Radiation of Dynamic Toroidal Moments.

Dynamic toroidal dipoles, a distinguished class of fundamental electromagnetic sources, receive increasing interest and participate in fascinating electrodynamic phenomena and sensing applications. As described in the literature, the radiative nature of dynamic toroidal dipoles is sometimes confounded, intermixing with static toroidal dipoles and plasmonic dark modes. Here, we elucidate this issue and provide proof-of-principle experiments exclusively on the radiation behavior of dynamic toroidal moments. Optical toroidal modes in plasmonic heptamer nanocavities are analyzed by electron energy loss spectroscopy and energy-filtered transmission electron microscopy supported by finite-difference time-domain numerical calculations. Additionally, their corresponding radiation behaviors are experimentally investigated by means of cathodoluminescence. The observed contrasting behaviors of a single dynamic toroidal dipole mode and an antiparallel toroidal dipole pair mode are discussed and elucidated. Our findings further clarify the electromagnetic properties of dynamic toroidal dipoles and serve as important guidance for the use of toroidal dipole moments in future applications.

Elementary electromagnetic (EM) sources with well-defined near- and far-field properties form the fundamentals to systematically describe and understand observed electromagnetic phenomena of objects and their interactions. Light–matter interaction is often explained via the decomposition of complex excitations versus those elementary sources—the so-called EM moments. The concept of elementary EM moments is of great help to understand EM systems in an ordered and predictable manner. Linear electric and magnetic multipoles are such well-known electromagnetic elementary sources. However, there are obviously more classes of moments. As an electric dipole can be formed by two charges with opposite sign and a magnetic dipole by a current loop, a toroidal dipole is yielded by poloidal currents on the surface of a torus.[1] More generally, toroidal multipoles are considered as the third family of elementary EM sources due to their distinct charge–current configuration and unique parity properties as compared to their electric and magnetic counterparts.[1,2] Fascinating phenomena have been shown to have a close association with toroidal excitations, such as optical activity[3] and electromagnetic-induced transparency.[4] Furthermore, toroidal excitations show a feasible potential in applications as nanophotonic devices.[5−7]

There have been two sets of toroidal multipoles under consideration in the literature, the so-called electric and magnetic toroidal multipoles.[8] This article is restricted to magnetic toroidal dipoles unless otherwise stated. The discovery of toroidal multipoles has a historical development from the static to the dynamic regime. The prototype of static toroidal dipoles was proposed as an anapole in the field of nuclear physics to explain the parity violation during electromagnetic interaction.[9] It was explicated by virtue of a classical solenoid bent into a torus with a constant current flowing helically around the surface. Due to internally confined fields, such an anapole belongs to the family of nonradiating sources, which at rest (zero speed) does not emit fields to the far-field.[10] Later, the concept of toroidal dipoles was extended in the context of electrodynamics.[8] They became oscillating in time (dynamic) by substituting the constant current by an alternating current in the classical solenoid model.[8] It brought dynamic toroidal dipoles to the horizons of researchers as a third fundamental point-like source in the family of electromagnetic multipoles. In contrast to a static toroidal dipole, a dynamic toroidal dipole radiates, with a radiation pattern identical to that of an electric dipole.[8,11] It encouraged the creation of the dynamic version of a nonradiating anapole by exploiting the destructive interference between a dynamic toroidal dipole and an electric dipole.[11]

Although the theoretical understanding of toroidal moments became sound, the experimental observation of the dynamic toroidal dipole response was ambiguous because it was often masked by more dominant electric and magnetic moments. In 2010, the significant response of dynamic toroidal dipoles was demonstrated with the aid of metamaterials within the microwave frequency range.[12] Metamaterials have boosted the experimental explorations on the toroidal moments’ EM properties in a wide spectrum from the GHz to the optical range.[12−15] Varieties of metamaterials involve the use of plasmonic materials.[16−19]

For the sake of understanding the further conclusions of this study, it is important to discuss plasmonic dark modes. Indeed, plasmonic dark modes, which are well-known nonradiating charge and current configurations, share much resemblance with the toroidal moments. Originally, plasmonic dark modes gained the name for their poor coupling to the far-field plane-waves under specific excitation or detection schemes.[20] Dark modes usually possess a vanishing or zero net dipole moment,[21] like the antisymmetry mode in plasmonic dimers/trimers,[22] the breathing mode in discs/triangles,[23] and the quadrupolar modes.[24,25] However, dark modes can be switched to bright modes by tuning the excitation angle of the incident light[26] or using fast electrons.[27] In addition, retardation can also cause the dark mode to radiate, while increasing the effective size of sustained structures.[28] Therefore, the term dark mode cannot be quantitatively linked to nanoscale nonstatic charge configurations, as from the above-mentioned observations.

Toroidal moments have been addressed as dark modes or subradiant modes mainly by reference to highly radiative electric or magnetic dipolar modes in the context of constructing Fano resonances.[29−32] However, the name dark mode implies difficulties in coupling with light in both excitation and radiation processes. Certainly, for the excitation process, a normal optical planar wave can hardly excite toroidal moments in most cases due to their complex and confined charge–current configuration. Toroidal moments are then dark in this sense, especially in contrast to other optically excited electric or magnetic dipolar modes. On the other hand, dynamic toroidal dipoles are intrinsically radiative. Its scattering field can be as strong as that of an electric dipole.[33,34] In the literature, there exists confusion about the properties of dynamic toroidal dipoles, partially from the conception interchange with the static toroidal dipoles and merging with a plasmonic dark mode. It is urgent to clarify this issue, as the research interest in dynamic toroidal moments is apparently increasing, even in the aspect of practical applications.[5,7,35] It is worth noting that the radiation properties of dynamic toroidal dipoles have been considered in a number of experiments.[34,36] However, there are no exclusive experiments in demonstrating the radiation of a single dynamic toroidal moment.

The aim of this article is to explicitly discuss this issue and provide exclusive experimental evidence as a proof-of-principle on the radiation of dynamic toroidal dipole moments. We adapt the plasmonic heptamer nanocavity in a silver thin film to host toroidal excitations. This structure exhibits D6 symmetry and has been proven elsewhere to support a toroidal dipole on the central hole with a head-to-tail vortex configuration of magnetic dipoles in the outer 6 nanoholes (inset at the top left in Figure ).[16] The associated magnetic field presents a characteristic circular distribution in the plane of nanoholes, whereas the related electric fields loop around the magnetic fields out of plane from the outer holes to the central hole. Note that toroidal dipole moments in oligomers cannot be excited by normal incident light but by normal incident fast electrons. As the first step of our two-fold investigation approach, toroidal excitations in plasmonic heptamer nanocavities are experimentally inspected by electron energy loss spectroscopy in a scanning transmission electron microscope (STEM-EELS) and by energy-filtered transmission electron microscopy (EFTEM). Then the corresponding far-field radiation behavior of the toroidal excitations is investigated by means of cathodoluminescence (CL) spectroscopy. Incident electrons have an electromagnetic near-field, which polarizes plasmonic structures and excites plasmonic modes with the conservation of energy and momentum. For EELS and EFTEM, the scattering fields of the excited modes in turn recoil upon the incident electrons along the trajectory, leading to energy losses.[37] For CL, the excited modes may couple to the far-field radiation and then become detected. EELS, EFTEM, and CL have been widely applied in characterizing plasmonic systems on the micro- and nanometric scale.[38,39] EFTEM and EELS measurements are only sensitive to the associated electric field of toroidal excitations projected along the electron trajectory. To confirm the toroidal excitations in our case, the characteristic vortex-like magnetic fields are revealed in parallel by finite-difference time-domain (FDTD) simulations. Interpretations on the experimental results are further supported by numerical calculations.

Figure 2

Simulated magnetic (a) and electric (b) fields, H and E, along the electron trajectory of the heptamer structure at 2.1 eV without time delay and at 2.5 eV with π/4 time delay. Insets above are the corresponding schematic illustration. Gray circles denote the nanoholes. The impact locations of the electron probe are indicated by the white dots. Red and green arrows denote magnetic and toroidal dipoles, respectively. (c) ZLP-normalized EFTEM images of the heptamer nanocavity at energy losses of 2.2 ± 0.1 and 1.7 ± 0.1 eV. The black triangular areas at the upper and lower left corners are beyond the acquisition area of CCD camera. Scale bars are (a,b) 200 nm and (c) 100 nm.

Results and Discussion

We first employed near-field measurements with EELS to probe the resonances of the supported toroidal modes in the heptamer structure with a selective excitation scheme along its symmetry axis (Figure a). In EELS, incident electrons, here with 200 keV kinetic energy, interact with the sample, resulting in collective excitations and simultaneous loss of energy. An electron spectrometer is used to measure the amount of energy loss. In theory, EELS is interpreted as the probability of the electron to lose quanta of the photon energy and is directly related to the electric field component projected along the trajectory of the electron (here normal to the heptamer surface).[37,40]

Figure 1

(a) High-angle annular dark-field (HAADF) image and zero-loss peak (ZLP)-normalized EEL spectra recorded along the green line of a fabricated heptamer nanocavity. (b) Image of the simulated heptamer cavity (left) and the corresponding EEL spectra along the cavity axis (green line). Scale bars are 100 nm. Vertical dashed red and black lines indicate the toroidal modes T1 and T2 at (a) 2.12 and 2.51 eV and (b) 2.2 and 2.58 eV, respectively. (c) Experimental (left) and simulated (right) EEL spectra of the investigated plasmonic heptamer cavity along the symmetric axis from the central to upper holes as depicted by colored spots in the inset.

Our plasmonic heptamer cavity structure was patterned on a free-standing silver thin film with a focused ion beam system (for details, see Methods). As shown in Figure a, the fabricated heptamer structure has a hole diameter of 80 ± 10 nm, and the thickness of the silver thin film is around 30 ± 15 nm. The incident electron beam direction is perpendicular to the structure, that is, the thin film surface, and parallel to the z-axis. First, the single toroidal dipole moment is easily recognized in the energy loss spectra by its characteristic excitation location. According to its field distribution, as described before, this mode has the electric field highly concentrated at the central hole and the nearby silver bridges,[16] where exactly the corresponding energy loss signal should be present (Figure , vertical dashed red line, named T1 mode). The extracted experimental EEL spectra at the positions depicted by the colored spots between the central and upper holes are displayed in Figure c, left column. It shows that the T1 mode is excited at both the central hole and the neighboring silver bridges. The corresponding EELS signal at the central hole is relatively weaker than that at the neighboring silver bridges, which will be explained later. Meanwhile, it has a broad-band feature with a maximum centered at around 2.12 eV. On the other hand, there is another pronounced resonance at 2.58 eV with the energy concentrated just at the silver bridges (Figure , vertical dashed black line, named T2 mode). Contrary to the T1 mode, this mode is absent at the central hole but is excited at the silver bridges and the rims of the holes (Figure c). Certainly, other cavity modes (radially and longitudinally polarized along the holes) can also be excited in this structure as reported previously in refs (16) and (41). However, here, we only focus on the toroidal moments. The zigzag curve right above the T2 mode is a camera artifact due to the afterglow of the zero-loss peak (ZLP), which does not affect the data interpretation.

Figure b shows the corresponding simulated EEL spectra via the numerical FDTD calculations. The individual spectra extracted between the central and upper holes are shown in Figure c, right column, with the same color code as that for the experimental spectra. Two pronounced resonances are observed at 2.1 and 2.5 eV (vertical dashed red and black lines, respectively). They show good agreement with the experimental T1 and T2 modes despite a slight energy shift of around 0.1 eV. By displaying the electromagnetic field distributions at 2.1 eV, it confirms the excitation of a single toroidal dipole moment with the clockwise rotation of the magnetic dipoles in the outer six nanoholes (Figure a and the inset on the left) and high electric field concentration at the central hole and the nearby silver bridges (Figure b, left). Very interestingly, the resonance at 2.5 eV demonstrates an antiparallel pair of toroidal dipoles (Figure a, right). The instant magnetic field distribution at the time of P/8, where P is the temporal duration of the optical cycle, shows a clockwise dipole loop in the upper four nanoholes (including the central hole) as well as a counterclockwise dipole loop in the lower four nanoholes (including the central hole), indicating an antiparallel pair of toroidal dipoles perpendicular to the x–y plane (right inset above Figure a). These two toroidal dipoles appear one after another in the time domain with a time lag of P/4 (for details, see Figure S1 in the Supporting Information). As shown on the right of Figure b, the corresponding electric field distribution reveals that the energy of this mode is mainly concentrated between the holes (silver bridges) along the symmetry axis. From the point of view of the structural symmetry, the upper four holes and the lower four holes can be mirrored through a horizontal plane across the center of the structure. Meanwhile, each upper or lower substructure is able to form an individual toroidal moment at its silver bridge. The conclusion of the E field calculations at 2.1 and 2.5 eV is consistent with the experimental observations that, at the center of the heptamer structure, the T1 mode is exclusively excited, whereas the T2 mode is not. Their clear difference at the central hole is also unambiguously captured by the ZLP-normalized EFTEM images (Figure c). Distinguished features of the T2 mode are its higher excitation energy and the lack of the EELS signal at the central hole, as also expected from the previously demonstrated EELS line scan (Figure a,b). It can serve as a fingerprint to distinguish radiation signals from these two modes later in CL spectra.

With the above knowledge on the toroidal moments in the investigated plasmonic heptamer nanocavity, we first calculated the corresponding CL spectra on the same structure along the symmetry axis in order to correlate it with the simulated EELS results (Figure a). The qualitative correlation between the simulated CL and EEL spectra on the same structure can be applied later to interpret the experimental CL data. In the following, we consistently use vertical dashed red and black lines for the notation of the T1 and T2 modes at their free-space wavelengths, respectively. The main feature of the simulated CL spectra is a strong emission spanning from 500 to 650 nm present on the silver bridges, whereas such signal is weaker inside the central hole (Figure a).

Figure 3

(a) Image of the simulated heptamer cavity (left) and the corresponding CL spectra collected along the cavity axis (green line). (b) Simulated EELS (solid curves) and CL (dashed curves) probabilities at the central hole and the center of silver bridge between holes, as the red and black crosses indicated in (a). The blue arrow with the red edge displays the blue-shift of the T1 mode to 510 nm in the CL simulation (vertical solid red line). The blue arrow with black edge indicates the same quantity of blue-shift as the T1 mode has and is applied on the T2 mode to 419 nm in the CL simulation (vertical solid black line). Hollow red and black crosses point out the calculated CL probabilities for the T1 and T2 modes, respectively. (c) HAADF image (left) and experimental CL spectra recorded along the green line of a fabricated heptamer nanocavity. Vertical dashed red and black lines mark the corresponding near-field resonances of toroidal T1 and T2 modes at (a,b) 585 and 494 nm or (c) 595 and 480 nm, respectively. Scale bars are 100 nm.

We compare the simulated CL and EELS spectra at the positions of the central hole and the neighboring silver bridge, which are the representative excitation locations for T1 and T2 modes, respectively (red and black crosses in Figure a). According to the EEL spectrum (solid red curve in Figure b) interpretation, only the T1 mode is excited in the center of the hole. The corresponding T1 mode signature in the CL spectrum is blue-shifted to 510 nm with respect to its EELS maximum (blue arrow with red edge and the dotted red curve in Figure b). It is mainly ascribed to the different light collection geometry used in the experiment and simulations (see Figure S2 in the Supporting Information), rather than electromagnetic dissipation of excited modes.[42] In addition, the T1 mode displays an intensity in the silver bridges higher than that in the central hole in both the CL (Figure a,c) and the EEL (Figure a,b) spectra. This feature is likely attributed to a higher coupling efficiency of inducing polarized currents on the material, rather than in the void in order to form the toroidal moments.

On the other hand, at the silver bridge, both T1 and T2 modes can be excited (solid black curve in Figure b). However, only the peak of the T1 mode at 510 nm is observed (hollow red cross) in the simulated CL spectrum (dotted black curve). Assuming that the simulated CL signal of the T2 mode is subjected to the same amount of blue-shift as the T1 mode has, it is then expected to see a peak at around 419 nm, as indicated by the vertical solid black line. However, no sharp peak but rather a very shallow rise is observed (dotted black curve). In fact, this shallow rise is also observed in the simulated CL spectrum at the center of the central hole (dotted red curve), at which the T2 mode is absent. Therefore, the shallow rise at 419 nm might be the signal of other cavity modes. Nevertheless, compared to the relative CL intensity of the T1 mode (hollow red cross), the CL probability at 419 nm is obviously lower (hollow black cross at the dotted black curve). This indicates a significant radiative behavior of the T1 mode (a single dynamic toroidal dipole) but only a weakly radiating behavior of the T2 mode (antiparallel toroidal dipole pair). The above observations via comprising the simulated EEL and CL spectra offer an important hint to interpret the corresponding experimental CL data.

Figure c shows the experimental CL spectra along the structural symmetry axis. Similar characters, such as a broad signal from 500 to 650 nm and an intensity at the silver bridges higher than that in the central hole, appear like in the simulated CL spectra. For further details, Figure a,b shows the simulated CL spectra together with the experimental spectra extracted from similar locations along the structural symmetry axis (color-coded locations in Figure c). Despite the similarity, the measured emission of the single toroidal T1 mode has almost no spectral shift with respect to its EELS resonance at 2.2 eV (vertical red dashed line at 564 nm in Figure b). This may imply a low damping rate of the single toroidal dipole mode in the investigated structure.[42] However, there are two small peaks appearing at 540 and 590 nm, highlighted by the inverse black and red triangles in Figure b. To further investigate the possible difference between them, we examined the corresponding spectral–spatial distribution at these three resonances (590, 564, and 540 nm) by displaying their chromatic CL maps (first row in Figure c). Interestingly, they all show the same distribution feature resembling to the electric field E of the T1 mode (Figure b, left). Therefore, this indicates the single excitation of the T1 mode and the broad-band emission feature of the single dynamic toroidal dipole moment. As a speculation, these small peaks may be the intensity variation resulting from the far-field interference between multiple plasmonic modes.

Figure 4

(a) Simulated and (b) experimental CL spectra extracted from the six color marked locations in Figure c. Each experimental spectrum is a sum over an area of 24 × 24 nm2. Smoothed curves are superimposed to the raw data (light gray). Vertical solid red line marks the simulated radiation peak of the T1 mode at 510 nm. Vertical red and black dashed lines correspond to the T1 and T2 modes at 595 and 480 nm, respectively. Vertical orange and green solid lines label the wavelengths of the emission at 400 nm and the silver bulk plasmon at 330 nm. Inverse black and red triangles highlight the emission peaks at 540 and 590 nm. (c) CL chromatic maps showing the spatio–spectral dispersion of the emissions at 590 ± 10, 564 ± 10, 540 ± 10, 480 ± 10, and 400 ± 30 nm. Scale bar is 100 nm.

Regarding the T2 mode, at its free-space wavelength of 480 nm, the experimental CL spectra do not show a pronounced peak, especially at the characteristic excitation positions of the T2 mode (brown and green curves in Figure b). The corresponding chromatic CL map (with the dashed black frame in Figure c) also confirms no characteristic spatial excitation of the T2 mode in relation to its electric field E, as shown in Figure b, right. This verifies the weakly radiating character of the T2 mode. As these two toroidal dipoles excited at the T2 mode are not exactly out of phase (i.e., a phase shift of π), their far-field interference should not be completely destructive. Therefore, the radiation of the T2 modes is, in principle, expected. However, another important factor has to be taken into account in this case, which is retardation. Retardation causes radiation when the size of the structure becomes larger than the resonance wavelength.[28] In our case, the effective size of the single toroidal dipole at the T2 mode is roughly the radius of the entire heptamer structure (∼175 nm), which is far smaller than the wavelength of the T2 mode (480 nm). Therefore, the radiation of the T2 mode was almost not observed, but stronger radiation is anticipated by increasing the size of the heptamer structure.

Furthermore, there is an evident peak at 400 nm (vertical orange line in Figure b). The corresponding chromatic CL map also is present in (c) with an orange frame. It clearly shows an excitation of a cavity mode, neither the T1 nor the T2 mode. The corresponding simulations of field distribution reveal that it is a radial electric dipole mode (see Figure S3 in the Supporting Information).

Conclusions

We have adapted the dynamic toroidal excitations in plasmonic heptamer nanocavities to exclusively investigate their radiation behaviors by means of EELS/EFTEM, CL, and FDTD numerical calculations. Not only the single dynamic toroidal moment but also an antiparallel toroidal dipole pair mode was observed having different radiation behaviors. Without a significant influence of retardation, the single dynamic toroidal dipole mode presented pronounced far-field CL signals, which unambiguously exhibits its inherent radiating character. In contrast, the antiparallel toroidal dipole pair mode showed much weaker radiation due to less radiation decay channels but strong absorption within the near-field region. Stronger radiation is anticipated if retardation plays a role. As the radiation decay scales exponentially with the propagation length of the fictitious photons, we anticipate that the radiation should become prominent by increasing the size of the structure. The above results serve as proof-of-principle experimental evidence to strengthen the radiative nature of electrodynamic toroidal dipole moments and to clarify the certain confusion of dark modes within the fields of metamaterials and plasmonics.

Methods

Sample Fabrication

The fabrication of plasmonic nanocavities in free-standing silver thin films involved the preparation of free-standing silver thin films and nanostructuring with a focused Ga+ beam, as described in a previous publication.[43] Silver discs with 3 mm diameter were electro-polished to have a hole in the center. The rim of the holes in silver discs was further thinned by ion milling. At the rim area, where the thickness was below 100 nm, heptamer cavity structures were then patterned with the focused Ga+ beam.

EFTEM and EELS Measurements

EFTEM and EELS measurements were performed with the subelectron volt subangstrom microscope (SESAM, Zeiss, Oberkochen, Germany) at an acceleration voltage of 200 kV. Energy-filtered images were acquired at energy loss steps of 0.2 eV between 0 and 5 eV. The full collection angle was 13 mrad. For the STEM-EELS measurements, the full convergence and collection angles were 14 and 13 mrad, respectively. The acquisition time for each EEL spectrum was 0.2 s. The local thickness was determined by the log–ratio method. Details can be referred to in ref (43).

CL Measurements

The CL measurements were conducted on a VG STEM machine operated at 100 kV with a probe current of around 1 nA. Data were acquired in the spectral imaging mode, in which the electron beam is raster-scanned over the region of interest, and the corresponding CL spectrum was collected simultaneously. The pixel size was 8 × 8 nm2, and the acquisition time per pixel was 50 ms.

Data Processing

To correct the acquired EEL spectra, the contribution of the zero-loss peak was subtracted by fitting a power-law function to the peak tail. Each ZLP-subtracted EEL spectrum was normalized to the maximum intensity of its own zero-loss peak.

For ZLP-normalized EFTEM images, each energy-filtered image was divided by the ZLP image averaged over −0.3 to 1.7 eV.

To process the acquired CL spectra, the first step was spectrum calibration and removal of spikes. Principle component analysis was then applied to reduce the noise of the spectra. The analysis was performed with the open source Python library HyperSpy V0.8.1.[44] The spectral response of the CL spectrometer at different wavelengths was sequentially corrected. Finally, background noise was subtracted from the spectra by taking the reference from the vacuum area.

FDTD Numerical Simulations

A charge broadening scheme, as described elsewhere, was introduced to mimic electron probes.[45] The whole simulation domain has been discretized by unit cells of 1.5 nm edge lengths. The permittivity of the silver is modeled by a Drude model in addition to two critical point functions. EEL spectra were calculated using the Fourier transformed electric field projected along the electron trajectory.[37]