1. Introduction

Spin-orbit interaction (SOI), ubiquitous in atomic and condensed matter physics, has received enormous attention in the realm of optics for the rich physics and promising application potentials in areas such as generating structured optical fields, enhanced optical detection and manipulation of particles, and optical fields manipulation at micro-/nano-scale [1,2]. Both spin angular momentum (SAM) and orbital angular momentum (OAM) interact with each other in nonparaxial optical waves, anisotropic and inhomogeneous medium [1,2]. Since SOI of light depends on properties of the medium where the interactions occur, the refractive index of the medium plays a vital role in SOI in optics as the energy potential does in condensed matter systems for the SOI of electrons. So far, research efforts are focused on studying and realizing SOI of light under various wavevector and polarization modulation schemes, or propagating in optical materials and structures with specially designed optical anisotropy and inhomogeneity. SOI has been studied in either classic scheme such as optical scattering through bulk random medium for science behind optical spin hall effect, or in novel materials such as two-dimensional metamaterials for their technical potential for integrated SOI devices [3–5]. However, all the inhomogeneity and anisotropy necessary for SOI to occur are created with materials of positive refractive index, SOI has not been studied in optical medium with negative refractive index.

Negative optical refractive index served as the onset of the endeavors on metamaterials, which are artificial structures with properties absent or hard to find in naturally occurring materials [6–8]. The main consequence of the negative refractive index is the reversal of phase propagation and energy propagation of light [9]. Negative index materials (NIM) provide novel methods to manipulate electromagnetic wave, and have induced many intriguing phenomena, such as negative refraction, electromagnetic cloaking, perfect lensing/imaging effects [9–15]. SOI is also the mechanism underlying novel phenomena such as optical Hall effect, photonic spin Hall effect, and topological photonics emerging recently [5,16–20]. Inspired by these breakthroughs in optical SOI and NIM, negative refraction of electrons and phonons have been actively studied in condensed matter systems and acoustic metamaterials [21–28], and investigations on SOI of phonons are also emerging [29,30]. While both SOI and NIM have been separately investigated extensively in the schemes of metamaterials and metasurfaces [3,19–25], yet studies on the effects of their combinations are still lacking. Understanding SOI in the context of optical NIM can deepen our understanding of their interplay in manipulating electromagnetic fields at the micro- and nanoscales, and could also stimulate the studies on this interplay in electronic and acoustic systems.

In this work, to our best knowledge, for the first time we study the SOI in a medium with a negative refractive index and observe some interesting and novel consequences with respect to how the negative refractive index modulates/controls the spin-to-orbit conversion process. We numerically study the SOI process with a circularly polarized illumination incident on plasmonic lens structures made of a whole circular slot and a circular arc slot milled in a thin silver film, respectively. Due to its reversing effect of energy propagating, we find that the presence of NIM within the center of the plasmonic vortex generator would reverse the in-plane energy flux while leaving the amplitude and phase distribution of the surface plasmon polariton (SPP) vortex field unaltered. Under the same circularly polarized excitations, the arc slot plasmonic lens would focus the SPP field to an inverse position in the NIM region with respect to that without NIM.

2. Results and discussion

Figure 1 illustrates the scheme employed in this work to study the interplay between SOI and the negative refractive index. We use the plasmonic vortex generator made of a circular slot milled in a 200 nm thick silver (Ag) film. The outer and inner radius of the circular slot is 5 and 4.9 um, respectively, leaving a 100 nm wide slot filled with air. As shown in Fig. 1(a), upon illumination with the wavelength of 633 nm and either left-handed-circular (LCP) or right-handed-circular polarization (RCP) from underneath the Ag film along the z-direction, excited SPP along the circular slot would bear geometric phase accumulation and experience SOI when propagating on the film surface (x-y plane) towards the center to form an SPP vortex. Thus, during this process, optical spin-to-orbital angular momentum conversion occurs. The formation and property of SPP vortex would depend on the interface between Ag and air (left) or NIM (right), where the NIM medium is considered as a cylinder shape centered on the Ag film relative to the circular slot.

 

Fig. 1. SOI in the presence of NIM. (a). Plasmonic lens based on a circular slot etched in a silver (Ag) film. The outer and inner radius of the circular slot is 5 and 4.9 um, respectively, leaving a 100 nm wide slot filled with vacuum. Circularly polarized excitations (LCP or RCP) illuminate from underneath on the slot along z direction, excited surface plasmon polariton propagate on the film surface (x-y plane) towards the center on the interfaces between Ag and vacuum (left) or NIM (right), where the NIM medium is a cylinder with 2 um radius centered on the Ag film relative to the circular slot. (b), the principle underlining the perfect focusing and imaging of the NIM lens. (c), imaging of the dipole source at the right-hand side of the NIM lens via perfect imaging. The width of the NIM lens is 2 um along the x-axis, and an electric dipole polarized along the z-axis at 1 um away from the left surface of the rectangular NIM lens.

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To investigate the influence of NIM on SOI, we numerically simulate the SOI process using the three-dimensional finite-difference time-domain method (Lumerical FDTD Solutions). The NIM is created using the FDTD component for magnetic and electric Lorentz medium. To account for material loss, both the dielectric permittivity and magnetic permeability of NIM is set to be −1 + 0.04i, thus resulting in a NIM with a refractive index close to −1 and a small imaginary part. Since the current work focuses on revealing the effect of NIM on SOI, this small material loss is chosen to facilitate the study. It is noted that, since the excited SPP could propagate into the NIM through different incident angles, the NIM in FDTD must be isotropic. Thus, to verify the isotropic property of the NIM, we modeled the well-known perfect lens function [21–28] of a rectangular NIM in FDTD simulation, as depicted in Fig. 1(b). In the perfect imaging process enabled by a NIM lens, a point object can be perfectly focused to the other side of a NIM lens. Figure 1(c) shows this simulation setting and the image of the dipole source at the right side of the NIM lens via perfect imaging. The verification of the perfect focusing effect of the NIM depicted in Fig. 1(c) shows that negative refraction from all angles of incidence can be realized in our FDTD simulation model, which is very important due to the spiral wavefront of the plasmonic vortex field to be studied in this model.

We then applied this simulation method and setting to study the influence of NIM on SOI in the system shown in Fig. 1(a). In all simulation settings, we set the simulation domain boundaries using perfectly matched layer condition mimicking open boundary encountered in realistic situations. We first studied the case without NIM for comparison. Figures 2(a)–2(c) and Figs. 2(g)–2(i) illustrate the amplitude of the dominant evanescent electric field component |E_z| of SPP field, its phase distribution, and the in-plane Poynting vector amplitude ${P_{xy}} = \sqrt {P_x^2 + P_y^2} $ overlapped with the energy flux direction distribution (indicated by arrows calculated with the real part of P_x and P_y) on a plane parallel to the air-Ag interface at z=10 nm for the LCP excitation σ₊ and the RCP excitation σ₋, respectively. Due to the evanescent nature of SPP field, the other in-plane components of electric field (E_x and E_y) and the associated out-of-plane Poynting vector P_z are very weak, thus not shown here. Generated by scattering the circularly polarized excitation off the nanoslot, there is a continuous distribution of local SPP sources along the slot, each of them is an infinitesimal slot section emitting SPP with the most efficiency only when excited by a perpendicular electric field polarization. Thus, within an optical period of the external circularly polarized excitation (LCP or RCP), the successive phase difference of all the local SPP sources would have a continuous distribution between 0 and 2π, with the phase spiraling direction dependent on the handedness of the excitation field. These excited SPPs propagate towards the center to form an SPP vortex bearing a dark center in the field profile, as shown in Figs. 2(a) and 2(g), and its phase in Figs. 2(b) and 2(h) has an anticlockwise (clockwise) distribution from 0 to 2π, indicating a topological charge of l_f =−1(+1). The donut shape of the Poynting vector amplitude ${P_{xy}}$ shown in Figs. 2(c) and 2(i) is in line with the electric field amplitude profile distribution shown in Figs. 2(a) and 2(g). Moreover, the in-plane distribution of the Poynting vector components shown in the insets of Figs. 2(c) and 2(i) resemble a vortex in the center with the same handedness of SPP vortex phase distribution in the insets of Figs. 2(b) and 2(h), indicating both the local wavevector and energy flow share the same direction. These results confirm that the spin-to-angular momentum conversion via SOI obeys the general rule, l_f = −σ_±, which was reported in previous studies on plasmonic vortex generation [31–33]. Note that we use the same convention for defining the handedness of circular polarization and helicity of vortex wavefront rotation, which is that σ and l is positive/negative when the left/right hand is used when pointing the thumb along beam propagating direction (+z, Fig. 1) and curving the other four fingers along the field vector rotating or wavefront rotating direction.

 

Fig. 2. SPP vortex induced by LCP (green arrows) and RCP excitations (blue arrows) with and without NIM. LCP excitation without NIM, (a) SPP vortex field |E_z| amplitude, (b) phase with inset of central spiral phase distribution, (c) in-plane Poynting vector amplitude overlapped with energy flux direction distribution (see inset). The white scale bar in the inset is 200 nm. (d,e,f), similar to (a,b,c) except with NIM indicated by a white dashed circle of 2 um radius. (g,h,i), same to (a,b,c) with RCP excitation instead. (j,k,l), same to (d,e,f) with RCP excitation instead. All results of |E_z| amplitude and Poynting vector amplitude are normalized by the maximum value in each figure.

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To reveal the influence of NIM on SOI, we also studied the formation of SPP vortex with a NIM cylinder, as shown in Fig.1(a) under both LCP and RCP excitations. Figures 2(d)–2(f) and Figs. 2(j)–2(l) present the corresponding amplitude of the evanescent electric field component |E_z|, its phase, and the in-plane Poynting vector on the NIM-Ag interface for the LCP excitation σ₊ and the RCP excitation σ₋, respectively. For SPP field and phase distribution both inside and outside the NIM region surrounded by a dashed white circle, they all have the concentric-ring shape with the same period as those without NIM, except the SPP field intensity are weaker due to NIM-induced reflection and absorption loss, and the phase experiences an overall rotation in the presence of NIM due to the refraction of SPP field at the air-NIM interface and through the NIM region. This means that the spin-to-orbital angular momentum conversion still obeys the same rule with SPP experiencing both normal and negative refractive index. The differences lie in the in-plane Poynting vector distributions for both LCP and RCP. By comparing the spiraling direction of Poynting vectors in the insets of Fig. 2(f) to Fig. 2(c) and that in Fig. 2(l) to Fig. 2(i), we found that the presence of air-NIM interface leads to a reversed rotation direction of energy flux in Figs. 2(f) and 2(l) compared to that in Figs. 2(c) and 2(i), respectively.

To further investigate the effect of NIM on the properties of the interaction between SAM and OAM, we extended the study to excitations with LCP (σ₊) and a different initial OAM topological charge of l_i which would induce the circular plasmonic cavity eigenmodes in the plasmonic lens [31,32],

Figure 3 presents similar cases to those shown in Fig. 2, except with l_i=−1 for the excitation in Figs. 3(a) and 3(d) and l_i=+1 in Figs. 3(g) and 3(j), as indicated in the insets of them. The negative refraction, as observed in Fig. 2, also has a minimal effect here on the overall amplitude and phase distribution of the SPP vortex field and still does not alter the angular momentum conversion process governed by the SOI, i.e., the converted SPP vortex field has a final topological charge of l_f= l_i −σ₊. Due to this rule, we note that both the field amplitude shown in Figs. 3(a), 3(d), 3(g), and 3(j), and phase distribution of the SPP field shown in Figs. 3(b), 3(e), 3(h), and 3(k), reflect the existence of a final topological charge l_f=−2 and 0 for the LCP excitation bearing an initial topological charge l_i=−1 and +1, respectively. The LCP excitation with l_i =−1 would induce an SPP vortex field with a typical donut shape of field amplitude distribution (Figs. 3(a) and 3(d)) and a 4π phase winding anticlockwise (insets of Figs. 3(b) and 3(e)), indicating a vortex of the topological charge of l_f= l_i −σ₊=−1−1=−2. Whereas, the LCP excitation with l_i=+1 would experience the SOI annihilating l_i=+1 by the angular momentum originated from the SOI process, i.e., l_f= l_i −σ₊=1−1 = 0, giving a bright peak in the center (Figs. 3(g) and 3(j)) and uniform phase distribution (insets of Figs. 3(h) and 3(k)) due to the lack of spiral wavefront. Again, the presence of NIM would transform the anticlockwise energy flux in normal refraction to clockwise energy flux during negative refraction, which is evident from comparing the insets of Fig. 3(f) to Fig. 3(c). In the insets of Figs. 3(i) and 3(j), the energy flux vectors are almost of zero magnitudes because of the absence of topological charge (l_f=0) and the interference of SPP fields with counteracted energy flux. The results from RCP excitations with initial orbital angular momenta l_i=±1 would be similar to the LCP cases shown in Fig. 3, and the final in-plane energy-flux can also be reversed by changing the initial orbital angular momenta to, e.g. l_i=+2, with LCP excitation as studied in the SOI in highly focused systems [37]. All these results support the spin-to-orbital angular momentum conversion rule l_f= l_i −σ_±, so they are not shown here.

 

Fig. 3. SPP vortex induced by LCP excitation (green arrows) with spiral phase front characterized by an initial topological charge of l_i=±1 in the presence/absence of NIM. With l_i=−1, (a) SPP vortex field |E_z| amplitude, the inset shows the handedness of the circularly polarized excitation field (LCP with green circular arrow) and the spiral phase distribution of an initial topological charge (background color map using middle colorbar on the last row). (b) phase with inset of central spiral phase distribution, (c) in-plane Poynting vector amplitude overlapped with central energy flux direction distribution (see inset). (d,e,f), similar to (a,b,c) except with NIM indicated by a white dashed circle of 2 um radius. (g,h,i), same to (a,b,c) with l_i=+1 instead. (j,k,l), same to (d,e,f) with l_i=+1 instead.

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Besides the formation of SPP vortex, we proceeded to study the influence of NIM and excitation helicity on the SPP focusing effect of a circular arc slot. This SPP focusing effect was initially demonstrated in semicircular grating serving as a spin-dependent plasmonic focusing lens [31,38]. Figure 4 presents the plasmonic focusing effect of an arc slot structure modified from the full circular slot shown in Fig. 1(a) with an opening angle of 120^o. Figures 4(a) and 4(c) show the SPP field distribution scattered from the arc slot by an incident LCP and RCP light. Due to SOI, the SPP field is focused to a spot shifted to the right and left side of the center for the LCP and RCP excitation, respectively. In contrast, under the influence of NIM, the focus spot is positioned to the left and right side of the origin for the LCP and RCP excitation, respectively. Since the SPP focus becomes weaker due to NIM-induced reflection and absorption loss, we plot in Fig. 4(e) the profile of |E_z| across the central focus along the x-axis to compare the focal positions of the four cases in Figs. 4(a)–4(d). We can note that SPP focus positions are reversed when NIM is present compared to those without NIM. This can also be attributed to the reversing effect of NIM on the in-plane SPP energy flux during the spin-to-orbital angular momentum conversion.

 

Fig. 4. SPP focusing by a circular arc plasmonic lens under excitations of LCP (green arrows) and RCP (blue arrows) in the presence and absence of NIM. The NIM region is indicated by a white dashed circle of 2 um radius. Under LCP excitation, SPP field |E_z| amplitude (a) without and (b) with NIM. Under RCP excitation, (c) without and (d) with NIM. All results of |E_z| amplitude are normalized by the maximum value in each figure. (e) Profile of SPP field amplitude across the central focus along the x-axis for the case with LCP (solid blue line), LCP with NIM (dashed blue line), RCP (solid green line), RCP with NIM (dashed green line). All curves are normalized by the maximum value in the focus.

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Finally, we theoretically studied the phase and in-plane energy flux direction distribution of the SPP vortex field in the absence/presence of NIM. The understanding of field and phase distribution of the SPP vortex field across the NIM region can be obtained by examining the Bessel function of the first kind in Eq. (1)

To explain the reversed energy flux direction of the SPP vortex field across the NIM region, we solve the Maxwell's equations with the proper boundary conditions at the air-NIM interface [8,39] and obtain the Poynting vectors

These understandings can be further validated with theoretical calculations of the phase and in-plane energy flux direction distribution of the SPP vortex field in the absence/presence of NIM, according to the Eqs. (1)–(3), which are illustrated in Fig. 5. For the scenarios that there is no NIM in the center of the plasmonic lens, Figs. 5(a) and 5(b) depict the phase distribution and in-plane energy flux direction distribution of the SPP vortex field induced by LCP excitation and RCP excitation without NIM, respectively. These distributions are in agreement with those FDTD simulation results presented in Figs. 2(b), 2(c), 2(h), and 2(i). The same distributions are shown in Figs. 5(c) and 5(d) for the SPP vortex field induced by LCP excitation with initial OAM l_i=−1 and l_i=+1, respectively, which again agree well with the results in Figs. 3(b), 3(c), 3(h), and 3(i). The theoretical calculations presented in Figs. 5(e)–5(h) have the same excitation conditions as those in Figs. 5(a-d), respectively, except in the presence of NIM. These calculations are also in agreement with those simulation results shown in Figs. 2(e), 2(f), 2(k), 2(l), 3(e), 3(f), 3(k), and 3(l). By comparing Figs. 5(a)–5(d) to Figs. 5(e)–5(h), it is obvious that the phase difference between the cases without and those with NIM, as illustrated in Figs. 2 and 3, are reproduced in the theoretical calculations using Eqs. (1)–(3).

 

Fig. 5. Theoretical calculations of SPP vortex field phase and in-plane energy flux direction distribution in the absence and presence of NIM. Color background and arrows indicates phase distribution and in-plane energy flux direction distribution, respectively, of SPP vortex field induced by LCP excitation in (a) and RCP excitation in (b) without NIM, induced by LCP excitation with initial OAM l_i=-1 in (c) and l_i=+1 in (d) without NIM. (e,f,g,h) illustrate the same quantities as in (a,b,c,d), respectively, with NIM instead.

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3. Conclusion

In this work, we studied the effect of NIM on the SOI using plasmonic lens structures to form SPP vortex and to focus SPP under circularly polarized excitations. We found that the presence of NIM would reverse the in-plane SPP energy flux direction while leaving the SPP vortex field and phase distribution almost untouched, except some weakening and distortion due to NIM-induced loss and interference. In the case of the plasmonic lens in the form of a circular arc slot, NIM would shift the SPP focus position to the opposite position to those in the absence of NIM. Our work reported in this article considers the two effects in the same scheme, which would provide insights into novel applications based on the regulatory effect of negative refractive materials on SOI.

In this work, although NIM is directly set and modeled in the FDTD simulation environment, in principle the results reported here are also applicable to other effective NIM realized in photonic crystals or multilayer metamaterials. We emphasize that here the small loss chosen for the NIM is to help reveal the interplay between NIM and SOI, nevertheless various practical realizations of NIM have been studied with meticulously designed experiments, this aspect of NIM in the context of SOI is not at the center of this work but could be a possible fruitful path to explore. In the respect of realizing NIM, some specific nanostructures or metamaterials have been designed, such as metal-dielectric-metal waveguide made of silver and gallium phosphide [40], and the application of gain material is also a viable way to achieve realistic NIM [41]. Future developments in this direction are expected to realize novel spin-orbit angular momentum control devices. The reversed rotation of in-plane energy flux can be employed to rotate particles and control the trajectories in plasmonic tweezers, working as tunable optical spanner or micromotor in fluid, and can also be used in on-chip optical signal processing, such as coupling optical signals into waveguides with reversed propagation direction [42–46]. Given the current research interests and efforts on NIM and SOI in electronic and acoustic systems, our work on the interplay between optical NIM and SOI could also help reveal the rich and intriguing physics involving hybridizations among photons, phonons, and electrons, which would have potentially important implications for novel quantum materials and devices [5,47,48].