We report the three-dimensional mapping of 150 nm gold metallic nanoparticles dispersed in a homogeneous transparent polyacrylamide matrix using second-harmonic generation. We demonstrate that the position of single nanoparticles can be well defined using only one incident fundamental beam and the harmonic photon detection performed at right angle. The fundamental laser beam properties are determined using its spatial autocorrelation function and used to prove that single nanoparticles are observed. Polarization resolved measurements are also performed allowing for a clear separation of the second-harmonic response of the single gold metallic nanoparticles from that of aggregates of such nanoparticles.
©2010 Optical Society of America
The optical properties of metallic nanoparticles are dominated by the Surface Plasmon resonances arising from the collective oscillation of the conduction electrons . The size and shape effects on the linear optical properties of metallic nano-objects are now widely understood. A deeper understanding has nevertheless been possible through the use of experimental set-ups allowing single nanoparticle measurements, preventing the averaging procedure over the morphology and orientation distributions [2–5]. These experimental set-ups include dark field spectroscopy, photothermal emission and spatial modulation techniques, the advantage of the latter being the determination of the absolute extinction cross-section. Nowadays, the ability to combine numerical simulations and 3D transmission electronic microscopy at the single particle level offers an ultimate development to characterize the linear optical properties of these particles .
The nonlinear optical properties of metallic nanoparticles have also been widely studied in the past decades. Second-harmonic generation (SHG), in particular, is an optical process very sensitive to the nanoparticles shape because it is forbidden in centrosymmetric media within the frame of the electric dipole approximation . Kauranen’s group and more recently Valev and associates have investigated the SHG from gold metallic nanoparticles with various non-centrosymmetric shapes produced by nanolithography. They especially discussed the local field effects and the multipole contributions to the nonlinear optical response [8–10]. Nanoparticles dispersed in solution were also studied by Hyper Rayleigh Scattering (HRS). The role played by the breaking of the centrosymmetry induced by the non perfect spherical shape of the nanoparticles was clearly demonstrated  as well as the symmetry cancellations occurring in the quadratic hyperpolarizability of non centrosymmetric gold decahedra . The SHG intensity from metallic nanoparticles is furthermore enhanced by the surface plasmon resonances leading to much higher hyperpolarizability magnitudes than the one reported for the best available molecular chromophores [13, 14]. The SHG efficiency can be increased further with core-shell nanocavities filled with a non centrosymmetric crystalline core . We have recently demonstrated that HRS is sensitive enough to reach the single nanoparticle detection level  opening the way to the first measurements of the second-harmonic generation of a single 150 nm gold nanoparticle embedded in a gelatin matrix . The main advantage of this configuration where the centrosymmetry of the surrounding medium is preserved around the nanoparticle studied is to allow the direct comparison with theoretical works, something which is no longer easily feasible in the case of nanoparticles deposited on a substrate . One can note that single gold nanoparticles can also be detected by Third Harmonic Generation enhanced by the Plasmon resonance .
Theoretical works have been devoted to the SHG response from a small perfect sphere, the diameter of which is much smaller than the wavelength of light, when the SH emission is due to retardation effects [20, 21] or to the presence of an inhomogeneous incident electric field [22, 23]. A nonlinear Mie theory has early been developed  (note that the set of boundary conditions have been corrected in ref. 25) and recently updated to describe sum-frequency scattering in the case of non-collinear incident beams [26, 27]. Finite Element Method (FEM) simulations  and classical electrodynamics where the quasi-free electrons inside the metal are approximated as a classical Coulomb-interacting electron gas  were also reported with the purpose of taking into account the SHG response from metallic nano-objects with non-spherical shapes. Comparison between FEM simulations and experimental results has in particular enabled us to determine the role played by the local and non-local effects in the SHG response from gold metallic nanoparticles using interferences between the selected dipolar and octupolar plasmons [17, 30, 31].
From a practical point of view, metallic nanoparticles are widely used in biological applications since they constitute both nano-sources of light and nano-sources of heat. Indeed, the optical and thermal properties of metallic nanoparticles are used for cancer cell imaging and therapy. The photo-thermal cancer therapy requires however that the position of the nanoparticles used as labels is well determined in the cancer cells . This can be carried out with nonlinear spectroscopic processes such as two-photon fluorescence . Then, the high absorption cross section of the nanoparticles can be used to locally heat up and kill the targeted cells. We have shown that the nonlinear optical process SHG allows the determination of single nanoparticles position in a transparent and homogeneous matrix . These results are promising for the use of the nonlinear optical properties in biological applications.
In this work, we study spherical gold nanoparticles the diameter of which is 150 nm dispersed in a transparent polyacrylamide (PAA) matrix. We show that the position of single nanoparticles could be well determined in the three directions of space by collecting at right angle their SHG signal driven by only one laser beam. Aggregation between the nanoparticles can occur during the sample preparation and their presence can be a serious impediment in single particle measurements. The clear separation of the responses of a single nanoparticle and an aggregate is shown to be readily accessible in polarization resolved SH measurements. The SH response from a single nanoparticle is shown to be quadrupolar as expected in this experimental configuration whereas the response from the aggregate is more complicated. The present report therefore shows that SHG is sensitive enough to symmetry to distinguish at the single particle level these two kinds of nano-objects.
2. Experimental set-up and sample preparation
A mode-locked Ti:sapphire laser tuned to a wavelength of 804 nm and delivering pulses of about 180 fs at a repetition rate of 76 MHz was used as the laser source. The pulse energy measured at the laser exit was 10 nJ. The laser beam was focused into a quartz cell with a microscope objective (X16, NA 0.32) and a low-pass filter was used to remove any residual light at the harmonic frequency generated prior to the cell. The SH photons were collected perpendicularly to the incident beam with a 25 mm focal length lens with a numerical aperture of 0.5. The scattered fundamental photons were removed by a high-pass filter placed before the monochromator. The polarization angle γ of the fundamental beam was selected with a rotating half-wave plate (for a vertically polarized field) and the polarization of the SH photons was selected by an analyzer for the polarization resolved measurements. The photon detection was performed by a sensitive cooled photomultiplier tube and the fundamental beam was chopped at 130 Hz to allow a gated photon counting regime removing the background photons.
The polymer matrix was prepared using a water solution (milliQ grade) of 10% acrylamide and bisacrylamide (ratio 98:2). After dispersion of the gold nanoparticles (150 nm, BBInternational), polymerisation occured in 30 minutes by adding ammonium peroxodisulfate ((NH4)2S2O8) and TEMED (1,2-Bis(dimethylamino)ethane). The refractive index of the matrix is close to that of pure water (taking into account 10% of PAA with a refractive index of 1.52 compared to 1.33 for water), leading to a broad dipolar plasmon resonance at around 650 nm, ie far from both the fundamental and the harmonic wavelengths. There is therefore no resonance effect in the present experiments.
3. Experimental results and discussion
The sample scans were carried out perpendicularly to the incident fundamental beam direction in order to obtained two-dimensional (2D) maps of the SH intensity versus cell position with 14 seconds acquisition time per pixels, following the experimental protocol described in ref 17, see Fig. 1 . The heterogeneous spatial distribution of the SH intensity reveals areas with low and high intensities. The background SH intensity corresponds to the signal recorded for the polymer matrix without particles and the low SH intensity areas correspond therefore to areas composed exclusively of the polymer matrix. This SH intensity produced by the PAA matrix is weaker than the one emitted by the gelatin matrix used in our previous single nanoparticles measurements  and allowed a higher signal-to-noise ratio. More precisely, the SHG signal from the matrix has been reduced down to 1.2 times that of water, in which the particles are dispersed. We have therefore almost reached the best contrast particle/matrix that can be obtained.
The high SH intensity zones in Fig. 1 are attributed to the presence of the gold metallic nanoparticles . In order to demonstrate the 3D imaging capability, successive 2D maps were recorded for different positions of the cell along the direction propagation of the incident beam with a distance of 8 µm from one map to the other (only four are shown here). The positions of the SH intensity maxima are well correlated between the different maps pointing out that the particles lateral positions are very well known. However, the amplitudes of these maxima evolve from one map to the other owing to the displacement of the nanoparticles along the laser beam. For example, one can note that four maxima appear on the right of the first map. Their amplitude increases as the sample is moved along the beam direction. This behaviour can be understood as follows: the nanoparticles are dispersed in the three dimensions in the sample. The nanoparticles are therefore moved in and out of the laser beam waist as the cell is moved along the incident beam.
The nonlinear efficiency of a given metallic nanoparticles can be obtained by subtracting the SH intensity from the maxima corresponding to a nanoparticle that of the background signal emitted by the polymer matrix, about 5 photons/s . The PAA polymer matrix can then be used as an internal reference to determine the quadratic hyperpolarizability of the 150 nm diameter gold nanoparticles. Preliminary experiment have demonstrated that the SH intensity recorded for the pure polymer matrix is identical to the one recorded for water in the same experimental configuration. The intensity radiated at the harmonic frequency for the matrix in the presence of the metallic particles can therefore be written as :14] are the concentration and the quadratic hyperpolarizability of water, respectively and the factor 1.2 stems for the relative efficiency of water and PAA matrix. G is a geometrical factor. In Eq. (1), we consider that the molecules are randomly oriented in a mesoscopic volume leading to an expression very similar to that used for hyper Rayleigh scattering. Note that a more detailed description taking into account all the components of the quadratic hyperpolarizability tensor might be found in Ref. 34. The quadratic hyperpolarizability of a single gold nanoparticle is given by and stems for the intensity of the fundamental beam illuminating the gold nanoparticles. In this case, the particle being fixed and isolated, there is no averaging over the particle orientations.
In Eq. (1), the spatial distribution of the square of the fundamental intensity has to be determined over the laser beam section. We use the PAA matrix as an internal reference for the determination of quadratic hyperpolarizability of the nanoparticles in the mapped area. For this purpose, the spatial autocorrelation map of Fig. 1(d) for which the amplitudes of the picks are maximum was calculated in order to determine the experimental conditions and is displayed in Fig. 2 .
The spatial autocorrelation for a real-valued function of two variables is given by:Fig. 1. The integration is performed over the whole map. The spatial autocorrelation map exhibits three specific length scales. First, the correlation of the noise itself with a length equal to one pixel contributes only to the origin. Hence, the high value of is due to the noise affecting the data during the acquisition runs and will not be further taken into account in the following analysis. The second characteristic length is related to the lateral dimensions of the laser beam. Indeed, the nanoparticles studied the diameter of which is 150 nm are much smaller than the waist of the fundamental laser beam. This implies that the number of pixels on the map corresponding to one nanoparticle is determined by the dimension of the laser beam in the sample and by the position of the nanoparticle in the beam. This contribution explains the maximum obtained at the centre even though the noise contribution has been rejected. This maximum has a gaussian profile with a standard deviation equal to 3.53 µm as determined by fitting procedure. This corresponds to the lateral resolution in the 3D imaging. This value is therefore related to the transverse dimension of the laser beam but is not the waist size of the fundamental beam. Indeed, SHG is a second order nonlinear process and the SH intensity is proportional to the fundamental intensity squared since two fundamental photons are necessary to create a single second-harmonic photon. In the present case, the SH intensity emitted by a nanoparticle is proportional to the squared local incident intensity at its position. In order to determine the waist size, the spatial autocorrelation of the 2D gold nanoparticles maps were compared with simple numeric calculations taking into account a random distribution of the nanoparticles and a gaussian beam with various waist sizes. The best agreement was found for a beam waist of about µm. The determination of the beam waist enables to compute the spatial distribution of both the intensity and the intensity squared, see Fig. 3 . It is obvious that the intensity squared decreases faster than the intensity itself. The volume probed by a two photons excited process is therefore much smaller than the one probed by a one photon excited process such as fluorescence for instance. This property is well known in microscopy and largely applied by imaging techniques in order to increase the spatial resolution . The evolution of the four maxima on the maps is in agreement with the spatial distribution of the fundamental intensity squared and its decrease along the incident beam propagation direction. The resolution of the 3D imaging along the incident beam propagation direction is therefore about 40 µm. This rather large value could be largely reduced by the use of a stronger focusing objective (larger numerical aperture). Finally, the last contribution length scale in Fig. 2 is due to the distribution of the gold nanoparticles in the probed area. The small maxima are related to the position of the gold nanoparticles which are randomly distributed in the sample.
Knowing the spatial distribution of the intensity squared, one is able to determinate the quadratic hyperpolarizability of the nanoparticles with the use of the Eq. (1). The background signal emitted by the matrix is due to the incoherent summation of the molecules forming the polymer matrix weighted by the local intensity squared integrated over the focussed beam crossection. The SH intensity emitted by a given nanoparticle therefore evolves on the different 2D SH maps and is maximum when the nanoparticle is in or very close to the beam waist. It is ensured that the nanoparticle is located within the beam focus when the SH intensity increases or decreases as the cell is moved along the laser beam. This is obviously the case for the nanoparticles on Fig. 1 d) where a maximum intensity of 450 counts is obtained. The broadband emission spectra for the matrix and a single nanoparticle were also recorded with a longer acquisition time than the one used per pixel for the different maps in order to improve the signal-to-noise ratio and to limit the uncertainty on the hyperpolarizability measured, see Fig. 4 . These spectra provide evidence for a collected signal intensity dominated by SHG response from the nanoparticles and not a photoluminescence background.
The quadratic hyperpolarizability for the single nanoparticle studied was found to be equal to esu. This value is in excellent agreement with ensemble measurements in solution where a value of esu was obtained . Here, the brackets stand for the size, shape and orientational average owing to the ensemble nature of the measurement. The value is also very similar to the one obtained for single nanoparticles in a gelatine matrix , namely esu, providing that i) the high intensity areas in Fig. 1 are associated with single particles and ii) the measure of the quadratic hyperpolarizability is almost not affected by the nature of the surrounding medium.
As described in the experimental section, the nanoparticles were synthesized in solution and then dispersed in the PAA polymer matrix. This process does not prevent the nanoparticles aggregation and it is therefore expected that aggregates can also be locally observed. Figure 5 shows a broad maximum, the amplitude of which is much higher than the one expected for a single nanoparticle. The latter maximum is therefore probably associated to the presence of an aggregate in the area scanned and may correspond to several nanoparticles instead of a single one . The amplitude of the four small maxima corresponds to 35 counts per second as expected for single gold nanoparticles. The two arrows show the aggregate (Agg) and the single nanoparticle (SP) studied below. The number of pixels corresponding to the aggregate Agg is larger than the one observed for single nanoparticles providing further evidence that its dimension is not negligible compared to the waist of the beam.
In order to demonstrate the presence of aggregation, the SH intensity for the vertically polarized harmonic field as a function of the fundamental polarization angle from the aggregate (Agg) and from the single nanoparticle (SP) was measured, see Fig. 6 . The SH intensity from the neat PAA polymer matrix was also recorded for comparison. The polarization resolved SH intensity plot from the single nanoparticle reaches maximum intensity values for 45°, 135°, 225° and 315° as expected. Indeed, the leading emission term in the SHG response from a spherical 150 nm diameter nanoparticle is a quadrupole as demonstrated by theoretical analyses [20, 21, 28], ensemble  and single nanoparticle measurements . The SH intensity is however not equal to zero for a vertically and horizontally polarized input beam and the four lobes are not exactly equal owing to the imperfect spherical shape of the nanoparticle under study , in contrast to the single nanoparticle reported in ref 17. In the case of the aggregate, the SH intensity is drastically larger for a vertically polarized incident electric field (0° and 180°) as compared to the other fundamental polarization states. This behaviour is related to the irregular shape of the aggregate leading to a specific electric field distribution inside and near the aggregate, dramatically modifying the nonlinear optical properties of the aggregate, in particular the SH scattered intensity and the polarization selection rules by symmetry breaking and near field interaction/coupling between the particles forming the aggregate [8, 34, 39, 40]. Furthermore, the pattern of the aggregate is very different from the one obtained for small but imperfectly spherical gold nanoparticles which has two lobes oriented at 0° and 180° characterizing a pure electric dipole emission [11, 28]. The SH intensity recorded as a function of the input polarization therefore allows the complete discrimination between single metallic nanoparticles with a shape close to the perfect sphere and aggregates of nanoparticles.
In summary, we have studied single gold metallic nanoparticles embedded in a homogeneous and transparent polyacrylamide (PAA) matrix having a low quadratic nonlinear optical efficiency allowing a high signal-to noise ratio despite the weak signal expected for single nanoparticles. The nanoparticles positions are accurately determined using only their SH signal excited by a single fundamental incident beam. The spatial autocorrelation function of the 2D maps obtained by scans perpendicular to the incident beam allows us to determine the waist of the beam which is equal to µm. This spatial autocorrelation procedure enables us to measure the quadratic hyperpolarizability of single gold metallic nanoparticles which are in good agreement with the value obtained by ensemble and previous single particle measurements. These results provide evidence that indeed single gold nanoparticles are studied. Finally, the SH intensity from a single gold nanoparticle and from an aggregate is recorded as a function of the polarization angle of the fundamental beam. These results demonstrate that single nanoparticles and aggregates of nanoparticles can be clearly discriminated by polarization resolved SHG measurements.
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