Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024
Advancing Society with Light, a special issue from general congress ICO25OWLS16DresdenGermany2022



Article Number  16  
Number of page(s)  14  
DOI  https://doi.org/10.1051/jeos/2024008  
Published online  29 April 2024 
Review Article
Spinorbit interactions of light: Fundamentals and emergent applications
^{1}
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina
^{2}
CONICETUniversidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Buenos Aires, Argentina
^{*} Corresponding author: gpuentes@df.uba.ar
Received:
1
November
2022
Accepted:
23
February
2024
We present a comprehensive review of recent developments in Spin Orbit Interactions (SOIs) of light in photonic materials. In particular, we highlight progress on detection of Spin Hall Effect (SHE) of light in hyperbolic metamaterials and metasurfaces. Moreover, we outline some fascinating future directions for emergent applications of SOIs of light in photonic devices of the upcoming generation.
Key words: SpinOrbit Interactions / Orbital Angular Momentum / Spin Hall Effect / Weak measurements / Photonic materials
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Light’s polarization degrees of freedom, also known as Spin Angular Momentum (SAM) and orbital degrees of freedom, also known as Orbital Angular Momentum (OAM), can be coupled to produce a wide variety of phenomena, known as SpinOrbit Interactions (SOIs) of light. Due to their fundamental origin and diverse character, SOIs of light have become crucial to a variety of active fields, such as singular optics, photonics, nanooptics and quantum optics, when dealing with SOIs at the singlephoton level. Among a variety of fascinating exotic phenomena, SOIs show the remarkable spindependent transverse shift in light intensity, also known as the Spin Hall Effect (SHE) and the Spin Orbit Conversion (SOC) of light.
The various planewave components of the beam that travel in slightly different directions and acquire slightly different complex reflection or transmission coefficients are what cause the regular SHE of light at a planar interface to form [1–4]. A handy quantumlike framework with generalized wavevectordependent Jonesmatrix operators at the interface, and expectation values of the position and momentum of light, provides a theoretical explanation of the photonic SHE [3–6]. Such a description also unifies the longitudinal (inplane) beam shifts connected to the Goos–Hanchen (GH) effect [3–6] and transverse SHE shifts, also known as the Imbert–Fedorov (IF) shifts in the case of the Fresnel reflection/refraction [7–10].
2D metamaterials, commonly referred to as metasurfaces, are an interdisciplinary area that encourages the use of substitute methods for light engineering based on spatially ordered metaatoms and subwavelengththick metasurfaces of different compositions. They display exceptional qualities in light manipulation in a 2D interphase. Metasurfaces can attain their 3D counterpart functions, such as invisibility cloaking and negative refractive index. In addition, they can eliminate some of the 3D metamaterial current restrictions, such as high resistivity or dielectric loss, for example. Additionally, the creation of metasurfaces via conventional methods for nanofabrication, such as electron beam lithography techniques, are readily available in the semiconductor industry.
In this review, we provide a summary of recent findings and future potentials for applications of SHE of light in photonic materials. As an optical equivalent of the solidstate spin Hall effect, SHE of light warrants promising opportunities for examination of innovative photonic materials and nanostructures physical characteristics, such as in figuring out the magnetic and metallic thin films’ material characteristics, or the optical characteristics of twodimensional atomically thin metamaterials, with unmatched spatial and angular precision resolution, a trait that SHE and other combined technologies can provide, utilizing quantum weak measurements and quantum weak amplification methods. Additionally, we provide a summary of recent developments in primary 2D metamaterials and metasurfaces applications for producing and manoeuvering Spin Angular Momentum (SAM) and Orbital Angular Momentum (OAM) of light, for applications in multicasting and multiplexing, spinbased metrology or quantum networks.
2 Spin Hall Effect of Light (SHEL)
SpinHall Effect of Light (SHEL) typically refers to a spindependent transverse yshift in the reflection or refraction of light at a sharp inhomogeneity of an isotropic optical interface [1–4]. An example of this effect is the socalled transverse Imbert–Fedorov beam shift, which occurs when a paraxial optical beam is reflected or refracted at a plane interface. A novel example of SHEL was recently reported, for transmission of light through a uniaxial crystal plate with tilted optical axis [11].
Using the terminology of the closely related work [11–13], we begin with the theoretical description of the problem. Namely, the SHE of light in a tilted photonic material. We note that what is tilted in the photonic material is the optic axis, relative to the propagation direction of the light. Figure 1a–1c depict the problem’s geometry. The normalized Jones vector describes the incident propagating paraxial beam’s polarization ψ〉 = (E_{x}, E_{y})^{ T } (T stands for the transposition operator), 〈ψψ〉 = E_{x}^{2} + E_{y}^{2} = 1. In the (x, z) plane, the photonic material is tilted so that its axis forms a θ angle (labeled ϑ for ease of notation in the following sections) with the zaxis and transmits mostly the ypolarization. The dichroic action of the photonic material in this geometry, and in the zeroorder approximation of the incident planewave field, can be characterized by the Jones matrix, of the form:(1)the Jones vector of the transmitted wave is . T_{x, y}, which can depend on θ, are the amplitude transmission coefficients for the x and ypolarized waves. While T_{x} = 0 and T_{y} = 1 for an ideal polarizer, we can assume that T_{x}/T_{y} ≪ 1 for real dichroic plates. Also take note of the fact that T_{x, y} = exp(∓iΦ/2) relates to the birefringent waveplate issue discussed in [11].
Figure 1 (a) General 3D geometry of the problem displaying the angle ϑ between the anisotropy axis of the plate and the beam axis z. (b) The wave vectors inplane deflections (Θ_{x}) cause the wellknown birefringence shift which is comparable to the GH shift, by altering the angle between k and the anisotropy axis. (c) View along the anisotropy axis of the crystal is shown. The transverse Θ_{y} deflections of the wave vectors rotate the corresponding planes of the wave propagation with respect to the anisotropy axis by the angle ϑ ≈ Θ_{y}/sin(Θ_{y}). This causes a new helicitydependent transverse shift , i.e., a spinHall effect similar to the IF shift. Further details are in the text. 
Considering that in the paraxial approximation the beam consists of a superposition of plane waves with their wavevector directions labeled by small angles Θ = (Θ_{x}, Θ_{y}) ≃ (k_{x}/k, k_{y}/k) (see Fig. 1a), the Jones matrix incorporates Θdependent corrections and can be written as [11]:(2)here(3)are the wellknown GoosHanschen (GH) and Spin Hall (SHEL) terms [11], the latter being the main focus of this review.
3 Quantum weak amplification techniques
A detailed explanation of the entire theory governing transverse shifts of light in tilted uniaxial crystals was elaborated in [11–13]. The anisotropic transverse shift, or SHE of light, is represented by the expectation value of the position operator , for an input state defined by a Jones vector ψ〉, and a transmitted state described by a Jones vector ψ ^{’}〉:(4)where k is the wavevector, ϕ_{0} refers to the difference in phase between ordinary and extraordinary waves as they propagate through a birefringent material, ϑ represents the tilt angle of the photonic medium, and (σ, χ, τ) represent the Stokes parameters for the light beam. The SHE can be measured directly, using the subwavelength shift of the beam centroid [1, 4, 14–16] (Eq. (3)). Various alternative techniques, such as quantum weak measurements [17–25] are also used. The latter technique enables substantial amplification, when employing almost crossed polarizers for preselection and postselection of the polarization input and output states, as they relate to the system.
The output polarizer corresponds to a postselected polarization state ψ〉 = (α ^{’},β ^{’})^{ T } whereas the input polarizer corresponds to a preselected state ψ〉 = (α,β)^{ T }, here (α(α ^{’}),β(β ^{’})) represent the input(output) polarization components of the input(output) beam (, and T stands for transposition operation. As opposed to traditional expectation values, weak values might display a quantum weak amplification effect and lie beyond the operator’s spectrum, moreover weak values can take imaginary values. We examine the quantum weak amplification of the SHE shift using an initial beam with e polarization ψ〉 = (1,0)^{ T }, and a nearly orthogonal polarization state ϕ〉 = (ϵ,1)^{T}, ϵ ≪ 1, for the postselection polarizer. In this configuration, the SHE weak value results in:(5)here z_{R} stands for the Rayleigh length. The second angular term, becomes dominant in the far field zone, and presents weak amplification due to two reasons: First, due to the fact that ϵ ≪ 1, and second due to the fact that z ≫ z_{R}, in the far field regime. It should be noted that the maximum weak amplification that can be achieved at ϵ ≈ (kω_{0})^{−1} is of the order of the beam waist ω_{0} z/z_{R}.
We compute the expected value of the centroid displacement based on the phase difference acquired by the beam (Eq. (3)) as it propagates through the photonic material to estimate the classical beam shift. This is how the phase difference accumulated during propagation in the photonic material is expressed:(6)where, is the refractive index for the extraordinary wave, and n_{o} is the refractive index for the ordinary wave, which propagate at the angle ϑ to the optical axis. The distances of propagation of the ordinary and extraordinary rays in the tilted plate are:(7)
Using equations (3)–(6) the expectation value of the spin Hall shift and its weak value can be derived, and contrasted with experimental findings.
4 SAM, OAM and TAM of Light
4.1 SAM of light
In optics, the Spin Angular Momentum (SAM) and Orbital Angular Momentum (OAM) of light can be observed separately. A SpinOrbital Angular Momentum decomposition for paraxial monochromatic beams is simple. This distinctive property, which inspires this review, explains in part the recent unrivaled development in photonic SAMOAM conversion in metasurfaces and 2D metamaterials. At the same time, the spin and orbital description of quantum electromagnetic field theories produce a variety of complexities, in the generic nonparaxial or nonmonochromatic angular momentum description [10, 11].
The SAM is connected to the polarization of light in the unified theory of angular momentum of light, which is based on canonical momentum and spin densities developed in [11]. Accordingly, RightHand Circular (RHC) and LeftHand Circular (LHC) polarizations of a paraxial beam correspond to positive and negative helicity σ = ±1. If the beam’s mean momentum (measured in units per photon) can be connected to its mean wave vector 〈k〉, then such beam carries the corresponding SAM 〈S〉 = σ〈k〉/k, where the helicity parameter σ is equivalent to the degree of circular polarization in the Jones formalism.
A plane wave is an idealized phenomenon that can extend to infinity. Extrinsic OAM cannot be carried by such a plane wave (like its mechanical counterpart L = r × p), because its position r is undefined. On the other hand, a circularlypolarized electromagnetic plane wave can carry SAM. In the canonical momentum representation, the vector describing the electric field, for a circularly polarized planewave propagating along the zdirection can be written as [26]:(8)where () are unit vectors and the helicity parameter σ = ±1 corresponds to the LHC and RHC polarizations, respectively. The wave number k, results from the dispersion relation for a plane wave, that is, k = ω/c. The electric field described in equation (8) represents the eigenmode of the zcomponent of the spin1 matrix operators with eigenvalue σ, of the form [26]. Where the spin1 operators (generators of the SO(3) vector rotations) are given by [26]:(9)
Therefore, the plane wave carries SAM density , defined as the local expectation value of the spin operator with the electric field equation (8).
4.2 OAM of light
In 1936, Beth made the first demonstration of the mechanical torque produced by the transmission of angular momentum from a circularly polarized light beam to a birefringent plate [27, 28]. In this experiment, a fiber suspended a fine quartz quarterwave plate. Such plate transforms RHC polarization, with spin component +ℏ, into LHC polarization, with spin component −ℏ, with a net SAM transfer of 2ℏ per photon to the birefringent plate. Beth measured torque, known as the measurement of SAM of the photon, agreed in sign and modulus with the quantum and classical expectations.
In reference [28], LaguerreGaussian modes with azimuthal angular dependency (exp[−ilϕ]) were shown to exist, which are eigenmodes of the momentum operator L_{z} and carry an Orbital Angular Momentum lℏ per photon. Using the vector potential, a proper representation of a linearly polarized TEM_{plq} laser mode can be obtained, in the Lorentz gauge [27]:(10)where u(r,ϕ,z) is the complex scalar field amplitude satisfying the paraxial wave equation and is the unit vector in xdirection. In the paraxial regime, du/dz is taken to be small compared to ku and second derivatives and the products of first derivatives of the electromagnetic field are ignored. The solutions describing the LaguerreGauss beam, for the cylindrically symmetric case u_{r,ϕ,z}, are of the form [28]:(11)
where w(z) is the radius of the beam, is the associated Laguerre polynomial, z_{R} is the Rayleigh range, C is a constant and the beam waist is taken at z = 0. Within this description, the time average of the real part of the Poynting vector ϵ_{0} E × B, results in:(12)where z is the unit vector in zdirection. When applying to a LaguerreGaussian distribution given in equation (4), the linear momentum density takes the form:(13)where are unit vectors. It may be seen that the Poynting vector (c ^{2} P) spirals along the direction of propagation along the beam. The z component relates to the linear momentum, the r component to the spatial dispersion of the beam, and the ϕ component generates the OAM.
5 Photonic materials: metamaterials and metasurfaces
5.1 SAM Control in Metasurfaces
Metasurfaces enable polarization conversion in the same way as ordinary wave plates by maneuvering two light eigenmodes that correspond to orthogonal polarization. The Jones vector of the incident field and the Jones vector of the desired output fields must be known in order to create a metasurface for polarization conversion (Fig. 2). With this information, it is possible to determine a Jones matrix J(x, y) for each spatial point (x, y) on the metasurface plane connecting the incident and output waves. Appropriate nanoantenna designs can be engineered to produce the calculated Jones matrix. Metasurfaces have been used to convert between linear and circular polarizations, between various linear polarization states, and between opposing circular polarization states [29–31]. Because circularly polarized input light is one of geometricphase metasurfaces main limitations, it is necessary to precisely match the geometric phase to the propagation phase in order to obtain unrestricted control of polarization states. Typically, hybrid nanoantenna patterns are used to achieve this.
Figure 2 (a) Representation in Poincare sphere of SpinOrbitConversion (SOC) via qplates. A state of circular polarization located in the poles of the Poincare sphere is mapped into its opposite state of circular polarization, while imprinting a fixed azimuthal phase ±2qϕ. (b) Representation in Poincare sphere of spinorbit conversion (SOC) via Jplates. Arbitrary states of elliptic polarization λ±〉 are mapped into its opposite state of elliptic polarization, while imprinting a tunable azimuthal phase (n, m)ϕ. 
In the context of this manuscript, the term metasurface refers to a nanostructured 2D metamaterial, the thickness of which is less than or comparable to the value of the spatial attenuation of surface evanescent waves. Thus, we do not refer to 2D nanostructures such as quantum wells or monolayers of solid materials, the physics of which is based precisely on the absence of a third spatial coordinate.
5.2 OAM control in metasurfaces
In order to create optical vortex beams with net topological charge or OAM of light (l), polarizationcontrol in metasurfaces that permit manipulation of the propagation phase and the geometrical phase can also be used [27–44]. The helixed wave front of such OAM beams, which is a direct result of the dependence of phase on azimuthal angle, is one of its most distinctive features. The number of twists a wavefront has, determines the integer l that represents the order of OAM (per unit wavelength). Since beams with different orders of OAM are orthogonal they do not interfere with each other, widespread acceptance of OAM as an unrestricted degree of freedom of light that can be used in fast freespace optical communication systems has been achieved [36–45].
Spatial Light Modulators (SLM), holograms, laser mode conversion, and Spiral Phase Plates (SPP) are examples of conventional techniques for producing OAM beams. On the other hand, by positioning nanoantennas with linearly increasing (or decreasing) phase shifts along the azimuthal direction, metasurfaces can produce helical wave fronts [46–52]. As a result, a metasurface can add an optical vortex to the incident light wave front, thus converting SAM into OAM, this transformation is also termed spintoorbit conversion (SOC) [29–31, 53, 54]. Due to the conservation of total angular momentum, the SOC typically facilitates the conversion of LHC and RHC polarization into states with opposing OAM (TAM). A metasurface can transform circular polarizations into states with independent values of OAM by adding an additional phase shift in the azimuthal direction. Recent years have seen the demonstration of the transformation of light with arbitrarily elliptical polarization states into orthogonal OAM vortex states [54, 55].
5.3 TAM control in metasurfaces
Precise control of the Total Angular Momentum (TAM) of light can also be achieved in metasurfaces. More specific, it is possible to introduce a controlled geometric phase and establish a link between polarization (SAM) and phase using phase elements with spatially variable orientations. According to [27], these devices are often constructed from periodic elements known as qplates. The precise transformation made by qplates can be written as [31, 32]:(14)where circular polarization LHC (L) and RHC (R) are mapped to its antiparallel spin state with an acquired OAM charge of ±2qℏ per photon. SpinOrbit Conversion (SOC) is the common name for this transformation. As introduced in the previous sections, the rotational elements (i.e., θ(x, y)) are often the only ones that change the angle spatially. As a result, the OAM output states are limited to conjugate values (±2qℏ). Additionally, the SOC operation carried out by qplates is restricted to SAM states of circular polarization as a result of the device symmetry. A more general device is needed in order to perform the SOC operations of different SAM states, including elliptic polarization. As will be discussed later, Jplates can carry out this arbitrary mapping (Fig. 2).
A Jplate (represented by the variable J) has the capacity to transfer two arbitrary TAM states into two arbitrary input SAM states, including but not limited to RHC or LHC. Any media that allows birefringence, absolute phase shift, and retarder orientation angles to vary spatially can be used to produce a Jplate. In other words, the relative phase shift between orthogonal spins (ϕ = ϕ(x, y)) should vary spatially in addition to the fast axes of rotating plates. Metasurfaces were used to construct such Jplates [31]. The Jplate’s precise transformation is best described as (Fig. 2):(15)where λ±〉 are arbitrary elliptical polarization expressed as [31]:(16)
the parameters (χ, δ) determine the polarizations state. Implementing the SOC operation reduces to determining the actual Jones matrix J(ϕ) that transforms J(ϕ)λ+〉 = e^{imϕ }(λ+)^{*}〉 and J(ϕ)λ−〉 = e^{inϕ }(λ−)^{*}〉. It can be demonstrated that the required spatially varying Jones matrix has the form:(17)
Unfortunately, the necessary control and subwavelength spatial variations in phase shift, birefringence, and orientation cannot be achieved with a conventional phase plate. On the other hand, subwavelength space metasurfaces and 2D metamaterials may provide such unprecedented control [56–69].
6 SHEL in photonic materials: experimental realizations
6.1 Experimental configurations
To confirm the above theoretical predictions for SHE in tilted photonic materials, we performed a series of experimental measurements using the setups shown in Figure 3 [12, 13]. While in reference [12] we use a sample of freestanding birefringent polymer foil, similar to the type Newport 05RP321064, as photonic material, in reference [13], we use a hyperbolic trench metamaterial structure. As a source of incident Gaussian beam, we employed a He–Ne laser (Melles Griot 05LHR111) of wavelength λ = 633 nm. The laser radiation was collimated using a microscope objective lens. We measure the anisotropic phase difference Φ_{0} versus the angle of the tilt ϑ via Stokes polarimetry methods [12, 13]. For this purpose we used the setup shown in Figure 3a.
Figure 3 Schematic depiction of experimental configurations for (a) polarimetric measurements and (b) quantum weak measurements. P1 and P2 represent double GlanLaser polarizers (Thorlabs GL10), QWP is quarter wave plate. L1 and L2 are used to identify lenses. He–Ne (Melles Griot 05LHR111) laser with a 633 nm emission wavelength; Thorlabs WFS1505C CCD camera type. The ordinary and extraordinary permittivities of the photonic material sample, and their corresponding axes, are denoted as ϵ_{o} and ϵ_{e}, respectively. 
The appropriate linearpolarization state in the incident beam was chosen using the twin GlanLaser polarizer (Thorlabs GL10) (P1). This was oriented at 45° polarization in the first experiment, equivalently in the formalism introduced in the previous sections. The Stokes parameters are then determined by passing the beam through the photonic material sample while utilizing a quarter wave plate (QWP) with a retardation angle of δ and a second polarizer P2 with a rotation angle of γ, as shown in Figure 3a. The Stokes parameters can be used to determine the phase difference using the expression:(18)where S_{2} = I(0°, 45°) − I(0°, 135°) is the normalized Stokes parameter in the diagonal basis, and S_{3} = I(90°, 45°) − I(90°, 135°) is the circular polarization’s normalized Stokes parameter, here the normalization factor S_{0} is given by the total intensity of the beam. The measured phase using equation (5) is wrapped in the range (−π, π). To calculate the unwrapped phase difference, we employ an unwrapping technique with a tolerance set to radians [12, 13].
6.2 SHEL in birrefringent polymers
In reference [12], a full experimental demonstration of SHE of light in birefringent polymers was provided. The schematic of tilted birefringent polymer film is depicted in Figure 4a. We observe a spinHall effect via Stokes polarimetry () using a 50 μm polymer film which is 10 times larger than the shift observed in reference [11], for a 1000 μm Quartz sample. We attribute this increase to the polymers’ greater effective birefringence. We also investigate the impact of tunable birefringence in the polymer film. By applying controllable voltage, leading to tunable stress in the polymer, it is possible to induce tunable birefringence in the polymer film, which can in turn induce controllable light shifts. In Figure 4b, we present numerical simulations of SHE for a set of stressinduced birefringence ranging from Δn = 0.009, 0.03, 0.06, 0.07 as a function of tilting angle ϑ, maximal tunability is achieved at ϑ = 0.3 radians [12].
Figure 4 (a) Paraxial beam’s transmission through a clear birefringent polymer slanted layer as shown in 3D geometry. The beam goes through transversal shift at the nanometer scale brought on by the Spin Hall Effect (SHE) of light. The paraxial angles (Θ_{x}, Θ_{y}) identify the propagation direction of the incident beam’s wave vectors k. (b) A tunable birefringent polymer results in enhanced SHE of light as shown in numerical simulations considering different birefringence: Δn 0.009 (gray, index difference in quartz), Δn 0.03 (green), Δn 0.06 (blue), and Δn 0.07 (red, index difference in stretched polymers [29]). Vertical dashed line at ν 0.3 (rad) is to show the tunability of beam shift by different birefringence. (c) Transverse intensity distributions (a.u.) in false scale for an opolarized beam transmitted through a tilted polymer plate and postselected in the almost epolarized state, with a tilt angle ϑ = 20°. Top: Postselected polarization state with ϵ = −1/83 × 10^{−2}. The beam centroid is shifted, resulting in a measurement of weak value . Middle: With crossed polarizers (ϵ = 0), a HermiteGaussian intensity distribution with peaks spaced apart from one another by ΔY = 1000 μm is created from a Gaussian distribution, corresponding to a weak amplification factor A = 200. Bottom: Postselected polarization state with ϵ = 1/83 × 10^{−2}, which corresponds to a weak value measurement of . 
Additionally, using the quantum weak measurement apparatus shown in Figure 3b, we measured the spin Hall shift weakly and observed the quantum weak amplification effect. A CCD camera (Thorlabs WFS1505C) is used to image the beam. To achieve this, two lenses (L1) and (L2) with a focal distance of f = 6 cm were implanted. Preselected and postselected polarization states are produced by polarizers GlanThompson Polarizers P1 and P2, respectively, having polarization states of ψ〉 and ψ ^{’}〉. The first lens (L1), which had a 6 cm focal length, generated a Gaussian beam with a waist of 30 μm and a Rayleigh range of z_{R} = 4.6 mm. Therefore, z/z_{R} = 10.86 is the propagation amplification factor for a CCD camera placed at a distance of z = 5 cm. The amplification factor due to crossed polarizers results 1/ϵ ≈ 1.83 × 10^{−2}. For , the overall weak amplification factor becomes = 200, this is confirmed in the experiment which involved a displacement between centroids of ΔY = 1000 μm between postselection polarizers oriented at ϵ = −1/1.83 × 10^{−2} (Fig. 4c, Top), and ϵ = +1/1.83 × 10^{−2} (Fig. 4c, Bottom) is measured at a tilt angle ϑ = 20°, consequently, the SHE is amplified by a factor A = 200. For crossed polarizers (ϵ = 0), a HermiteGaussian distribution is created from the input Gaussian beam (Fig. 4c, Middle), and the two centroids are roughly separated from one another by approximately ΔY = 1000 μm.
In conclusion, we experimentally demonstrated the fine lateral circular birefringence of a tilted birefringent polymer, the first instance of the SHEL in a polymer material [12]. We revealed experimental results of this nanometerscale phenomena and found a quantum weak amplification factor of 200 using Stokes polarimetry and quantumweak measurement techniques. Because the polymer’s birefringence may be controlled using mechanical stress in the case of stressinduced birefringence or voltage in the case of liquid crystals, this lateral shift could be utilized as an optical switch at the nanoscale scale. Numerous cuttingedge applications in photonics, nanooptics, quantum optics, and metamaterials might become possible as a result. Such emergent application of SHEL are discussed in the following sections.
6.3 SHEL in metamaterials
In reference [13], we present experimental findings for SHE of light in Hyperbolic Metamaterials (HMM). In particular, we experimentally show enhanced spinHall effect of light in HMMs in the visible region and exceptional angle sensitivity. As shown in Figure 5a, the effect is shown in the transmission configuration using an HMM that is a few hundred nanometers thick and made up of alternating layers of metal and dielectric as illustrated in Figure 5a. With a change of ≈0.003 rad (≈0.17°) in the angle of incidence, the transverse beam shift in the HMM configuration can alter dramatically, from essentially no beam shift to a few hundred microns. Therefore, it is to be expected that compact spin Hall photonic devices can take advantage of the huge photonic spin Hall enhancement in such a tiny structure, with a great angular sensitivity to manipulate photons via polarization.
Figure 5 (a) Scheme of the Spin Hall Effect (SHE) of light in hyperbolic metamaterials. The incidence angle, indicated by θ_{i}, switches the transverse beam shift along the yaxis denoted as . The HMM structure’s unit cell is made up of Al_{2}O_{3}(10 nm)–APTMS(1 nm)–Au(10 nm)–APTMS(1 nm). Eight periods totaling 176 nm in thickness make up the HMM. (b) According to the effective media approximation, the effective permittivities of a multilayer HMM structure made up of a unit cell of Al_{2}O_{3}(10 nm)–APTMS(1 nm)–Au(10 nm)–APTMS(1 nm) were calculated. The structure exhibits type II (ϵ_{o} < 0 and ϵ_{e} > 0) HMM behavior in the visible wavelength range. The vertical dashed lines indicate the wavelengths λ = 520 nm and 633 nm at which experiments were conducted. Measured and calculated spin Hall transverse shifts for (c) λ = 633 nm and (d) λ = 520 nm, respectively, are shown in solid lines. Beam waist wo is fitted to the simulation results in (a) and (b). Note that there is an estimated beam divergence of Δθi = 0.01 rad for λ = 633 nm and Δθi = 0.02 rad for λ = 520 nm indicated as lateral error bars, respectively. Further details are in the text 
The HMM sample has eight goldalumina periods placed on a 500 μm thick glass substrate for a total thickness of 176 nm. Al_{2}O_{3}(1 nm)–APTMS(1 nm)–Au(10 nm)–APTMS(1 nm) are the four layers that make up one period of the HMM structure. Amino Propyl Tri Methoxy Silane, often known as APTMS, is an almost lossfree adhesion layer that is beneficial for highly confined propagating plasmon modes [13]. Al_{2}O_{3} layer was deposited via atomic layer deposition, upon Au layer sputtering. Calculated as [18] are the HMM’s ordinary and extraordinary permittivities, indicated as ϵ_{o} and ϵ_{e}, respectively.
Based on the effective media approximation [13], HMMs are treated as homogenized uniaxial mediums with effective permittivities. The thicknesses of individual layers are assumed to be deeply subwavelength within the effective media approximation [13]. The wavelength range of λ = 500 – 700 nm, normalized by the unit cell Λ = 22 nm, yields the ratio of Λ/λ = 1/22.7 − 1/31.8. Therefore, it is justified to use the effective media approximation in our scenario.
Calculated as described in [18], the HMM’s ordinary and extraordinary permittivities, are indicated as ϵ_{o} and ϵ_{e}, respectively:(19) (20)where f_{m} and f_{d} are the volume fractions of metal and dielectric, respectively, and ϵ_{m} and ϵ_{d} are the permittivities of metal and dielectric, respectively. The Drude model with the thicknessdependent correction is used to describe the Au film’s permittivity, ϵ_{m} [19].
APTMS has a refractive index of 1.46 [20]. The dispersion of the HMM effective permittivities in the visible region is seen in Figure 5b. With λ = 500 nm as the zero crossing wavelength for our HMM structure, type II HMMs (ϵ_{o} < 0 and ϵ_{e} > 0) are formed when red wavelengths are approached.
We ran simulations based on the theory developed by Tang et al. [13] in order to comprehend how the spin Hall beam shift behaves in HMMs, with realistic parameters (Fig. 5b). Air serves as the ambient medium for the entire HMMSiO_{2} substrate structure, as depicted in Figure 5a. We consider that the incident light is impinging on the HMM structure with an incident angle θ_{i} in the y–z plane. Since ϵ_{1} = ϵ_{2} = ϵ_{5} = 1 (air), ϵ_{4} corresponds to the SiO_{2} substrate, and ϵ_{3} corresponds to the HMM, the relative permittivities of the media in regions 15 are indicated by ϵ_{i} (i = 1, 2, 3, 4, 5) respectively. The HMM is considered to be uniaxially anisotropic, nonmagnetic, and to have a relative permittivity tensor (ϵ_{3}):(21)
Taking into account the input waist Gaussian beam ω_{0}:(22)we can define the transverse beam shifts after transmission through the structure in the form:(22)where the transverse shifts for the Right Hand Circular (RHC) and Left Hand Circular (LHC) polarizations are indicated by η ^{±}, respectively. The terms for the transverse shifts contain zdependent and zindependent terms, which, respectively, represent angular and spatial transverse shifts [21]. In this case, our attention is on the spatial transverse shift of light transmitted by the HMM waveguide, which has the following form:
(24)where t_{s,p} are the transmission amplitudes for the s, p modes, respectively [13], θ_{t} is the transmission angle, and with n1 = 1 (air) are all constants. We assume transmission along the laser beam axis, thus for a large beam waist and θ_{t} = 0f or a large beam waist.
We carried out a number of characterizations with the help of the polarimetric setup depicted in Figure 3a to highlight the angular sensitivity of the photonic SHE in the HMM structure. Figure 5a. Polarimetric and quantum weak measurement techniques can be used to determine the transverse beam shift [12–14]. We used a He–Ne laser with a wavelength of λ = 633 nm and a diode laser with a wavelength of λ = 520 nm, as sources of an incident Gaussian beam. A microscope objective lens was used to collimate the laser light. Using Stokes polarimetry, we calculated the anisotropic phase difference Φ0 vs angle θi. A Quarter Wave Plate (QWP) and then a GlanThompson polarizer (P1) are used to produce the input polarization state (RHC). The Stokes parameters can be used to determine the phase difference using the expression:(25)where the normalized Stokes parameter for circular polarization is defined as S_{3} = I(90°, 45°) − I(90°, 135°), and the normalized Stokes parameter for diagonal basis is defined as S_{2} = I(0°, 45°) − I(0°, 135°), where normalization factor S_{0} is determined by the total beam intensity. The retardation angle of the quarter wave plate QWP2 and the rotation angle of the polarizer P2 are represented by δ and α of I(δ,α), respectively. The range (−π, π). encompasses the measured phase obtained using equation (5). We employ an unwrapping algorithm [12] to calculate the unwrapped phase difference with a 0.01 rad tolerance set.
The transverse beam shift, , is found from the measured phase, Φ_{0},(26)where the Stokes parameters for the circular and diagonal bases, respectively, are σ and χ. They are given by σ = 2Im(α ^{*} β), and χ = 2Re(α ^{*} β) from the Jones vector ψ〉 = (α,β) of the incident beam, respectively. The incident beam has , which corresponds to RHC polarization.
The experimental beam shift points for wavelengths λ = 520 nm and λ = 633 nm are contrasted with the findings of simulations that account for the HMM’s losses in Figure 5c and 5d. ϵ_{o} = −2.9460 + 1.2163i and ϵ_{e} = 5.5063 + 0.3399i for λ = 633 nm and ϵ_{o} = −0.3257 + 1.3664i and ϵ_{e} = 5.8847 + 1.9963i for λ = 520 nm, respectively, are the effective permittivities.
In Figure 5, we can see the characteristics of the photonic spin Hall effect in transmissionconfigured HMM systems. The transverse beam shift in HMMs is quite sensitive to the incident angle; for λ = 633 nm, changes from θi = 0 rad to just θi = 0.003 rad (0.17°) (Fig. 5c) causes a massive beam shift of several hundred microns while demonstrating milliradianlevel sensitivity. The angular variation from θi = 0.003 rad to above also significantly changes the beam shift from to merely , which is almost one order of magnitude difference. Given that we observed the same result in dielectric media [14], the significant anisotropy of HMMs is believed to be the cause of the beam shift’s strong peak. With a wider beam diameter and wavelength λ = 520 nm (Fig. 5d). The peak shift of is reached by the incident angle change of 0.001 rad (0.057°) only (≈4700 μm/°). This means that the beam shift exhibits even sharper resonance and, as a result, greater angular sensitivity. The beam shift dramatically decreases to and less when the incident angle increases, for example, θ_{i} = 0.01 rad (0.57°).
The simulation results, which account for alignment error and beam divergence, are quantitatively consistent with the experimental results. According to estimates, the He–Ne laser’s beam divergence is approximately Δθi = 0.01 rad (0.57°) while the green diode laser’s beam divergence is approximately Δθi = 0.02 rad (1.14°). Additionally, the rotating mount’s inaccuracy of around 0.0017 rad (0.1°) limits the angular resolution. Given that λ = 633 nm, the experimental transverse beam shift under normal incidence approaches (Fig. 5d).
However, the transverse beam shift drastically decreases to when the incidence angle is inclined by just θi = 0.035 rad (≈2°). Such findings show the great angular sensitivity of the SHE in HMMs and prove that the transverse beam shift may be controlled within an order of magnitude range by minor angular adjustments. Experimental measurements were utilized to determine the waist (w_{0} = 100 μm) for the simulations.
According to Figure 5d, with λ = 520 nm, we observe for θi = 0 rad. At θi = 0.035 rad (≈2°), the beam shift decreases to = 5 m. The previously observed spin Hall effect in dielectric anisotropic media, such as a quartz crystal (≈150 m/5° = 30 μm/°) and polymer film (approximately 250 μm/20° = 12.5 μm/°) [14], is strikingly different from the extremely high angular sensitivity of this new device. When the beam shift is two orders of magnitude larger and occurs within a few degrees of sample tilting. Additionally, the HMM has a thickness of only 176 nm as contrasted to the dielectric materials, which have thicknesses of 50 μm for polymer films and 1 mm for quartz plates.
In summary, we experimentally proved the photonic SHE in a hyperbolic metamaterial at visible wavelengths for the first time. The incidence angle has a significant impact on the tranverse beam shift in the transmission arrangement. We found that a little difference of a few milliradians can affect the beam shift by two orders of magnitude, going from a few hundreds of microns to a few microns, for example. A HMM that is two hundreds of nanometers thick achieves this tremendous angular tunability. Such sensitivity can result in small and compact spin Hall devices, such as switches, filters, and sensors, that control light at the nanoscale by varying the wavelength, incidence angle, and spin.
7 Emergent applications of SOI in photonic materials
SOIs may be engineered to modify they way in which an artificial substance disperses. SAM and OAM can therefore both be used as a means of control for light. The multifunctional spindependent element is made available by the geometric phase design, which also permits the spinbased optical devices. In addition, unlike conventional devices based on dynamic phase, the manipulation of cumulative geometric phase is essentially the control of light polarization. In conventional settings, a phase distribution is created by varying the reflective indexes or the thickness of optical materials. However, planarization and miniaturization of components are necessary for the advancement of integration optics. The photonic SHE may be developed in conjunction with the functions of conventional components to create infinitesimally tiny and multifunctional devices. Photonic SHE devices that provide the basic optical components variation for emergent applications of SOIs will be briefly outlined in this section.
7.1 Sensing mechanical and optical properties of 2D materials and metasurfaces
The transverse and spindependent shifting of light is referred to as the optical Spin Hall Effect (SHE). As a result, it has been enthusiastically advocated for a variety of sensing applications, including biosensing, material interface studies, polarizationdependent sensors, and refractive index spectroscopy [70–72]. SHE biosensors usually include a graphene sheet, or some alternative 2D material, an Au film, and a BK7 glass. A change in the concentration of biomolecules in the sensing medium will cause a localized change in the refractive index close to the graphene surface. The photonic SHE’s spindependent shifts are sensitive to changes in the sensing medium’s refractive index. More specifically, it is possible to infer a quantitative link between the spindependent splitting and the sensing medium’s refractive index. Furthermore, by varying the refractive index of the sensing medium, the spindependent splitting in the biosensor may be studied. Similar SHEbased sensing applications have been documented [70] (Fig. 6a and 6c). Weak measurement amplification methods can be used to further improve such sensing applications.
Figure 6 SHE of light applications in precision measurements of (a) mechanical properties of photonic materials, (b) quantum network applications, (c) optical/electronic properties of 2D materials, (d) optical switching for spinbased metrology and spintronics. Further details are in the text. 
7.2 Spintronics based on photonic SHE
The creation of a spin current perpendicular to the direction of the charge current flow constitutes the Spin Hall Effect itself. The optical SHE based on polaritons, and consisting of a separation between real space and momentum space, was recently predicted for laserinduced spinpolarized excitonpolaritons, in a semiconductor microcavity, due to a combination of structural disorderbased dispersion of excitonpolaritons, and an effective magnetic field resulting from polarization splitting of the polariton states. The excitonic spin current is controlled by the linear polarization of the laser pump. The first experimental evidence for this effect was reported in [73–75]. It was stated that polariton spin currents might travel over 100 μm distances in a superior GaAs/AlGaAs quantum microcavity. By rotating the laser pump polarization plane, it is feasible to switch the spin currents directions, opening the door to a host of amazing applications in optical spin switching and spintronics, in addition to further emergent spinbased metrology functionalities (Fig. 6d).
7.3 Applications in quantum information networks
A polarizing beam splitter or spindependent splitter can separate the orthogonally polarized components of a beam into different propagation directions. Since the beam is divided into several spatial modes according to its polarization, the photonic SHE generator may intuitively be thought of as a polarization beam splitter. Metasurfaces have recently been shown to be able to replace bulk optical components and be extended into the singlephoton quantum optical regime. For use in quantum network applications, the decreased propagation losses brought on by metasurfaces make it possible to realize a spindependent splitter at the single photon level. In particular, SHE switches and decoders are a fundamental building block for Quantum Information Networks where polarization dependent operations, such as CNOT and SWAP gates, are daily required [22, 76, 77] (Fig. 6b).
7.4 Photonic multiplexing and muticasting devices
Metasurfaces can contribute to a new class of onchip scalable devices, which are expected to revolutionize nanophotonic and optoelectronic circuitry through smart integration of multiple functions in metallic, dielectric, or semiconductor building blocks. Metasurfaces among artificially produced materials offer considerable promise to succeed in technologically relevant applications because of their 2D nature, which is a key benefit for waferscale manufacturing and integration. Here, we will concentrate on applications of metasurfaces for highspeed data transmission in integrated OAM multiplexing and multicasting devices [22, 76, 78].
In order to distinguish between the many orthogonal channels, OAM division multiplexing is an experimental technique for boosting the transmission capacity of electromagnetic signals, as stated in [77]. It corresponds to transmission over a few kilometers in OAMmaintaining fibers equivalent to Wavelength Division Multiplexing (WDM), Temporal Division Multiplexing (TDM), or Polarization Division Multiplexing (PDM). The extremely bulky optical components needed for OAM generation and OAM detection are one of the key constraints for scalable OAM multiplexage in addition to the lower transmission range [30], in the region of space that OAM of light covers. While OAM multiplexing can access a theoretically endless collection of states and as a result can offer an infinite number of channels for multiplexage, SAM or polarization multiplexing only offers two orthogonal states that correspond to the two states of circular polarization.
Although 2.5 Tbit/s transmission rates in MIMO systems have been reported, OAM multiplexing still remains an experimental approach and has only been tested in the lab so far, over relatively short distances of a few km over OAM maintaining fibers. Nevertheless, it promises very significant improvements in bandwidth. The extremely bulky optical components needed for OAM generation and OAM detection are one of the key constraints for scalable OAM multiplexage in addition to the lower transmission range. Recently [79–83], the first experimental demonstration of an OAM multiplexing technique based on singlelayer metasurfaces in the Terahertz (THz) band was realized. In particular, OAM multiplexing with four channels is made possible by the developed structure’s ability to produce four focused phase vortex beams with various topological charges (or OAM number l) when illuminated by a Gaussian beam.
The OAM signal is demultiplexed when a single vortex beam is employed as the incident light because only one channel is recognized and extracted as a focal spot. The subwavelengthlevel thickness of the metasurface structure expands the range of viable methods for the integration and downsizing of THz communication systems. Excellent agreement between theoretical predictions and practical results can be seen in the performance of the developed OAM multiplexing and demultiplexing device, proving its suitability for scaled ultra highspeed THz communications.
8 Conclusions
In conclusion, we presented a thorough review of recent developments in Spin Orbit Interactions (SOIs) of light in photonic materials. In particular, we highlighted progress on detection of Spin Hall Effect (SHE) of light in hyperbolic metamaterials and metasurfaces via polarimetric measurements, reporting unprecendented angular resolution at visible wavelength. Moreover, we outlined some fascinating future directions for emergent applications of SOIs of light in photonic devices of the upcoming generation. As a rapidly expanding interdisciplinary field, SOIs of light in 2D metamaterials and metasurfaces has important emergent applications in nanophotonics, biosensing, plasmonics, quantum optics, and telecommunication. SOIs in metamaterials and metasurfaces largely guarantees exceptional performance and versatility in the exact control of optical fields. Moreover, applications made possible by optical metasurfaces decreased dimensionality are significantly different from those made possible by bulk metamaterials. In general, metasurfaces can offer a novel tool for scalable OAM generation and conversion with minimal losses. This feature can encourage many applications in integrated onchip OAM generation, such as multiplexing and multicasting approaches, which may hold the promise of boosting transmission capacity and resolving scalability problems beyond the state of the art [12, 13, 22, 76–78, 83, 84].
Acknowledgments
This review is intended as a contribution to the advancement of scientific knowledge, for the benefit of the entire society, and its future generations. The author is grateful to Konstantin Bliokh, Andrei Lavrinenko, and Ricardo Depine for many helpful discussions. The author ackowledges Osamu Takayama and Radu Malureanu for providing the metamaterial samples used in reference [13]. This work was supported by ANPCyT via grant PICT Startup 2015 0710 and UBACYT PDE 2017.
Funding
G.P. acknowledges financial support via grant PiCT start up 2015 0710.
Conflicts of Interest
The author declares no conflict of interest.
Data availability statement
The data and numerical codes will be available upon request.
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All Figures
Figure 1 (a) General 3D geometry of the problem displaying the angle ϑ between the anisotropy axis of the plate and the beam axis z. (b) The wave vectors inplane deflections (Θ_{x}) cause the wellknown birefringence shift which is comparable to the GH shift, by altering the angle between k and the anisotropy axis. (c) View along the anisotropy axis of the crystal is shown. The transverse Θ_{y} deflections of the wave vectors rotate the corresponding planes of the wave propagation with respect to the anisotropy axis by the angle ϑ ≈ Θ_{y}/sin(Θ_{y}). This causes a new helicitydependent transverse shift , i.e., a spinHall effect similar to the IF shift. Further details are in the text. 

In the text 
Figure 2 (a) Representation in Poincare sphere of SpinOrbitConversion (SOC) via qplates. A state of circular polarization located in the poles of the Poincare sphere is mapped into its opposite state of circular polarization, while imprinting a fixed azimuthal phase ±2qϕ. (b) Representation in Poincare sphere of spinorbit conversion (SOC) via Jplates. Arbitrary states of elliptic polarization λ±〉 are mapped into its opposite state of elliptic polarization, while imprinting a tunable azimuthal phase (n, m)ϕ. 

In the text 
Figure 3 Schematic depiction of experimental configurations for (a) polarimetric measurements and (b) quantum weak measurements. P1 and P2 represent double GlanLaser polarizers (Thorlabs GL10), QWP is quarter wave plate. L1 and L2 are used to identify lenses. He–Ne (Melles Griot 05LHR111) laser with a 633 nm emission wavelength; Thorlabs WFS1505C CCD camera type. The ordinary and extraordinary permittivities of the photonic material sample, and their corresponding axes, are denoted as ϵ_{o} and ϵ_{e}, respectively. 

In the text 
Figure 4 (a) Paraxial beam’s transmission through a clear birefringent polymer slanted layer as shown in 3D geometry. The beam goes through transversal shift at the nanometer scale brought on by the Spin Hall Effect (SHE) of light. The paraxial angles (Θ_{x}, Θ_{y}) identify the propagation direction of the incident beam’s wave vectors k. (b) A tunable birefringent polymer results in enhanced SHE of light as shown in numerical simulations considering different birefringence: Δn 0.009 (gray, index difference in quartz), Δn 0.03 (green), Δn 0.06 (blue), and Δn 0.07 (red, index difference in stretched polymers [29]). Vertical dashed line at ν 0.3 (rad) is to show the tunability of beam shift by different birefringence. (c) Transverse intensity distributions (a.u.) in false scale for an opolarized beam transmitted through a tilted polymer plate and postselected in the almost epolarized state, with a tilt angle ϑ = 20°. Top: Postselected polarization state with ϵ = −1/83 × 10^{−2}. The beam centroid is shifted, resulting in a measurement of weak value . Middle: With crossed polarizers (ϵ = 0), a HermiteGaussian intensity distribution with peaks spaced apart from one another by ΔY = 1000 μm is created from a Gaussian distribution, corresponding to a weak amplification factor A = 200. Bottom: Postselected polarization state with ϵ = 1/83 × 10^{−2}, which corresponds to a weak value measurement of . 

In the text 
Figure 5 (a) Scheme of the Spin Hall Effect (SHE) of light in hyperbolic metamaterials. The incidence angle, indicated by θ_{i}, switches the transverse beam shift along the yaxis denoted as . The HMM structure’s unit cell is made up of Al_{2}O_{3}(10 nm)–APTMS(1 nm)–Au(10 nm)–APTMS(1 nm). Eight periods totaling 176 nm in thickness make up the HMM. (b) According to the effective media approximation, the effective permittivities of a multilayer HMM structure made up of a unit cell of Al_{2}O_{3}(10 nm)–APTMS(1 nm)–Au(10 nm)–APTMS(1 nm) were calculated. The structure exhibits type II (ϵ_{o} < 0 and ϵ_{e} > 0) HMM behavior in the visible wavelength range. The vertical dashed lines indicate the wavelengths λ = 520 nm and 633 nm at which experiments were conducted. Measured and calculated spin Hall transverse shifts for (c) λ = 633 nm and (d) λ = 520 nm, respectively, are shown in solid lines. Beam waist wo is fitted to the simulation results in (a) and (b). Note that there is an estimated beam divergence of Δθi = 0.01 rad for λ = 633 nm and Δθi = 0.02 rad for λ = 520 nm indicated as lateral error bars, respectively. Further details are in the text 

In the text 
Figure 6 SHE of light applications in precision measurements of (a) mechanical properties of photonic materials, (b) quantum network applications, (c) optical/electronic properties of 2D materials, (d) optical switching for spinbased metrology and spintronics. Further details are in the text. 

In the text 
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