1. Introduction

The localized surface plasmon (LSP) resonance generated in metallic nanostructures induces a highly enhanced electric field, which has been employed to achieve a variety of functions, such as highly sensitive biosensing [1], hot carrier generation [2], and the optical trapping of nanoparticles [3]. LSP-related effects are expected to contribute not only to creating next-generation nanotechnologies but also to the understanding of fundamental physics by probing nanoscale optical responses. One of the peculiar effects in the optical response induced by LSP resonance is the breakdown of long-wavelength approximation (LWA) at the single molecular level, wherein optical forbidden transition occurs owing to the spatial variation of the electric field at the nanometer-scale. For example, the excitation of quadrupole-like polarization of single-walled carbon nanotubes [4] and the 2s-3d forbidden transition in a hydrogen atom [5] have been reported. Among the various types of electronic transitions in nanostructures, most transitions are optically forbidden, whereas the allowed transitions are particular ones whose associated polarization patterns are nearly structureless. However, in various types of optical processes and photochemical reactions, the electronic levels that cannot be reached by the one-photon optical transition presumably play significant roles. Therefore, unveiling and controlling the forbidden optical transitions of nanostructures or at the single-molecular level will enhance the degrees of freedom in the design of the optical functions of materials.

To understand the optical response of individual nanostructures, it is crucial to elucidate the microscopic interactions between the near-field light and matter. Scanning near-field optical microscopy (SNOM) [6–9], tip-enhanced Raman spectroscopy (TERS) [10,11], and two-photon-induced photoluminescence (TPL) [12,13] are powerful tools for investigating the near-field distributions beyond the diffraction limit of light and the optical processes of individual nanostructures. However, these techniques observe the interaction of near-field light with the targeted materials through the far field in the scattered light, i.e., they do not directly detect the near-field light itself. Further, the resolution of aperture SNOM and TPL employing the SNOM system is limited owing to the collection and propagation losses in the optical fiber. By contrast, photoinduced force microscopy (PiFM) [14,15] detects optical interactions at the nanometer scale as local forces in the near-field, which do not suffer from any type of photo-signal attenuation. Thus, this technique is promising for the direct detection of matter near-field interactions at high resolutions. Moreover, PiFM has no difficulties regarding the resolution, such as background scattered light or the aforementioned losses. Theoretical and experimental studies on cantilever dynamics revealed the nanometer-scale sensitivity of photoinduced forces to the tip-sample distance owing to the gradient force component [16]. Furthermore, it was demonstrated that the combination of mechanical eigenmodes and laser modulation frequency provides images with clearer contrast [17]. Recently, the heterodyne frequency modulation method [18,19] was used for PiFM measurements in ultrahigh vacuum to remove photothermal oscillations. Additionally, PiFM can acquire information on samples through their specific linear and nonlinear optical responses. For example, the detection of nonlinear optical signals through optical force measurements by using atomic force microscopy (AFM) has been theoretically studied [20]. Further, the selective detection of particular chemical species in a sample [21] and the analysis of the features of buried components [22] were performed through the resonant optical responses of samples.

Notably, PiFM can potentially acquire information on photo-excitation processes, including forbidden optical transitions. As discussed in theoretical studies [23,24], multifaceted information on the excited states of nanostructures can be obtained as a spatial map of induced forces in high resolution under an electronic resonance condition by using the degrees of freedom of incident light, such as polarization, frequency, and wave-vector (incident angle). Presently, more elaborate studies on such capabilities are desired, considering modern PiFM techniques. Thus, the purpose of this study is to elucidate the possible sensitivity, resolution, and information we can acquire when the state-of-the-art PiFM technique [18,19] is applied to observe the forbidden optical transitions. Using the discrete dipole approximation (DDA) method [25–27], we calculated the total response electric field self-consistently with induced polarization and investigated the forces acting on the tip, treating the highly enhanced LSP in the metallic nanogap region. Consequently, we successfully reproduced the frequency-dependent three-dimensional force vector map that reflected the functions of the targeted sample and elucidated the mechanism of achieved resolution in the above experiment. In this study, we extend our model system to measure the composite molecules and discuss the information obtained for the relevant cases.

2. Model and theory

Here, we assume that the dimer molecule has four kinds of optical transitions based on the type of interaction between its dipoles; in order from lower to higher energy, these are the allowed bonding (AB), forbidden bonding (FB), allowed antibonding (AA), and forbidden antibonding (FA) states (See Fig. 1(a)-(c)). Optical-allowed transitions are associated with the states in which the dipoles on each molecule are in the same phase, and conversely, the optically forbidden transitions are associated with the states wherein the dipoles are in the opposite phase. The latter transitions cannot be induced optically under the LWA condition. However, when the dimer molecule is in the vicinity of the metal nanogap, the transitions become possible owing to a steep electric field gradient [28]. We clarify the incident photon energy dependence of PiFM measurements for dimer molecules and reveal that completely different PiFM images can be obtained for each transition under non-LWA conditions. The different images reflect specific spatial structures of the three-dimensional polarization for different transitions enhanced by LSP, induced between the tip and the substrate.

We calculated the self-consistent total response electric field induced in a nanogap made of a metal tip, a nanoparticle, and a metal substrate by using the DDA. The following equations were used for the calculation:

 

Fig. 1. (a) Model geometry of the dimer used in the calculation. (b) The dimer is represented by two aligned cells whose susceptibilities are given by the Lorentz model in DDA. (c) Spectra of the induced field intensity of a monomer (black line) and a dimer (red line) in vacuum. The induced field intensity was normalized by the incident field intensity. Vertical lines represent the energies of the allowed bonding (AB), forbidden bonding (FB), allowed antibonding (AA), and forbidden antibonding (FA) states of the dimer. In vacuum, the dimer can be excited at the energies of the AB and AA states under the LWA condition. The circles above the spectra represent the dimers, and the arrows in the circles indicate the directions of the dipoles. (d) Schematic of the PiFM model. We approximate the metal-coated tip to the gold sphere.

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We assume that the gold-coated tip is a gold sphere with a diameter of 19 nm, and the gold substrate is a thin gold film that was $143\times 143\times 9$ nm$^{3}$, as depicted in Fig. 1(d). The metallic structures are assumed to have a Drude-type susceptibility with the parameters of Au,

Regarding the nanoparticles, we assign one cell in the DDA to one particle with a size of 1 nm$^3$ that assumes the following Lorentzian-type susceptibility:

The photoinduced force acting on the tip is obtained by the following formula: [30]

We used the following parameters for the gold structures: $\epsilon _{\textrm{Au}}=12.0$, $\epsilon _{0}=1$, $\hbar \omega _\textrm {Au}=8.958\ \textrm {eV}$, $\gamma _\textrm {bulk}=72.3\ \textrm {meV}$, $\hbar V_\textrm {F}=0.922\ \textrm {nm} {\cdot} \textrm{eV}$, and $L_\textrm {eff}=20{\ {\rm nm}}$ [31,32]. We assumed that p-polarized and 10 kW/cm$^2$ plane wave light illuminates the sample with 70$^\circ$ incident angle as the laser.

The absolute magnitude of the force is determined by many elements that are not relevant to the physical results of the present study. By contrast, the geometrical structures of the maps obtained through PiFM are determined by the tip apex. Therefore, in the present study, we design the theoretical model by focusing on the latter issue. In other words, we assume a much smaller tip than that used in the realistic experiment to model only the tip apex. The dependence of the photoinduced force on the diameter of the tip apex is discussed in appendix B.

3. Sensitivity and resolution of PiFM

To reveal the sensitivity of PiFM, we calculated the photoinduced force spectrum and the force curve. We consider nanoparticles whose resonant energies are $\hbar \omega _\textrm {np}=2.0$ and 2.1 eV, and the dipole moment of 10 debye, which is similar to the value for porphyrin-based dye molecules [33,34], with parameters $\epsilon _{\textrm{np}}=1.5$, $\gamma _\textrm {np}=5$ meV, and $V_\textrm {np}=1$ nm$^{3}$. We refer to the nanoparticles as NP1 and NP2, respectively.

In Fig. 2(a), the red and green solid lines represent the spectra of the absorption cross-section for the presence of only one nanoparticle, i.e., NP1 and NP2, in the free space, respectively. The absorption cross-section was previously defined by [35]

 

Fig. 2. Spectra. (a) Spectra of the absorption cross-section of NP1 (red line) and NP2 (green line) in free space. (b) Photoinduced force spectra for the tip on NP1 (red line), NP2 (green line), and the gold substrate (black dashed line). (c) The ratios of the photoinduced force spectra of NP1 and NP2 ($F$) to the force when the tip is just above the substrate ($F_0$). When $F/F_0$ is greater (less) than 1, the contribution from the nanoparticle acts in the direction of the attractive (repulsive) force. (d) Enlarged figure of (c).

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3.1 Wavelength selectivity of PiFM

Figure 2(b) shows the photoinduced force spectra detected by the PiFM tip. The black line in Fig. 2(b) represents the spectrum on the gold substrate, i.e., in the absence of nanoparticles. The plasmonic resonance of gold yields a broad spectrum. The red and green lines represent the spectra for the presence of only one nanoparticle, i.e., NP1 and NP2, on the substrate, respectively. The center of the tip is just above the nanoparticle and tip-sample distance, $d$, defined as the distance between the tip surface and nanoparticle surface, at 1 nm.

The peak positions of the photoinduced force are 1.990 and 2.088 eV. They are red-shifted and broadened owing to the strong self-interaction via the interaction with the LSP induced at the metal nanogaps. As the force obtained in the PiFM measurement is the force acting on the tip, the real part of the tip (gold) is dominant in the PiFM spectrum. As expressed in Eq. (6), the contribution from the nanoparticle is included in the electric field because the photoinduced force can be written as the product of the tip polarization and the electric field gradient. Figure 2(c) depicts the ratios of the photoinduced force spectra of NP1 and NP2 to the force when the tip is just on the substrate ($F_0$). At a lower resonance energy of the nanoparticles, the photoinduced force is very strong owing to the synergistic enhancement of the polarization of the nanoparticles and the LSP. By contrast, at a higher resonance energy of the nanoparticles, the polarization of the nanoparticles and LSP are in opposite phases and weaken each other. In this case, the polarization of the nanoparticles is so small that it can weaken the LSP only slightly. Therefore, the peak in the attraction direction is large, and the dip in the repulsion direction is small. This holds true, regardless of the position of the LSP peak. Figure 3 presents the force curve of the photoinduced force at laser photon energy $\hbar \omega = 1.990$ eV. The horizontal axis depicts the distance between the position of the center of the tip, $z_\textrm {tip}$, and the nanoparticle, $z_\textrm {np}$. The red, green, and black markers represent the force curve on the NP1, NP2, and gold substrate, respectively. The diameters of the tip and the nanoparticle were 19 nm and 1 nm, respectively, and they contacted at $z_\textrm {tip}-z_\textrm {np}=10$ nm ($d=0$ nm). However, because the cell size of the DDA is set to 1 nm$^3$, $d$ cannot be less than 1 nm in this calculation. For a long distance, the force curve follows $z^{-2}$ lows owing to the Coulomb force. The photoinduced gradient force is proportional to $z^{-4}$ by approximating the tip and sample as two point dipoles [16,36]. In this situation, the laser photon energy is close to the resonant energy of NP1, and the force curve of NP2 is proportional to $z^{-4}$, as in the case of the gold substrate (absence of nanoparticles) for a short distance. By contrast, the force curve of NP1 does not follow the $z^{-4}$ law and becomes even steeper in the near range ($z_\textrm {tip}-z_\textrm {np}<13$ nm) owing to multiple enhancements in the field between the LSP in the nanogap and the polarization of the NP1 resonance. This indicates that PiFM has high sensitivity for the resonance of target materials. In this study, as we assume a much smaller tip than that used in the realistic experiment to model only the tip apex, the calculated force is smaller than that expected in experiment.

 

Fig. 3. Force curves of the PiFM. The attractive force is plotted on a logarithmic graph as positive. The red and green markers represent the force curve on NP1 and NP2, respectively. The black markers represent forces on the gold substrate, which are overlapped by green markers. The photoinduced force is proportional to $z^{-4}$ for short distances and $z^{-2}$ for long distances. For NP1, the force curve does not follow the $z^{-4}$ law because of the influence of the resonance, and it becomes even steeper in the near range.

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3.2 Distinguishing different nanoparticles

We demonstrate that PiFM is advantageous in detecting the presence of nanoparticles, as well as the structure and physical properties of the surface of the sample. This leads to the identification of defects and impurities within the sample and its surface, as well as the chemical composition and optical response of the sample molecules.

We obtained the PiFM images, as shown in Fig. 4. The photoinduced force was calculated at each point of the tip, and the PiFM images were drawn through raster scanning. It should be noted that the tip was fixed at a height of 1 nm from the nanoparticle and scanned parallel to the substrate to obtain the PiFM images. The results obtained through this scanning mode exhibit the essential properties of experimental results because PiFM uses an optical microscope to observe the spatial structure of the LSP gradient, although generally, in PiFM experiments, the tip-sample distance is maintained at the same value as that used in AFM. Figure 4(a) presents a PiFM image of one nanoparticle (NP1) with laser photon energy $\hbar \omega =1.990$ eV. The incident direction of the laser is from positive to negative on the $y$-axis at an incident angle of 70$^\circ$ and p-polarization. The red square in the figure represents the position of the nanoparticle. The color map represents the z-axis component of the photoinduced force acting on the tip. The force profile on the line in Fig. 4(a) is shown by the red line in Fig. 4(e). The full width at half maximum is approximately 10 nm, and this value corresponds to the resolution in this model (see Appendix C). This resolution is proportional to the tip diameter. Figure 4(b) presents a PiFM image in the case where two same kinds of nanoparticles (NP1) are aligned. The distance between the two nanoparticles is approximately 4.24 nm (diagonal length of three cells). The force profile on the line is shown by the yellow line in Fig. 4(e). As mentioned above, the resolution is approximately 10 nm, such that the force profiles overlap, and PiFM cannot differentiate between the two nanoparticles.

 

Fig. 4. PiFM images. (a) Single nanoparticle PiFM image. The red squares represent the positions of the nanoparticles. (b) Image of two aligned nanoparticles. (c) and (d) present the images for different types of nanoparticles with different resonant energies aligned. The incident direction of the laser is from positive to negative on the y-axis at an incident angle of 70$^\circ$ and p-polarization. The laser photon energies are 1.990 eV for (a), (b), and (c) and 2.088 eV for (d). (e) depicts the force profiles of (a) and (b), and (f) depicts the force profiles of (c) and (d). The pale-colored bands of red, yellow, and green represent the positions of the nanoparticles. The PiFM images and line profiles are normalized by the force when the tip is just above the substrate ($F_0$).

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However, if two different types of nanoparticles are aligned, we can differentiate between them through PiFM. Figures 4(c) and (d) present PiFM images of two different kinds of nanoparticles with laser photon energies of $\hbar \omega =1.990$ and 2.088 eV, respectively. The red square represents NP1, and the green square represents NP2. In the case of laser photon energy of $\hbar \omega =2.088$ eV, NP1 can be measured (Fig. 4(c)). By contrast, in the case of laser photon energy of $\hbar \omega =1.990$ eV, NP2 is measured (Fig. 4(d)). Figure 4(f) presents the force profiles of each laser photon energy. These results imply that PiFM can distinguish between different types of nanoparticles beyond the resolution of the tip diameter by tuning the wavelength of the incident laser.

The wavelength selectivity of PiFM can be used to help identify chemical modifications on the surface of protein molecules [37]. PiFM also has the advantage of being able to observe buried materials [22], which could play a role, for example, in detecting interlayer impurities in nanoparticle multilayered optical devices, assessing the structure of the underlying layers, and identifying chemical species adsorbed within the porous medium.

3.3 Damping constant dependence of sensitivity

We present the damping constant dependence of the photoinduced force spectra in Fig. 5. Their resonance energies are $\hbar \omega _\textrm {np}=2.0$ eV, and the damping constants are $\gamma _\textrm {np}=$ 1, 5, 10, 50, and 100 meV. As the decay constant of the nanoparticles decreases, the peaks become larger and narrower.

Even when the damping constant is large, if the two energies are sufficiently far apart such that the peaks of the photoinduced force spectrum do not overlap, they can be observed selectively, albeit with low sensitivity. Thus, it is desirable that the line width of the sample be sharp for the PiFM measurement with high sensitivity and high selectivity. At low temperatures, the scanning tunnel luminescence peaks of the dye molecules have line widths of a few µeV [29]. In this case, PiFM allows us to distinguish between various molecules and observe various vibrational levels.

 

Fig. 5. Damping constant of nanoparticle dependence of the photoinduced force spectrum. The dashed black line represents the absence of nanoparticles, and the solid colored lines depict nanoparticles with damping constants of $\gamma _\textrm {np=}$ 100, 50, 10, 5, and 1 eV. The resonance energy of the nanoparticles is at 2.0 eV, and as the damping constant decreases, the peak appears.

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4. Optical response of allowed and forbidden transitions of a dimer

In this section, we analyze the PiFM measurements of a dimer. We model the dimer as two aligned monomers, whose susceptibilities are described by the Lorentz model Eq. (4) with dipole moment $\mu$ = 10 debye and resonance energy $\hbar \omega _\textrm {np}$ = 2.0 eV. Two monomers are coupled with each other via dipole interaction and form a dimer that can assume four excited transition states: AB, FB, AA, and FA [28]. For clear energy splitting, we assume the damping constant in the susceptibilities to be $\gamma _\textrm {np}$ = 0.1 meV. We present the absorption cross-section of the dimer in free space in Fig. 1(c). The bonding and antibonding states of the dimer appear symmetrically split from the resonance state of the monomer. In the LWA, the states with the same phase of dipoles are allowed and forbidden when the dipoles are in the opposite phase. We show that completely different PiFM images can be obtained for each transition energy of the incident laser. These PiFM images reflect the three-dimensional structure of the local electric field vectors.

4.1 PiFM spectra of the dimer

We calculated the photoinduced force spectra of the dimer to reveal the PiFM measurement for the allowed and forbidden transitions. We illustrate the calculation model in Fig. 6(a). The dimer is on the gold substrate, and the white dots represent the center of the tip. The incident direction of the laser is from positive to negative on the $y$-axis at an incident angle of 70$^\circ$ and p-polarization. The tip is just 1 nm above the dimer. Figure 6(b) presents the photoinduced force spectra normalized by $F_0$ for the different positions of the tip illustrated in Fig. 6(a). As the tip moves away from the dimer, the peaks corresponding to the AA state become smaller. By contrast, the peaks that correspond to the AB and FB states appear around the tip position of 2.8-4.2 nm; nevertheless, they are not observed at 0 nm. This can be explained by the structure of the electric field response. The vertical polarization of LSP is strongly induced at the nanogap. Therefore, as shown in Fig. 6(d), the peak of the AA state is broadened because of the strong interaction with LSP. As the interaction of the dimer with the LSP is the strongest just under the tip, the peak for the AA state becomes red-shifted as the dimer approaches the tip. By contrast, the FB state arising from the electric field gradient and the AB state arising from the horizontal component of the electric field have narrower peaks and smaller shifts than the AA state because the energy exchange through the interaction with the LSP is smaller than that of the AA state. Fig. 6(c) shows the spatial distribution of the response electric field vectors around the tip without the dimer. The field vectors are normalized by the magnitude of the incident field ($E_0$). Here, we demonstrate a case where the photon energy of the laser is 2.0 eV. Owing to the wide range of plasmon resonance peaks, the vector field hardly changes with a difference of a few tens of meV. Because the $z$-component of the LSP field is strongly enhanced at the nanogap between the tip and the substrate, the AA state can be excited just under the tip (Fig. 6(d)). On the contrary, because of the steep electric field gradient, the FB state can be excited around the tip. Owing to the shape of the tip, the lateral component of the LSP field appears; as a result, the AB state can be excited at 1.4 nm or more.

4.2 PiFM images of dimer

We demonstrated that the excitation selectivity of the dimer depends on the spatial structure of the localized response electric field. We obtained different PiFM images of the dimer with the incident photon energies corresponding to each transition state. In this section, we show that the PiFM images reflect the spatial structure of the intensity and polarization of the localized response field.

 

Fig. 6. (a) Schematic illustration of dimer on the gold substrate and tip position. The center of the tip is represented by white dots. The grid on the substrate represents the cells in DDA. The incident direction of the laser is from positive to negative on the $y$-axis at an incident angle of 70$^\circ$ and p-polarization. (b) Photoinduced force spectra. The lines represent the dimer, and each tip position is shown in (a). Each force spectrum is normalized by $F_0$. As the tip moves away from the dimer, the peak corresponding to the AA state becomes smaller, whereas the peaks corresponding to the AB and FB states appear strongly in the range of 2.8-4.2 nm. (c) Vector map of the response field around the tip. The color of the vectors represents the magnitude of the response field in the absence of the dimer. The vector scales are normalized by the magnitude of the incident electric field ($E_0$). (d) When the dimer is just under the tip, the AA state can be excited. (e) When the dimer is away from the tip, the FB state can be excited owing to the gradient of the field. (f) The AB state can be excited by the appearance of the lateral component of the field.

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First, the PiFM image for the AA state is shown in Fig. 7(a). Similar to Fig. 4, the tip was fixed at a height of 1 nm from the nanoparticle and scanned parallel to the substrate to obtain the PiFM images. The AA state excitation can be understood in terms of the LWA and depends on the intensity of the electric field perpendicular to the direction in which the dimers are aligned (i.e., $\boldsymbol{E}_{\perp }\equiv (0,E_{y},E_{z})$), as shown in Fig. 6(d). The intensity of the electric field $\boldsymbol{E}_{\perp }$ distribution on the substrate around the tip is shown in Fig. 7(d), where the tip position is represented by the yellow dashed-line circle. The incident photon energy is $\hbar \omega =2.0$ eV; however, the metallic structure without any nanoparticles creates a wide range of plasmon resonance peaks, and hence, it hardly changes when the photon energy changes by a few tens of meV. In the PiFM system, the response electric field is remarkably enhanced and localized in the $z$-direction, which is the direction of the gap between the tip and the substrate. The electric field is slightly distorted owing to the asymmetry caused by the incident direction of the laser ($y$-direction). This is reflected in the PiFM image (Fig. 7(a)).

 

Fig. 7. (a), (b), and (c) depict PiFM images of the dimer for the AA, FB, and AB states, respectively. The PiFM images are normalized by $F_0$. The dimer is represented by green squares. (d), (e), and (f) represent the response field distributions under the tip that is absent from the dimer. $\boldsymbol{E}_0$ represents the incident electric field. The incident photon energy is $\hbar \omega =2.0$ eV. In the absence of the dimer, the metallic structure creates a wide range of plasmon resonance peaks, which hardly changes with a difference of a few tens of meV. The yellow dashed-line circles represent the tip position and diameter. (d) is the intensity of the $z$-component of the field, (e) is the $x$-direction gradient of the $z$-component of the field, and (f) is the intensity of the $x$-component of the field. The incident direction of the laser is from positive to negative on the $y$-axis at an incident angle of 70$^\circ$ and p-polarization.

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Subsequently, in Fig. 7(b), we present the PiFM image for the FB state. Considering the dipole structure of the dimer, $\boldsymbol{E}_{\perp }$ elements contribute to FB state excitation as well as the AA state, as shown in Fig. 6(e). However, this is a plasmonic phenomenon beyond the LWA, and it occurs owing to the steep gradient along the $x$-direction of the electric field $\boldsymbol{E}_{\perp }$ on the size scale of the dimer, that is, $\partial \boldsymbol{E}_{\perp }/\partial x$, as shown in Fig. 7(e). This effect results in the PiFM image around the dimer, as shown in Fig. 7(b). Because of the size of the dimer itself, the dimer feels a certain electric field gradient, even just below the tip. Our result of the bright edges in the PiFM image replicate the experimental results well. Finally, the PiFM image for the AB state is shown in Fig. 7(c). AB state excitation is a phenomenon that occurs in the LWA. However, in contrast to AA and FB, it is contributed by the electric field in the direction of the dimer alignment (i.e., $\boldsymbol{E}_{\parallel }\equiv (E_{x},0,0)$), as shown in Fig. 6(f). In PiFM, the electric field in the $z$-direction, which is the direction of the gap, is remarkably enhanced, as mentioned above. Because the bottom of the tip is relatively flat, the horizontal component of the electric field is hardly induced just under the tip. However, the horizontal electric field is generated away from the nadir of the tip owing to the inclination of its curved surface. We present the distribution of the intensity of the electric field of $\boldsymbol{E}_{\parallel }$ in Fig. 7(f). This distribution reflects in the PiFM image shown in Fig. 7(c). A previous experimental study [38] observed the LSP created by the horizontal component of the electric field polarization using PiFM. In the present study, we obtained bright edges in the PiFM image, thereby providing a good insight into this observation.

We assumed a tip diameter of 19 nm and stated that the resolution was approximately 10 nm. However, in the PiFM measurement of the dimer, the PiFM images can provide information on localized electric field structures beyond the resolution determined by the tip radius. As described above, PiFM uses a unique optical microscope that not only reveals the geometrical structure and optical property of the sample but also provides the three-dimensional structure of the localized electric field, including the spatial distribution and polarization, based on the microscopic interaction between the localized electric field and the sample at the nanometer scale.

5. Summary and conclusion

We theoretically analyzed the photoinduced force measurement on nanoparticles through PiFM. The interaction between the polarization of nanoparticles and the LSP induced at the metal nanogap between the gold tip and the gold substrate are self-consistently considered, yielding the total response electric field through DDA. We clarified that PiFM exhibits high sensitivity to the resonance energy of nanoparticles. We obtained the PiFM images of nanoparticles on the gold substrate. Because of the high sensitivity to the resonance energy, different PiFM images can be obtained by tuning the wavelength of the incident laser. Even if different kinds of nanoparticles exist beyond the resolution determined by the tip diameter, PiFM allows the differentiation of these nanoparticles depending on their spectral sharpness.

In this study, we simulated the PiFM measurement for a dimer molecule. Forbidden optical transition excitations are observed by PiFM. The near-field is induced at the nanogap between the PiFM gold tip and gold substrate. The three-dimensional structure of the near-field can excite various optical transitions, including transitions that are forbidden under LWA. When the dimer is just under the tip, the polarization of the near field is directed to the gap mode, and AA transition is excited. Around the tip, the steep electric field gradient enables the FB transition excitation. Further, owing to the tip shape, the lateral component of the polarization of the near field appears, which makes it possible to excite the AB transition. Owing to PiFM sensitivity, the PiFM images corresponding to the respective transitions are acquired completely differently, and they reflect the three-dimensional structure of the near-field PiFM images for AA, FB, and AB corresponding to the vertical component, the lateral gradient of the vertical component, and the lateral component of the near field, respectively.

Unlike other microscopy techniques, such as SNOM [6], TERS [39], and TPL [13], PiFM does not require a photo-signal propagating to the detectors, as in PiFM, the induced force is measured between the dipoles of the sample and the tip. Therefore, this method does not suffer from any signal attenuation during its propagation. This feature enables a significantly higher resolution than other optical microscopes. Although SNOM is a powerful tool for observing forbidden transitions, the aperture type used in SNOM predominantly observes the horizontal polarized electric field as light propagating through an optical fiber, and it is difficult to observe the vertical polarized electric field. By contrast, PiFM directly observes the gradient of the electric field, including both the vertical and horizontal components, as a force. In the present study, we have demonstrated that PiFM can possibly obtain the information of spatial structures of induced polarization reflecting electronic transitions, including optical forbidden ones; however, it is difficult to obtain such information through STM emission. As shown in [29], the state-of-the-art technique of STM emission can obtain the spatial structures of the density of states of electrons and the lattice vibrations inside a single organic molecule. Therefore, the complementary use of PiFM with SNOM and STM emission will considerably improve our understanding of the interaction between light and matter at the nanoscale. Unlike conventional AFM, PiFM can acquire information on electronically excited states through the observation of induced polarization that directly reflects three-dimensional spatial structures of the electric field in the vicinity of the samples. A particularly notable point is the possibility of observing optically forbidden transitions. If we make complete use of the degrees of freedom of light, including the frequency, polarization, and wavevector, we can obtain multifaceted information on microscopic interaction between nanomaterials and light. The picocavity effect [40] may play an essential role in the high resolution of PiFM at cryogenic temperatures. We can treat the picocavity effect by using the multi-sized cell DDA method developed in [4] to consider the atomic-scale spatial structures on the tip and the sample. In the present study, we did not use this method to conduct the study with wide parameter regions, avoiding the heavy computational load. However, in a following study, we will use the multi-sized cell DDA combined with elaborated calculations of electronic states of molecular systems to demonstrate what we can see by the observation of a single molecule with PiFM, which would reveal the potential capability of PiFM to observe the excitation processes of molecular systems.

Appendix A: method for calculating the electric field

In the present study, we calculate the self-consistent electric field by using the DDA method [25–27], in which the entire space is divided into small cubic cells that are approximated as point dipoles. The interaction of those dipoles with each other through the electric field can be obtained by solving Maxwell’s equations.

Assuming a time-harmonic electric field, the following Helmholtz equation is obtained from Maxwell’s equation:

(7)$$\nabla\times\nabla\times\boldsymbol{E}(\boldsymbol{r})-k^2\boldsymbol{E}(\boldsymbol{r})=4\pi k^2\boldsymbol{P}(\boldsymbol{r}),$$

where $k$ is the wave number. $\boldsymbol{E}(\boldsymbol{r})$ and $\boldsymbol{P}(\boldsymbol{r})$ are the electric field and the polarization at $\boldsymbol{r}$, respectively. To solve this differential equation, the dyadic Green’s function must generally be found; this function can be written as

(8)$$(\nabla\times\nabla\times{-}k^2)\boldsymbol{G}(\boldsymbol{r},\boldsymbol{r}^{\prime})=4\pi k^2 \boldsymbol{I}\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime}).$$

Green’s function can be obtained as

(9)$$\boldsymbol{G}(\boldsymbol{r},\boldsymbol{r^{\prime}})=(k^2\boldsymbol{I}+\nabla\nabla)\frac{\exp(ik|\boldsymbol{r}-\boldsymbol{r^{\prime}}|)}{|\boldsymbol{r}-\boldsymbol{r^{\prime}}|}.$$

Using Green’s function, Eq. (7) can be rewritten as

(10)$$\boldsymbol{E}(\boldsymbol{r})=\boldsymbol{E}_{0}(\boldsymbol{r})+\int d\boldsymbol{r^{\prime}}\boldsymbol{G}(\boldsymbol{r},\boldsymbol{r^{\prime}})\boldsymbol{P}(\boldsymbol{r^{\prime}}),$$

where $\boldsymbol{E}_{0}$ is the incident electric field.

Dividing the entire space into small cubic cells with volume $V$ and regarding the inside of the cells as homogeneous, Eq. (10) can be written as

(11)$$\boldsymbol{E}(\boldsymbol{r}_{i})=\boldsymbol{E}_{0}(\boldsymbol{r}_{i})+\sum_{j}V\boldsymbol{G}(\boldsymbol{r}_{i},\boldsymbol{r}_{j})\boldsymbol{P}(\boldsymbol{r}_{j}),$$

where $i$ and $j$ represent the cell numbers in the DDA calculation.

Appendix B: tip diameter dependency of the photoinduced force

In Fig. 8, we present the dependence of the tip diameter on the magnitude of the force, maintaining the distance between the tip surface and the NP surface as $d=$ 1 nm. If the tip diameter is 50 nm, the force is observed to be tens of times greater than that when the diameter is 19 nm.

 

Fig. 8. Dependence of the tip diameter on the magnitude of the photoinduced force. The red and green markers represent the force on nanoparticles NP1 and NP2, respectively. The black markers represent the force just above the gold substrate.

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Appendix C: PiFM resolution in this model

The resolution of the imaging technique is the minimum distance, $s$, by which two features are separated in the image. To determine the resolution in this model, we calculate the PiFM images and their line profiles when $s$ is varied. In the PiFM images shown in Fig. 9(a) - (c), the two nanoparticles are separated by distances of $s=$ 7.07, 9.90, and 12.73 nm (or 5, 7, and 9 cells), respectively. These distances are obtained the diagonal length of the cells in DDA. The line profiles shown in Fig. 9(d) - (e), corresponding to (a) - (c), respectively, reveal that two nanoparticles can be observed in isolation at $s=$ 9.90 nm. As explained in subsection 3.2, as the full width at half maximum of each nanoparticle is approximately 10 nm, the distance, $s$, at which the overlaps are resolved should be the resolution.

 

Fig. 9. PiFM images and their line profiles with laser energy $\hbar \omega =1.990$ eV. (a) - (c) present the PiFM images of the two nanoparticles where distance $s=$ 7.07, 9.90, and 12.73 nm, respectively. The white squares in the PiFM images represent the positions of the nanoparticles. (d) - (f) represent the line profiles of (a) - (c), respectively.

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Funding

Japan Society for the Promotion of Science (JP16H06504).

Acknowledgments

This work was supported in part by JSPS KAKENHI, Grant Number JP16H06504, for Scientific Research on Innovative Areas "Nano-Material Optical-Manipulation".

Disclosures

The authors declare no conflicts of interest.

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