Abstract

The theory of second harmonic generation (SHG) in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials is developed by means of multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency and second harmonic, as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other distributions of the metallic spheres. It is shown that the linear and nonlinear optical response of all ensembles of metallic spheres is strongly influenced by the excitation of localized surface plasmon-polariton resonances. The physical origin for such a phenomenon has also been analyzed.

© 2012 OSA

1. Introduction

During the past two decades,nonlinear optical responses of small particles have been subject of intensive studies due to the development of nanofabrication techniques [16]. In particular, second harmonic generation (SHG) has provided a powerful tool for probing physical and chemical properties of the surface and interface of these materials [712]. Thus, many experimental and theoretical studies have been done for the SHG of the nanostructures [112].

Recently, some novel metamaterials have been fabricated successfully [1316]. In these materials, the localized electromagnetic excitations induced by the plasmonic oscillations of the conduction electrons inside the metal can play important role in the process of nonlinear optical responses. Therefore, a renaissance of scientific interests appears in the study on the SHG of these materials. The SHGs from different geometric configurations such as sharp metal tips [17, 18], imperfect spheres [19, 20], split-ring resonators [21, 22], metallodielectric multilayer photonic-band-gap structures [23], T-shaped and L-shaped nanoparticles [9, 2426], and “fishnet” structures [27] have been observed experimentally. Corresponding to the experimental investigations, some theoretical methods with various kinds of approaches for the SHG have been developed [2837]. Very recently, a numerical method for two-dimensional metamaterials consisting of centrosymmetric cylinders, which can rigorously describe the general case, has been provided by using the multiple scattering matrix algorithm [38, 39].

In this work, we present a theory of the SHG in three-dimensional structures consisting of arbitrary distributions of metallic spheres made of centrosymmetric materials by means of the multiple scattering of electromagnetic multipole fields. The electromagnetic field at both the fundamental frequency (FF) and second harmonic (SH), as well as the scattering cross section, are calculated in a series of particular cases such as a single metallic sphere, two metallic spheres, chains of metallic spheres, and other ordered distributed arrays of spheres.

2. Theory and method

We consider an ensemble of N spheres (the nth sphere is marked by n, n=1,2......N), embedded in a background medium with electric permittivity εb and magnetic permeability μb. The nth sphere has radius a, relative permittivity εs(ω), relative permeability μs(ω) and second-order nonlinear susceptibility P(2ω). The geometry of the problem is shown in Fig. 1 . When the ensemble of spheres is irradiated by a plane wave, the scattering field at the FF and the SH can be obtained by means of the multiple scattering method.

 

Fig. 1 Geometry of the scattering problem for an ensemble of N spheres consisting of centrosymmetric materials. Here θn, ϕn and rn are the coordinate of the nth sphere in the spherical coordinate system, Ω=(θ,ϕ) represents the solid angle of an arbitrary point P. The x, y and z represent three axis directions in corresponding Cartesian coordinate system.

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2.1 Calculation of the fields at the fundamental frequency

The multiple scattering method to calculate the scattering field at the FF has been discussed in many literatures [4047], which is summarized in the following. For an arbitrary initial incident time-harmonic plane wave (E0eiωt) with complex wave vector Ki and polarization , the electric field component E0=eiKir can be expanded in the vector spherical waves around the nth sphere as follows [40]:

En0(xn)=lm{anlm0,EJElm(xn)+anlm0,HJHlm(xn)}
with
JElm(r)=ikb×jl(kbr)Xlm(r^),JHlm(r)=jl(kbr)Xlm(r^),
and
[anlm0,Hanlm0,E]=4πeiKirnil[Xlm*(Ωi)Xlm*(Ωi)(×Κι)/kb],
where Ωi is the polar angle of Ki and xn=rrn,rn is the center coordinate of the nth sphere with radius a, kb is the wave number of the background, jl represents the first-kind spherical Bessel function, Xlm are vector spherical harmonics [48]. In fact, the total incident field of the nth sphere consists of two distinct parts, firstly the initial incident wave, and secondly the sum of all the scattered waves from other spheres (nn). The total incident field that strikes the surface of the nth sphere Eninc(xn), the scattered field Ensc(xn), and the internal field Enin(xn) can also be expanded as series of the vector spherical harmonics in the coordinate frame center at the nth sphere:
Eninc(xn)=lm{anlminc,EJElm(xn)+anlminc,HJHlm(xn)},
Ensc(xn)=lm{anlmsc,EHElm(xn)+anlmsc,HHHlm(xn)},
Enin(xn)=lm{anlmin,EJElms(xn)+anlminc,HJHlms(xn)}
with
JElms(r)=iks×jl(ksr)Xlm(r^),JHlms(r)=jl(ksr)Xlm(r^),
HElm(r)=ikb×hl(kbr)Xlm(r^),HHlm(r)=hl(kbr)Xlm(r^),
where ks is the wave number inside the sphere, hl is the first-kind spherical Hankel function. The expansion coefficients are determined by the following boundary conditions:

Etinc+Etsc=Etin,
Drinc+Drsc=Drin,
Htinc+Htsc=Htin,
Brinc+Brsc=Brin.

Here the subscripts r and t refer, respectively, to components perpendicular and parallel to the surface of the sphere. The fields Dinc, Dsc and Din are related to the Einc, Esc and Ein by the usual linear response. The corresponding magnetic fields Hinc, Hsc and Hin are related to the Binc, Bsc and Bin, respectively. Solution of the boundary value problem allows one to express the relations between the expansion coefficients as

TlE=anlmsc,E/anlminc,E,TlH=anlmsc,H/anlminc,H,ClE=anlmin,E/anlminc,E,ClH=anlmin,H/anlminc,H,
where TlE, TlH, ClE and ClH represent the Mie coefficients for a single sphere, which their explicit forms can be found elsewhere [28, 41]. Thus, with the use of the addition theorem [49], the expansion coefficients anlminc,E and anlminc,H can be expressed as [50]
anlminc,E=anlm0,E+nnlm(Ωnlm,nlmEEanlmsc,E+Ωnlm,nlmEHanlmsc,H),
anlminc,H=anlm0,H+nnlm(Ωnlm,nlmHEanlmsc,E+Ωnlm,nlmHHanlmsc,H),
where Ωnlm,nlmPP(P,P=E,H) represent the free-space propagator functions, which their explicit forms can be found in [50]. Inserting Eqs. (11) and (12) into Eq. (10), we obtain

nlm[δnnδllδmmTlE(Ωnlm,nlmEEanlmsc,E+Ωnlm,nlmEHanlmsc,H)]=TlEanlm0,E,
nlm[δnnδllδmmTlH(Ωnlm,nlmHEanlmsc,E+Ωnlm,nlmHHanlmsc,H)]=TlHanlm0,H.

The above introduction about the method for the FF only focuses on the case that all spheres have the same radius. In fact, it is convenient to discuss the scattering problem with different radii of spheres by using such a method. The detailed descriptions about solving the linear system of the above equations have been given in [4047]. After solving Eqs. (13) and (14), the expansion coefficients anlmsc,E and anlmsc,H are obtained. Then, the fields inside the spheres can be determined by using Eqs. (10) and (6).

2.2 Calculation of the fields at the second harmonic

After the FF field in the cluster with N spheres is obtained, we calculate the electromagnetic field at the SH. We first determine the source of the field at the SH, namely, the nonlinear polarization induced by the field at the FF. The nonlinear polarization vector (P(2ω)) includes surface (Psurface(2ω)(r)) and bulk (Pb(2ω)(r)) nonlinear polarization components. For some metal systems, the SHG is dominated by the surface component and the bulk nonlinear source polarization can be neglected [51]. Here we treat the nonlinearity of the interface as a sheet of current or, equivalently, as a sheet of nonlinear source polarization according to Refs [2,3,52],

Psurface(2ω)(r)=Ps(2ω)(θ,φ)δ(ra)=χs(2):E(ω)(r)E(ω)(r)δ(ra),
where χs(2) is the surface second-order susceptibility tensors. The nonlinear polarization vector (Ps(2ω)(θ,φ)) can be expanded in terms of the spherical harmonics Ylm(θ,φ) and vector spherical harmonics,
Ps(2ω)(θ,φ)=lmGr,nlmYlm(θ,φ)r^+GM,nlmXlm(θ,φ)+GE,nlmr^×Xlm(θ,φ),
where Gr,nlm, GM,nlm and GE,nlm are the expansion coefficients. Similar to the case of the FF, the SH source field inside the nth sphere can be expanded as
Enint,(2ω)(xn)=lm{Anlmint,EJElms,(2ω)(xn)+Anlmint,HJHlms,(2ω)(xn)},
where JElms,(2ω)(xn) and JHlms,(2ω)(xn) are given by Eq. (7) by replacing kswith Ks=2ωcεs(2ω)μs(2ω). The SH source field outside the nth sphere are expressed as
Enout,(2ω)(xn)=lm{Anlmout,EHElm(2ω)(xn)+Anlmout,HHHlm(2ω)(xn)},
where HElm(2ω)(xn) and HHlm(2ω)(xn) are given by Eq. (8) by replacing kbwith Kb=2ωcεb(2ω)μb(2ω). The expansion coefficients in Eqs. (17) and (18) Anlmint,E, Anlmint,H, Anlmout,E and Anlmout,H are determined by the following four boundary conditions [2]

Etout,(2ω)Etin,(2ω)=4πε1(2ω)tPs,r(2ω),
Drout,(2ω)Drin,(2ω)=4πtPs(2ω),
Htout,(2ω)Htin,(2ω)=4πi2ωcr^×Ps(2ω),
Brout,(2ω)Brin,(2ω)=0.

Inserting Eq. (16) into Eqs. (19)a)-(19d), we obtain,

Anlmint,E=εb(2ω)Kbhl(1)(Kba)Anlmout,E+4πiGE,nlmεb(2ω)jl(K1a)/K1,
Anlmint,H=hl(1)(Kba)jl(K1a)Anlmout,H,
where Anlmout,E and Anlmout,H are given in [52]. If we make no assumptions about the nature of the interface other than that it exhibits isotropic symmetry with a mirror plane perpendicular to the interface, the surface nonlinear susceptibility tensor χs(2) then has three nonvanishing and independent elements: χ,χ, χ=χ,where and refer to the local spatial components perpendicular and parallel to the surface. The susceptibility χs(2) can be written, in terms of the unit vectors for the spherical coordinate system, as the triadic [5]

χs(2)=χr^r^r^+χr^(θ^θ^+φ^φ^)+χ(θ^r^θ^+φ^r^φ^+θ^θ^r^+φ^φ^r^).

Hence, the nonlinear source polarization for the surface is

Ps(2ω)=r^(χEr(ω)Er(ω)+χEt(ω)Et(ω))+2χEr(ω)Et(ω).

Inserting Eq. (23) into Eq. (16), the expansion coefficients Gr,nlm, GM,nlm and GE,nlm can be obtained, which are given in [52].

Similar to the linear scattering problem, the total incident SH field of the nth sphere can be viewed as consisting of two distinct components: a source field from other spheres without being scattered by any of the spheres, and the sum of all the SH scattered waves from other spheres. As for the SH scattered field (Ensca,(2ω)(xn)) of the nth sphere, it can be expanded as

Ensca,(2ω)(xn)=lm{Anlmsca,EHElm(2ω)(xn)+Anlmsca,HHHlm(2ω)(xn)},
where Anlmsca,E and Anlmsca,H are the expansion coefficients. The total incident SH field(Enloc,(2ω)(xn)) of the nth sphere is written as
Enloc,(2ω)(xn)=nn[Enout,(2ω)(xn)+Ensca,(2ω)(xn)],
with

Ensca,(2ω)(xn)=lm{Anlmsca,EHElm(2ω)(xn)+Anlmsca,HHHlm(2ω)(xn)},

Here Enout,(2ω)(xn) and Ensca,(2ω)(xn) correspond to the source field and the SH scattered field from the nth sphere, respectively. Inserting Eqs. (18) and (26) into Eq. (25), the total incident SH field of the nth sphere is rewritten as

Enloc,(2ω)(xn)=nnlm{(Anlmout,E+Anlmsca,E)HElm(2ω)(xn)+(Anlmout,H+Anlmsca,H)HHlm(2ω)(xn)}.

By inserting in Eq. (26) the Graf formula [49], Eloc(2ω)(xn) can be expressed as

Enloc,(2ω)(xn)=nlm{Anlmloc,EJElm(2ω)(xn)+Anlmloc,HJHlm(2ω)(xn)},
where

Anlmloc,E=nnlm{Ωnlm,nlmEE(Anlmout,E+Anlmsca,E)+Ωnlm,nlmEH(Anlmout,H+Anlmsca,H)},
Anlmloc,H=nnlm{Ωnlm,nlmHE(Anlmout,E+Anlmsca,E)+Ωnlm,nlmHH(Anlmout,H+Anlmsca,H)}.

At the same time, the total incident and scattered SH fields of the nth sphere still satisfy the following relations:

Anlmsca,E=TlEAnlmloc,E,
Anlmsca,H=TlHAnlmloc,H.

From Eqs. (29)-(32), we obtain [52]

nlm[δnnδllδmmTlE(Ωnlm,nlmEEAnlmsca,E+Ωnlm,nlmEHAnlmsca,H)]=nlmTlE[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H],
nlm[δnnδllδmmTlH(Ωnlm,nlmHEAnlmsca,E+Ωnlm,nlmHHAnlmsca,H)]=nlmTlH[Ωnlm,nlmEEAnlmout,E+Ωnlm,nlmEHAnlmout,H].

Solving the self-consistent equations, Eqs. (33) and (34), we can obtain the SH scattered coefficients Anlmsca,E and Anlmsca,H. Then the SH field can be calculated from the following relations:

E(2ω)(r)=nlm(Anlmout,E+Anlmsca,E)HElm(2ω)(rn)+(Anlmout,H+Anlmsca,H)HHlm(2ω)(rn).

The above description about the method focuses on plane-wave sources. If we use focused beams or near-field dipole sources, the theory is also applicable. Similar to the case of the FF, the multiple scattering method for the SH is accurate and can reach rapid convergence in the undepleted-pump approximation. Based on such a method, it is convenient to study the scattering problem for the systems consisting of non overlapping spheres. The method allows one to determine not only the spatial distribution of the electromagnetic field but also a series of important physical quantities, such as the scattering cross section.

2.3 Calculation of scattering cross sections

The field distribution provides the essential information regarding the properties of the optical near field, the scattering cross sections characterize the process of energy transfer from the incident wave into the far-field. The total scattering cross section, Cs(ϖ), is defined as

Cs(ϖ)=qs(ϖ;Ω)dΩ,
where Ω is the solid angle, qs(ϖ;Ω) is the differential cross section, which is expressed as
qs(ϖ;Ω)=limrr2Re{[Es(ϖ)×Hs(ϖ)]r^},
where Es and Hs represent the FF (ϖ=ω) or the SH (ϖ=2ω) scattered field. As r, they can be written as
limrEs(r)=f(Ω)eikrr,
limrHs(r)=r^×f(Ω)eikrr.
Here
f(Ω)=neikbr^rnil1lm[Xlm(Ω)anlmH/kr^×Xlm(Ω)anlmE/k],
where k=kb, anlmE=anlmsc,E and anlmH=anlmsc,H are for the FF; k=Kb, anlmE=Anlmsca,E and anlmH=Anlmsca,H for the SH. Inserting Eqs. (38) and (39) into Eq. (37), the differential cross section is expressed as

qs(ϖ;Ω)=|f(Ω)|2.

Based on Eqs. (36), (40), (41) and the multiple scattering formula, the total scattering cross section can be obtained by performing numerical calculations.

3. Numerical results and discussion

In this section, we present numerical results for the linear and nonlinear scatterings from a set of homogeneous centrosymmetric metallic spheres.

3.1 A single metallic sphere

We first consider the case of a single metallic sphere. This is an important case because it has an analytical solution, which allows us to validate our numerical method. Figure 2 (I) and (II) show the calculated results of the scattering cross sections for the FF and the SH scattering from the single metallic sphere as a function of the FF photon energy. In the calculation, a circularly polarized incident beam (=(1,i)/2) is taken and the wave vector of the incident wave is assumed along x axis (Ωi=(π2,0)). The permittivity of the metallic sphere is described by the following Drude model

εs(ω)=1ωp2ω(ω+iν),
where ωp and ν are the plasma and damping frequency, respectively. As specific values for these parameters we choose ωp=1.35×1016rad/s and ν=0.03ωp, which correspond to the metal Ag [53]. Because the SHG is dominated by the surface components for the present systems and the contributions of the bulk nonlinear polarization can be neglected, in the calculations we take the surface second-order susceptibility χ^=2.79×1018m/V,χ^=χ^=3.98×1020m/V and χ^=0 [54]. Different sizes of spheres are considered. The solid line, dashed line and dotted line correspond to a = 10nm, 30nm and 50nm, respectively. It can be seen in the figures that new plasmon resonances appear in the SH spectra comparing with those in the FF spectra. The bigger the radius of the sphere is, the more resonance peaks appear. The above results are only for the case with the circularly polarized incident beam. If the linearly polarized incident beam is used, similar phenomena can be observed. The comparative study for two kinds of incident beam had been provided in the previous investigations [3, 5, 6]. Here our calculated results are in agreement with them. These results are for the case of the permittivity of the material being taken as Drude model. We have also performed calculations with different form of the permittivity such as measured values, the similar phenomena can also be observed.

 

Fig. 2 The spectra of the scattering cross sections for metal spheres of radii a = 10nm (dashed line), 30 (solid line), and 50 nm (dotted line). (I): the FF; (II): the SH.

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The properties of resonance peaks can be revealed, in part, by calculating the distribution of the scattering field. The panels A and B in Fig. 3 display the spatial distribution of the amplitude of the electric field (a = 10 nm) for the FF and the SH at ω=5.0eV, respectively, which corresponds to one of the maxima in Fig. 2(I) and (II). The amplitudes of the electric field around the sphere exhibit local maxima for both the FF and the SH, which is a signature of the excitation of localized surface plasmon-polariton (SPP) modes. This is a type of SPP resonances excited at the SH and the FF simultaneously. On the other hand, the SPP resonances at low frequency at ω=2.7eV in the panel D of Fig. 3, are due to the excitation of SPPs at the SH, with no such localized modes being excited at the FF (see panel C in Fig. 3). Here the panels A and B correspond to the dipolar excitation, and D to the quadrupolar excitation as has been described in [3, 5]. The distinction between the two types of SPP resonances also appears in more complex scattering geometries as well.

 

Fig. 3 (Color online) The spatial profile of the amplitude of the electric field, calculated at ω=5.0eV (panels A and B) and ω=2.7eV (panels C and D). The radius of the sphere is a = 10 nm. A and C correspond to the FF; B and D to the SH.

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3.2 A spherical metallic dimer

Figure 4 shows the scattering cross sections of two-sphere systems for the different separation distances at the FF ((I) and (III)) and the SH ((II) and (IV)), respectively. Here the radii of two spheres are taken as a = 10nm. Figure 4(I) and (II) correspond to the separation distance d = 22nm, while (III) and (IV) to d = 35nm. The solid lines and dashed lines represent the cases at the incident direction of the circularly polarized wave perpendicular or parallel to the longitudinal axis of the system, respectively. Similar to the case of the single sphere, the SPP resonances at the SH can be also divided into two types, those induced by the resonant excitation of SPP modes at the FF and those that are associated with the excitation of SPP modes solely at the SH.

 

Fig. 4 The spectra of the scattering cross sections for the metallic two-sphere system. The radii of two spheres are taken as a = 10 nm. (I) and (II) correspond to the separation distance d = 22nm, (III) and (IV) to d = 35nm. Solid line and dashed line correspond to the case at the incident direction of the wave perpendicular or parallel to the longitudinal axis of the system, respectively. (I) and (III): the FF; (II)and (IV): the SH.

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In contrast to the case of the single sphere, the spectra of the two-sphere system exhibit richer phenomena. On the one hand, the excitation spectra for the FF and the SH are related to the angle of incidence. This can be found from the comparison between the solid lines and dashed lines in the figures. For example, there is no localized SPP excitation for the parallel incidence at ω=4.1eV and d = 22nm, whereas strong excitations for both the FF and the SH can be observed for the normal incidence (A and B points in Fig. 4(I) and (II)).

On the other hand, the excitation spectra also depend on the separation distance between two spheres. When the distance between two spheres is big, the coupling between them is weak. In such a case the excitation feature for the two-sphere system is similar to the case of the single sphere. This can be seen more clearly from the panels C and D in Fig. 5 . The panels C and D in Fig. 5 show the electric field distributions of the two-sphere system at the normal incidence. The panel A for the FF and B for the SH correspond to the case at ω=4.1eV and d = 22nm, while C and D to that at ω=4.1eV and d = 35nm. Comparing them, we find that different excitation features appear when the distance between two spheres decreases and the coupling becomes stronger. The corresponding cases for the parallel incidence are shown in the panels E, F, G and H of Fig. 5. The changes of the excitation modes with the separation distance are observed again, which means that the coupling between two spheres play an important role in the excitation spectra. Due to the parallel incidence, some asymmetric properties of the field distribution around two spheres are also observed (H in Fig. 5 shows more clearly). In fact, the excitation spectra of the systems are also related to the polarization of the incident beam. The above discussions only focus on the circularly polarized incident beam. If we use other polarized incident beams such as linearly polarized beams (perpendicular or parallel to the longitudinal axis of the systems), the change in the excitation spectra will take place. However, the coupling phenomena exhibited between two spheres are similar.

 

Fig. 5 (Color online) The spatial profile of the amplitude of the electric field for the two-sphere system, calculated at ω=4.1eV for the normal incidence with d = 22nm (panels A and B) and d = 35nm (panels C and D); calculated at ω=6.54eV for the parallel incidence with d = 22nm (panels E and F) and d = 35nm (panels G and H). The radii of the spheres are a = 10 nm. A, C, E and G correspond to the FF; B, D, F and H to the SH. Here A, B, C, D, E, F, G and H correspond to the points marked in Fig. 3.

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3.3 A chain of metallic spheres

Now we apply our numerical method to study the SHG in chains of coupled metallic spheres. Such nanostructures can find important technological applications to subwavelength active optical waveguides and optical nanoantennae [55, 56]. This is because the coupling effect among the spheres leads to the long-range interactions along the chain. Such an interaction plays an important role in determining the global optical response of the structure.

The panels A and B in Fig. 6 depict the electric field patterns along a sphere chain with N = 12 for the FF and the SH, respectively, corresponding to the peaks at ω=5.3eV of the scattering cross sections as shown in Fig. 6(I) and (II). Here the circularly polarized incident beam is taken. From the distributions of the field, we can observe clearly the coupling characteristics of the field among the spheres. In some cases, sphere plasmon resonance modes can couple each other to form the coupled modes (propagating modes) along the chain. This can be demonstrated by comparing the field distributions inside the chain with those outside the sphere chain [38]. The panels A and B in Fig. 6 show such cases for both the FF and SH. Such a phenomenon depends on the excitation frequency. In some frequency regions, the coupling effects do not occur. For example, the panel C in Fig. 6 shows the electric field distribution of the FF along the sphere chain at ω=2.82eV. There is no coupling effect among the spheres and the field is excited solely around the single sphere. In contrast, the coupling effect for the SH at such a case is observed as shown in the panel D of Fig. 6. This is a kind of case that the chain of metallic spheres supports propagating modes only at the SH. In addition, the new excitations around 4.8eV for both the FF and the SH have also been observed owing to the coupling effect of spheres inside the chain.

 

Fig. 6 (Color online) The top two panels show the spectra of the scattering cross section corresponding to a chain of N = 12 metallic spheres under the parallel incidence. The radius of the sphere is a = 10 nm and the separation distance is d = 22nm. (I): the FF; (II): the SH. The spatial profile of the amplitude of the electric field, calculated at ω=5.3eV ((A) and (B)) and ω=2.82eV ((C) and (D)), is presented in the bottom panels. The panels A and C correspond to the FF, whereas the panels B and D correspond to the SH. Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

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The results shown in Fig. 6 are only for the case at the incident direction of the wave parallel to the axis of the chain of spheres. For the case at the normal incidence, the phenomena exhibited are more remarkable. Figure 7 shows the corresponding calculated results for such a case. In contrast to no coupling effect (panel C in Fig. 7), the standing waves in the chain of spheres are excited for the FF (panel A in Fig. 7) and the SH (panels B and D in Fig. 7) due to the long-range interactions among the metallic spheres, which are also in agreement with the spectra of the scattering cross sections as shown in Fig. 7(I) and (II).

 

Fig. 7 (Color online) The same as in Fig. 6, but for the normal incidence. The spatial profile of the amplitude of the electric field, calculated at ω=5.4eV ((A) and (B)) and ω=2.59eV ((C) and (D)). Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

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3.4 A cluster of metallic spheres

In the following we consider a cluster consisting of multi-layer spheres as shown in Fig. 8(I) . This is a typical 3D structure consisting of three layers, every layer includes 4×4 metallic spheres. The radii and the separation distances of spheres are taken as a = 10 nm and d = 22nm, respectively. The calculated results of the scattering cross sections for such a structure with various layers under the incidence of the circularly polarized plane wave from the left side of the sample are plotted in Fig. 8(II) and (III) for the FF and the SH, respectively. The dot-dashed line, dashed line and solid line correspond to the case with one-layer, two-layer and three-layer, respectively. Comparing them, we find that more excited modes appear with the increase of the layer number. This means that the interlayer coupling leads to new excitations. Such excitations also include two types: the resonant excitation of SPP modes at both the SH and the FF, and the excitation of SPP modes solely at the SH. The features of the excitations in such a structure exhibit 3D coupling effects because of three-direction interactions, which can be seen more clearly from the field distributions inside the sample.

 

Fig. 8 (I) The 3D sample and cross sections. (II) and (III) The spectra of the scattering cross sections corresponding to the cluster as shown in (I) under the incidence of the plane wave from the left side of the sample with a = 10 nm and d = 22nm. (II): the FF; (III): the SH.

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Without loss of generality, we take two cross-sections inside the sample as shown in Fig. 8(I) to analyze the distributions of the FF and SH fields. The first cross-section goes through the middle “vacant space” between two layers as illustrated by the shaded plane and the second goes across the centers of the spheres inside one layer as shown by dot-dashed lines (white color plane). The FF and SH fields at the first cross-section with ω=2.21eV (corresponding to the points A and B in Fig. 8(II) and (III)) are plotted in Fig. 9(a) and (b) , respectively, while the corresponding fields at the second cross-section are described in Fig. 9(c) and (d). The calculated results for the other systems have shown that the excitations of SPPs at the SH can be observed when the localized modes at the FF are not excited. The motivation to choose the energy ω=2.21eV is to test whether or not the phenomenon exists in such a complex structure. Due to the transverse and the longitudinal coupling interactions, the twist patterns for the SH (see Fig. 9(b) and (d)) are observed clearly. In contrast, there is no such a phenomenon for the FF (see Fig. 9(a) and (c)). This means that the collective SH excitations along different directions in 3D cluster can be realized, even the FF excitation does not exist.

 

Fig. 9 The spatial profiles of the amplitude of the electric field, calculated at ω=2.21eV (corresponding to the points A and B in Fig. 7 (II) and (III)). (a) and (b) correspond to the FF and the SH fields at the cross-section as shown by shadow in Fig. 7 (I), respectively, whereas (c) and (d) correspond to the FF and the SH fields at the cross-section as shown by dot-dashed lines in Fig. 7 (I). The other parameters are identical with those in Fig. 7.

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4. Summary

To summarize, we have developed in this paper a theory of SHG, based on the multiple scattering method, for studying the linear and nonlinear scattering effects in the 3D metamaterials made of centrosymmetric spheres. Based on such a method, the electromagnetic field at both the FF and the SH, as well as the scattering cross section, are calculated in a series of particular cases, a single metallic sphere, metallic two-sphere system, a chain of metallic spheres, and a finite 3D array of spheres. Our results show that the linear and nonlinear optical responses of all ensembles of metallic spheres were strongly influenced by the excitation of SPP resonances. The physical origin for such a phenomenon has also been elucidated and discussed. Finally, we would like to point out that our method can apply to any 3D structures consisting of the spheres, even for random systems, although our calculations focus on several special 3D structures.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No.10825416) and the National Key Basic Research Special Foundation of China under Grant 2007CB613205.

References and links

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

2. T. F. Heinz, Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991) Chap. 5.

3. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999). [CrossRef]  

4. H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996). [CrossRef]  

5. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004). [CrossRef]  

6. G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011). [CrossRef]  

7. K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006). [CrossRef]   [PubMed]  

8. M. D. McMahon, R. Lopez, R. F. Haglund Jr, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006). [CrossRef]  

9. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007). [CrossRef]   [PubMed]  

10. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010). [CrossRef]   [PubMed]  

11. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010). [CrossRef]   [PubMed]  

12. J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010). [CrossRef]   [PubMed]  

13. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef]   [PubMed]  

14. C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007). [CrossRef]   [PubMed]  

15. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]  

16. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef]   [PubMed]  

17. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003). [CrossRef]   [PubMed]  

18. C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005). [CrossRef]  

19. J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005). [CrossRef]  

20. J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006). [CrossRef]  

21. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006). [CrossRef]   [PubMed]  

22. M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007). [CrossRef]   [PubMed]  

23. M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008). [CrossRef]  

24. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008). [CrossRef]   [PubMed]  

25. B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007). [CrossRef]   [PubMed]  

26. H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008). [CrossRef]  

27. E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008). [CrossRef]  

28. Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004). [CrossRef]  

29. C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21(1), 36–44 (2004). [CrossRef]  

30. A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009). [CrossRef]  

31. J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009). [CrossRef]  

32. Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009). [CrossRef]  

33. L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009). [CrossRef]  

34. S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010). [CrossRef]  

35. R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010). [CrossRef]  

36. E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007). [CrossRef]   [PubMed]  

37. V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010). [CrossRef]   [PubMed]  

38. C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010). [CrossRef]  

39. C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010). [CrossRef]   [PubMed]  

40. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995). [CrossRef]   [PubMed]  

41. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998). [CrossRef]  

42. X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993). [CrossRef]   [PubMed]  

43. A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995). [CrossRef]   [PubMed]  

44. K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996). [CrossRef]  

45. F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999). [CrossRef]  

46. W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001). [CrossRef]   [PubMed]  

47. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005). [CrossRef]  

48. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

49. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 363.

50. V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007). [CrossRef]  

51. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009). [CrossRef]  

52. J. Xu and X. Zhang, “Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles,” Opt. Express 19(23), 22999–23007 (2011). [CrossRef]   [PubMed]  

53. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef]   [PubMed]  

54. D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004). [CrossRef]  

55. S. A. Maier, Plasmonics: Fundamental and Applications (Springer, New York, 2007).

56. J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009). [CrossRef]  

References

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  1. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, Berlin, 1995).
  2. T. F. Heinz, Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991) Chap. 5.
  3. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
    [CrossRef]
  4. H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
    [CrossRef]
  5. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
    [CrossRef]
  6. G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
    [CrossRef]
  7. K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006).
    [CrossRef] [PubMed]
  8. M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
    [CrossRef]
  9. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
    [CrossRef] [PubMed]
  10. Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
    [CrossRef] [PubMed]
  11. J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
    [CrossRef] [PubMed]
  12. J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
    [CrossRef] [PubMed]
  13. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
    [CrossRef] [PubMed]
  14. C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
    [CrossRef] [PubMed]
  15. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
    [CrossRef]
  16. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
    [CrossRef] [PubMed]
  17. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
    [CrossRef] [PubMed]
  18. C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
    [CrossRef]
  19. J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
    [CrossRef]
  20. J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
    [CrossRef]
  21. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
    [CrossRef] [PubMed]
  22. M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
    [CrossRef] [PubMed]
  23. M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
    [CrossRef]
  24. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
    [CrossRef] [PubMed]
  25. B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
    [CrossRef] [PubMed]
  26. H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
    [CrossRef]
  27. E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
    [CrossRef]
  28. Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
    [CrossRef]
  29. C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21(1), 36–44 (2004).
    [CrossRef]
  30. A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
    [CrossRef]
  31. J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
    [CrossRef]
  32. Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
    [CrossRef]
  33. L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
    [CrossRef]
  34. S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
    [CrossRef]
  35. R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
    [CrossRef]
  36. E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
    [CrossRef] [PubMed]
  37. V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
    [CrossRef] [PubMed]
  38. C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
    [CrossRef]
  39. C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010).
    [CrossRef] [PubMed]
  40. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995).
    [CrossRef] [PubMed]
  41. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
    [CrossRef]
  42. X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
    [CrossRef] [PubMed]
  43. A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995).
    [CrossRef] [PubMed]
  44. K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
    [CrossRef]
  45. F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
    [CrossRef]
  46. W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
    [CrossRef] [PubMed]
  47. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
    [CrossRef]
  48. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  49. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 363.
  50. V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
    [CrossRef]
  51. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
    [CrossRef]
  52. J. Xu and X. Zhang, “Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles,” Opt. Express 19(23), 22999–23007 (2011).
    [CrossRef] [PubMed]
  53. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985).
    [CrossRef] [PubMed]
  54. D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
    [CrossRef]
  55. S. A. Maier, Plasmonics: Fundamental and Applications (Springer, New York, 2007).
  56. J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
    [CrossRef]

2011 (2)

G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

J. Xu and X. Zhang, “Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles,” Opt. Express 19(23), 22999–23007 (2011).
[CrossRef] [PubMed]

2010 (8)

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
[CrossRef]

C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010).
[CrossRef] [PubMed]

2009 (6)

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
[CrossRef]

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

2008 (5)

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

2007 (7)

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
[CrossRef]

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

2006 (4)

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006).
[CrossRef] [PubMed]

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

2005 (3)

C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
[CrossRef]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

2004 (5)

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[CrossRef]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
[CrossRef]

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, “Second-harmonic generation in the scattering of light by an infinite cylinder,” J. Opt. Soc. Am. B 21(1), 36–44 (2004).
[CrossRef]

2003 (1)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

2001 (1)

1999 (2)

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

1998 (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

1996 (2)

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
[CrossRef]

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

1995 (2)

Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573–4588 (1995).
[CrossRef] [PubMed]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995).
[CrossRef] [PubMed]

1993 (1)

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

1985 (1)

Ahorinta, R.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Aktsipetrov, O. A.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Albers, W. M.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Alexander, R. W.

Andersen, U. L.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Bachelier, G.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

Bai, B.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Belardini, A.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Bell, R. J.

Benichou, E.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

Bhat, R. D. R.

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

Biris, C. G.

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
[CrossRef]

C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010).
[CrossRef] [PubMed]

Bloemer, M. J.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Borguet, E.

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

Brevet, P. F.

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Brevet, P.-F.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

Bunton, P. H.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

Butet, J.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

Canfield, B. K.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Cao, L.

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

Cappeddu, M. G.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Cassagne, D.

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

Centeno, E.

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

Centini, M.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Chan, C. T.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

Chipouline, A.

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

Chui, S. T.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

Dadap, J. I.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

Dai, H.-L.

G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

de Beer, A. G. F.

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
[CrossRef]

de Ceglia, D.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Du, J.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

Eisenthal, K. B.

K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006).
[CrossRef] [PubMed]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

Elser, D.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Fazio, E.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Felbacq, D.

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

Feth, N.

Fürst, J. U.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

García de Abajo, F. J.

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Gillijns, W.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Gonella, G.

G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

Grange, R.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Haglund, R. F.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

Hansson, T.

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

Harmon, B. N.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

Heinz, T. F.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

Hsieh, C.-L.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Hubner, W.

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
[CrossRef]

Husu, H.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Jonin, C.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Kapshai, V.

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

Kauranen, M.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Kim, E.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Kivshar, Y. S.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Klein, M. W.

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

Koch, S. W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

Krause, D.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Kuittinen, M.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Kujala, S.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Larciprete, M. C.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Lassen, M.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Laukkanen, J.

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Lederer, F.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Leuchs, G.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Lin, Z.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

Lin, Z. F.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

Liu, J.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

Liu, S.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

Long, L. L.

Lopez, R.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

Marquardt, C.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

McMahon, M. D.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

Mendez, E. R.

Mendoza, B. S.

Modinos, A.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Moloney, J. V.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

Moroz, A.

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995).
[CrossRef] [PubMed]

Moshchalkov, V. V.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Nappa, J.

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Neacsu, C. C.

C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
[CrossRef]

Ng, J.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

Ohtaka, K.

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
[CrossRef]

Ordal, M. A.

Osgood, R. M.

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

Panoiu, N. C.

C. G. Biris and N. C. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express 18(16), 17165–17179 (2010).
[CrossRef] [PubMed]

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
[CrossRef]

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

Pavlyukh, Y.

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

Petschulat, J.

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

Psaltis, D.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Pu, Y.

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

Querry, M. R.

Raschke, M. B.

C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
[CrossRef]

Ray, E. A.

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

Reider, G. A.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
[CrossRef]

Revillod, G.

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Rodríguez, F. J.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Rogers, C. T.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Roke, S.

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
[CrossRef]

Russier-Antoine, I.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

Scalora, M.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Shan, J.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21(7), 1328–1347 (2004).
[CrossRef]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

Shen, Y. R.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Sheng, P.

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

W. Y. Zhang, C. T. Chan, and P. Sheng, “Multiple scattering theory and its application to photonic band gap systems consisting of coated spheres,” Opt. Express 8(3), 203–208 (2001).
[CrossRef] [PubMed]

Sibilia, C.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

Silhanek, A. V.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Sipe, J. E.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

Stefanou, N.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Stiopkin, I.

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

Strekalov, D. V.

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

Svirko, Y.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Tanabe, Y.

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
[CrossRef]

Teplin, C. W.

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

Tünnermann, A.

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

Turunen, J.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipolar analysis of second-harmonic radiation from gold nanoparticles,” Opt. Express 16(22), 17196–17208 (2008).
[CrossRef] [PubMed]

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Valencia, C. I.

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Valev, V. K.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

van der Meulen, P.

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

Van Dorpe, P.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Vandenbosch, G. A. E.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Verbiest, T.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Verellen, N.

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Viarbitskaya, S.

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

Vitanov, N. V.

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
[CrossRef]

Wang, F.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Wang, F. X.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

Wang, H.

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

Wang, X. D.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

Wegener, M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

M. W. Klein, M. Wegener, N. Feth, and S. Linden, “Experiments on second- and third-harmonic generation from magnetic metamaterials,” Opt. Express 15(8), 5238–5247 (2007).
[CrossRef] [PubMed]

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wu, W.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Xu, J.

Xu, Y. L.

Yan, E.

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

Yannopapas, V.

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

Yu, Q. L.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

Yu, Z.

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Zeng, Y.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, W. Y.

Zhang, X.

J. Xu and X. Zhang, “Negative electron energy loss and second-harmonic emission of nonlinear nanoparticles,” Opt. Express 19(23), 22999–23007 (2011).
[CrossRef] [PubMed]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Zhang, X.-G.

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

Zi, J.

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

H. Husu, B. K. Canfield, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Chiral coupling in gold nanodimers,” Appl. Phys. Lett. 93(18), 183115 (2008).
[CrossRef]

Chem. Phys. Lett. (1)

H. Wang, E. Yan, E. Borguet, and K. B. Eisenthal, “Second harmonic generation from the surface of centrosymmetric particles in bulk solution,” Chem. Phys. Lett. 259(1-2), 15–20 (1996).
[CrossRef]

Chem. Rev. (1)

K. B. Eisenthal, “Second harmonic spectroscopy of aqueous nano- and microparticle interfaces,” Chem. Rev. 106(4), 1462–1477 (2006).
[CrossRef] [PubMed]

Comput. Phys. Commun. (1)

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113(1), 49–77 (1998).
[CrossRef]

J. Appl. Phys. (1)

D. Krause, C. W. Teplin, and C. T. Rogers, “Optical surface second harmonic measurements of isotropic thin-film metals: Gold, silver, copper, aluminum, and tantalum,” J. Appl. Phys. 96(7), 3626–3634 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Soc. Jpn. (1)

K. Ohtaka and Y. Tanabe, “Photonic band using vector spherical waves. 1. Various properties of Bloch electric fields and heavy photons,” J. Phys. Soc. Jpn. 65(7), 2265–2275 (1996).
[CrossRef]

Nano Lett. (1)

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in non-centrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007).
[CrossRef] [PubMed]

Nat. Photonics (1)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[CrossRef]

Nature (1)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008).
[CrossRef] [PubMed]

Opt. Express (5)

Phys. Rev. A (6)

J. Du, S. Liu, Z. Lin, J. Zi, and S. T. Chui, “Guiding electromagnetic energy below the diffraction limit with dielectric particle arrays,” Phys. Rev. A 79(5), 051801 (2009).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77(1), 013809 (2008).
[CrossRef]

J. Shan, J. I. Dadap, I. Stiopkin, G. A. Reider, and T. F. Heinz, “Experimental study of optical second-harmonic scattering from spherical nanoparticles,” Phys. Rev. A 73(2), 023819 (2006).
[CrossRef]

J. Petschulat, A. Chipouline, A. Tünnermann, and T. Pertsch, “Multipole nonlinearity of metamaterials,” Phys. Rev. A 80(6), 063828 (2009).
[CrossRef]

S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A 81(5), 053850 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. S. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A 81(2), 023820 (2010).
[CrossRef]

Phys. Rev. B (14)

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009).
[CrossRef]

L. Cao, N. C. Panoiu, R. D. R. Bhat, and R. M. Osgood, Jr., “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B 79(23), 235416 (2009).
[CrossRef]

A. G. F. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B 79(15), 155420 (2009).
[CrossRef]

E. Kim, F. Wang, W. Wu, Z. Yu, and Y. R. Shen, “Nonlinear optical spectroscopy of photonic metamaterials,” Phys. Rev. B 78(11), 113102 (2008).
[CrossRef]

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70(24), 245434 (2004).
[CrossRef]

C. C. Neacsu, G. A. Reider, and M. B. Raschke, “Second-harmonic generation from nanoscopic metal tips: Symmetry selection rules for single asymmetric nanostructures,” Phys. Rev. B 71(20), 201402 (2005).
[CrossRef]

J. Nappa, G. Revillod, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Electric dipole origin of the second harmonic generation of small metallic particles,” Phys. Rev. B 71(16), 165407 (2005).
[CrossRef]

G. Gonella and H.-L. Dai, “Determination of adsorption geometry on spherical particles from nonlinear Mie theory analysis of surface second harmonic generation,” Phys. Rev. B 84(12), 121402(R) (2011).
[CrossRef]

M. D. McMahon, R. Lopez, R. F. Haglund, E. A. Ray, and P. H. Bunton, “Second-harmonic generation from arrays of symmetric gold nanoparticles,” Phys. Rev. B 73(4), 041401(R) (2006).
[CrossRef]

V. Yannopapas and N. V. Vitanov, “Electromagnetic Green’s tensor and local density of states calculations for collections of sphererical scatters,” Phys. Rev. B 75(11), 115124 (2007).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 233402 (2009).
[CrossRef]

J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005).
[CrossRef]

C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B 81(19), 195102 (2010).
[CrossRef]

F. J. García de Abajo, “Multiple scattering of radiation in clusters of dielectrics,” Phys. Rev. B 60(8), 6086–6102 (1999).
[CrossRef]

Phys. Rev. B Condens. Matter (2)

X. D. Wang, X.-G. Zhang, Q. L. Yu, and B. N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B Condens. Matter 47(8), 4161–4167 (1993).
[CrossRef] [PubMed]

A. Moroz, “Density-of-states calculations and multiple-scattering theory for photons,” Phys. Rev. B Condens. Matter 51(4), 2068–2081 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (8)

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[CrossRef] [PubMed]

Y. Pu, R. Grange, C.-L. Hsieh, and D. Psaltis, “Nonlinear optical properties of core-shell nanocavities for enhanced second-harmonic generation,” Phys. Rev. Lett. 104(20), 207402 (2010).
[CrossRef] [PubMed]

J. U. Fürst, D. V. Strekalov, D. Elser, M. Lassen, U. L. Andersen, C. Marquardt, and G. Leuchs, “Naturally phase-matched second-harmonic generation in a whispering-gallery-mode resonator,” Phys. Rev. Lett. 104(15), 153901 (2010).
[CrossRef] [PubMed]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010).
[CrossRef] [PubMed]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90(1), 013903 (2003).
[CrossRef] [PubMed]

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83(20), 4045–4048 (1999).
[CrossRef]

E. Centeno, D. Felbacq, and D. Cassagne, “All-angle phase matching condition and backward second-harmonic localization in nonlinear photonic crystals,” Phys. Rev. Lett. 98(26), 263903 (2007).
[CrossRef] [PubMed]

V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. E. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “Asymmetric optical second-harmonic generation from chiral G-shaped gold nanostructures,” Phys. Rev. Lett. 104(12), 127401 (2010).
[CrossRef] [PubMed]

Science (3)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313(5786), 502–504 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

Other (5)

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

T. F. Heinz, Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (North-Holland, Amsterdam, 1991) Chap. 5.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 363.

S. A. Maier, Plasmonics: Fundamental and Applications (Springer, New York, 2007).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the scattering problem for an ensemble of N spheres consisting of centrosymmetric materials. Here θ n , ϕ n and r n are the coordinate of the nth sphere in the spherical coordinate system, Ω=(θ,ϕ) represents the solid angle of an arbitrary point P. The x, y and z represent three axis directions in corresponding Cartesian coordinate system.

Fig. 2
Fig. 2

The spectra of the scattering cross sections for metal spheres of radii a = 10nm (dashed line), 30 (solid line), and 50 nm (dotted line). (I): the FF; (II): the SH.

Fig. 3
Fig. 3

(Color online) The spatial profile of the amplitude of the electric field, calculated at ω=5.0 eV (panels A and B) and ω=2.7 eV (panels C and D). The radius of the sphere is a = 10 nm. A and C correspond to the FF; B and D to the SH.

Fig. 4
Fig. 4

The spectra of the scattering cross sections for the metallic two-sphere system. The radii of two spheres are taken as a = 10 nm. (I) and (II) correspond to the separation distance d = 22nm, (III) and (IV) to d = 35nm. Solid line and dashed line correspond to the case at the incident direction of the wave perpendicular or parallel to the longitudinal axis of the system, respectively. (I) and (III): the FF; (II)and (IV): the SH.

Fig. 5
Fig. 5

(Color online) The spatial profile of the amplitude of the electric field for the two-sphere system, calculated at ω=4.1 eV for the normal incidence with d = 22nm (panels A and B) and d = 35nm (panels C and D); calculated at ω=6.54 eV for the parallel incidence with d = 22nm (panels E and F) and d = 35nm (panels G and H). The radii of the spheres are a = 10 nm. A, C, E and G correspond to the FF; B, D, F and H to the SH. Here A, B, C, D, E, F, G and H correspond to the points marked in Fig. 3.

Fig. 6
Fig. 6

(Color online) The top two panels show the spectra of the scattering cross section corresponding to a chain of N = 12 metallic spheres under the parallel incidence. The radius of the sphere is a = 10 nm and the separation distance is d = 22nm. (I): the FF; (II): the SH. The spatial profile of the amplitude of the electric field, calculated at ω=5.3 eV ((A) and (B)) and ω=2.82 eV ((C) and (D)), is presented in the bottom panels. The panels A and C correspond to the FF, whereas the panels B and D correspond to the SH. Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

Fig. 7
Fig. 7

(Color online) The same as in Fig. 6, but for the normal incidence. The spatial profile of the amplitude of the electric field, calculated at ω=5.4 eV ((A) and (B)) and ω=2.59 eV ((C) and (D)). Here A and B correspond to the dipolar excitation, and D to the quadrupolar excitation [3, 5].

Fig. 8
Fig. 8

(I) The 3D sample and cross sections. (II) and (III) The spectra of the scattering cross sections corresponding to the cluster as shown in (I) under the incidence of the plane wave from the left side of the sample with a = 10 nm and d = 22nm. (II): the FF; (III): the SH.

Fig. 9
Fig. 9

The spatial profiles of the amplitude of the electric field, calculated at ω=2.21 eV (corresponding to the points A and B in Fig. 7 (II) and (III)). (a) and (b) correspond to the FF and the SH fields at the cross-section as shown by shadow in Fig. 7 (I), respectively, whereas (c) and (d) correspond to the FF and the SH fields at the cross-section as shown by dot-dashed lines in Fig. 7 (I). The other parameters are identical with those in Fig. 7.

Equations (48)

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E n 0 ( x n )= lm { a nlm 0,E J Elm ( x n )+ a nlm 0,H J Hlm ( x n ) }
J Elm ( r )= i k b × j l ( k b r ) X lm ( r ^ ), J Hlm ( r )= j l ( k b r ) X lm ( r ^ ),
[ a nlm 0,H a nlm 0,E ]=4π e i K i r n i l [ X lm * ( Ω i ) X lm * ( Ω i )( × Κ ι ) / k b ],
E n inc ( x n )= lm { a nlm inc,E J Elm ( x n )+ a nlm inc,H J Hlm ( x n ) } ,
E n sc ( x n )= lm { a nlm sc,E H Elm ( x n )+ a nlm sc,H H Hlm ( x n ) } ,
E n in ( x n )= lm { a nlm in,E J Elm s ( x n )+ a nlm inc,H J Hlm s ( x n ) }
J Elm s ( r )= i k s × j l ( k s r ) X lm ( r ^ ), J Hlm s ( r )= j l ( k s r ) X lm ( r ^ ),
H Elm ( r )= i k b × h l ( k b r ) X lm ( r ^ ), H Hlm ( r )= h l ( k b r ) X lm ( r ^ ),
E t inc + E t sc = E t in ,
D r inc + D r sc = D r in ,
H t inc + H t sc = H t in ,
B r inc + B r sc = B r in .
T l E = a nlm sc,E / a nlm inc,E , T l H = a nlm sc,H / a nlm inc,H , C l E = a nlm in,E / a nlm inc,E , C l H = a nlm in,H / a nlm inc,H ,
a nlm inc,E = a nlm 0,E + n n l m ( Ω nlm, n l m EE a n l m sc,E + Ω nlm, n l m EH a n l m sc,H ) ,
a nlm inc,H = a nlm 0,H + n n l m ( Ω nlm, n l m HE a n l m sc,E + Ω nlm, n l m HH a n l m sc,H ) ,
n l m [ δ n n δ l l δ m m T l E ( Ω nlm, n l m EE a n l m sc,E + Ω nlm, n l m EH a n l m sc,H ) ] = T l E a nlm 0,E ,
n l m [ δ n n δ l l δ m m T l H ( Ω nlm, n l m HE a n l m sc,E + Ω nlm, n l m HH a n l m sc,H ) ] = T l H a nlm 0,H .
P surface (2ω) ( r )= P s (2ω) ( θ,φ )δ(ra)= χ s (2) : E (ω) ( r ) E (ω) ( r )δ(ra),
P s (2ω) ( θ,φ )= lm G r,nlm Y lm (θ,φ) r ^ + G M,nlm X lm (θ,φ)+ G E,nlm r ^ × X lm (θ,φ) ,
E n int,(2ω) ( x n )= lm { A nlm int,E J Elm s,(2ω) ( x n )+ A nlm int,H J Hlm s,(2ω) ( x n ) } ,
E n out,(2ω) ( x n )= lm { A nlm out,E H Elm (2ω) ( x n )+ A nlm out,H H Hlm (2ω) ( x n ) } ,
E t out,(2ω) E t in,(2ω) = 4π ε 1 (2ω) t P s,r (2ω) ,
D r out,(2ω) D r in,(2ω) =4π t P s (2ω) ,
H t out,(2ω) H t in,(2ω) =4πi 2ω c r ^ × P s (2ω) ,
B r out,(2ω) B r in,(2ω) =0.
A nlm int,E = ε b (2ω) K b h l (1) ( K b a) A nlm out,E +4πi G E,nlm ε b (2ω) j l ( K 1 a) / K 1 ,
A nlm int,H = h l (1) ( K b a) j l ( K 1 a) A nlm out,H ,
χ s (2) = χ r ^ r ^ r ^ + χ r ^ ( θ ^ θ ^ + φ ^ φ ^ )+ χ ( θ ^ r ^ θ ^ + φ ^ r ^ φ ^ + θ ^ θ ^ r ^ + φ ^ φ ^ r ^ ).
P s (2ω) = r ^ ( χ E r (ω) E r (ω) + χ E t (ω) E t (ω) )+2 χ E r (ω) E t (ω) .
E n sca,(2ω) ( x n )= lm { A nlm sca,E H Elm (2ω) ( x n )+ A nlm sca,H H Hlm (2ω) ( x n ) } ,
E n loc,(2ω) ( x n )= n n [ E n out,(2ω) ( x n )+ E n sca,(2ω) ( x n ) ] ,
E n sca,(2ω) ( x n )= lm { A n lm sca,E H Elm (2ω) ( x n )+ A n lm sca,H H Hlm (2ω) ( x n ) } ,
E n loc,(2ω) ( x n )= n n lm { ( A nlm out,E + A n lm sca,E ) H Elm (2ω) ( x n )+( A nlm out,H + A n lm sca,H ) H Hlm (2ω) ( x n ) } .
E n loc,(2ω) ( x n )= nlm { A nlm loc,E J Elm (2ω) ( x n )+ A nlm loc,H J Hlm (2ω) ( x n ) } ,
A nlm loc,E = n n l m { Ω nlm, n l m EE ( A n l m out,E + A n l m sca,E )+ Ω nlm, n l m EH ( A n l m out,H + A n l m sca,H ) } ,
A nlm loc,H = n n l m { Ω nlm, n l m HE ( A n l m out,E + A n l m sca,E )+ Ω nlm, n l m HH ( A n l m out,H + A n l m sca,H ) } .
A nlm sca,E = T l E A nlm loc,E ,
A nlm sca,H = T l H A nlm loc,H .
n l m [ δ n n δ l l δ m m T l E ( Ω nlm, n l m EE A n l m sca,E + Ω nlm, n l m EH A n l m sca,H ) ] = n l m T l E [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] ,
n l m [ δ n n δ l l δ m m T l H ( Ω nlm, n l m HE A n l m sca,E + Ω nlm, n l m HH A n l m sca,H ) ] = n l m T l H [ Ω nlm, n l m EE A n l m out,E + Ω nlm, n l m EH A n l m out,H ] .
E (2ω) ( r )= nlm ( A nlm out,E + A nlm sca,E ) H Elm (2ω) ( r n )+( A nlm out,H + A nlm sca,H ) H Hlm (2ω) ( r n ) .
C s ( ϖ )= q s ( ϖ;Ω )dΩ ,
q s ( ϖ;Ω )= lim r r 2 Re{ [ E s ( ϖ )× H s ( ϖ ) ] r ^ },
lim r E s ( r )= f ( Ω ) e ikr r ,
lim r H s ( r )= r ^ × f ( Ω ) e ikr r .
f ( Ω )= n e i k b r ^ r n i l1 lm [ X lm ( Ω ) a nlm H /k r ^ × X lm ( Ω ) a nlm E /k ] ,
q s ( ϖ;Ω )= | f ( Ω ) | 2 .
ε s (ω)=1 ω p 2 ω( ω+iν ) ,

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