1. Introduction

Metal nanostructures and nanoparticles possess the unique ability to spatially localize electric fields in sub-wavelength regions at the vicinity of their surface. In the last decades a large amount of work has been dedicated to characterize these plasmon confinements and take advantage of their underlying fields’ enhancements, for efficient light control nano-devices or highly sensitive molecular sensors. These localizations are specifically related to the excitation field polarization, which makes it a unique tool to control electromagnetic fields at the near field scale. Several techniques have been developed to probe the spatial characteristics of electric field localizations in sub-micrometric metallic nanoparticles. Near-field maps have been achieved using near-field scanning optical microscopy [1], electron energy-loss spectroscopy [2], photoemission electron microscopy [3–5], or cathode luminescence imaging [6]. Those techniques however suffer from complex implementation and in some cases indirect interpretations. Purely optical methods offer an interesting alternative since it provides in situ analysis of optically induced plasmons. Recent developments have reported sub-diffraction resolution scale mapping of fields’ localizations by recording indirect emission signatures from nearby single molecules responses [7, 8] or more directly using the two-photon luminescence (TPL) emission from metal nanostructures [9, 10]. Most of these works have underlined the inherent relation between the polarization state of the incident electromagnetic excitation and the sub-wavelength spatial localization of fields’ enhancements, which are essentially governed by the particle size and shape [1, 3, 4, 11]. While TPL provides direct monitoring of the spatial features of nanoparticles’ enhancements, its emission characteristics however fail in reporting the vectorial properties of local fields in particles of complex shapes. TPL is indeed emitted incoherently and scrambled by depolarization due to fast dephasing and thermalization of the photoexcited electrons [12]. In contrast, the coherent nonlinear process Second Harmonic Generation (SHG) preserves phase information, and is therefore capable of revealing vectorial properties of confined optical fields in isolated structures [13] or interference mechanisms in coupled nanoparticles [14]. In the dipolar approximation, SHG is allowed only at the nanoparticles’ surface where centrosymmetry is locally broken, therefore exhibiting superior sensitivity to local hot spots [15, 16]. SHG imaging of individual metallic nanoparticles has been performed on centro-symmetric rods [17] or nano-spheres [13], on which numerous studies have enlighted the role of nonlocal electric dipole and local electric quadrupole modes excitations [18–21]. However due to their symmetry, the SHG efficiency of these structures is very weak and very sensitive to shape deviations or surface defects [17, 19, 22]. Recently new non-centrosymmetric geometries have shown enlarged SHG efficiencies, based on three dimensional nano-cups [23], chiral shape nanostructures assemblies [24, 25], or local nano-gaps [14, 26, 27]. In all these SHG investigations, the role of the fields’ spatial localization and its intrinsic coupling to the incident light polarization has not been fully investigated nor exploited.

In this work, we use far field SHG microscopy to probe the nano-scale coupling mechanisms between polarization excitation and localization as well as vectorial properties of local fields in individual nanoparticles. As a model system, we use branched, three-fold symmetry gold nanoparticles named nano-stars (Fig. 1). These structures exhibit a non-centrosymmetric shape and are resonant for the incident optical excitation wavelength, two favorable features for efficient SHG emission. By monitoring the spatial and intensity dependencies of the far-field SHG emission from individual nanoparticles upon a varying incident polarization, we provide information on the vectorial symmetry characteristics and the near-field spatial extend of the local fields in the nanoparticles, which are found to follow common trends with their shape observed by scanning electron microscopy (SEM) images. We show that such information is not accessible using TPL.

 

Fig. 1 (a) FDTD calculations of the electric field at the nano-stars particles/glass interface, at the incident fundamental wavelength 800 nm, for two incident polarization directions (white arrows). The scale represents the total field enhancement produced by the nano-star. (b) Similar calculation (horizontal polarization) for an incident wavelength at 400nm. (c) Vectorial map of the induced nonlinear dipoles generated from (a) for two incident polarization directions (nano-star positioned with its center on the optical axis). (d) Deduced SHG intensity map in the object plane (scale in arbitrary units).

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2. Modelling SHG scanned imaging of a single nano-star

Nano-stars are made of gold planar nanoparticles (150nm length, 50nm height) fabricated on a glass substrate by electron beam lithography (combined with scanning electron microscopy (SEM) in a Nanometer Pattern Generation System), in arrangements of typically 5 × 5 nanoparticles distant by 1.5μm, avoiding inter-particles optical interactions. Each nanoparticle is measured individually by both SEM and SHG microscopy. The nano-stars have been designed to exhibit a plasmon resonance at 800 nm (fundamental wavelength), characteristic of the dipolar mode of the particles, which coincides with the excitation laser used for SHG microscopy.

Finite Difference Time Domain (FDTD) calculations (Rsoft, FullWave) have been first carried out on these nano-stars to evaluate the possible sensitivity of SHG imaging to their shape and size. The simulated nano-star (Fig. 1) is 50 nm high (in the longitudinal Z direction) and its contour (in the microscopy (X, Y) sample plane) is interpolated from Scanning Electron Microscopy (SEM) images in a parametric polar form of third order symmetry given by (in nm):

The structure is made of gold (index n= 0.1808–5.1173i [28]), lying on a 105 nm high coverglass (refraction index at 800 nm and 400 nm: 1.5) with air above (refraction index: 1.0). The simulated area have 350 nm × 300 nm × 150 nm dimensions along the X, Y and Z directions and the mesh grid is set to 1 nm in all directions. A 50 nm wide perfectly matched layer (PML) with reflectivity of 10⁻¹⁰ surrounds the simulated area. A monochromatic planar wave at a wavelength 800 nm (fundamental wavelength) or 400 nm (SHG wavelength), linearly polarized, is launched into the structure along the optical axis Z. The amplitude and the phase of the total field (incident field and field scattered by the nano-star) are then extracted in the (X, Y) plane perpendicular to the optical axis, and located at mid-height of the nano-star. Simulations can be performed at any incident linear polarization direction.

These calculations show that for an excitation wavelength close to their plasmon resonance, the particles locate the scattered electric field E^ω in a region confined close to the tip of an arm when the incident polarization lies along this arm (Fig. 1(a)). Rotating this polarization by 90° delocalizes the scattered field along the two other arms, similarly to recent electron microscopy observations in three fold symmetry structures [5, 29]. This spatial behavior is expected to have important consequences on the SHG scanning image of single nano-stars, as seen below. Note that under excitation at 400 nm, the electric field is not enhanced and not favored for any specific arm direction, leading to a weak, homogeneously distributed response around the structure contour parts normal to the excitation direction (Fig. 1(b)). The nanostructure thus does not significantly affect the symmetry behavior of the harmonic fields radiation, which will be later calculated in the vaccuum in a first approximation.

The simulated fundamental local field is used as a source for non-linear currents within the nano-star. Rigorously, two contributions are responsible for SHG emission in a metal nano-particule: a surface and a bulk contribution. Locally, the surface contribution is proportional to the square of the fundamental field (at 800 nm) whereas the bulk contribution is proportional to gradient of the square of this field [20, 30]. Inspection of the simulated fields maps for these two quantities show a spatial overlap for any polarization state (data not shown). As a consequence, the surface and bulk contributions are predicted to show the same role in terms of symmetry properties of the spatial distribution of the nonlinear sources, therefore only the surface contribution is kept in this model. Following this first approximation, we therefore model the local nonlinear dipoles, sources of SHG radiation, as induced by a local nonlinear susceptibility contribution ${\chi }_{nnn}^{\left(2\right)}$ along the direction n of the normal to the nano-star surface (in this model, off-diagnoal tensorial terms are neglected, which is a reasonable approximation [30]):

This information is then used to model the image of a nano-star produced by SHG scanning microscopy in a epi-mode detection, where the same objective lens is used for excitation at the fundamental wavelength and collection at the harmonic wavelength. The focused excitation beam is first decomposed as a collection of plane waves [31] refined for Gaussian beams [32]. The nano-star is placed into the focused excitation field pattern at a given position which can be changed in the sample plane, mimicking the SHG scanned imaging mode which is used experimentally. For each position of the nano-star in its plane, its local field map is computed. This computation takes into account the spatial distribution of the focused field as well as the local enhancement related to the plasmon resonance, extracted from the FDTD maps. To avoid edge effects and to smooth the finite size of the mesh grid in the FDTD simulations, the local field is calculated 3 nm above the nano-star surface and then locally binned over an area of 2×2 nm². Its amplitude and phase are finally extracted and its local normal projection on the metal-air interface is used in Eq. (2), leading to a nonlinear dipole vectorial map as shown in Fig. 1(c).

To produce a SHG image, the harmonic fields emitted by this collection of non-linear induced dipoles around the nano-star countour are then computed for each wave-vector direction contained in the aperture cone of the collection objective lens, following a similar procedure as in [33]. A subsequent coherent summation permits to extract the collected SHG intensity for each position of the nano-star in its sample plane, thus building-up SHG images. Note that the dipoles located on the upper and lower surfaces are not accounted for: these dipoles are parallel to the optical axis and are therefore only slightly excited by the Z components of the incident focused field. In addition their radiated harmonic field is mostly emitted along directions that are not comprised within the aperture cone of the collection objective lens.

Simulated SHG scanned images obtained from this model are shown in Fig. 1(d). When the incident polarization is along one arm, the SHG response is clearly stronger with its image spot size almost limited by diffraction and slightly displaced towards the arm tip. Rotating this polarization by 90° leads to a SHG response which is not only spatially displaced towards the other two arms (Fig. 1(d)) but also more spread in shape, leading to a lower intensity. These properties are visibly the consequence of phase delay effects between local nonlinear dipoles, which here contribute to a weak coherent build up of the nonlinear signal. Our simple numerical approach finally shows that SHG imaging under a varying incident polarization has the potential to provide rich information on near-field scale behaviors of nano-particules, since both SHG spots center position and shape can be measurable quantities. It also provides a good qualitative knowledge of the nano-particle shape’s symmetry features (in particular symmetry defects) at scales below the diffraction limit. Such information is not reachable using un-polarized imaging or ensemble measurements, such as more traditionally done using Harmonic Light Scattering [20].

3. Experimental polarization-resolved SHG imaging of single nano-stars

The polarization-resolved SHG set-up used to image individual nano-stars is based on an inverted two-photon excitation microscope [34]. A Ti:Sapphire laser (wavelength 800 nm, pulse duration 150 fs, repetition rate 80 MHz) is reflected on a polarization distortion-free dichroic mirror and focused onto the sample by a high numerical aperture objective lens (×40, NA 1.15 water immersion), reaching a lateral optical resolution of about 300 nm. A SHG image (Fig. 2(a)) is obtained by scanning the incident focused beam in the sample plane using galvanometric mirrors, at a typical rate of 100 μs per pixel. The low average incident power at the focal spot (typically 0.4 mW) ensures sufficient signal to noise conditions and stable signals, below the photo-damage threshold of the nanoparticles. The SHG emission from the nanoparticles is collected by the same objective lens and spectrally filtered at 400 nm (the absence of photoluminescence leakage in this spectral region is ascertained by a separate emission spectrum measurement). The SHG signal is recorded by two photomultipliers working in the photon counting mode, along two perpendicular polarization directions separated by a polarizing beam splitter. The recorded intensities are denoted I_X and I_Y (X and Y are the sample plane axes). For polarization resolved measurements, the linear incident polarization is rotated in the sample plane using a half-wave plate mounted in a step motor holder at the entrance of the microscope. Images are recorded for each linear incident polarization angle α, between 0° and 180° relative to the X horizontal sample direction in 32 steps.

 

Fig. 2 (a) SHG scanning image of an individual nano-star for two different incident polarizations (upper image: 0° along one branch, lower image: 90°). Pixel size: 60 nm. The maximum intensity is 10⁵ counts/s (above) and 5.10⁴ counts/s (below). (b) Experimentally measured locations of the SHG spots reported in the object plane for a single nano-star, at different incident polarization angles indicated by different colors. The colored lines correspond to the measured positions relative to their global barycenter (four measurements are performed per incident polarization angle). The crosses indicate the positions expected from FDTD calculations.

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Figure 2(a) depicts SHG intensity images of an isolated nano-star (with one branch along the X direction) at α = 0° and 90°. The observed deformation of the SHG spot into a more elliptical shape at 90° excitation resembles Fig. 1(d), which confirms the sensitivity of SHG imaging to spatial near-field scale features in nanoparticles. To measure furthermore the spatial displacement of SHG spots experimentally, we implemented a polarimetric imaging measurement on a single nano-star initially centered on the optical axis for the incident polarization angle α = 0°. In this experiment, the SHG images are formed using a piezo-electric sample stage scanning of the particle, which is more precise than using galvanometric scanner imaging. Images of a single nano-star are recorded for a polarization angle varying between 0° to 180°, every 30° step, four to six times in a row. The integration time per pixel is set to 10 ms in order to increase the total number of photons recorded and therefore the accuracy of spot center determination. The center position of each SHG spot is retrieved by a Gaussian fit of a spot image profiles along its main symmetry axes. SHG image spots of round shape exhibit typically 170 nm standard deviation (340 nm FWHM), with a total number of photons of ranging between 500 to 1500 depending on the nano-star and on the incident polarization. The accuracy of the center positioning is estimated to be 10 nm, as derived from the theoretical calculation in super-resolution imaging [35], accounting for the measured number of photons, the background noise (1 photon) and the pixel size (60 nm) used in the experiment. Thanks to the multiple data set taken for each incident polarization, the mechanical drift of the particle in the sample plane could be identified by following the trajectory of the sliding barycenter point taken for each 0–180° angular range of the incident polarization. Typically the nanoparticle is seen to mechanically drift by about 60 nm between two polarimetric measurements, along both X and Y directions. All effective spot displacement due to the polarization rotation can then be directly measured by plotting the distance between the measured position and the corresponding local barycenter (Fig. 2(a)). The SHG spot center is seen to exhibit a displacement correlated with the incident polarization angle (Fig. 2(b)), with distances up to about 50 nm from the barycenter of all the recorded spot locations, which we assimilate to the nano-star center. This displacement map, although smaller in magnitude than the position shift expected from FDTD (crosses in Fig. 2(b)), follows a threefold symmetry feature and is of measurable magnitude even though the particle is smaller than the diffraction limit.

4. Nano-star shape deviations probed by polarization resolved SHG responses

In addition to influencing the spatial distribution of local fields, the incident polarization also strongly governs their vectorial nature, which symmetry properties can be probed by polarization resolved SHG responses [13, 34]. Examples of polarization responses I_X (α) and I_Y (α) averaged over 10×10 pixels regions around the SHG image spot center are depicted in Figs. 3(a,b). While SHG polarization responses show excitation and emission characteristics, the corresponding TPL responses exhibit excitation specificity but complete emission depolarization with identical signals along the X and Y analysis directions.

 

Fig. 3 (a,b) Experimental SHG and TPL polarization responses $I_X^{SHG,TPL}\left(\alpha \right)$ (green markers) and $I_Y^{SHG,TPL}\left(\alpha \right)$ (red markers) from two different individual nano-stars, with their corresponding polarization resolved SHG fits (continuous black lines). The corresponding SEM images are shown on the right. (c) Phenomenological model used for polarization responses fitting, represented by three independent nonlinear induced dipoles. (d) The characteristics of the nonlinear induced dipoles deduced from the polarization responses fits are represented schematically with their orientation and amplitude, together with the information of their distance H to the nano-star center. Fitting parameters values for star (a)/(b) (see text) ${\chi }_1^{\left(2\right)}=62.2/36.2$ (a.u.); ${\chi }_2^{\left(2\right)}=67.2/34.3$ (a.u.); ${\chi }_3^{\left(2\right)}=78.6/42.5$ (a.u.); ν₁ = 130° / 109°; ν₂ = 120° / 112°, φ = −6° / −10°, H = 65nm / 122nm.

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To relate the measured SHG polarization responses of nano-stars and to their local fields’ vectorial information, a simple model of the nonlinear coupling occuring in this structure is drawn by placing three nonlinear induced dipoles (which orientation forms ideally a three fold symmetry), at a given distance H from the center of the nano-star (Fig. 3(c)). Although phenomenological, this model is able to report, using only a few parameters (see below), both spatial and symmetry specificities of each individual nano-star, without the need to infer complex dipoles contour calculations such as in Fig. 1(c). For each incident polarization angle α in the sample plane, the nano-stars’ SHG polarization responses can be deduced by calculating the induced dipoles’ radiation through the microscope used in this study, and the total SHG intensity integrated over the nanoparticle image. This model is used to analyze the SHG polarization responses from single nano-stars such as in Fig. 3(a,b), by fitting simultaneously the polarization responses I_X (α) and I_Y (α) with different parameters left free (defined in Fig. 3(c)): the distance of the dipoles to the nanoparticle center (H), their relative amplitudes ( ${\chi }_1^{\left(2\right)}$, ${\chi }_2^{\left(2\right)}$ and ${\chi }_3^{\left(2\right)}$), their relative orientations (angles ν₁ and ν₂), and finally the global orientation of the structure (φ). The fact that two independent polarization responses are measured simultaneously brings stringent constraints and allows these parameters to be determined without ambiguity. The polarization responses show deviations from particle to particle, however for all of them the phenomenological model is seen to be in good agreement with the experimental data. A majority of the nano-stars exhibit four-lobes shape polarization responses for both I_X (α) and I_Y (α) (Fig. 3(a)), and therefore almost polarization-independent total SHG responses, which can be expected from a pure three fold symmetry system in the dipolar approximation [36]. For other particles showing more pronounced two-lobe shapes such as in Fig. 3(b), the derived orientations of the nonlinear induced dipoles (ν₁ and ν₂) do not follow a pure three fold symmetry. This is also reflected in the geometrical shape of the particle SEM image, which exhibits an arm of slightly different length/thickness than the two other ones (note that the particles shapes measured by SEM are identical before and after SHG microscopy measurements). Finally, the spatial distance between induced nonlinear dipoles is found to range between H = 60 nm and 125 nm depending on the particle. Although H is related to a rather simplified representation of the nonlinear vectorial map represented in Fig. 1(c), its experimental values fall in the same range as obtained by FDTD calculations and experimental displacement values reported above. This model, although it does not include the whole complexity of local fields enhancements, reveals finally the essential parameters of the phenomenon: nonlinear induced vectorial properties reflecting the symmetry of the nanoparticle, and spatial phase delay effects reflecting its size. It further presents the advantage of being extendable to various structures geometries made of multiple branches.

5. Conclusion

In conclusion, we have shown that far field SHG polarization resolved microscopy imaging of metal nanoparticles can reveal quantitative vectorial and spatial information occurring at the near-field scale. This technique, combining far field microscopy imaging and polarization resolved analyzes, is a unique tool to probe the strong interplay between local field’s enhancements and excitation polarization in nanoparticles of complex shapes.