† Corresponding author. E-mail:
Project supported by the National Key R&D Program of China (Grant Nos. 2017YFA0205700, 2015CB932403, and 2017YFA0206000) and the National Natural Science Foundation of China (Grant Nos. 21790364, 11374023, 61422501, 11674012, 61176120, 61378059, 6097701, and 61521004).
We investigate the dynamic quantities: momentum, spin and orbital angular momenta (SAM and OAM), and their conversion relationship in the structured optical fields at subwavelength scales, where the spin–orbit interaction (SOI) plays a key role and determines the behaviors of light. Specifically, we examine a nanostructure of a Ag nanoparticle (Ag NP) attached on a cylindrical Ag nanowire (Ag NW) under illumination of elliptically polarized light. These dynamic quantities obey the Noether theorem, i.e., for the Ag nanoparticle with spherical symmetry, the total angular momentum consisting of SAM and OAM conserves; for the Ag NW with translational symmetry, the orbital momentum conserves. Meanwhile, the spin-to-orbital angular momentum conversion is mediated by SOI arising from the spatial variation of the optical potential. In this nanostructure, the conservation of momentum imposes a strict restriction on the propagation direction of the surface plasmon polaritons along the Ag NW. Meanwhile, the orbital momentum is determined by the polarized properties of the excitation light and the topography of the Ag NP. Our work offers insights to comprehend the light behaviors in the structured optical fields in terms of the dynamic quantities and benefits to the design of optical nano-devices based on interactions between spin and orbital degrees of freedom.
Nowadays, the spin–orbit interaction (SOI) of light has attracted great attention in optics, owing to its fundamental physical significance and potential applications in novel photonic devices.[1–17] It turns out that most of conventional optical processes, such as propagation, reflection, focusing, scattering, are associated with the SOI,[5,7,18,19] although they have very weak manifestation, and are often neglected in traditional macroscopic geometrical optics. For example, in the spin Hall effects of light,[4,13,20] the SOI deviates photons with opposite spins–left-handed or right-handed circularly polarized light (LCP or RCP)–propagating in different directions with a transverse subwavelength shift when they are incident on the interface of different media. While in nano-optics, by engineering SOI using metamaterials, geometrical phases, and plasmonic nanostructures at subwavelength scales, it was demonstrated that spin-related optical phenomena induced by SOI are more distinct than those occurred in macroscopic optics. In the subwavelength-sized structured optical fields, the propagation direction of light varies rapidly with the variation of the medium permittivity and permeability, which leads to the spin change due to the correlation of the electric field E and the wavevector k by the transversality condition k⋅ E(k) = 0. In most cases, the SOI effects are described in terms of the electromagnetic field quantities, which represent the wave properties of light. When concerning the SOI, it is also reasonable to use the dynamic quantities, such as momentum, spin and orbital angular momenta (SAM and OAM), to describe the behaviors of light in structured optical fields, which can better clarify the physical processes as does in the electron counterparts. Recently, Bliokh et al.[21,22] gave a detailed theoretical derivation about the representation of these dynamic quantities in the canonical Minkowski approach,[23] which is very valid to describe structured optical fields in dispersive inhomogeneous media. Therefore it is of significance to present a specific structured optical field for revealing the behaviors of the dynamic quantities.
Here we present a prototypical nanostructure constructed by a spherical Ag nanoparticle (Ag NP) attached on a cylindrical Ag nanowire (Ag NW) under illumination of elliptically polarized light. A clear picture shows that the conversion of spin-to-orbital angular momentum occurs through the SOI arising from the spatial variation of the optical potential. Due to the symmetry of the nanostructure, the total angular momentum consisting of SAM and OAM conserves under the spherical symmetry; the orbital momentum conserves under the translational symmetry. The conservation of momentum imposes a strict restriction on the propagation direction of the surface plasmon polaritons along the Ag NW.
In subwavelength-scale structured optical fields, the Dirac-like form of Maxwell equations is more appropriate to describe the dynamical behaviors of light, which is[6]
![](cpb_29_6_067301/cpb_29_6_067301_ieqn1.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn2.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn3.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn4.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn5.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn6.gif)
![](cpb_29_6_067301/cpb_29_6_067301_ieqn7.gif)
Based on Eq. (
Firstly, we consider a spherical Ag nanoparticle with the radius of 100 nm under illumination of a circularly polarized light, as schematically shown in Fig.
As schematically shown in Fig.
Only considering the Ag NW, it is cylindrically symmetric, the optical potential presents the form of V(ρ), where ρ is the polar coordinate in x–z plane. In y-direction along the NW, V(ρ) is translationally invariance. Taking into account the Hamiltonian H with cylindrically symmetric V(ρ), we have
As mentioned above, the PO direction in the gap is mainly governed by the incident light helicity. By changing the ratio of different helicities in the incident light, we can vary the PO direction, and then adjust the amount of SPPs propagating along one side of the Ag NW. For this purpose, we use elliptically polarized light to illuminate the above Ag NP and NW system. The complex electric field of this incident light can be written as
Through simulation and comparison, we choose four different elliptically polarized light, as shown in Figs.
The topography of the Ag NP can also affect the direction of PO in the gap. Without loss of generality, the Ag NP is changed from the aforementioned sphere to an ellipsoid, as shown in Fig.
In experiments, chemically synthesized Ag NWs and NPs were deposited simultaneously on a silicon substrate. By seeking under scanning electron microscope, we obtained a desired sample showing that a NP adhered on almost the middle of a NW [Fig.
In summary, we have investigated the evolution of momentum, spin and orbital angular momenta, and their SOI in the structured optical fields that are creased by a spherical Ag NP attached on a cylindrical Ag NW. It is found that in the case of a spherical symmetric Ag NP, the conservation of angular momenta (including SAM and OAM) is obeyed; in the case of a translational symmetric Ag NW, the conservation of momentum is obeyed. Spin-to-momentum conversion is mediated by the SOI which arises from the spatial variation of the optical potential. The helicity of the polarized light is able to control the direction of the orbital momentum PO at the gap between the Ag NP and NW, which can govern the propagation direction of SPPs excited on the Ag NW based on the conservation of momentum. The behavior analysis of the dynamic quantities in the structured optical fields offers a new approach to assist the design of optical nano-devices.
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