Spin-dependent optics with metasurfaces

3.1 Transverse beam splitting with gradient-index metasurfaces

When light is refracted from air to a dielectric medium at an angle, the direction of light is changed, and the transversality of light imposes a requirement on the transportation of the polarization according to the equation of motion in the last section. The different k-components of the light evolve with an additional spin-dependent geometric phase, giving rise to a split of the beam center in the transverse direction in the early demonstrations, as shown in Figure 2A [18], [19]. With the recent introduction of metasurfaces, a completely refreshed look can be taken in OSHE as the metasurfaces provide a new way in bending light. To manifest the OSHE effect through a metasurface, we require the violation of inversion symmetry and the respect of the time-reversal symmetry. The time-reversal symmetry is naturally respected without involving magnetic optics or extrinsic OSHE. An inversion symmetry breaking metasurface, in its simplest version, is a single layer of “V-shaped” metamaterial atoms, which generate a linear gradient of local transmission phase profile [45], [46], [47]. Such a linear gradient of transmission phase profile, albeit a polarization conversion in the transmission, bends the incident light from one side of the metasurface to the other side. Such a bending can have an arbitrarily designed angle much larger than the one given by an air-glass interface, giving rise to a spin-split of the beam center in the transverse direction. Such a scheme is summarized in Figure 2B with the beam helicity (the S₃ Stokes parameter) plotted as different colors representing the two spins [56]. Although the splitting may still be in the order of wavelength, it is much larger than the one given by previous approaches. Now, it can be easily detected by the measurement in the far-field regime and it can occur at normal incidence using a metasurface. By performing a quantum weak measurement [57], [58], [59] on the light field with the preselected and postselected position and helicity, one can further amplify the OSHE effect and the associated transverse transport of the light field [19], [60]. It is also worth noting that the pure electronic readout of OSHE has been recently proposed and demonstrated through the momentum transfer from the light field to the collective electron motion in the transverse direction [61]. In addition, the detection of the OSHE can be further improved using dielectric metasurfaces instead of the lossy metallic ones [62], [63]. Symmetry and symmetry breaking play a key role in OSHE. A respected time-reversal symmetry and a broken inversion symmetry of the discussed metasurface introduce a transverse motion of light beam, which has been mostly discussed in the literature. In addition, for a Gaussian beam with radial symmetry, the transverse motion can also be introduced in the azimuthal [64] and radial [65] directions by carefully engineering the spatial variation of the light-bending metasurfaces.

Figure 2:

OSHE in giving transverse beam-splitting.

(A) Spin-splitting of beam center from air to glass at oblique incidence due to spin-dependent geometric phase picked up by different k-components. Reprinted from Ref. [19]. (B) OSHE at normal incidence using a metasurface, with the beam helicity (S₃ Stokes parameter) plotted on the right. Reprinted from Ref. [56].

3.2 Linear and angular momentum splitting using geometric-phase metasurfaces

A move in the k-space, with a spin-dependent geometric phase to bend light, can be regarded as an OSHE. Similarly, a change of polarization also induces a geometric phase and it can be used to do an equivalent job. The polarization state of light can be easily altered by an anisotropic material, with its orientation angle α (see Figure 1B) to control the phase of the transmitted light. For an LCP incident light being converted to an RCP light, light propagation with two different orientation angles (α) follow two different paths from the north to the south pole of the Poincaré sphere. The phase difference between the two routes is simply half of the enclosed area, being called the PB phase. This fact can be simply derived by considering the rotation of coordinate frame between the laboratory coordinates (x,y) and the principal coordinates (u,v) of the anisotropic material. An incident LCP can be written in the material frame as x^+iy^=eiα(u^+iv^), whereas the transmitted RCP can be written as x^−iy^=e−iα(u^−iv^). Therefore, the transmission phase from LCP to RCP in the laboratory frame can be simply understood as the corresponding one in the material frame with an addition of geometric phase 2α [66], [67]. This geometric phase only occurs for the cross-polarized transmission and has its sign flipped for RCP to LCP conversion. It can be summarized as 2σα, where σ=+1 with LCP incidence and σ=−1 for RCP incidence. Instead of transforming the different k-components with a geometric-phase profile in a parallel fashion in the last section, now we can directly transform the wave front at different positions, again in a parallel fashion, by having a spatially inhomogeneous profile of the material orientation, as long as the material response is anisotropic, giving a cross-polarized transmission. For example, one can design the orientation profile to be a linear profile σ=(∆k)·x/2+α₀, where ∆k is interpreted as the momentum imparted by the inhomogeneous material. According to the generalized Snell’s law of a metasurface [45], [46], [47], the transmitted light travels to two off-normal directions depending on the input spin by

where k_x(k_x,0) is the transverse wave-number of the transmitted (incident) wave. Here, the upper (lower) sign indicates the situation for an LCP (RCP) incidence. The OSHE is then revealed as a symmetric spin-splitting of the propagating direction, or linear momentum, due to the opposite geometric phases of the two spins. Figure 3A shows the first experiment in revealing such an OSHE using a chain of localized plasmonic coaxial nanoaperture. In this case, the anisotropy comes from the coupling between neighboring particles. The tangent line between neighboring particles represents the previously introduced orientation α. The corresponding spin-splitting of the linear momentum is shown in Figure 3A [53]. Conventionally, the different orientations of local anisotropy can be obtained using quasi-periodic diffraction grating structures [27], [28], [29], [30], [31]. Using metasurfaces, the local orientations of the metamaterial atoms (the anisotropy) can now be individually controlled, leading to a more intuitive local control of the geometric-phase profile. The inset of Figure 3B shows a monolayer of subwavelength plasmonic nanorods with strong local anisotropy on a metasurface. The linear geometric-phase profile is obtained by locally rotating the rods, of orientation angle α, also in a linear fashion [67], [68], [69], [70], found as Φ_B=±2α. As the phase shift only relies on direction of local nanorods, the spin-dependent ∆k is wavelength independent, whereas the outgoing angle depends on different wavelengths (see Figure 3B). This type of metasurfaces can be called geometric-phase metasurfaces.

Figure 3:

Spin splitting in linear momentum space.

(A) Isotropic coaxial nanoapertures chain for SHE. Left: Transmission spectra of the plasmonic chain showing anisotropy for linear polarization along two orthogonal directions due to the coupling of neighboring particles. Right: Spin-dependent momentum redirection for the OSHE. Inset: SEM image of a curved chain. Reproduced from Ref. [53]. (B) Anisotropic nanorod array for OSHE. Left (Right): Measured reflection angle versus incident angle for σ=1 (σ=−1) incidence, respectively. Inset: SEM image of nanorods fabricated on an ITO-coated glass. Reproduced from Ref. [68].

Apart from demonstrating a spin-splitting of linear momentum, such a geometric phase in polarization space can also be used to manipulate the orbital angular momentum (OAM) of the transmitted light. A beam with OAM possesses a helical wave-front comprising an azimuthal profile ∝exp(ilΦ). Here, Φ is the azimuthal angle and l is an integer, simply called the OAM of the beam [71], [72], [73], [74]. Early works on generating OAM relied on using locally anisotropic dielectric half-wave plates [28], [29]. Similar to the previous case, spin is flipped in the transmission together with an additional geometric-phase profile ±2α but now is designed as a linear profile of the azimuthal angle: α=qΦ+α₀. q is called the topological charge of the material and α₀ is the initial orientation at zero angle. The orientation profile α then impacts the opposite geometric-phase profile (±2α) for the two spins on the incident wave-front, which now acquires a change of OAM by

For example, when a light with zero OAM passes through a q=1/2 plate, the transmitted beam acquires an OAM of l=±1 (see Figure 4A) [75]. These two beams of different OAMs only have difference in their phase but not the amplitude profile. They can be analyzed by an additional refractive helical phase plate, in which a unit of OAM is added to the output, giving resultant l=0 and 2 for the two different incident spins, as shown in the same figure. A signature of the nonzero OAM is a singularity spot at the beam center of undefined phase and zero amplitude. In fact, if we just want to show the spin-splitting to reveal the OSHE, we can use the residual beam, the copolarization transmission without geometric phase, as a reference beam. In this case, the metasurface consists of slits with q=1, which does not need to behave locally as half-wave plates [76], [77]. The resultant interference pattern from a linear polarized incident beam (consisting of both spins) shows a spin-splitting in the far-field, as illustrated in the measured Stokes parameter S₃ in Figure 4B. The two different colors represent the two different spins in the figure. The observed orbital rotation from the x-y axes results from the nonzero α₀ at zero azimuthal angle, as the slits are actually zigzag in shape. In these examples, the spin angular momentum is transferred to the OAM of the transmitted beam. Based on these simple versions of metasurfaces, more spin-dependent phenomena in orbital momentum were proposed [63], [68], [78], [79], [80], [81], such as, for example, far-field spin-dependent focal point splitting [79], controllable OSHE with large OAM [80], OSHE with rotational symmetry breaking [63], OAM-focusing lens [81].

Figure 4:

Spin splitting in angular momentum space.

(A) Measuring an l=±1 vortex beam from a diffraction q-plate using a spiral wave plate. Measured intensity distributions are shown on the right. Reproduced from Ref. [75]. (B) Metasurface for spin-induced manipulation of OAM. Measured (left column) and numerical (right column) Stokes parameter S₃ with RCP incident light (top row) and linear polarized incident light (bottom row). Inset: SEM image of metasurface sample. Reprinted from Ref. [76]. (C) Vortex beam generator using spin-to-orbit coupling through a metasurface. Reprinted from Ref. [77].

As metasurfaces are much thinner than a wavelength, diffraction from neighboring “pixels” is not severe; therefore, the resolution of the control of the geometric-phase profile can be very high, only limited by the size of individual metamaterial atoms. On the contrary, the efficiency of converting to the cross-polarization can be easily controlled by tuning the resonating response of the individual atoms, alongside the high resolution of the geometric-phase control. The residual beam, the one not carrying geometric phase, is therefore minimized in its intensity. With optimal design in the transmission geometry, almost 25% efficiency can be achieved by a single-layer nonmagnetic metasurface (see Figure 5A) [82]. Based on the use of three anisotropic sheet metasurfaces (see Figure 5B) [83], [84] cascaded along the direction of propagation, the control of vector Bessel beams is proposed with very high efficiency (a beam generator with 20 dB) due to the added magnetic response in the geometric-phase metasurface. In the reflection geometry with a ground plane, magnetic response is already embedded and almost 100% efficiency (see Figure 5C and D) can be obtained in steering the reflected beam against the specular reflection of the residual beam [85], [86]. These works based on plasmonic particles and metasurfaces have paved a unique way to achieve OSHE in both the visible and the infrared regimes, resulting from the discussed opposite geometry phases for the two incident spins.

Figure 5:

High efficiency geometric-phase metasurfaces.

(A) High-efficiency geometric-phase metasurface in transmission geometry. Reprinted from Ref. [82]. (B) High-efficiency vector Bessel beams generator. Reprinted from Ref. [83], [84]. (C and D) Almost perfect OSHE in reflection geometry working in microwave and tetrahertz region. (C and D) Reprinted from Refs. [85], [86], respectively.

4.1 Surface plasmon generation with a single scatterer

Interestingly, OSHE in the near field can occur when the incident light illuminated even on a single isotropic scatterer [87]. Figure 6A shows a circular slit that is used to scatter a circularly polarized light in the far field into an outgoing surface plasmon, which can be described by a scalar wave E_z. By considering a rotation of the coordinate frame of the incident polarization, one can easily derive that the surface wave bears an angular phase profile e^iσΦφ. This spin-dependent geometric-phase profile can be regarded as an optical vortex with unit OAM. Figure 6B shows the measured spin-dependent fringes by interference with a reference surface-wave generated from a long rectangular slit at the same time [87].

Figure 6:

SPP generation by circular slit.

(A) Circular nanoslot in generating spin-dependent outgoing surface waves. Interference fringes between the surface wave with a traveling plane wave for the two CP incidences are shown on the right. Reprinted from Ref. [87]. (B) Semicircular nanoslot in focusing surface waves. The spin-splitting of the focal spot is shown on the right. Reprinted from Ref. [54].

The converse situation will be scattering on a circular nanoslot in focusing the surface wave into its center, as shown in Figure 6B. It is a summation of surface waves from each local portion of the slit at an orientation angle α in a radiating direction perpendicular to the slit. A rotation of coordinate frame then connects these different cases together, giving rise to a geometric phase of ±α. This is different from the geometric phase of ±2α in the far-field situation. By putting α=Φ to represent the orientation of the slits at different azimuthal angles, we immediately arrive that the focusing surface wave should be e^iσΦ, with an OAM of ±1. When we consider a semicircular arc for the slot instead of a full circle, this angular profile of geometric phase induces a transverse shift of the focal point, either λ_SPP/2 or −λ_SPP/2, depending on the incident spin, where λ_SPP is the wavelength of the surface wave [54], [55].

It is interesting to note that the spin angular momentum of the light is completely converted into the OAM of the surface waves, or viewed as a conservation of the total angular momentum, for the case of isotropic defect with rotational symmetry. More spin-dependent phenomena can be achieved by employing chiral plasmonic structures with embedded topologic charge. Figure 7A shows the schematic diagram of the Archimedes spiral [88], [89], [90], [91], [92], [93], [94], [95]. The topological charge q=−1 for the structure is defined by the rate change of radius against azimuthal angle: r=r₀−λ_SPPqΦ/2π. Because each local part of the spiral nanoslot (located at r) can be approximated to a circular structure with radius r (like in the last subsection), the surface plasmon propagating towards the center of spiral has a phase profile of σΦ. In addition, the surface wave from different portions of the spiral slit arrives at the center with different spin-independent dynamic phases due to the increasing radius. As a result of these two effects, the total phase gained from this consideration is Φ_SPP, which can be expressed as

Figure 7:

Spiral structures for OAM control in near field.

(A) Schematic diagram of the Archimedes spiral in generating surface plasmon. (B) Electric field of the generated surface plasmon on a metal surface. Reprinted from Ref. [88]. (C) Near-field measurement of the OAM of the generated surface plasmon. Reprinted from Ref. [89]. (D) Spin filter with each pixel consisting a spiral with a coaxial nanoaperture. Measured spin-dependent transmission through the array is shown on the right. Reprinted from Ref. [90].

where l_SPP is the OAM of surface plasmon arriving the center. In this case, the additional angular momentum is provided by the topological charge of the spiral plate. However, unlike the far-field examples in Section 3.2, the contribution of the topological charge is spin independent. Figure 7B illustrates the amplitude and the phase of the surface wave when the structure is excited by a circularly polarized plane wave. The OAM of the surface wave is modified by the topological charge according to Eq. (6), now with a phase singularity and a dark spot at the center. We also note that the value of OAM is related to the radius of the dark spot, which can be directly measured by the standard near-field measurement technique (see Figure 7C) [55], [89]. Such a spin-dependent SPP profile can be further exploited to achieve a spin-dependent transmission filter through an array of coaxial nanoapertures [90], as shown in Figure 7D. In this case, the spiral has a topological charge q=−2, being added to the incident angular momentum σ=±1, to excite the surface waves. The OAM of the surface wave becomes l_SPP=−1 (−3) for LCP (RCP) incidence and it has to match with the coaxial aperture beneath and at the center of each spiral, which only allows a mode of OAM of ±1 to pass through. The combined effect is shown in Figure 7D. The word “SPIN” is obtained in transmission for one kind of circular polarization incidence but not the other with a huge contrast due to the mismatch of angular momentum [90].

4.2 Surface plasmon generation with geometric-phase metasurfaces

The previous structure with definite topological charge (e.g. circular or spiral nanoslots) provides a spin-orbit coupling between the incident spin and the generated surface plasmon. Actually, such a spin-orbit coupling can also occur even for a scatterer with other shapes in scattering the incident light to surface plasmon. Figure 8A shows an example of a single slit illuminated by a tightly focused linearly or elliptically polarized light beam [96]. The shift in either the focal point or the direction of the generated surface waves is measured using a “weak measurement” approach. Spin-splitting can also be directly observed using circularly polarized light as incidence as an asymmetric excitation of surface plasmon propagating towards the left- and right-hand sides of the slit. When an appropriate angle of incidence is chosen, the asymmetry is optimal and is revealed as a spin-dependent unidirectional excitation of the surface plasmon (see Figure 8B and C) [97], [98], [99]. Moreover, such asymmetric effects can be demonstrated for even a single particle placed on the surface of metal surface [98], [99], [100]. As a direct implication, the particle is optically pushed in opposite directions for the incidence of different circular polarizations as demonstrated as the mechanical effects from spin-orbit coupling [101], [102].

Figure 8:

OSHE on a single nanoslot.

(A) Observation of the spin-dependent shifting of scattered surface plasmon from a single nanoslot using the weak measurement approach. Left: Scheme of the experimental set-up. Right: SPP field distributions in real (top row) and Fourier spaces (bottom row) with different incident polarizations are shown in cyan. Reproduced from Ref. [96]. (B) Exciting surface plasmon with different circular polarization on a long slit. Reproduced from Ref. [97]. (C) Asymmetric SPP emission from a single nanoslot and nanoparticle (top row). Reproduced from Ref. [98].

When the slit becomes much smaller than a wavelength, it reradiates as a localized dipole moment [100]. In this case, a circular polarization incidence (x^+iσy^) generates a “spin-flipped” dipole moment p→=ei2σα(x^−iσy^), where α is the orientation of the slit measured from the x-axis. The surface plasmon reradiated from this dipole, a scalar wave (E_z), can be simply obtained by changing x^/y^ to cosβ/sinβ, where β is the azimuthal angle between the x-axis and the location of the surface wave to be evaluated sufficiently far away from the dipole (see Figure 9A). Therefore, the total phase carried by the surface-wave radiation for the spin-flipped component is

Figure 9:

Spin-dependent trajectories enabled by nanoslots array.

(A) Schematic diagram in generating surface plasmon by a dipole source on metal surface. (B and C) Unidirectional propagation achieved by nanoslots, where B (C) employs coupled (single) nanoslots in one unit cell. (B and C) Reprinted from Refs. [103], [104], respectively. (D) Scheme of flexible coherent control of plasmonic SHE. Left: Independent generation of local orbitals for the two incident spins. Right: Motion picture generated by rotating a linear incident polarization in writing a letter “b”. Reprinted from Ref. [105].

where 2σα can be regarded as the same PB phase term with that in the far field. Here, by setting the radiation angle always perpendicular to the nanoslot (β=α), the phase of spin-flipped component will become Φ=σα. Such a geometric phase can also be interpreted by the Coriolis effect of light [54], where light suffers from both a momentum redirection and a polarization change when coupled into surface wave. On the contrary, for the radiation without spin-flipping, the geometric-phase factor 2α in the above formula is absent. As both terms radiate from the same dipole, the total radiation can be written as e^iσβ+e^iσ(2α−β)⁼2e^iσα cos(α−β) where the σα term is the geometric phase arisen from an interference of the “spin-flipped” and “spin-conserved” component. Figure 9B shows an array of slits. Each unit cell consists of two columns of nanoslits, one at orientation α=π/4 and another at orientation α=−π/4, with combined radiation in the horizontal direction [103]. Suppose it is an LCP (σ=1) incidence. The right (left)-hand column of slits radiates with phase −π/4 (+π/4), according to Eq. (7). Together with a designed dynamic phase separation of π/2 between the two columns, the radiations are destructively (constructively) interfered to the right (left) in the horizontal direction. For an RCP incidence, the unidirectional propagation is in the opposite direction (see Figure 9B). The radiation pattern for such a unit cell is very anisotropic. The orientations of the slits are chosen to have cosα invariant for the two horizontal directions so that the interference is between waves of the same magnitude. Such a slit (with dipolar radiation profile) array is very useful in tailoring the propagation of surface plasmon polaritons, for example, generating Cherenkov surface plasmon [106] and achieving spin-dependent transmission through decorated subwavelength aperture on plasmonic metal [107]. The appearance of the spin-flipping geometric-phase term involving 2σα in Eq. (7) is actually playing the key role in the phenomenon. The structure can be further simplified to a unit cell consisting of only one slit. In this case, the orientation profile α is designed as a linear function of x, as illustrated in Figure 9C [104]. For a periodicity of α in the x-direction, a unidirectional propagating surface plasmon can only be excited for a chosen incident spin σ by matching the momentum of surface waves: k_{x, SPP}=2σ∂_xα+m 2π/a, where the integer m is the chosen working diffraction order. As there is a mirror symmetry breaking in the geometric-phase profile 2α, the matching can only be fulfilled in a single direction. Such a consideration can also lead to more dispersion-related phenomena (e.g. Rashba-like effect in thermal radiation) [108], [109], [110], [111], [112], which will be introduced in Section 5.

The opposite geometric phases for the two spins are related to the various opposite spin-dependent shifts in the previous sections. These include a transverse shift of beam center, a shift in the linear momentum or the OAM of the incident beam, and a shift of focal spot from a circular grating. For surface-wave excitation, this simple relationship between the two incident spins can actually be relaxed if we are only interested in generating local orbitals in a finite region. Figure 9D shows a platform with slits of a tailor-made orientation profile around a central slit-free region [105]. Another way to interpret Eq. (7) is that if we require neighboring slits contribute to a surface plane wave with phase Φ_B in a direction of angle β=arg(k_x+ik_y), we should set α=(σΦ_B+β)/2. Therefore, by taking the holographic approach to generate a target profile E_z using constructive interference from all the slits, we change Φ_B to argE_z and k_x(y) to ∂_x(y). Neighboring slits then contribute to the required local plane waves and we obtain α=±12arg((∂x±i∂y)Ez) with an arbitrary global additive constant. Because we are only interested in the standing waves generated in the center region, we only require E_z to share in common with your actual target profile on the inward radiation part. We can therefore set α=±12arg((∂x±i∂y)(Ez++(Ez−)∗)), where E_z⁺ and E_z⁻ represent the target profiles, which only include the inward radiation parts for the two spins separately. The target profiles for the two spins within the central region can then be specified individually without a simple relationship between them. Figure 10A illustrates the independent and flexible SHE: a cross (triangle) for LCP (RCP) incidence. Such flexible control of SHE enables the two spins to work together in a coherent way [116], [117], [118]. Motion pictures with a series of picture frames can be assembled and played by rotating a linear polarization as incidence, as shown in the same figure.

Figure 10:

OAM generators and detectors.

(A and B) Geometric-phase metasurface in sorting different polarizations. Reprinted from Refs. [113], [114], respectively. (C) Generating optical angular momentum by metasurface. Reprinted from Ref. [115].

5.1 OAM generator and detector

Usually, the spin angular momentum and intrinsic orbital momentum of light are not easy to be measured. Conventional approaches use bulky free space components, such as wave-plates and polarizing analyzers, diffraction gratings [119], [120], spatial light modulator [121], [122], [123], and interference measurement [124], [125], [126], [127]. Spin-dependent metasurfaces, on the contrary, show a strong capability to generate and detect the angular momentum of light [113], [114], [128], [129], [130]. For example, a detector metasurface designed by holographic approach was proposed [128]. Only the incident light carrying the designed OAM can launch the particular surface plasmon with constructive interference. Similarly, using geometric metasurfaces with rotating slits or bars, a composite light beam with different spins or polarization states can be sorted into different directions in either reflection (see Figure 10A) and transmission (see Figure 10B) geometry [113], [114]. The ellipiticity of the incident light can also be analyzed. The geometric-phase route has an advantage of being robust against fabrication impurities. On the contrary, the generation of vortex beams is critical for fascinating applications ranging from super-resolution imaging techniques such as stimulated emission depletion (STED) microscope to high-bandwidth quantum information processing. Space-variant metasurfaces have been widely adopted to generate and manipulate the OAM of light in a precise and versatile manner (see Figure 4C). More realistic approaches were proposed based on dielectric metasurfaces made of silicon (or TiO₂) cut-wires to obtain a high conversion efficiency and a low ohmic loss (Figure 10C) [115].

5.2 Geometric phase enabled planar lens and holograms

Although OSHE has been observed for decades, only simple splitting in the real space and in the momentum space has been demonstrated until not long ago [19], [53], [54], [55], [56]. This is partly due to the fact that only simple structures and materials can be employed in fabrication previously. The resultant geometric-phase profile is relatively simple. Recently, metasurfaces with anisotropic nanostructures provide a straightforward approach to achieve arbitrary geometric-phase profiles with high resolution simply by rotating the nanostructures [67], [68], [69], [70]. The associated OSHE opens a new gateway to more flexible spin-dependent optics. For example, a spin-dependent metasurface lens can be fabricated with an orientation of the structures with quadratic profile. Such a lens can be switched from a focusing to a diverging lens simply by changing the incident light from one spin to another [131], [132], [133]. To achieve more generic applications, a holographic approach was proposed to generate a desired complex wave-front, which is flexible enough to realize either a 2D or 3D hologram [134]. Comparing to conventional approach of making holograms using discrete depths of materials, the generated phase can be continuously varied in the geometric-phase approach. The holograms generated can now depend on the incident spin and its efficiency can reach nearly 80% by employing a reflection geometry of the metasurface, as shown in Figure 11B and C [134], [135], [136], [137].

Figure 11:

Geometric phase enabled flat optics and hologram.

(A) Dual-polarity plasmonic flat lens for visible light. Reprinted from Ref. [131]. (B) Geometric phase enabled holographic scheme and reconstruction procedure. Reprinted from Ref. [135]. (C) Helicity multiplexed metasurface hologram with dual holographic images. Reprinted from Ref. [136].

5.3 Symmetry-related applications with spin-orbit interaction

Symmetry plays a key role in OSHE and symmetry-breaking metasurfaces have shown significant spin-dependent transportation through the strong light-structure interactions. In this way, metasurfaces are providing a very flexible platform to investigate structural symmetry effects. The Rashba effect denotes a splitting of spin-degenerate parabolic bands into bands of opposite spins in the dispersion diagram [138], [139], [140]. Such an effect arises from spin-orbit interactions with spatial inversion symmetry broken while the time-reversal symmetry can still be respected. In Section 4, we have seen numerous examples of using a geometric-phase profile to induce a spin-dependent coupling between the incident light and the material. A very useful and alternative perspective is a spin-splitting of the dispersion diagram of the material itself. By exploiting geometric-phase metasurfaces with rotated microstructures [103], [104], [106], [107], an optical counterpart of the Rashba effect can be observed [109], [110], [111], [112]. Figure 12A shows a 1D case, in which the structure is rotated from one cell to the next. It causes a spin-splitting of the surface-wave (phonon-polariton) dispersion, which in turns causes a spin-splitting in the absorption spectrum and the thermal radiation bands (see Figure 12A) [108], [109]. A more Rashba-like band structure with spin-split parabolic bands in two dimensions can be obtained using a nanoslot array with rotated orientation angle in a 2D kagome lattice (see Figure 12B) [110]. This spin-dependent coupling between the incident light and the material can be applied to structures of even lower symmetry (e.g. a quasi-crystal) to achieve this Rashba effect [111], [112].

Figure 12:

Symmetry effect with spin-orbit interaction.

(A) Spin-split dispersion relation measured by S₃ Stokes parameter of thermal radiation from the rotating slits in one dimension. Reprinted from Ref. [108]. (B) Schematic set-up for the spin-projected dispersion based on 2D Rashba-like metasurface. Reprinted from Ref. [110]. (C) Metasurfaces with two- and four-fold symmetry allows strong third harmonic generation [141]. (D) Geometric-phase profile in selecting the outgoing angle of the second harmonic signal.

Nonlinear optical processes are also highly sensitive to the symmetry of the nonlinear materials under investigation. For example, second harmonic generations also rely on the absence of inversion centers in the materials or structures. Clearly, the OSHE and the nonlinear optical processes on a 2D metasurface will be intrinsically intertwined. The nonlinear susceptibility tensors of the metasurfaces can be easily tailored by the structural design, and the spin state and spectra of the nonlinearly converted light beam can be modulated. Typical examples includes L-shaped gold nanoparticle [142], [143], T-shaped nanoparticle [144], split-ring resonator (SPR) with electric or magnetic dipole resonance [145], [146], [147], and silver triangular nanoprisms and nanoholes [148], [149]. A symmetry-breaking metasurface not only significantly enhances its nonlinear optical response but also shows strong spin-selectivity in the nonlinearly generated light [150]. Polarization-dependent selection rules for high harmonic generations on a metasurface have also been extensively studied [151]. As shown in Figure 12C, metasurfaces with two- or four-fold symmetry allow strong third harmonic generation while the process is prohibited in a three-fold symmetric system using a consideration of the geometric phase carried by the plasmonic particle [141], [152]. When these nonlinear particles are arranged into an array with a geometric-phase profile, the spin-orbit interaction dictates the nonlinear susceptibility and alter the spin state and the direction of the nonlinearly converted light through a geometric-phase metasurface (see Figure 12D). Last but not least, secure quantum communication and information processing rely on parametrically generated photon pairs with entangled polarizations states, more explicitly, the entangled spin and/or orbital angular momentum. Metasurfaces with high efficiency in manifesting and detection of photon spin and OAM freedom shall find their applications in quantum information storage, processing, and computation based on the angular and OAM of light.