Imaging of Plasmonic Eigen Modes in Gold Triangular Mesoplates by

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Imaging of Plasmonic Eigen Modes in Gold Triangular Mesoplates by Near-Field Optical Microscopy Keisuke Imaeda,† Seiju Hasegawa,‡ and Kohei Imura*,†,‡ †

Research Institute for Science and Engineering and ‡Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan S Supporting Information *

ABSTRACT: We investigated the spectral and spatial characteristics of plasmons induced in chemically synthesized triangular gold nano- and microplates by aperture-type scanning nearfield optical microscopy. Near-field transmission images taken at plasmon resonance wavelengths showed two-dimensional oscillating patterns inside the plates. These spatial features were well reproduced by the square moduli of calculated eigen functions confined in the two-dimensional triangular potential well. From the irreducible representations of the eigen functions, it was found that both the out-of-plane modes and in-plane modes were clearly visualized in the near-field images. We compared near-field transmission images of a triangular nanoplate to those of a truncated one with a similar dimension and revealed that the fine details of the geometrical shape of the apex on the plate strongly influence the experimentally observed eigen mode structures. We also performed near-field transmission measurements of micrometer-scale triangular plates and found that wavy patterns were observed along the edges of the plates. The wavy features can be interpreted as the superposition of eigen modes with similar eigen energy. These findings prove that near-field transmission imaging enables one to directly visualize plasmonic eigen modes confined in the particle and provide fruitful information not only for a deeper understanding of plasmons but also for the application of the design and active control of plasmonic optical fields.

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INTRODUCTION Noble metallic nanoparticles have attracted much attention during the past few decades because of their remarkable optical properties arising from the collective oscillations of free electrons called surface plasmons. Surface plasmons confine electromagnetic waves spatially as well as temporally and consequently induce intense optical fields in the vicinity of the particle.1,2 In particular, when nanoparticles are brought into close proximity to form an assembly, the assembly exhibits distinctive properties such as Fano resonances3 and ultrasmall cavity modes,4 resulting in dramatically enhanced fields. Because of these unique properties, plasmons induced in metallic nano- and microparticles have great potential for applications in many fields. For example, plasmonic-enhanced fields interact strongly with molecules and induce unique phenomena such as fluorescence enhancement,5,6 surfaceenhanced Raman scattering,7 and surface-enhanced infrared absorption.8 The plasmonic fields also enable one to induce various nonlinear optical processes, such as second harmonic generation,9 two-photon induced photoluminescence,10 and multiphoton-induced photochemical reactions,11,12 with relatively low light excitation intensity. The spatial distributions of plasmonic fields are directly related to the spatial characteristics of plasmon modes and vary depending on the resonance wavelengths. Therefore, detailed knowledge of the spatial and spectral characteristics of plasmon modes is indispensable not only for a deeper understanding of the fundamental properties of plasmons but also for their © XXXX American Chemical Society

practical applications. The spectral features of single nanoparticles were extensively studied by scattering spectroscopy with a dark-field microscope. However, because the spatial scale of plasmon modes induced in a nanoparticle is essentially smaller than the diffraction limit of light, conventional optical microscopy is not applicable for visualization of plasmon modes. For the purpose of optical imaging of plasmon modes with high spatial resolution, numerous studies with the use of nonlinear optical microscopy,13,14 scattering-type scanning near-field optical microscopy (SNOM),15,16 and photoemission electron microscopy17,18 have been reported so far. In addition to these methods, novel microscopic methods using plasmonenhanced fluorescence19−21 or photochemical reactions22−24 have also been developed. Although these studies provide detailed knowledge on plasmon modes, a monochromatic laser source is often used in these methods, and thus, it is practically difficult to obtain the spectral features of the sample. Electron energy loss spectroscopy (EELS) with highresolution scanning transmission electron microscopy (STEM) is a powerful tool for measurements of the spatial and spectral characteristics of plasmons.25,26 In this method, in addition to the spatial features of plasmon modes, positiondependent spectral features can be obtained. Moreover, this method enables one to observe optically dark modes, which Received: January 20, 2018 Revised: March 13, 2018

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DOI: 10.1021/acs.jpcc.8b00678 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

Figure 1. (a) SEM image of a gold triangular nanoplate (edge length ∼ 500 nm, thickness ∼ 40 nm). (b) Near-field extinction spectra of a gold triangular plate. The red and blue curves represent the spectra measured at the red and blue positions in (a), respectively. (c, d) Near-field transmission images observed at 715 and 800 nm, respectively. The dotted white lines represent the approximate shape of the triangular plate. The scale bars are 200 nm.

and spatial characteristics of plasmons excited in gold triangular plates with dimensions ranging from several hundred nanometers to a few micrometers (called as mesoplates). The spatial distributions of plasmons were visualized in the near-field transmission images taken at the plasmon resonances. We also performed near-field transmission measurements of single truncated mesoplates for a deeper understanding of the tip truncation effects on plasmon modes. From experimental and theoretical studies, it was known that the resonance peak of the dipole plasmon mode is very sensitive to the tip truncation.50,51 In contrast, the tip truncation effect on higher-order plasmon modes has not been demonstrated experimentally so far. In this study, we compared near-field images between triangular and truncated mesoplates and unraveled that the spatial patterns of plasmon modes were strongly influenced by the tip truncation. To reveal the physical origin of the observed near-field images, we calculated the eigen functions confined in a two-dimensional triangular potential well. From comparisons of near-field images with the square moduli of the eigen functions, we revealed that the plasmon modes in near-field images were comprehensively interpreted as the square moduli of plasmonic eigen functions confined inside the boundary of the plates.

cannot be directly excited by far-field irradiation. A number of studies utilized the STEM−EELS method for visualization of the plasmon modes excited in various plasmonic nanostructures.27−34 For two-dimensional nanostructures, it was reported that the breathing mode and the edge mode were selectively excited depending on the excitation position of the electron beam on the sample.35−41 Near-field transmission measurements using aperture-type SNOM have also been established as a nano-imaging method. In a similar manner to the STEM−EELS method, this method allows us to measure the local near-field spectra in the nanostructure.42−44 Theoretical and experimental studies of SNOM demonstrated that SNOM visualizes the photonic local density of states (LDOS) on the sample.45,46 Under the resonance condition, the LDOS is approximately equal to the square amplitude of the eigen functions of the elementary excitations induced in materials.47,48 Therefore, SNOM enables one to optically observe the square moduli of plasmonic eigen functions excited in metallic nanoparticles.42,49 Moreover, SNOM can obtain the polarization characteristics of plasmons, which are not accessible by the conventional STEM−EELS method.44 In this study, we performed near-field transmission measurements using aperture-type SNOM to investigate the spectral B


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EXPERIMENTAL SECTION Sample Preparation and Characterization. Gold triangular mesoplates were chemically synthesized in solution according to the previously reported method.52 Briefly, 3 mL of an aqueous 9.5 × 10−4 mol/dm3 sodium citrate solution was heated to 85 °C. Then, 2 mL of an aqueous solution containing 1.25 × 10−3 mol/dm3 of HAuCl4 and 0.01 mol/dm3 of hexadecyltrimethylammonium chloride was poured into the heated sodium citrate solution. The mixture was kept at 85 °C for 15 min and then cooled to room temperature without any disturbance for 18 h. The gold mesoplates were purified by removing the supernatant solution containing the spherical particles and then redispersed into 5 mL of deionized water. The sample was prepared by dispersing an aqueous solution containing gold mesoplates on a coverslip. After completely drying, the coverslip was spin-coated by a drop of poly(vinyl alcohol) (PVA) solution (0.5 wt %, ∼2000 rpm). The thickness of the PVA film was estimated to be ∼8 nm from a Dektak surface profiler (Dektak XT-S; Bruker). Because the refractive index of PVA film is almost the same as that of glass substrate, gold plates are considered to be embedded in the homogeneous medium. The dimensions of the gold mesoplates were determined by scanning electron microscopy (SEM, S3400N; Hitachi) and topography measurements by the SNOM. SNOM Measurements. We employed a home-built aperture-type SNOM for optical characterization of the sample. Details of the SNOM setup have been described previously.53,54 An apertured near-field optical fiber probe (JASCO Corp.) was used as the local near-field light source. The sample substrate was mounted on a piezo-driven stage for lateral scanning. The distance between the tip of the near-field probe and the sample was maintained at ∼10 nm by a shear-force feedback method. For near-field transmission measurements, we used a halogen lamp as light source. Gold mesoplates were locally illuminated by the near-field light at the aperture of the probe, and the transmitted light after interaction with the sample was collected by an objective lens (N.A. = 0.85, CFI Plan Fluor 60×; Nikon) beneath the coverslip and directed into a polychromator equipped with a charge-coupled device (PIXIS256E; Princeton Instruments) detector. The transmitted intensity spectrum was measured at each position in a whole scanned area, and the near-field extinction, −(I − I0)/I0, was obtained from the transmitted intensity observed on the gold mesoplate, I, and on the bare glass substrate, I0.

transmission images of the triangular mesoplate observed at the resonance peaks ∼715 and ∼800 nm, respectively. The white dotted lines represent the approximate shape of the plate. The dark parts in the transmission images correspond to the reduction of transmitted light due to the extinction by the sample and indicate that the excitation probability of the plasmon is high. As evident in Figure 1c,d, near-field transmission images exhibit unique spatial patterns different from the morphological shape. We also found that the spatial features of the near-field images varied depending on the observation wavelength. For instance, the near-field image observed at 715 nm exhibits four extinction spots at the center and near the apices of the triangular plate, as shown in Figure 1c. On the other hand, the transmission image at 800 nm in Figure 1d shows three extinction spots inside the plate. The results indicate that the number of extinction spots increases as the resonance wavelength becomes shorter. To reveal the origin of the observed spatial features, the spatial distributions of the LDOS exited in a triangular plate are of great use. We calculated the geometrical LDOS of a triangular plate by solving the Schrödinger equation for a particle confined in an equilateral two-dimensional triangular potential well.55 Boudarham et al. studied the correlation between the geometrical LDOS and electromagnetic LDOS in the presence of dissipation and reported the link between the near-field optical microscopy and the mapping of the geometrical LDOS.56 In this study, the coherent length of plasmons under study is longer than the particle size used, and hence, the influence of the dissipation on the spatial structures of the electromagnetic LDOS is less significant. It should be noted that this method does not provide a quantitative spectroscopic feature but a qualitative spectroscopic feature. Therefore, the eigen energy calculated in this method provides order of the plasmon modes excited in the plate. We also employed group theory to assign the irreducible representations to the eigen functions (see ref 55 for details). It has been reported that the irreducible representations provide the optical selection rules of plasmon modes for an external vector field.57,58 The triangular plate on a substrate is classified as the C3v point group (Table 1). As shown in Figure 2a, the plate Table 1. Characteristic Table for the C3v Point Group

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RESULTS AND DISCUSSION An SEM image of a gold triangular mesoplate is shown in Figure 1a. The edge length of the mesoplate was estimated to be approximately 500 nm from the SEM image. The thickness of the mesoplate was evaluated to be ∼40 nm from the line profile of the topographic image by the SNOM. The red and blue curves in Figure 1b show the near-field extinction spectra taken at the red and blue positions in Figure 1a, respectively. The extinction spectrum taken at the red position (red curve) exhibits two positive peaks at ∼590 and ∼850 nm. These extinction peaks are assignable to the plasmon resonances of the gold mesoplate.20,42,44 In addition to these two peaks, an additional resonance peak appeared near ∼715 nm in the extinction spectrum taken at the blue position (blue curve in Figure 1b). This observation indicates that the excitation probability of the plasmon varies depending on the observation position on the mesoplate. Figure 1c,d shows near-field

C3v

E

2C3

3σv

A1 A2 E

1 1 2

1 1 −1

1 −1 0

z Rz (x, y); (Rx, Ry)

x2 + y2, z2 (x2 − y2, xy); (xz, yz)

plane and the threefold rotation axis (C3 axis) are taken as the xy plane and z-axis of the Cartesian coordinates, respectively. Figure 2b shows the square moduli of the eigen functions confined in a two-dimensional triangular potential well and the corresponding irreducible representations. In Figure 2b, (p, q) represents the index of the calculated eigen function (called the mode index).55 The bright spots in the image represent that the probability of finding the particle is high, meaning that the excitation probability is high. Because the near-field transmission images reflect the LDOS, the bright spots in the calculated image correspond to the dark spots in the near-field transmission images. It is also to be noted that the eigen modes (6) and (11) in Figure 2b are energetically degenerate, whereas these are not classified as identical irreducible representations (details of the mode degeneration are given in Figure S1 in C


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Figure 2. (a) Schematic illustration of the sample. The plate plane and threefold rotation (C3) axis are taken as the xy plane and z-axis, respectively. (b) Squared moduli of the eigen functions calculated for an equilateral triangle potential well, where (p, q) indicate the mode index of the calculated functions. The corresponding irreducible representations and eigen energies are shown for each eigen mode. E0 denotes the eigen energy of the lowest eigen mode (1).

in the visible to near-infrared regions. Most of these peaks are qualitatively reproduced by the simulated far-field spectrum (see Figure S2 in Supporting Information). Figure 3c−e shows near-field transmission images taken at these resonance peaks (observation wavelength: ∼740 nm for (c), ∼800 nm for (d), and ∼880 nm for (e)). These images show two-dimensional oscillating spatial patterns in the plate plane, and the oscillating period becomes shorter with the decrease in the observation wavelength. The spatial patterns of these images were smaller and more complicated than those shown in Figure 1c,d, indicating that higher-order plasmon modes were excited due to the increase in the plate size. Figure 3f−h shows the square moduli of the calculated eigen functions corresponding to the observed images in Figure 3c−e. Here, the eigen modes in Figure 3f−h are equal to those labeled (10), (7), and (5) in Figure 2b, respectively. As evident in Figure 3c−h, the spatial features in the near-field images are qualitatively reproduced by the calculated mode patterns. We should note that higher-order plasmon modes showing very complicated spatial structures were clearly visualized by the near-field optical microscope. We found from the irreducible representations of the C3v point group that the out-of-plane mode (Figure 3d) and the in-plane mode (Figure 3c,e) were clearly visualized by the near-field imaging. We also found that the eigen modes labeled (9), (8), and (6) were not observed in the near-field images, although these modes were expected to be excited from the viewpoint of the eigen energy shown in Figure 2b. This result suggests that all of the eigen modes are not experimentally visualized, but

Supporting Information). As shown in Figure 2b, the number of local maxima in the eigen modes increases with an increase in the eigen energy, which is consistent with the observation mentioned above. We found that the observed spatial patterns in Figure 1c,d are in good agreement with the calculated eigen modes (3) and (2), respectively. The agreements noted above strongly support that near-field transmission imaging visualizes the plasmon modes excited in the triangular plate. As shown in Figure 2b, eigen mode (2) shows the irreducible representation E, which indicates that this mode is assignable to the in-plane mode. According to the optical selection rule, this mode is optically allowed by an external electric field with a component in the xy plane. On the other hand, the irreducible representation of eigen mode (3) is A1, meaning that this mode belongs to the out-of-plane mode and is accessible with an electric field polarized along the z-axis. Therefore, when a plane wave is irradiated from the normal incidence, only mode (2) is optically excited and mode (3) is not. In the case of the aperture-type SNOM, the near-field is polarized both parallel and perpendicular to the aperture plane.59 Thereby, both eigen modes (2) and (3) are excited by the near-field and are visualized in the near-field transmission images. Spatial patterns of plasmon modes vary depending on the edge length of the plate. We performed near-field transmission measurements for a larger triangular plate shown in Figure 3a. The edge length and thickness of the plate were determined to be ∼810 and ∼30 nm, respectively. The near-field extinction spectrum shown in Figure 3b exhibits multiple resonance peaks D


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Figure 3. (a) SEM image of a gold triangular nanoplate (edge length ∼ 810 nm, thickness ∼ 30 nm). (b) Near-field extinction spectrum measured at the red position in (a). (c−e) Near-field transmission images observed near the plasmon resonances. Observed wavelength: (c) ∼740 nm, (d) ∼800 nm, and (e) ∼880 nm. The black dotted lines represent the approximate shape of the triangular plate. The scale bars are 200 nm. (f−h) Squared moduli of the eigen functions confined in an equilateral triangular well. The modes are identical to those labeled (10), (7), and (5) in Figure 2b.

measurements on a truncated gold plate (edge length ∼ 820 nm, thickness ∼ 30 nm) shown in Figure 4a. It should be noted that the edge length and the thickness of the truncated plate are comparable to those of the triangular plate in Figure 3a. Therefore, comparisons between the near-field images in Figures 3a and 4a would reveal the tip truncation effect on the spatial characteristics of plasmon modes. The near-field extinction spectrum of the truncated plate in Figure 4b shows multiple peaks attributable to plasmon resonances. The nearfield transmission images taken near these resonances are shown in Figure 4c−e (observation wavelength: (c) ∼690 nm,

some of the specific modes are dominantly visualized in nearfield observation. There are two possible reasons for the observed mode selectively. First, the resonance peaks of the unobserved modes are energetically too close to separate from those of the observed ones. Second, the excitation probability of the unobserved modes is smaller than that of the observed ones. Because plasmon mode structure is significantly affected by the geometrical shape of the plate, we explored the influence of the triangular plate tip truncation on the spatial characteristics of the plasmon modes. We performed near-field transmission E


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Figure 4. (a) SEM image of a truncated gold nanoplate (edge length ∼ 820 nm, thickness ∼ 30 nm). (b) Near-field extinction spectrum measured at the red position in (a). (c−e) Near-field transmission images observed near the plasmon resonances. Observed wavelength: (c) ∼690 nm, (d) ∼750 nm, and (e) ∼810 nm. The black dotted lines represent the approximate shape of the truncated plate. The scale bars are 200 nm. (f−h) Squared moduli of the eigen functions in an equilateral triangular plane. The modes are identical to those labeled (12), (9), and (6) in Figure 2b.

(d) ∼750 nm, and (e) ∼810 nm). In these images, twodimensional oscillating patterns are observed and are strongly dependent on the observed wavelength. We calculated eigen modes of the truncated plate by solving the Schrödinger equation for a particle confined in a tip-truncated triangular potential well and found that the spatial patterns of the eigen modes for a truncated plate are almost the same as those for a triangular plate as far as the snipping is less significant. Therefore, we compared the near-field images of the truncated plate with the eigen modes of a triangular plate. The spatial features of near-field images of the truncated plate are in good

agreement with those of the calculated eigen modes shown in Figure 4f−h, which are identical to those labeled (12), (9), and (6) in Figure 2b. We should also note that the observed images of the truncated plate are found to be entirely different from those of the triangular plate shown in Figure 3c−e, even though the edge length and the thickness are comparable to each other. For instance, eigen modes (9) and (6) were only observed in the truncated plate but not in the triangular plate. Contrarily, eigen modes (7) and (5) were observed in the triangular plate but not in the truncated plate. The calculated eigen modes and the near-field images observed in the triangular and truncated F


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thickness ∼ 35 nm) shown in Figure 6a. The red and blue curves in Figure 6b show near-field extinction spectra observed at the red and blue positions in Figure 6a, respectively. These spectra show multiple resonance peaks, which alternately appear in one curve then in the other in the near-infrared spectral region (700−950 nm). Figure 6c−e shows the nearfield transmission images observed at these resonance peaks (observation wavelength: (c) ∼720 nm, (d) ∼760 nm, and (e) ∼820 nm). In the near-field images, wavy patterns were observed at the edges of the plate, whereas the clear spatial features were not visualized at internal points in the plate. This observation is similar to the edge modes reported in EELS microscopy.41 In Figure 6b, the number indicates an index l determined from the number of antinodes observed at the plate edge in the near-field images. We found from Figure 6b that the index l increases as the resonance wavelength becomes shorter. This tendency is analogous to a Fabry−Pérot cavity mode induced in a one-dimensional nanostructure.30 We also found from Figure 6b that odd and even modes are selectively excited at the red and blue positions on the plate edge, respectively. This observation implies that the red and blue positions in Figure 6a correspond to the antinodes of the odd and even plasmon modes excited at the plate edge, respectively. This result again supports that the spatially resolved spectra observed by the SNOM are useful for revealing the plasmon characteristics in detail. We considered that the spatial features observed in Figure 6c−e are interpreted as a superposition of the eigen modes with similar eigen energy. The eigen modes with high eigen energy exhibit highly oscillating features, and the oscillation patterns for these modes are entirely different from each other, as shown in Figure 6f−h. Thereby, when the eigen modes with similar eigen energy are superimposed, interference between the modes should occur. We found that destructive interference occurs at the internal part of the triangle, whereas constructive interference occurs at the edges, as illustrated in Figure 6i. As a consequence, superimposed images in Figure 6j−l exhibit higher intensity at the edges compared to the center (details are provided in Figure S3 in the Supporting Information). These spatial features imply that the excitation probabilities at the plate edges should be higher than that at the internal part of the plate, which are qualitatively consistent with the observations shown in Figure 6c−e. We analyzed a dispersion feature of the plasmon modes from the observed results. The plasmon resonance energy was determined from the resonance wavelength in the near-field extinction spectrum in Figure 6b. We derived a pseudoplasmon wavenumber kp‑sp from the relation kp‑sp = 2π/λp‑sp, where λp‑sp/2 was determined from the distance between two adjacent antinodes observed along the plate edge in near-field images. We performed this analysis for several plates with different dimensions (one of them is shown in Figure S4 in the Supporting Information) and obtained a pseudo-plasmon dispersion relation by plotting resonance energy against wavenumber, as shown in Figure 7. We found from the analysis that all data points follow a single dispersion curve as far as the plates with identical thickness are concerned. The dispersion shows a converging feature similar to that observed for gold nanorods.42,60 We compared the observed dispersion with the simulated one for a thin gold film (thickness: 35 nm; refractive index of a surrounding medium: 1.5; the black dashed curve in Figure 7)61 and found that the near-field data points exhibit a lower curvature compared to the thin film. This

plates are summarized in Figure 5. We found from Figure 5 that the eigen modes arranged from bottom to top were

Figure 5. Left column: the calculated eigen modes and the corresponding eigen energies. Middle column: SEM image and nearfield transmission images of the triangular nanoplate shown in Figure 3. Right column: SEM images and near-field transmission images of the truncated nanoplate shown in Figure 4.

alternatively visualized in the triangular and truncated plates. This finding is closely related to the spatial distribution of the eigen modes and the geometrical shape of the plate. We found that for the truncated plate the eigen modes with bright spots near the apices were not excited, but those at the middle of the triangle were preferentially excited. In the triangular plate, on the other hand, the eigen modes with bright spots near the apices were predominantly excited. These findings imply that the observed mode selectivity can be explained by the geometrical shape of the plate. We also performed near-field transmission measurements on a micrometer-scale triangular plate (edge length ∼ 1900 nm, G


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Figure 6. (a) SEM image of a micrometer-scale gold triangular plate (edge length ∼ 1900 nm, thickness ∼ 35 nm). (b) Near-field extinction spectra measured at the red and blue positions in (a). (c−e) Near-field transmission images observed at the plasmon resonances. Observed wavelength: (c) ∼720 nm, (d) ∼760 nm, and (e) ∼820 nm. The black dotted lines represent the approximate shape of the plate. The scale bars are 500 nm. (f−h) Square moduli of the calculated eigen modes with similar eigen energy. Eigen energy: E = 25E0 for (f), E = 25.3E0 for (g), and E = 26.3E0 for (h). (i) Line profiles along the white dotted lines in (f−h). The black dotted lines indicate that the local maximum is located at the same position along the triangle edge. (j−l) Superposition of multiple eigen modes with similar eigen energies. The details of the superposition are shown in Figure S3 in the Supporting Information.

discrepancy can be explained as follows. The plasmon wavelength in Figure 7 was evaluated from the adjacent local maxima in the observed near-field image. As mentioned previously, the image does not reflect a single eigen mode but the superposition of a few modes. Consequently, the estimated plasmon wavelength is shorter than that for a single mode (see Figure S5 in the Supporting Information). This is the origin of the lower curvature observed in Figure 7. As discussed so far, near-field imaging visualizes the plasmon modes for the nanoplate. For the microscale plate, very-highorder plasmons are excited, and the resonance energies of the modes are very close to each other. Consequently, the superposition of multiple modes was visualized. Therefore, careful analysis is required to interpret the observed spatial features.

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CONCLUSIONS In conclusion, we studied the spectral and spatial characteristics of the plasmons induced in chemically synthesized gold nanoand microplates by near-field transmission microscopy. The near-field transmission spectra of the plates show multiple plasmon resonances in the visible to near-infrared spectral region. Near-field transmission images taken at the resonances show two-dimensional oscillating patterns, which vary dramatically depending on the dimensions of the plates and on the observed wavelength. The spatial structures visualized in nearfield images for the nanoplates are well reproduced by the square moduli of the eigen functions confined in a two-

Figure 7. Dispersion relation of a pseudo-plasmon mode excited in micrometer-scale gold plates. Edge length: ∼1900 nm (circle); ∼1750 nm (square); ∼1550 nm (triangle). The dashed curve is the simulated dispersion relation for a gold thin film (thickness: 35 nm; refractive index of surrounding medium: 1.5).

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dimensional triangular potential well. The irreducible representations of the eigen functions in the C3v point group were utilized to classify the observed plasmon modes as in-plane and out-of-plane characteristics. We found that both in-plane and out-of-plane modes were clearly visualized by the near-field optical imaging. We also found that not all of the eigen modes but some of the specific modes were selectively visualized, and this mode selectivity was closely related to the geometrical shape of the apex in the plate. We performed near-field transmission imaging of micrometer-scale triangular plates and found that the near-field images of the plates showed wavy patterns along the edge. The observed oscillating features were attributed to the superposition of multiple eigen modes with similar eigen energy. We also obtained the pseudo-dispersion relation from the observed near-field images and spectra and found that the dispersion curve had lower curvature than the simulated dispersion curve for the thin film. This difference was partly explained by the spatial and spectral overlap of multiple eigen modes with similar eigen energy. We described the applicability of the near-field optical imaging to directly visualize the plasmonic eigen modes confined inside the plates. Because the LDOS is directly related to the spontaneous relaxation of the emitter in the vicinity of the plasmonic nanoparticle, the spatial distributions of the plasmonic eigen modes visualized in this study are indispensable for plasmonenhanced optical processes, energy-efficient light harvesting, and various photochemical applications. Furthermore, in the micrometer-scale gold plates, as the multiple plasmon modes are simultaneously excited, coherent control of the plasmons allows to establish novel spatial modulation schemes for designing the plasmonic fields on the nanoscale.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b00678. Mode degeneration in a triangular plate; far-field scattering spectra of the triangular and truncated nanoplates simulated by the discrete dipole approximation method; superposition of multiple eigen modes with similar eigen energies, near-field transmission spectra and images of a micrometer-scale triangular plate; and line profiles of the eigen and superimposed modes (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Keisuke Imaeda: 0000-0001-8877-1085 Kohei Imura: 0000-0002-7180-9339 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Nos. JP26107001, JP26107003, JP26620018, JP16K13939, and JP16H04100 in Scientific Research on Innovative Areas “Photosynergetics”. I


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Imaging of Plasmonic Eigen Modes in Gold Triangular Mesoplates by

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