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We comparatively study two schemes of multiple polarization channels (from orthogonal two channels to twelve channels) by using birefringent dielectric metasurfaces, as shown in Fig. 1. We show that polarization and angle-multiplexed holograms can be realized by utilizing nanofins with different cross-sections but without rotation (Fig. 1a). Two sets of off-axis holographic images can be generated with two orthogonal states. However, by considering the flexibility for designing the entire Jones matrix, we can achieve more multiplexed functionalities with complete control of polarization and phase (Fig. 1b). Three independent images (we chose the words "holography", "meta", and "surface" as the reconstructed images) can be reconstructed successfully with high resolution and high fidelity for different combinations of input and output polarization channels. All combinations of thse three images in a total of twelve different polarization channels can be observed ("meta" + "holography", "meta" + "surface", "surface" + "holography" and "meta" + "surface" + "holography") with satisfactory efficiencies. In addition, the reconstructed images exhibit a vectorial feature for the polarization state of the different words in analogy to optical vector beams.
Fig. 1 Principle of our designed metasurface hologram.
Schematic illustrations of polarization multiplexed holograms based on dielectric metasurfaces. The red and blue arrows indicate the polarization of the incident light and the transmission axis of the polarizer placed behind the metasurface sample. The red, blue, and green color of the reconstructed images (the words "holography", "meta", and "surface") represent the x, y, and RCP (or LCP, with the same helicity as incident light) component of the output light, respectively. a Two channel polarization and an angle multiplexed hologram based on metasurfaces composed of nanofins with different cross-sections but fixed orientation angles, which can be used to reconstruct two sets of off-axis images. b Multichannel polarization multiplexed hologram based on metasurfaces composed of nanofins with different cross-sections and orientation angles, which can be used to reconstruct three independent images and all combinations of these images (12 channels in total)For achieving multichannel vectorial holography, we have to explore suitable building blocks for the metasurfaces. As is well known, the general relation between the electric field of the input (Ein) and output (Eout) waves at each pixel can be expressed using the Jones matrix description:
$$ E^{{{out}}} = { {TE}}^{ {in}}\,{{where}}\,T = \left[ {\begin{array}{*{20}{c}} {T_{xx}} & {T_{xy}} \\ {T_{yx}} & {T_{yy}} \end{array}} \right] $$ (1) Indeed, any desired symmetric and unitary Jones matrix can be realized using a birefringent metasurface if the polarization-dependent phase shift (ϕx, ϕy) and the orientation angle θ can be chosen freely33:
$$ T = V\left[ {\begin{array}{*{20}{c}} {e^{i\phi _x}} & 0 \\ 0 & {e^{i\phi _y}} \end{array}} \right]V^T = R(\theta )\Delta R( - \theta ) $$ (2) where Δ indicates the eigenvalue matrix of the Jones matrix T and the real unitary matrix V can be treated as a rotation matrix R. Crucially, the desired phases (combining both dynamic and geometric phases) can be extended to impart on any set of orthogonal polarization states.
We first consider the two-channel polarization multiplexed holograms as the simplest case. By formulating the Jones matrix of each pixel as:
$$ T{{ = }}\left[ {\begin{array}{*{20}{c}} {e^{i\varphi _1}} & 0 \\ 0 & {e^{i\varphi _2}} \end{array}} \right] $$ (3) We can easily tailor the desired output light to $E_{x,{\kern 1pt} { {inc}}}^{ {out}} = \left[ {\begin{array}{*{20}{c}} {e^{i\varphi _1}} \\ 0 \end{array}} \right]$ in each pixel according to the profile of the hologram 1 while guaranteeing the phase distribution of $E_{y,{\kern 1pt} { {inc}}}^{ {out}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {e^{i\varphi _2}} \end{array}} \right]$. Therefore, the original image 1 and image 2 for orthogonal incident polarizations can be reconstructed and switched. These two channels for polarization multiplexed metasurface holograms can be easily realized by generating a polarization-dependent phase shift in the x and y directions but with an identical orientation angle for all nanofins. Note that to further enlarge the information capacity, one can also encode angular or distance multiplexed holographic images for each phase profile of φ1 and φ2.
More complex multiplexed functionalities can be realized by utilizing the design freedom of the rotation matrix R. Assuming that the Jones vector of the output light is described by $E_{x,{\kern 1pt} { {inc}}}^{ {out}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{*{20}{c}} {e^{i\varphi _1}} \\ {e^{i\varphi _2}} \end{array}} \right]$ when the incident light is x-polarized light. Now, the x and y components of the output light can be designed to reconstruct the original holographic image 1 and image 2 with equal intensity and high contrast, respectively. More importantly, the reconstructed images can be switched by selecting the desired transmission polarization with negligible polarization cross-talk. For such a situation, the Jones matrix of each pixel can be derived as:
$$ T{{ = }}\frac{{\sqrt 2 }}{2}\left[ {\begin{array}{*{20}{c}} {e^{i\varphi _1}} & {e^{i\varphi _2}} \\ {e^{i\varphi _2}} & {e^{i(2\varphi _2 - \varphi _1 + \pi )}} \end{array}} \right] $$ (4) From the above Jones matrix, one can recognize that when the incident light polarization is changed to y-polarized light, the Jones vector of the output light will be expressed by $E_{y,{\kern 1pt} { {inc}}}^{ {out}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{*{20}{c}} {e^{i\varphi _2}} \\ {e^{i(2\varphi _2 - \varphi _1 + \pi )}} \end{array}} \right]$. This additional phase relation between φ1 and φ2 for the y-component of the output field opens up the possibility of encoding an additional holographic image. By utilizing a suitable algorithm for the generation of holograms, holographic image 3 can be reconstructed by satisfying the relation φ3 = 2 × φ2 − φ1 + π. Hence, the y-component of the output light results in reconstruction of image 3 by illumination with y-polarized light. Meanwhile, the concept can be extended further by utilizing circular polarization states for the incident wave. For circularly polarized input light, the output field can be derived as:
$$ \begin{array}{*{20}{c}} E_{L(R){\kern 1pt} ,{\kern 1pt} {\kern 1pt} { {inc}}}^{L(R){\kern 1pt} ,{\kern 1pt} { {out}}} = \frac{{e^{i\varphi _1} + e^{i(2\varphi _2 - \varphi _1 + \pi )}}}{4}, \\ E_{L(R){\kern 1pt} ,{\kern 1pt} {\kern 1pt} { {inc}}}^{R(L){\kern 1pt} ,{\kern 1pt} { {out}}} = \frac{{e^{i\varphi _1} + 2e^{i(\varphi _2 \pm \pi /2)} + e^{i(2\varphi _2 - \varphi _1)}}}{4} \end{array} $$ (5) where the super/subscript in the brackets denote the alternative case for orthogonal left-handedness (L) or right-handedness (R) circular polarization. Based on the phase relation of the output fields, one can find that three different independent images (1-3) and all combinations of these images (1 + 2, 1 + 3, 2 + 3, 1 + 2 + 3) can be reconstructed from each item using the above equations, while a constant phase difference of ±π/2 and 0/π in Eq. 5 will only result in the same image. Hence, we obtain 12 channels in total (different combinations of incident/output polarization: Txx, Txy, Tyx, Tyy, Tll, Tlr, Trl, Trr, Exout, Eyout, ELout, ERout) and seven combinations of holographic images within one single metasurface, without any additional spatial multiplexing.
To acquire the target phase profiles to fulfill Eq. 4 (we can view Eq. 3 as a special case without rotation), both the hologram generation algorithm and suitable design of the meta-atoms are key points. We build a new hologram algorithm based on a modified Gerchberg-Saxton (GS) scheme by considering the quantified relations of each other (φ1, φ2, and φ3 = 2φ2 − φ1 + π) to adapt to each polarization channel. As a phase retrieval method, the desired phase distribution obtained with the GS algorithm is calculated by constructing an iterative loop between the object plane and the hologram via a Fourier transform37. Crucially, the phase φ3 has a quantified relation with both other phases φ1 and φ2, while the latter two phase profiles can be altered independently. Therefore, we used another serial loop by connecting all of these three phase profiles. A flowchart with details can be found in the Supplementary Material. Meanwhile, we use the birefringence as well as the rotational property (adapted from Eq. 2) of the simplest rectangular nanofins with a tailorable cross-section and azimuthal angles. Such structures provide a suitable platform for the realization of multichannel vectorial holograms. To realize the desired phase shifts in each polarization channel, one has to guarantee that the phase shift of either φ1 or φ2 can cover the entire 0-2π range while still allowing all possible combinations of (φ1, φ2). Furthermore, we have to maintain a uniform amplitude distribution with high efficiency. A 2D parameter optimization by using a rigorous coupled wave analysis (RCWA) method was carried out. The simulation results for the amplitude and phase distribution of Si nanofin for the orthogonal polarization state illumination are shown in Fig. 2 (see Methods).
Fig. 2 Simulated amplitude and phase of the transmission coefficients of our designed silicon nanofin.
a Schematic illustration of an amorphous silicon nanofin positioned on a glass substrate. The metasurface will be composed of a periodic arrangement of such unit-cells. b-e Simulation results for the amplitude and phase of the transmission coefficients txx and tyy shown for a 2D parameter optimization by using a rigorous coupled wave analysis method. The length and width of the nanofin are both swept in the range of 80-280 nm at an incident wavelength of 800 nmIn the following, we fabricated several designed dielectric silicon metasurfaces on top of a glass substrate by using a plasma etching process, followed by electron beam lithography for patterning (see Methods). The experimental setup and two typical scanning electron microscopy images of the samples with and without rotation are shown in Fig. 3. For the specific simple case of no rotation (Eq. 3), we encode four separate original images into φ1 and φ2 by considering polarization and angular multiplexing, respectively, as shown in Fig. 4. This metasurface yields off-axis holographic images of a cartoon tiger and a snowman with high fidelity and high resolution when illuminated by x-polarized light. By switching to y polarization for the incident light, the reconstructed images are changed to a teapot and a cup. Note that in this case, there are only two polarization channels, with both pairs of the holographic image reconstructed and made to disappear simultaneously by rotating the polarizer behind the sample. For the polarization of the incident light with an angle of 45° or 135°, all four original images can be observed simultaneously. The experimental results are in good agreement with the simulation results, which confirms the basic design principle. In our experiment, we define two types of diffraction efficiencies. The diffraction efficiency of each polarization channel (Txx, Txy, Tyx, and Tyy) is defined as the ratio of the power of the output light in the different channels to the power of the incident light. These diffraction efficiencies of our fabricated samples are retrieved from spectral analysis (transmission spectra). The corresponding experimental results are shown in the Supplementary Material. The net diffraction efficiency for the holography is defined as the ratio of the intensity of the single reconstructed image to the power of the incident light (the residual power in the central area is not included). For the four reconstructed images, the net diffraction efficiencies (cartoon tiger, snowman, teapot, cup) are 15.37%, 10.87%, 13.08%, 10.92%, respectively (see the video and further efficiency analysis in the Supplementary Materials).
Fig. 3 Experimental setup and scanning electron microscopy images of the fabricated metasurface samples.
a The experimental setup for the observation of the holographic images. The two linear polarizers (LP1, LP2) and two quarter-wave plates (QWP1, QWP2) are used to set the precise polarization combination for the incident/transmitted light. The lens images the back focal plane of the microscope objective lens (×40/0.6) to a CCD camera. b-e Scanning electron microscopy images of two typical fabricated silicon metasurface samples shown with a top and side view. The metasurface holograms are composed of 1000 × 1000 nanofins with different cross-sections and orientation anglesFig. 4 Simulated and experimental results for the two-channel polarization and angle-multiplexed hologram.
a-f Two sets of off-axis images are reconstructed, which can be switched by changing the polarization of the incident light (denoted by the red arrows)Furthermore, we fabricate another dielectric metasurface hologram that can utilize additional polarization channels by considering the additional design freedom afforded by the orientation angles of the nanofins. In such a way, more complex multiplexing functionalities can be obtained. From the linear polarization experimental results (top two rows), we observe that when the incident light is x-polarized, the x ($E_{x,\;{\kern 1pt} { {inc}}}^{x,\;{\kern 1pt} { {out}}}$) and y ($E_{x,{\kern 1pt} { {inc}}}^{y,{\kern 1pt} { {out}}}$) components of the output field will contribute to the reconstruction of the words "holography" and "meta", respectively. Hence, the total electromagnetic field of the output light ($E_{x,{\kern 1pt} { {inc}}}^{{ {norm}},{\kern 1pt} { {out}}}$) results in reconstruction of both words ("holography" and "meta") simultaneously, but with orthogonal 'vectorial' properties for each word (Fig. 5g-i). Upon switching to the incident light with y polarization, the x ($E_{y,{\kern 1pt} { {inc}}}^{x,{\kern 1pt} { {out}}}$) and y ($E_{y,{\kern 1pt} { {inc}}}^{y,{\kern 1pt} { {out}}}$) components of the output light will be switched to reconstruct the words "meta" and "surface", respectively (Fig. 5j-l). We further carried out verification of the multiplexed holograms under illumination with circularly polarized light (bottom two rows). When the incident light is LCP/RCP, we can observe the words "holography" and "surface" simultaneously by selecting the LCP ($E_{L,{\kern 1pt} inc}^{L,{\kern 1pt} out}$)/RCP ($E_{R,\;{\kern 1pt} { {inc}}}^{R,\;{\kern 1pt} { {out}}}$) components of the output light (Fig. 5w, s). All three images can be obtained by using the RCP ($E_{L,{\kern 1pt} inc}^{R,{\kern 1pt} out}$)/LCP ($E_{R,\; { {inc}}}^{L,\; { {out}}}$) components of the output light (Fig. 5v, t). Similarly, we can determine that the vectorial nature of the reconstructed images of the words "holography" and "surface" is linearly polarized from Eq. 5. While for the word "meta", it only appears in the orthogonal circular polarization channels (Fig. 5x, u), as can be derived from Eq. 5. Therefore, the polarization state of the reconstructed word "meta" is right (left) handedness circularly polarized. Further analysis of arbitrary elliptical input/output polarization combinations can be carried out that will result in a much more complex "vectorial" nature for the reconstructed holographic images. We determined the net diffraction efficiencies for the reconstructed images ("holography", "meta", and "surface") to be 15.97%, 8.03%, and 9.91% for the case of a linearly polarized combination, respectively. Therefore, three independent images and all the combinations of these images (12 channels in total with seven different combinations) can be reconstructed with high contrast by our proposed method. Additional results for demonstrating the feasibility of achieving a dynamic holographic display and efficiency analysis can be found in the Supplementary Material.
Fig. 5 Simulated and experimental results for the multichannel vectorial holography (linear and circular channels).
a-x The two white arrows in the corners indicate the input (first arrow) and output (second arrow) polarization of light. In total, twelve polarization channels and seven different image combinations are demonstrated. The vectorial nature of each word can be analyzed from Eqs. 4 and 5In addition, the method also enables encrypting different original images that can be superimposed at the same spatial location. Such superposition has the ability to convey a different meaning in the reconstructed image and can be used to provide alterable information content for the image, i.e., encryption, as shown in Fig. 6. Here, we chose a dice as our original image. The dice has six faces that are represented by the number of pips on the surface. Interestingly, by using a suitable selection of different combinations of input/output polarization states, a different number of pips (from one to six) can be observed. This kind of illusion for viewing different sides of the dice results from the increased multiplexing capability of our method that can encode up to six different images for the various combinations of polarization states (see the video in the Supplementary Materials).
Fig. 6 Simulated and experimental results for the reconstruction of a dice with different exposed faces.
The points one to six are reconstructed by using different combinations of input/output polarization. The two white arrows in the corners indicate the input (first arrow) and output (second arrow) light polarization. For numbers 5 and 6, no polarization analyzer was used