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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23386–23399
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Boundary element method for surface nonlinear optics of nanoparticles

Jouni Mäkitalo, Saku Suuriniemi, and Martti Kauranen  »View Author Affiliations


Optics Express, Vol. 19, Issue 23, pp. 23386-23399 (2011)
http://dx.doi.org/10.1364/OE.19.023386


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Abstract

We present the frequency-domain boundary element formulation for solving surface second-harmonic generation from nanoparticles of virtually arbitrary shape and material. We use the Rao-Wilton-Glisson basis functions and Galerkin’s testing, which leads to very accurate solutions for both near and far fields. This is verified by a comparison to a solution obtained via multipole expansion for the case of a spherical particle. The frequency-domain formulation allows the use of experimentally measured linear and nonlinear material parameters or the use of parameters obtained using ab-initio principles. As an example, the method is applied to a non-centrosymmetric L-shaped gold nanoparticle to illustrate the formation of surface nonlinear polarization and the second-harmonic radiation properties of the particle. This method provides a theoretically well-founded approach for modelling nonlinear optical phenomena in nanoparticles.

© 2011 OSA

1. Introduction

Second-harmonic generation (SHG) is a nonlinear optical phenomenon, in which a field frequency component oscillating at double the frequency of the exciting field is generated. In the electric dipole approximation of material response, SHG vanishes in the bulk of materials with inversion symmetry [1

1. Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989). [CrossRef]

]. At the interface, this symmetry is broken, which gives rise to surface SHG, which is very sensitive tool to probe surfaces.

Bulk SHG is allowed in centrosymmetric media if magnetic dipole and electric quadrupole responses are considered. The surface and bulk contributions to SHG are not fully separable in the sense that part of the bulk response behaves as the surface contribution [2

2. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987). [CrossRef]

]. This inseparable contribution can be included in the surface response, which is then described by an effective surface susceptibility. In general the surface and bulk responses of centrosymmetric media can be of similar magnitude. However, theoretical considerations estimate that for materials with high permittivity, such as metals, the surface contribution can be an order of magnitude greater than bulk contribution [3

3. P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986). [CrossRef]

]. Also, experimental measurements suggest that the effective surface suceptibility is sufficient to describe the total SHG from gold structures [4

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

].

The development of nanofabrication processes has enabled the study of nanoscale structures and their optical properties. From subwavelength structures, it is possible to construct metamaterials with tailored optical properties not found in nature. These properties depend on the size, shape and material of the particles, the properties of the surrounding medium [5

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

] and the possible array structure [6

6. H. Husu, J. Mäkitalo, R. Siikanen, G. Genty, H. Pietarinen, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Spectral control in anisotropic resonance-domain metamaterials,” Opt. Lett. 36, 2375–2377 (2011). [CrossRef] [PubMed]

]. Metal nanostructures exhibit plasmon resonances, which result in high local fields near particle surfaces. This can further amplify any nonlinear processes. Because SHG is sensitive to inversion symmetry, the observable second-harmonic (SH) far field also depends on the symmetry properties of the particles and the particle array.

The aforementioned methods all either approximate the linear optical response or are based on a field expansion suitable to a very restricted class of geometries. There have been attempts to model SHG from arbitrarily shaped 3-D particles by employing numerical schemes directly to partial differential equations. These include the Finite-Difference Time-Domain method [22

22. W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B 78, 195416 (2008).

, 23

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

] and the Finite Element Method [24

24. G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25, 955–960 (2008). [CrossRef]

]. In these methods one resorts to artificial absorbing boundaries to simulate an unbounded domain.

Although modelling nonlinear electromagnetic phenomena usually requires time-domain formulations, these present one significant drawback. They lead to cumbersome descriptions of the material dispersion and one is required to use simplified dispersion models to perform response convolutions. Thus it is difficult to use measured material parameters directly. Similar drawback applies to modelling the nonlinear response as one needs to use full time-domain models and ab-initio parameters even though only time-harmonic responses are often of interest. In most practical cases harmonic generation can be decomposed into coupled linear problems, so that frequency-domain methods can be used and thus measured material parameters can be directly utilized.

Recently, frequency-domain integral operator methods have gained popularity in the study of the linear response of nanoparticles. Depending on the electromagnetic properties of the media, these can be formulated as volume integral or surface integral methods. The finite-dimensional form of surface integral operator based scattering problems is usually referred to as the Boundary Element Method (BEM) or Surface Integral Equation method. BEM has been used for solving scattering from dielectric and ideally conducting bodies for a long time [25

25. K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986). [CrossRef]

], but recently it has been applied also to scattering by plasmonic nanostructures [26

26. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003). [CrossRef]

31

31. B. Gallinet, A. Kern, and O. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]

].

In this work, we develop a BEM formulation for solving the surface SHG problem in lossy dielectric particles of virtually arbitrary shape. The formulation builds upon the undepleted-pump approximation, which allows the SH fields to be solved in three steps: 1) Solve a linear scattering problem at the fundamental frequency. 2) Determine the locally-varying source polarization at the second-harmonic frequency from the fundamental fields. 3) Solve a linear scattering problem at the SH frequency by using the polarization as a source. This treatment is valid in the case of low SHG conversion efficiency, which is the usual case for nanoparticles as verified by measurements. Note that although the problems are linear, as a whole they model a nonlinear phenomenon. The formulation allows an arbitrary excitation source, e.g., a plane-wave, focused Gaussian beam or an oscillating dipole. An advantage over reciprocity based approach is that the whole radiation pattern is obtained by solving a single scattering problem and also the SH near fields can be obtained. The integral operator formulation of the problem is developed in Section 2. We then present a finite-dimensional approximate representation in Section 3. For validation purposes, we also develop a multipole solution in Section 4, so that we may test our BEM method for the case of a spherical particle. In Section 5 we bring all this together and show comparison between BEM results and the multipole method and characterize the SH radiation properties of an L-shaped particle. We discuss the properties of the developed method in Section 6 and conclude in Section 7 with reference to future work.

2. Problem statement and integral operators

The solution domain of our SH scattering problem is illustrated in Fig. 1(a). The domain of the electric field E and magnetic field H is ℝ3 and it is divided into an unbounded exterior domain V1 and a compact interior domain V2. The compact parts of the domain boundaries are denoted ∂Vi and they are oriented such, that their respective normal vectors ni point out of Vi. The surface of the scatterer is denoted by S and it is assumed to be piece-wise smooth and oriented with surface normal n pointing into V1. The linear electromagnetic properties of the domains are characterized by the permittivity ɛi ∈ ℂ and permeability μi ∈ ℝ, which also define the wave impedance ηi = (μi/ɛi)1/2.

Fig. 1 (a) Solution domains of the second-harmonic scattering problem and the interface S, over which the nonlinear source is defined. (b) RWG-basis function. Arrows indicate surface current density. (c) Icosahedral triangle mesh. (d) Triangular mesh of the L-shaped particle cut by a symmetry plane.

The SHG problem involves fields oscillating at two different frequencies: the fundamental fields ei and hi at frequency ω and the SH fields Ei and Hi at frequency Ω = 2ω with i = 1,2 denoting the domains. We assume that the surface SH sources may be described in terms of a surface polarization distribution defined as PS = ɛ0χ(2) : e2e2 over the surface S of the particle [3

3. P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986). [CrossRef]

]. Of special interest are locally isotropic surfaces i.e. surfaces with local Cν symmetry, so that the effective susceptibility tensor χ(2) has only seven non-vanishing components: χnnn(2), χnss(2)=χntt(2), χssn(2)=χsns(2)=χttn(2)=χtnt(2), where n refers to local normal and s,t refer to the two mutually orthogonal tangent vectors.

Our formulation builds upon the undepleted-pump approximation, where the SH fields do not couple back to the fundamental field. This is justified by the fact that the measured SH signals are always orders of magnitude weaker than the source at the fundamental frequency. The problems at both frequencies are then linear and we can first solve the fundamental fields and then calculate the polarization PS, which acts as a source for the SH fields.

The time-dependence of harmonic fields is taken to be exp(−iωt). The fundamental fields in the domains Vi satisfy the Helmholtz equation ∇ × ∇ × fki(ω)2f = 0, where ki(ω)2 = ω2ɛi(ω)μi(ω) and f ∈ {e,h}. The expressions for the SH fields are the same with substitutions eE, hH and ω → Ω. In the SH problem, the surface polarization implies the following interface conditions over S for tangential components of the SH fields [34

34. T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman (Elsevier, Amsterdam, 1991) p. 353.

]:
(E1E2)tan=1ɛSPnS,
(1)
(H1H2)tan=iΩPS×n,
(2)
where PnS=nPS and ɛ′ is called the selvedge region permittivity [35

35. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980). [CrossRef]

]. This model assumes that the medium in the infinitely thin layer between ∂V1 and ∂V2 can be modelled macroscopically. In practice it may not be possible to determine this permittivity and one needs to resort to ad hoc values. The problems of ɛ′ are not in the scope of this work.

By assuming that the media in domains Vi are homogeneous, we may express the fields at any point by specifying the fields over the compact domain boundaries ∂Vi. This is governed by the Stratton-Chu equations [36

36. J. Stratton, Electromagnetic theory (New York and London: McGraw-Hill, 1941).

]. To express the equations in our particular case, we define the Green’s function Gl(r,r′) = exp(iklR)/(4πR) with R = ||rr′||2 and the following integro-differential linear operators:
𝒟lf(r)=iΩμlVlGl(r,r)f(r)dS1iΩɛlVlGl(r,r)f(r)dS,
(3)
𝒦lf(r)=Vl[Gl(r,r)]×f(r)dS.
(4)
The fields in Vi can now be expressed as
Ei=𝒟ini×Hi𝒦iEi×ni,
(5)
Hi=𝒦ini×Hi1ηi2𝒟iEi×ni.
(6)
Here the Silver-Müller radiation conditions are imposed on the fields in V1 so that the fields represent purely outgoing waves [37

37. D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).

]. The nonlinear source is not present in the integrals, because the source is introduced only in the interface conditions in Eqs. (1) and (2) and does not affect the representation of the fields in V1 and V2.

It is customary to introduce the equivalent surface current densities, which are defined as JiS=(ni)×Hi and MiS=Ei×(ni). Unlike in the linear problem, we now have four surface current densities to solve for due to the discontinuities in the tangential fields on S.

We take the limit where the boundaries ∂Vi approach S and enforce the interface conditions. By using the above representations of the fields, we obtain the infinite-dimensional problem: Given PS, seek JiS, MiS:S3, such that Eqs.
(𝒟1J1S+𝒦1M1S𝒟2J2S𝒦2M2S)tan=1ɛSPnS,
(7)
(𝒦1J1S+1η12𝒟1M1S+𝒦2J2S1η22𝒟2M2S)tan=iΩPS×n
(8)
J1S+J2S=iΩPtanS,
(9)
M1S+M2S=1ɛn×SPnS
(10)
hold. Unfortunately, solving this problem in closed form is impossible for anything but the simplest of surfaces S. Note that we could substitute e.g. J2S and M2S from the Eqs. (9) and (10) into Eqs. (7) and (8), but this would result in numerically cumbersome integrals.

A similar formulation for the fundamental fields can be obtained by setting PS = 0 and adding the tangential components of the incident fields to the right-hand-side of Eqs. (5) and (6) for V1. This leads to the traditional formulation with only two unknown surface current densities as shown in e.g. [29

29. A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009). [CrossRef]

]. Denoting the current densities in this case with lower case letters, it follows that j1 = −j2 and m1 = −m2. The polarization PS is then evaluated from the field e2, whose components are given directly by the relations to corresponding equivalent surface current densities: m2 = −e2 × n2 and ∇S · j2 = −iωɛ2n2 · e2.

3. Finite dimensional formulation: the Method of Moments

To obtain approximate solutions to the SH problem, we employ the Method of Moments [38

38. R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993). [CrossRef]

]. We seek such solutions from a finite-dimensional space that ensures proper continuity. A good choice for such a space is the one spanned by the Rao-Wilton-Glisson (RWG) functions, which are divergence-conforming affine functions that implicitly enforce the continuity of surface current and the conservation of charge [39

39. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982). [CrossRef]

]. These functions are geometrically attributed to the edges of a triangular mesh and their support, denoted by Sn, is a pair of adjacent triangles, as is illustrated in Fig. 1(b) (by the support of a function f we mean the set Sf = {xS|f(x) ≠ 0}). The unknown surface current densities are expanded using the RWG basis functions fn as
J1S=n=1Nαnfn,M1S=n=1Nβnfn,J2S=n=1Nγnfn,M2S=n=1Nδnfn,
where we have 4N unknowns αn, βn, γn, δn ∈ ℂ, that can be arranged into a vector x = (α1,...,αN, β1,...,βN, γ1,...,γN,δ1,...,δN)T. To obtain these coefficients, we use Galerkin’s testing [38

38. R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993). [CrossRef]

] with the inner-product 〈f,g〉 = ∫Sf · gdS.

The testing procedure leads to a linear system of equations, which can be expressed as Zx = b, where we have the system matrix Z:
Z=(D(1)K(1)D(2)K(2)K(1)1η12D(1)K(2)1η22D(2)F0F00F0F).
(11)
The matrix representations of the operators are:
Dmn(l)=iΩμlSmdSfm(r)SndSfn(r)Gl(r,r)1iΩɛlSmdS[Sfm(r)]SndS[Sfn(r)]Gl(r,r),
(12)
Kmn(l)=SmdSfm(r)SndS[Gl(r,r)]×fn(r),
(13)
Fmn=SmSndSfm(r)fn(r),
(14)
where the constitutive parameters and Gl are evaluated at Ω. By enforcing both the electric and magnetic field interface conditions (the PMCHW testing), we avoid the internal resonance problem of BEM and thus also ensure robustness against plasmonic resonances [29

29. A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009). [CrossRef]

].

The nonlinear surface polarization appears only in the source vector b:
b=(bn1T,bt1T,bt2T,bn2T)T,
(15)
bmn1=1ɛSmSfmPnSdS=1ɛ±Sfm±Tm±PnSdS,
(16)
bmt1=iΩSmfmPS×ndS,bmt2=iΩSmfmPSdS
(17)
and Tm± denotes the associated triangles. The element bn2 reduces to a contour integral by the identity
Sf×FndS=SfFdrSf×FndS.
(18)
For a triangular patch surface n is discontinuous, which implies that this identity should be applied piece-wise over each triangle. The curl of RWG-basis functions vanishes over their support and we obtain
bmn2=1ɛSmfmn×SPnSdS=1ɛSmSPnS×fmndS=1ɛ±Tm±PnSfmdr.
(19)
The integrals over the edges common to Tm+ and Tm do not necessarily vanish. The obtained source elements bn1 and bn2 clearly vanish if PnS is constant, which is expected as the source in its original form depends on the surface gradient of PnS.

In case only the component PnS is considered significant, we have bt1 = bt2 = 0 and Eq. (9) implies that J2S=J1S. Then the problem is reduced to:
Z=((D(1)+D(2))K(1)K(2)(K(1)+K(2))1η2D(1)1η22D(2)0FF),b=(bn10bn2),
(20)
which is computationally less arduous.

In the presented formulation, all the integrals can be evaluated with high precision by utilizing the singularity subtraction technique [40

40. I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006). [CrossRef]

] and by using a high-order Gaussian quadrature for the resulting integrals with smooth kernels. In our calculations, a two-term singularity subtraction and 13-point Gauss-Legendre quadrature were used.

4. Solution in multipoles

To validate our BEM implementation, we develop an analytic solution for the same problem for the special case of a spherical particle by using the multipole expansion. This has been done previously in the small particle limit with the same interface conditions as here [13

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

] and for an arbitrarily large sphere with different interface conditions [12

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

]. An indirect far field solution has also been developed by using the Lorentz reciprocity [10

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

].

5. Numerical results

5.1. The spherical particle

We take a moderately sized spherical gold nanoparticle of radius a = 50 nm in vacuum, whence a plasmonic resonance takes place at the wavelength of λ = 520 nm. The excitation source is an x-polarized plane wave propagating in z-direction. The multipole solution of the fundamental field is sufficiently accurate with lmax = 4 (m = ±1) and the SH solution with lmax = 3 (all m-values included). We choose as the only nonzero tensor component χnnn(2)=1.

For the BEM we used two regularly triangulated icosahedral meshes, with 1280 and 5120 triangles and a more irregular mesh with 1454 triangles generated by GMSH’s MeshAdapt algorithm. The first mesh is shown in Fig. 1(c).

The computed radiated power per unit solid angle σ and the relative errors between the multipole and BEM solutions are shown in Fig. 2. The radiated power, defined by the far fields, is very accurate, the relative error being practically sub one percent. Surprisingly, the finest regular mesh yields the largest error, while the irregular mesh results in almost lowest errors in general. Thus the irregularity of the mesh does not deteriorate far field accuracy, but the maximum obtainable accuracy might be limited.

Fig. 2 (a) SH power radiated per unit solid angle σ for two different azimuthal angles ϕ. The incident field at frequency ω is polarized in x-direction and propagates in z-direction. The results given by the multipole method are virtually indistinguishable from the BEM results. The inset shows the whole radiation pattern. (b) The relative errors in σ. Solid lines correspond to ϕ = 0° and dashed lines to ϕ = 90°. Blue, red and olive correspond to meshes with 1280, 5120 and 1454 triangles, respectively.

The SH electric field’s x-component amplitude near the sphere and its relative errors are shown in Fig. 3. Again the relative error is mostly below one percent, peaking at the point of highest field enhancement. Now the finest regular mesh yields markedly most accurate results and the irregular mesh is practically on par with the regular mesh of slightly lower triangle density. Overall the mesh refinement does not significantly remedy the high error at the point of highest enhancement.

Fig. 3 (a) SH E-field’s x-component amplitude as a function of position along x-axis through the sphere. Black, blue and red lines depict the multipole solution and BEM solutions with 1280 and 5120 triangle meshes, respectively. Olive line depicts the BEM solution with irregular mesh. The inset shows |Ex| on the z = 0 plane and the dashed line shows the actual plot line. (b) The relative error in |Ex| as a function of position. Colors match those of (a).

We note that while the error in the far field quantity is largest for the densest mesh, it is already very low. Because the total size of the system is in the deep sub-wavelength regime, the far field can be very insensitive to small variations in the near field and the error cannot be expected to display a regular behaviour when the mesh is changed. The behaviour of the error is specific to each problem, this being just one particular case.

The normal component of the fundamental field en is represented in the method with piecewise constant functions, and the SH response depends on the surface gradient of en2, which is only differentiable in the weak sense. Considering this, the results are surprisingly accurate, although it is quite well known that the BEM can produce highly accurate far fields even if using very coarse meshes.

5.2. The L-shaped particle

The second-order nonlinear properties of L-shaped gold nanoparticles have been extensively studied experimentally (e.g. [43

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

]). Although SHG from these particles has been measured, the simulations have been limited to modelling the linear response [44

44. H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express 18, 16601–16606 (2010). [CrossRef] [PubMed]

] and making indirect estimations of SHG. Here we apply our method to characterize the surface second-harmonic response of a single gold L-shaped nanoparticle in vacuum at plasmon resonances.

Fig. 4 (a) The extinction cross-sections of the L-shaped particle for two incident plane wave polarizations. The inset illustrates the particle and the chosen cartesian coordinate system. (b) The amplitude of the nonlinear surface polarization evaluated with the measured relative susceptibility tensor. The two plots correspond to incident plane-wave of x- and y-polarization and they are normalized separately. The amplitudes are normalized with respect to the y-polarized case.

The effective second-order susceptibility components for gold have been measured from thin films, and their relative values are χnnn(2)=250, χntt(2)=1 and χttn(2)=3.6 when the polarization is evaluated using the fields inside the particle [4

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

]. By using these values, the nonlinear surface polarization was computed by using the two polarizations for the incident wave at the corresponding resonance wavelengths. The results are plotted in Fig. 4(b). It is evident, that the nonlinear source polarization is driven into the corners of the particle. This suggests that only a small fraction of the particle surface gives rise to significant SHG, although the whole particle can affect the formation of the polarization in the first place. Small defects in these corners could drastically alter the SHG. The localization also implies that one must take care that the discretization of the problem is sufficiently smooth around sharp edges and corners. For this reason we used a mesh where the edges are carefully rounded as depicted in Fig. 1(d). As a last remark, the polarizations are practically symmetrical with respect to the z = 0 plane. It will be worthwhile to investigate if e.g. altering the height of the particle will induce phase retardation to the fundamental fields and give rise to less symmetrical source polarization.

The surface polarization does not directly tell how the system actually radiates second-harmonic waves. The full radiation patterns were computed with the developed method for the two incident polarizations at their resonance wavelengths. We attempt to gain insight into the importance of the different susceptibility components by calculating the radiated power by using the full susceptibility tensor and each tensor component separately. The results are shown in Fig. 5, where it can be seen, that at least in this case, χnnn(2) clearly dominates the response. It is also clear that approximately the highest power per unit solid angle goes to forward and backward directions in the case of x-polarized input, but this is not the case for y-polarized input. The plots are symmetrical with respect to the plane x = 0, which is required for a valid solution to the problem. Symmetry considerations also dictate that SHG intensity in the forward and backward directions must be the same and this was verified to hold within 1 % relative error margin, except for the case of Fig. 5(g), for which the error was 2.8 %. The actual intensity in this case is, however, very weak compared to the other cases.

Fig. 5 Radiated second-harmonic power per unit solid angle from an L-shaped gold nanoparticle. In plots (a)–(d) the incident wave is x-polarized and in (e)–(h) y-polarized. The tensor components used in the computations are indicated under the plots. The numbers above the plots denote the maximum power per unit solid angle normalized to the full tensor case separately for both incident polarizations. The ratio of maximum power per unit solid angle of cases (a) and (e) is 0.046.

In the measurements, one is usually interested in the x- and y-components of the second-harmonic signal in the forward direction. Symmetry considerations dictate that only y-component can be nonzero in the case of an ideal particle. The fulfilment of this condition has been of considerable interest in the measurements, and it has been observed that small defects can easily brake this rule. For validation purposes, we made sure that our method gives rise to SHG that obeys this symmetry rule

6. Discussion

The developed method is applicable to a very general class of particle geometries: it is required that the particle is topologically homeomorphic with a sphere. The method is also applicable to a broad range of frequencies. This range is bounded by the low-frequency breakdown of BEM at very low frequencies and by the computational cost at very high frequencies. The frequency domain formulation allows the direct use of measured permittivity and second-order susceptibility values, which is a great advantage in scattering problems. The low refractive index contrast limit of [9

9. A. de Beer, S. Roke, and J. Dadap, “Theory of optical second-harmonic and sum-frequency scattering from arbitrarily shaped particles,” J. Opt. Soc. Am. B 28, 1374–1384 (2011). [CrossRef]

] is obtained by making the Born approximation in Eqs. (5) and (6) i.e. by operating on incident fields to obtain an explicit solution.

It is also important to consider the computational burden of new numerical techniques. The presented BEM yields a dense, non-Hermitian system matrix. Thus, if we have N basis functions, the matrix build time and memory requirements scale proportionally to N2. However, due to the block structure of the matrix, the matrix build time is not essentially different from the case of the linear scattering problem. If direct methods are used to solve the linear system of equations, the time complexity is of the order O(N3). The general ”rule of thumb” is that at least ten basis functions are needed per wavelength, but sharp geometrical features may require locally denser mesh. However, BEM tends to yield high far field accuracy with very few basis functions [45

45. A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996). [CrossRef]

]. Real samples usually consist of arrays of particles on a substrate. BEM can be conveniently generalized for modelling also this type of systems by using a periodic Green’s function, which can be efficiently evaluated by the Ewald’s method. It is also possible to significantly reduce the memory requirements and solution time by utilizing the Adaptive Cross Approximation or the Fast Multipole Method. The latter can reduce matrix-vector product time complexity to N logN in iterative solution methods.

As has been pointed out before, it is possible to use the Lorentz reciprocity for obtaining the SH scattered far field. Assume that we want to find out u* · E (u* denotes complex conjugate of u) far away for some unit vector u. We first solve the linear scattering problem at frequency ω for given excitation to obtain the nonlinear surface polarization PS. Then we solve another linear problem at frequency Ω, where the excitation is a plane wave incident from the observation direction with polarization u yielding solution E′. The Lorentz reciprocity [46

46. L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

] then states that
u*E=iΩSPSEdS.
(28)
The reciprocity relation is convenient, because it does not depend on the linear polarization of the materials induced by the second-harmonic source and thus we avoid the need to solve the nonlinear scattering problem directly. This method demands approximately one fourth of the memory than BEM for a general second-order susceptibility tensor χ(S) and is at least four times less time consuming if only a single scattering direction is of interest. If one desires to solve the whole radiation pattern, then BEM will be superior. This can be useful e.g. when we wish to simulate the SHG signal collected by an objective with large numerical aperture in nonlinear microscopy of nanostructures. Also, attempting to obtain the SH fields near the particle by using the reciprocity is not very convenient and even for the far fields, only relative scattering amplitude is obtained. As a final note, because in BEM one solves directly the fields on the boundary of the particle, the integral (28) can be evaluated conveniently in closed form in the case of RWG-basis.

7. Conclusion

We presented a BEM formulation for solving surface second-harmonic generation from particles of arbitrary shape and material. The comparison of the results to accurate multipole solutions for a sphere revealed that the developed method has potential in modelling SHG in complicated structures. Both near and far fields exhibit relative errors in the range 0.1 %–10 % when using a practically feasible number of basis functions.

We also characterized the SHG response of an L-shaped gold nanoparticle, whose second-order nonlinear properties have been studied experimentally. The calculations suggest that the nonlinear surface polarization is driven into the sharp edges of the particle and that the second-order nonlinear susceptibility component χnnn(2) dominates the second-harmonic response as suggested by its large relative magnitude.

Although the treatment here was focused on surface SHG, it is in principle possible to extend the method for modelling bulk SHG that originates from higher microscopic multipoles. This would be done by considering the general Stratton-Chu equations with source volume current densities, which depend on gradients of the fundamental electric field. The accurate evaluation of these gradients near the particle surface is nontrivial since it requires proper treatment of hypersingular integral kernels and thus requires further analysis.

The treatment presented here could also be easily extended to modelling e.g. sum-frequency generation and higher harmonic generation. It is also possible to give up the undepleted-pump approximation and seek a solution to a fully coupled problem. This will give rise to a large nonlinear system of equations, so that it will become necessary to exploit geometrical symmetry, advanced matrix compression or possibly the Fast Multipole Method.

The BEM can also be extended to modelling spatially periodic structures by employing periodic Green’s functions, which can be rapidly evaluated by e.g. the Ewalds method. Also scattering problems consisting of multiple bodies of different media are straight-forward to implement in BEM. These additional features should be implemented before a realistic comparison to measurements can be carried out.

To conclude, the presented method enables accurate simulation of nonlinear phenomena in plasmonic nanostructures. This can be used in the design of new kinds of nanostructures and metamaterials with special nonlinear optical properties.

Acknowledgments

JM acknowledges support from the Graduate School of Tampere University of Technology. We thank Stefan Kurz from the Department of Electronics of Tampere University of Technology for helpful discussions about theoretical details.

References and links

1.

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989). [CrossRef]

2.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987). [CrossRef]

3.

P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254 (1986). [CrossRef]

4.

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

5.

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

6.

H. Husu, J. Mäkitalo, R. Siikanen, G. Genty, H. Pietarinen, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Spectral control in anisotropic resonance-domain metamaterials,” Opt. Lett. 36, 2375–2377 (2011). [CrossRef] [PubMed]

7.

J. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987). [CrossRef]

8.

S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70, 115106 (2004). [CrossRef]

9.

A. de Beer, S. Roke, and J. Dadap, “Theory of optical second-harmonic and sum-frequency scattering from arbitrarily shaped particles,” J. Opt. Soc. Am. B 28, 1374–1384 (2011). [CrossRef]

10.

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

11.

J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 75–84 (1996). [CrossRef] [PubMed]

12.

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

13.

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

14.

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

15.

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

16.

C. Biris and N. Panoiu, “Excitation of linear and nonlinear cavity modes upon interaction of femtosecond pulses with arrays of metallic nanowires,” Appl. Phys. A pp. 1–5 (2011).

17.

L. Cao, N. Panoiu, and R. Osgood Jr, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B 75, 205401 (2007). [CrossRef]

18.

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

19.

W. Nakagawa, R.-C. Tyan, and Y. Fainman, “Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation,” J. Opt. Soc. Am. A 19, 1919–1928 (2002). [CrossRef]

20.

B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B 24, 1105–1112 (2007). [CrossRef]

21.

T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for analyzing higher harmonic generation in multilayer nanostructures,” J. Opt. Soc. Am. B 27, 1118–1130 (2010). [CrossRef]

22.

W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B 78, 195416 (2008).

23.

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

24.

G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B 25, 955–960 (2008). [CrossRef]

25.

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986). [CrossRef]

26.

J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett. 90, 57401 (2003). [CrossRef]

27.

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. D. Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14, 9988–9999 (2006). [CrossRef] [PubMed]

28.

G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. 8, 631–636 (2008). [CrossRef] [PubMed]

29.

A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009). [CrossRef]

30.

B. Gallinet and O. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Phot. Nano. Fund. Appl. 8, 278–284 (2010). [CrossRef]

31.

B. Gallinet, A. Kern, and O. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). [CrossRef]

32.

A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B 27, 408–416 (2010). [CrossRef]

33.

M. Centini, A. Benedetti, C. Sibilia, and M. Bertolotti, “Coupled 2D Ag nano-resonator chains for enhanced and spatially tailored second harmonic generation,” Opt. Express 19, 8218–8232 (2011). [CrossRef] [PubMed]

34.

T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman (Elsevier, Amsterdam, 1991) p. 353.

35.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389 (1980). [CrossRef]

36.

J. Stratton, Electromagnetic theory (New York and London: McGraw-Hill, 1941).

37.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).

38.

R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993). [CrossRef]

39.

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982). [CrossRef]

40.

I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res. 63, 243–278 (2006). [CrossRef]

41.

J. Jackson, Classical electrodynamics (John Wiley & Sons inc., 1999).

42.

P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972). [CrossRef]

43.

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

44.

H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express 18, 16601–16606 (2010). [CrossRef] [PubMed]

45.

A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag. 44, 500–503 (1996). [CrossRef]

46.

L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(190.2620) Nonlinear optics : Harmonic generation and mixing
(240.4350) Optics at surfaces : Nonlinear optics at surfaces
(250.5403) Optoelectronics : Plasmonics
(290.5825) Scattering : Scattering theory
(310.6628) Thin films : Subwavelength structures, nanostructures

ToC Category:
Nonlinear Optics

History
Original Manuscript: September 20, 2011
Revised Manuscript: October 14, 2011
Manuscript Accepted: October 14, 2011
Published: November 1, 2011

Citation
Jouni Mäkitalo, Saku Suuriniemi, and Martti Kauranen, "Boundary element method for surface nonlinear optics of nanoparticles," Opt. Express 19, 23386-23399 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23386


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References

  1. Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature337, 519–525 (1989). [CrossRef]
  2. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B35, 9091–9094 (1987). [CrossRef]
  3. P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B33, 8254 (1986). [CrossRef]
  4. F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B80, 233402 (2009). [CrossRef]
  5. K. Kelly, E. Coronado, L. Zhao, and G. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B107, 668–677 (2003). [CrossRef]
  6. H. Husu, J. Mäkitalo, R. Siikanen, G. Genty, H. Pietarinen, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Spectral control in anisotropic resonance-domain metamaterials,” Opt. Lett.36, 2375–2377 (2011). [CrossRef] [PubMed]
  7. J. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B4, 481–489 (1987). [CrossRef]
  8. S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70, 115106 (2004). [CrossRef]
  9. A. de Beer, S. Roke, and J. Dadap, “Theory of optical second-harmonic and sum-frequency scattering from arbitrarily shaped particles,” J. Opt. Soc. Am. B28, 1374–1384 (2011). [CrossRef]
  10. A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B79, 155420 (2009). [CrossRef]
  11. J. Dewitz, W. Hübner, and K. Bennemann, “Theory for nonlinear Mie-scattering from spherical metal clusters,” Zeitschrift für Physik D Atoms, Molecules and Clusters37, 75–84 (1996). [CrossRef] [PubMed]
  12. Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B70, 245434 (2004). [CrossRef]
  13. J. Dadap, J. Shan, and T. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B21, 1328–1347 (2004). [CrossRef]
  14. C. Biris and N. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B81, 195102 (2010). [CrossRef]
  15. C. Biris and N. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express18, 17165–17179 (2010). [CrossRef] [PubMed]
  16. C. Biris and N. Panoiu, “Excitation of linear and nonlinear cavity modes upon interaction of femtosecond pulses with arrays of metallic nanowires,” Appl. Phys. A pp. 1–5 (2011).
  17. L. Cao, N. Panoiu, and R. Osgood, “Surface second-harmonic generation from surface plasmon waves scattered by metallic nanostructures,” Phys. Rev. B75, 205401 (2007). [CrossRef]
  18. L. Cao, N. Panoiu, R. Bhat, and R. Osgood Jr, “Surface second-harmonic generation from scattering of surface plasmon polaritons from radially symmetric nanostructures,” Phys. Rev. B79, 235416 (2009). [CrossRef]
  19. W. Nakagawa, R.-C. Tyan, and Y. Fainman, “Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation,” J. Opt. Soc. Am. A19, 1919–1928 (2002). [CrossRef]
  20. B. Bai and J. Turunen, “Fourier modal method for the analysis of second-harmonic generation in two-dimensionally periodic structures containing anisotropic materials,” J. Opt. Soc. Am. B24, 1105–1112 (2007). [CrossRef]
  21. T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for analyzing higher harmonic generation in multilayer nanostructures,” J. Opt. Soc. Am. B27, 1118–1130 (2010). [CrossRef]
  22. W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B78, 195416 (2008).
  23. Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B79, 235109 (2009). [CrossRef]
  24. G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. Brevet, “Multipolar second-harmonic generation in noble metal nanoparticles,” J. Opt. Soc. Am. B25, 955–960 (2008). [CrossRef]
  25. K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag.34, 758–766 (1986). [CrossRef]
  26. J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. Garcia de Abajo, “Optical properties of gold nanorings,” Phys. Rev. Lett.90, 57401 (2003). [CrossRef]
  27. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. D. Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express14, 9988–9999 (2006). [CrossRef] [PubMed]
  28. G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett.8, 631–636 (2008). [CrossRef] [PubMed]
  29. A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A26, 732–740 (2009). [CrossRef]
  30. B. Gallinet and O. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Phot. Nano. Fund. Appl.8, 278–284 (2010). [CrossRef]
  31. B. Gallinet, A. Kern, and O. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A27, 2261–2271 (2010). [CrossRef]
  32. A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B27, 408–416 (2010). [CrossRef]
  33. M. Centini, A. Benedetti, C. Sibilia, and M. Bertolotti, “Coupled 2D Ag nano-resonator chains for enhanced and spatially tailored second harmonic generation,” Opt. Express19, 8218–8232 (2011). [CrossRef] [PubMed]
  34. T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman (Elsevier, Amsterdam, 1991) p. 353.
  35. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21, 4389 (1980). [CrossRef]
  36. J. Stratton, Electromagnetic theory (New York and London: McGraw-Hill, 1941).
  37. D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, vol. 93 (Springer Verlag, 1998).
  38. R. Harrington, Field computation by moment methods (Wiley-IEEE Press, 1993). [CrossRef]
  39. S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag.30, 409–418 (1982). [CrossRef]
  40. I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Elec. Res.63, 243–278 (2006). [CrossRef]
  41. J. Jackson, Classical electrodynamics (John Wiley & Sons inc., 1999).
  42. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370 (1972). [CrossRef]
  43. S. Kujala, B. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett.98, 167403 (2007). [CrossRef] [PubMed]
  44. H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express18, 16601–16606 (2010). [CrossRef] [PubMed]
  45. A. F. Peterson, D. R. Wilton, and R. E. Jorgenson, “Variational nature of Galerkin and non-Galerkin moment method solutions,” IEEE Trans. Antennas Propag.44, 500–503 (1996). [CrossRef]
  46. L. D. Landau and E. M. Lifshits, The electrodynamics of continuous media (Pergamon, Oxford, 1960).

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