Two possible setups are proposed here for in-plane coherent control of plasmon resonances through changing the symmetry of in-plane illumination. One is based on a fiber-waveguide interferometer (Fig. 1a), where completely symmetrical in-plane illumination is constructed by coupling two polarized coherent beams from the single-mode fiber-connected objective lens into the input waveguides without phase delay, whereas asymmetrical in-plane illumination can be realized by blocking one input beam (half-illumination) or introducing a phase delay between two beams. This setup can be widely used to study the in-plane coherent control of plasmon resonances in all kinds of plasmonic nanostructures with axial symmetry. The out-of-plane far-field signal carrying the LSPR information will be collected by an objective lens. However, this kind of setup still faces challenges in experiments, including accurate phase control required between two input fibers and high coupling efficiency required between fibers and waveguides. As a comparison, the widely used dark-field (DF) confocal microscopy is more convenient (Fig. 1b), where the polarized light is focused onto the sample through a condenser lens with an annular aperture and the backward scattering light is collected by an objective lens in a confocal setup. In this setup, completely symmetrical in-plane illumination can be easily satisfied once the input light is focused onto the center of the sample, whereas asymmetrical in-plane illumination (e.g., quarter illumination) can be constructed by blocking three-fourth of the area of the annular aperture. It is clear that such a DF illumination setup still contains out-of-plane wave vector components; thus, a relatively large numerical aperture (NA) of the condenser lens is required to reach the grazing incident condition. This setup is suitable for studying plasmonic nanostructures with sizes comparable to the focused spot size of the incident light beam.
Based on the fiber-waveguide interferometer, we theoretically studied the in-plane coherent control of plasmon resonances in gold nanodisks. Figure 2a, b shows the calculated absorption spectra of gold nanodisk monomers on SiO2/Si substrate with diameter ranging from 140 to 200 nm. For each gold nanodisk, the s-polarized plan wave was introduced from the right side (dashed line), generating one fundamental LSPR ("F") peak and one retardation-induced high-order plasmon resonance ("H") peak, both of which continuously red-shift with increasing diameters of gold nanodisks41. When symmetrical in-plane illumination (solid line) was applied without phase delay, the "F" mode was weakened and the "H" mode was apparently enhanced42. As a comparison, when a phase delay of π was applied in the symmetrical in-plane illumination, an opposite weakened/enhanced interference was observed for the two modes.
Fig. 2 In-plane coherent control of plasmon resonances in gold nanodisk monomers.a, b Calculated normalized absorption spectra of gold nanodisk monomers with a diameter ranging from 140 to 200 nm for s-polarized in-plane plan wave coming from the right side (dashed line) or both sides (solid line) without phase delay, or with a phase delay of π. "F" and "H" represent fundamental and high-order plasmon resonances. c-e The corresponding spatial distributions of electric-field amplitude |E|, real part Re(Ez), and imaginary part Im(Ez) for the "F" and "H" modes (square and circle signs) of the representative gold nanodisk monomer (D = 160 nm) under asymmetrical and symmetrical in-plane illumination. Under symmetrical in-plane illumination, we can observe phase delay-dependent destructive/constructive interference for the "F" and "H" modes
The weakened and enhanced phenomena are caused by the destructive and constructive plasmon resonances, which can be clearly demonstrated by the spatial distributions of electric-field amplitude for the gold nanodisk with a diameter of 160 nm (Fig. 2c and Supplementary Movie 1). The destructive and constructive interference can be further explained by the distributions of real and imaginary parts of electric-field component Ez for the "F" and "H" modes (Fig. 2d, e). More specifically, when s-polarized light comes only from the right side, the corresponding Re(Ez) and Im(Ez) distributions are identical ("ID") and opposite ("OP") for the "F" and "H" modes, respectively. Considering that the nanodisk is axisymmetric, when s-polarized light comes from the left side without phase delay, the corresponding Ez distribution will be mirrored with a negative sign along the central axis of the nanodisk (x = 0), i.e., Ez(− x) = − Ez(x). As a result, the "F" mode with "ID" Re(Ez) and Im(Ez) distribution caused by right-side illumination will cancel with its negative mirror caused by left-side illumination and result in destructive interference, whereas the "H" mode with "OP" Re(Ez) and Im(Ez) distribution will superpose with its negative mirror and lead to constructive interference. As a comparison, when s-polarized light comes from the left side with a phase delay of π, the result of using a mirror operator is Ez(− x) = Ez(x). As a result, under symmetrical illumination with a phase delay of π, the "F" and "H" modes will experience constructive and destructive interference, respectively. For the case of p-polarized light, the phase delay-dependent mirror operators are consistent with the s-polarized light under symmetrical illumination. However, when p-polarized light comes only from the right side, the corresponding Re(Ez) and Im(Ez) distributions are opposite ("OP") and identical ("ID") for the "F" and "H" modes, respectively. Therefore, the total phase delay-dependent destructive/constructive interference for the "F" and "H" modes are opposite to the case of s-polarization (Fig. S1 in Supplementary Information). The destructive/constructive interference of plasmon resonances was rechecked by the electromagnetic multipole theory, based on which the calculated absorption spectra of a 200nm gold nanodisk monomer under symmetrical illumination matched well with the simulation results (Fig. S2 and Supplementary Note 1 in Supplementary Information). We have also studied the in-plane coherent control of plasmon resonances in gold nanodisk dimers, which show opposite phase delay-dependent destructive/constructive interference for the "F" and "H" modes as compared with the monomers (Fig. S3 in Supplementary Information).
Table 1 summarizes the distribution rule for all three electrical-field components under in-plane symmetric illumination, where the rule of destructive/constructive interference is always identical for Ey and Ez, but opposite for Ex. We used the polarization-sensitive scattering-type scanning near-field optical microscopy (s-SNOM) to verify the electric-field distribution rule. The s-SNOM measurement was conducted by a focused laser (λ = 633 nm) coming from one side with an incidence angle of 30° with respect to the plane of the substrate, corresponding to an s-s/s-p geometry scheme (Fig. 3a). For s-s measurement, we recorded Es-s which is equal to Ey. In s-p measurement, we detected the total electric field Es-p, written by (Ex2 + Ez2)[1/2], where the calculated amplitude of Ez is much larger than Ex and thus the total electric field Es-p is roughly equal to Ez. As an example, s-s and s-p excitation-collection measurements were carried out on a 200 nm gold nanodisk monomer and dimer with a gap size of 30 nm. From Fig. 3b, we can find that the "H" mode at incidence angle 30° is located near the excitation wavelength, and both the monomer and dimer show constructive interference of "H" mode under symmetrical illumination without phase delay, which implies that both Re(Ey) and Re(Ez) should satisfy an "OP" distribution, according to Table 1. Figure 3d shows the measured and simulated amplitude, phase, and real part of Ey for the s-s excitation-collection measurement, whereas the corresponding results of the Ez component for s-p measurement are shown in Fig. 3e. Both measured and simulated results give strong evidence to demonstrate the "OP" distribution for both Re(Ey) and Re(Ez), where single and coupled quadrupole-like plasmon resonance can be observed for the monomer and dimer, respectively.
Phase delay s-/p-polarization Re/Im(Ex) Re/Im(Ey) Re/Im(Ez) 0 Destructive OP ID ID 0 Constructive ID OP OP π Destructive ID OP OP π Constructive OP ID ID
Table 1. Electric-field distribution rule for destructive and constructive plasmon resonances under symmetrical inplane illumination
Fig. 3 Demonstration of electrical-field distribution rule for the 200 nm gold nanodisk monomer and dimer by s-SNOM.a Schematic of the s-SNOM measurement for s-s and s-p excitation-collection configurations. The wavelength of the excitation laser is 633 nm and the incidence angle with respect to the plane of the substrate is 30°. b Calculated normalized absorption spectra of 200 nm gold nanodisk monomer and dimer at incidence angle 30° under asymmetrical (dashed line) or symmetrical (solid line) illumination without phase delay. The gap size in the dimer is 30 nm. c Atomic-force microscopic (AFM) images of gold nanodisk monomer and dimer for s-s and s-p measurements. The red arrow represents the incidence direction of the laser and the blue dashed line represents the central axis of the nanodisk. d, e Experimental and simulated spatial distributions of the amplitude |A|, phase ϕ, and real part of electric-field component Ey in s-s measurement and Ez in s-p measurement for 200 nm gold nanodisk monomer and dimer. The scale bar is 200 nm
One promising application based on the in-plane coherent control of plasmon resonances is plasmonic switching. Here we employed the DF confocal microscopy (Fig. 1b) to experimentally demonstrate the plasmonic switching effect in a gold nanodisk monomer and dimer. During the measurement, a condenser lens with a high NA (0.9) was used to provide sufficient in-plane illumination and an opaque tape was used to construct the quarter illumination. Both s- and p-polarization components of excitation were considered, and the collection was unpolarized. We experimentally observed the plasmonic switching effect in a 200 nm gold nanodisk monomer (Fig. 4a), i.e., constructive interference (area I) for the "H" plasmon resonance and destructive interference (area II) for the "F" plasmon resonance. Such a plasmonic switching phenomenon is clear and consistent for both y- and x-polarization under full and quarter illumination, and is in good agreement with the simulated scattering and absorption spectra (Fig. 4b). The polarization diagrams for full and quarter illumination (Fig. 4e) indicate that the y- and x-polarization of excitation are identical for the gold nanodisk monomer, resulting in an isotropic plasmonic switching effect. As a comparison, the plasmonic switching effect only occurs for the x-polarization of excitation in the gold nanodisk dimer, as shown in Fig. 4c, d. The anisotropic plasmonic switching effect in the gold nanodisk dimer is due to the superposition of anisotropic destructive and constructive plasmon resonances contributed by the in-plane wave vectors along the long and short axis, as shown in Fig. 4f.
Fig. 4 Demonstration of plasmonic switching by DF scattering measurement of gold nanodisk monomer and dimer.a Normalized DF scattering spectra of gold nanodisk monomer with a diameter of 200 nm (SEM image) under full and quarter illumination. b The corresponding normalized simulated scattering and absorption spectra. c, d Normalized measured and simulated DF scattering spectra of gold nanodisk dimer with a diameter of 200 nm and a gap size of 30 nm (SEM image) under full and quarter illumination. The red solid curves in Fig. 4c are the smoothing results. The scale bar in SEM images is 200 nm. e, f Polarization diagrams of full and quarter illumination in the DF scattering measurement and simulation for gold nanodisk monomer and dimer. In both experiment and simulation, the excitation is s-or p-polarized and the collection is unpolarized. The black and red double-headed arrows represent the initial polarization and the polarization after focusing, respectively
Another possible application based on the in-plane coherent control of plasmon resonances is plasmonic encoding. Here we propose a plasmonic encoding strategy in a series of gold nanodisk chains consisting of different numbers of 140 nm nanodisks with a separation distance of 30 nm. Figure 5a shows the absorption spectra of gold nanodisk chains under in-plane asymmetric (dashed line) or symmetrical (solid line) s-polarized illumination without phase delay. The "F" mode for gold nanodisk chains with number N from 2 to 6 always shows constructive interference under the symmetrical illumination, whereas the "H" mode shows alternate constructive and destructive interference from nanodisk monomer to dimer, trimer, tetramer, and pentamer. Figure 5b shows the spatial distributions of electric-field amplitude for the "F" mode under symmetrical illumination, where the gold nanodisk chains show quite different propagating coupling behavior compared with that under asymmetric illumination. More specifically, under asymmetric illumination, the plasmonic coupling and propagation are along the long axis of the gold nanodisk chain and decrease in intensity (Fig. S4 and Supplementary Note 2 in Supplementary Information)42. Under symmetrical illumination, the plasmonic coupling and propagation are no longer continuous. The nanodisk in the center of chains with odd numbers of N always shows completely destructive interference, whereas the two nanodisks in the center of chains with even numbers of N always show constructive interference (see Supplementary Movie 2). The spatially related destructive and constructive interference inside the gold nanodisk chains can be used for plasmonic encoding, as shown in the sliced electric-field amplitude distributions in Fig. 5c, where signals 0, 1−, and 1+ can be generated based on the relative intensity along the nanodisk chain's edge. The plasmonic encoding can be well explained by the superposition of Re(Ez) and Im(Ez) distributions under left- and right-side illumination, as shown in Fig. 5d-g, where the destructive and constructive interference always occurs when the nanodisk satisfies "OP" and "ID" distributions, respectively. Besides the number of gold nanodisk chains, the input phase delay can also affect the plasmonic encoding. The signals 0 and 1+ inside the gold nanodisk pentamer can be step-by-step shifted to the adjacent nanodisks by introducing an increasing phase delay from 0 to 3π/2, with an interval of π/2 (Fig. S5 and Supplementary Movie 3).
Fig. 5 Plasmonic encoding in gold nanodisk chains.a Calculated absorption spectra of gold nanodisk chains consisting of different numbers of nanodisks illuminated by the s-polarized in-plane plan wave coming from right side (dashed line) or both sides (solid line). The diameter of the nanodisk is 140 nm and the separation distance is 30 nm. The destructive and constructive plasmon resonances are represented by green and red colors, respectively. b Spatial distributions of electric-field amplitude |E| for the "F" plasmon resonances (peak position) under symmetrical illumination. c Sliced electric-field amplitude distributions along the chain's edge (the white dashed line in Fig. 5b). d-g Corresponding spatial distributions of real and imaginary part of Ez when the s-polarized in-plane plan wave comes from the left side (d, f) and right side (e, g) respectively