1. Introduction

Several optical [1–14] and electron [15–19] microscopy techniques have been developed to visualize the plasmonic local field around a nanostructure [20]. In particular, the scattering-type scanning near-field optical microscopy (s-SNOM) [1–10, 12–14] provides not only the intensities but also the phases of the field with ~10 nm resolution. One popular scheme of the s-SNOM is to excite the nanostructure and the tip with s-polarized (parallel to the sample plane and perpendicular to the tip axis) light and detect the p-polarized (approximately perpendicular to the sample plane and parallel to the tip axis) component of scattering (cross-polarized s-SNOM) [9]. The method relies on the fact that tips with conical or pyramidal shapes are dominantly polarizable along their axes (z) [9], and this anisotropic response of tip enables the mapping of the out-of-plane field (E_z) of plasmons with minimum tip-sample coupling. While this scheme has been proven useful, it has also provided the general impression that the E_z component may be the only component that can be recorded by the s-SNOM. A few recent studies have reported in-plane field (E_x) measurements using the s-SNOM technique; Lee et al. [11] and Olmon et al. [6] employed nanoparticle-terminated tips with near-isotropic polarizabilities, and they successfully measured the E_x components. While it is obvious that commercially available tips with conical or pyramidal shapes will have less sensitivities for the E_x components than the nanoparticle-terminated tips [6, 11], these conventional tips are likely to provide better spatial resolution with less effort of tip-fabrication, thereby allowing for more routine operations. Quite recently, Schnell et al. [2, 3] employed a pyramidal Si tip to image the interstitial field (which is mostly in-plane polarized) formed at the gap of infrared antennas, which suggests significant in-plane polarizability (α_xx) of the tip at infrared frequency. Their results are mostly restricted to the gap-fields at infrared frequencies (λ ~10 μm). Further investigations on the E_x-field mapping with a pyramidal tip are needed because the polarization-specific response of the tip may be drastically different for infrared and visible frequencies, and for different plasmonic structures (e. g., gap-structures vs isolated nanostructures). In addition, the polarization anisotropy (ratio of out-of-plane (α_zz) and in-plane (α_xx) polarizabilities) of the tip is largely unknown.

Here we carried out the s-SNOM measurements on a gold nanoprism to investigate the E_x-field sensitivities of a pyramid-shaped tip at visible frequency. We aim to answer the following two questions: (1) do the pyramidal tips possess sufficient E_x-sensitivities at visible frequencies? ; (2) If they do, what are their effective polarization anisotropies? The two questions are particularly relevant to the plasmonics of small metallic nanoparticles: small polyhedral nanoparticles (such as cubes, octahedrons, and prisms) support degenerate resonances at visible frequencies, and possess sharp vertexes that lead to strong local field enhancements. If the s-SNOM could provide complete vector field maps (intensities and phases of E_x, E_y, and E_z) of plasmon modes of a nanoparticle, it will allow one to establish the most direct connection between the shape of a nanoparticle and its plasmonic responses. We indeed found that the tip has sufficient α_xx for measuring the E_x-distribution at visible frequency. In addition, because of the non-zero value of α_xx, the cross-polarized s-SNOM images may contain contributions from the E_x-distribution of an out-of-plane plasmon mode, as well as the E_z-distribution of an in-plane mode, with the relative importance of the two being determined by the anisotropy of the tip. By comparing the experimental s-SNOM images with a scattering model that takes into account the anisotropy of the tip, we determined the polarization anisotropies (|α_zz / α_xx|) of Si-tips and metal-coated Si-tips.

2. Experimental method

The details of the s-SNOM instrument (Fig. 1 ) and its basic operating principles are already described elsewhere [7, 19]. Throughout our study, we use pyramid-shaped atomic force microscopy (AFM) tips made of silicon and PtIr-coated silicon (Nanosensors, PointPlus® Probe; the radii of curvature are ~10 nm and ~30 nm for Si- and PtIr-tips, respectively). The tip is dithered near the resonance frequency of the cantilever (~300 kHz) with the full cantilever oscillation amplitude of 20 – 50 nm above the sample surface. Linearly polarized light from a laser (HeNe laser, 633 nm) is focused onto the tip-sample junction with an angle of θ = 30° with respect to the sample surface via an objective lens (NA of 0.42). Back-scattered light from the tip-sample junction is collected by the same lens and homodyne-amplified by a Michelson interferometer. The polarization direction of the excitation beam is controlled by a half-wave plate (HWP). A quarter-wave plate (QWP) placed in the reference arm of the interferometer controls the linear polarization of the reference field. By rotating the quarter-wave plate, the s- and p- components of the scattered light are selectively amplified and detected in the far-field. The far-field background is rejected by the 3rd harmonic demodulation via a lock-in amplifier to give separate intensity (|s₃|²) and phase (ϕ₃) information of the scattered field. When measuring the weak s-SNOM signals for s-polarized excitation and s-polarized detection, we placed a polarizer in front of the detector to further reject the far-field background. The images are acquired by raster-scanning the sample and recording the optical (s-SNOM) and topographic (AFM) signals simultaneously. The sample comprises single crystalline Au nanoprisms (edge length and thickness of 120 – 140 nm and 60 – 80 nm, respectively) that are naturally present (~5% population) in commercial nanoparticle colloidal solutions (BBI international). The nanoprisms are dispersed on a flat Si-substrate.

 

Fig. 1 The schematic of s-SNOM setup. M = mirror, QWP = quarter-wave plate, HWP = half-wave plate, BS = 50 / 50 beam splitter, Pol = polarizer, PD = photodiode, L = objective lens. The directions of incident and scattered light are drawn as solid and dashed arrows, respectively. Also shown are the sample coordinates and polarization directions (s and p): the x-axis is parallel to the s-polarization direction, z-axis is perpendicular to the sample plane, y-axis is parallel to the sample-plane projection of the propagation vector of the incident light. The z’-axis is parallel to the electric field vector of a p-polarized excitation light, and is tilted by θ = 30° from the z-axis.

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3. Scattering model of s-SNOM

Here we employ the general scattering theory of SNOM by Sun et al. [21] to model the polarization dependent s-SNOM contrasts (see Fig. 2 ). In the original formulation by Sun et al., the tip and the sample are assumed to be strong and weak scatterers, respectively. In our model, however, the two are reversed because the sample is a plasmon-resonant nanoparticle (strong scatterer) whereas the tip is made of dielectric or off-resonant metallic structure (weak scatterer). This changes the definitions of the scattering operators (see Appendix), yet major formulation remains the same. In addition, we set the excitation and detection directions to be anti-parallel to each other (back scattering geometry), which allows us to apply the optical reciprocity theorem [22]. The tip is modeled as a uniaxial spheroid (α_xx = α_yy) with a diagonal dipole polarizability tensor, α. A plane wave E₀excites both the dipole of the tip and the plasmon modes of the sample. The lowest order scattering, E_scat, detected at far field can be expressed using the scattering operators of the sample (S) and the tip (T) [21]:

 

Fig. 2 Tip-sample interaction model. (a) E₀ = incident field, E_scat = scattered field, α = polarizability tensor of the tip. Also shown is the sample coordinate system. (b) The ST and TS scattering processes in the cross-polarized s-SNOM of a nanoparticle: the polarization states of the incident (black) and scattered (blue) light are shown as double-sided arrows (red). The two double-sided arrows that connect the panels indicate the reciprocal relationships of the possible scattering events.

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The 2^nd order scattering model is usually sufficient for describing the s-SNOM of isolated nanoparticles. For structures that generates very strong local fields, such as gap-plasmon structures, one may need to take into account higher order contributions such as the STS-process [14]. The tip-only or sample-only scattering contributions are filtered out by the 3^rd harmonic demodulation in s-SNOM, and hence, they are not included in Eq. (1). The first term, called the TS process, describes the following scattering sequence: laser (L) → sample (S) → tip (T) → detector (D). The term is proportional to the sample local field, E_loc(r_tip), at the position of the tip-end (r_tip), which is given as: TSE₀ = bαE_loc(r_tip),where the b denotes a proportionality constant set by the geometry of the experimental setup. The second term, the ST process, describes another scattering sequence: L → T → S → D. It describes the local point-dipole (i.e., the tip-dipole) excitation of the sample plasmon modes, and the far-field detection of the radiation from the excited sample. In general, the TS and ST processes do not obey the reciprocity theorem, because the reciprocity holds only for a primary source and a primary receiver [22]; note that the sample and the tip are not primary sources or receivers [23]. However, in our particular experimental setup, we use the same objective lens to excite and collect the back-scattering, thereby rendering the positions of the laser (a primary source) and the detector (a primary receiver) to be nearly indistinguishable from the viewpoint of the tip-sample system. Therefore, the ST and TS processes are essentially reciprocal pairs in our experiment. With the excitation light polarized along ${\widehat{e}}_{\text{i}}$ and the detection polarizer oriented along ${\widehat{e}}_{\text{j}}$, the polarization-selected s-SNOM amplitude, A_i,j, is:

In particular, Eq. (5) shows that the polarization anisotropy of the tip (α_xx : α_zz) influences what is recorded in the cross-polarized s-SNOM images. With an anisotropic tip that is dominantly polarizable along the z-direction (|α_zz| >> | α_xx|), the A_s,p and A_p,s components are essentially the ${\text{E}}_{\text{z}}^{(\text{x})}$component, unless the magnitude of ${\text{E}}_{\text{x}}^{(\text{z})}$ is much larger than that of the ${\text{E}}_{\text{z}}^{(\text{x})}$component. With an isotropic tip (|α_zz| = |α_xx|), on the other hand, the A_s,p and A_p,s amplitudes are an equal mixture of the two field components. No such mixing occurs for A_s,s and A_p,p irrespective of the anisotropy of the tip. Thus, unless we specifically fabricate a tip that is mainly polarizable along the in-plane direction, we cannot measure the pure${\text{E}}_{\text{x}}^{(\text{z})}$component with any excitation and detection polarization combinations. When both the tip and the sample are isotropically polarizable nanospheres, the model is essentially the same as the one described by Deutsch et al. [10] For infrared or terahertz nano-antennas with micrometric lateral lengths and nanometric thicknesses [2, 6], only the in-plane modes are active at the excitation wavelength and therefore, the tip-dependent mixing of field components does not occur.

In the current experiment, the excitation and detection directions are tilted by an angle of θ = 30° from the sample surface (see Fig. 1). The scattering amplitudes under this geometry are given by:

4. Experimental results

Figure 3 shows AFM topography, s-SNOM images, the simulated far-field scattering spectra, scanning electron microscopy (SEM) images, and simulated field distributions of a gold nanoprism (edge length of 120 nm and height of 65 nm, see also inset of Fig. 3(b)). From the spectra (Fig. 3(b), calculated with the finite-difference time domain, FDTD, method), we expect that both the in-plane and out-of-plane dipole modes can be resonantly excited with light with a wavelength of 633 nm. Figures 3(d)–3(k) show the intensity (|s₃|²) and phase (ϕ₃) s-SNOM images obtained using s and p excitation and detection polarizations (hereafter denoted as excitation / detection). The phase images shown in Figs. 3(h)-3(k) are corrected for the linear phase gradient [24, 25] of 0.49° / nm along the y-axis, which arises from the oblique-illumination geometry of the experiment and is independent of the excitation and detection polarizations.

 

Fig. 3 s-SNOM images, simulated field distributions, spectra, and structure of a gold nanoprism. (a) AFM topography. The vertices v₁, v₂, and v₃ point to the positions of top vertices of the nanoprism. (b) Simulated far-field scattering spectra of a nanoprism and the electron-microscopy images of a representative nanoprism (inset). The spectra show both the in-plane (thin black trace) and out-of-plane (thick black line) dipolar resonances. The red arrow points to the excitation wavelength (633 nm) employed in s-SNOM measurements. (c) Excitation and detection geometry and the coordinates used (see also the bottom right of (k) for another coordinate reference). (d)–(k) experimental s-SNOM intensity (|s₃|²) and phase (ϕ₃) maps. The phase images are corrected for the illumination phase gradients (see the main text). To the left of the images, the excitation/detection polarizations (s or p) for each image are shown. (l)–(q) the simulated local field intensity (|E|²) and phase (ϕ) maps sampled 10 nm above the top surface of the nanoprism. To the right of the images, the names of field components are shown. In both the experimental and simulated images, the dashed triangles show the approximate shape of the nanoprism and the positions of vertices. (r) The line-profiles of (f) and (j) (marked as dashed lines). (s) The line profiles of (n) and (q). The abrupt phase-jump (red region) in (f) that appears on the left vertex of the nanoprism arise from the jitter of the AFM feedback loop that occurred during the scan.

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All four intensity images show hotspots localized near the vertices of the nanoprism (marked as v₁, v₂, and v₃ in AFM topography image, Fig. 3(a)), which agrees with the general expectation that sharp vertices of polyhedral nanoparticles strongly localize the electric field. Typically, the s/s-intensity image is ~100 times weaker than that of the p/p-image. Based on Eq. (6) and (7), we roughly estimate the polarization anisotropy of the tip to be |α_zz / α_xx| ~10. The phase images show near in-phase oscillations for the p/p and s/s-polarizations, and out-of-phase oscillations for the s/p and p/s-polarizations, respectively. Particularly for the s/s image, the hotspots in the intensity map (Fig. 3(f)) are located in positions similar to those in the s/p image (Fig. 3(e)); however, the phase distributions are distinctly different for the two images. The close similarity between the p/s and s/p images shows the reciprocity of the excitation and detection as described above, and both images carry the same information.

Besides the overall agreement between the experiment and the simulation, it is also noteworthy that the s/s intensity deviates from the simulation near the center of the nanoprism (compare Figs. 3(f), 3(n), 3(r), and 3(s)). We expect the E_x intensity to be nearly zero above the center of the nanoprism, whereas the experimental image shows appreciable intensities at this position.

This deviation is likely to originate from the dipole-image dipole coupling between the tip and locally flat surface of the nanoprism; this coupling has not been taken into account in the analysis. This dipole-image dipole coupling does not depolarize the scattering, and hence, it only contributes to the s/s and p/p images. The numerically simulated E_z and E_x distributions for p- and s-polarized excitations (Figs. 3(l)–3(q)) agree well with the respective experimental images, thereby confirming the interpretation that the p/p, s/p (also p/s), and s/s images represent the E_z component of an out-of-plane mode ($E_z^{(z\text{'})}\cong E_z^{(z)}$), the E_z component of an in-plane mode ($E_z^{(x)}$), and the E_x component of an in-plane mode ($E_x^{(x)}$), respectively. This agreement also shows that the α_xx is non-zero, yet its magnitude is much smaller than that of α_zz.

To further investigate the influence of polarization anisotropy of the tip on s-SNOM images, we compare the experimental s/p-image (Fig. 4(a) ) and the $A_{s,p}$-images (Figs. 4(e)-4(i)) evaluated for tip anisotropies (r = |α_zz / α_xx|) of 50, 10, 5, 1, and 0.1. In general, the$E_z^{(x)}$,$E_x^{(z\text{'})}$, and$E_y^{(x)}$ field components (Figs. 4(b)-4(d)) contributes to the $A_{s,p}$, and their relative contributions change with illumination angle (θ) and the polarization anisotropy. (see Eq. (8)). For the particular case of $A_{s,p}$ with θ = 30° and with isotropic tip, the relative importance of the three components is 1: 4: 0.3 (amplitudes) for $E_z^{(x)}$,$E_x^{(z\text{'})}$, and $E_y^{(x)}$, respectively. Thus, the two dominating components in $A_{s,p}$are $E_z^{(x)}$and$E_x^{(z\text{'})}$. With r > 10, the $|A_{s,p}{|}^2$-images are essentially the $|E_z^{(x)}{|}^2$-distributions. As the anisotropy is reduced, the image gradually changes from$|E_z^{(x)}{|}^2$to $|E_x^{(z\text{'})}{|}^2$ (the most distinguishing feature of the two field components is the relative intensities of hotspots near the v₂ and v₃; see Figs. 4(a)-4(c)). In fact, we notice that $|A_{s,p}{|}^2$ is almost the same as the $|E_x^{(z\text{'})}{|}^2$with r $\le$ 5. This indicates that cross-polarized s-SNOM mapping with nanosphere-functionalized tip (r ~1) would have resulted in the images of $|E_x^{(z\text{'})}{|}^2$instead of$|E_z^{(x)}{|}^2$. The $|A_{s,p}{|}^2$simulations with r $\ge$10 agree well with the experiment, which sets an upper bound of 0.1|α_zz| for the |α_xx| of the particular Si-tip employed.

 

Fig. 4 Possible influence of polarization anisotropy on s-SNOM images. (a) Experimental intensity (|s₃|²) image of a nanoprism obtained with the s/p-polarization. (b), (c), and (d) Intensity distributions of FDTD-simulated field components, $E_z^{(x)}$, $E_x^{(z\text{'})}$, and$E_y^{(x)}$, respectively. (e)-(i) Simulated cross-polarized s-SNOM images (|A_s,p|²) calculated with Eq. (8) and the field components shown in (b)-(d), evaluated for tip anisotropies (r = | α_zz / α_xx |) of 50, 10, 5, 1, and 0.1.

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Figure 5 shows another set of s-SNOM images of a nanoprism obtained with a PtIr-coated Si-tip. The PtIr-tip has the same pyramidal shape as the Si-tip, except for the increased radius of curvature of the tip-end. We were able to obtain s/s-images of a nanoprism (Figs. 5(d) and 5(g)), demonstrating that metal-coated tips also possess significant in-plane polarizability. Although the qualitative features of experimental p/p, s/p and s/s-images (Figs. 5(b)-5(g)) agree with the simulated$E_z^{(z\text{'})}$,$E_z^{(x)}$ and $E_x^{(x)}$field components (Figs. 5(i)-5(k) and 5(m)-5(o)), the detailed agreement is less satisfactory than in the case of Si-tip. As is well known, part of the deviation may be attributed to the higher-order scattering (3rd and higher) in s-SNOM signals caused by the strongly scattering metallic tip [26].

 

Fig. 5 s-SNOM images obtained with a PtIr-coated Si-tip with similar pyramidal shape. (a) AFM topography. (b)–(g) experimental s-SNOM intensity (|s₃|²) and phase (ϕ₃) maps. To the left of the images, the excitation/detection polarizations (s or p) for each image are shown. (h)-(o) the simulated local field components (intensity, |E|² ; phase, ϕ) sampled 10 nm above the top surface of the nanoprism. To the right of the images, the names of field components are shown. (p)-(s) Simulated cross-polarized s-SNOM images (|A_s,p|²) calculated with Eq. (8) and the field components shown in (h), (l), (j), and (n), evaluated for tip anisotropies (r = |α_zz / α_xx|) of 100, 3.3, 2, and 0.5. The v₁ in (a), and arrows in (b), (c), (q), and (r) points to the vertex position of the nanoprism mentioned in the text. Dashed triangles in the images show the shape of the top-surface of the nanoprism.

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One particularly notable deviation between the experiment and the simulation is the location of a hotspot near the top vertex of the nanoprism (v₁, Fig. 5(a)): while the hotspot in Fig. 5(j) is precisely positioned at the vertex, the one in the s/p-image is shifted to the right side of the vertex (Fig. 5(c), marked as an arrow). Such shift does not occur in the p/p-image (Fig. 5(b), marked as an arrow) obtained with the same tip, indicating that the shift is associated with specific s/p-polarization combination, rather than with the left-right asymmetry of the tip shape. Figures 5(p)-5(s) show corresponding |A_s,p|² maps similar to the ones in Figs. 4(e)-4(i). We note that the shift in hotspot position can be fully reproduced by the |A_s,p|²-map with tip anisotropy of r = 2 ~3 (arrows in Figs. 5(q) and 5(r)). In other words, significant in-plane polarizability of the metallic tip causes the mixing of $E_z^{(x)}$ and $E_x^{(z\text{'})}$components in the scattered field, and thus leads to the observed changes in s/p-image.

The above analysis indicates that the Si-tip is more anisotropic than the metal-coated Si-tip with similar shapes. Currently, it is not clear whether or not the difference is associated with the presence and absence of metallic coating on the tip, or with the simple difference in the radii of curvatures of the tip-end (R_c = 7 nm and 25 nm for Si- and PtIr / Si-tips, respectively). Nevertheless, the above results do show the importance of tip-dependent mixing of two or more plasmon modes in the cross-polarized s-SNOM images.

5. Conclusion

To conclude, we found that pyramid-shaped tips (both dielectric and metal-coated dielectric) have sufficient in-plane polarizability for measuring the E_x-distributions with s-SNOM at visible frequency. Because of the non-zero in-plane polarizability, the cross-polarized s-SNOM images may contain contributions from the E_x-distribution of an out-of-plane plasmon mode, as well as from the E_z-distribution of an in-plane mode, with the relative importance of the two being determined by the polarization anisotropy of the tip. By comparing the cross-polarized s-SNOM images with a scattering model, we estimated the anisotropy of the tip (| α_zz / α_xx |) to be $\ge$10 for the Si-tip, and 2 ~3 for the PtIr-coated Si-tip. The results suggest an important requirement the tip for the reliable vector-field mapping of degenerate, or quasi-degenerate plasmon modes of a nanoparticle: the α_xx should be large enough to detect the in-plane field components, yet it should be much smaller than α_zz to avoid the mixing of different vector field components in cross-polarized s-SNOM signals.

6. Note

During the process of manuscript preparation, the authors were informed that similar tip anisotropy effects in s-SNOM is considered by the research groups of Drs. Rainer Hillenbrand (CIC NanoGUNE, San Sebastian, Spain) and Javier Aizpurua (DIPC San Sebastian, Spain) in order to explain the s-SNOM images of coupled infrared nano-antennas [14].

7. Appendix: Definitions of scattering operators of sample (S) and tip (T)

More in-depth formulation of the theory is described in Sun et al. [21], and only the key aspects that are relevant to s-SNOM are shown here. For an incident field E₀, the scattering (E_scat) from the tip-nanoparticle system can be expressed as a series of multiple scattering events between the tip and the sample, as follows:

The terms $G(r,r\text{'})$, $\eta (r\text{'})$and$\chi (r\text{'})$ denote the Green’s dyadic propagator, the dipole susceptibility of the tip, and the dipole susceptibility of the sample, respectively. The parameter k₀ indicates the magnitude of the wave-vector of incident and scattered light. In usual s-SNOM measurements, the tip is a weak scatterer (for example, Si-tip), whereas the sample is a strong scatterer (plasmon resonant Au or Ag nanoparticle). Thus, we truncate the summation in Eq. (11) at the first order in$\chi$:

This new sample operator takes into account the multiple “intra-particle” scattering among the volume elements within the sample. As mentioned above, the contributions that do not vary with tip-sample distance (such as TE₀, T²E₀, SE₀ and S²E₀ terms) are filtered out by harmonic demodulation. Thus, the equation for E_scat is simplified to:

We assume that the TS and ST terms are much larger than the STS term for the case of isolated dipolar plasmons.