† Corresponding author. E-mail:
Localized surface plasmonic resonance has attracted extensive attention since it allows for great enhancement of local field intensity on the nanoparticle surface. In this paper, we make a systematic study on the excitation of localized surface plasmons of a graphene coated dielectric particle. Theoretical results show that both the intensity and frequency of the plasmonic resonant peak can be tuned effectively through modifying the graphene layer. Furthermore, high order localized surface plasmons could be excited and tuned selectively by the Laguerre Gaussian beam, which is induced by the optical angular orbital momentum transfer through the mutual interaction between the particle and the helical wavefront. Moreover, the profiles of the multipolar localized surface plasmons are illustrated in detail. The study provides rich potential applications in the plasmonic devices and the wavefront engineering nano-optics.
Localized surface plasmon resonance in metal nanoparticles that results in the resonant absorption, scattering, and near field enhancement around the particle can be tuned across a broadband range from visible to far-infrared wavelength band. The resonant enhancement allows for a wide range of scientific and technological applications, such as optical tweezers,[1–3] surface-enhanced Raman scattering (SERS),[4–6] nano-antennas,[7–10] plasmonic devices,[11] and biosensing.[12,13]
Recently, graphene with two-dimensional Dirac fermions has attracted intensive investigation[14,15] due to its excellent physical properties.[16,17] Graphene decorated nanomaterials and nanostructures have widely been used in the photonic and optoelectronic fields such as photovoltage device,[18] photodetectors,[19] optical modulators,[20] and medical science applications. Since the experimental fabrication of curved and spherical graphene sheets is demonstrated,[21] graphene modulated localized surface plasmon resonance and SERS on nanoparticles are realizable. In this paper, utilizing Lorenz–Mie theory, we make a systematical study on the multipolar localized surface plasmons of the graphene coated dielectric particle excited by Gauss beams with complex wavefront. The detailed results are shown in the following.
In the Lorenz–Mie theory, the general electromagnetic wave can be expanded into vector wave functions in the spherical coordinate system with the origin coincide with particle center[22]
![](cpb_29_5_057302/cpb_29_5_057302_ieqn1.gif)
The following procedure is used to obtain the electromagnetic wave solution for the scattering and internal fields of the graphene coated dielectric particle by applying proper boundary conditions at the surface of the particle. In the study, the graphene layer coating is considered as a surface current surround the spherical particle. The boundary condition at the particle surface is that the tangential component of the electric field E is continuous and the discontinuity of the tangential component of the magnetic field H is proportional to the surface current density σ(ω) Ep,[24]
![](cpb_29_5_057302/cpb_29_5_057302_ieqn2.gif)
The optical conductivity of the graphene layer σ(ω) can be derived from the Kubo formula.[26–28] For simplicity, we only consider the low-temperature conditions where the absolute value of Fermi level EF of graphene is much larger than kBT, where kB is the Boltzmann constant and T is the absolute temperature. The optical conductivity σ (ω) is the summation over the intraband σintra(ω) and interband σinter(ω) contributions: σ(ω) = σintra(ω) + σinter(ω). The Drude conductivity describes the intraband conductivity part
The optical extinction, scattering, and absorption spectra of a nano-particle irradiated by an electromagnetic wave can be obtained by the Mie expansion coefficients
The schematic of scattering diagram is shown in Fig.
![]() | Fig. 1. Excitation of the localized surface plasmon on a graphene-coated polystyrene spherical particle with radius R by Gaussian beam. |
As observed in Fig.
However, as demonstrated above, the high order plasmonic mode cannot be excited through modifying the properties of the graphene coating. In view of this limitation, we proceed to explore the optical resonance of the graphene coated dielectric particle interact with beams with complex wavefront, the Laguerre Gaussian (LG) beam, to further investigate the high order plasmonic resonant mode response. The LG beam possesses the cylindrical symmetry and helical wavefront with a special vortex phase exp(isφ), where s is the azimuthal mode index.[29] Suppose the LG beam is polarized along x direction and propagating along z direction, the vector potential A can be expressed as
![](cpb_29_5_057302/cpb_29_5_057302_ieqn3.gif)
![](cpb_29_5_057302/cpb_29_5_057302_ieqn4.gif)
Figures
![]() | Fig. 3. Modal profile of cross section fps(ρ) of LG beam of order p = 5, s = 5: (a) exp(isθ), (b) cos(sθ). Beam waist ω0 = 5 μm. |
The optical extinction spectra of a graphene coated polystyrene particle excited by the LG electromagnetic wave described by Eqs. (
The investigation above demonstrates that the multi-polar localized surface plasmons of the graphene coated dielectric particle could be excited and tuned selectively by modifying the helical wavefront of the LG beam. The underlying physics can be understood from the mutual interaction of the particle and the helical wavefront of the high order LG beam which carries angular orbital momentum (OAM) of sℏ per photon.[29] The OAMs of the high order LG beam are transferred to the graphene coated dielectric particle under the resonant absorbing condition, exciting the corresponding mode of localized surface plasmons.[31,32] The selective multi-polar surface plasmons excitation through the wavefront engineering of optical beams could open a new path for the study on enhancing light–matter interaction.[33]
In further study, we illustrate the profile of multipolar mode of the electric field of the localized surface plasmon of the graphene coated dielectric particle excited by the LG beam in Fig.
Besides the local field distribution, we also examine the electric charge distribution of the graphene coated dielectric particle excited by the LG beam, since the charge pattern can offer a clear picture of the nature of a particular plasmonic wave mode at resonance within the particle.[34,35] In particular, the graphene coating is considered as a surface current surround the particle. The polarized electric charge distribution of the graphene coated dielectric particle excited by the LG beam for different plasmonic mode is shown in Fig.
We have made a systematic investigation on the excitation of localized surface plasmon of the graphene coated dielectric particle. It is found that both the intensity and resonant wavelength of the plasmonic peak can be tuned effectively through modifying the graphene layer. Furthermore, high order localized plasmons could be excited and tuned selectively by an LG beam, which is induced by the mutual interaction of the particle with the helical wavefront. Simultaneously, the multipolar mode profile of the localized surface plasmon is illustrated in detail. These investigations can find rich potential applications in the plasmonic devices and the wavefront engineering of optical beams for enhancing the light–matter interaction.
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