Spin angular momentum (SAM) and orbital angular momentum (OAM) are inherent nature of light. These dynamical properties are determined by the polarization helicity σ = ±1, which corresponds to the right-hand and left-hand circular polarizations, and the topological charge l (that is the azimuthal phase increment, modulo 2π), respectively [1]. The extraordinary features of angular momentum (AM) have been applied in a variety of situations ranging from classical optics to quantum realm, such as optical tweezers [2] and spanners [3], rotational Doppler metrology [4,5], quantum key distribution (QKD) [6,7], and photonic crystal slabs (PhCSs) using bound states in the continuums (BICs) [8,9]. In contrast to SAM, OAM beams carrying infinite eigenstates, i.e., l = 0, ±1, ± 2, ±3…, have extensively diverse degrees of freedom, which are preferable in the application of high-capacity optical communications [10]. Though the viability of OAM encoding or multiplexing has been demonstrated in free space [11] and in fiber [12], it is still an open challenge to detect the arbitrary state-space of SAM and OAM beams without ambiguous degeneracy.

Based on the optical Spin–Hall effect [13,14] and the Pancharatnam–Berry phase [15,16] principles, a number of striking studies have accomplished SAM demultiplexing. However, contrary to SAM characteristic, the twisted OAM beams are polarization independent. As a consequence, the previous approaches of solely tuning spin-dependent diffraction and deconstructive interference in the evanescent field, e.g., chiral-light deflection [17–19] and spin-controllable unidirectional coupling [20], are invalid to distinguish the OAM of interest. Moreover, macroscale techniques such as onefold fork grating [21], q-plate [22], Mach–Zehdner interferometer [23], and transformation-optics hologram [24] are considerably incompatible with many miniaturized applications.

The cumbersome issue of apparatus dimensions beyond the scale of the free-space wavelength can be harnessed through the confinement of nanoplasmonics. Ren et al. demonstrated a remarkable work in which four distinctive SAM and OAM states were successfully distinguished by a nano-ring aperture-structured (NRA) chip [10]. Despite being spin–orbit sensitive, the NRA chip inevitably suffers from the degeneracy of the total AM quantum number. The fundamental reason is attributed to those undecorated nano-ring structures that only respond to the mutual interaction of discretized SAM and OAM states simultaneously. In case of equivalent total AM, it would fail to discriminate orthogonally involved SAM and OAM states. Thus, the detection of arbitrary SAM and OAM without the degeneracy of total AM still remains unsolved completely.

In this Letter, we capitalize on a nanoplasmonic metachain to distinguish any chirality and topological charge of incident light, which is different from a recent effort that breaks SAM and OAM degeneracy in the far-field [25]. The proposed method is competitive in terms of being effectively integrated on plasmonic chips and having dynamic and reconfigurable advantages. It is noteworthy that the metachain simultaneously breaks both the SAM and the total AM degeneracy. We fundamentally interpret that it originates from two mechanisms: the spin-to-orbital conversion via the Pancharatnam–Berry phase and the effect of the inherent spiral phase of OAM on the metachain curvature. The metadevice has the advances of broadband and ultra-compact. Hence, it potentially has a wide range of applications in topological photonics and will evoke new physics related to spin–orbit interactions.

The conceptual scheme of nanoplasmonic metachain to distinguish any chirality and topological charge of incident light is shown in Fig. 1. An effective AM sorter that can divide the OAM and SAM modes of incoming light into various subwavelength focus spots is accomplished by a metachain structure. Specifically, the metachain structure is made up of an ensemble of perforated slits in a gold thin film, which is different from the assembled nanopillars into a meta-atom reported in Ref. [25]. It is more advantageous in terms of design and implementation. The specific structural design will be analyzed and discussed in detail later. Different incident vortex beams (l = 0, ±1, ±2, ±3, ±4, ±5…) with different left ($\sigma ={-} 1$)- or right ($\sigma ={+} 1$)-handed circularly polarized states illuminate the metachain structure vertically and excite the surface plasmon polaritons (SPPs). These phase-modulated plasmons are focused into spatially separated subwavelength spots for different vortices due to the spin–orbit interaction. It is worth mentioning that the SAM response can be achieved by the metachain that consists of the multiple columns in a regular arrangement and the overall circle structural sizes are subwavelength [20]. In addition, the coherent superposition of the SPPs was excited by individual columns and finally achieved the shift focus point. This strategy offers a practical solution for on-chip discrimination of AM in various applications, including optical communications and integrated optics, because of the independence of the ring geometry.

 

Fig. 1. Schematic diagram of the nanoplasmonic metachain to distinguish arbitrary chirality and topological charge of incident light. When the incident light with varying chirality ($\sigma ={\pm} 1$) and topological charges (l = 0, ±1, ±2, ±3, ±4, ±5…) excites the nanoplasmonic metachain structure, it produces a regular spatial shift of SPP focus that can be distinguished. The degeneracy breaking of total AM is further realized.

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According to the Huygens–Fresnel principle, a circularly polarized optical vortex beam (e.g., $\vec{E} = {e^{i({l + \sigma } )\varphi }}({{{\vec{e}}_r} + i\sigma {{\vec{e}}_\varphi }} )$) illuminates the semicircle metachain, which carries a uniformly distributed radial electric field along the azimuthal direction. l is the number of the topological charge, $\sigma $ represents the handedness of the circular polarization (+1 for right-handed and −1 for left-handed), $\varphi $ is the angle coordinate, ${\vec{e}_r}$ is the radial unit vector, and ${\vec{e}_\varphi }$ is the angular unit vector. The electric field at the surface can be described as follows [26]:

 

Fig. 2. Schematic of the semicircle metachain under different topological charges of the optical vortex illumination. (a) Circularly polarized state of incident beam is fixed (e.g., $\sigma ={+} 1$), and the full-wave calculation simulates the shifted focusing positions of different OAM modes (l = 0, ±1, ±2, ±3, ±4) using an analytical model. (b) Mechanism of the OAM mode division structure.

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Specifically, when the circular polarization state of the incident light is given by $\sigma ={+} 1$, the initial phase P($\varphi $) = (1 + l)$\varphi $ is only dependent on the OAM. The finite-difference time-domain (FDTD) simulation full-wave calculation results of the intensity profile for different OAM modes are shown in Fig. 2(a). Separated focal locations are shown along the x direction by the red dotted line. The original point O corresponds to the focal point of the incident beam (Gaussian beam) with a total angular momentum being 0 (l = −1). On the left side of point O, the incident beams with positive order l = +1, +2, +3, +4 are shifted and arranged in that order. Because the direction of the phase gradient is the mirror symmetry, the negative l = −2, −3, −4 are focused on the right side of point O, respectively.

The geometric phase describes an extra phase gradient that is related with the polarization of the incident light and the rotation of nanoantennas. As the surface plasmons are focused by the semicircular geometry of the sample, the phase modifying the wavefront brings about the transverse shift of the resulting focal spot. Each SPP focal point displays a uniform arrangement of subwavelength dimensions in the x direction. Using the paraxial approximation theory, we can simplify the position ${x_j}$ at the peak focal point from Eq. (1) as follows [26]: ${x_j} \propto sgn({\sigma + l} ){\mathrm{\lambda }_{spp}}/2\pi $. The focal positions are totally determined by the topological charge l, handedness of the circular polarization $\sigma ,$ and wavelength ${k_{spp}}$ of SPPs.

The FDTD solutions are adopted to simulate the focal points’ performance of the semicircle metachain structure. The wavelength of the incident beam is 633 nm with the SAM of $\sigma ={+} 1$ and −1, respectively. The results of the normalized intensity profile along the x direction can be shown in Figs. 3(a)–3(b). The peak focal point distance between any two adjacent OAM modes is about 125 nm, which can distinguish the separation of different OAM modes. The distinctive helical phase of an optical vortex beam can be used to analyze the focal spot’s displacement. The incident optical vortex beam’s inherent helical phase is transmitted to the initial phase difference of SPP waves coming from various regions of the slit, as they travel through the subwavelength semicircle metachain. Consequently, the superposition of these initial SPPs creates the plasmonic field close to the focus point.

 

Fig. 3. Normalized field distribution intensity along x different positions under various OAM modes. (a) $\sigma ={+} 1$; (b) $\sigma ={-} 1$; the results are obtained via full-wave calculations using the FDTD method. (c) The positions of the SPP focal spot versus total angular momentum (AM) $j = \mathrm{\sigma } + l$. (The red and gray dotted lines correspond to the positions of the SPP focal spot versus j in case (a) and case (b), respectively.) The overlapping blue boxes indicate that the same position corresponds to the total AM degeneracy.

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Furthermore, the results of the total angular momentum j at different focusing positions on the x-axis are shown in Fig. 3(c). When the total angular momentum (AM) $j = \mathrm{\sigma } + l$ is considered, the results show that at some fixed focusing point position, there corresponds a merger of the j value. For instance, at point x = 0 nm, $j$ = 0; however, the SAM mode is merged at this time. It is not possible to determine whether the excitation is RCP or LCP. Similar cases take place at other positions such as x = ${\pm} 250$ nm or ${\pm} 375$ nm. As a result, there are two cases that correspond to the $j\; $ value at the same position point; the AM cannot be identified. Therefore, the cross talk is increased. This is primarily due to the fact that it is incomplete to realize the mode separation of OAM by a single shift of the SPP focusing point, because the chirality of the incident beam (SAM, $\sigma ={\pm} 1$) will bring about the unambiguous states, resulting in the inability to fully realize the mode separation of the AM.

We further optimize the structure to realize the AM mode division, which discriminatively couple arbitrary input SAM and OAM components to an extrinsic orbital AM. The optimized structure design is shown in Fig. 4(a). A nanoplasmonic spiral metachain structure consists of two (upper O₁ and lower O₂) semicircles with diameters of T₁= 7.875 µm and T₂= 8 µm, resulting in a misalignment of ΔR = 125 nm. Specifically, two mutually perpendicular metamolecule spacing with a distance S = $\lambda$_spp/4 = 150 nm (S < $\lambda$_spp) are made up of apertures with the width of W = 40 nm and the length of L = 200 nm (W << L < $\lambda$_spp). We performed the design on a structure consisting of two semicircles for operation at $\lambda$=633 nm in a 200 nm Au film on a glass substrate. The narrow aperture in the Au film selectively scatters the incident light that is polarized perpendicular to it, giving rise to spin-dependent directional SPPs. The SPP emission patterns of a subwavelength aperture are surface waves that propagate perpendicularly toward either side of the metamolecule. The offset by the distance S along their axes reduces near-field coupling and scattering of the SPPs by neighboring apertures. The apertures of the first and second columns are oriented at angles 45° and 135° with respect to the x-axis. This pair of metamolecule is arranged along the track of the semicircle to form the upper and lower nanoplasmonic spiral metachain structures, and the metamolecules of the upper and lower semicircles are symmetrical along the central axis.

 

Fig. 4. (a) Schematic of the AM mode-division nanoplasmonic metachain. The columns of the nanoslits couple to the field components $\overrightarrow {{E_1}} \; $ and $\overrightarrow {{E_2}} \; $ of the incident field $\vec{E}$. (b) The diagram of the combined SPP waves propagating away from the semicircular path under RCP and LCP, respectively. (c) FDTD simulation of the real part of near-field intensity (E_z) of various $\sigma $ and l ($\sigma ={\pm} 1$ and $l ={\pm} 1$). The white solid frame and solid arrows indicate the actual inward focusing direction. The white dotted line indicates the actual outward excitation direction. (d) Numerical calculations of the wave propagation with the phase fronts (white dashed lines) disturbed by the geometric phase.

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As a result, the two circularly polarized states thus respond to two distinct channels that can selectively excite each side of the coupler. By adjusting the handedness of the incident light, switchable unidirectional connection is made feasibly. Due to the fact that the SPP fields propagating to the up and the down correspond to the circularly polarized components of the incident light, the spiral metachain mocks up a spin-dependent beam splitter with two SPP output channels. The amplitude and relative phase of the SPPs then completely encode the polarization state of the input light. Specifically, when the circular polarization state of the incident light is given by the RCP $\sigma ={+} 1$, the direction of the combined electric field $\vec{E}$ is excited upward. Ortherwise, when the circular polarization state of the incident light is given by the LCP $\sigma ={-} 1$, the direction of the combined electric field $\vec{E}$ is excited downward, as shown in Fig. 4(b).

To analyze the interaction between various SAM–OAM combinations and spiral metachain, we carried out FDTD simulations of the nanoplasmonic metachain patterned on a gold surface. This structure was designed for operation at $\mathrm{\lambda }$=633 nm, corresponding to ${\mathrm{\lambda }_{spp}} \cong \; $606 nm. Our proposed method is not strictly bound to a specific working spectrum. For different working wavelengths, the period of the structure should be adjusted according to the SPP wavelengths. The results of the real part of the near-field intensity ($R\textrm{e}({\overrightarrow {Ez} } )$) show a good agreement with the predictions in Fig. 4(c). Further simulation results in other different values of $\sigma $ and l are reliable. Figure 4(d) shows the numerical calculations of the corresponding phase change. The phase modifying the wavefront brings about the transverse shift of the resulting focal spots. This effect originates from the polarization-dependent geometric phases of the confined field in the focusing SPPs.

Furthermore, when the different vortex beams with different handed circular polarizations illuminate the metachain structure vertically, the normalized intensity profile along the x direction can be shown in Fig. 5(a). The FDTD simulation results show that the peak focal point distance between any two neighboring different AM modes can be distinguished at about 65 nm. The results of the total angular momentum j at different focusing positions on the x-axis are shown in Fig. 5(b).

 

Fig. 5. (a) Normalized field distribution intensity along different x positions under various AM modes. (b) The focal spots of SPPs versus total angular momentum (AM) $j = \mathrm{\sigma } + l$. (The two red dotted lines correspond to the positions of the SPP focal spot versus j in the case of $\sigma ={+} 1$ and $\sigma ={-} 1$, respectively. ① and ② indicate the two adjacent AM modes in different $\sigma $.) The AM mode can be identified in a one-to-one correspondence in different positions. (c) FDTD simulation of the near-field intensity shifted focusing positions of different AM modes in cases ① and ②. The adjacent focal points (corresponding to different RCP and LCP modes, respectively) can be clearly distinguished.

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The results show that there is no AM mode degeneracy and that each SPP focusing position point corresponds to a single, distinct j. The result of full-wave simulation is shown in Fig. 5(c). Any two adjacent AM modes can be distinguished and measured. Total AM discrimination is guaranteed by the stable sorting function. Our method can separate any chirality and topological charge of the incident light directly and spatially. It is worth mentioning that the generated SPPs have a partial loss that is positively related with the propagating distance. Hence, the diameter of the metachain cannot be infinitely larger. In our scheme, the diameter of the metachain is 8 µm and achieves an efficient resolution of 16 AM modes in total. Generally, the energy loss of SPPs on the Au film can be acceptable within a distance of 20 µm. Based on this assumption, the diameter of the metachain can be expanded to the condition of ∼20 µm [27]. According to the rational prediction, the approximate number of discriminative modes could be up to ∼40 AM states. By the way, the limit is very likely to be further enhanced if the signal-to-noise can be optimized.

In summary, we proposed and demonstrated a nanoplasmonic metachain made up of decorated nanoslits that can discriminate arbitrary spin state and topological charge of incident light. The considerable sorting interval of 65 nm between nearby AM modes could pave the way for nano-scale discrimination and manipulation of quantum modes of light. The proposed method is based on the unique and flexible way to modulate SPPs in a dynamic and reconfigurable fashion, which may lead to many benefits across fields of plasmonic optics, integrated optics, and nano-photonics. Additionally, it will potentially inspire novel physics about spin–orbit interactions, and it has a wide variety of possible applications in topological photonics.