Chirality is a research topic of longstanding interest in light–matter interactions [1–3]. Circularly polarized light is a typical chiral light field with left- and right-handed helicities [4] which have been widely used in biological science and pharmacies for chiral-elective interactions and sensing of biomolecules [5], e.g., DNA/RNA, protein amino acids, etc. [6]. On the other hand, as the contactless and label-free advantages, lateral optical forces (non-conservative forces perpendicular to the direction of light propagation) arising from the helicity gradient of light fields with non-uniform spin angular momentum (SAM), namely, spin-structured light fields, have been demonstrated as a remarkable tool to sort chiral objects [7–11], e.g., the chiral-sensitive separation force of chiral particles in light fields with one-dimensional variant helicity [12–14].

Many efforts have been devoted to the construction of such light fields with laterally modulated helicity by manipulating the superposition of two orthogonally polarized components with elaborate phase structures [12–15]. In free space, these kinds of synthetic light fields are realized by optical setups composed of traditional polarization and phase modulators with high efficiencies, while those bulky devices have disadvantages in miniaturization and integration [14, 15]. To overcome this issue, evanescent waves such as surface plasmonic polaritons (SPPs) have been utilized to engineer the surface chirality structures [16–18] which exhibited significantly enhanced light–matter interaction effects and spatial frequency of the chirality structure [19, 20]. On the other hand, the near-field local characteristic, however, greatly limits the practical application [21, 22]. Anisotropic metasurfaces, which work as nanoscale birefringent phase retarders, provide an elegant solution to these difficulties [23]. These artificially ultra-thin materials consist of two-dimensional arrays that enable the simultaneous control of polarization, amplitude, and the phase of the wavefront in a sub-wavelength scale [24–27], facilitating the development of functional nanodevices such as lenses [28], wave plates [29], holograms [30], polarization detectors [31], and multiplexers [32, 33]. Importantly, elaborate metasurfaces have been developed in typical applications such as three-dimensional polarization modulation [33–36] and polarization-encoding vectorial holography [37, 38].

In this paper, we show that spin-structured focal fields with laterally variant helicities can be conveniently generated by a minimalist metalens. The metalens is designed according to a developed hybrid vector-focusing model based on the combined control of the polarization and phase of the wavefront [39, 40]. It consists of two segments with orthogonal polarization modulations and two-dimensional hybrid-focusing phases, which enable polarization-dependent field focusing and mixing to generate laterally oscillating SAM in the focal plane. Under the illumination of linear polarizations, the optical helicity gradient is valid to induce chirality-sensitive lateral forces for the sorting of chiral particles. Our scheme might supply an integrated and compact platform for the application of optical tweezers.

Figure 1A illustrates the basic principle of the metalens. This device consists of two segments divided by the y axis, which transforms the uniform incident polarization into two orthogonal states with two-dimensional hybrid phase retardations, of which on-axis distributions are shown in Figure 1B. The phase retardation function is defined as φ(x,y) = -k ₀(-f ) -k ₀ xsinα. Here, k ₀ is the wave number, f denotes the focal length corresponding to the cylindrical lens along the y axis, α = tan⁻¹ (R/2f) denotes the gradient of the phase along the x axis, and R depicts the aperture of the metalens. The hyperbolic and linear phases exhibit focusing properties as cylindrical and conical lenses along the y and x directions, respectively.

We study the focusing dynamics of two segments by using the vectorial diffraction integration theory, which is valid beyond the paraxial approximation [41]. Supposing that the metalens is illuminated by a normally incident plane wave and the transmitted field is expressed as E₀ = [E _x E _y ]^T, with E _x and E _y depicting the complex amplitudes of two basic states, considering the x < 0 segment, the focal field can be written as [41, 42] E = − i f k e − i k f 2 π ∫ − θ max θ max E x θ cos ⁡ α 0 − sin ⁡ α + E y θ 0 cos ⁡ θ sin ⁡ θ ⋅ e i Φ θ cos ⁡ θ d θ , (1) where Φ θ = k x ⁡ tan ⁡ α + k y ⁡ tan ⁡ θ + k z / 1 + tan   2 ⁡ α + tan   2 ⁡ θ with θ denoting the y-axis-focusing angle. Derivation details of this focusing model are provided in Note 1 and Supplementary Figure S1 of the Supplementary Material. In the square brackets, these two vectors depict the polarization conversion arising from focusing in two coordinate directions. It is noteworthy that the polarization conversion evokes a longitudinal component, of which the intensity is dependent on the numerical apertures (NAs) along the two coordinates. Moreover, the asymmetric polarization conversions along x and y axes allow the construction of remarkable polarization and SAM structures in the focal field.

We first consider that the transmitted field has normalized intensity in a paraxial condition so that polarization is expressed as E ₀ = [cosχ sinχe^iδ ]^T, with χ and δ determining the polarization. The electric component of the focal field can, thus, be appropriately rewritten as E x , y , z ∝ sin ⁡ c L y λ f cos ⁡ χ ⁡ cos ⁡ α sin ⁡ χ e i δ 0 e − i k x ⁡ sin ⁡ α + i k z ⁡ cos ⁡ α , (2) where the sinc (∙) function depicts the focal profile generated from the y-axis-focusing angle and L is the aperture size of the metalens. Eq. 2 indicates that the focal field presents a sinc² function profile along the y axis but a uniform intensity along the x axis, as the experimental result shown in Figure 1C. Likewise, the focal field corresponding to the x > 0 segment presents an identical intensity profile but the opposite wave vector in the lateral plane.

By engineering the polarization and phase modulation of the two segments, one can flexibly control the superposition state. Here, we lock the y-axis focal length with the x-axis-tilted phase. As a result, two focal components completely overlap with each other and thereby present a uniform intensity along the x axis. However, the diffraction arising from the knife edge between the two segments decreases the uniformity of the total focal field. Furthermore, in a tightly focusing case, an intriguing polarization state such as lateral SAM would be constructed because of the strong coupling from the transverse component to the longitudinal component.

We then select polycrystalline silicon (Poly-Si) to design the metalens, of which the meta-atom is schematically shown in the inset of Figure 1A. It consists of a SiO₂ substrate and a Poly-Si rectangular nanopillar [40]. According to the effective waveguide theory, the Jones matrix corresponding to the equivalent birefringence effect of a meta-atom can be expressed as J = T e i φ 0 [ cos δ 2 + i ⁡ sin δ 2 cos 2 ϕ i ⁡ sin δ 2 sin 2 ϕ i ⁡ sin δ 2 sin 2 ϕ cos δ 2 − i ⁡ sin δ 2 cos 2 ϕ ] , (3) where δ = φ _x -φ _y , φ ₀ = (φ _x +φ _y )/2, and T = T _x = T _y , with T _{x, y} and φ _x,y being the transmission amplitudes and phases of the two eigenstates; ϕ depicts the rotation angle of the rectangular nanopillar.

As a proof of principle, we select the |H⟩ and |V⟩ states on the Poincaré sphere as basic states. In this case, we set the input polarization as E _in = [1 1]^T, and δ = π, ϕ ₁ = 22.5°, and ϕ ₂ = 67.5°, with ϕ _1,2 depicting the rotation angles of the meta-atom in the x < 0 and x > 0 regions. The transmitted fields can thus be correspondingly expressed as E ₁ = Texp (iφ ₀₁)[1 0]^T and E ₂ = Texp (iφ ₀₂)[0 1]^T. Substituting two transmitted fields into Eq. 2, the normalized distributions of Stokes parameters in the focal plane are approximately expressed as S ₁ = −sin² α/2, S ₂ = cosαcos (2kxsinα), and S ₃ = -cosαsin (2kxsinα). Under a paraxial condition, the superposition field presents oscillating helicity (quantitatively expressed by S ₃) along the x axis with a spatial frequency of f _x = ksinα/π, of which the polarization variation trajectory is shown as the blue meridian on the Poincaré sphere in Figure 2A. The two-dimensional HSV image in Figure 2B gives the normalized intensity and optical helicity density distributions of the focal field, i.e., S ₀ and S ₃. Therein, the brightness and color values correspond to S ₀ and S ₃, respectively. Obviously, the focal field presents a bright linear focus with the one-dimensional gradient of optical helicity.

We design an all-dielectric metalens by using the electron beam exposure and plasma-etching process [40]. It should be noted that since the propagation phase φ ₀ actually works as the modulation phase, we select eight geometries with similar transmission amplitudes but linearly variant propagation phases to design the metalens, according to the meta-atom libraries of polarization and phase modulation in Note 2 of Supplementary Material. The transmission amplitude T, propagation phase φ _x , and phase retardation δ of selected geometries are shown in Figure 2C in conditions of λ = 670nm, p = 450nm, and H = 550 nm. The complex refractive index of Poly-Si is n = 3.25710 + i0.0107. Figure 2D gives a local scanning electron microscope (SEM) image near the y axis. This metalens has 1,000 × 1,000 nanopillars. Figure 2E depicts the experimental setup. The λ = 670 nm laser output from a supercontinuum laser (SC-Pro, YSL Photonics) after acoustic-optic filtering is transformed into 45° polarization, which then normally illuminates the metalens. A microscope composed by an objective (×20), tube lens, and CCD is employed to observe the focal field. This system is located on a translation stage to observe the focusing dynamics. A quarter wave plate and polarizer are used to measure the Stokes parameters.

Figure 3 displays the experimental results of NA = 0.2 and 0.5 metalenses. Here, NA is defined by the focusing phase along the y axis, so the focusing parameters of the two metalenses are f ₁ = 1.1mm, α ₁ = 5.83°, and f ₂ = 0.39mm, α ₂ = 16.1°, respectively. As shown in Figures 3A, D, the focal fields present a bright line focus with the oscillating helicity density along the x axis. The generation efficiency of this type of metalens is about 71%, which is obtained by measuring the power ratio between the focal field and input field. Figures 3B, E illustrate the measured Stokes parameter distributions in a period, where the red curves and points depict the theoretical trajectories and experimental results, respectively. Figures 3C, F show the experimentally measured focusing process of metalens in the y–z plane, and the insets are the numerically calculated results. As shown, the experimental results are consistent with the theory. Clearly, these optical helicity intensities in the focal plane oscillate in the form of sinusoidal functions with periods of about 3.2 μm and 1.2 μm, which are consistent with the theoretical predictions.

We remark that the optical helicity structure is insensitive to the input polarization orientation, namely, the oscillating property of the optical helicity intensity is invariant for the incidence of arbitrary linear polarization. According to Eq. 3, for the incidence of a linear polarization, the transmitted polarizations are always orthogonal, i.e., E ₀₁⋅E ₀₂≡0. This means that the variational trajectory of the x-dependent superposition state is locked by a meridian. On the other hand, the x-dependent phase difference is invariable for a constant NA. Therefore, the optical helicity density and its gradient of the superposition field are robust under the illumination of linear polarization. This stability is extremely important for related applications of particle manipulation.

This metalens platform is also suitable for the generation of other polarization oscillating light fields. We next design another metalens by choosing two spin states as bases. In this case, the transmitted polarizations consequently are E ₀₁ = [1 i]^T and E ₀₂ = [1 -i]^T, respectively. Substituting these two states into Eqs. 1, 2, one obtains the Stokes parameters of the focal field as S ₁ = [−2sin² α+(1 + cos² α) sin (2kxsinα)]/2, S ₂ = cosαcos (2kxsinα), and S ₃ = 0. Under the paraxial condition, S ₁ = sin (2kxsinα) and S ₂ = cos (2kxsinα); this means that the superposition state oscillates along the equator of the Poincaré sphere, as per the red trajectory shown in Figure 4A. Since the polarization azimuthal angle is 2ϕ' = tan⁻¹(S ₂/S ₁); therefore, the polarization orientation varies as ϕ' = π/4-kxsinα along the x direction. As a proof-of-principle, we fabricate an NA = 0.2 metalens with f = 1.1 mm and α = 5.83°. The meta-atom has parameters δ = π/2, ϕ ₁ = 0°, and ϕ ₂ = 90°. Figure 4B shows the transmission properties of the selected geometries, of which the geometric sizes are shown in Note 2 and Supplementary Table S2 of Supplementary Material. The SEM image of the metalens is shown in Figure 4C. Figures 4D, E give the experimentally measured polarization orientation and Stokes parameters under the illumination of an 808-nm laser with [1 1]^T polarization. Because of the challenge on selecting a nanopillar with suitable phase retardation, we fabricate and operate the experiment at a wavelength of 808 nm (the refractive index of Poly-Si at this wavelength is n = 3.23479 + i0.00102); the efficiency of this type of metalens is about 78% at this wavelength. As shown, the experimental result is consistent with the theory that the polarization orientation oscillates along the x direction with a period of 3.9 μm.

We further analyze the optical force induced by this intriguing optical helicity gradient, which exhibited promising application potential in chirality-related deflection [12, 13]. We use the scattering property of a spherical chiral particle to analyze this typical force [9, 13], which has been derived as F = ∇ U + σ c S − I m α e m ∇ × S + c σ e ∇ × L e + c σ m ∇ × L m + ω γ e L e + ω γ m L m + c k 0 4 12 π I m [ α e e α m m * ] I m E × H * , (4) where U=(Re [α _ee ]|E|² + Re [α _mm ]|E|²-2Re [α _em ]Im|H∙E ^* |)/4 is the term due to the light–matter interaction; ⟨S⟩ = Re [E×H ^* ]/2 is the time-averaged Poynting vector; ⟨L _e⟩ = ε ₀ Im [E×E ^* ]/(4ωi) and ⟨L _m⟩ = μ ₀ Im [H×H ^* ]/(4ωi) are the time-averaged spin densities; σ _e = k ₀Im [α _ee ]/ε ₀, σ _m = k ₀Im [α _mm ]/μ ₀, and σ = σ _e +σ _m -c ² k4 0(Re[α _ee α ^* _mm ]+α _em α ^* _em ); γ _e = -2ωIm [α _em ]+ ck4 0Re [α _ee α ^* _em ]/(3πε ₀), and γ _e = -2ωIm [α _em ]+ ck4 0Re [α _mm α ^* _em ]/(3πμ ₀); α _ee , α _mm , and α _em denote the polarizability of the particle.

In the focal plane, because of the uniform intensity distribution along the x direction, the lateral component associated with the gradient force, that is, the first term on the right side of Eq. 4, only presents the y component, i.e., F^g _y . However, the radiation pressure, vortex forces, and the second and third terms in Eq. 4 are neglectable because of the uniform phase distributions. Moreover, the last term of Eq. 4 is about 10^–6 smaller than that of the spin density force [9]. We, hence, focus on the terms related to the SAM density ⟨L⟩. In this case, we calculated the lateral forces on a spherical dipolar chiral particle (r = 50 nm) characterized by ε _r = 2.53, μ _r = 1, and chirality parameter κ = 1, when a |A⟩ polarized laser with λ = 670 wavelength illuminates an NA = 0.5 metalens [9, 13]. The force distributions along the x axis are shown in Figures 5A, B. Since the polarization conversion, the enhanced z-polarized component induces curl-spin (the fourth and fifth terms) and spin-density (the sixth and seventh terms) forces along the y and x directions, i.e., F ^c and F ^s , respectively. Because of the cylindrical focusing, the gradient force F ^g dominates the y component of the lateral force, as shown in Figure 5C. The lateral forces F _x and F _y exerted on the dipole chiral particle in the focal plane are shown in Figure 5D. The gradient force F^g _y acts as always as the restoring force toward y = 0. In the presence of the helicity gradient, the spatially modulated F _x component induces chiral-selective transverse motion along the x axis, that is, this transverse force derives particles with chirality of κ > 0 from the unstable equilibrium points indicated by the blue cross to stable equilibrium points indicated by the red cross. However, this force derives the particles with opposite chirality (κ < 0) along the direction depicted by the dotted line to their stable equilibrium points indicated by the blue cross.

In conclusion, we reported a monolayer dielectric metalens that can generate focal fields with oscillating optical SAM to induce a chiral-sensitive lateral optical force. We introduced a hybrid vector model to construct the focal field with a laterally tailored polarization structure and to design the metalens. Under the illuminations of linearly polarized light fields, we experimentally demonstrated the capability of constructing light fields with one-dimensional oscillating optical helicity and analyzed the lateral force induced by the gradient optical helicity affecting chiral particles. This work provides a platform for the design of integrated and compact devices to conveniently control the polarization structure of light fields for optical trapping and manipulation.