Plasmonic nanostructures, which attracted extensive studies in the past decades, enable unparalleled manipulation of electromagnetic waves at the nanoscale [1–3]. Metallic nanoparticles can exhibit strong modulation of visible light due to the large optical cross section induced in the far field and the strong local field enhancement in the near field by surface plasmons when resonantly excited. Such modulation relies on the resonance modes and couplings of local surface plasmons. Compared with dielectric nanoparticles whose multipole electric and magnetic resonances can be selectively excited and tuned, plasmonic resonances rely on the collective oscillation of free electrons, which mainly exhibit electric resonances and demonstrate stronger coupling with light. Local surface plasmon resonances (LSPRs) stimulated by sources without optical orbital angular momentum (OAM) have been intensively studied. Single nanoparticle plasmon excitations involve dipole and multipole electric plasmon mode [4,5], magnetic mode [6,7], standing waves of surface plasmon polariton [2,8,9], toroidal mode [10,11], as well as chiral plasmon mode [12,13]. Complex plasmon excitations by near-field couplings between adjacent plasmonic nanoparticles give rise to more interesting optical phenomena, which have been modeled as in-phase and out-of-phase couplings [14,15], plasmon hybridization [16–18], Fano resonance [19–23], electromagnetically induced transparency [24,25], and charge transfer plasmon [26,27].

In recent years, the rapid development of laser technology has enabled us to explore light–matter interactions with more degrees of freedom. Structured light with specially designed amplitude, distribution, duration, polarization, and phase expands the spectroscopies with more degrees of freedom [28–31]. In particular, the optical vortex includes a new physical dimension of OAM [32–52], besides the better-known spin angular momentum (SAM). While the SAM of light is associated with the helical electrical vectors or the photon spin, the OAM of light is related to the helical phase structure of the beam or the photon phase distribution. Twisted photons with the phase structure eilϕ carry an OAM of lℏ per photon, where l is an integer related to the azimuthal topological charge number, ϕ is the azimuthal angle, and ℏ is the Planck constant. The optical OAM dimension with complete and unbounded sets of eigenfunctions not only provides an efficient dimension for optical multiplexing [53–61] but also creates a complex electric field for light–matter interactions [62–66]. OAM beams have been used to rotate and manipulate small particles since more than 25 years ago [67–70]. In an atom or ions, photons carrying OAM enable high-order electron transitions, which are forbidden using only SAM of light due to the lack of angular momentum (AM) required for the transition [63,64]. Fueled by OAM source technology, nanoplasmonics displays a new vitality. In nanoparticles, OAM beams can induce high-order resonances, which typically result from the retardation effect and are themselves rare when the light does not carry OAM [71,72]. The underlying physical mechanism is AM transfer from the photon to the electron. Recently, excitations of surface plasmon multipoles and dark modes by OAM beams have been demonstrated [72–74]. Plasmonic nanoantennas can also show OAM dichroism [75]. In addition, the OAM of light has been used to control the relative oscillation phases of nanoantennas in a nanoarray, and the interference of radiations from each element in the far field contributes an OAM-modulated phased array nanoantenna [76]. The optical responses of plasmonic nano-oligomers have also been analyzed with symmetry-imposed selection rules using group theory [77,78]. Our work reveals the optical property of oligomers from the perspective of coupling of the individual elements. To the best of our knowledge, there are no prior studies that have specifically explored the effect of photon OAM and SAM on plasmon couplings in the near field, especially Fano resonances stimulated by a vortex beam.

In this investigation, we systematically investigate the excitation and coupling of surface plasmons with the OAM physical dimension. Single plasmonic nanoparticle shows different excitations when it is located at different points of the OAM beams. For single particles illuminated on-center of the light beam, the mode multiplicity of the excited plasmon matches the sum of the quantum number of SAM and OAM. When SAM and OAM are equal in magnitude and opposite in direction, there will be a pure longitudinal polarization along the light propagation direction. This investigation provides a quantitative analysis of the longitudinal polarization state under the influence of a vortex beam. When the nanorod is off-center, its polarization direction is determined by the SAM of light, while the OAM of light influences its oscillation phase. Neighboring nanorods under OAM beams show anomalous couplings. Nanorods located end-to-end (side-by-side) symmetrically along the polarization direction of the OAM beams couple out of phase (in phase), and the spectrum peak blueshifts (redshifts), which appears counterintuitive compared with the non-OAM case. Furthermore, plasmonic Fano resonances occur through the coupling between a bright mode of the nanorod along the propagation direction of light and a dark mode of the nanorod perpendicular to the propagation direction. The dark mode and bright mode giving rise to such a Fano resonance are acted by opposite elements using a non-OAM source. The investigation includes an additional OAM physical dimension to the field of plasmonics and proposes different plasmon excitation and coupling properties which are forbidden under non-OAM sources. Such plasmon behaviors enrich nanoplasmonics and may boost its broad applications in optical modulation, sensing, and computation.

It has been demonstrated in experiments that both the SAM and the OAM of light can be transferred to materials. It is easy to accept that optical SAM can cause a particle rotation or polarization around the axis of the particle. Optical OAM can also rotate or polarize a nanoparticle, but around the axis of the optical beam. The schematic illustration of a nanoparticle polarized by an OAM beam in a spin–orbital state of |σ,l⟩=|–1,1⟩ is shown in Fig. 1(a). OAM of a Laguerre–Gaussian (LG) beam is characterized by the helical wavefront in Fig. 1(a) or by the azimuthal phase gradient in Fig. 1(b). Nanoparticles’ polarization is determined both by SAM and OAM.

Now we consider a gold nanorod that shows a longitudinal dipole eigenstate around 640 nm and a transverse dipole eigenstate around 505 nm. When the gold nanorod is located on the center of the beam, OAM and SAM polarize the nanorod around the same axis, so the difference of the total AMs possessed by each of the surface plasmons of the initial and final states must equal the sum of SAM and OAM of the absorbed photon. The surface plasmons excited by photons in |–1,1⟩ state will not carry AM. But it does not mean that surface plasmon could not be excited. The optical scatterings of |–1,1⟩ light by a gold nanorod orientated perpendicular to and along the propagating direction of light are compared in Fig. 2. When the nanorod is orientated perpendicular to the propagating direction, only the transverse plasmon mode of the nanorod is excited. The phenomenon is much different from the non-OAM case where both transverse and longitudinal excitations are expected under circularly polarized illumination. When the nanorod is oriented along the propagation direction, the longitudinal plasmon mode of the nanorod is excited. The phenomenon also seems counterintuitive to the non-OAM case, when only transverse is expected. As indicated by the surface charge distributions in the insets, both polarizations are along the propagation direction of the beam. These counterintuitive plasmon behaviors can be understood if we realize the special field profiles of the vortex beams. The local electric field has a component along the propagation direction of the |–1,1⟩ beam, Ez, as explained in Appendix A. Ez arises from the partial derivatives of the electric field distributions with respect to x and y independently in paraxial approximation. Ez dominates the local polarization of the on-center nanoparticles, which are much smaller than the waist radius of the beam. So, on the focal center of |–1,1⟩ beams, the plasmon excitation is not along the global polarization direction which defines the overall polarization state of the entire beam, but along the propagation direction of light which is the local polarization direction of the beam. The conclusion also applies to a nanosphere located at the center of an OAM beam or radially polarized beams that have an Ez component [76]. By utilizing an anisotropic nanorod, it is possible to determine in the far field which physical quantity (electric field, magnetic field, or the Poynting vector) governs the polarization of the nanorods. This information can further aid in the investigation of nanorod couplings.

There are four different situations when the nanorod is located off the center of a linearly polarized vortex beam (|σ,l⟩ equals |0,1⟩), depending on the orientation and offset directions of the nanorod, as schematically shown in Fig. 3(a). Here, we neglect the discussion of a longitudinally oriented nanorod, whose optical spectrum will hardly show any difference when the nanorod is moved within the slowly varying Ez field. When the nanorod is oriented along the electric field (Ex), as shown in Figs. 3(b) and 3(c), an offset from the beam center will “unlock” the longitudinal plasmon mode, and the longitudinal intensity grows when the offset becomes larger. The results can be understood from the perspective of the complex vortical force field. When the nanorod is on the beam center of a linearly polarized vortex beam, the x-z plane divides the nanorod into two identical parts. These two parts suffer equal forces of opposite directions along the electric field (x axis), so the longitudinal plasmon mode is hardly excited. However, when there is an offset, no matter along the x or y axis, there will be a pure electric dipole momentum along the electric field, which grows when the offset becomes larger, and then “unlocks” the longitudinal plasmon mode. When the nanorod is oriented along the magnetic field Hy, there will be no longitudinal plasmon excitation, no matter whether there is an offset or not. When there is no offset, the transverse plasmon mode is excited by Ez and when there is an offset, the transverse plasmon mode is excited by Ex. To conclude Figs. 3(b)–3(d), when the nanoparticle is on-center, the nanoparticle polarization mode is determined both by OAM and SAM; and when the nanoparticle is off-center, the nanoparticle polarization mode is mainly determined by SAM, while OAM only puts a phase on the polarization.

Surface plasmons supported by the adjacent nanoparticles will couple through the near field. It has been shown in Figs. 2, 3(b), and 3(c) that longitudinal plasmon modes can be excited when the nanorod is oriented along the light propagation direction or when an off-center nanorod is along the electric field direction of a vortex beam of |0,1⟩. Generally, the couplings of such nanorods depend on the spatial distribution of the nanorods: end-to-end or side-by-side. The spectra of two symmetrically arranged nanorods oriented along the electric field direction are shown in Fig. 4(a). Compared with the spectrum of the single nanorod, the resonance of the end-to-end configuration shows a blueshift, while the resonance of the side-by-side configuration shows a redshift. The corresponding surface charge distributions at the resonance wavelengths are shown in the insets. The two nanorods in the end-to-end configuration oscillate out of phase, resembling an anti-bounding molecular state σ*, which is at a higher energy level. The oscillation in the side-by-side configuration resembles a bounding molecular state π which is at a lower energy level. Such couplings are exactly the opposite of the non-OAM case, which generally exhibits σ and π* states. This is because the vortex beam creates a non-uniformly polarized field, which polarizes the two nanorods in opposite directions. The spectra of two nanorods oriented along the propagation direction and stacked side-by-side and end-to-end are shown in Fig. 4(b). The dimer shows a π* resonance at the high energy level and a σ resonance at the low energy level, which agrees with the non-OAM case. The polarizations of the two nanorods are determined by Ez, which is almost uniform around the nanorod, so the two nanorods oscillate in phase and show a typical plasmon coupling.

The couplings of nanorods in different orientations in the x-y plane under an x-polarized vortex beam are shown in Fig. 5. The longitudinal plasmon mode of the nanorod oriented along the electric field direction (x axis) can be excited by the vortex beam when the nanorod is offset from the center. The nanorod oriented along the y axis exhibits no longitudinal excitation, no matter whether the nanorod is offset or not. The configuration of a pair of perpendicular nanorods located at different positions of a vortex beam is schematically shown in Fig. 5(a). The nanorods are moved in the second and fourth quadrants [Fig. 5(a) left] or in the first and third quadrants [Fig. 5(a) right], and their optical scatterings are shown in Figs. 5(b) and 5(c), respectively. The spectra exhibit a Fano line shape: two peaks divided by a dip at the original resonance wavelength of a single nanorod. The nanorod along the x axis can couple directly with the incident light, while the nanorod along the y axis cannot, as discussed in Fig. 3. So, the nanorods along the x axis and y axis act as the bright mode and dark mode for the Fano resonance, respectively. The phenomenon we observe is the opposite of plasmonic Fano resonance induced by non-OAM beams. In the case of Fano resonance induced by non-OAM beams, the nanorod oriented along the electric field axis behaves as the bright mode, while the nanorod oriented along the propagation direction behaves as the dark mode [20,21]. The surface charge distributions of the nanorods at the two peaks are shown in the insets of Fig. 5(b). At the shorter wavelength, the two nanorods show an out-of-phase state; and at the longer wavelength, the two nanorods show an in-phase state. This further confirms the Fano resonance in the two perpendicular nanorods system under vortex beams. The distribution and relative intensities of the two peaks are not determined by the quadrant where the nanorods are located. The peak intensities rely on how much the x-oriented nanorod is offset. The scattering spectrum shows the highest intensity when there is a largest offset, and the scattering disappears when the x-oriented nanorod is on-center, as shown by the blue line in Fig. 5(d).

Fano resonance can also happen to a dimer consisting of a vertical nanorod (along the light propagation direction) and a horizontal nanorod (along the electric field or magnetic field) under a linearly polarized vortex beam |0,1⟩. The horizontal nanorod can be along the magnetic or electric field direction, as illustrated in the insets of Figs. 6(a) and 6(b). When the horizontal nanorod is along the magnetic field, Fano profile appears in the scattering spectrum, which hardly shows any difference when the dimer is moved around the beam center. The Fano resonance results from the interference between the bright mode of the vertical nanorod and the dark mode of the horizontal nanorod. The bright mode and dark mode are not influenced by the offset, so the Fano line shape does not change. When the horizontal nanorod is along the electric field, the Fano line shape changes with the offset of the dimer. This is because the horizontal nanorod is no longer a dark mode when it is offset. The longitudinal plasmon mode of the horizontal nanorod becomes stronger when it is more offset, so the Fano resonance gradually evolves into an in-phase coupling at the low energy state. The Fano resonance is highly tunable according to the location of the dimer.

In conclusion, we theoretically investigate the anomalous plasmon excitation and coupling behavior under vortex beams. For a single nanoparticle system, we reveal the interplay between OAM and SAM: when the nanoparticle is on-center, the nanoparticle polarization mode is determined both by OAM and SAM; when the nanoparticle is off-center, the nanoparticle polarization multiplicity is only determined by SAM, while OAM only puts a phase on the polarization. For an on-focal-plane nanodimer whose oscillation phases are controlled by OAM of light, we find anomalous plasmon coupling behaviors of an end-to-end anti-bounding high-energy mode and a side-by-side bounding low-energy mode, which are exactly the opposite of the non-OAM case. In addition, we observe a contrary of the typical plasmonic Fano resonance in a nanorod dimer: one rod aligned with light propagation direction as the bright mode and the other aligned with light polarization direction as the dark mode. By controlling the location of the nanorod dimer, the couplings show high tunability from a Fano type to an in-phase coupling type, determined by how much the dark mode is excited. Our results offer insights into a fundamental question regarding the coupling of surface plasmons under a complex field. The plasmon behaviors enabled by OAM enrich the field of nanoplasmonics and have the potential to drive its continuous development and application.