Optical Spin Sorting Chain

Transverse spin angular momentum of light is a key concept in recent nanophotonics to realize unidirectional light transport in waveguides by spin-momentum locking. Herein we theoretically propose subwavelength nanoparticle chain waveguides that efficiently sort optical spins with engineerable spin density distributions. By arranging high-refractive-index nanospheres of different sizes in a zigzag manner, directional optical spin propagation is realized. The origin of the efficient spin transport is revealed by analyzing the dispersion relation and spin angular momentum density distributions. In contrast to conventional waveguides, the proposed asymmetric waveguide can spatially separate up- and down-spins and locate one parity inside and the other outside the structure. Moreover, robustness against bending the waveguide and its application as an optical spin sorter are presented. Compared to previous reports on spatial engineering of local spins in photonic crystal waveguides, we achieved substantial miniaturization of the entire footprint down to subwavelength scale.

engineering of local spins in photonic crystal waveguides, we achieved substantial miniaturization of the entire footprint down to subwavelength scale.
Advances in the field of nanophotonics have paved the way toward highly integrated photonic circuits by miniaturizing optical components. Recently, recognition of the spinmomentum locking (SML) of light, in analogy to the quantum spin Hall effect, is revolutionizing the field. 1 SML is the key concept of chiral light-matter interactions, where the direction of momentum is fundamentally locked by the polarization state of light. 2 Using this idea, spin-photon interfaces are now capable of uniquely determining the information transfer direction from or to a quantum emitter according to its spin. There are versatile fields that gain benefits from chiral lightmatter interactions, ranging from valleytronics to quantum information processing. 3,4 In particular, quantum network designs would be substantially accelerated through the development of basic device elements that works as qubits, memories, nodes, paths, routers, isolators, and so on.
SML is a universal phenomenon accompanied by evanescent waves; 2 in principle, no complicated structures are required to realize chiral interactions. Indeed, SML has been demonstrated in various photonic structures, such as metal surfaces [5][6][7] , fibers 8 , strips [9][10][11] , and photonic crystal waveguides 12,13 . In such symmetric one-dimensional (1D) waveguides, SML is typically achieved by positioning an individual scatterer beneath the waveguides or by embedding or placing a light source (e.g., quantum dots) at a position offset from the propagation so that spins with different parities have different spatial or energetical distribution. This however requires sophisticated and accurate positioning of the emitter since the spin densities are equally distributed in the same material and a slight displacement in the material can flip the SML direction. On the other hand, in the case of photonic crystal waveguides, the local spins in the waveguide itself have 3 been spatially engineered. 12 Although its large degree of freedom to engineer optical properties is useful, the two-dimensional (2D) nature of photonic crystals can be a limitation in highly integrated photonic circuits since the entire footprints spanning over dozens of periods (typically >10λ) are required to constitute photonic bands.
Herein, we propose a novel 1D waveguide in the form of a chain, which can highly miniaturize the SML waveguides with engineerable spin densities down to a subwavelength scale.
The design of the waveguide is based on linearly aligned 1D nanoparticle chains made of highindex dielectrics. In such dielectric chain structures, it is known that light propagates with low loss (~5 dB/mm 14 ) by couplings of Mie resonances resulting from the confinement of light within each nanoparticle. [14][15][16][17][18][19][20][21][22] They are one of the promising waveguides because of various prominent features: bending stability to 90° corner 19 , all-optical modulation with a response time of 50 ps 16 , and slow light with group velocity down to 0.03 of the speed of light 17 . Our idea in this work is to introduce asymmetry to the 1D chain waveguide by a side chain to realize an SML waveguide that spatially separates up and down local spins predominantly inside and outside waveguides. In contrast to conventional 1D waveguide geometries, the spatial distributions of local spins can be tailored by simple structural parameters, such as the particle size, position, and gap. Furthermore, the discontinuous nature of the chain structure offers a facile connection to various functional photonic components including isolators and routers, which we also demonstrate in this letter.  31, if not specified. The circular dipole is composed of linearly polarized two dipoles oriented along x-and y-axes, respectively. By setting a π/2-phase difference between these orthogonal dipoles, the source has σ ± polarizations ( = 0 ( ± ) ). At first, the source is situated at the center of a larger sphere at the middle of the chain (the 16th sphere when N = 31). The spin-sorting performance of the chain is evaluated by light transmission at the rightmost (TR) and leftmost (TL) spheres. We define a starter structural parameter set (R1, R2, g) = (120 nm, 80 nm, 5 nm), which we call the "basic parameter set". These spheres have the dipolar Mie resonances (i.e., electric and magnetic dipole modes) around red to near-infrared spectral range. Further details of the simulation method are provided in the Supporting Information.  Note that the highest ∆T wavelength slightly differs depending on N because of the formation of a band within a finite number of periods ( Figure S1). Even after averaging the transmission over N (i.e., N dependent factors are eliminated), 97% of the transmitted light reaches the right at the highest ∆T wavelength. Besides, the operating wavelength can be readily tuned by the sphere size because of the scaling nature of the Mie resonance ( Figure S2). An electric field profile at λ = 0.76 μm is shown on the top panel of Figure 1d and clearly depicts unidirectional propagation of light towards the right without significant decay. Because the system does not break time-reversal symmetry, the propagating direction can be switched oppositely under σ + dipole excitation as shown on the bottom panel. These behaviors unambiguously evidence the existence of SML.
To further understand the effect of symmetry breaking by the side chain, we present the side chain size (R2) dependence of the transmission spectrum with a fixed large chain size (R1 = 120 nm) and gap length (g = 5 nm) in Figure 2. Figures 2a and b show two-dimensional (2D) transmission maps when R2 is changed from 0 nm (a linear 1D chain without the side chain) to 120 nm (R1 = R2, that is, a symmetric zigzag chain). The corresponding 2D map of ∆T is shown in  transmission in the first band. In this configuration, the directionality originates from the asymmetrically positioned excitation source. 9 Although the total transmission is comparable to the asymmetric zigzag chain for the second band, the ∆T (~ −0.2) is much smaller than the asymmetric chain. As another example of structural design, the gap length dependence is provided in the Supporting Information ( Figure S3). Besides, we remark that arrays of nanodisks, which can be produced by conventional nanofabrication approaches, have a much more degree of freedom to tailor chracteristics than those of spheres ( Figure S4).
Next, we discuss the origin of the SLM behavior in the asymmetric zigzag chain. To this end, we present dispersion relations of the linear and asymmetric zigzag chains in Figure 3. The ordinate and abscissa are a free-space wavenumber (k0 = 2π ∕ λ) and a Bloch wavenumber (β), respectively, normalized by a periodicity (a) and π. They are simulated for infinite chains using   To obtain more physical insight into the SML in this system, we now discuss the spatial distribution of the transverse optical spin associated with the LE mode. The spin of structured optical fields can be described by spin angular momentum (SAM) densities in the following formalism 23,24 :  The electric field profile of the LE' mode (λ = 0.76 μm) of the asymmetric zigzag chain in the same geometry is shown in the top panel of Figure 4b, which resembles the field profile under the circular dipole excitation (Figure 1d). In stark contrast to the linear chain, the SAM density of the asymmetric zigzag chain in the bottom panel is not symmetrically distributed with respect to the xz-plane. On the right side of the chain (i.e., for light propagating to the right), down-spins dominate the large sphere, while the gap region is dominated by up-spins. This elucidates the observed directional SAM transport in Figure 1, that is, the light from the σ − dipole positioned in the large sphere selectively propagates in the right direction. Moreover, the electric field maximum nicely overlaps the SAM density maximum at the center of the sphere. Thus, adding the side chain enables us to realize a spatial overlap between the high SAM density and field intensity (electromagnetic density of states), resulting in the efficient chiral coupling of the rotating electric dipole and the propagating mode.
In addition, the asymmetric zigzag chain offers accessible spots of efficient chiral lightmatter couplings in the gap region between large and small spheres with highly enhanced electric fields as well as the SAM density. As shown in Figure S7, we see equivalently high transmission and directionality under the excitation at the gap, while the transport direction is opposite. This provides an efficient and easily accessible coupling site to place light sources outside the structure, circumventing technical difficulties to embed quantum dots inside dielectric materials. 9,11,12,25 Finally, we demonstrate the functionalities of the proposed chain. Figure  In addition to the SML waveguiding, the introduction of asymmetry offers new functional elements that cannot be recognized in symmetric systems, for instance, an optical spin sorter (oneway splitter). As shown in Figure 5b, we design a star connection spin-sorter consisting of a linear chain as an input line and two lines of asymmetric zigzag chains as output ports. To generate the selectivity in light propagation depending on the transverse optical spins of impinging light, the asymmetric chains are arranged oppositely for upper and lower branches. The light source is a circular dipole in a linear chain, and σ − polarized light is injected into the junction. Due to the asymmetric distribution of the SAM density in the asymmetric zigzag chain, light is sorted into the upper branch. By switching the rotation of the source, the sorter branch is flipped to the bottom one. Similar junctions that selectively transport circular light into a certain direction have been reported in photonic crystal systems using an approach based on the topology. 26,27 The asymmetric zigzag chains miniaturize such devices down to a sub-wavelength scale, offering great compatibility with integrated photonics. In addition to this routing element, an optical spin isolator, that attenuates injected light of a certain parity (either up or down), can be realized by placing a damping element at one end.
In conclusion, we proposed an asymmetric zigzag chain made of silicon nanospheres as a novel platform for chiral light-matter interactions. Unlike previously studied waveguides whose symmetry is broken by the position of a light source or by an additional coupler, the chain inherently possesses SAM of light. Its large degree of freedom to design structure enables engineering of SAM densities for efficient couplings of local light sources at the electric field maxima. Furthermore, we proposed a novel design concept of spin device elements, such as an optical spin sorter (and router), which offers significant miniaturization of device footprints compared to previously reported similar systems based on photonic crystal waveguides (from > 10λ to ~ 0.5λ). To the best of our knowledge, this is the first report showing the SML phenomenon in chain systems. The proposed chain may be fabricated by assembling colloidal solutions of silicon nanospheres. [28][29][30][31] The presented strategy works as well for nanodisk arrays which are readily producible using standard nanofabrication technologies. Moreover, we expect the current