k_{ℓ}} \frac{e_{n m}^{q} ξ_{n}^{p - q} ρ_{30}^{p - q} f^{p - q}}{(p - q)!}] {\overset{⃑}{n}}_{n m}} + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{p - 1} \sum_{n, m} [\frac{d_{n m}^{q} δ_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) + \frac{e_{n m}^{q} Δ_{n}^{p - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!}] o_{n m},$

(34)

$\frac{{(ρ {\overset{⃑}{H}}_{i t})}^{(p)} ρ_{10}^{p} f^{p}}{p!} + \sum_{q = 0}^{p} \sum_{n, m} χ_{1} [\frac{a_{n m}^{q} α_{n}^{p - q + 1} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m} + \frac{b_{n m}^{q} α_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m}] + (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(H_{i r})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + χ_{1} \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} γ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m}] = κ \sum_{q = 0}^{p} \sum_{n, m} χ_{2} [\frac{c_{n m}^{q} β_{n}^{p - q + 1} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m} + \frac{d_{n m}^{q} β_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m}] + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{p - 1} \sum_{n, m} χ_{2} \frac{c_{n m}^{q} δ_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m},$

(35)

$\frac{{(ρ E_{i r})}^{(p)} ρ_{10}^{p} f^{p}}{p!} + \sum_{q = 0}^{p} \sum_{n, m} \frac{b_{n m}^{q} ζ_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} n (n + 1) o_{n m} - \frac{\partial f}{\partial θ} [\frac{{(E_{i θ})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} ζ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} (γ_{n}^{p - 1 - q} + ζ_{n}^{p - q}) ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m}] - \frac{1}{sin θ} \frac{\partial f}{\partial φ} [\frac{{(E_{i φ})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} - \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} ζ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} (γ_{n}^{p - 1 - q} + ζ_{n}^{p - q}) ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m}] = \sum_{q = 0}^{p} \sum_{n, m} [\frac{k_{1}}{k_{2}} \frac{d_{n m}^{q} Ω_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} n (n + 1) + k_{1} \frac{e_{n m}^{q} (ξ_{n}^{p - q + 1} + Δ_{n}^{p - q}) ρ_{30}^{p - q} f^{p - q}}{(p - q)!}] o_{n m} - \frac{\partial f}{\partial θ} ρ_{10} \sum_{q = 0}^{p - 1} \sum_{n, m} {[\frac{c_{n m}^{q} Ω_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \frac{d_{n m}^{q} (δ_{n}^{p - 1 - q} + Ω_{n}^{p - q}) ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \frac{e_{n m}^{q} \nabla_{n}^{p - 1 - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} k_{ℓ} t_{n m}] - \frac{1}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} [\frac{- c_{n m}^{q} Ω_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \frac{d_{n m}^{q} (δ_{n}^{p - 1 - q} + Ω_{n}^{p - q}) ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \frac{e_{n m}^{q} \nabla_{n}^{p - 1 - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} k_{ℓ} u_{n m}]} .$

(36)
Furthermore, we rewrite the forms of Eqs. (34, 35, 36) as

$\sum_{n, m} [(a_{n m}^{p} α_{n}^{0} - κ c_{n m}^{p} β_{n}^{0}) {\overset{⃑}{m}}_{n m} + (b_{n m}^{p} α_{n}^{1} - κ d_{n m}^{p} β_{n}^{1} - k_{1} e_{n m}^{p} Δ_{n}^{0}) {\overset{⃑}{n}}_{n m}] = {\overset{⃑}{A}}_{p},$

(37)

$\sum_{n, m} [(χ_{1} a_{n m}^{p} α_{n}^{1} - κ χ_{2} c_{n m}^{p} β_{n}^{1}) {\overset{⃑}{n}}_{n m} + (χ_{1} b_{n m}^{p} α_{n}^{0} - κ χ_{2} d_{n m}^{p} β_{n}^{0}) {\overset{⃑}{m}}_{n m}] = {\overset{⃑}{B}}_{p},$

(38)

$\sum_{n, m} (b_{n m}^{p} ζ_{n}^{0} - \frac{k_{1}}{k_{2}} d_{n m}^{p} Ω_{n}^{0}) n (n + 1) o_{n m} - \sum_{n, m} k_{1} e_{n m}^{p} (ξ_{n}^{1} + Δ_{n}^{0}) o_{n m} = C_{p},$

(39)
where ${\overset{⃑}{A}}_{p}$ , ${\overset{⃑}{B}}_{p}$ , and $C_{p}$ are

${\overset{⃑}{A}}_{p} = κ \sum_{q = 0}^{p - 1} \sum_{n, m} {\frac{c_{n m}^{q} β_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m} + [\frac{d_{n m}^{q} β_{n}^{p - q + 1} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} + \frac{k_{2}}{k_{ℓ}} \frac{e_{n m}^{q} ξ_{n}^{p - q} ρ_{30}^{p - q} f^{p - q}}{(p - q)!}] {\overset{⃑}{n}}_{n m}} + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{p - 1} \sum_{n, m} [\frac{d_{n m}^{q} δ_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) + \frac{e_{n m}^{q} Δ_{n}^{p - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!}] o_{n m} - \sum_{q = 0}^{p - 1} \sum_{n, m} [\frac{a_{n m}^{q} α_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m} + \frac{b_{n m}^{q} α_{n}^{p - q + 1} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m}] - \frac{{(ρ {\overset{⃑}{E}}_{i t})}^{(p)} ρ_{10}^{p} f^{p}}{p!} - (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(E_{i r})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} γ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m}],$

(40)

${\overset{⃑}{B}}_{p} = κ \sum_{q = 0}^{p - 1} \sum_{n, m} χ_{2} [\frac{c_{n m}^{q} β_{n}^{p - q + 1} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m} + \frac{d_{n m}^{q} β_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m}] + ρ_{10} (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) \sum_{q = 0}^{p - 1} \sum_{n, m} χ_{2} \frac{c_{n m}^{q} δ_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m} - \frac{{(ρ {\overset{⃑}{H}}_{i t})}^{(p)} ρ_{10}^{p} f^{p}}{p!} - (\frac{\partial f}{\partial θ} {\hat{e}}_{θ} + \frac{1}{sin θ} \frac{\partial f}{\partial φ} {\hat{e}}_{φ}) [\frac{{(H_{i r})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + χ_{1} \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} γ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} n (n + 1) o_{n m}] - \sum_{q = 0}^{p - 1} \sum_{n, m} χ_{1} [\frac{a_{n m}^{q} α_{n}^{p - q + 1} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{n}}_{n m} + \frac{b_{n m}^{q} α_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} {\overset{⃑}{m}}_{n m}],$

(41)

$C_{p} = \sum_{q = 0}^{p - 1} \sum_{n, m} [\frac{k_{1}}{k_{2}} \frac{d_{n m}^{q} Ω_{n}^{p - q} ρ_{20}^{p - q} f^{p - q}}{(p - q)!} n (n + 1) + k_{1} \frac{e_{n m}^{q} (ξ_{n}^{p - q + 1} + Δ_{n}^{p - q}) ρ_{30}^{p - q} f^{p - q}}{(p - q)!}] o_{n m} - \frac{\partial f}{\partial θ} ρ_{10} \sum_{q = 0}^{p - 1} \sum_{n, m} {[\frac{c_{n m}^{q} Ω_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \frac{d_{n m}^{q} (δ_{n}^{p - 1 - q} + Ω_{n}^{p - q}) ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \frac{e_{n m}^{q} \nabla_{n}^{p - 1 - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} k_{ℓ} t_{n m}] - \frac{1}{sin θ} \frac{\partial f}{\partial φ} ρ_{10} [\frac{- c_{n m}^{q} Ω_{n}^{p - 1 - q} ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \frac{d_{n m}^{q} (δ_{n}^{p - 1 - q} + Ω_{n}^{p - q}) ρ_{20}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \frac{e_{n m}^{q} \nabla_{n}^{p - 1 - q} ρ_{30}^{p - 1 - q} f^{p - 1 - q}}{(p - 1 - q)!} k_{ℓ} u_{n m}]} - \frac{{(ρ E_{i r})}^{(p)} ρ_{10}^{p} f^{p}}{p!} - \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} ζ_{n}^{p - q} ρ_{10}^{p - q} f^{p - q}}{(p - q)!} n (n + 1) o_{n m} + \frac{\partial f}{\partial θ} [\frac{{(E_{i θ})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} ζ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} (γ_{n}^{p - 1 - q} + ζ_{n}^{p - q}) ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m}] + \frac{1}{sin θ} \frac{\partial f}{\partial φ} [\frac{{(E_{i φ})}^{(p - 1)} ρ_{10}^{p} f^{p - 1}}{(p - 1)!} - \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{a_{n m}^{q} ζ_{n}^{p - 1 - q} ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} t_{n m} + \sum_{q = 0}^{p - 1} \sum_{n, m} \frac{b_{n m}^{q} (γ_{n}^{p - 1 - q} + ζ_{n}^{p - q}) ρ_{10}^{p - q} f^{p - 1 - q}}{(p - 1 - q)!} u_{n m}] .$

(42)
After some complicated calculation for Eqs. (37, 38, 39), the simultaneous equations of these four coefficients $(a_{n m}^{p}, b_{n m}^{p}, c_{n m}^{p}, d_{n m}^{p})$ are given by the matrix form

$(\begin{matrix} α_{n}^{0} & 0 & - κ β_{n}^{0} & 0 \\ χ_{1} α_{n}^{1} & 0 & - κ χ_{2} β_{n}^{1} & 0 \\ 0 & α_{n}^{1} & 0 & - κ β_{n}^{1} \\ 0 & χ_{1} α_{n}^{0} & 0 & - κ χ_{2} β_{n}^{0} \end{matrix}) (\begin{matrix} a_{n m}^{p} \\ b_{n m}^{p} \\ c_{n m}^{p} \\ d_{n m}^{p} \end{matrix}) = \frac{1}{C_{n m}} \int d Ω (\begin{array}{l} {\overset{⃑}{A}}_{p} \cdot {\overset{⃑}{m}}_{n m}^{*} \\ {\overset{⃑}{B}}_{p} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{A}}_{p} \cdot {\overset{⃑}{n}}_{n m}^{*} \\ {\overset{⃑}{B}}_{p} \cdot {\overset{⃑}{m}}_{n m}^{*} \end{array}) + (\begin{matrix} 0 \\ 0 \\ k_{1} e_{n m}^{p} Δ_{n}^{0} \\ 0 \end{matrix}),$

(43)
and hence the following equation is obtained:

$(\begin{matrix} a_{n m}^{p} \\ b_{n m}^{p} \\ c_{n m}^{p} \\ d_{n m}^{p} \end{matrix}) = \frac{1}{C_{n m}} \int d Ω (\begin{matrix} \frac{- κ χ_{2} β_{n}^{1}}{- κ χ_{2} β_{n}^{1} α_{n}^{0} + κ β_{n}^{0} χ_{1} α_{n}^{1}} & \frac{κ β_{n}^{0}}{- κ χ_{2} β_{n}^{1} α_{n}^{0} + κ β_{n}^{0} χ_{1} α_{n}^{1}} & 0 & 0 \\ 0 & 0 & \frac{- κ χ_{2} β_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} & \frac{κ β_{n}^{1}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \\ \frac{- χ_{1} α_{n}^{1}}{- κ χ_{2} β_{n}^{1} α_{n}^{0} + κ β_{n}^{0} χ_{1} α_{n}^{1}} & \frac{α_{n}^{0}}{- κ χ_{2} β_{n}^{1} α_{n}^{0} + κ β_{n}^{0} χ_{1} α_{n}^{1}} & 0 & 0 \\ 0 & 0 & \frac{- χ_{1} α_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} & \frac{α_{n}^{1}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \end{matrix}) \times (\begin{matrix} {\bar{A}}_{p} \cdot {\bar{m}}_{n m}^{*} \\ {\bar{B}}_{p} \cdot {\bar{n}}_{n m}^{*} \\ {\bar{A}}_{p} \cdot {\bar{n}}_{n m}^{*} \\ {\bar{B}}_{p} \cdot {\bar{m}}_{n m}^{*} \end{matrix}) + (\begin{matrix} 0 \\ \frac{- κ χ_{2} β_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} k_{1} e_{n m}^{p} Δ_{n}^{0} \\ \frac{- χ_{1} α_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} k_{1} e_{n m}^{p} Δ_{n}^{0} \end{matrix}) .$

(44)
The following equation is obtained after imposing the ABC [Eq. (24)]:

$[b_{m n}^{p} ζ_{n}^{0} - \frac{k_{1}}{k_{2}} d_{m n}^{p} Ω_{n}^{0}] n (n + 1) - k_{1} e_{n m}^{p} (ξ_{n}^{1} + Δ_{n}^{0}) = \frac{1}{C_{n m}^{'}} \int d Ω C_{p} o_{n m}^{*},$

(45)
where $C_{n m}^{'} = (\frac{4 π}{2 n + 1}) \frac{(n + m)!}{(n - m)!}$ .
Using Eqs. (44, 45), the coefficient $e_{n m}^{p}$ is obtained as the following form:

$e_{n m}^{p} = \frac{\frac{1}{C_{n m}^{'}} \int d Ω C_{p} o_{n m}^{*} - n (n + 1) \frac{1}{C_{n m}} \int d Ω {[A_{1} {\bar{A}}_{p} \cdot {\bar{n}}_{n m}^{*} + A_{3} {\bar{B}}_{p} \cdot {\bar{m}}_{n m}^{*}] ζ_{n}^{0} - [A_{2} {\bar{A}}_{p} \cdot {\bar{n}}_{n m}^{*} + A_{4} {\bar{B}}_{p} \cdot {\bar{m}}_{n m}^{*}] \frac{k_{1}}{k_{2}} Ω_{n}^{0}}}{[A_{1} ζ_{n}^{0} - A_{2} \frac{k_{1}}{k_{2} Ω_{n}^{0}}] n (n + 1) - k_{1} (ξ_{n}^{1} + Δ_{n}^{0})},$

(46)
where $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_{4}$ are

$\begin{matrix} A_{1} = \frac{- κ χ_{2} β_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \\ A_{2} = \frac{- χ_{1} α_{n}^{0}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \\ A_{3} = \frac{κ β_{n}^{1}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \\ A_{4} = \frac{α_{n}^{1}}{- κ χ_{2} β_{n}^{0} α_{n}^{1} + κ β_{n}^{1} χ_{1} α_{n}^{0}} \end{matrix}} .$

(47)
Hence the general solutions to the coefficients for each order of η are obtained via Eq. (44) and Eq. (46). Again, we can solve five coefficients $(a_{n m}, b_{n m}, c_{n m}, d_{n m}, e_{n m})$ by using Eq. (28).

4. SUMMARY AND CONCLUSION

In this work, we have developed mathematical formulations for light scattering from a nearly spherical particle including nonlocal effect via the perturbation method. When we put

e_{n m}^{p} = 0

, Eq. (43) reduces to Eq. (16) for local optical response. For the case of nonlocal optical response, not only transverse (described by

a_{n m}, b_{n m}, c_{n m}, d_{n m}

) but also longitudinal (described by

e_{n m}

) surface plasmon modes of the particle are excited. The nonlocal dielectric function can be chosen from the following forms as deduced by Lindhard [20

J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).

ε_{t} (k, ω) = 1 - \frac{ω_{p}^{2}}{ω (ω + i Γ)} \frac{3}{2 α^{2}} (\frac{1 + α^{2}}{α} {tan}^{- 1} α - 1),

ε_{ℓ} (k, ω) = 1 - \frac{ω_{p}^{2}}{ω (ω + i Γ)} \frac{3}{α^{2}} (1 - \frac{{tan}^{- 1} α}{α}) {\times [1 + \frac{i Γ}{ω} (1 - \frac{{tan}^{- 1} α}{α})]}^{- 1},

(48)

where

α^{2} = - \frac{k^{2} v_{F}^{2}}{{(ω + i Γ)}^{2}},

(49)

ω_{p}

is the plasma frequency,

v_{F}

is the Fermi velocity, and Γ is the damping factor of the electrons. Alternatively, we can choose the hydrodynamics model, and the dielectric functions are given by [29

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008). [CrossRef]

]

ε_{t} (ω) = 1 - \frac{ω_{p}^{2}}{ω (ω + i Γ)},

ε_{ℓ} (k, ω) = 1 - \frac{ω_{p}^{2}}{ω (ω + i Γ) - \frac{3}{5} v_{F}^{2} k^{2}} .

(50)

Quantum effects of the particle can be considered in Eqs. (48, 49, 50) because these equations include the term of the Fermi velocity

(v_{F})

[18

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

]. Furthermore, we can substitute other nonlocal dielectric functions in our mathematical formulations by using the more general quantum mechanical formulation obtained by electronic structure calculations [22

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002). [CrossRef]

]. The surface characteristic function

f (θ, φ)

plays an important role in controlling higher-order modes of this scattering problem for a non-spherical particle including the nonlocal effects. It is worth mentioning that we can use this idea to predict changes in the higher-order modes caused by different surface characteristic functions

f (θ, φ)

by considering our analytical results for the expansion coefficients. Furthermore, our analytical formulations provide an alternative method to investigate the optical behaviors of the surface plasmon of non-spherical nanoparticles with nonlocal optical response. For example, the coefficients

a_{n m}

and

b_{n m}

are associated with the optical properties in far field, and these properties are expressed in terms of the scattering and extinction cross sections. Besides the far-field optical properties, we can also investigate the near-field optical properties. For example, we can construct the mathematical formulations for the decay rate of a molecule near a non-spherical particle and the energy transfer rate between two molecules (a donor and an acceptor) in the vicinity of a non-spherical particle, including the nonlocal effect. Such information will be useful in analyzing the surface enhanced Raman spectroscopy, biosensing systems, and plasmonic devices.

ACKNOWLEDGMENTS

This work was supported by Academia Sinica and by the National Science Council of the Republic of China under Contract No. NSC 98-2112-M-001-022-MY3. We thank Dr. Hai-yan Guo for fruitful discussions.

References and links

1.	C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
2.	Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002). [CrossRef] [PubMed]
3.	J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]
4.	T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]
5.	J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]
6.	L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005). [CrossRef] [PubMed]
7.	H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006). [CrossRef] [PubMed]
8.	J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003). [CrossRef]
9.	K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]
10.	U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
11.	H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008). [CrossRef] [PubMed]
12.	E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef] [PubMed]
13.	G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908). [CrossRef]
14.	S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]
15.	V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]
16.	J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]
17.	W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986). [CrossRef]
18.	P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).
19.	G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979). [CrossRef]
20.	J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).
21.	Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009). [CrossRef] [PubMed]
22.	G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002). [CrossRef]
23.	R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987). [CrossRef]
24.	R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988). [CrossRef]
25.	R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006). [CrossRef] [CrossRef]
26.	R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006). [CrossRef] [CrossRef]
27.	H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009). [CrossRef]
28.	R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]
29.	V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008). [CrossRef]
30.	R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005). [CrossRef]
31.	A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005). [CrossRef]
32.	J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009). [CrossRef] [PubMed]

Fig. 1 Geometry of the scattering problem of a non-spherical particle. The dashed line denotes an unperturbed sphere

(r_{s} = a)

, and the solid line describes the surface shape function

[r_{s} = a (1 + η f (θ, φ))]

of the particle considered.

Fig. 2 Schematic diagrams for the electric fields associated with (a) local and (b) nonlocal optical responses of a non-spherical particle.

{\overset{⃑}{E}}_{i}

denotes the incident electric field,

{\overset{⃑}{E}}_{s}

denotes the scattered electric field,

{\overset{⃑}{E}}_{t}

denotes the transverse electric field, and

{\overset{⃑}{E}}_{ℓ}

denotes the longitudinal electric field.

OCIS Codes
(290.0290) Scattering : Scattering
(290.5850) Scattering : Scattering, particles
(290.5880) Scattering : Scattering, rough surfaces

ToC Category:
Scattering

History
Original Manuscript: May 19, 2010
Manuscript Accepted: July 29, 2010
Published: October 18, 2010

Citation
Huai-Yi Xie, Ming-Yaw Ng, and Yia-Chung Chang, "Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect," J. Opt. Soc. Am. A 27, 2411-2422 (2010)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-27-11-2411

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References

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002). [CrossRef] [PubMed]
J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]
T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]
J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]
L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005). [CrossRef] [PubMed]
H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006). [CrossRef] [PubMed]
J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003). [CrossRef]
K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]
U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008). [CrossRef] [PubMed]
E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef] [PubMed]
G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908). [CrossRef]
S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]
V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]
J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]
W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986). [CrossRef]
P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).
G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979). [CrossRef]
J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).
Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009). [CrossRef] [PubMed]
G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002). [CrossRef]
R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987). [CrossRef]
R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988). [CrossRef]
R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006). [CrossRef]
R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006). [CrossRef]
H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009). [CrossRef]
R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]
V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008). [CrossRef]
R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005). [CrossRef]
A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005). [CrossRef]
J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009). [CrossRef] [PubMed]

Asano, S.
Barbic, M.
Barton, G.
Bohren, C.
Brandl, D. W.
Carney, P. S.
Chang, R.
Chang, S. H.
Chang, Y. -C.
Chen, H.
Chung, H. Y.
Claro, F.
Coronado, E.
Duyne, R. P. V.
Ekardt, W.
Erma, V. A.
Feldmann, J.
Fuchs, R.
Gray, S. K.
Grosse, S.
Halas, N. J.
Halevi, P.
Hao, E.
Huffman, D.
Keller, O.
Kelly, K. L.
Klar, T.
Kottmann, J. P.
Kou, X.
Kreibig, U.
Le, F.
Leung, P. T.
Lindhard, J.
Martin, O. J. F.
McMahon, J. M.
Mie, G.
Mock, J. J.
Moroz, A.
Ni, W.
Nordlander, P.
Onida, G.
Penzar, Z.
Perner, M.
Reining, L.
Rojas, R.
Rubio, A.
Ruppin, R.
Rutherford, J. M.
Schatz, G. C.
Schoonover, R.
Schultz, D. A.
Schultz, S.
Sherry, L. J.
Smith, D. R.
Spirkl, W.
Sun, Y. G.
Tsai, D. P.
Vielma, J.
Vollmer, M.
von Plessen, G.
Wang, H.
Wang, J.
Wiley, B. J.
Xia, Y.
Xia, Y. N.
Xie, H. Y.
Yamamoto, G.
Yang, Z.
Yannopapas, V.
Zhao, L. L.

Ann. Phys. (Leipzig)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908). [CrossRef]

Ann. Phys. (N.Y.)

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005). [CrossRef]

Appl. Opt.

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975). [PubMed]

J. Chem. Phys.

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007). [CrossRef] [PubMed]
E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004). [CrossRef] [PubMed]
J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]

J. Phys. Chem. B

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). [CrossRef]

J. Phys. Condens. Matter

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008). [CrossRef]

K. Dan. Fidensk. Selsk. Mat. Fys. Medd.

J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).

Langmuir

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008). [CrossRef] [PubMed]

Nano Lett.

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005). [CrossRef] [PubMed]
H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006). [CrossRef] [PubMed]
J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003). [CrossRef]

Phys. Lett. A

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005). [CrossRef]

Phys. Rev.

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969). [CrossRef]

Phys. Rev. B

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001). [CrossRef]
W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986). [CrossRef]
R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987). [CrossRef]
R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988). [CrossRef]
R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006). [CrossRef]
R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006). [CrossRef]
H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009). [CrossRef]
R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975). [CrossRef]

Phys. Rev. Lett.

Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009). [CrossRef] [PubMed]
J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009). [CrossRef] [PubMed]
T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998). [CrossRef]

Rep. Prog. Phys.

G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979). [CrossRef]

Rev. Mod. Phys.

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002). [CrossRef]

Science

Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002). [CrossRef] [PubMed]

Other

C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

2009, Chang, Phys. Rev. Lett.
2009, Xie, Phys. Rev. B
2009, McMahon, Phys. Rev. Lett.
2008, Yannopapas, J. Phys. Condens. Matter
2008, Chen, Langmuir
2007, Vielma, J. Chem. Phys.
2006, Wang, Nano Lett.
2006, Chang, Phys. Rev. B
2006, Chang, Phys. Rev. B
2005, Sherry, Nano Lett.
2005, Schoonover, Phys. Lett. A
2005, Moroz, Ann. Phys. (N.Y.)
2004, Hao, J. Chem. Phys.
2003, Mock, Nano Lett.
2003, Kelly, J. Phys. Chem. B
2002, Sun, Science
2002, Mock, J. Chem. Phys.
2002, Onida, Rev. Mod. Phys.
2001, Kottmann, Phys. Rev. B
1998, Klar, Phys. Rev. Lett.
1995, Kreibig,
1992, Halevi,
1988, Rojas, Phys. Rev. B
1987, Fuchs, Phys. Rev. B
1986, Ekardt, Phys. Rev. B
1983, Bohren,
1979, Barton, Rep. Prog. Phys.
1975, Asano, Appl. Opt.
1975, Ruppin, Phys. Rev. B
1969, Erma, Phys. Rev.
1954, Lindhard, K. Dan. Fidensk. Selsk. Mat. Fys. Medd.
1908, Mie, Ann. Phys. (Leipzig)

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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

Analytical solutions to light scattering by plasmonic nanoparticles with nearly spherical shape and nonlocal effect

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Abstract

1. INTRODUCTION

2. MATHEMATICAL DESCRIPTIONS OF A PARTICLE OF ARBITRARY SHAPE

3. MATHEMATICAL FORMULATION FOR LIGHT SCATTERING

3A. Case (i): Local Optical Response

3B. Case (ii): Nonlocal Optical Response

4. SUMMARY AND CONCLUSION

ACKNOWLEDGMENTS

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Langmuir

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