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We start by considering a simple case with a one-to-two mapping between the intensity of light and the orientation of each nanostructure. When incident LP light (the polarization direction is α1) with an intensity of I0 passes through an anisotropic nanostructure with an in-plane orientation θ, the intensity of the transmitted light can be derived as
$$ I_1 = I_0\left[ {A^2\cos ^2(\theta - \alpha _1) + B^2\sin ^2(\theta - \alpha _1)} \right] $$ (1) where A and B are the complex transmission coefficients for incident light LP along the long and short axes of the nanobrick, respectively. (See Section 1 of the Supplementary materials for details of the equation derivation.) Specifically, when the nanostructure acts as an ideal polarizer (i.e., A = 0 and B = 1), we can simplify the intensity of the transmitted light as
$$I_1 = I_0{\mathrm{sin}}^2(\theta - \alpha _1) $$ (2) As a result, for incident light polarized along a given direction (assuming α1 is fixing), a continuous greyscale pattern can be formed at the surface by configuring the orientation angles θ of the nano-polarizer array. Meanwhile, in the defined orientation angle interval [0, π], each nano-polarizer has two orientation options to produce the same output intensity, which can be used to form a two-step GEMS. More details about the expressions of the two different orientation angles are summarized in Table S1 of the Supplementary materials.
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A two-step GEMS cannot avoid the issue of generating twin images in principle, and the diffraction efficiency is only 40.5% theoretically40. To increase the quantization steps of GEMSs, we propose a one-to-four mapping scheme: only by simply inserting a bulk analyser into the same optical setup can a four-step GEMS be attained while maintaining the other conditions unchanged. Specifically, when incident LP light successively passes through a nano-polarizer and a bulk analyser, the intensity of the transmitted light can be expressed as
$$I_2 = I_0\left[ {\frac{{A - B}}{2}\cos \left( {2\theta - \alpha _2 - \alpha _1} \right) + \frac{{A + B}}{2}\cos \left( {\alpha _2 - \alpha _1} \right)} \right]^2 $$ (3) where α2 represents the direction of the bulk analyser transmission axis. In particular, by setting α2α1 + π/2, Eq. (3) can be rewritten as
$$I_2 = I_0\left( {\frac{{A - B}}{2}} \right)^2\cos ^2\left( {2\theta - 2\alpha _1 - {\pi}/2} \right) $$ (4) From Eq. (4), one finds that there are four orientation angles in the defined interval [0, π] that can produce the same output intensity (more details about the expressions of the four different orientation angles are summarized in Table S1 of the Supplementary materials). This provides the capability for metasurfaces to manipulate the PB phase under the illumination of CP light. Therefore, on the basis of the one-to-four mapping scheme, we can employ not only continuous intensity manipulation but also four-step phase manipulation without complicating the design and fabrication of the nanostructure.
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According to Eq. (2), when LP light with polarization orientation angle α1 = π/2 passes through a metasurface, the corresponding intensity of transmitted light can be simplified as I1 = I0 cos2θ, which indicates that each nano-polarizer with orientation angle θ or π − θ can produce equal intensity of transmitted light but different phase delays (Fig. 1a). The freedom provided by the one-to-two mapping scheme allows us to record a continuous greyscale pattern (intensity manipulation) on the nanostructure surface and simultaneously produce a two-step phase-only holographic image (phase manipulation) in the far field (Fig. 1b). Here, we utilize silver nanobrick arrays sitting on a glass substrate to form the Malus metasurface. By elaborately designing the geometric size, each silver nanobrick can function as a nano-polarizer and transmit the incident beam polarized along the nanobrick short axis while reflecting that polarized along the long axis.
Fig. 1 Schematic of Malus metasurfaces based on one-to-two mapping and two independent information channels for intensity and phase manipulation.
a Each nanobrick polarizer has two orientation angle options to produce the same transmitted intensity but different geometric phase delays. Here, α1 = π/2. b A continuous greyscale pattern with intensity manipulation is recorded on the metasurface under incident LP light (observed with an optical microscope), while a Fourier holographic image with phase manipulation is reconstructed in the far field under laser source illumination (observed with the naked eye or a camera).Based on the above designed nanobrick, we encode two independent target patterns (one for the greyscale pattern and another for the holographic image) into a single metasurface. First, according to the greyscale distribution of the target pattern, we calculate all possible combinations of nanobrick orientation angles. Since each pixel cell has two options for the nanobrick orientation, a simulated annealing algorithm41 is applied to optimize the orientation distribution (more details are presented in Fig. S1), and then, the Fourier GEMS hologram is formed (the phase is exactly twice the orientation angle).
To demonstrate the feasibility and flexibility of the Malus metasurfaces based on one-to-two mapping, three different samples (labeled A–C) were fabricated with electron beam lithography. Figure S2 shows scanning electron microscopy (SEM) images of a partial region of one metasurface sample. For samples A and B, we encode the same near-field greyscale pattern but different far-field holographic images. Samples A and C are designed to generate the same far-field holographic image but different near-field greyscale patterns. Figure 2a, b shows the experimental setup used for characterizing the metasurfaces. As a result, a greyscale pattern of a "cat" with high resolution and fidelity can be observed by an optical microscope (the central wavelength is 633 nm) under LP light illumination. In another measurement for the same sample, when sample A is normally illuminated by a laser source, a holographic image containing English letters and Chinese characters appears in the far field (300 mm from the sample). Sample B shows the same greyscale pattern as sample A but a totally different holographic image (letters of "SOS" with a centrosymmetric design), as shown in Fig. 2d, g. Meanwhile, sample C shows a different near-field pattern of a "dog" but the same far-field holographic image as sample A, as shown in Fig. 2e, h. All results are in good accordance with our design.
Fig. 2 Experimental setup and results of the Malus metasurfaces based on one-to-two mapping.
a An optical microscope (Motic BA310Met) is used to capture the greyscale pattern when LP light illuminates the sample. b The transmitted holographic image can be generated in the far field when an LP light beam emitted by a supercontinuum laser illuminates the metasurface sample. Second row c–e: experimental greyscale patterns under the illumination of LP light (the operating wavelength is 633 nm). Third row f–h: experimental Fourier holographic images under LP light illumination. Samples A and B can generate the same greyscale pattern of a "cat" in the near field but different holographic images in the far field. Samples A and C can generate different greyscale patterns of a "cat" and a "dog" in the near field but the same holographic image of English letters and Chinese characters in the far field. Samples A–C are designed with dimensions of 150 × 150 μm2. The scale bar is 30 μm. The holographic images of samples A–C are designed to create wide image angles of 59° × 66°, 50° × 50°, and 59° × 66°, respectively. -
A four-step metasurface can be designed to project complex holographic images while eliminating the inevitable "twin-image" problem of two-step metasurfaces. For the one-to-four mapping (Eq. (4)), if α2 = α1 + π/2 = π/4, that is, an LP beam with polarization orientation angle α1 = −π/4 passes through a nanobrick polarizer and then an analyser with polarization orientation angle α2 = π/4, the simplified formula $I_2 = \frac{{I_0}}{4}\cos ^2\left( {2\theta } \right)$ can be utilized for describing the intensity of the transmitted light. As a result, there are four orientation angles of θ, π/2 + θ, π/2 − θ, and π − θ that can produce equal transmitted beam intensity but different PB phase delays of 2θ, π + 2θ, π − 2θ, and 2π − 2θ, as shown in Fig. 3a. Therefore, we can obtain a continuous intensity manipulation as well as a four-step phase manipulation with only one metasurface.
Fig. 3 Schematic of Malus metasurfaces based on one-to-four mapping and two independent information channels for intensity and phase manipulation.
a Each nanobrick polarizer has four orientation angle options to produce equal transmitted beam intensity but different geometric phase delays. Here, α1 = −π/4, and α2 = π/4. b Under LP light illumination, a continuous greyscale pattern with intensity manipulation can be encoded at the sample surface. When a QWP is inserted into the optical path to convert incident LP light into CP light, a four-step holographic image with phase manipulation can be reconstructed in the far field.With the same nanostructural parameters and Fourier hologram design algorithm, we further design and fabricate three different samples (labeled D–F). The arrangement of the three samples is the same as that of samples A–C, i.e., samples D and E have the same greyscale pattern but different holographic images, whereas samples D and F have the same holographic image but different greyscale patterns. As shown by the experimental results in Fig. 4, sample D can not only encode a high-resolution greyscale pattern of a "dog doll" in the near field but also generate a holographic image with high fidelity (Einstein's portrait) in the far field, which indicates the feasibility of Malus metasurfaces based on one-to-four mapping. More interestingly, the twin-image that the two-step hologram always suffers from disappears (only very weak residual images exist because of algorithm residuals and sample and polarizer fabrication errors), as shown in Fig. 4d–f. For samples E and F, the experimental results have good consistency with our design, as shown in Fig. 4b, e, c, f. In summary, all experimental results show that the two information channels are independent, so we can design target patterns for meta-nanoprinting and meta-holography at will.
Fig. 4 Experimental results of the Malus metasurfaces based on one-to-four mapping.
First row a–c: experimental greyscale patterns under the illumination of LP light when the operating wavelength is 633 nm. Second row d–f: experimental holographic images under CP light illumination. Samples D and E can generate the same greyscale pattern ("dog doll") in the near field but different holographic images (Einstein and Mona Lisa's portraits, respectively) in the far field. Samples D and F can generate different greyscale patterns of a "dog doll" and a "duck doll" but the same holographic image of Einstein's portrait. Samples D–F are designed with dimensions of 150 × 150 μm2. The scale bar is 30 μm. The holographic images of samples D–F are designed to create wide image angles of 85° × 41°, 34° × 38°, and 85° × 41°, respectively.The measured hologram efficiency, defined as the ratio of the power of the transmitted holographic image to the power of the incident beam, is 7% at the operating wavelength of 633 nm. The efficiency could be improved by applying more precise fabrication procedures, using low-loss dielectric materials (such as TiO2) or reducing the coverage angles of the holographic image. More details about the efficiency calculation and measurement are discussed in Figs. S3 and S4 of the Supplementary materials.
The proposed Malus metasurfaces have great technical advantages. Generally, most optical devices can only work in either reflection or transmission. A significant feature of the Malus metasurfaces is that they can work not only in transmission but also in reflection, which greatly facilitates practical applications since the working modes can be chosen at will. Supplementary Fig. S5 shows all experimental results of samples A–F working in reflection. Compared with their transmission counterparts, the contrast of the reflection greyscale patterns has a slight reduction. The reason is that the unwanted reflectivity Ry is slightly higher than the unwanted transmissivity Tx when the operating wavelength is 633 nm (Fig. 6b shows the numerical simulation results). Meanwhile, reflective far-field holographic images with high fidelity can be observed, as shown in Fig. S5d–f, j–l.
Fig. 6 Schematic illustration of a nanobrick-based polarizer.
a Schematic illustration of a Ag nanobrick. b Simulated results (Rl, Rs, Tl, Ts) versus incident wavelength.To explore the broadband response characteristics of the Malus metasurfaces in the near field, we acquired the greyscale patterns under illumination by a quartz halogen lamp using an optical microscope, as shown in Supplementary Fig. S6. The greyscale patterns obtained in reflection and transmission show very clear visual effects. Then, to explore the spectral response in the far field, we utilized a supercontinuum laser (the wavelength varied within the range of 480–680 nm) to illuminate samples A and D. From Supplementary Fig. S7, one can see that all the holographic images possess high fidelity in both transmission and reflection. Benefiting from the broadband response characteristics of the metadevice we propose, the observation requirements in practical applications will be greatly reduced.
It is noted that the proposed one-to-four mapping scheme does not require the nanostructure to act as a perfect polarizer or a half-wave plate; that is, this scheme can be realized by using any nanostructure with anisotropy (A ≠ B). According to Eq. (4), when the polarization directions of normally incident LP light and the bulk analyser are orthogonal to each other (i.e., α2 = α1 + π/2), any desired intensity of the transmitted beam can be achieved by arranging the orientation angles of the anisotropic nanostructures. More details about the general concept of the one-to-M mapping scheme are discussed in Section 1 of the Supplementary materials. This characteristic is significant because a large fabrication error of metasurfaces is acceptable in principle, which may greatly improve the fabrication tolerance for mass production in industrial applications.
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The proposed one-to-M mapping strategy destroys the overall consistency between the greyscale pattern and the polarization orientation angle of incident light. On the other hand, we can benefit from this characteristic to increase the security of the greyscale pattern. For a Malus metasurface based on one-to-two mapping (sample A), we rotate the polarization orientation angle of incident LP light from the designed π/2 to 3π/4, 0 and π/4 and present the experimental results in the first row of Fig. 5. When the polarization orientation angle is zero degrees, the brightness of the greyscale pattern is complementary to the result at π/2, and the result can be easily interpreted by Malus's law. It is interesting that when the polarization orientation angles are π/4 and 3π/4, the greyscale patterns are almost lost in the background noise. For the metasurfaces based on the one-to-four mapping design, a similar phenomenon is observed (shown in the second row of Fig. 5).
Fig. 5 Experimental results for greyscale patterns with various polarization directions of incident LP light and the analyser.
The first row shows the experimental results of sample A under the illumination of LP light with polarization orientation angles of π/2, 3π/4, 0, and π/4. The second row presents the results of sample D for LP light with polarization orientation angles of −π/4, −π/8, 0, and π/8, and the analyser has corresponding polarization orientation angles of π/4, 3π/8, π/2, and 5π/8, respectively. The black and red arrows represent the polarization orientation angles of the normally incident LP light and the analyser, respectively. The scale bar is 30 μm.The destruction of the greyscale patterns can be explained by observing the mathematical functions cos2θ and cos22θ, which determine the intensity of transmitted light. When the polarization orientation angles of the incident light have deviation values of π/4 and π/8, respectively, the intensity of transmitted light is proportional to cos2(θ − π/4) and cos2(2θ − π/8), respectively. Since each orientation angle θ has two or four "random" options depending on the geometric phase design, the intensity of the transmitted light will become rather irregular, which leads to destruction of the greyscale pattern. Such a design is quite different from conventional one-to-one mapping, in which LP light with any polarization orientation angle can form different but recognizable greyscale patterns42-45. As a comparison, we simulate greyscale patterns with one-to-one, one-to-two, and one-to-four mapping designs, and the results are shown in Supplementary Fig. S8. Notably, this distinctive polarization-dependent characteristic can be used to increase the security of greyscale patterns and make them difficult to decode and duplicate.