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Bachelier et al. Vol. 25, No. 6 /June 2008/J. Opt. Soc. Am. B 955

Multipolar second-harmonic generation in noblemetal nanoparticles

Guillaume Bachelier, Isabelle Russier-Antoine, Emmanuel Benichou, Christian Jonin, and Pierre-François Brevet

Laboratoire de Spectrométrie Ionique et Moléculaire, Université Claude Bernard Lyon 1—CNRS UMR 5579,43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France

*Corresponding author: guillaume.bachelier@lasim.univ-lyon1.fr

Received November 15, 2007; revised February 26, 2008; accepted March 24, 2008;posted April 7, 2008 (Doc. ID 89723); published May 20, 2008

Second-harmonic generation from noble metal nanoparticles with a noncentrosymmetrical shape is theoreti-cally investigated by using finite element method simulations. The relative weight of the dipolar and quadru-polar responses is investigated in terms of both light polarization and size dependence of the harmonic scat-tered intensity. It is shown that, even for small deformations as compared with purely spherical particles, thedipolar response dominates and scales as the nanoparticle surface area squared. The difference between goldand silver metal nanoparticles is also addressed. © 2008 Optical Society of America

OCIS codes: 190.2620, 190.4350, 240.4350, 240.6680.

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. INTRODUCTIONoble metal nanoparticles have received increasing at-

ention over the past years due to their optical propertiesoverned by surface plasmon polaritons, the collectivelectronic oscillation coupled to the electromagnetic field,n the visible domain. These optical properties are highlyensitive not only to their environment but also to thearticle morphology [1,2]. Recently, the convergence ofransmission electron microscopy characterization, singleanoparticle detection, and numerical simulations hasriggered an extensive interest in designing various par-icle shapes such as nanorods, nanocubes, nanorices, oranostars to adjust the morphology to obtain the desiredpectral responses [3–5]. Indeed, one of the main specifici-ies of noble metals is their ability to support electric fieldocalization at the nanometer scale owing to the evanes-ent character of the surrounding waves. This allows theventual guiding of light below the diffraction limit [6].his localization is also responsible for very large field en-ancements of first interest in spectroscopic applicationsuch as Raman scattering, and more particularly surfacenhanced Raman scattering [7] or nonlinear optical pro-esses such as two-photon luminescence or wave mixing8]. This localization effect is furthermore strengthened atough surfaces or random materials with the appearancef hot spots, leading to very large enhancements due tohe overlap of fundamental and harmonic eigenmodes9–11]. However, the spatial distribution of these hotpots being hardly predictable, another approach was pro-osed to tailor the electric field either by using sharpetal tips for near-field applications [12] or nanolenses

omposed of self-similar chains of nanoparticles [13].Among all nonlinear optical phenomena, second har-onic generation (SHG) is the simplest, but like all even-

rder processes it is also forbidden within the electric di-ole approximation in centrosymmetrical bulk materialsuch as noble metals like gold and silver [14–19]. Foranosized noble metal structures, this selection rule is

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till valid if one assumes a volume material structuredentical to the bulk material one. Hence, in such struc-ures, contributions arising from the breaking of cen-rosymmetry at the surface must be taken into account,ince they may not be negligible anymore. However, ifhese structures also possess centrosymmetrical shapes,he surface SHG response will again vanish in the electricipole approximation. Contributions involving quadrupo-ar surface plasmon polaritons associated with retarda-ion effects either at the excitation or the radiation stagere therefore expected as well as contributions arisingrom nonlocal nonlinear sources. They are both respon-ible for a nonvanishing SHG intensity scaling with thequare of the particle volume [14–19]. As a matter of fact,everal experiments were performed recently on sphericalold nanoparticles showing a clear dipolar response forhe smaller sizes together with a SHG intensity scalingith the particle surface area squared [20–22]. These re-

ults are at variance with the theoretical expectations forarticles with centrosymmetrical shapes. Deviations fromuch regular shapes therefore had to be introduced to cor-ectly account for the experimental data [21,22]. This ap-roach has been further confirmed in recent results ob-ained with L-shaped nanoparticles, where the nonlinearptical properties are largely dominated by dipolelike con-ributions [23,24]. The aim in this work is thus to addresshe problems of the deviation from centrosymmetricalhapes and the retardation effects in the nonlinear prop-rties of metal nanoparticles to clarify the origin of the ob-erved discrepancies between experimental and theoreti-al results.

. MODELuadrupolar SHG is the expected response for spherical,nd more generally centrosymmetrical, particles, and itsize and light polarization dependence is now well estab-ished [14–19]. Deviations from these predictions may be

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956 J. Opt. Soc. Am. B/Vol. 25, No. 6 /June 2008 Bachelier et al.

ue to either heterogeneities in the particle volume,amely, a noncentrosymmetrical material structure, or toeviations from the spherical shape. At this point it muste stressed that the deviations could arise either from aenuine morphological deviation from the sphericalhape, among other centrosymmetrical shapes like ellip-oids, for instance, or from an inhomogeneous surface ad-orption layer, for example. It has to be pointed out,hough, that the first case, with an origin arising fromolume defects of the material structure, would lead toonlinear sources distributed randomly in the particleolume and thus to a SHG intensity scaling with the par-icle volume squared. This is not in agreement with thexperimental results [21,22]. For this reason, only the de-iation of the particle shape from that of a perfect spheres considered here.

To obtain a noncentrosymmetrical deformation, the topalf-sphere of the particle was rescaled along the x axissee Fig. 1) in such a way that the deformation can be pa-ameterized with a single parameter, namely, �R /R,hich corresponds to the relative increase of the particle

adius along the x axis. This done, an analytical approachs no longer possible, and therefore finite element methodFEM) simulations were performed by using a commercialoftware (COMSOL Multiphysics). The electric field�r ,�� at the fundamental frequency was computed in

he framework of the scattered field formulation by usingerfectly matched layers to avoid spurious reflections athe surrounding medium boundaries [25]. The dielectriconstants used for the metal particle were taken from26].

The nonlinear sources of the SH field are usually recastnto local and nonlocal responses, the latter involving notnly the electric fields but also the electric field gradients.or centrosymmetrical materials, the appropriate nonlin-ar response is that composed of the surface local term

Psurf�r,2�� = �Jsurf:E�r,��E�r,�� �1�

nd the bulk nonlocal terms

ig. 1. Schematic of the geometrical configuration used for theimulations: � is the angle of polarization of the linearly polar-zed incident beam Ein�r ,�� propagating along the z axis (waveector kin). �R corresponds to the deformation applied to the up-er half-sphere part of the particle. The vertically polarized elec-ric field EV�r ,2�� scattered at the harmonic frequency is col-ected at a right angle in the y axis direction.

Pbulk�r,2�� = ���E�r,�� · E�r,��� + ��E�r,�� · ��E�r,��,

�2�

here P is the polarization vector, �Jsurf the second-orderonlinear susceptibility, and � and � two parameters de-ning the nonlocal nonlinear response [18,19]. The secondulk nonlocal term may dominate at the particle surface,ut it has also been shown that it could be incorporatednto the surface response [27]. The nonlinear currents, ef-ectively the true sources of the electric field E�r ,2�� athe harmonic frequency, were deduced in a second steprom the spatial distribution of the polarization vector Phrough the well-known relation J�r ,2��=�P�r ,2�� /�t.fter the Maxwell equations were solved by using theeak formulation, the far-field component of the electriceld at the harmonic frequency E�r ,2�� was computed

rom the near-field component by using the Stratton–Chuormula [25]. This procedure allows us to evaluate a scat-ered SHG intensity that is, to some extent, comparableith the experimental data.While solving Maxwell equations, we take into account

ll multipolar components in the sense of the scatteringheory of both the fundamental and harmonic electricelds. However, for the sizes and the spherical shape de-iations studied in the present work, the SHG scatteredntensity is dominated mainly by the dipolar and quadru-olar terms. The aim in this work is therefore to weighthe dipolar and quadrupolar responses driven by retarda-ion and deformation effects. In this respect, the generalehavior of all nonlinear sources, both surface and vol-me, were found to be similar, and thus the discussionill be limited to the surface local term only:

Psurf,��r,2�� = �surf,���Eex,�2 �r,��, �3�

here the nonlinear surface susceptibility tensor is re-uced to the single element �surf,���. Psurf,��r ,2�� andex,��r ,�� are the surface nonlinear polarization and the

undamental electric field components perpendicular tohe particle surface, respectively. The normal componentf the electric field being discontinuous at the particle sur-ace, the nonlinear current Jsurf,��r ,2���Psurf,��r ,2�� /�t was arbitrarily computed by using thexternal fundamental field.

. RESULTS AND DISCUSSION. Single Particle Responsehe geometrical configuration used to compute the lightcattered at the harmonic frequency is given in Fig. 1.he linear polarization of the incident electric field propa-ating along the z axis is defined with the angle �, and theertically polarized SHG intensity along the x axis is col-ected at the right angle in the y axis direction for directomparison with the experimental setup used in a previ-us study [21,22]. The scattered harmonic intensity islotted on a polar plot as a function of the incident polar-zation angle �. For a perfectly spherical nanoparticle,ypically 5 nm in radius, this plot reveals a typical four-obe pattern corresponding to a pure quadrupolar re-ponse of a centrosymmetrical particle [14–19]. This pat-ern simply arises from the fact that for �=0�� /2�, the

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Bachelier et al. Vol. 25, No. 6 /June 2008/J. Opt. Soc. Am. B 957

Oz plane is a symmetry plane for the nonlinear sources.herefore, the component of the nonlinear electric fieldv�r ,2�� perpendicular to the yOz plane vanishes in this

ase. When the particle is deformed in a noncentrosym-etrical way following the procedure described above (seeig. 1), the yOz plane is no longer a symmetry plane, anddominant dipolelike pattern is observed; see Fig. 2(a). It

s interesting to note that the two major lobes are notymmetric with respect to the horizontal axis in Fig. 2,ince the +� and −� polarization angles are no longerquivalent. The SHG response is thus not purely dipolar,ontaining a quadrupolar contribution as well; otherwiset would have been fully symmetrical with only two lobes.o account for the full � dependence of the vertically po-arized SHG intensity ISHG

V ���, the following expressionas therefore used:

ISHGV ��� = aV cos4��� + bV cos2���sin2��� + cV sin4���

+ dV cos3���sin��� + eV cos���sin3���, �4�

here aV, bV, and cV are the usual parameters regularlyntroduced in previous works, whereas dV and eV accountor the asymmetry of the polar graphs introduced by thengle � [21,22]. These latter parameters are necessary toccount for the incomplete orientational averaging proce-ure over the orientations taken by the particles [28–30].s shown in Figs. 2(a)–2(c), this expression perfectly ac-ounts for the data points calculated with the FEM simu-ations.

. Orientation Averaging Effecthe SHG experiments on metal nanoparticles are usuallyerformed through hyper-Rayleigh scattering in solutionshere the particle orientation is random [21,22]. The

ig. 2. Normalized polar plots of the vertically polarized SHGntensity as a function of the incident polarization angle � for dif-erent orientation of the particle in the xOy plane determined byhe angle �: (a) �=0, (b) �=� /4, (c) �=� /2, (d) averaged. �=0oincides with the right-hand side of the horizontal axis, and �ncreases anticlockwise. The radius of the gold particle is 5 nm,nd the deformation is �R /R=15%.

cattered intensity thus has to be averaged randomly overll particle orientations. To highlight the importance ofuch an averaging procedure, Figs. 2(a) and 2(c) show theesults obtained when the particle is rotated around the zxis by the angles �=� /4 and �=� /2, respectively. For=� /2, the yOz plane is again a plane of symmetry of theonlinear sources for an incident polarization angle �0�� /2�. In this case, the polar graph recovers the four-quivalent-lobes pattern, as is the case for a perfectphere. In contrast, the � dependence of the SHG inten-ity for �=� /4 is highly distorted owing to the rotation by5° of the particle x axis. Averaging over all values of therientation angle � [Fig. 2(d)], the polar graph recovers aully symmetric pattern, leading to a vanishing value forhe parameters dV and eV. The observed four-lobe re-ponse with nonvanishing intensity at �=0�� /2� is char-cteristic of the coexistence of both a dipolar and a qua-rupolar contribution in the SHG response from thearticle: the remaining intensity at �=0�� /2� is due to theure dipolar response only.A realistic comparison to the hyper-Rayleigh scattering

xperiments would require an averaging procedure overll 3D random orientations but also over the deformationsf the particles. This is beyond the scope of the presentork and the capacities of the simulations performed in

his work. Hence, only the averaging involving a rotationbout the z axis of Fig. 1 will be discussed to illustrate thealient features of the orientational effects.

. Deformation Effecthe relative weight of the dipolar and the quadrupolarontributions strongly depends on the amplitude of theeformation applied to the particle as shown in Fig. 3.he larger the deviation from a perfect spherical shape,

ig. 3. Normalized polar plots of the vertically polarized SHGntensity as a function of the incident polarization angle � for dif-erent deformations of a 5 nm radius gold particle. �=0 coincidesith the right-hand side of the horizontal axis, and � increasesnticlockwise. The scattered intensity is averaged over the orien-ations of the particles in the xOy plane determined by the angle. (a) �R /R=0%, (b) �R /R=10%, (c) �R /R=20%, (d) �R /R=40%.

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he higher the dipolar contribution, as expected. To quan-ify the ratio between the two contributions, the weight-ng parameter V is introduced as [21,22,31]

V = �bV − �aV + cV�

bV � . �5�

ased on the fact that aV=cV=0 for a pure quadrupolarattern and aV+cV=bV for a pure dipolar one, V rangesrom zero for a pure dipolar response to unity for a pureuadrupolar one. This is shown in Fig. 4 for a gold nano-article with a 5 nm radius and an increasing relative de-ormation �R /R. Despite the highly symmetrical deforma-ion introduced—in particular this deformation possessesn axial symmetry leading to a rather regular shape—atrong dipolar response can already be achieved for a de-ormation of �R /R=30%, i.e., for a diameter increase ofnly 15% along the deformation axis. For more realisticoncentrosymmetrical deformations, including faceting,n even stronger dipolar contribution is expected, ex-laining why the second-harmonic response of small goldanoparticles is essentially of dipolar type [21,22].

. Size Effectow focusing on the size dependence of the vertically po-

arized SHG intensity for a given deformation of the par-icle, namely, �R /R=30%, a clear transition from a dipo-ar to a quadrupolar response is observed while thearticle size is increased (see Fig. 5), despite the noncen-rosymmetrical shape of the particle. This is attributed tohe retardation effects taking place for larger nanosizedarticles, enhancing the quadrupolar contribution withespect to the dipolar one. This transition can be easilyollowed by using the weighting parameter V as shown inig. 4. The general trend of the V size dependence is inood agreement with the experimental data reported pre-iously [21,22], although it is not possible to quantita-ively reproduce the size at which the transition occurs

ig. 4. Plot of the V parameter for a gold nanoparticle as aunction of the relative deformation �R /R (circles) for a fixed par-icle size (5 nm radius) and as a function of the particle size (dia-onds) for a fixed deformation ��R /R=30% �. The scattered in-

ensity used for the computation of V is averaged over therientations of the particles in the xOy plane determined by thengle �. The lines are guides for the eye.

V=0.5�, about 6 nm in radius in the present study inontrast to the 35 nm in radius reported experimentally.his originates from the rather smooth deformation usedere for the FEM simulation as compared with the realorphology of the particles experimentally studied. It is

hus not surprising that the agreement is only qualita-ive.

ig. 6. Size dependence of the vertically polarized SHG inten-ity for a gold particle with a relative deformation of �R /R30%. The scattered intensity is averaged over the orientationsf the particles in the xOy plane determined by the angle �. Theotted and dashed curves correspond to a surface area squaredR4� and volume squared �R6� dependence, respectively (log–loglot).

ig. 5. Normalized polar plots of the vertically polarized SHGntensity as a function of the incident polarization angle � for dif-erent a gold particle sizes with a fixed relative deformation ofR /R=30%. �=0 coincides with the right-hand side of the hori-ontal axis, and � increases anticlockwise. The scattered inten-ity is averaged over the orientations of the particles in the xOylane determined by the angle �. (a) R=2.5 nm, (b) R=5 nm, (c)=10 nm, (d) R=20 nm.

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Bachelier et al. Vol. 25, No. 6 /June 2008/J. Opt. Soc. Am. B 959

For a given deformation, the relative weight betweenhe dipolar and the quadrupolar contributions to the SHGntensity is directly governed by the size dependence ofhese two contributions. As shown in Fig. 6, the SHG in-ensity is found to be proportional to the particle surfacerea squared �R4� when the dipolar response dominatest small radii, whereas it is proportional to the particleolume squared �R6� when the SHG response is domi-ated by the quadrupolar response at large radii. Theresent FEM simulations clearly support the interpreta-ions of the experimental data reported recently [21,22],hich deviates from the common belief that the SHG in-

ensity from nanosized quasi-spherical particles alwayscales as the particle volume squared. It is also interest-ng to note in Fig. 6 that for the largest particles a devia-ion from the volume-squared dependence toward higherowers of the particle radius is observed owing to the in-uence of higher multipoles.Finally, the case of silver nanoparticles (data not shown

ere) has been investigated as well. The same general be-avior is observed except that the retardation effectsrise at smaller particle sizes as compared with gold par-icles, as is the case for the linear optical properties. As aonsequence, the quadrupolar contribution leading to theour-lobe pattern and the particle surface-area-squaredependence of the SHG intensity are observed at smallerizes, in good qualitative agreement with the experimen-al observation reported recently [21,22].

. CONCLUSIONn conclusion, FEM simulations have been performed tonvestigate the effects of a noncentrosymmetrical defor-

ation on the SHG response from noble metal nanopar-icles. It is found that (i) the SHG response from smallarticles is dominated by the dipolar response even formall deformations; (ii) the dipolar contribution to theHG response of a quasi-spherical particle is proportionalo the surface area squared and not to the volumequared, as is the case for the quadrupolar response; (iii)he light scattered at the harmonic frequency is domi-ated by the quadrupolar response for large particlesven for noncentrosymmetrical particles owing to retar-ation effects; and (iv) the retardation effects responsibleor the quadrupolar SHG contribution are observed atmaller sizes for silver particles than for gold ones, simi-arly to the linear optical properties. All these results areupported by the experimental results published recently.

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