1. Introduction

The visualization of surface plasmon waves[1] of noble metal nanostructures is important for developing and optimizing chemical and biochemical sensors that are based on local field enhancements[2] and plasmon resonance shifts[3] associated with these nanostructures. While propagating plasmon waves are rather routinely imaged by fiber-based near-field scanning optical microscopy (NSOM),[4] high-resolution mapping of plasmon waves localized on small (<200 nm) metallic nanostructures is not currently feasible with fiber-NSOM because of the limited spatial resolution (~100 nm) inherent to the fiber probes.

Recently, Hillenbrand, et al., [5, 6] imaged a longitudinal vector component (parallel to the tip axis) of the localized plasmonic field around gold nanospheres and nanodiscs, using a heterodyne-amplified, scattering-type apertureless NSOM technique (hereafter called ANSOM for short). The technique was recently proven to be useful for imaging propagating plasmons and phonons[7, 8] as well. In this technique, a metallic or dielectic tip scatters the near-field surrounding a nanostructure and the Z-component of the scattered field is recorded by a detector in the far-field region. Not only does the technique provide a high resolution (~10 nm) map of the field intensity, but it also accesses a map of the local field phase.

Here, we report that polarization-selective detection of ANSOM scattering can be used to image not only the longitudinal (Z), but also the transverse (perpendicular to the tip axis, Y) components of the near-field around a single gold nanoparticle. The observation and interpretation are further confirmed by a model calculation based on self-consistent field Green’s dyadic propagator method. [9]

2. Experimental method

The details of the ANSOM instrument (Fig. 1) and its operating principles are already described elsewhere.[10, 11] The Pt-Ti coated silicon atomic force microscopy (AFM) tip is dithered near the resonant frequency of the cantilever (Ω ~300 kHz) with a full amplitude of 20–100 nm (the amplitudes of oscillation used are different for different tip) above the sample surface. The ANSOM images are obtained under the constant gap condition (constant vibration amplitude and constant average tip sample distance). Linearly polarized light from a laser (HeNe, 632.8 nm) is focused onto the tip-sample junction with an angle of 30° with respect to the sample surface via an objective lens. Back-scattered light from the tip-sample junction is collected by the same lens and homodyne-detected to give separate intensity and phase information of the scattered field. In the following, we exclusively use the third harmonic components (n=3) of the scattered signals unless mentioned otherwise. The phase represents the relative phase shift between the incident and the scattered radiation and contains the information of the sign of the near-field vector. The far-field background is rejected by the third harmonic (3Ω ~900 kHz) demodulation technique via a lock-in amplifier. The polarization direction of the excitation beam is fixed along the Z-direction in the tip-sample frame (see Fig. 1) throughout the experiment. A quarter-wave plate placed in the reference arm of the interferometer controls the linear polarization angle, θ, of the reference field, E _ref. With θ=0° and θ=90°, the sample frame Y- and Z- components of the scattered light are detected in the far-field, respectively. The optical and AFM topography images are simultaneously acquired by scanning the sample and recording the optical and topographic signals. The gold nanosphere samples are prepared by drop-casting a dilute solution of colloidal gold nanospheres (nominal diameters of 50–90 nm, British Biocell International) onto MPTMS (3-mercapto-propyl trimethoxy silane) treated silicon substrates.

The diffraction-limited illumination spot is much larger than the size of the tip apex. Therefore, the back-scattered field (E_bs) not only contains weak scattered field from the tip (E_scat) but also strong far-field reflection from the sample surface and the tip shaft (E_ff), and they add coherently (E_bs=E_scat+E_ff). Therefore, the direct (non-interferometric) lock-in detection of the back-scattered light intensity always yield self-homodyned (interference) signals,

where the Φ is the phase difference between E_{scat, n} and E_ff. The n denotes n’th harmonic component. See below). Unfortunately, the phase difference (Φ) between the E_scat,n and E_ff is ill-defined and is topography-dependent. Consequently, this causes serious ambiguity in separating out the tip scattered intensity (|E_scat,n|²) information from the images. In this work, we use the interferometric homodyning [11] to remove this topography-dependent phase (Φ), and to record the intensity (|E_scat,n|²) and phase (δ) information independently. The magnitude of the reference field (E_ref) is typically 10 times larger than that of the back-scattered field (E_bs), such that |E_ref|≫|E_ff|≫|E_scat|. Under this condition, lock-in amplitude signal at n’th harmonic frequency of the tip oscillation, S_n, is given by:

,where the δ_n is the phase of the tip scattered field and the Γ is the (controllable) phase term caused by the path difference between the two arms of the interferometer. In other words, use of strong reference field overrides the ill-defined self-homodyning between the sample reflection and the tip-scattering. For the acquisition of the ANSOM images, the same line in image is scanned twice with two reference-beam phases differing by π/2 (Γ₁ and Γ₂=Γ₁+π/2) and stored separately in the computer. The two images with different reference phases are later processed offline to extract the ANSOM intensity (I_n) and phase (δ_n) according to the following relation:

 

Fig. 1. Schematic diagram of ANSOM instrument and coordinate system used throughout this work. The unprimed coordinate (sample frame) and primed coordinate (polarization frame) are rotated by 30°. PZT=piezo transducer, M=mirror, BS=50/50 beam splitter, PD=photodiode, QWP=quarter-wave plate, E_ref=reference field and its polarization, E_scat=back-scattered field.

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3. Harmonic demodulation technique

The use of harmonic demodulation technique is essential for filtering out the far-field background in ANSOM experiment. At the same time, the harmonic demodulation places additional complications in interpreting the ANSOM images of plasmonic local field [5,6]. Here we briefly outline how the demodulated ANSOM signals and the local field distributions are related, following the arguments of Hillenbrand, et al., [5,6]. In general, the n’th harmonic demodulated signal (intensity and phase) corresponds to the spatial Fourier component of the local field, E_loc, along the tip vibration direction (z), according to the following equation.

where the I_n,j and δ_n,j are the intensity and phase of the n’th harmonic component (E_n,j) of the j’th (j=x, y, and z) vector component of the local field, E_loc,j, respectively. The z is the (vertical) tip-sample distance, and the A(z) is the amplitude of tip oscillation for a given z. The x and the y are the lateral position of the tip, and the ϕ is the instantaneous phase of tip oscillation. The E_n,j is approximately the z-coordinate partial derivative of the local field (this correspondence becomes rigorously correctly when the A(z) is negligibly smaller than the radius of curvature of the tip-end). In our experiment, the ANSOM image is obtained the constant tip-sample distance condition (constant z and constant A(z)). Therefore, the n’th harmonic signal is approximated as

As can be seen in Eqs. (1) and (2), the E_n,j is not exactly the same as E_loc,j. Instead, we need a complete set of all harmonic components {E_n,j} in order to fully reconstruct an accurate three-dimensional map of the local field. However, lower harmonic components of ANSOM signals (n=1 or 2) are usually overwhelmed by the far-field background, which makes the rigorous local field reconstruction not possible. Therefore, we further assume that the z-coordinate of the field is locally separable near the surface (z=z₀):

,where the Q_j(x,y) is the lateral local field profiles sampled at constant tip-surface distance (z=z ₀), and the Z_j(z) represents the vertical decay of the local field. The AFM operates under constant gap mode (CGM) condition. The feedback loop maintains the constant oscillation amplitude A(z) and the constant tip-sample distance, z₀ while scanning. Therefore, the ${\partial }^n{Z}_j\left(x,y,z\right)\partial z^n{\mid }_{{z=z}_0}$ term is constant and we can relate the harmonic component, E_n, to the lateral field profile, Q_j(x,y) of the local field:

While the general validity of the assumptions is still an open question, Hillenbrand, et al., [5] recently demonstrated that the demodulated ANSOM images show close similarity to the theoretical field distribution, which confirms that the approximations are qualitatively valid for plasmonic nanoparticles such as nanospheres and nanodiscs. Throughout this work, we qualitatively interpret our demodulated images as the local field distribution.

4. Results and discussion

 

Fig. 2. ANSOM intensity (I ₃) approach curves, collected at the 3^rd harmonic of the tip oscillation frequency with the tip placed above a gold nanosphere. (a) Detector polarization at θ=90° (Z), and (b) detector polarization θ=0° (Y).

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Figure 2 displays ANSOM approach curves (3^rd harmonic demodulation signal intensity, I ₃=|E ₃|², as a function of tip-particle distance) above the center of a gold nanosphere, obtained with Z- and Y-detection polarizations. The approach curve is obtained with the feedback loop temporarily disabled. The zero position (z=0) is defined as the point where the AFM tapping amplitude begins to decrease abruptly by the physical contact of the tip and the particle surface. Both of the curves show a fast decrease of signals with z distance (~ 10 nm scale) without any oscillatory component. The decrease and fluctuation in ANSOM signals observed at z<0 regions are caused by the abrupt reduction in tapping amplitude, A(z) upon physical contact of the tip and the particle (The demodulated signal intensity is reduced with reduced oscillation amplitude. See Eq. (4). The absence of the oscillatory component in z>0 region ensures that the demodulation process successfully isolates the near-field component and rejects the far-field background from the detected signal.

 

Fig. 3. The normalized ANSOM intensity (a, c, and e), ANSOM phase (b, d, f), and topography (g) images of 50 nm a gold nanosphere on Si substrate, obtained with detection angles θ=90° (a, b), θ=0° (c, d), and θ=45° (e, f). See also Fig. 1 for polarization coordinates. The bars on the images represent the same 80 nm length scales. Also shown is the sample coordinate system.

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Figure 3 displays topography (g), ANSOM intensity (a,c, and e), and ANSOM phase (b, d, and f) images of a single gold nanoparticle (~50 nm diameter) obtained with parallel (Z, θ=90°), perpendicular (Y, θ=0°), and intermediate (YZ, θ=+45°) detection polarization geometries while exciting the particle with parallel (Z, 30° from the tip-axis) polarization. The intensity ratios for each detector polarization configuration are typically Z: YZ: Y=1: 0.23: 0.14 (the color scales for the YZ and Y-intensity images are enhanced by factors of 4.3 and 7.1, respectively, in order to show the details of the images).

The Z-polarization images show a unimodal intensity distribution [Fig. 3(a)] and a prominent linear phase gradient of 1.3°/nm along the illumination direction (Fig. 3(b)]. The Y-polarization images, on the other hand, show a c-shaped intensity distribution [Fig. 3(c)], with two intensity maxima that are 180° out of phase with each other (Fig. 3(d)]. Also present is the linear phase gradient of ~0.9°/nm along the x-direction that is similar to the one observed in Z-polarization image. In all of the nanosphere images we have examined with Z- and Y- detector polarizations, we consistently observe that the +X sides of the particles always appear brighter than the -X side. The YZ-polarization (θ=+45°) image [Figs 3(e) and 3(f)] shows a notable intensity asymmetry along the Y-direction (+Y side is brighter than the -Y side) that are not present in either Y- or Z- polarization images. With θ=-45° detector polarization, we obtained an image (not shown) that is similar but with reversed sense of asymmetry (-Y side is brighter than the +Y side). All of the features of the ANSOM images mentioned in the above are observed in every nanosphere (more than 10) we have examined with different tips used on different days. Therefore, the features in the intensity and phase images are not likely caused by the specific shapes of the tips used or a fine-structure of a specific nanoparicle.

We interpret that the ANSOM images obtained with Y- and YZ-detector polarization directions are the maps of the Y- and YZ- field vector components around the metallic nanostructures, in analogy with the interpretation of Z-polarization images[5, 6].

 

Fig. 4. Simulated electric field (a, d, and g), ANSOM intensity (b, e, and h) and ANSOM phase (c, f, i) images of a gold nanosphere (40 nm radius) on Si substrate, obtained with detection polarization angles Z (θ=90°; a, b, and c), Y(θ=0°; d, e, and f), and YZ (θ=45°; g, h, and i). See also Fig. 1 for polarization coordinates. The simulated area corresponds to 200 nm×200 nm. The simulated intensities (a, c, and e) are normalized to show the details of the images. The color scale for (f) is -100° to 100°. The dashed lines in (a), (d), and (e) indicate the center position of the gold nanosphere. (j) Expected field distributions as seen in XZ- and YZ- planes.

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More specifically, we assert that the features mentioned in the previous paragraph are a direct measure of the shape and symmetry of the dipolar near-field patterns that are generated by the optically driven gold nanospheres. Before we proceed to analyze the features of ANSOM images in detail, it is worthwhile to mention two general concerns regarding the role of the tip in ANSOM field imaging. Firstly, one may argue that the tip is sensitive only to the Z-component of the field because of the lightning-rod effect [5, 6, 12, 13] and that the transverse (Y) component may not be detectable. Secondly, use of a metallic tip may significantly perturb the near-field distribution of the nanoparticles to be studied via the strong tip-sample coupling. We show in the theoretical model below that the tip also has an appreciable sensitivity toward the transverse-component (Y), and that the Pt-Ti coated tip does not alter the near-field distribution significantly.

A rigorous simulation of the ANSOM images requires electrodynamics simulation of the local field around the particle and the conically-shaped metal tips using numerical procedures such as finite-difference time-domain (FDTD) method. In addition, complete tip-sample distance information is needed in order to accurately simulate the lock-in demodulation process, which is beyond the scope of the current work. Instead, we carried out a simple simulation of the ANSOM images and local fields using the self-consistent Green’s dyadic propagator method [9]. The main purpose of the simulation is to determine if the experimental images correctly capture the essential features (symmetries and phases) of the dipolar field distribution excepted from a sphere. In the simulation, a monochromatic plane wave (632.8 nm) drives the coupled oscillation of the tip dipole (Pt sphere with radius 20 nm) and the nanosphere dipole (Au sphere with radius 40 nm), and the Y-, Z-, and YZ polarization components of the scattered field are evaluated in the far-field region as a function of the position of the tip. Both the tip and the nanosphere take on isotropic polarizability tensors, as derived from the quasi-static approximation of the polarizability of a sphere. We explicitly take into account the mutual perturbation (coupling) between the tip and the particle by the Green’s dyadic propagator. The presence of Si-substrate is also taken into account via the indirect Green propagator method [9]. However, we find that the presence of the substrate causes only a minor change in local field distribution and the simulated ANSOM images. The lock-in demodulation process is modeled by a simple difference between the two scattering signals calculated at z = 5 nm (tip close to the surface) and at z = 100 nm (tip away from the surface) positions. A rigorous simulation of the ANSOM images requires complete information on the tip-sample distance dependence of the scattering and subsequent Fourier transformation according to the Eq. (4). However, we find that a simple two-point difference of the simulated field is sufficient to qualitatively simulate the images.

Figure 4 compares the unperturbed local field intensities (|E_Z|², |E_Y|², and |E_YZ|², hereafter called EM-field for short) and the corresponding ANSOM simulation signals (intensity and phase) around a gold nanosphere calculated at each tip position scanned under the constant-gap condition. Overall, qualitative agreement among the EM-field distributions, simulated ANSOM, and experimental ANSOM images establishes that the interpretation of the ANSOM image is valid. From the comparison of the theory with the observed ANSOM distribution, we can immediately identify that the ANSOM images of a gold nanosphere correspond to a simple field pattern expected from a point dipole aligned along the Z-direction: E_Z-component is maximal near the center of the particle and maintains the same sign above the nanosphere, whereas the E_Y-component shows an intensity node at the center of the particle and assumes opposite signs (180° phase shift) on either side of the node [see Fig. 4(j)].

We find that the asymmetric side-illumination geometry renders subtle influence in the ANSOM images as well. Firstly, the tilt in excitation polarization (30° from surface normal) leads to the tilt in particle dipole vector. This dipole orientation is reflected in the asymmetry in E_z-field [Fig. 4(a) and 4(j)] and the corresponding ANSOM images [Fig. 4(b) and Fig. 3(a)]. Secondly, the asymmetric illumination causes the nanospheres to experience different phases of excitation for different nanosphere positions with respect to the tip position, which accounts for the nearly linear phase gradient [6, 14] along the x-direction in the images shown in Figs. 3(b), 3(e) (experiment: ~1°/nm) and Fig. 4(c) (simulation: ~0.6 °/nm). We are unable to account for the small, yet consistent, difference between the experimental and simulated phase gradient currently. We speculate that the deviation is caused by the multipole contribution of the tip and the particle that are not included in the current simulation.

Our simulation does not take into account the extended cone structure around the tip-end and we are currently unable to evaluate the role of the extended structure on ANSOM contrast. We speculate that the extended structure of the tip can also interact with the nanoparticle weakly and cause slight reduction in spatial resolution. Indeed, Stebounova, et al., [15] found that the long range tail of the ANSOM approach curves could only be explained by including the extended structure of the tip, which suggest measurable, although minor, effect of the extended structure.

The qualitative agreement of the YZ polarization images reveals a particularly interesting insight into the polarizability tensor of the metal-coated tips employed. A strongly anisotropic tip would generate ANSOM images that are strongly biased toward a particular vector component (e.g., Z-direction) of the near-field. Our YZ-ANSOM image, however, bares qualitative similarity to the corresponding (unbiased) EM-field distribution and the simulated ANSOM image based on isotropic tip polarizability. The qualitative agreement among experimental images and EM-field distribution does not necessarily implicate the isotropic polarizability of the tip. However, it does indicate that the tip employed has comparable, if not isotropic, polarizability tensor components along the Y and Z directions at the visible frequencies. Lastly, it must be noted that our conclusion may not be applicable to ANSOM experiments in infrared or terahertz wavelength regions, or experiments with etched metal wire or carbon-nanotube probes. Increased electrical conductivity in infrared frequency regions and narrow cone-angle of the metal wire or nanotube probes may have strongly anisotropic polarizability along the tip axis and therefore can lead to qualitatively different ANSOM images.

3. Conclusion and summary

We showed that the polarization-selective ANSOM technique can image the transverse as well as the longitudinal vector components of the plasmonic near-field around a single metallic nanostructure in a phase-sensitive manner. The information on local field vector directions as well as the local phase is becoming particularly important for research in photonic crystals and nano-plasmons, yet no experimental technique is available for obtaining this information to date. With the field intensity and the phase maps of each coordinate being available, ANSOM can be used to reconstruct a vector field around a nanostructure provided that (1) the tip polarizability tensor is completely known, (2) the image intensities for each polarization are properly calibrated, and (3) the relative phase relationships between the images are known.

Currently, comparison of the approach curves such as the ones shown in Fig. 2 and their fit to the dipole – dipole interaction model typically yield effective radius of the tip to be 20 − 100 nm, which allows us to estimate the magnitude of polarizability of 10⁴ − 10⁵ nm³. However, the polarizability tensor components of the tip are not fully characterized and our intensity images for different polarization components are not accurately calibrated with each other. We believe that ellipsometric methods such as the ones by Lee, et al., [16] should allow better characterization of the tip polarizability tensors. As mentioned in the above, the harmonic demodulation not only eliminates the far-field background but also induces a few complications in quantitative interpretation of the images. In particular, relative image contrast is influenced by the tip oscillation amplitude, which often prevents one from extracting quantitative information from the demodulated images. In this regard, systematic studies on the relationship between the harmonic demodulation parameters and the detailed image contrast would also help establishing the ANSOM for a routine application.

Lastly, we expect that a set of images of X, Y, and Z polarization components will be achievable with a more advanced illumination/detection schemes, such as epi-illumination combined with radially polarized excitation light source [17]. Also, use of less perturbing (such as a bare silicon tip [18], or a carbon nanotube functionalized tip [5]) and more isotropically polarizable tip (such as gold nanosphere functionalized tip[10, 19, 20]) should minimize the tip-induced distortion and normalization uncertainty in the vector field reconstruction.