100
K. Imaeda and K. Imura
be seen that spectral complementarity is not feasible for wavelengths range shorter
than 620 nm, which can be attributed to the interband transition of gold.
As depicted in Fig. 6.4b, the spatial mode patterns excited in one-dimensional
mesostructures are systematically described by the Fabry–Pérot modes of a onedimensional cavity resonator [44]. By contrast, the spatial mode patterns excited in
two-dimensional mesoplates are much more complex because of the higher dimensionality [48–51]. So far, several research groups have reported high-resolution
imaging of the spatial patterns of plasmon modes excited in metal mesoplates using
scanning transmission electron microscopy (STEM) combined with electron energy
loss spectroscopy (EELS) [52–59]. Although these results provide crucial information on the spatial features of plasmon modes, visualization by optical microscopy
is highly desirable for photophysical and photochemical applications. Recently, we
have successfully visualized the spatial patterns of plasmon modes induced in mesoplates of various shapes by adopting the near-field optical imaging method and
revealed that the spatial patterns observed in the near-field images can be comprehensively interpreted based on the eigenfunctions of a particle confined inside the mesoplates [60]. Figure 6.6a shows an SEM image of a triangular gold mesoplate (edge
length ~810 nm, thickness ~30 nm). We obtained the near-field spectral and spatial
characteristics of the mesoplate by performing near-field transmission measurements
using a halogen lamp as a light source. Figure 6.6b shows the near-field extinction
spectrum taken near the apex of the mesoplate. Several plasmon resonance peaks can
clearly be observed in the visible to near-infrared region of this spectrum. We obtained
near-field transmission images by mapping the transmitted light intensity at the resonances, as shown in Fig. 6.6d–f. The bright areas in these images represent the reduction of the transmitted light intensity, indicating high excitation probability of plasmons [25, 60]. As seen in these images, a variety of spatial patterns can be observed
inside the mesoplate, depending on the observation wavelengths. To elucidate the
physical origin of the observed spatial patterns, we calculated the eigenfunctions and
eigenenergies of the mesoplate by solving the Schrödinger equation for a particle
confined in a two-dimensional triangular potential well [61]. Figure 6.6g–i show the
square moduli of the calculated eigenfunctions (which are henceforth referred to as
eigenmodes). The bright areas of the eigenmodes correspond to regions where the
existence probability of a particle is high, that is, the bright spots of the calculated
eigenmodes correspond to those in the near-field images. As seen in Fig. 6.6d–i,
the spatial patterns of the calculated eigenmodes are evidently in good agreement
with those in the near-field transmission images. This agreement implies that the
spatial patterns in the near-field transmission images can be interpreted as plasmonic
standing waves confined inside the boundary of the mesoplate. We also found a
linear correlation between the observed plasmon resonance energies and the calculated eigenenergies, as shown in Fig. 6.6c. This means that the plasmon resonance
energy can be qualitatively evaluated by calculating the eigenenergies.
We also adopted the group theory to classify the irreducible representations of each
eigenfunction [61]. A triangular mesoplate on a glass substrate belongs to the C 3v
point group shown in Table 6.1. The eigenmodes in Fig. 6.6g, i exhibit the irreducible
representation E, that is, these modes have the same symmetrical characteristics as the
K. Imaeda and K. Imura
be seen that spectral complementarity is not feasible for wavelengths range shorter
than 620 nm, which can be attributed to the interband transition of gold.
As depicted in Fig. 6.4b, the spatial mode patterns excited in one-dimensional
mesostructures are systematically described by the Fabry–Pérot modes of a onedimensional cavity resonator [44]. By contrast, the spatial mode patterns excited in
two-dimensional mesoplates are much more complex because of the higher dimensionality [48–51]. So far, several research groups have reported high-resolution
imaging of the spatial patterns of plasmon modes excited in metal mesoplates using
scanning transmission electron microscopy (STEM) combined with electron energy
loss spectroscopy (EELS) [52–59]. Although these results provide crucial information on the spatial features of plasmon modes, visualization by optical microscopy
is highly desirable for photophysical and photochemical applications. Recently, we
have successfully visualized the spatial patterns of plasmon modes induced in mesoplates of various shapes by adopting the near-field optical imaging method and
revealed that the spatial patterns observed in the near-field images can be comprehensively interpreted based on the eigenfunctions of a particle confined inside the mesoplates [60]. Figure 6.6a shows an SEM image of a triangular gold mesoplate (edge
length ~810 nm, thickness ~30 nm). We obtained the near-field spectral and spatial
characteristics of the mesoplate by performing near-field transmission measurements
using a halogen lamp as a light source. Figure 6.6b shows the near-field extinction
spectrum taken near the apex of the mesoplate. Several plasmon resonance peaks can
clearly be observed in the visible to near-infrared region of this spectrum. We obtained
near-field transmission images by mapping the transmitted light intensity at the resonances, as shown in Fig. 6.6d–f. The bright areas in these images represent the reduction of the transmitted light intensity, indicating high excitation probability of plasmons [25, 60]. As seen in these images, a variety of spatial patterns can be observed
inside the mesoplate, depending on the observation wavelengths. To elucidate the
physical origin of the observed spatial patterns, we calculated the eigenfunctions and
eigenenergies of the mesoplate by solving the Schrödinger equation for a particle
confined in a two-dimensional triangular potential well [61]. Figure 6.6g–i show the
square moduli of the calculated eigenfunctions (which are henceforth referred to as
eigenmodes). The bright areas of the eigenmodes correspond to regions where the
existence probability of a particle is high, that is, the bright spots of the calculated
eigenmodes correspond to those in the near-field images. As seen in Fig. 6.6d–i,
the spatial patterns of the calculated eigenmodes are evidently in good agreement
with those in the near-field transmission images. This agreement implies that the
spatial patterns in the near-field transmission images can be interpreted as plasmonic
standing waves confined inside the boundary of the mesoplate. We also found a
linear correlation between the observed plasmon resonance energies and the calculated eigenenergies, as shown in Fig. 6.6c. This means that the plasmon resonance
energy can be qualitatively evaluated by calculating the eigenenergies.
We also adopted the group theory to classify the irreducible representations of each
eigenfunction [61]. A triangular mesoplate on a glass substrate belongs to the C 3v
point group shown in Table 6.1. The eigenmodes in Fig. 6.6g, i exhibit the irreducible
representation E, that is, these modes have the same symmetrical characteristics as the