## Boundary element method for surface nonlinear optics of nanoparticles |

Optics Express, Vol. 19, Issue 23, pp. 23386-23399 (2011)

http://dx.doi.org/10.1364/OE.19.023386

Acrobat PDF (1969 KB)

### 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

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]

*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]

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]

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]

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]

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]

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]

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]

## 2. Problem statement and integral operators

**E**and magnetic field

**H**is ℝ

^{3}and it is divided into an unbounded exterior domain

*V*

_{1}and a compact interior domain

*V*

_{2}. The compact parts of the domain boundaries are denoted

*∂V*and they are oriented such, that their respective normal vectors

_{i}**n**

*point out of*

_{i}*V*. The surface of the scatterer is denoted by

_{i}*S*and it is assumed to be piece-wise smooth and oriented with surface normal

**n**pointing into

*V*

_{1}. The linear electromagnetic properties of the domains are characterized by the permittivity

*ɛ*∈ ℂ and permeability

_{i}*μ*∈ ℝ, which also define the wave impedance

_{i}*η*= (

_{i}*μ*/

_{i}*ɛ*)

_{i}^{1/2}.

**e**

*and*

_{i}**h**

*at frequency*

_{i}*ω*and the SH fields

**E**

*and*

_{i}**H**

*at frequency Ω = 2*

_{i}*ω*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

**P**

*=*

^{S}*ɛ*

_{0}

*χ*

^{(2)}:

**e**

_{2}

**e**

_{2}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]

*C*symmetry, so that the effective susceptibility tensor

_{∞}_{ν}*χ*

^{(2)}has only seven non-vanishing components:

*n*refers to local normal and

*s,t*refer to the two mutually orthogonal tangent vectors.

*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

**P**

*, which acts as a source for the SH fields.*

^{S}*iωt*). The fundamental fields in the domains

*V*satisfy the Helmholtz equation ∇

_{i}*×*∇ ×

**f**–

*k*(

_{i}*ω*)

^{2}

**f**=

**0**, where

*k*(

_{i}*ω*)

^{2}=

*ω*

^{2}

*ɛ*(

_{i}*ω*)

*μ*(

_{i}*ω*) and

**f**∈ {

**e**,

**h**}. The expressions for the SH fields are the same with substitutions

**e**→

**E**,

**h**→

**H**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]: where

*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]

*∂V*

_{1}and

*∂V*

_{2}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.

*V*are homogeneous, we may express the fields at any point by specifying the fields over the compact domain boundaries

_{i}*∂V*. This is governed by the Stratton-Chu equations [36]. To express the equations in our particular case, we define the Green’s function

_{i}*G*(

_{l}**r**,

**r**′) = exp(

*ik*)/(4

_{l}R*πR*) with

*R*= ||

**r**

*–*

**r**′||

_{2}and the following integro-differential linear operators: The fields in

*V*can now be expressed as Here the Silver-Müller radiation conditions are imposed on the fields in

_{i}*V*

_{1}so that the fields represent purely outgoing waves [37]. 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

*V*

_{1}and

*V*

_{2}.

*equivalent*surface current densities, which are defined as

*S*.

*∂V*approach

_{i}*S*and enforce the interface conditions. By using the above representations of the fields, we obtain the infinite-dimensional problem: Given

**P**

*, seek*

^{S}*S*. Note that we could substitute e.g.

**P**

*=*

^{S}**0**and adding the tangential components of the incident fields to the right-hand-side of Eqs. (5) and (6) for

*V*

_{1}. 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]

**j**

_{1}= −

**j**

_{2}and

**m**

_{1}= −

**m**

_{2}. The polarization

**P**

*is then evaluated from the field*

^{S}**e**

_{2}, whose components are given directly by the relations to corresponding equivalent surface current densities:

**m**

_{2}= −

**e**

_{2}×

**n**

_{2}and ∇

*·*

_{S}**j**

_{2}= −

*iωɛ*

_{2}

**n**

_{2}·

**e**

_{2}.

## 3. Finite dimensional formulation: the Method of Moments

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]

*S*, is a pair of adjacent triangles, as is illustrated in Fig. 1(b) (by the support of a function

_{n}*f*we mean the set

*S*= {

_{f}*x*∈

*S*|

*f*(

*x*) ≠ 0}). The unknown surface current densities are expanded using the RWG basis functions

**f**

*as where we have 4*

_{n}*N*unknowns

*α*,

_{n}*β*,

_{n}*γ*,

_{n}*δ*∈ ℂ, that can be arranged into a vector

_{n}**x**= (

*α*

_{1},...,

*α*,

_{N}*β*

_{1},...,

*β*,

_{N}*γ*

_{1},...,

*γ*,

_{N}*δ*

_{1},...,

*δ*)

_{N}*. To obtain these coefficients, we use Galerkin’s testing [38*

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

*f,g*〉 = ∫

*·*

_{S}f*g*d

*S*.

**x**=

**b**, where we have the system matrix Z: The matrix representations of the operators are: where the constitutive parameters and

*G*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

_{l}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]

**b**: and

**b**

^{n2}reduces to a contour integral by the identity 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 The integrals over the edges common to

*b*

^{n1}and

*b*

^{n2}clearly vanish if

**b**

^{t1}=

**b**

^{t2}=

**0**and Eq. (9) implies that

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]

## 4. Solution in multipoles

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]

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

*a*and size parameter

*x*=

*k*

_{1}

*a*. We restrict ourselves here to the case of polarization

*B*-expansion coefficient of Eq. (21) of the fundamental problem in the interior domain. We use the identity ∇

*= −(*

_{S}*i*/

*r*

^{2})

**r**×

**L**to conveniently express the source function

**S**: where

*ψ*(

_{l}*x*) =

*xj*(

_{l}*x*) and

*ξ*(

_{l}*x*)=

*xh*

^{(1)}(

*x*) and the relative refractive index

*N*=

*n*

_{2}(Ω)/

*n*

_{1}(Ω).

## 5. Numerical results

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

*ɛ*′ =

*ɛ*

_{0}.

### 5.1. The spherical particle

*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

*l*

_{max}= 4 (

*m*= ±1) and the SH solution with

*l*

_{max}= 3 (all

*m*-values included). We choose as the only nonzero tensor component

*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.

*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.

*e*is represented in the method with piecewise constant functions, and the SH response depends on the surface gradient of

_{n}### 5.2. The L-shaped particle

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]

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]

*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.

*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.

*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

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]

*N*basis functions, the matrix build time and memory requirements scale proportionally to

*N*

^{2}. 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*(

*N*

^{3}). 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]

*N*log

*N*in iterative solution methods.

**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

**P**

*. Then we solve another linear problem at frequency Ω, where the excitation is a plane wave incident from the observation direction with polarization*

^{S}**u**yielding solution

**E**′. The Lorentz reciprocity [46] then states that 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

## Acknowledgments

## References and links

1. | Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature |

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 |

3. | P. Guyot-Sionnest, W. Chen, and Y. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B |

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 |

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 |

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. |

7. | J. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B |

8. | S. Roke, M. Bonn, and A. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B |

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 |

10. | A. de Beer and S. Roke, “Nonlinear Mie theory for second-harmonic and sum-frequency scattering,” Phys. Rev. B |

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 |

12. | Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B |

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 |

14. | C. Biris and N. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B |

15. | C. Biris and N. Panoiu, “Nonlinear pulsed excitation of high-Q optical modes of plasmonic nanocavities,” Opt. Express |

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 |

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 |

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 |

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 |

21. | T. Paul, C. Rockstuhl, and F. Lederer, “A numerical approach for analyzing higher harmonic generation in multilayer nanostructures,” J. Opt. Soc. Am. B |

22. | W. Schaich, “Second harmonic generation by periodically-structured metal surfaces,” Phys. Rev. B |

23. | Y. Zeng, W. Hoyer, J. Liu, S. Koch, and J. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B |

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. | K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. |

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. |

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 |

28. | G. Bryant, F. de Abajo, and J. Aizpurua, “Mapping the plasmon resonances of metallic nanoantennas,” Nano Lett. |

29. | A. Kern and O. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A |

30. | B. Gallinet and O. Martin, “Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation,” Phot. Nano. Fund. Appl. |

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 |

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 |

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 |

34. | T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in |

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 |

36. | J. Stratton, |

37. | D. Colton and R. Kress, |

38. | R. Harrington, |

39. | S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. |

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. |

41. | J. Jackson, |

42. | P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

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. |

44. | H. Husu, J. Mäkitalo, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Particle plasmon resonances in L-shaped gold nanoparticles,” Opt. Express |

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. |

46. | L. D. Landau and E. M. Lifshits, |

**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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