Resonant excitation and manipulation of complex interactions among two or more resonances in high-index dielectric nanostructures provide great opportunities for engineering novel optical phenomena and applications. However, difficulties often arise when interpreting the observed spectra because of the overlap of the broad resonances contributed by many factors such as particle size, shape, and background index. Therefore, selective excitation of resonances that spectrally overlap with each other provides a gateway towards an improved understanding of the complex interactions. Here, we demonstrate selective excitation and enhancement of multipolar resonances of silicon nanospheres using cylindrical vector beams (CVBs) with different diameters of nanospheres and numerical apertures (NAs) of the excitations. By combining single particle spectroscopy and electrodynamic simulations, we show that the radially polarized beam can selectively excite the electric multipoles, whereas the azimuthally polarized beam can selectively excite the magnetic multipoles even though multipolar resonances are convoluted together due to their spectral overlap. Moreover, focusing the CVBs with high NA can lead to a dominant longitudinal polarization of the electric or magnetic field. We show that the enhanced longitudinal polarization with increasing NA of the radially and azimuthally polarized beams can selectively enhance the electric and magnetic multipolar resonances, respectively. Our approach can be used as a spectroscopy tool to enhance and identify multipolar resonances leading to a better understanding of light-matter interactions in other dielectric nanostructures as well as serve as a first step toward excitation of dark mode and Fano resonances in dielectric oligomers by breaking the symmetry of the nanostructures.
Resonant optical excitation of dielectric nanoparticles offers unique opportunities for future optical and nanophotonic devices because of their reduced dissipative losses and large enhancement of both the electric and magnetic near fields.1 In principle, dielectric nanostructures can exhibit many of the same features as plasmonic nanostructures, including enhanced scattering, high-frequency magnetism, and negative refractive index, with superior performance in comparison to their lossy plasmonic counterparts.1 As a result, there is a growing interest in understanding the optical resonances of high-index dielectric nanostructures that can facilitate enhanced light-matter interaction beyond the diffraction limit for future optical and nanophotonic devices.
Although the Mie theory predicted the resonant behavior of high-index subwavelength particles over a century ago,2 the possibility of enhanced optical magnetic response of high-index nanoparticles in the visible frequency range was first discussed theoretically in 2010.3 Subsequently, the scattering properties of high-index nanoparticles, such as silicon (Si), germanium (Ge), etc., have been studied in detail that include experimental realization of the concept of “magnetic light” in the visible frequency range.4,5 In high-index nanoparticles, the Mie resonance spectra can exhibit both the electric and magnetic resonances of comparable strengths leading to a number of novel electromagnetic phenomena, such as the Kerker-type directional scattering,6 magnetic hotspot enhancing the magnetic transition rates of molecules or quantum emitters (Purcell effect),7,8 surface enhanced Raman scattering,9 electrodynamic anapole excitations,10–12 nanoantenna,13,14 etc. Furthermore, resonant behavior of high-index nanoparticles can enable the realization of low-loss nonplasmonic metamaterials and metasurfaces15,16 with rich optical functionalities17,18 as well as nonlinear19–21 light manipulation.
On a fundamental level, it has been shown that the multipolar resonances of dielectric nanostructures scale with the refractive index n of the nanostructures;1 for n > 2, all main multipoles [magnetic dipole (MD), electric dipole (ED), magnetic quadrupole (MQ), and electric quadrupole (EQ)] are visible in the scattering spectra with the scattering efficiency increasing with increasing n.3,5,22 The spectral positions of the resonances depend on the ratio of the wavelength inside the particle (λ/n) to its diameter, d. Multipolar resonances can be achieved not only for spheres but also for spheroids,6,23 disks and cylinders,24 rings,25 and many other geometries.26
Since multipolar resonances excited in nanoparticles are dependent on their size, shape, refractive index, etc., the optical spectra of high-index nanostructures can be quite complex,3–6,22–26 especially if the resonances are broad and at the same time overlap with each other. Therefore, the interpretation of the optical spectra can be quite challenging! In general, identification of the multipolar resonances and understanding the optical resonances of high-index dielectric nanostructures rely on comparing the experimental results with the numerical and analytical calculations. To this end, it would be desirable to have access to a spectroscopy tool that can help identify the multipolar resonances by selective excitation of multipolar resonances leading to a better understanding of the optical responses of the dielectric nanostructures.
In this article, we use focused cylindrical vector beams (CVBs) to selectively excite and enhance multipolar resonances in single Si nanospheres with different diameters and numerical apertures (NAs) of the excitations. CVBs are solutions of Maxwell's equations that possess spatially varying polarization with cylindrical symmetry in both amplitude and phase.27 The CVBs can have the electric field aligned in the radial [radially polarized (RP) beam, Fig. 1(a)] or azimuthal [azimuthally polarized (AP) beam, Fig. 1(b)] directions, while the magnetic field is aligned in the azimuthal or radial directions with respect to the optical axis.28 Such beams have been previously used to tailor multipolar electric and magnetic resonances,11,29–31 excite dark mode and Fano resonances32–37 in metallic and dielectric nanostructures, and perform optical imaging38,39 and optical trapping40 of plasmonic particles. Here, we perform systematic experimental measurement and electrodynamic simulation of the excitation of single Si nanospheres with different diameters and numerical apertures of the CVBs. Our results show that the radially polarized beam can selectively excite the electric multipoles (ED and EQ), whereas the azimuthally polarized beam can selectively excite the magnetic multipoles (MD and MQ) even though the multipolar resonances are convoluted together because of their spectral overlap under linear beam excitation. Moreover, we also show that one can selectively enhance the electric or magnetic multipolar resonances by increasing the NA of the radially and azimuthally polarized beams, respectively.
In addition to being a spectroscopy tool that can selectively enhance and excite multipolar resonances in high-index dielectric nanostructures, our results will pave the way for engineering excitation of dark mode and Fano resonances32–37,41,42 in symmetric dielectric oligomers by breaking the symmetry of the nanostructures.
II. EXPERIMENTAL AND THEORETICAL METHODS
We illuminated a colloidally synthesized single Si nanosphere with focused radially and azimuthally polarized beams as schematically shown in Figs. 1(a) and 1(b), respectively. The top panels in Figs. 1(a) and 1(b) show the calculated electric field distributions of a single Si nanosphere under excitation with radial and azimuthal beams, respectively. For the azimuthal beam, the rotating electric field creates an induced circulating displacement current inside the nanosphere giving rise to magnetic resonances according to Maxwell-Faraday's law of induction .
The electric field distribution of CVBs can assume a doughnut shape with null intensity at the center, which can be expressed as a superposition of orthogonally polarized Hermite–Gauss (HG) and modes, given by28,43,44
Here, (x, y, z) is related to the fundamental mode E according to
where n and m denote the order and degree of the beam, and w0 denotes the radius of the beam waist.
Moreover, focusing the CVBs with a very high NA can cause spatial separation of the electric and magnetic fields at the focus leading to a finite longitudinal (z)-component of the magnetic field (Hz) (for the azimuthal beam) or electric field (Ez) (for the radial beam).28,43,44 For excitation of nanospheres with CVBs, due to the cylindrical symmetry around the optical axis of both the focal fields of the CVBs and the nanosphere, only Hz and Ez contribute to the far-field scattering.11 As a result, since the azimuthally polarized beam exhibits finite Hz, but no Ez along the beam axis, the excitation of a nanosphere using the azimuthally polarized beam excites the magnetic modes only. On the contrary, for radial beam excitation, only the electric modes are excited inside the nanosphere.
For our experiments, we used a liquid-crystal based polarization converter (ARCoptix) to generate azimuthally and radially polarized CVBs.45 For this purpose, the output of a spatially coherent broadband (λ = 420–800 nm) continuum (Leukos SMPH 4.0) was sent though the polarization converter as schematically shown in Fig. 1(c). The azimuthally polarized beam coming out of the polarization converter was then coupled to an inverted optical microscope equipped with a 100× oil immersion objective with NA ≤ 1.4. The back-reflected scattering spectra of single Si nanospheres were recorded by a CCD connected to an imaging spectrometer coupled to the side port of the microscope via a home-built achromatic 4f relay system [Fig. 1(c)]. Figure 1(d) shows the images of the broadband radially (upper panel) and azimuthally (lower panel) polarized beams that were generated experimentally. The linear orthogonal components associated with the HG01 and HG10 modes were detected by inserting an analyzer in front of the detector at different angles as shown by white arrows in Fig. 1(d). The intensity distribution of the radial and azimuthal beams and decomposition of the linear orthogonal components confirmed the generation of vector beams in our setup. Note that to obtain a fully azimuthal or radial beam, there is a phase compensator in the vector beam generator, which permits compensation of the λ/2 phase step between the upper and lower halves of the polarization converter (theta-cell).45 The phase compensation depends on the voltage applied to the phase compensator, which depends on the wavelength. To account for this, we calibrated the relation between voltage and the corresponding wavelength range, in which a high-quality vector beam, i.e., a nice doughnut shape, is observed. In postprocessing data analysis, we “stitched” these spectra according to the voltage-wavelength relation using a procedure that we described elsewhere.30
For our experiments, the Si nanospheres were synthesized colloidally from Si-rich borophosphosilicate glass (BPSG) as reported previously.46 The solutions of Si nanospheres were drop-cast and dried on a glass substrate before measurement. Numerical calculations were performed by finite-difference time-domain (FDTD) electrodynamic simulations using the Lumerical “FDTD Solutions” software package.47 The single dielectric nanospheres were illuminated by a linearly or radially or azimuthally polarized light for the wavelength range of λ = 420–800 nm. A nonuniform mesh with a maximum grid size of 2 nm was used to calculate the spectra. The dielectric function of Si was modeled with the Palik data,48 and the background index (nb) was set to that of air (nb = 1) or oil (nb = 1.48).
III. RESULTS AND DISCUSSION
Scattering spectra of Si nanospheres under linear beam excitation: The experimentally measured and the corresponding calculated scattering spectra of single Si nanospheres with different diameters under linear beam excitation are shown in Fig. 2. The scattering spectra were measured and calculated for single Si nanospheres with three different diameters, d = 125, 170, and 180 nm for two different background indices. Initially, the spectra were measured for single Si nanospheres in air (nb = 1), situated on a glass substrate. However, the glass−air interface may introduce additional scattering and can influence the backscattering intensity.30,31 To overcome the effect of the glass–air interface, we used an oil medium (nb = 1.48) to nearly index match the oil−substrate interface and reduce unwanted reflections in the experiment. The diameter of the single nanospheres was determined by the correlated Scanning Electron Microscope (SEM) images; the insets in Figs. 2(a)–(c) show the correlated SEM images of the exact same nanospheres whose scattering spectra were measured. The details of the identification and characterization of an Si nanosphere at the single nanoparticle level is described in the Sec. I in the supplementary material.
The experimentally measured scattering spectrum of a single Si nanosphere with d = 125 nm and nb = 1 shows two distinct peaks at wavelengths, λ ∼ 465 and 532 nm [Fig. 2(a)]. From knowledge gained from a previous study,46 we can assign the 532 nm peak to be due to the excitation of MD and the 465 nm peak to be the ED. For nb = 1.48, the two peaks completely overlap with each other and become a broad single peak [Fig. 2(a)]. For d = 170 and 180 nm, multiple distinct peaks appear in the scattering spectra [Figs. 2(b) and 2(c)]. Interestingly, for nb = 1 and 1.48, the scattering spectra appear to be very different from each other. At this point, interpreting the experimental spectra becomes extremely challenging! To analyze these results and identify the nature of the resonances, we have performed FDTD calculations and calculated the total scattering spectra, and subsequently performed multipolar decomposition. The calculated total scattering spectra are shown in Figs. 2(d)–2(f) for d = 125, 170, and 180 nm. The calculated scattering spectra for d = 150 nm for nb = 1 and 1.48 are shown in Sec. II in the supplementary material. The calculated total scattering spectra qualitatively agree with the experimentally measured spectra except that the relative intensities of some of the peaks are different. We attribute this discrepancy to the different geometries used in the experiment and calculation and the influence of the glass substrate. Note that in the experiments, the spectra were measured in the backscattered geometry for a specific angular range determined by the NA of the microscope objective (NA ∼ 1.2), whereas the scattering spectra were calculated for the entire hemisphere. Moreover, the simulations did not take into account the presence of the glass substrate, which can influence the relative intensities of the various modes in a nanosphere.4
Multipolar decomposition of the total scattering spectra: To identify the nature of the peaks, we have performed multipolar decomposition of the total scattered spectra by numerically computing multipole moments from the internal electric field distributions at each frequency using a three-dimensional FDTD method.47,49 After computing the local field distributions inside the nanosphere, we can obtain the induced current distribution at each frequency using the following relation:50
where J and E are the induced current density and electric field at internal points r = (x, y, z) in a Cartesian coordinate system, ω is the angular frequency of the excitation, ε0 is the dielectric permittivity of free space, and n is the complex refractive index of the material in the nanostructure. Once the current distribution in the nanosphere is obtained, we can compute the first four multipole moments using the expressions summarized in Table I,51 where p, m, Qαβ, and Mαβ indicate the electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ) moments, respectively. The quadrupoles are expressed as tensorial components, where indices α and β indicate the Cartesian axis, x, y, or z (e.g., has nine components). Based on the knowledge of radiating multipole moments, we can calculate the corresponding far-field scattered power contributions for each multipole moment and their interactions as summarized by the expressions listed in Table I.52
|Type .||Multipole expression .||Far-field scattering power .|
|Type .||Multipole expression .||Far-field scattering power .|
The results of the multipolar decomposition as depicted in Fig. 3 show that for nanospheres with the diameter d = 125 nm [Figs. 3(a) and 3(d)], only the ED and MD modes are excited, whereas for d = 170 nm [Figs. 3(b) and 3(e)] and 180 nm [Figs. 3(c) and 3(f)], the ED, MD, EQ, and MQ modes are excited. The result of the multipolar decomposition for d = 150 nm and nb = 1.48 is shown in Fig. S2 in Sec. II of the supplementary material. The distributions of the calculated local fields associated with the ED, MD, EQ, and MQ modes for d = 180 nm and nb = 1.48 are shown in Figs. 3(g)–3(j), respectively. The results also show that for nb = 1.48, the electric modes (both dipole and quadrupole) are redshifted as well as broadened, whereas the magnetic modes (both dipole and quadrupole) are only broadened. These multipolar contributions spectrally overlap with each other resulting in complex total scattering spectra as measured experimentally (Fig. 2) depending on the size and background index of the nanospheres.
Selective excitation of multipolar resonances of single Si nanospheres under vector beam excitation: From the above discussion, it is evident that because of the overlap of the broad multipolar resonances in the scattering spectra, the identification of the resonance peak positions can be difficult. Therefore, the interpretation of the experimental results (Fig. 2) completely relies on multipolar decomposition as shown in Fig. 3. Next, we investigate if we could selectively excite the electric and magnetic multipolar resonances with radial and azimuthal beam excitations, respectively. Figure 4 shows the experimental [Figs. 4(a)–4(c)] and the corresponding calculated [Figs. 4(d)–4(f)] results of the excitation of single Si nanospheres with the diameter d = 125, 170, and 180 nm under linear, radial, and azimuthal beam excitations, respectively, for nb = 1.48. The calculated results for d = 150 nm under linear, radial, and azimuthal beam excitations are shown in Fig. S2 in Sec. II in the supplementary material. The results show that for radial and azimuthal beam excitations, different peaks are selectively excited. Comparing these results with the multipolar decomposition of Fig. 3, we can conclude that for radial beam excitation, only the electric modes (both ED and EQ) are excited, whereas for azimuthal beam excitation, only the magnetic modes (MD and MQ) are excited. The selective excitation of electric (magnetic) modes by a focused radially (azimuthally) polarized beam can be explained by the multipole expansion of the local field,53
where and are the vector spherical harmonics associated with the electric and magnetic modes, and and are the strengths of the electric and magnetic components, respectively. For a radially polarized focused beam, , that is, all the magnetic multipole components are zero, whereas for an azimuthally polarized focused beam, , that, is, all the electric multipole components are zero.
Note that the selective excitation of the electric and magnetic modes occurs when the nanosphere is placed at the focal plane of the CVBs; away from the focus, the excitation symmetry is broken. As a result, CVBs have nonzero values of both the electric and magnetic fields away from the focus, resulting in the excitation of both electric and magnetic modes in the nanospheres. The experimental spectra in Figs. 4(a)–4(c) were obtained by scanning the single nanospheres in the x, y, and z directions and placing the nanospheres at the focal plane to make sure the excitation symmetry is preserved for the selective excitation of the electric and magnetic modes.
In light of these results, we briefly discuss how such a scheme could be implemented to excite dark modes and Fano resonances in dielectric oligomers. Dark modes are described by out-of-phase coupling of the dipole moments that can lead to reduced radiative damping resulting in a stronger enhancement of the electromagnetic fields in the near-field region and high-quality factors.34,35 Moreover, interference between spectrally overlapping narrow dark modes and broad bright modes can also lead to Fano resonances.36 Excitation of dark modes usually relies on breaking the symmetry of the nanostructures; however, it has been shown that CVB illumination could be used to efficiently excite dark modes and Fano Resonance in metal nanoparticle oligomers by selective excitation of electric and magnetic modes.33,37,41 Having demonstrated that we could selectively excite multipolar resonances in single dielectric nanospheres, our approach could easily be extended to excite dark modes and Fano resonances in dielectric oligomer structures.
Enhancement of the multipolar resonances of single Si nanospheres with increased NA: It has been shown that focusing CVBs with a microscope objective with a finite numerical aperture in the nonparaxial limit causes spatial separation of the electric and magnetic fields at the focus. This can lead to dominant longitudinal polarization of the electric or magnetic field.28,43 The upper panel in Fig. 5(a) shows the calculated x, y, and z components of the electric field for the radial beam with NA = 0.4. Under very high-NA focusing, the longitudinal component of the magnetic (electric) field can exceed the transverse component of the magnetic (electric) field in the focal region whereas the electric (magnetic) field component is purely transverse for azimuthal (radial) polarized beams. This is shown in the bottom panel of Fig. 5(a) by plotting |Hz|/|Hρ| and |Ez|/|Eρ| vs NA for azimuthal and radial beams, respectively, where and . Since the Ez and Hz components are responsible for excitation of the multipolar resonances under CVB excitation, we should be able to control the strength of these modes by varying the NA of the excitation. Figure 5 shows the experimental and the corresponding calculated scattering spectra of the Si nanosphere with d = 170 nm for azimuthal [Fig. 5(b)] and radial beam [Fig. 5(c)] excitations. As expected, with increasing NA, the strengths of the multipolar resonances increase because of the enhancement of Ez and Hz components for radial and azimuthal beams, respectively.
In conclusion, resonant excitation of high-index dielectric nanostructures can produce resonance peaks related to all main multipoles in the scattering spectra with a dependence on their size, background refractive index, etc. As a result, understanding the optical responses of high-index dielectrics can be challenging, especially if the resonances are broad and, at the same time, overlap with each other. Here, we have taken advantage of field distributions of the azimuthally and radially polarized cylindrical vector beams to selectively excite magnetic and electric multipolar resonances, respectively, of single high-index dielectric nanospheres. More specifically, in combination with numerical calculations, we showed that our approach could be used to resolve spectrally overlapped broad multipolar resonances spectra in silicon nanospheres. Furthermore, we also showed that one could selectively enhance the electric and magnetic multipolar resonances in a silicon nanosphere by increasing the numerical aperture of the radially and azimuthally polarized beams, respectively. Hence, our approach can be used as a spectroscopy tool in identifying multipolar resonances in complex spectra leading to a better understanding of the optical responses of dielectric nanostructures that offer unique opportunities for future optical and nanophotonic devices. As a next step, our approach could be extended to excite dark modes and Fano resonances in symmetric dielectric oligomers; we are currently working on demonstrating this possibility.
See the supplementary material for details on the identification and characterization of single Si nanospheres at the single nanoparticle level, and simulation results showing scattering spectra, multipolar decomposition, and selective excitation of multipolar resonances for a nanosphere diameter of 150 nm.
This material is based on the work supported by the National Science Foundation (NSF) under Grant No. ECCS-1809410. H.S. and M.F. acknowledge KAKENHI Grant Nos. 18K14092 and 18KK0141.