Abstract

Second Harmonic Generation (SHG) is a widely used tool to study surfaces. Here we investigate SHG from spherical nanoparticles consisting of a dielectric core (radius 100 nm) and a metallic shell of variable thickness. Plasmonic resonances occur that depend on the thickness of the nanoshells and boost the intensity of the Second Harmonic (SH) signal. The origin of the resonances is studied for the fundamental harmonic and the second harmonic frequencies. Mie resonances at the fundamental harmonic frequency dominate resonant effects of the SH-signal at low shell thickness. Resonances excited by a dipole emitting at SH frequency close to the surface explain the enhancement of the SHG-process at a larger shell thickness. All resonances are caused by surface plasmon polaritons, which run on the surface of the spherical particle and are in resonance with the circumference of the sphere. Because their wavelength critically depends on the properties of the metallic layer SHG resonances of core-shell nanoparticles can be easily tuned by varying the thickness of the shell.

© 2013 OSA

1. Introduction

Surface science relies on probing and characterization techniques. Second Harmonic Generation (SHG) has proven to be a surface sensitive measurement tool, hence it has been used for in situ investigations for several decades [1]. For centrosymmetric materials SH is generated at the surface of nano-particles, because symmetry is broken at that interface as Wang has demonstrated experimentally for the first time [2]. SHG has been applied to study the surface and buried micro-structure of colloidal nano-particles [313] or as probing technique in bio-physical experiments [1418]. We have used SHG to determine the nonlinear susceptibility χ(2) of a thin layer of Malachite Green [19], to study the layer of oriented water near polystyrene particles [20] and the organization of polyelectrolyte chains, depending on the salt concentration of the solvent [21].

SHG has also been used to investigate the surface of metals, where surface plasmons can be excited. First experimental and theoretical studies were performed on plane surfaces [2227]. Recently studies on SHG or Hyper Rayleigh [28] scattering were performed on colloidal metallic nanoparticles [2936] as well as on particles embedded in a matrix [37,38] and more complex plasmonic structures [3942].

There exist several theoretical models for SHG from spherical nano-particles. The response of small particles can be modeled by Second-Harmonic Rayleigh Scattering [4346]. For small particles of low refractive index contrast, which do not significantly perturb the incident field, also the Rayleigh-Gans-Debye approximation can be used [4751]. Larger spheres or such with a high refractive index contrast require the application of a full Mie theory, based on which several numerical models have been developed for solid particles [18,19,5254]. Only recently, a nonlinear Mie model for core-shell particles has been applied to describe the wavelength-dependence of SHG on nanoparticles with up to 100 nm outer diameter and with fixed gold and silver shells of 10 nm thickness [55,56]. Simulations were used to investigate the dependence of the SH signal on the refractive indices of the core and the surrounding.

Here, we apply a refined nonlinear Mie model [19] to investigate in detail the nature of the various plasmonic resonances, which influence the SHG process from dielectric-metallic core-shell particles with variable shell thickness. We analyze the resonant behavior of the Second Harmonic intensity when varying the geometry and the material parameters of the nanoparticle. We find that even the nonlinear response is solely dominated by plasmonic resonances at fundamental harmonic (FH) and the second harmonic (SH) frequencies.

As the aim of our paper is to enable an analysis of experimentally obtained scattering data a typical experimental set-up is introduced at the beginning. This allows introducing experimentally relevant parameters, which will later be discussed in more detail. After that the general behavior of SHG and its dependence on particle parameters is discussed. Because plasmonic resonances play a decisive role in SHG we discuss the excitation and propagation of plasmons.

2. Nonlinear Mie model

In most other nonlinear Mie models second harmonic generation is assumed to take place in a nonlinear core or at an interface, which all have spherical symmetry. Solutions for the scattering coefficients are than obtained by decomposing nonlinear products of vector spherical harmonics into the basis functions of the Mie model and by evaluating respective boundary conditions [52,53]. Here we use a different approach and start from a molecular Mie model that we have introduced earlier [19]: The nonlinear polarization

Pi(2)j,kχijk(2)EjFHEkFH
is calculated from the fundamental harmonic fieldsEFH and the surface susceptibility χ(2), which depends on the materials. Similar to the Rayleigh-Gans-Debye-theory, where the polarization is modeled as a dipole density, we interpret P(2) as an ensemble of dipoles on a discrete grid rd. The second harmonic is driven by this ensemble of nonlinear dipoles. . rd. Although the distribution of the dipoles need not obey spherical symmetry all electric fields can be calculated using classical Mie theory with a plane wave as incident field. The model also takes into account the correct influence of the sphere on the emission of the nonlinearly induced dipoles into the SH. The effective fields of the individual dipoles are calculated using generalized Mie theory, with the field of a dipole, placed at any position rd [57]. One advantage of our approach is that it is in general not limited to nonlinear configurations with spherical symmetry. For example it can also account for particles, which are covered with incomplete layers of second harmonic generating molecules. Only the linear index distribution must have spherical symmetry.

Mie theory, which is the basis of our model, relies on the expansion of all fields into a set of vector spherical harmonics Mm,n(j) and Nm,n(j) j=1j=3with expansion coefficients am,n, bm,n, cm,n, and dm,n:

EFH(r)=n=1Nm=nnam,nNm,n(1)(r)+bm,nNm,n(1)(r)+cm,nNm,n(3)(r)+dm,nMm,n(3)(r).
Each vector component of these is a product of Bessel function (j = 1) or Hankel function (j = 3) of r, associated Legendre function of θ and exponential function of ϕ. Only the Bessel or Hankel functions that describe the radial dependence finally enter the continuity-conditions. Inside the sphere, Hankel functions diverge, so that cm,n and dm,n are zero. The scattered field outside the particle is described as an outgoing wave, so that only Hankel functions are used (fields with j=3) are used.

Standard Mie theory only evaluates one boundary condition for the incident electric field and scattered fields inside and outside the sphere

Eθ,ϕinc(R)+Eθ,ϕsc, out(R)=Eθ,ϕsc, in(R)
and a respective continuity condition for the magnetic fields. To apply the Mie model to a sphere with L layers, the continuity conditions have to be modified. For several layers, denoted by l=1L+1, there exist L continuity conditions for the scattered electric fields Esc, l in the layers l and l+1 at the boundaries with radius Rl:
Eθ,ϕsc, l(Rl)=Eθ,ϕsc, l+1(Rl).
The Mie scattering process of the fundamental harmonic fields involves a plane wave Epw(r) that enters the boundary condition at the outer radius RL [58]:
Eθ,ϕsc, l(RL)=Eθ,ϕsc, l+1(RL)+Eθ,ϕpw(RL).
The 4L expansion coefficients in the different layers l=1L+1 are obtained from two sets of linear equations with 2L×2L matrices that have to be solved for every expansion order n [58].

For metallic particles it is essential, that some modifications are added to Mie theory for stability [59,60]. The refractive index, which has a dominant imaginary part in case of metals, enters the equations through the argument of the Bessel functions. Bessel functions of an imaginary argument diverge exponentially and therefore make Mie scattering unstable for highly absorbing particles. In the original equations for the boundary conditions in Mie-Theory, the Bessel functions and their derivatives are therefore replaced by ratios of Bessel functions and logarithmic derivatives for which stable recursion formulas exist [58].

The polarization P(2)(rd) is resembled by dipoles at discrete positions rd. The electric fields of a dipole at position rd are calculated from a dipole at the origin EDipole(r)=N0,1(j=3)(r) by a translation theorem [61]. This results in different expansions, depending on r:

EDipole(r)={E(j=3),Dipole(r)=m,nam,n(j=3)Nm,n(j=3)(r)+bm,n(j=3)Nm,n(j=3)(r)if|r|>if|rd|.E(j=1),Dipole(r)=m,nam,n(j=1)Nm,n(j=1)(r)+bm,n(j=1)Nm,n(j=1)(r)if|r|<|rd|
If the dipole is positioned inside layer l (Rl1<|rd|<Rl), EDipole(r) enters the boundary conditions at Rl1 and Rl:

Eθ,ϕsc, l1(Rl1)=Eθ,ϕDipole,(1)(Rl1)+Eθ,ϕsc, l(Rl1),
Eθ,ϕsc, l(Rl)+Eθ,ϕDipole,(3)(Rl)=Eθ,ϕsc, l+1(Rl).

3. Results and discussion

In typical experiments [20] the SH fields, generated at the surface of nanoparticles, are measured as a function of the scattering angle θ and for the polarizations of the incident and the detected light being either parallel (p) or perpendicular (s) to the scattering plane (Fig. 1). As high power pulses are required we assume the FH to be generated by a Ti:Sa-Laser at 800 nm wavelength. The intensity and the polarization of the scattered SH light at 400 nm is detected. The respective intensities are given for example by Isp(θ), if the incident fundamental harmonic is s-polarized and the p-polarized second harmonic is detected under an angle θ with respect to the exciting laser pulse. Ipp(θ) is measured if both beams are p-polarized. Due to symmetry reasons Ips(θ) and Iss(θ) are identical to zero for spherical particles.

 

Fig. 1 a) typical experimental setup for angle-resolved SHG from colloidal core-shell particles. b) Angular scattering pattern Isp(θ) (FH s-polarized, SH p-polarized) and Ipp(θ) (FH and SH p-polarized) for a silica particle (radius 100 nm) with a silver shell of 6.3 nm thickness (compare Fig. 2).

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Here we simulated the SH-intensities Ipp(θ) and Isp(θ) for a nano-particle consisting of a silica core (radius 100 nm) and a silver shell, which is suspended in water. Dielectric constants are taken from Johnson and Christy [62]. In addition we first assume a dominating χzzz(2) [55,6365] for the nonlinear surface susceptibility of the silica-silver and the silver-water interfaces. We will later investigate the effect of the nonlinear coefficient on the SH-signal.

A typical angular dependent emission spectrum is displayed in Fig. 1. For symmetry reasons the SH emission in forward direction is always zero (but a nonzero intensity might be measured in the experiment due to a finite aperture and detector size), but characteristic peaks emerge for higher angles. Varying the thickness of the shell between 0 nm and 30 nm causes changes in the angular emission pattern, but mainly influences the total efficiency of SHG (Fig. 2). Resonances occur e.g. at 4 nm, 7 nm or 25 nm thickness and are even more pronounced, if an idealized material without losses is assumed for the metallic shell. When plasmon losses are included (ϵshell400 nm=4.4+0.2i, ϵshell800 nm=30.8+0.4i), the resonant enhancement is reduced and the resonance at 25 nm shell thickness even disappears (Fig. 2).

 

Fig. 2 Angle-resolved SH-intensity Ipp(θ) and Isp(θ) (a,b) and mean intensity, averaged over the detection angle (c,d) for a growing silver shell on a silica particle. For the metallic shell either idealized silver without plasmon losses ϵshell400 nm=4.4, ϵshell800 nm=30.8 (a,b,c) or realistic values, including plasmon losses ϵshell400 nm=4.4+0.2i, ϵshell800 nm=30.8+0.4i, are used (d).

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To understand these resonances, we have to keep in mind that SHG takes place in two steps. The first one includes the formation of the nonlinear polarization at the inner and outer interface of the core-shell particle. The nonlinear polarization is directly proportional to the square of the scattered FH wave that excites the particle. Hence, plasmonic resonances at FH frequency lead to a higher SH-intensity due to the increased field strength at the two interfaces.

The second step is calculating the electromagnetic field at SH frequency that is produced by the nonlinear polarization. To this end we decompose the nonlinear polarization into an ensemble of dipolar point sources each emitting an electromagnetic field at SH frequency. This emission process is again heavily determined by resonances of the particle, but now at SH frequency.

Field enhancement at such Mie resonances happens for both the radial |Er|2 and azimuthal |Eϑ|2, |Eϕ|2 components of the electric field simultaneously. Also the inner (core-shell) and the outer (shell-surrounding) interfaces of the shell (Fig. 3) experience qualitatively the same field strength. Strength and width of resonances are heavily influenced by the shell thickness. FH resonances occur for a shell thickness below 10 nm. Their spectral position is equally reproduced by peaks of the SH-intensity. In contrast the linear extinction spectrum, which is usually measured does not show such a significant dependence on the shell thickness because the farfield is much less sensitive to resonant effects compared with the the fundamental harmonic nearfield at the surface of the sphere which finally causes the SH emission. For metals with higher plasmon losses, like gold, this difference between linear extinction spectra and field enhancement at the surface of the sphere becomes even more pronounced.

 

Fig. 3 Mean electric field Er(ϑ)ϑ induced by a scattered plane wave (fundamental harmonic wavelength) at the inner (rin) and outer (rout) interface of a sphere with silica core of radius 100 nm and a silver shell of variable thickness with (ϵshell800 nm=30.8+0.4i) and without (ϵshell800 nm=30.8) damping. Additionally, the extinction at λ=800 nm is given as a function of the shell thickness.

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A second possible reason for enhancement of SHG is the excitation of a resonance at SH frequency. The emission of a dipole is influenced by the sphere (Fig. 4). This influence can be very dramatic in the case of a plasmonic resonance: The emitting dipole is hardly visible and all radiation seems to emerge from the particle in a star-like pattern. For dipoles with radial orientation the generated tangential field distribution shows a minimum at the position of the dipole and on the opposite site of the sphere. When the dipole is parallel to the surface, maxima and minima switch places, so that a maximum is observed at the position of the dipole and on the opposite site of the sphere. Obviously the near-field of the dipole is imaged onto the opposite side of the sphere, showing that a spherical surface has perfect imaging properties even on a sub wavelength scale.

 

Fig. 4 Field distribution log10|E|2 emitted by a dipole radiating at 400 nm wavelength (a) and respective field distribution log10|Eϑ|2 modified by the presence of a sphere (b,c). The sphere has a silica core of radius 100 nm, and silver shell of thickness D=25 nm. The shell thickness is tuned for the resonance K=5 with 10 lobes. For the metallic shell either idealized silver without plasmon losses ϵshell400 nm=4.4 (b) or realistic ϵshell400 nm=4.4+0.2i is used (c).

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The situation of a single dipole close to a sphere is similar to single molecule spectroscopy performed near a metallic surface in order to profit from the enhanced local fields as they are employed in surface enhanced Raman scattering (SERS). Here, the environment influences the molecular emission properties [66]. Kuhn et al. [67] even used a spherical gold nanoparticle as an optical nanoantenna to enhance the fluorescence signal.

Although FH resonances are excited by a plane wave, but SH resonances by localized dipoles the respective field pattern are quite similar. In both cases each resonance exhibits a star-like pattern with a certain number of lobes. For decreasing shell thickness two more lobes arise at every new resonance (Fig. 5). This suggests that resonances are caused by surface plasmon polaritons running around the sphere. A resonance with 2K lobes occurs, whenever a multiple of the effective wavelength of the plasmon fits to the circumference of the sphere: 2πR=KλSPP. Hence, resonances depend on both the diameter of the sphere and the effective wavelength of the plasmon. The latter one is mainly determined by the shell thickness. In our case the resonance corresponding to the longest plasmon wavelength at SH frequency occurs for D=25 nm and explains the respective enhancement of the total SHG. For smaller shell thicknesses the plasmon wavelength decreases and further resonances are observed, at least for the lossless case (Fig. 5). But, plasmons are guided waves and do only interact with the farfield due to the curvature of the surface. The thinner the shell, the shorter is the plasmon wavelength and the stronger is the guidance. Therefore resonances at smaller shell thickness can be excited by nonlinear dipoles, but they are only visible close to the surface of the sphere (Fig. 5), but do not significantly contribute to an emission to the farfield.

 

Fig. 5 Tangential component of the intensity of a dipole (second harmonic wavelength) placed at the surface of a spherical particle with silica core of radius 100 nm and a silver shell (with and without damping) of variable thickness (compare Fig. 4 for the geometry). a) Intensity |Eϑ(ϑ)|2 of a dipole at the outer surface (without plasmonic losses, ϵshell400 nm=4.4). The intensity is detected 20 nm above the surface of the sphere and shown as a function of the scattering angle ϑ. b, c: Intensity |Eϑ(ϑ)|2ϑ of a dipole, which is placed at the inner and outer interface of a sphere, without (b) and with (c) plasmonic losses (ϵshell400 nm=4.4+0.2i), detected in 2 cm distance.

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To test our assumption that resonances are caused by running plasmons we shortly discuss the case of a solid metallic particle because the wavelength of plasmons propagating at a pure metal interface is known analytically as

λSPP=λDipoleεSphere+εSurroundingεSphereεSurrounding
Resonance positions in fact occur almost exactly when the circumference of the sphere coincides with multiples of λSPP even though the interface is highly curved (Fig. 6).

 

Fig. 6 Resonances with 2K lobes occurring at certain values of the permittivity εNSphere of a solid metallic sphere in comparison to the model based on the planar plasmon for R=100 nm and λDipole=400 nm.

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Hence, any enhancement of SHG on core-shell nanoparticles is solely based on resonances at FH or SH wavelength, which both are caused by plasmons running along the metallic film. As the plasmon wavelength critically depends on the shell thickness SHG provides an extremely sensitive tool to determine the thickness of nanoshells by in situ scattering experiments and by evaluating the spectral position of respective peaks (Fig. 7). This sensitivity is based on the strong response of SHG on the local near fields at the surface of the sphere. As discussed earlier conventional linear extinction spectra cannot provide this information with a comparable contrast in particular in case of metals with higher losses.

 

Fig. 7 SH-intensity (a) and extinction spectrum (b) as a function of the fundamental harmonic wavelength. The dashed lines show the SH-scattering cross section Cextm,n(2n+1)k2(|am,nSH|2+|bm,nSH|2). Three systems of silica particles with silver shell (including plasmon losses) of 5 nm, 7.5 nm, and 10 nm thickness are shown.

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Up to now we have assumed the nonlinear coefficient to have only a nonvanishing χzzz(2) component, also because the various tensor elements characterizing the quadratically nonlinear response of metals are not well known. As spectral resonance positions are solely determined by the linear optical response the above obtained results do not depend on this choice. We now include further nonvanishing tensor components into the calculations. In all simulation χ(2) of the inner interface is identical but inverted to χ(2) of the outer interface. The different surface areas of the inner and outer interface are taken into account. All calculations show that if χzzz(2) is present it dominates the nonlinear response (see Fig. 8). A pure χxxz(2) or χzxx(2) yield only a negligible SH-intensity. Varying the various components of the surface susceptibility χ(2) affects the angular response. The position of the first maximum in pp-polarization (θpp) is only slightly affected by the actual choice of the nonlinear coefficients, θsp and the ratio of the intensities of the first maxima Ipp/Isp show a larger dependence, which potentially allows to extract more information about nonlinear coefficients from scattering data.

 

Fig. 8 Effect of the variation of χ(2) on the position of the first maximum in pp- and sp-polarization (θpp, θsp), the ratio of the intensities of the first maximum (Ipp/Isp) and the sum of the intensities of the first maxima in sp- and pp-polarization (Ipp+Isp) for a silica particle (R=100 nm) with a silver shell (including plasmon losses) of R=15 nm thickness. The triangular parameter space displays the relative amount of the three independent components of χ(2): χzzz(2), χzxx(2), and χxxz(2). In the corners of the triangle, only the component written next to the corner is nonzero. Along the edges the two components are nonzero, their relative magnitude becomes larger the closer the point lies to the respective corner [19,20].

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4. Conclusion

We have successfully refined the nonlinear Mie model to account for metallo-dielectric core-shell particles. We found that for certain shell thicknesses SHG is extremely enhanced. We attribute this to the occurrence of plasmonic resonances at either the FH or SH frequency. Resonances at a low shell thickness arise from Mie-resonances of the nanoshells-particle excited the scattered FH plain wave. Enhancement of SHG at larger shell thicknesses is caused by resonances excited by the nonlinear dipoles oscillating at SH frequency. All resonances are caused by surface plasmons, which run on the surface of the spherical particle and are in resonance with the circumference of the sphere. Because the plasmon wavelength critically depends on the properties of the metallic layer SHG resonances of core-shell nanoparticles can be easily tuned by varying the thickness of the shell. In turn SHG might therefore become a convenient tool to study the growth and the evolution of the thickness of metallic shells on particles with much higher sensitivity than it can be provided by linear scattering experiments. Our simulations have also shown that the actual composition of the nonlinear tensor mainly influences the strength of the interaction with χzzz(2) playing the dominant role, but does not influence the resonance positions. In contrast the angular distribution of the scattered SH field is affected, which in principal allows to deduce also individual tensor elements from angularly resolved SHG scattering data.

Acknowledgments

The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German National Science Foundation (DFG) in the framework of the excellence initiative

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39. N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett. 33(17), 1975–1977 (2008). [CrossRef]   [PubMed]  

40. F. B. P. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a GaAs substrate,” Opt. Lett. 34(13), 1997–1999 (2009). [CrossRef]   [PubMed]  

41. N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: The role of separation and relative orientation,” Opt. Express 18(7), 6545–6554 (2010). [CrossRef]   [PubMed]  

42. H. Lu, X. Liu, R. Zhou, Y. Gong, and D. Mao, “Second-harmonic generation from metal-film nanohole arrays,” Appl. Opt. 49(12), 2347–2351 (2010). [CrossRef]   [PubMed]  

43. G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun. 41(6), 499–501 (1982). [CrossRef]  

44. 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]  

45. W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B 68(8), 085318 (2003). [CrossRef]  

46. B. S. Mendoza and W. L. Mochán, “Second harmonic surface response of a composite,” Opt. Mater. 29(1), 1–5 (2006). [CrossRef]  

47. J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A 55(6), 4520–4525 (1997). [CrossRef]  

48. S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett. 91(25), 258302 (2003). [CrossRef]   [PubMed]  

49. 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]  

50. S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C 114(10), 4302–4308 (2010). [CrossRef]  

51. J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B 78(20), 205322 (2008). [CrossRef]  

52. D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D 28, 169–175 (1993). [CrossRef]  

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

54. S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B 70(11), 115106 (2004). [CrossRef]  

55. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B 29(8), 2213–2221 (2012). [CrossRef]  

56. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C 117(2), 1172–1177 (2013). [CrossRef]  

57. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969).

58. M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir 3(1), 85–93 (1987). [CrossRef]  

59. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop. 13(3), 302–313 (1969). [CrossRef]  

60. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19(9), 1505–1509 (1980). [CrossRef]   [PubMed]  

61. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag. 19(3), 378–390 (1971). [CrossRef]  

62. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

63. 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]  

64. G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82(23), 235403 (2010). [CrossRef]  

65. G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett. 3(19), 2877–2881 (2012). [CrossRef]  

66. K. H. Drexhage, “IV Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1974), Vol. 12, pp. 163–232.

67. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef]   [PubMed]  

References

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    [CrossRef]
  45. W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68(8), 085318 (2003).
    [CrossRef]
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    [CrossRef]
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  49. S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A81(5), 053850 (2010).
    [CrossRef]
  50. S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
    [CrossRef]
  51. J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B78(20), 205322 (2008).
    [CrossRef]
  52. D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
    [CrossRef]
  53. Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B70(24), 245434 (2004).
    [CrossRef]
  54. S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70(11), 115106 (2004).
    [CrossRef]
  55. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B29(8), 2213–2221 (2012).
    [CrossRef]
  56. J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
    [CrossRef]
  57. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, 1969).
  58. M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir3(1), 85–93 (1987).
    [CrossRef]
  59. J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop.13(3), 302–313 (1969).
    [CrossRef]
  60. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.19(9), 1505–1509 (1980).
    [CrossRef] [PubMed]
  61. J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag.19(3), 378–390 (1971).
    [CrossRef]
  62. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
    [CrossRef]
  63. 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. B80(23), 233402 (2009).
    [CrossRef]
  64. G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
    [CrossRef]
  65. G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
    [CrossRef]
  66. K. H. Drexhage, “IV Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1974), Vol. 12, pp. 163–232.
  67. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett.97(1), 017402 (2006).
    [CrossRef] [PubMed]

2013

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[CrossRef]

2012

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B29(8), 2213–2221 (2012).
[CrossRef]

G. Gonella, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
[CrossRef]

S. Roke and G. Gonella, “Nonlinear light scattering and spectroscopy of particles and droplets in liquids,” Annu. Rev. Phys. Chem.63(1), 353–378 (2012).
[CrossRef] [PubMed]

2011

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. B84(12), 121402 (2011).
[CrossRef]

S. Wunderlich, B. Schürer, C. Sauerbeck, W. Peukert, and U. Peschel, “Molecular Mie model for second harmonic generation and sum frequency generation,” Phys. Rev. B84(23), 235403 (2011).
[CrossRef]

B. Schürer, M. Hoffmann, S. Wunderlich, L. Harnau, U. Peschel, M. M. Ballauff, and W. Peukert, “Second harmonic light scattering from spherical polyelectrolyte brushes,” J. Phys. Chem. C115(37), 18302–18309 (2011).
[CrossRef]

W. Gan, B. Xu, and H.-L. Dai, “Activation of Thiols at a Silver Nanoparticle Surface,” Angew. Chem. Int. Ed. Engl.50(29), 6622–6625 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

L. H. Haber, S. J. J. Kwok, M. Semeraro, and K. B. Eisenthal, “Probing the colloidal gold nanoparticle/aqueous interface with second harmonic generation,” Chem. Phys. Lett.507(1-3), 11–14 (2011).
[CrossRef]

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

2010

A. G. F. de Beer and S. Roke, “Obtaining molecular orientation from second harmonic and sum frequency scattering experiments in water: Angular distribution and polarization dependence,” J. Chem. Phys.132(23), 234702 (2010).
[CrossRef] [PubMed]

C.-L. Hsieh, R. Grange, Y. Pu, and D. Psaltis, “Bioconjugation of barium titanate nanocrystals with immunoglobulin G antibody for second harmonic radiation imaging probes,” Biomaterials31(8), 2272–2277 (2010).
[CrossRef] [PubMed]

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

B. Schürer, S. Wunderlich, C. Sauerbeck, U. Peschel, and W. Peukert, “Probing colloidal interfaces by angle-resolved second harmonic light scattering,” Phys. Rev. B82(24), 241404 (2010).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

N. Feth, M. König, M. Husnik, K. Stannigel, J. Niegemann, K. Busch, M. Wegener, and S. Linden, “Electromagnetic interaction of split-ring resonators: The role of separation and relative orientation,” Opt. Express18(7), 6545–6554 (2010).
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H. Lu, X. Liu, R. Zhou, Y. Gong, and D. Mao, “Second-harmonic generation from metal-film nanohole arrays,” Appl. Opt.49(12), 2347–2351 (2010).
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S. Viarbitskaya, V. Kapshai, P. van der Meulen, and T. Hansson, “Size dependence of second-harmonic generation at the surface of microspheres,” Phys. Rev. A81(5), 053850 (2010).
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S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
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2009

F. B. P. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a GaAs substrate,” Opt. Lett.34(13), 1997–1999 (2009).
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M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (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. B80(23), 233402 (2009).
[CrossRef]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
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J. Griffin, A. K. Singh, D. Senapati, E. Lee, K. Gaylor, J. Jones-Boone, and P. C. Ray, “Sequence-Specific HCV RNA Quantification Using the Size-Dependent Nonlinear Optical Properties of Gold Nanoparticles,” Small5(7), 839–845 (2009).
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2008

M. Subir, J. Liu, and K. B. Eisenthal, “Protonation at the Aqueous Interface of Polymer Nanoparticles with Second Harmonic Generation,” J. Phys. Chem. C112(40), 15809–15812 (2008).
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N. Feth, S. Linden, M. W. Klein, M. Decker, F. B. P. Niesler, Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, J. V. Moloney, and M. Wegener, “Second-harmonic generation from complementary split-ring resonators,” Opt. Lett.33(17), 1975–1977 (2008).
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J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B78(20), 205322 (2008).
[CrossRef]

2007

J. Jiang, K. B. Eisenthal, and R. Yuste, “Second harmonic generation in neurons: electro-optic mechanism of membrane potential sensitivity,” Biophys. J.93(5), L26–L28 (2007).
[CrossRef] [PubMed]

2006

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
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B. S. Mendoza and W. L. Mochán, “Second harmonic surface response of a composite,” Opt. Mater.29(1), 1–5 (2006).
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S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett.97(1), 017402 (2006).
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2005

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. B71(16), 165407 (2005).
[CrossRef]

2004

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Y. Pavlyukh and W. Hübner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B70(24), 245434 (2004).
[CrossRef]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70(11), 115106 (2004).
[CrossRef]

2003

W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68(8), 085318 (2003).
[CrossRef]

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett.91(25), 258302 (2003).
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2002

R. C. Johnson, J. Li, J. T. Hupp, and G. C. Schatz, “Hyper-Rayleigh scattering studies of silver, copper, and platinum nanoparticle suspensions,” Chem. Phys. Lett.356(5-6), 534–540 (2002).
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E. C. Hao, G. C. Schatz, R. C. Johnson, and J. T. Hupp, “Hyper-Rayleigh scattering from silver nanoparticles,” J. Chem. Phys.117(13), 5963 (2002).
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2001

Y. Liu, E. C. Y. Yan, and K. B. Eisenthal, “Effects of bilayer surface charge density on molecular adsorption and transport across liposome bilayers,” Biophys. J.80(2), 1004–1012 (2001).
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2000

H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
[CrossRef]

1999

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

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
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E. C. Y. Yan, Y. Liu, and K. B. Eisenthal, “New method for determination of surface potential of microscopic particles by second harmonic generation,” J. Chem. Phys. B102(33), 6331–6336 (1998).
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F. W. Vance, B. I. Lemon, and J. T. Hupp, “Enormous hyper-Rayleigh scattering from nanocrystalline gold particle suspensions,” J. Chem. Phys. B102(50), 10091–10093 (1998).
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1997

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A55(6), 4520–4525 (1997).
[CrossRef]

1996

H. Wang, E. C. Y. 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]

1993

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
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1987

M. Sitarski, “Internal heating of multilayered aerosol particles by electromagnetic radiation,” Langmuir3(1), 85–93 (1987).
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1986

Y. Shen, “Surface 2nd harmonic-generation - a new technique for surface studies,” Annu. Rev. Mater. Sci.16(1), 69–86 (1986).
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1982

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun.41(6), 499–501 (1982).
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1980

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt.19(9), 1505–1509 (1980).
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J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B21(10), 4389–4402 (1980).
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1977

D. L. Mills, “Theory of second harmonic generation by surface polaritons on metals,” Solid State Commun.24(9), 669–671 (1977).
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1976

H. J. Simon and J. K. Guha, “Directional surface plasmon scattering from silver films,” Opt. Commun.18(3), 391–394 (1976).
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1975

H. J. Simon, D. E. Mitchell, and J. G. Watson, “Second harmonic generation with surface plasmons in alkali metals,” Opt. Commun.13(3), 294–298 (1975).
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1974

H. J. Simon, D. E. Mitchell, and J. G. Watson, “Optical second-harmonic generation with surface plasmons in silver films,” Phys. Rev. Lett.33(26), 1531–1534 (1974).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1971

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag.19(3), 378–390 (1971).
[CrossRef]

J. Rudnick and E. A. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B4(12), 4274–4290 (1971).
[CrossRef]

1969

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop.13(3), 302–313 (1969).
[CrossRef]

Abid, J.-P.

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. S. Jha, “Theory of second harmonic generation at a metal surface with surface plasmon excitation,” Solid State Commun.41(6), 499–501 (1982).
[CrossRef]

Aguiar, H. B.

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

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. B80(23), 233402 (2009).
[CrossRef]

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. B80(23), 233402 (2009).
[CrossRef]

Antoine, R.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Bachelier, G.

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

Ballauff, M. M.

B. Schürer, M. Hoffmann, S. Wunderlich, L. Harnau, U. Peschel, M. M. Ballauff, and W. Peukert, “Second harmonic light scattering from spherical polyelectrolyte brushes,” J. Phys. Chem. C115(37), 18302–18309 (2011).
[CrossRef]

Benichou, E.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B29(8), 2213–2221 (2012).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
[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. B71(16), 165407 (2005).
[CrossRef]

Bennemann, K. H.

D. Östling, P. Stampfli, and K. H. Bennemann, “Theory of nonlinear optical properties of small metallic spheres,” Z. Phys. D28, 169–175 (1993).
[CrossRef]

Berg, O.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

Bonn, M.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

S. Roke, M. Bonn, and A. V. Petukhov, “Nonlinear optical scattering: The concept of effective susceptibility,” Phys. Rev. B70(11), 115106 (2004).
[CrossRef]

S. Roke, W. G. Roeterdink, J. E. G. J. Wijnhoven, A. V. Petukhov, A. W. Kleyn, and M. Bonn, “Vibrational sum frequency scattering from a submicron suspension,” Phys. Rev. Lett.91(25), 258302 (2003).
[CrossRef] [PubMed]

Borguet, E.

H. Wang, E. C. Y. 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]

Brevet, P. F.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Brevet, P.-F.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B29(8), 2213–2221 (2012).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

J. Nappa, I. Russier-Antoine, E. Benichou, Ch. Jonin, and P.-F. Brevet, “Second harmonic generation from small gold metallic particles: From the dipolar to the quadrupolar response,” J. Chem. Phys.125(18), 184712 (2006).
[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. B71(16), 165407 (2005).
[CrossRef]

J.-P. Abid, J. Nappa, H. H. Girault, and P.-F. Brevet, “Pure surface plasmon resonance enhancement of the first hyperpolarizability of gold core-silver shell nanoparticles,” J. Chem. Phys.121(24), 12577–12582 (2004).
[CrossRef] [PubMed]

Broyer, M.

R. Antoine, M. Pellarin, B. Palpant, M. Broyer, B. Prevel, P. Galletto, P. F. Brevet, and H. H. Girault, “Surface plasmon enhanced second harmonic response from gold clusters embedded in an alumina matrix,” J. Appl. Phys.84(8), 4532–4536 (1998).
[CrossRef]

Brudny, V. L.

W. L. Mochán, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny, “Second-harmonic generation in arrays of spherical particles,” Phys. Rev. B68(8), 085318 (2003).
[CrossRef]

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering of EM waves by spheres part I: multipole expansion and ray-optical solutions,” IEEE Trans. Antenn. Propag.19(3), 378–390 (1971).
[CrossRef]

Buitenhuis, J.

S. Roke, O. Berg, J. Buitenhuis, A. van Blaaderen, and M. Bonn, “Surface molecular view of colloidal gelation,” Proc. Natl. Acad. Sci. U.S.A.103(36), 13310–13314 (2006).
[CrossRef] [PubMed]

Busch, K.

Butet, J.

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
[CrossRef]

J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Nonlinear Mie theory for the second harmonic generation in metallic nanoshells,” J. Opt. Soc. Am. B29(8), 2213–2221 (2012).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P.-F. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B82(23), 235403 (2010).
[CrossRef]

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

Chandra, M.

M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Corbalán, R.

J. Martorell, R. Vilaseca, and R. Corbalán, “Scattering of second-harmonic light from small spherical particles ordered in a crystalline lattice,” Phys. Rev. A55(6), 4520–4525 (1997).
[CrossRef]

Dadap, J. I.

J. I. Dadap, “Optical second-harmonic scattering from cylindrical particles,” Phys. Rev. B78(20), 205322 (2008).
[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, W. Gan, B. Xu, and H.-L. Dai, “The effect of composition, morphology, and susceptibility on nonlinear light scattering from metallic and dielectric nanoparticles,” J. Phys. Chem. Lett.3(19), 2877–2881 (2012).
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W. Gan, B. Xu, and H.-L. Dai, “Activation of Thiols at a Silver Nanoparticle Surface,” Angew. Chem. Int. Ed. Engl.50(29), 6622–6625 (2011).
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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. B84(12), 121402 (2011).
[CrossRef]

S.-H. Jen, H.-L. Dai, and G. Gonella, “The effect of particle size in second harmonic generation from the surface of spherical colloidal particles. II: The nonlinear Rayleigh−Gans−Debye model,” J. Phys. Chem. C114(10), 4302–4308 (2010).
[CrossRef]

H. Wang, T. Troxler, A.-G. Yeh, and H.-L. Dai, “In situ, nonlinear optical probe of surfactant adsorption on the durface of microparticles in colloids,” Langmuir16(6), 2475–2481 (2000).
[CrossRef]

Das, P. K.

M. Chandra and P. K. Das, “Size dependence and dispersion behavior of the first hyperpolarizability of copper nanoparticles,” Chem. Phys. Lett.476(1-3), 62–64 (2009).
[CrossRef]

M. Chandra and P. K. Das, ““Small-particle limit” in the second harmonic generation from noble metal nanoparticles,” Chem. Phys.358(3), 203–208 (2009).
[CrossRef]

Dave, J. V.

J. V. Dave, “Scattering of electromagnetic radiation by a large, absorbing sphere,” IBM J. Res. Develop.13(3), 302–313 (1969).
[CrossRef]

de Aguiar, H. B.

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
[CrossRef] [PubMed]

de Beer, A. G. F.

H. B. de Aguiar, M. L. Strader, A. G. F. de Beer, and S. Roke, “Surface structure of sodium dodecyl sulfate surfactant and oil at the oil-in-water droplet liquid/liquid interface: a manifestation of a nonequilibrium surface state,” J. Phys. Chem. B115(12), 2970–2978 (2011).
[CrossRef] [PubMed]

R. Vácha, S. W. Rick, P. Jungwirth, A. G. F. de Beer, H. B. de Aguiar, J.-S. Samson, and S. Roke, “The orientation and charge of water at the hydrophobic oil droplet-water interface,” J. Am. Chem. Soc.133(26), 10204–10210 (2011).
[CrossRef] [PubMed]

A. G. F. de Beer and S. Roke, “Obtaining molecular orientation from second harmonic and sum frequency scattering experiments in water: Angular distribution and polarization dependence,” J. Chem. Phys.132(23), 234702 (2010).
[CrossRef] [PubMed]

H. B. Aguiar, A. G. F. de Beer, M. L. Strader, and S. Roke, “The interfacial tension of nanoscopic oil droplets in water is hardly affected by SDS surfactant,” J. Am. Chem. Soc.132(7), 2122–2123 (2010).
[CrossRef] [PubMed]

A. G. F. de Beer, H. B. de Aguiar, J. F. W. Nijsen, and S. Roke, “Detection of buried microstructures by nonlinear light scattering spectroscopy,” Phys. Rev. Lett.102(9), 095502 (2009).
[CrossRef] [PubMed]

Decker, M.

Duboisset, J.

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P.-F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010).
[CrossRef] [PubMed]

Eisenthal, K. B.

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J. Butet, I. Russier-Antoine, C. Jonin, N. Lascoux, E. Benichou, and P.-F. Brevet, “Effect of the dielectric core and embedding medium on the second harmonic generation from plasmonic nanoshells: tunability and sensing,” J. Phys. Chem. C117(2), 1172–1177 (2013).
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Figures (8)

Fig. 1
Fig. 1

a) typical experimental setup for angle-resolved SHG from colloidal core-shell particles. b) Angular scattering pattern I sp ( θ ) (FH s-polarized, SH p-polarized) and I pp ( θ ) (FH and SH p-polarized) for a silica particle (radius 100 nm) with a silver shell of 6.3 nm thickness (compare Fig. 2).

Fig. 2
Fig. 2

Angle-resolved SH-intensity I pp ( θ ) and I sp ( θ ) (a,b) and mean intensity, averaged over the detection angle (c,d) for a growing silver shell on a silica particle. For the metallic shell either idealized silver without plasmon losses ϵ shell 400 nm =4.4 , ϵ shell 800 nm =30.8 (a,b,c) or realistic values, including plasmon losses ϵ shell 400 nm =4.4+0.2i , ϵ shell 800 nm =30.8+0.4i , are used (d).

Fig. 3
Fig. 3

Mean electric field E r ( ϑ ) ϑ induced by a scattered plane wave (fundamental harmonic wavelength) at the inner ( r in ) and outer ( r out ) interface of a sphere with silica core of radius 100 nm and a silver shell of variable thickness with ( ϵ shell 800 nm =30.8+0.4i ) and without ( ϵ shell 800 nm =30.8 ) damping. Additionally, the extinction at λ=800 nm is given as a function of the shell thickness.

Fig. 4
Fig. 4

Field distribution log 10 | E | 2 emitted by a dipole radiating at 400 nm wavelength (a) and respective field distribution log 10 | E ϑ | 2 modified by the presence of a sphere (b,c). The sphere has a silica core of radius 100 nm, and silver shell of thickness D=25 nm . The shell thickness is tuned for the resonance K=5 with 10 lobes. For the metallic shell either idealized silver without plasmon losses ϵ shell 400 nm =4.4 (b) or realistic ϵ shell 400 nm =4.4+0.2i is used (c).

Fig. 5
Fig. 5

Tangential component of the intensity of a dipole (second harmonic wavelength) placed at the surface of a spherical particle with silica core of radius 100 nm and a silver shell (with and without damping) of variable thickness (compare Fig. 4 for the geometry). a) Intensity | E ϑ ( ϑ ) | 2 of a dipole at the outer surface (without plasmonic losses, ϵ shell 400 nm =4.4 ). The intensity is detected 20 nm above the surface of the sphere and shown as a function of the scattering angle ϑ. b, c: Intensity | E ϑ ( ϑ ) | 2 ϑ of a dipole, which is placed at the inner and outer interface of a sphere, without (b) and with (c) plasmonic losses ( ϵ shell 400 nm =4.4+0.2i ), detected in 2 cm distance.

Fig. 6
Fig. 6

Resonances with 2K lobes occurring at certain values of the permittivity ε N Sphere of a solid metallic sphere in comparison to the model based on the planar plasmon for R=100 nm and λ Dipole =400 nm .

Fig. 7
Fig. 7

SH-intensity (a) and extinction spectrum (b) as a function of the fundamental harmonic wavelength. The dashed lines show the SH-scattering cross section C ext m,n (2n+1) k 2 (| a m,n SH | 2 +| b m,n SH | 2 ) . Three systems of silica particles with silver shell (including plasmon losses) of 5 nm, 7.5 nm, and 10 nm thickness are shown.

Fig. 8
Fig. 8

Effect of the variation of χ ( 2 ) on the position of the first maximum in pp- and sp-polarization ( θ pp , θ sp ), the ratio of the intensities of the first maximum ( I pp / I sp ) and the sum of the intensities of the first maxima in sp- and pp-polarization ( I pp + I sp ) for a silica particle ( R=100 nm ) with a silver shell (including plasmon losses) of R=15 nm thickness. The triangular parameter space displays the relative amount of the three independent components of χ ( 2 ) : χ zzz ( 2 ) , χ zxx ( 2 ) , and χ xxz ( 2 ) . In the corners of the triangle, only the component written next to the corner is nonzero. Along the edges the two components are nonzero, their relative magnitude becomes larger the closer the point lies to the respective corner [19,20].

Equations (9)

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P i (2) j,k χ ijk ( 2 ) E j FH E k FH
E FH ( r )= n=1 N m=n n a m,n N m,n ( 1 ) ( r )+ b m,n N m,n ( 1 ) ( r ) + c m,n N m,n ( 3 ) ( r )+ d m,n M m,n ( 3 ) ( r ).
E θ,ϕ inc ( R )+ E θ,ϕ sc, out ( R )= E θ,ϕ sc, in ( R )
E θ,ϕ sc, l ( R l )= E θ,ϕ sc, l+1 ( R l ).
E θ,ϕ sc, l ( R L )= E θ,ϕ sc, l+1 ( R L )+ E θ,ϕ pw ( R L ).
E Dipole ( r ) ={ E (j=3),Dipole ( r ) = m,n a m,n (j=3) N m,n (j=3) ( r )+ b m,n (j=3) N m,n (j=3) ( r ) if| r |>if| r d | . E (j=1),Dipole ( r ) = m,n a m,n (j=1) N m,n (j=1) ( r )+ b m,n (j=1) N m,n (j=1) ( r ) if| r |< | r d |
E θ,ϕ sc, l1 ( R l1 )= E θ,ϕ Dipole,(1) ( R l1 )+ E θ,ϕ sc, l ( R l1 ),
E θ,ϕ sc, l ( R l )+ E θ,ϕ Dipole,(3) ( R l )= E θ,ϕ sc, l+1 ( R l ).
λ SPP = λ Dipole ε Sphere + ε Surrounding ε Sphere ε Surrounding

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