Topological insulators (TI) are new phases of matter with topologically protected surface states (SS) possessing novel physical properties such as spin-momentum locking. Coupling optical angular momentum to the SS is of interest for both fundamental understanding and applications in future spintronic devices. However, due to the nanoscale thickness of the surface states, the light matter interaction is dominated by the bulk. Here we propose and experimentally demonstrate a plasmonic cavity enabling both nanoscale light confinement and control of surface plasmon-polariton (SPP) spin angular momentum (AM) – towards coupling to topological-insulator SS. The resulting SPP field components within the cavity are arranged in a chess-board-like pattern. Each chess-board square exhibits approximately a uniform circular polarization (spin AM) of the local in-plane field interleaved by out-of-plane field vortices (orbital AM). As the first step, we demonstrate the predicted pattern experimentally by near-field measurements on a gold-air interface, with excellent agreement to our theory. Our results pave the way towards efficient optical access to topological-insulator surface states using plasmonics.
© 2015 Optical Society of America
A new paradigm in fundamental physics and device engineering has recently emerged - based on the abrupt change in topological invariants at the interface between two media . This topological phase of matter has been demonstrated to exist at the surface of certain materials with strong spin-orbit coupling, resulting in topologically protected surface state (SS) with novel physical properties including relativistic dispersion, spin locked to momentum, and dissipationless spin currents, having various potential applications including low-power electronics, quantum technologies and spintronics . Trying to understand this phenomenon by mimicking the underlying physics, the field of optical topological insulators has emerged, where the focus is on studying various photonic lattices supporting topological, scattering-free optical edge modes [3–5]. The most widely used experimental approach to directly study electronic topological insulators (TIs) is by angle-resolved photoemission spectroscopy, enabling the extraction of various surface state (SS) properties such as energy, momentum and spin, and the control over the photo-emitted spin polarization texture of the SS using light polarization [6, 7]. Nevertheless, this powerful technique requires sophisticated large-scale equipment and has limited spectral resolution due to ultraviolet excitation . Approaches based on infrared (IR) light could greatly enhance spectral resolution. Moreover, potential optoelectronic, spintronic, and photovoltaic devices based on TIs should be able to operate in the IR regime. Recently, generation of photocurrents in TIs, dependent on circular polarization of near IR illumination was demonstrated . However, in available TI materials it is difficult to distinguish the contribution of the bulk to the photocurrent from that of the surface states (SSs). The challenge therefore is to obtain optical access to the SSs of TIs, while avoiding the bulk of the material. In a system with a perfect Dirac cone dispersion, such as the idealized TI SSs, the spins of the electrons lie within the plane of their motion. However, the dispersion of actual TIs deviates from the cone shape, in particular as the energy increases from the Dirac point towards optical frequencies [9, 10] giving rise to an out-of-plane (OP) spin component [11–13]. Therefore, optical access to the TI SS can benefit significantly from nanoscale photonic in-plane (IP) fields highly confined to the TI surface which exhibit ‘circular polarization’. Surface plasmon polaritons (SPPs)  evanescently decay away from the interface between a metal and a dielectric, providing nanoscale confinement of the electromagnetic (EM) field to the interface, and certain structures have been shown to support SPPs carrying out-of-plane (OP) orbital angular momentum [15–18]. Yet, to the best of our knowledge, the excitation of in-plane (IP) circularly polarized SPPs, with out-of-plane (OP) spin angular momentum (AM) has not been studied.
Here we propose to employ spin-patterned SPPs on a gold-TI interface in order to confine EM fields to TI SSs in the near IR regime, thus enhancing the surface-to-bulk ratio of the interaction. We propose, implement and experimentally demonstrate a plasmonic square cavity which couples circularly-polarized light into SPPs carrying spatially dependent (localized) out-of-plane (OP) spin angular momentum (AM). This allows coupling to the OP spin component of the TI SS electrons. We demonstrate that this spatial SPP spin AM distribution depends on the handedness of the illumination polarization, paving the way to experiments allowing controllable optical probing of the TI SS.
The bulk of most available TIs behaves as an anisotropic dielectric with exceptionally high permittivity values [19, 20]. To describe metal-TI SPPs we employed a first-order model where only the bulk properties of the TI govern the SPP characteristics. The model takes in account the anisotropy of the TI . Optimal interaction of the SPPs with the TI SS requires: (1) Tight confinement to the interface; (2) Low losses to allow sufficient SPP propagation; (3) Considerable in-plane (IP) field components to interact with the out-of-plane (OP) spin of TI SS; (4) Controllable spin AM for the IP fields. The strong confinement of the SPP field is facilitated by the typical high permittivity values of TIs, such as Bi2Se3 e.g . (Figs. 1 (a) and (b)) maximizing the interaction of the EM field with the TI SS. We show that the plasmonic figure of merit, FOM = Re(kspp)/2πIm(kspp), (kspp the SPP propagation constant) is high throughout the near-IR regime due to the low bulk losses in TIs, given by the imaginary part of the permittivity, This demonstrates significant SPP propagation length >10 SPP wavelengths (Fig. 1(d)). The out-of-plane (OP) spin AM will be related to the in-plane (IP) SPP field components; therefore an important requirement is having a relatively large IP field component inside the TI. The field components ratio EIP⁄EOP for the selected case is presented in Fig. 1(c) showing that as much as 25% of the total SPP energy is concentrated in the IP component. To demonstrate we can achieve control over the spin AM of the SPP field, we employ a square SPP cavity under circularly polarized illumination. The spatial field pattern, formed as a consequence of the interferences within the square, is of high importance to our goal of coupling photons to helical TI states. The functionality of the square cavity, can be interpreted by considering first two parallel vertical slits, spaced at a distance h and illuminated by horizontally polarized light (Fig. 2(a)). Omitting the anisotropy for simplicity of the following expressions and without loss of generality, the SPP field generated by a single vertical slit over an isotropic dielectric (z<0) under horizontally polarized illumination is given by [14, 22],
A slit acts as a polarization filter coupling to SPPs only the polarization component normal to the slit. For two parallel vertical slits the IP field of the counter propagating SPPs forms a horizontal interference pattern, . Similarly two horizontal parallel slits (Fig. 2(b)) with vertically polarized illumination will yield a vertical interference pattern, .
A square cavity is comprised of both sets of parallel slits. By means of the superposition principle we obtain the field inside the square cavity under circularly polarized illumination (Fig. 2(c)). The circular polarization can be expressed in terms of the linear polarization basis as. The vertical and horizontal slits will filter the and components respectively with the relative phase maintained by the linear system, resulting in a 2D vector IP field pattern given by, f(x) ± if(y). As can be deduced from the above expression, wherever f(x)≈f(y) is fulfilled, the resulting local field is essentially circularly polarized. That is, at a given spatial point the field rotates in time within the plane of the cavity. The handedness of the rotation is determined by the relative sign between the and components at each location. Throughout the cavity we obtain SPP fields dissected into small squares – in each of the latter the phase between the IP components, is a binary value of ± π⁄2 (Fig. 3). Each region of constant phase indicates a global rotation direction of the IP field in that region resulting in an array of alternating right ( + ) or left (-) rotating field regions. Thus, every square region of well-defined rotation handedness carries out-of-plane (OP) spin AM, with counter rotation compared to its nearest neighbors.
For each chess-board square we calculated the circular polarization purity factor ,
We performed a full finite difference time domain (FDTD) study of the square cavity milled in gold on top of Bi2Se3 (experimental data given in ) using Lumerical Solutions, Inc. simulator. The IP SPP field components (top row of Fig. 3) form a 2D standing wave (Fig. 3(c)) yielding a chess-board-like phase pattern (Figs. 3(a) and 3(b)) comprised of square cells with alternately left or right handed rotating IP field components (solid white arrows), carrying predominantly “up” or “down” SPP spin AM as described above. Moreover, switching the handedness of the input polarization switches the sign of the SPP spin in all chess-board squares (Figs. 3(a) and 3(b)).
The overall field pattern is more intricate due to the presence of the OP field, Ez∝sin(kspp(x + h⁄2)) ± i∙sin(kspp(y + h⁄2)) obtained using the same superposition approach. Taylor expansion of the above expression around any of the zero points of the field, r0 = (x0,y0), results in, Ez(r0)≈(x-x0) ± i(y-y0)≡. The phase of the OP field in the vicinity of each of its vanishing points, given by ϕ = atan(/), defines a phase singularity at that point. Moreover, encircling the singularity results in a 2π phase accumulation, characteristic of a scalar vortex of charge +/−1 . The spatial field distribution of the vortices is clearly visible in the FDTD simulation (bottom row of Fig. 3); Figs. 3(d) and 3(e) show the array of vortices (dashed black arrows) formed inside the square cavity. Each vortex contributes additional OP orbital AM with respect to its origin. However, the coupling of optical orbital AM to electrons is negligible compared to the spin AM contribution . Overall, within each of the squares comprising the field pattern, the local AM is spin-like, allowing coupling to a specific branch of the spin-helical topological state – with a reversed direction in adjacent squares. Only at the boundaries of each square the AM is contributed by the vortices of the OP field which is predominantly orbital. Finally, it is confirmed in Figs. 3(a) and 3(b) that switching the handedness of light polarization switches the sign of the SPP spin in all chess-board squares as well as the charge sign of the vortices (Figs. 3(d) and 3(e)), thus flipping the total AM.
Following our theoretical modelling, we fabricated the proposed cavity by focused ion beam engraving in a 150nm thick gold layer that was sputtered onto a glass substrate (Fig. 2(d)). Concentric squares, with λspp spacing, were added to the cavity design to increase the SPP generation efficiency in a Bragg-like manner . The near-field pattern formed by the SPPs on the gold-air interface, was recorded by a transmission mode aperture-less phase resolved near-field scanning optical microscope (Neaspec GmbH). The sample was placed on a moving stage and illuminated from the substrate direction with a circularly polarized beam at λ0 = 671nm wavelength using a semiconductor laser. To maintain an approximately uniform spatial phase and amplitude of the illumination over the scanning area, the illuminating beam was weakly focused to a 30μm spot. A metallic tip scattered predominantly the OP field component into a pseudo heterodyne detection scheme  that was used to remove the background and provide phase information of the field obtaining full information on the complex OP field distribution. The experimental results in Figs. 4(a) and 4(c) show the array of counter rotating vortices as discussed above. Both phase and amplitude measurements remarkably match our prediction and the detailed FDTD simulations in Figs. 4(b) and 4(d). The bottom right lobe of the experimental interference pattern in Fig. 4(c) appears brighter than the rest of the lobes and the corresponding location on the phase measurement in Fig. 4(a) contains a local distortion of the phase pattern.
Despite the high resolution of the measurement (10-20nm) compared to the SPP wavelength the rapidly varying azimuthal phase in the immediate vicinity of the singular vortex points is slightly smeared comparing to the simulation results of Fig. 4(b). Direct measurement of the IP SPP is not feasible by our experimental configuration as it is ~15 times weaker than the OP field and the scattering of the IP field by the metallic tip is very small. Nevertheless, the obtained high resolution OP field measurements, and their underlying straightforward functional connection to the IP components through the Gauss law of Maxwell’s equations, provide sufficient experimental verification of our design.
In conclusion, we proposed plasmonics of metal-TI interfaces as a tool for optical access to the TI SS in the near IR regime. We demonstrated the feasibility of the approach on a gold-Bi2Se3 interface using experimental data of their optical dielectric constants. Highly confined SPPs are obtained throughout the near IR range with significant FOM localizing the interaction of the SPPs to the interface. As a first step towards full realization of the approach, we designed and demonstrated experimentally a SPP square cavity that couples circularly polarized illumination into a chess-board-like pattern of standing SPPs. Each chess-board square is comprised of SPP with circularly polarized IP field components carrying OP spin AM allowing coupling to the spin-momentum locked TI SS. The orientation of the AM in each cell was shown to be controlled externally, by the handedness of the illumination polarization and is to be used as a control parameter in future spectroscopic pump-probe experiments. Our experimental results of the near-field vortex arrays exhibit excellent agreement with the performed FDTD simulations. Our results open new directions in the future investigation of TIs. By enabling the sculpting of complex near-field excitations at the surface using readily available fabrication techniques, we believe we could access the spin-momentum locked dispersion of TI’s, investigate their band structure and generate spin photocurrents. We believe that pursuing this approach can potentially bring the incorporation of TI SSs into the fields of optoelectronics and spintronics operating in the near IR regime.
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