As a technique to fully engineer the wavefronts of light, holography has demonstrated remarkable modulation abilities for both free-space beams [1–4] and surface waves [5–7], and facilitated applications of wavefront shaping, data storage, three-dimensional display, and so on. However, for optical holographic video display and complex optical encryption, the bandwidth is still limited, and undesired diffraction orders exist in using traditional approaches. The metasurface, proposed as a novel artificial planar element with subwavelength units, has become a powerful platform for hologram recording in recent years. It can overcome the above limitations and also provide unprecedented spatial resolution and a large field of view (FOV). Over the past decade, delicately designed meta-atoms have shown flexible light manipulation properties, such as amplitude [6,8–13], phase [8–15], polarization [14,16–21], orbital angular momentum (OAM) [22–26], and frequency [12,25,27–31]. Through coding the meta-atoms with diverse holograms in different optical channels, holographic information capacity can be further augmented, and multiplexed functions are available [8,10–14,17–27,31–33].

Among the fundamental light properties, polarization records the vectorial nature of light containing rich invisible information to human eyes. Due to the unique advantages of metasurfaces in designing anisotropic optical response, polarization multiplexed holography based on various metasurface design strategies has emerged [8–12,14,18–21,25,34–37], and greatly promotes potential applications such as optical document security and optical switching devices. However, most schemes are dependent on orthogonal polarization states [14,19,21,24], rather than using all states on the full Poincaré sphere, which decreases the versality and information capacity of polarization multiplexed holography. As one of the most widely used strategies, the spatial multiplexing method [8,20,34,35] relying on supercells suffers from low conversion efficiency and undesired diffraction orders. Especially for supercell metasurfaces tailored by the detour phase [8,14,36], oblique incidence becomes necessary and does not adapt to applications working under normal illumination. So far, progress has put forward challenging requirements for an advanced encoding method and more delicate metasurface design to achieve greater polarization modulation possibilities, more information capacity, and high conversion efficiency.

Here, we propose a novel metasurface encoding method to achieve polarization multiplexed holography, which can realize diverse holographic mappings from one full-Stokes space to another efficiently with subwavelength units. As shown in Fig. 1, based on metasurfaces encoded with vectorial holograms with unbounded possibilities, we can selectively address the intensity distributions in the transmitted field according to input and output polarization states, which greatly improves the availability of polarization modes and information capacity. The delicately designed metasurfaces have high polarization conversion in the circular polarization channels, and can achieve independent phase modulations that combine dynamic phase and geometric phase. With the help of a hybrid genetic algorithm, we generated phase-only holograms to synthesize multiple vectorial fields based on the designed meta-atoms. By loading the time-varying polarization channels in sequence, a holographic video and complex optical encryption with double secret keys have been experimentally demonstrated in real time. Such a scheme opens new avenues for multi customizable polarization modulations, and is expected to be used in dynamic display, dynamic optical manipulation, optical encryption, and anticounterfeiting.

Mathematically, at a given input polarization $|p_0^{\mathrm{in}}⟩$, the polarization multiplexed hologram can be expressed as the superposition of sub-holograms with diverse polarization states: $H^{\mathrm{mul}}=\sum A_j{e}^{i{\psi }_j}|p_j^{\mathrm{out}}⟩$, where $A_j$, $e^{i{\psi }_j}$, and $|p_j^{\mathrm{out}}⟩$ represent the amplitude, phase, and polarization state of the $j$th hologram. Here, we utilize the interaction of the input polarized plane wave and metasurface to construct $H^{\mathrm{mul}}$, which satisfies the relationship $H^{\mathrm{mul}}=J|p_0^{\mathrm{in}}⟩$, and $J$ is the $2\times 2$ Jones matrix of metasurfaces at each pixel. As we build the vectorial images in the far field, the reconstructed field $E_{\mathrm{rec}}$ can be regarded as the Fourier transform of $H^{\mathrm{mul}}$, namely, $E_{\mathrm{rec}}=\Im (\sum A_j{e}^{i{\psi }_j}|p_j^{\mathrm{out}}⟩)$, where $\Im$ is the Fourier transform operator. Polarization state $|p_j^{\mathrm{out}}⟩$ connected with the $j$th hologram is independent of the spatial coordinate and not involved in the process of Fourier transform, so we can recast the reconstructed field as $E_{\mathrm{rec}}=\sum \Im (A_j{e}^{i{\psi }_j})|p_j^{\mathrm{out}}⟩$ according to the linear property of Fourier transform. By imposing polarization selection $|p_s⟩$ to the whole field with a spatial inhomogeneous vectorial nature, we can modulate the optical field $E$ and intensity distribution $I$ to $\sum \Im (A_j{e}^{i{\psi }_j})⟨p_s|p_j^{\mathrm{out}}⟩|p_s⟩$ and ${|\sum \Im (A_j{e}^{i{\psi }_j})⟨p_s|p_j^{\mathrm{out}}⟩|}^2$, respectively. Clearly, the intensity of each vectorial component is governed by the polarization correlation, and the polarization of the sub-image with maximum intensity distribution is consistent with $|p_s⟩$. Thus, a holographic mapping from customized $|p_0^{\mathrm{in}}⟩$ to diverse $|p_s⟩$ is available for optical data storage and information processing. By simplifying the $2\times 2$ Jones matrix of nanostructures into two off-diagonal components, we can further build the holographic connections between diverse input polarization states and specific output polarization selections clearly.

Next, we describe the recording rules of vectorial holography to achieve dynamic modulations by diverse input/output polarization channels. Since arbitrary $|p_j^{\mathrm{out}}⟩$ is described by two orthogonally polarized vectors, at least two polarization channels are required to digitize the vectorial holography. We decomposed the vectorial holography into two circularly polarized holograms, which can be recorded with metasurfaces in subwavelength pixels commendably. The complex-amplitude holograms of right circular polarization (RCP) and left circular polarization (LCP) channels can be expressed as $H_R=\sum A_j{e}^{i{\psi }_j}⟨R|p_j^{\mathrm{out}}⟩$ and $H_L=\sum A_j{e}^{i{\psi }_j}⟨L|p_j^{\mathrm{out}}⟩$, respectively, where $⟨R|$ and $⟨L|$ are the left vector of RCP and LCP, respectively. On the other hand, the phase-only holograms of RCP and LCP channels can be described as $H_R^{\prime }=\exp (i{\psi }_r)$ and $H_L^{\prime }=\exp (i{\psi }_l)$, respectively, where ${\psi }_r=\arg (\sum A_j{e}^{i{\psi }_j}⟨R|p_j^{\mathrm{out}}⟩)$ and ${\psi }_l=\arg (\sum A_j{e}^{i{\psi }_j}⟨L|p_j^{\mathrm{out}}⟩)$ in some cases. To efficiently reconstruct the desired field, the final phase-only holograms $H_R^{\prime }$ and $H_L^{\prime }$ are usually provided by suitable optimization algorithms. Obviously, the reconstructed field in the far field decomposed as $E_{\mathrm{rec}}=E_R|R⟩+E_L|L⟩$ satisfies the conditions as $E_R=\Im (H_R)$ and $E_L=\Im (H_L)$, where $E_R$ and $E_L$ are the complex amplitude fields of RCP and LCP channels in the far field, respectively. We purposely designed the metasurface as a transmitted half-wave plate with sufficient phase modulation capacity, whose Jones matrix can be described as the following equation in the circular polarization basis: $$J=\left[\begin{array}{cc}{t}_{rr}& t_{rl}\\ t_{lr}& t_{ll}\end{array}\right]=\left[\begin{array}{cc}0& e^{i{\phi }_r}\\ e^{i{\phi }_l}& 0\end{array}\right],$$(1)where $t_{rl}$ is the transmission coefficient of $|L⟩$ input / $|R⟩$ output and so are others, and ${\phi }_r$ and ${\phi }_l$ correspond to the phase response of the modulation channels $t_{rl}$ and $t_{lr}$, respectively. In this case, the meta-atoms have total circular polarization conversion and full-range phase modulation. According to $H^{\mathrm{mul}}=J|p_0^{\mathrm{in}}⟩$, we can get the relationship between the holograms with the input as $H_R=e^{i{\phi }_r}⟨L|p_0^{\mathrm{in}}⟩$ and $H_L=e^{i{\phi }_l}⟨R|p_0^{\mathrm{in}}⟩$, where $⟨L|p_0^{\mathrm{in}}⟩$ represents the LCP component of $|p_0^{\mathrm{in}}⟩$ and so is $⟨R|p_0^{\mathrm{in}}⟩$. Here, we set the initial input polarization $|p_0^{\mathrm{in}}⟩$ as $x$ polarized; in this case, the metasurface can simultaneously record both phase-only holograms in $t_{rl}$ and $t_{lr}$ polarization channels. That is, $e^{i{\phi }_r}=H_R^{\prime }$ and $e^{i{\phi }_l}=H_L^{\prime }$, which can be obtained by a specific optimization algorithm. In a sense, we can use $H_R$ and $H_L$ to express the off-diagonal components of the Jones matrix approximately, which can help to better analyze the modulation effect of the input polarized field.

When we use another polarized input $|p_{\kappa }^{\mathrm{in}}⟩$ to interact with metasurfaces, the output RCP and LCP holograms become $H_R⟨L|p_{\kappa }^{\mathrm{in}}⟩$ and $H_L⟨R|p_{\kappa }^{\mathrm{in}}⟩$ approximately, and the synthetic multiplexed hologram $H_{\kappa }^{\mathrm{mul}}$ at input $|p_{\kappa }^{\mathrm{in}}⟩$ can be expressed as $$H_{\kappa }^{\mathrm{mul}}\approx \sum A_j{e}^{i{\psi }_j}\left[\begin{array}{c}⟨R|p_j^{\mathrm{out}}⟩⟨L|p_{\kappa }^{\mathrm{in}}⟩\\ ⟨L|p_j^{\mathrm{out}}⟩⟨R|p_{\kappa }^{\mathrm{in}}⟩\end{array}\right]=\sum A_j{e}^{i{\psi }_j}|p_{j\kappa }^{\mathrm{out}}⟩,$$(2)where $|p_{j\kappa }^{\mathrm{out}}⟩$ represents the generated output polarization state of the $j$th hologram at input polarization state $|p_{\kappa }^{\mathrm{in}}⟩$. After the far-field reconstruction and output polarization selections, the output field changes to $\sum \Im (A_j{e}^{i{\psi }_j})⟨p_s|p_{j\kappa }^{\mathrm{out}}⟩|p_s⟩$, and the intensity distribution changes to ${|\sum \Im (A_j{e}^{i{\psi }_j})⟨p_s|p_{j\kappa }^{\mathrm{out}}⟩|}^2$. Therefore, a bridge of polarization multiplexed holography has been built between the input polarization states and output polarization selections. Note that, albeit the polarization mapping is flexible between the two polarization spaces, the mapping of the same circular polarization is forbidden, which results from the zero energies in $t_{rr}$ and $t_{ll}$ polarization channels based on the designed Jones matrix. On the other hand, the multiplexing capacity has been greatly enhanced, and novel applications are expected.

Next, we introduce the design strategy of metasurface digitalization. First, we developed an efficient approach to design meta-atoms with separated and sufficient phase modulations in $t_{rl}$ and $t_{lr}$ channels. As the most popular method of metasurface designs, geometric phase, also known as Pancharatnam–Berry (PB) phase, utilizes polarization evolution to realize continuous phase modulation ability related to azimuthal rotations. For meta-atoms with rotation angle $\theta$, a phase response $\pm 2\theta$ can be introduced to $t_{rl}$ and $t_{lr}$, respectively. To make the phase modulation of $t_{rl}$ and $t_{lr}$ independent, we integrated geometric phase together with dynamic phase. We chose amorphous silicon (α-Si) nanofins with rectangular cross sections as modulated units, which have excellent optical response in the near infrared band and the same dynamic phase response ${\theta }_0$ in $t_{rl}$ and $t_{lr}$ channels resulting from geometric symmetry. That is, the ${\phi }_r$ and ${\phi }_l$ of each modulated unit are equal to ${\theta }_0+2\theta$ and ${\theta }_0-2\theta$, respectively, where ${\theta }_0$ can be tailored by the change of geometric cross sections. We conducted a parameter sweep of $\mathrm{\alpha }$-Si nanofins with diverse cross sections by the rigorous coupled wave analysis (RCWA) method to determine suitable structures. The period and height are fixed at 400 nm and 600 nm, respectively, the refractive index we used for calculation is $3.85+0.01i$ at a wavelength of 800 nm, and the sweeping range of the width and length is from 80 nm to 240 nm. The amplitude response and the calculated dynamic phase of the $t_{rl}$ channel are shown in Fig. 2(b). Among the nanofins with amplitude response $|t_{rl}|$ above 0.83, 56 structures were selected to cover a full phase range at the same time. Afterwards, by rotating the selected nanofins, we got an effective database to encode each separate hologram.

Then we designed a hybrid genetic algorithm to calculate the phase-only holograms of RCP and LCP channels based on the structural database. As shown in Fig. 2(c), we first generated the initial metasurface by randomly setting the lengths, widths, and rotations based on the above nanofins. The constructed fields of both circularly polarized channels $E_R$ and $E_L$ can be calculated based on diffraction theory. To reconstruct the vectorial images, we have three target parameters: intensity distribution $I_j$, polarization components $⟨R|P_j^{\mathrm{out}}⟩$ and $⟨L|P_j^{\mathrm{out}}⟩$ of all images, and the design degrees of freedom (DOFs) containing only two phase distributions $H_R^{\prime }$ and $H_L^{\prime }$. In this case, we nested an alternative selection in the iterative loops to manipulate more parameters by a few ones. The idea is that if the number of target parameters $M$ is larger than the number of DOFs $Q$, at least $M/Q$ (a nature number rounded up) alternative selections are nested in the iterative loop. Here, we need to nest the selections only twice. The complete loop begins with the odd iteration number (iterative number $n=2q-1$; $q$ is positive integer), and in this case, we selected the absolute phase of $E_R(n)$ as the imaging phase $\phi (n)$. With the later replacement of target intensities $I_j$ and polarization distributions $⟨R|P_j^{\mathrm{out}}⟩$ and $⟨L|P_j^{\mathrm{out}}⟩$, the next holograms can be generated by inverse Fourier transform and nanofin digitalization. The iteration number becomes even ($n=2q$), and in this selection process, we chose the absolute phase of calculated $E_L(n)$ as $\phi (n)$, which has the characteristic of previous $E_R(2q-1)$ and target fields. The replacement process was kept the same as in the previous step. Through the hybrid iteration, both characteristics of the two polarization channels can be preserved based on the above database of nanofins. Therefore, holography recording based on metasurfaces is algorithmically realized.

We designed and fabricated two metasurfaces to demonstrate our method. As described in Fig. 1, the first is the time sequence holographic display of a prose poem, and the second is an optical encryption composed of a metasurface and two separate keys. Nano fabrication depends on the standard electron beam lithography and inductively coupled plasma reactive ion etching process. Scanning electron microscopy images of sample 1 (dynamic display) and sample 2 (optical encryption) are shown in Fig. 3(a) and Fig. 3(b). Both metasurfaces have a footprint of $480\mathrm{\mu m}\times 480\mathrm{\mu m}$. By using the experimental setup shown in Fig. 3(c), we collected holographic images at diverse input/output polarization selections.

To facilitate practical application, we encoded the time-dependent frames with linearly polarized input that orbits around the equator of the Poincaré sphere, and the analyzer in the output space was fixed without any adjustment. For the $j$th word of the poem, we designed various phase delays $\pm {\phi }_j$ into the RCP and LCP holograms, which helps to create the multiplexed sequence. At the initial $x$-polarized incidence, the output poem is composed of diverse linearly polarized words $\sum _j^N\sqrt{I_j(x,y,z)}|p_j^{\mathrm{out}}⟩$ before modulation of the analyzer, where $I_j$ is the intensity distribution of the $j$th part, $N$ is the total number of words and equals 16, and $|p_j^{\mathrm{out}}⟩=\frac{1}{\sqrt{2}}{[\begin{array}{cc}{e}^{i{\phi }_j}& e^{-i{\phi }_j}\end{array}]}^T$ in the circularly polarized basis $[|R⟩,|L⟩]$. We changed the input linear polarization angle and described it with $\alpha$ (that is, $|p_{\kappa }^{\mathrm{in}}⟩={[e^{i\alpha },e^{-i\alpha }]}^T$), and the linear polarization angle of the analyzer was set as $\beta$ ($|p_s⟩={[e^{i\beta },e^{-i\beta }]}^T$). The collected intensity of the $j$th vectorial component at this input/output polarization channel can be calculated based on the above principles, which can be expressed as $$I_{s\kappa ,j}=I_j{\cos }^2({\phi }_j-\alpha -\beta ).$$(3)

Here, we make $\beta$ be zero, and ${\phi }_j$ in the poem ranges from 0° to 180°. Thus, when the incident $\alpha$ changes from 0° to 180°, the word with maximum intensity can be selectively addressed in sequence. The designed metasurface is composed of $1200\times 1200$ nanofins, with different cross sections and rotations optimized by the hybrid genetic algorithm. As shown in Fig. 3(c), we used ${\mathrm{P}}_1$ and a half-wave plate to generate varied linearly polarized light, and in the output, we removed the quarter-wave plate and fixed ${\mathrm{P}}_2$ as $x$ polarized. The simulated and experimental results shown in Fig. 4(c) are consistent with the theoretical design; the poem is dynamically displayed by varying the incident polarization angle $\alpha$, following the order of designed ${\phi }_j$. In addition, the collected profile of the $|L⟩$ input / $|R⟩$ output channel shows uniform distribution of each word, and the detected signals of $t_{rr}$ and $t_{ll}$ channels are relatively weak (see Fig. 6 in Appendix A).

Above all, the dynamic display of holographic videos with high performance has been experimentally realized. It does not require adjustment of the analyzer at the output end, which is more practical for system integration. By increasing the multiplexed number, the movie can be encoded with more information. Furthermore, different display requirements are also available by the polarization multiplexed strategy. For example, for videos played in linearly polarized input/output channels, the display time of sub-images can be precisely customized according to the ellipticity, and real-time editing, rewinding, and inserting of the movie can be easily realized based on diverse polarization selections. Compared with other multiplexed schemes, such as utilizing incident wavelength, OAM, and nonlinear frequencies as multiplexing parameters, the polarization multiplexed strategy shows the advantage of simple manipulation. Various functionalities can be realized by modulating the input/output polarizations. In contrast, nonlinear frequency multiplexing needs high incident intensity, large nonlinear susceptibility, and a complex design to meet phase matching conditions. OAM multiplexing relies on a spatially distributed phase profile generated from spiral plates or spatial light modulators. Furthermore, polarization multiplexing based on our method can facilitate applications of compact devices with the integration of liquid crystal.

The first design mainly depends on linear polarization multiplexing of input/output channels. In the second design, we extended the polarization modulation range to the full-Stokes space, and a demonstration of optical encryption with high security is provided as follows. The secret information is hidden in multiple polarization multiplexed images, which has many possibilities due to distinct input/output polarization selections. As shown in Fig. 5(b), the designed pattern of a dove with five wings and leaves has seven polarization states, which can convert an arbitrary incident polarization state (except for circular polarization states) into different polarization states in the output full-Stokes space. Upon $x$-polarized incidence, we describe the vectorial dove as $\sum _j^7\sqrt{I_j(x,y,z)}|p_j^{\mathrm{out}}⟩$, where $|p_j^{\mathrm{out}}⟩={[\cos {\chi }_j{e}^{i{\phi }_j},\sin {\chi }_j{e}^{-i{\phi }_j}]}^T$, and ${\chi }_j$ and ${\phi }_j$ imply the changes of amplitude and phase in circularly polarized channels, respectively. Next, we change input polarization to another state $|p_{\kappa }^{\mathrm{in}}⟩$, and the constructed vectorial field becomes $\sum \sqrt{I_j(x,y,z)}{[\cos {\chi }_j{e}^{i{\phi }_j}⟨L|p_{\kappa }^{\mathrm{in}}⟩,\sin {\chi }_j{e}^{-i{\phi }_j}⟨R|p_{\kappa }^{\mathrm{in}}⟩]}^T$. That is, the generated vectorial field and the output intensity of each sub-image change upon different illuminance conditions except for linearly polarized input. With further decoding with an analyzer that can be expressed as $|p_s⟩⟨p_s|$, we can get a reliable tool to encode the secret information into polarization multiplexed images. The intensity distribution of the $j$th vectorial component at the $|p_{\kappa }^{\mathrm{in}}⟩$ input / $|p_s⟩$ output polarization channel can be expressed as follows: $$I_{s\kappa ,j}=I_j{(\cos {\chi }_j{C}_{s\kappa }+\sin {\chi }_j{C}_{s\kappa }^{\prime })}^2,$$(4)where $C_{s\kappa }$ and $C_{s\kappa }^{\prime }$ are two coefficients related to polarization selections $C_{s\kappa }=⟨p_s|R⟩⟨L|p_{\kappa }^{\mathrm{in}}⟩$ and $C_{s\kappa }^{\prime }=⟨p_s|L⟩⟨R|p_{\kappa }^{\mathrm{in}}⟩$, respectively. We set ${\chi }_j$ and ${\phi }_j$ of different vectorial parts as [90°, 0°, 52.5°, 45°, 45°, 45°, 67.5°] and [0°, 0°, 45°, 0°, 60°, 90°, $-45°$] from top to bottom, respectively, which is shown in Fig. 5(b). Thus, the leaves in the upper right corner are the RCP state, and the upper left wing is the LCP state. The first two columns of experimental results in Fig. 5(e) demonstrate the encoding of phase-only holograms in circular polarization channels, which are essential for constructing multiple polarized images in momentum space.

For the use of optical encryption, we set the first secret key as $x$-polarized incidence. When using the first secret key to interact with the metasurface, all vectorial patterns are displayed in Fig. 5(c). The second secret key is set as $y$ polarized, which is expected to show brighter third and fourth wings and the absence of second wings; the experimental result in the third column of Fig. 5(e) is consistent with the theoretical design. Furthermore, much deceptive information can be realized at other input/output polarization channels (results at more input/output polarization selections are provided in Figs. 7 and 8, Appendix A). Since the proposed encoding rules are based on the linear additivity of Fourier transformation, it is more suitable for large capacity information multiplexing compared with other schemes. The polarization states of input and output as secret keys have innumerable possibilities in the full-Stokes space, and the complexity and security of such optical encryption are extremely high, especially when the three elements (metasurfaces and two secret keys) are preserved and transferred separately.

In conclusion, we proposed and realized polarization multiplexed holography based on metasurfaces optimized by a hybrid genetic algorithm, which demonstrated the viability and versatility of time sequence dynamic display and optical encryption. Through full-Stokes polarization transformations and delicate metasurface design, multiple and independent input/output polarization modulations can be customized flexibly in two full-Stokes spaces. The subwavelength feature of the nanofin without coherent pixels or any spatial multiplexing not only decreases fabrication difficulties, but also ensures a larger FOV. Based on the efficient and independent phase encoding of a circular polarization basis, the output vectorial wavefront can be freely customized within an observable numerical aperture that reaches 0.80 (see Appendix A, Note 2) and conversion efficiency reaches 40.24%. The designed holographic videos with large information capacity have more displayed possibilities by using different polarization combinations. The optical encryption shows significantly improved performance compared with other polarization encryption schemes in security and data capacity. This method is expected to be used in dynamic display and beam shaping, polarization detection, holographic tweezers, optical encryption/anticounterfeiting, and so on.

Fabrication of metasurfaces. We chose $\mathrm{\alpha }$-Si as the optical material and fabricated dielectric metasurfaces on quartz substrates. The nanofin arrays have diverse cross sections and rotations. First, we deposited the $\alpha$-Si film with a thickness of 600 nm by using plasma-enhanced chemical vapor deposition. Second, we made a resist layer by a spin coating method, and patterned the structures by a standard electron beam lithography process. After a process of development, a 100-nm-thick chromium layer was patterned through electron beam evaporation. Then, we used hot acetone to remove redundant overlayers, and transferred the desired pattern from chromium to silicon through inductively coupled plasma reactive ion etching. Thus, the desired metasurfaces were fabricated successfully.