High-efficiency vectorial holography based on ultra-thin metasurfaces

Tong Liu; Changhong Dai; Dongyi Wang; Che Ting Chan; Lei Zhou

doi:10.1117/1.AP.7.5.056004

1 Introduction

Holography, an optical technology to record and reconstruct a target light field, has attracted intensive attention recently for its numerous practical applications such as imaging, virtual reality, data encryption, and storage.1^–3 Traditional approaches to implement holography require complex optics interference setups, and the generated holograms usually exhibit bulky sizes,4^–6 all being unfavorable for optical integration. Such a limitation is partially addressed after the Gerchberg–Saxton (GS) algorithm was developed to retrieve the optical properties of a hologram by computer without doing realistic interference experiments. However, holographic images generated are typically scalar ones, which exhibit uniform polarization distributions.

Metasurfaces, ultra-thin metamaterials composed of subwavelength planar microstructures (e.g., meta-atoms) arranged in certain sequences, have emerged as a powerful platform to manipulate light waves. Many fascinating effects were demonstrated based on metasurfaces, including polarization control,7^–11 propagating-wave to surface-wave coupling,12^–14 meta-lensing,15^–18 cloaking,19 and so on.20^–23 In particular, choosing meta-atoms exhibiting reflection/transmission phases in accordance with those retrieved from the GS algorithm, one can construct a metasurface which, under appropriate external illumination, can generate a predesigned holographic image.24^–28 In contrast to conventional technology, such metadevices are flat, ultra-compact, easy to realize, and exhibit better lateral resolutions, all being highly desired in integration optics. Despite the exciting progress, however, most holographic images realized with metasurfaces so far are scalar ones exhibiting uniform polarization distributions. Although certain efforts have been recently devoted to generating vectorial holographic images based on metasurfaces,29^–32 unit phase-bits adopted to construct meta-holograms usually consist of two distinct Pancharatnam–Berry (PB) resonators working for two circular-polarization (CP) channels. Under illumination of incident light, anomalous scatterings by two subsets of PB resonators can generate CP beams with opposite helicity, and their interference can form the target vectorial image. However, vectorial images generated by such a scheme are of low efficiency because normal scatterings by the PB resonators inevitably exist, and the interference nature of the scheme sets certain restrictions on the incident polarization, which must contain two CP components (i.e., the incident polarization cannot be a pure CP). Very recently, a simple scheme was proposed to construct vectorial meta-holograms with rectangle-shaped rod-resonators exhibiting anisotropic resonant phases.33 However, employing merely resonant phases without PB phases, such a scheme33 lacks the necessary degrees of freedom to produce generic vectorial holographic images with arbitrary polarization distributions. A general method for designing vectorial meta-holograms, with no restrictions on both incident polarization and the polarization distributions of generated patterns, remains unexplored.

In this work, we propose a generic approach, working for arbitrary incident polarizations, to realize arbitrary predesigned vectorial holographic images based on high-efficiency metasurfaces constructed by deep-subwavelength single-structure meta-atoms. We first establish an efficient approach to retrieve the wave-scattering properties (e.g., reflection phases and polarization-conversion capabilities) of meta-atoms needed to construct a metasurface for generating a target vectorial image, combining the GS algorithm and the wave-decomposition technique. We next design a series of subwavelength single-structure meta-atoms exhibiting tailored reflection Jones matrices and experimentally characterize their wave-scattering properties. Compared to previous work, our designed meta-atoms exhibit reflection phases of both resonant and PB origins, thus providing the necessary degrees of freedom to generate vectorial holographic images without restrictions on polarization distributions. With these building blocks at hand, we finally design and fabricate a set of metadevices and experimentally demonstrate that they can generate predesigned complex vectorial holographic images, under external illuminations at the wavelength of 1064 nm.

2 Results

2.1 Generic Strategy for Designing Vectorial Meta-Holograms

We now establish a generic strategy to design a metasurface, which, as illuminated by a light beam with polarization $| σ_{0} ⟩$ , can generate a target vectorial holographic image in the far field (FF) exhibiting tailored distributions of intensity and polarization on its wave-front (see Fig. 1). As shown in the inset to Fig. 1(a), an arbitrary polarization state corresponds to a point on Poincaré’s sphere characterized by two spherical coordinates, i.e., the polar angle $Θ$ and the azimuthal angle $Ψ$ , respectively. Specifically, $\cos Θ$ defines the ellipticity of the polarization state, whereas $Ψ / 2$ represents the polar angle of the polarization state. To characterize an FF vectorial beam, it is customized to set targets in the $\vec{k}$ -space and use $A_{tar}^{h} (\vec{k})$ and $| σ_{tar}^{h} (\vec{k}) ⟩$ to describe, respectively, the complex amplitude and polarization of a wave-component inside the beam, with superscript h representing the FF holographic plane. It is worth noting that $| A_{tar}^{h} (\vec{k}) |^{2}$ and $| σ_{tar}^{h} (\vec{k}) ⟩$ precisely represent the intensity and polarization distributions of the field pattern measured on the focal plane of a lens, commonly adopted in FF characterization experiments to image the vectorial beam. Therefore, our task is to find an algorithm to retrieve the near-field (NF) scattering properties of the target metasurface from the FF vectorial image described by $A_{tar}^{h} (\vec{k})$ and $| σ_{tar}^{h} (\vec{k}) ⟩$ . Our approach is in general a two-step process [see Fig. 1(b)]: (1) we first decompose the target FF vectorial beam into a left-circular-polarized (LCP) beam and its right-circular-polarized (RCP) counterpart and then retrieve two NF planar sources for generating these two beams by standard FF-NF transformation iterations, and (2) we next construct the final vectorial NF planar source via linear superposition of two retrieved planar sources and finally obtain the distribution of scattering Jones matrix for our target metasurface to generate the required NF, under illumination of a light beam with polarization $| σ_{0} ⟩$ .

Figure 1.Schematics of vectorial meta-holograms and its design process. (a) Shining a metasurface composed of single-structure metaatom arrays with tailored wave-scattering properties (including reflection phases and polarization-coversion capabilities) by an incident light with certain polarization, a vectorial holographic image can be generated in the far field exhibiting inhomogeneous distributions of both intensity and polarization. (b) Flow chart of the design process: (1) for a holographic image with intensity and polarization distributions given by $A_{tar}^{h} (\vec{k})$ and $| σ_{tar}^{h} (\vec{k}) ⟩$ , one can employ the generalized GS algorithm to retrieve the phase distributions of two planar sources for two circular-polarization channels; (2) linear superposition of two planar sources $Φ_{\pm}^{m} (\vec{r})$ yields the final near-field source, which further helps us sort out meta-atoms located at different positions with arbitrarily given incident polarization. The superscript h indicates the far-filed holographic plane, and m indicates the near-field metasurface plane.

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First, consider step 1. The polarization distribution of a target vectorial beam can be generally written as $| σ_{tar}^{h} (\vec{k}) ⟩ = (\begin{matrix} e^{- i Ψ_{tar}^{h} (\vec{k}) / 2} \cos (Θ_{tar}^{h} (\vec{k}) / 2) \\ e^{+ i Ψ_{tar}^{h} (\vec{k}) / 2} \sin (Θ_{tar}^{h} (\vec{k}) / 2) \end{matrix})$ , where ${Θ_{tar}^{h} (\vec{k}), Ψ_{tar}^{h} (\vec{k})}$ denotes a specific point on Poincaré’s sphere representing the polarization state of the wave component with $\vec{k}$ . This arbitrarily polarized vectorial wave can then be decoupled into two CP bases, $A_{tar}^{h} (\vec{k}) | σ_{tar}^{h} (\vec{k}) ⟩ = A_{tar, +}^{h} (\vec{k}) {| + ⟩}_{\vec{k}} + A_{tar, -}^{h} (\vec{k}) {| - ⟩}_{\vec{k}}$ , where ${| + ⟩}_{\vec{k}}$ and ${| - ⟩}_{\vec{k}}$ represent LCP and RCP states, respectively. Here, ${\begin{matrix} A_{tar, +}^{h} (\vec{k}) = A_{tar}^{h} (\vec{k}) e^{- \frac{i Ψ_{tar}^{h} (\vec{k})}{2}} \cos (\frac{Θ_{tar}^{h} (\vec{k})}{2}) \\ A_{tar, -}^{h} (\vec{k}) = A_{tar}^{h} (\vec{k}) e^{+ \frac{i Ψ_{tar}^{h} (\vec{k})}{2}} \sin (\frac{Θ_{tar}^{h} (\vec{k})}{2}) \end{matrix},$ (1)denote the complex amplitudes of two components in the wave with $\vec{k}$ . In general, for different $\vec{k}$ -components, LCP and RCP bases (i.e., ${| + ⟩}_{\vec{k}}$ and ${| - ⟩}_{\vec{k}}$ ) are distinct. However, assuming that the target vectorial beam is a paraxial one, we can safely replace these individual bases ( ${| + ⟩}_{\vec{k}}$ and ${| - ⟩}_{\vec{k}}$ ) by a set of global bases ${| + ⟩}_{\vec{K}}$ and ${| - ⟩}_{\vec{K}}$ , where $\vec{K}$ represents the central wave vector of the beam. Therefore, under the paraxial approximation, the target vectorial FF beam can be decoupled into two nonvectorial LCP and RCP beams, with $\vec{k}$ -space expansion coefficients given by Eq. (1). We emphasize that these two beams are not fully independent but exhibit well-defined phase differences $\arg (A_{tar, -}^{h} (\vec{k})) - \arg (A_{tar, +}^{h} (\vec{k})) = Ψ_{tar}^{h} (\vec{k})$ for different $\vec{k}$ components. Interference of these two beams with appropriate phase differences reconstructs the target vectorial image. We note that such a composite source construction approach is not confined to LCP/RCP bases but can be extended to any other orthogonal bases (such as two linear-polarization bases), as long as the considered beam fulfills the paraxial approximation.

We next retrieve two planar NF sources, placed on the $x y$ -plane at $z = 0$ , which can separately generate these two coherent opposite-helicity beams in the FF. Because the NF-to-FF transformation preserves polarization within the paraxial approximation, we can employ the standard GS algorithm to retrieve the phase distribution $Φ_{+}^{m} (\vec{r})$ of the LCP NF source from the LCP FF intensity distribution $I_{+} (\vec{k}) = | A_{tar, +}^{h} (\vec{k}) |^{2}$ and its RCP counterpart $Φ_{-}^{m} (\vec{r})$ from the RCP FF intensity distribution $I_{-} (\vec{k}) = | A_{tar, -}^{h} (\vec{k}) |^{2}$ . Meanwhile, these two beams exhibit $\vec{k}$ -dependent phase differences given by $Ψ_{tar}^{h} (\vec{k})$ , which is taken into consideration as a constraint in our two-channel FF-to-NF retrieval process (see Sec. 1 in Supplementary Material for details). After certain rounds of iterations, we finally obtain a stable solution of ${Φ_{+}^{m} (\vec{r}), Φ_{-}^{m} (\vec{r})}$ , which fulfills our requirements, where the superscript m indicates the near-field metasurface plane. We note that the iterations discard the variation of NF local amplitudes retrieved from the inverse Fourier transform in each round to fit the requirement of local energy conservation in the totally reflective MIM meta-atom. Only phases of orthogonal components ${Φ_{+}^{m} (\vec{r}), Φ_{-}^{m} (\vec{r})}$ are left in the optimized results, setting no constraints on synchronized control of amplitude and phase.

We now proceed to step 2 to construct the final NF source via coherently combining the two retrieved planar sources. We note that this step is our key improvement over the previous design strategy, which enables our metadevices to be formed by single-structure meta-atoms instead of composite meta-atoms containing two resonators. However, the traditional GS algorithm yields no information for the amplitudes of the retrieved LCP and RCP NF sources. Considering the conservation of polarization throughout the NF to FF process and that the total powers of the two generated beams can be calculated via $I_{\pm} = \iint I_{\pm} (\vec{k}) d \vec{k}$ , we can set the amplitudes of LCP and RCP sources as $\frac{\sqrt{I_{+}}}{\sqrt{I_{+} + I_{-}}}$ and $\frac{\sqrt{I_{-}}}{\sqrt{I_{+} + I_{-}}}$ , respectively, in consistency with energy conservation. Therefore, the final planar NF source, which is a coherent linear superposition of the retrieved LCP and RCP sources, must exhibit the following field distribution on the $x y$ -plane at $z = 0$ : ${\vec{E}}_{N F} (\vec{r}) = \frac{1}{\sqrt{I_{+} + I_{-}}} (\sqrt{I_{+}} e^{i Φ_{+}^{m} (\vec{r})} | + ⟩ + \sqrt{I_{-}} e^{i Φ_{-}^{m} (\vec{r})} | - ⟩) = e^{i Φ_{tar}^{m} (\vec{r})} (\begin{matrix} e^{- \frac{i Ψ_{tar}^{m} (\vec{r})}{2}} \cos (\frac{Θ_{tar}^{m} (\vec{r})}{2}) \\ e^{+ \frac{i Ψ_{tar}^{m} (\vec{r})}{2}} \sin (\frac{Θ_{tar}^{m} (\vec{r})}{2}) \end{matrix}),$ (2)where ${\begin{cases} Φ_{tar}^{m} (\vec{r}) = \frac{Φ_{+}^{m} (\vec{r}) + Φ_{-}^{m} (\vec{r})}{2} \\ Θ_{tar}^{m} (\vec{r}) = 2 \arctan (\frac{\sqrt{I_{-}}}{\sqrt{I_{+}}}) \\ Ψ_{tar}^{m} (\vec{r}) = Φ_{-}^{m} (\vec{r}) - Φ_{+}^{m} (\vec{r}) \end{cases}$ (3)are three characteristic quantities that unambiguously dictate the scattering properties of a metaatom located at position $\vec{r}$ on the metasurface. We note that $| + ⟩$ and $| - ⟩$ defined in Eq. (2) are the global LCP and RCP bases corresponding to the central wave vector, where the subscript $\vec{K}$ is neglected for convenience.

Equation (2) has a very clear physical meaning: at an arbitrary point $\vec{r}$ on the source plane, the required NF takes a phase $Φ_{tar}^{m} (\vec{r})$ and a polarization $| σ_{tar}^{m} (\vec{r}) ⟩ = (\begin{matrix} e^{- i Ψ_{tar}^{m} (\vec{r}) / 2} \cos (Θ_{tar}^{m} (\vec{r}) / 2) \\ e^{+ i Ψ_{tar}^{m} (\vec{r}) / 2} \sin (Θ_{tar}^{m} (\vec{r}) / 2) \end{matrix})$ with ${Θ_{tar}^{m} (\vec{r}), Ψ_{tar}^{m} (\vec{r})}$ denoting its position on Poincaré’s sphere. Therefore, the scattering properties of a metasurface under such design are also clear: it should be able to generate the required NF [Eq. (2)] on its surface, as illuminated by a light beam with polarization $| σ_{0} ⟩$ . To facilitate our readers’ understandings of the whole design procedures, we present a concrete example (see Fig. S1 in the Supplementary Material) to illustrate how to retrieve three distributions ${Φ_{tar}^{m} (\vec{r}), Θ_{tar}^{m} (\vec{r}), Ψ_{tar}^{m} (\vec{r})}$ required to design the metadevice for the generation of a desired FF vectorial image. One thing to emphasize is that with the employment of a parallel GS algorithm, we utilize brand-new FF to NF logics in this work, well-distinguished from previous attempts on vectorial field manipulation works based on NF to FF logics20^,21^,23^,34 by breaking the constraints on wavefront/polarization distribution symmetries. The systematic design guideline proposed in this work, particularly the closed-form formulas to help explicitly design single-structure meta-atoms to construct meta-holograms, is the key contribution of the present work over previous ones35^–43 employing a similar concept.

2.2 Basic Meta-Atoms and Experimental Characterizations

We now discuss how to design a series of single-structure meta-atoms to construct our metadevices. Obviously, such meta-atoms should exhibit the required reflection phases $Φ_{tar}^{m} (\vec{r})$ and polarization conversion capabilities $| σ_{0} ⟩ \to | σ_{tar}^{m} (\vec{r}) ⟩$ , as shined by a light beam carrying a certain polarization $| σ_{0} ⟩$ . We find that meta-atoms in a metal-insulator-metal (MIM) configuration are appropriate candidates to satisfy our requirements. As shown in the inset to Fig. 2(b), a typical MIM metaatom consists of a metallic cross (with two bars of lengths $L_{u}$ and $L_{v}$ , respectively) rotated by an angle $ξ$ , a continuous metal film, and a dielectric spacer separating two metallic layers. Shining the metaatom with a normally incident light with $E$ field polarized along the $\hat{u}$ -axis (or $\hat{v}$ -axis), anti-parallel currents are induced on two metallic layers, forming a magnetic resonance with a resonant frequency determined by $L_{u}$ (or $L_{v}$ ). Due to the continuous metal film on the back, the metaatom does not allow transmission, and thus, we only need to consider its reflection properties. The reflection Jones’ matrix of the meta-atom, expressed in linear-polarization (LP) bases in the $u - v$ coordinate system, can be generally written as $R = (\begin{matrix} | r_{u u} | e^{i Φ_{u}} & 0 \\ 0 & | r_{v v} | e^{i Φ_{v}} \end{matrix})$ , where $Φ_{u}$ and $Φ_{v}$ are reflection phases dictated by two magnetic resonances, and in turn, by $L_{u}$ and $L_{v}$ , correspondingly. This method can be easily extended to transmissive systems employing dielectric meta-atoms with enough modulation degrees of freedom, designed according to the transmission Jones’ matrix $T = (\begin{matrix} | t_{u u} | e^{i Φ_{u}} & 0 \\ 0 & | t_{v v} | e^{i Φ_{v}} \end{matrix})$ with $Φ_{u}$ and $Φ_{v}$ denoting transmission phases. In the ideal lossless condition, we have $| r_{u u} | = | r_{v v} | \equiv 1$ . With losses in realistic materials taken into consideration, the reflection amplitudes are no longer exactly 1 but can still be quite close to 1 via careful design (see Sec. 2 in the Supplementary Material). Therefore, we only consider reflection phases in our design. We define two new parameters, $Φ_{res} = (Φ_{u} + Φ_{v}) / 2 - π / 4$ and $Δ Φ = Φ_{v} - Φ_{u}$ , representing the average resonance phase and the cross-polarization phase difference, respectively. Figures 2(a) and 2(b) depict how $Φ_{res}$ and $Δ Φ$ of the metaatom (periodically repeated to form an array) vary against its geometric parameters $L_{u}$ and $L_{v}$ , calculated at the wavelength of 1064 nm, with other geometric parameters fixed as: $h_{Str} = 30 nm$ , $h_{Ins} = 125 nm$ , $h_{Sub} = 125 nm$ , $w = 80 nm$ , and $P_{x} = P_{y} = 600 nm$ . Here, we choose Ag as the metallic material and ${SiO}_{2}$ as the dielectric material for constructing our spacer and substrate, respectively.

Figure 2.Designs and characterizations of meta-atoms. (a) Resonance phase $Φ_{R es}$ and (b) cross-polarization phase difference $Δ Φ$ of MIM metaatom arrays (see inset for the structure of a typical metaatom) with different $L_{u}$ and $L_{v}$ , calculated by FDTD simulations at 1064 nm. Other geometric parameters of the MIM meta-atoms are fixed as: $h_{Str} = 30 nm$ , $h_{Ins} = 125 nm$ , $h_{Sub} = 125 nm$ , $w = 80 nm$ , and $P_{x} = P_{y} = 600 nm$ . Dashed lines denote the $Δ Φ = \pm π$ contours, with Nos. 1, 2, and 3 labeling three typical meta-atoms functioning as quarter, half, and 2/3 waveplates, which are experimentally characterized. Measured polarization-filtered intensity patterns of (c) incident light with $E$ polarized along the 45-deg angle in-between $u$ and $v$ axes, and light beams reflected by metasurfaces composed by meta-atoms labelled with (d) No. 1, (e) No. 2, and (f) No. 3, respectively. The working wavelength is 1064 nm. Solid lines in panels (d)–(f) are FDTD calculated results.

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Further considering the rotation operation, we find that the reflection Jones’ matrix in CP bases of the laboratory system becomes $\tilde{R} = SM (ξ) {RM}^{- 1} (ξ) S^{- 1}$ , where $S$ and $M (ξ)$ represent the LP to CP transformation and the rotation operation, respectively. Given an incidence polarization $| σ_{0} ⟩ = (\begin{matrix} e^{- i \frac{Ψ_{0}}{2}} \cos \frac{Θ_{0}}{2} \\ e^{+ i \frac{Ψ_{0}}{2}} \sin \frac{Θ_{0}}{2} \end{matrix})$ , we can derive the wave scattered by such a rotated metaatom as $\tilde{R} | σ_{0} ⟩ = e^{i (Φ_{res} + \frac{π}{4})} (\begin{matrix} e^{- i \frac{Ψ_{0}}{2}} \cos \frac{Δ Φ}{2} \cos \frac{Θ_{0}}{2} - i e^{- 2 i ξ} e^{+ i \frac{Ψ_{0}}{2}} \sin \frac{Δ Φ}{2} \sin \frac{Θ_{0}}{2} \\ e^{+ i \frac{Ψ_{0}}{2}} \cos \frac{Δ Φ}{2} \sin \frac{Θ_{0}}{2} - i e^{+ 2 i ξ} e^{- i \frac{Ψ_{0}}{2}} \sin \frac{Δ Φ}{2} \cos \frac{Θ_{0}}{2} \end{matrix}) .$ (4)

Comparing Eq. (4) with Eq. (2), we can thus capture the three characteristic parameters ${Φ_{tar}^{m}, Θ_{tar}^{m}, Ψ_{tar}^{m}}$ of a metaatom in terms of another set of parameters ${Φ_{res}, Δ Φ, ξ}$ with the incident polarization fixed.

The role of rotation angle $ξ$ can be more clearly seen from the discussions below. In LP bases, we can derive from Eq. (4) that $\tilde{R} | σ_{0} ⟩ = ({\hat{e}}_{x}, {\hat{e}}_{y}) (\begin{matrix} E_{tar, x}^{m} \\ E_{tar, y}^{m} \end{matrix}) = \frac{e^{i (Φ_{res} + \frac{π}{4})}}{\sqrt{2}} ({\hat{e}}_{x}, {\hat{e}}_{y}) (\begin{matrix} e^{- \frac{1}{2} i (4 ξ + Ψ_{0})} (e^{2 i ξ} \cos \frac{Δ Φ}{2} (\cos \frac{Θ_{0}}{2} + e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2}) - i \sin \frac{Δ Φ}{2} (e^{4 i ξ} \cos \frac{Θ_{0}}{2} + e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2})) \\ i e^{- \frac{i Ψ_{0}}{2}} \cos \frac{Θ_{0}}{2} (\cos \frac{Δ Φ}{2} + {i e}^{2 i ξ} \sin \frac{Δ Φ}{2}) + e^{- 2 i ξ + \frac{i Ψ_{0}}{2}} \sin \frac{Θ_{0}}{2} ({- i e}^{2 i ξ} \cos \frac{Δ Φ}{2} + \sin \frac{Δ Φ}{2}) \end{matrix})$ with the amplitudes and phases of $x$ and $y$ components being $| E_{tar, x}^{m} | = \sqrt{\frac{1}{2} (1 + (\cos \frac{Δ Φ}{2})^{2} \cos Ψ_{0} \sin Θ_{0} + \cos (4 ξ - Ψ_{0}) (\sin \frac{Δ Φ}{2})^{2} \sin Θ_{0} + \cos Θ_{0} \sin Δ Φ \sin 2 ξ)},$ $| E_{tar, y}^{m} | = \sqrt{\frac{1}{2} (1 - (\cos \frac{Δ Φ}{2})^{2} \cos Ψ_{0} \sin Θ_{0} - \cos (4 ξ - Ψ_{0}) (\sin \frac{Δ Φ}{2})^{2} \sin Θ_{0} - \cos Θ_{0} \sin Δ Φ \sin 2 ξ)},$ and $Ψ_{tar, x}^{m} = - i \ln (\frac{\begin{matrix} \sqrt{2} e^{- \frac{1}{2} i (4 ξ - 2 Φ_{res} + Ψ_{0})} ((1 + i) e^{2 i ξ} \cos \frac{Δ Φ}{2} (\cos \frac{Θ_{0}}{2} + e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2}) + (1 - i) \sin \frac{Δ Φ}{2} (e^{4 i ξ} \cos \frac{Θ_{0}}{2} + e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2})) \end{matrix}}{\begin{matrix} \sqrt{4 + \sin (Θ_{0} - Ψ_{0}) + 4 \sin 2 ξ (\cos Θ_{0} \sin Δ Φ + \cos Δ Φ \sin Θ_{0} \sin (2 ξ - Ψ_{0})) + \sin (Θ_{0} + 4 ξ - Ψ_{0}) + \sin (Θ_{0} + Ψ_{0}) + \sin (Θ_{0} - 4 ξ + Ψ_{0})} \end{matrix}}),$ $Ψ_{tar, y}^{m} = - i \ln (\frac{(- 1)^{1 / 4} e^{- \frac{1}{2} i (4 ξ - 2 Φ_{res} + Ψ_{0})} ({i e}^{2 i ξ} \cos \frac{Δ Φ}{2} (\cos \frac{Θ_{0}}{2} - e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2}) - \sin \frac{Δ Φ}{2} (e^{4 i ξ} \cos \frac{Θ_{0}}{2} - e^{{i Ψ}_{0}} \sin \frac{Θ_{0}}{2}))}{\sqrt{1 - 2 \cos Θ_{0} \cos ξ \sin Δ Φ \sin ξ - \sin Θ_{0} (\cos 2 ξ \cos (2 ξ - Ψ_{0}) + \cos Δ Φ \sin 2 ξ \sin (2 ξ - Ψ_{0}))}}),$ respectively. We find that the two amplitudes satisfy the energy conservation $| E_{tar, x}^{m} |^{2} + | E_{tar, y}^{m} |^{2} = 1$ . Interestingly, we find that the rotation angle $ξ$ effectively modulates the amplitudes $| E_{tar, x}^{m} |$ and $| E_{tar, y}^{m} |$ , whereas $Φ_{res}$ and $Δ Φ$ mainly affect the two phases $Ψ_{tar, x}^{m}$ , $Ψ_{tar, y}^{m}$ . Therefore, with three independent degrees of freedom ${ξ, Φ_{res} and Δ Φ}$ , our rotated MIM meta-atoms can effectively realize amplitude/phase synchronized controls over locally scattered waves subject to energy conservation, thus exhibiting the capability to generate arbitrary vectorial fields.

Under LCP or RCP incidence (i.e., $Ψ_{0} = 0$ , $Θ_{0} = 0$ or $π$ ), we have ${\begin{cases} Θ_{tar}^{m} = \frac{1 - σ π}{2} + σ Δ Φ \\ Φ_{tar}^{m} = Φ_{res} + 2 σ ξ \\ Ψ_{tar}^{m} = 0 \end{cases},$ (5)in the case of $Δ Φ = \pm π$ ( $Φ_{res}$ is defined to be $Φ_{u}$ for these cases20), and ${\begin{cases} Θ_{tar}^{m} = \frac{1 - σ π}{2} + σ Δ Φ \\ Φ_{tar}^{m} = Φ_{res} + σ ξ \\ Ψ_{tar}^{m} = 2 ξ - σ \frac{π}{2} \end{cases},$ (6)in the case of $Δ Φ \neq \pm π$ , with $σ = \pm 1$ denoting LCP and RCP states, respectively. Under the incidence of arbitrary polarizations, ${Φ_{tar}^{m}, Θ_{tar}^{m}, Ψ_{tar}^{m}}$ exhibit rather complicated dependences on ${Φ_{res}, Δ Φ, ξ}$ , and we provide their general expressions in Sec. 1 of the Supplementary Material. Therefore, given any NF functions ${Φ_{tar}^{m} (\vec{r}), Θ_{tar}^{m} (\vec{r}), Ψ_{tar}^{m} (\vec{r})}$ retrieved from a target vectorial holographic image, we can solve from Eqs. (5) and (6) [or Eqs. (S13)–(S15) in Sec. 1 of the Supplementary Material] the optical properties ${Φ_{res}, Δ Φ, ξ}$ of meta-atoms at positions $\vec{r}$ , and in turn, obtain from Figs. 2(a) and 2(b) the geometric parameters ( $L_{u}$ and $L_{v}$ ) of our metaatoms. With distributions of these three geometrical parameters ${L_{u} (\vec{r}), L_{v} (\vec{r}), ξ (\vec{r})}$ determined, the metasurface can be correspondingly designed.

It is helpful to compare our scheme, based on rotated MIM meta-atoms, with the previously proposed scheme based on unrotated rectangle-shaped rod-resonators.33 Without rotations, the rod-resonators adopted in Ref. 33 only exhibit resonant phases without PB phases, thus discarding an important degree of freedom to manipulate light. In fact, setting $ξ \equiv 0 °$ in our formulas, we find that three parameters ${Φ^{m}, Θ^{m}, Ψ^{m}}$ cannot be freely and independently modulated. In such cases, the polarization distributions of generated vectorial holographic images exhibit certain restrictions. By contrast, with $ξ$ added into our scheme, such a restriction on polarization distribution is lifted (see Supplementary Material for details of comparison).

We experimentally characterize the wave-scattering properties of three typical meta-atoms labeled by Nos. 1, 2, and 3 in Figs. 2(a) and 2(b), which have different values of $Δ Φ$ ( $Δ Φ = \frac{π}{2}, π, \frac{2 π}{3}$ to be specific). We fabricate three metasurface samples using standard electron-beam-lithography (EBL) technology, each containing a periodical array of meta-atoms with geometric parameters given in Figs. 2(a) and 2(b). Figure S3 in Sec. 3.2 of the Supplementary Material illustrates the scanning electron microscopy (SEM) images of three fabricated samples. In our experiments, shining the metasurfaces by LP light with $E$ -field polarized along the 45-deg direction in-between $\hat{u}$ and $\hat{v}$ axes, we measure the intensities of the reflected light with a linear polarizer oriented at different angles placed in front of the detector. Such polarization-filtered pattern can well capture the polarization state of a light beam. Figure 2(c) depicts the measured polarization-filtered pattern of the incident light. The “8”-shaped pattern with intensity zeros appearing at the angles of 135 deg and 315 deg is direct evidence to show that the incident light is a linearly polarized one with $E$ -field oriented along the 45-deg direction. Meanwhile, the measured polarization-filtered patterns of light beams reflected by three metasurfaces are depicted in Figs. 2(d)–2(f), respectively. Analyzing these patterns carefully, we find that the reflected light beams exhibit circular polarization [Fig. 2(d)], 135-deg-oriented linear polarization [Fig. 2(e)], and elliptical polarization [Fig. 2(f)], respectively, all being consistent with theoretical predictions. We note that dielectric structures inevitably suffer from the trade-off between system thicknesses and efficiencies,38^,39^,44^–47 yet with MIM meta-atoms as our building blocks, meta-holograms constructed in the following sections can simultaneously exhibit ultra-thin thicknesses and high efficiencies, showcasing clear advantages over previous ones.

2.3 Applicability of the Design Strategy for Arbitrary Incident Polarization

In addition to high efficiency, one important merit of our design strategy is that it can be applied to arbitrary incident polarizations. This is in sharp contrast to previously proposed strategy of merging two subsets of PB meta-atoms,29^–32 which can only work under restricted polarizations. Given an FF vectorial holographic image as shown in Fig. 3(b), we now illustrate how to design appropriate metasurfaces to generate the same target holographic image, as the incident polarization $| σ_{0} ⟩$ changes along a longitude line of Poincaré’s sphere.

Figure 3.Design of meta-holograms for achieving identical vectorial images under excitations of incident light with distinct polarizations. (a) Three incident polarizations $| σ_{0} ⟩$ with ${Θ_{0} = \frac{π}{4}, \frac{π}{2}, \frac{13 π}{20}; Ψ_{0} = 0}$ selected in the designs, as marked by red stars on Poincaré’s sphere. (b) The target holographic image contains 9 parts exhibiting different polarizations indicated by green segments (representing linear polarization) and red/blue circles (representing left/right circular polarizations). Top-view structures of three meta-holograms designed under incident polarizations with (c) $Θ_{0} = \frac{π}{4}$ , (e) $Θ_{0} = \frac{π}{2}$ , and (g) $Θ_{0} = \frac{13 π}{20}$ , respectively, with insets depicting zoomed-in views at the four corners of the samples. Calculated far-field images generated by three samples under illuminations of normally incident light with polarizations (d) $Θ_{0} = \frac{π}{4}$ , (f) $Θ_{0} = \frac{π}{2}$ , and (h) $Θ_{0} = \frac{13 π}{20}$ , respectively. Segments and circles indicate the local polarization states.

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Selecting three different points on the longitude line with $Θ_{0} = \frac{π}{4}, \frac{π}{2}, \frac{13 π}{20}$ and $Ψ_{0} = 0$ , respectively, we employ Eqs. (S13)–(S15) in Sec. 1 of the Supplementary Material to retrieve three sets of ${Φ_{res} (\vec{r}), Δ Φ (\vec{r}), ξ (\vec{r})}$ distributions. We then determine the ${L_{u} (\vec{r}), L_{v} (\vec{r}), ξ (\vec{r})}$ distributions from the above retrieved functions, based on which we design three metasurfaces. Figures 3(c), 3(e), and 3(g) depict the top-view structures of three metasurfaces, designed for different incident polarizations. We next employ the Fourier transformation to reconstruct the holographic images generated by these metasurfaces, under illuminations of normally incident light with different polarizations corresponding to the points marked on Poincaré’s sphere. Figures 3(d), 3(f), and 3(h) clearly illustrate that all reconstructed FF holographic images are well-consistent with the target vectorial image.

We emphasize that here identical vectorial holographic images are generated, as our designed metasurfaces are shined by incident light beams with different polarizations. This is in sharp contrast with previous incident-polarization-dependent holography generations, where different images are generated as the incident polarization is varied48^–50 with distinct designs. In addition, with $Δ Φ$ employed as an additional degree of freedom, in our scheme, the incident polarization can also be arbitrary EPs possessing distinct $Θ_{0}$ , which further extends the applicable range of incident polarization.

2.4 Metadevices for Generating Simple Vectorial Holographic Images with Rotational Symmetries

As a benchmark test of our general strategy, we first design and fabricate two metasurfaces and experimentally demonstrate that they can generate predesigned vectorial images with simple patterns. Two target holographic images are shown in Figs. 4(b) and 4(h), both containing a set of double-headed arrows representing the orientations of local $E$ -fields, located at different azimuthal positions of a circle. We note that these vectorial images generally exhibit certain symmetries, serving as ideal benchmark tests of our general theory. Following the design processes introduced in last subsection, we first retrieve the required distributions from the target vectorial images, then attain the distributions of ${Φ_{Res} (\vec{r}), Δ Φ (\vec{r}), ξ (\vec{r})}$ from Eqs. (5) and (6) under LCP incidence, and eventually obtain the parameter distributions ${L_{u} (\vec{r}), L_{v} (\vec{r}), ξ (\vec{r})}$ to design two metasurfaces.

Figure 4.Experimental set-up and characterizations on the first series of vectorial meta-holograms. (a) Experimental set-up for characterizing vectorial meta-holograms, with P, QWP, L, and O denoting polarizer, quarter-wave plate, lens, and optical lens, respectively. The inset illustrates an SEM picture and zoomed-in views of the first meta-hologram sample. (b) The first target holographic image and (c) experimentally observed pattern as the first meta-hologram sample is illuminated by LCP light at 1064 nm. Panels (d)–(g) depict the polarization filtered patterns with a rotatable polarizer placed at different angles in front of the CCD, as the first meta-hologram sample is illuminated by LCP light at 1064 nm. (h) The second target holographic image and (i) experimentally observed pattern as the second meta-hologram sample is illuminated by LCP light at 1064 nm. Panels (j)–(m) depict the polarization filtered patterns with a rotatable polarizer placed at different angles in front of the CCD, as the second meta-hologram sample is illuminated by LCP light at 1064 nm. Here, segments and circles denote local polarization states of the target holographic images.

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We fabricate two metasurfaces according to the designs, with the inset of Fig. 4(a) depicting the SEM image of the first sample (see Fig. S7 in the Supplementary Material for the SEM image of the second sample). To better satisfy the periodic boundary approximation utilized in obtaining a single metaatom database and minimize the effect of mutual couplings between adjacent meta-atoms, we purposely employ $2 \times 2$ identical meta-atoms to form a single phase-bit in constructing our metasurfaces. We next set up a micro-imaging system [Fig. 4(a)] to experimentally characterize the holographic images generated by these two metadevices. In the experimental system, the incident polarization is converted to LCP using a linear polarizer P1 combined with a quarter-wave plate QWP1. The fabricated metadevices are placed on the focal plane of lens L1 (focal length 175 mm) and are shined by the incident LCP light at an angle of $θ = 22 \deg$ . A high numerical-aperture (NA = 0.55) condenser lens O, together with another lens L2 (focal length 100 mm), is employed to collect and focus the vectorial holographic images onto a CCD camera placed on the focal plane of L2. Figures 4(c) and 4(i) depict the recorded patterns on the imaging plane, as two metadevices are illuminated by incident LCP light at 1064 nm, respectively. The generated patterns are in excellent agreement with the designated target holographic images as shown in Figs. 4(b) and 4(h).

To further reveal the vectorial properties of the generated holographic images, we place a rotatable polarizer P2 in front of the CCD to record the polarization-filtered patterns. Figures 4(d)–4(g) and 4(j)–4(m) show, respectively, the polarization-filtered patterns recorded on the image plane as P2 is rotated at different angles (see white arrows in the figures), for the two metadevices shined by incident LCP light at 1064 nm (see Secs. 4.2 and 4.4 in the Supplementary Material for more results). In both cases, we find intensity zeros appearing at different angles, “killing” those arrows perpendicular to the polarizer P2. These are direct evidence to demonstrate the vectorial nature of generated images, which are consistent with theoretical expectations.

2.5 Metadevices for Generating Complex Vectorial Holographic Images without Rotational Symmetries

To showcase the full power of our generic design approach, we now experimentally demonstrate three metadevices to generate vectorial images without any rotational symmetry and with more complex polarization distributions. The target vectorial holographic images are shown in Figs. 5(a), 5(g), and 5(m), which are a clock, a five-petal flower, and a flying bird, respectively. In addition, different parts of the target images exhibit distinct polarization states, represented by segments and circles, correspondingly. Obviously, these vectorial images exhibit both complex intensity and polarization distributions without any rotational symmetry, so they are perfect candidates to demonstrate the full power of our proposed strategy.

Figure 5.Characterizations on the second series of vectorial meta-holograms. Target holographic images to be reconstructed: (a) a vectorial clock, (g) a vectorial flower, and (m) a vectorial flying bird. Here, segments and circles denote local polarization states of the images. Panels (b), (h), and (n) depict the experimentally observed patterns as three fabricated meta-hologram samples are illuminated by LCP light at 1064 nm, respectively. Panels (c)–(f), (i)–(l), and (o)–(r) depict the polarization filtered patterns with a rotatable polarizer placed at different angles in front of the CCD, as three meta-hologram samples are illuminated by LCP light at 1064 nm, respectively.

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With target vectorial images given, we employ the strategy discussed in Sec. 2 to design three metasurfaces, which, under illuminations of LCP light at 1064 nm, can generate the corresponding vectorial images in the FF. We fabricate three metasurfaces according to the designs, with Figs. S9, S11, and S13 in the Supplementary Material depicting their SEM images. We next employ the same experimental setup as shown in Fig. 4(a) to characterize the holographic images generated by these metadevices. Indeed, without employing the polarizer P2, we obtain three image patterns [Figs. 5(b), 5(h), and 5(n)] as three metadevices are shined by the incident LCP light at 1064 nm. The obtained images are in nice agreement with the predesigned patterns.

We next use the same technique as in the last subsection to reveal the vectorial properties of these generated images. Putting a polarizer P2 in front of the CCD, the observed patterns are shown in Figs. 5(c)–5(f), 5(i)–5(l), and 5(o)–5(r) for three different metadevices, as the polarizer is placed along different orientation angles (see white arrows on the right-up corner). Along with the rotation of polarizer P2, a particular part inside each generated image diminishes, which signifies a linear polarization perpendicular to P2. Meanwhile, for those parts designed to exhibit circular polarizations, the observed brightness does not change as P2 rotates (see Sec. 5.2, 5.4, and 5.6 in the Supplementary Material for more results). All these features unambiguously reveal the vectorial nature of the generated images, in consistent with the predesigned targets. Interestingly, we note that three series of images recorded in Fig. 5 can tell different stories. Figures 5(c)–5(f) show a vectorial holographic clock with different parts “illuminated” in turn as if time passes, Figs. 5(i)–5(l) demonstrate a vectorial holographic flower with different petals “picked off” in turn as the analyzer rotates, and Figs. 5(o)–5(r) illustrate a vectorial holographic bird is “flying” when the polarizer is rotating continuously. We emphasize that the target FF holographic images exhibit strong inhomogeneities and asymmetries. As shown in Figs. 5(a), 5(g), and 5(m), different regions of the vectorial clock, different petals of the vectorial flower, and different parts of the vectorial bird can carry distinct polarizations. These highly inhomogeneous vectorial patterns cannot be generated by our previous scheme20^,21^,23^,34 and can only be achieved with the present generic approach for arbitrarily complex vectorial images. The zero-order reflection of incident CP light still exists in the presented experimental characterizations for the consideration of polarization-structure preservation. We believe that in our future work with target vectorial image propagating along a nonspecular direction, the imaging quality can be further improved by filtering out these zero-order reflections.

Finally, we experimentally demonstrate that our scheme exhibits high working efficiencies. Considering the difficulty of directly measuring the power carried by an arbitrary vectorial beam, we specially design and fabricate a metadevice for the generation of a simple ring-shaped vectorial pattern in the FF and experimentally characterize its performance. Our experiment shows that the working efficiency of such a benchmark metadevice reaches 67.9%, which is much higher than the previously realized meta-holograms (see Sec. 5.7 in the Supplementary Material).29^,30

3 Discussion and Conclusion

We establish a generic strategy to design metasurfaces with single-structure meta-atoms, which can generate predesigned vectorial holographic images under illuminations of light with a particular polarization. Compared with previous schemes based on “merging” two sets of PB meta-atoms, the current approach exhibits the advantages of high efficiency and fine lateral resolution (see Secs. 5.7 and 6 in the Supplementary Material for details) and can work for arbitrary incident polarization, i.e., capable of generating identical vectorial holography for differently polarized incidences. In addition, with PB phases added in designing our meta-atoms, our scheme can generate vectorial holographic images exhibiting unrestricted polarization distributions. After designing a series of single-structure meta-atoms and experimentally characterizing some of them in the telecom frequency domain ( $\sim 1064 nm$ ), we employ our strategy to design two sets of meta-holograms, which can generate vectorial holographic images with and without rotational symmetries, respectively. Experimental characterizations on these samples in the telecom frequency regime show that the generated holographic images exhibit desired inhomogeneous distributions of both intensity and polarization, agreeing well with the pre-designed targets. In our experiments, we employ a rotatable polarizer as the analyzer to characterize the polarization distribution on the holographic plane. However, such a characterization scheme only allows us to easily distinguish LPs and CPs, and thus, we only realized holographic images with local polarizations being either LPs or CPs. Nevertheless, we emphasize that the proposed method is generic enough to generate arbitrary polarization distributions with local polarization states being EPs as well. Based on ultra-thin, super-compact, and rotatable single-structure meta-atoms, the results presented in this work may inspire numerous future applications in integration optics: the break of polarization distribution constraints provides more possibilities in channel complexing, which may benefit optical information encryption and data storage compatible with the existing on-chip photonics platforms;51^–55 and the polarization-independent design may stimulate the next generation of anti-counterfeiting measurements, etc. The proposed meta-platform is generic enough to work for other wavelength regimes and for the transmissive mode. In addition, replacing metallic meta-atoms with all-dielectric ones can further improve the working efficiencies of fabricated devices. All these possibilities may stimulate more realistic applications in the future.

4 Appendix: Materials and Methods

4.1 Numerical Simulation

In our finite-difference time-domain simulations, the permittivity of Ag is described by the Drude model $ε_{r} (ω) = ε_{\infty} - \frac{ω_{p}^{2}}{ω (ω + i γ)}$ , with $ε_{\infty} = 5$ , $ω_{p} = 1.367 \times 10^{16} s^{- 1}$ , $γ = 1.474 \times 10^{14} s^{- 1}$ , obtained by fitting with experimental results. The ${SiO}_{2}$ spacer is considered as a lossless dielectric with permittivity $ε = 2.25$ . Additional losses caused by surface roughness and grain boundary effects in thin films as well as dielectric losses are effectively considered in the fitting parameter $γ$ .

4.2 Sample Fabrications

All MIM samples are fabricated using standard thin-film deposition and EBL techniques. In the first step, we sequentially deposit 125-nm-thick Ag and a 125-nm-thick ${SiO}_{2}$ dielectric layer onto a silicon substrate using magnetron DC sputtering (Ag) and RF sputtering ( ${SiO}_{2}$ ). Then, we lithograph the cross structures with EBL, employing an $\sim 100 - nm$ -thick PMMA2 layer at an acceleration voltage of 100 keV. After development in a solution of methyl isobutyl ketone and isopropyl alcohol, a 3-nm Ti adhesion layer and a 30-nm Ag layer are subsequently deposited using thermal evaporation. The Ag patterns are finally formed on top of the ${SiO}_{2}$ film after a lift-off process using acetone.

4.3 Experimental Setup

We use a near-infrared (NIR) microimaging system to characterize the performance of all designed metaatoms. A broadband supercontinuum laser (Fianium SC400) source and a fiber-coupled grating spectrometer (Ideaoptics NIR2500) are used in the FF measurements. A beam splitter, a linear polarizer, and a CCD are also used to measure the reflectance and analyze the polarization distributions.

Acknowledgments

Acknowledgment. D. Wang acknowledges support from Prof. Shuang Zhang, Department of Physics, University of Hong Kong. L. Zhou acknowledges technical support from the Fudan Nanofabrication Laboratory for sample fabrication. This work was funded by the National Key Research and Development Program of China (Grant No. 2022YFA1404700), the National Natural Science Foundation of China (Grant Nos. 12221004, 62192771), and the Natural Science Foundation of Shanghai (Grant No. 23dz2260100). Work done in Hong Kong was supported by the Research Grants Council (RGC) of Hong Kong (Grant Nos. AoE/P-502/20 and CRS_HKUST601/23).

Tong Liu received his BS and PhD degrees, both in physics, from Fudan University in 2016 and 2022, respectively. He is currently a post-doctoral research fellow in Department of Physics of HKUST. His current research is focused on metamaterials/metasurfaces, plasmonics and nanophotonics.

Changhong Dai received her bachelor’s degree from Anhui University of Technology in 2014 and her PhD from Fudan University in 2025. During her doctoral studies, she has been engaged in cutting-edge research in the fields of metasurfaces and plasmonics, with a particular emphasis on complex field manipulation using high-efficiency metasurfaces.

Dongyi Wang received her BS and PhD degrees, both in physics, from Fudan University in 2017 and 2022, respectively. She is currently a post-doctoral research fellow in Prof. Shuang Zhang’s Group, at the University of Hong Kong and the new-media editor of the journal Nanophotonics. Her current research is focused on metamaterials/metasurfaces, plasmonics, nanophotonics, topology in classical wave systems and non-Hermitian physics.

Che Ting Chan received his BSc degree from the University of Hong Kong in 1980 and his PhD from the University of California at Berkeley in 1985. He is currently a chair professor of physics at HKUST and was the interim director of HKUST IAS. His primary research interest is the theory and simulation of material properties. He is now working on the theory of a variety of advanced materials, including photonic crystals, metamaterials and nano-materials.

Lei Zhou received his BS and PhD degrees, both in physics, from Fudan University in 1992 and 1997, respectively. He is currently a vice president of Fudan University and a full professor in the Department of Physics. His current research is focused on metamaterials/metasurfaces, plasmonics, and nanophotonics.

Category: Research Articles

Received: May. 13, 2025

Accepted: Jul. 25, 2025

Posted: Jul. 25, 2025

Published Online: Aug. 20, 2025

The Author Email: Dongyi Wang (physwang@hku.hk), Lei Zhou (phzhou@fudan.edu.cn)

DOI:10.1117/1.AP.7.5.056004

CSTR:32187.14.1.AP.7.5.056004