1. Introduction

Due to the ability of recording a light field with entire information (both amplitude and phase), optical holography is an attractive technology for various significant applications such as three-dimensional imaging [1,2] and optical data storage [3]. Conventional holograms record the intensity of the interference fringes formed by a scattered beam from an object and a reference light wave, which results in variation of either transmissivity or reflectivity of the recording media [4,5]. Although holograms record the intensity of interference fringes, the amplitude information and phase information of the object have been both encoded into the hologram, leading to the complete 3D image reconstruction.

With the development of computer technology and emergence of spatial light modulator (SLM), computer-generated holograms (CGHs) obtained by numerical computation provide a new way to record holographic information. However, since most of optical elements (such as SLMs, DOEs and so on) only modulate amplitude or phase [6–8], the incomplete modulation cannot completely reconstruct the original 3D object, but only produce an image that resembles the target object.

In recent years, metasurfaces [9–14], characterized as planar metamaterials with subwavelength dimensions, provide infinite flexibility in manipulating the electromagnetic field for versatile applications, such as holography [15–22], nanoprinting [23–29], a hybrid of holography and nanoprinting [30–33], metalens imaging [34,35], data storage [36] and so on. Owing to the unique ability of light manipulation, metasurfaces represent a new route for 3D holography [37–44]. However, previous researches either utilize the metasurfaces with full complex-amplitude modulation [38–41] or employ phase-only [42–44] metasurfaces assisted with kinds of design algorithms (such as Gerchberg−Saxton (GS) algorithm [42,43], Fienup algorithm [44] and so on) to realize 3D holographic display. The former either requires meta-atoms with different dimensions and orientations [38–40], which is difficult to achieve continuous complex-amplitude manipulation because of the limitation of nanofabrication technology, or employs double-nanoblock unit cells [41], bringing difficulty to the design and fabrication of metasurfaces. In addition, the latter cannot realize the accurate reappearance of the original object wave due to not recording the complete complex-amplitude information.

Here, inspired by the generation of conventional holograms and CGHs, we propose an extremely simple method to realize on-axis 3D holographic display with extinction of zero-order light, by employing an amplitude-only meta-hologram consisting of orientation-variant nanostructures with identical dimensions. We mathematically simulate the process of interference between object light wave and a plane reference (PR) light wave by computer and obtain the intensity distribution of the interference fringes. Then the transmittance distribution of the meta-hologram is obtained after directly removing the first two non-interference terms from the intensity, which forms the zero-order light of the 3D hologram [45]. As each orientation-variant nanostructure could realize continuous amplitude modulation [46], by carefully configuring the orientations of nanostructures, the transmittance distribution containing complex-amplitude information of the 3D object is translated into an amplitude-only meta-hologram.

To demonstrate the feasibility of the method, we fabricate a 3D meta-hologram, with target images of three letters “W”, “H” and “U” located in the center of three different z-planes. Under normal incidence of linearly polarized (LP) light, three letters clearly appear at their target image planes. Moreover, as the amplitude modulation originates from the orientation of the nanostructure, our proposed 3D meta-hologram is independent of wavelengths. We irradiate the meta-hologram with operating wavelengths ranging from 570 nm to 660 nm and the experimental results show that holographic images are well reconstructed at their z-planes. With simple meta-atom and broadband response, this skillful approach towards 3D display by using amplitude-only metasurfaces can significantly decrease the complexity of design and manufacturing, and is promising in optical information storage, virtual reality, and other applications based on 3D display.

2. Continuous amplitude modulation via orientation-variant nanostructures

Schematic of an amplitude-only meta-hologram for 3D holographic display is shown in Fig. 1(a). The meta-hologram consists of an array of orientation-variant nanostructures with identical dimensions (width W, length L, height H and cell size C). Each nanostructure is made up of a rectangle silver nanobrick with orientation angle θ (defined as the angle between the short axis of the nanobrick and the x-axis) sitting on a square silica substrate, as shown in Fig. 1(b). When such a nanostructure is illuminated by the normal incident x-LP light (polarized along the x-axis), the nanostructure acts as a subwavelength amplitude modulation unit and the output light can be written as

 

Fig. 1. (a) Schematic of the amplitude-only meta-hologram for 3D holographic display. The three letters “WHU” are reconstructed on different z-planes: z₁-plane for “W”, z₂-plane for “H” and z₃-plane for “U”. (b) The structure diagram of one pixel of the amplitude-only meta-hologram. Each nanostructure behaves as an amplitude modulator, governed by Malus law. (c) The amplitude modulation curve of the orientation-variant nanostructure and the simulated result of |t_xx - t_yy| versus wavelength (400 ∼ 700 nm).

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Similarly, in the case of y-LP light (polarized along the y-axis) illumination, the transmitted light passing through the nanostructure can be expressed as:

From Eqs. (1) and (2), it can be seen that the amplitude of the cross-polarized light (x-LP in, y-LP out; or y-LP in, x-LP out) follows the same modulation rules below [Eq. (3)], whether employing x-LP or y-LP incident light.

It is obvious that the amplitude of the incident light can be modulated continuously (from positive values to negative values) by rotating the nanobrick within the xy-plane (see the pink curve plotted in Fig. 1(c)). In our design, when t_xx=1 and t_yy=0, the silver nanobrick acts as a perfect nano-polarizer which can reflect most of the x-LP light and transmit most of the y-LP light.

A set of geometric parameters of the nanostructure are optimized by using the CST Microwave Studio software, and the simulated result of |t_xx - t_yy| is shown in Fig. 1(c). With L of 130 nm, W of 80 nm, H of 70 nm and C of 400 nm, the peak value of |t_xx - t_yy| can reach 90% at a design wavelength of 633 nm. If the incident wavelength deviates from 633 nm, the |t_xx - t_yy| will decrease, reducing the efficiency of the meta-hologram consisting of the nanostructures.

3. Design of on-axis amplitude-only meta-holograms for 3D display

As a proof of concept, we design an on-axis meta-hologram for 3D display to demonstrate the amplitude modulation capability of orientation-variant nanostructure. Theoretically, a 3D object could be regarded as the superposition of multiple parallel flat objects located at different planes [47]. As an example, three layers containing three letters of “W”, “H” and “U” respectively are employed as the target flat images. The flat images are designed to reconstruct at three different z-planes along the propagation direction within the Fresnel diffraction range.

The design procedure of the amplitude-only meta-hologram for 3D display is shown in Figs. 2(a)–2(d), including four steps (more details are provided in the Appendix A). The complex-amplitude distributions of the three target flat object are described as U₁(x₁, y₁, z₁), U₂(x₂, y₂, z₂) and U₃(x₃, y₃, z₃). Firstly, based on the Huygens-Fresnel diffraction theory [40], computing separately the complex field distributions [U_H1(x₀, y₀, 0), U_H2(x₀, y₀, 0) and U_H3(x₀, y₀, 0)] that propagate to the meta-hologram plane according to the three target flat images and their corresponding diffraction distances. As shown in Fig. 2(a), let the meta-hologram plane be z₀ = 0 µm, and three image planes are z₁ = 700 µm, z₂ = 900 µm and z₃ = 1100 µm, respectively. Secondly, synthesizing the complex fields calculated in the previous step. The accumulated field distribution is the complex field of the 3D object light wave at the hologram plane. Thirdly, superimposing the object light wave and a plane reference (PR) light wave to form the interference fringes, the normalized intensity distribution is shown in Fig. 2(b). In this way, the complex amplitude information of the object light wave is converted into intensity information. Note that the proposed nanostructure could realize positive or negative amplitude control, thus we subtract the non-interference terms (the first and second terms that represent the intensity of zero-order light) [45] from the intensity distribution of the interference fringes to eliminate the zero-order light. The normalized transmittance distribution of the meta-hologram is obtained after eliminating zero-order light, as shown in Fig. 2(c). Finally, transferring the transmittance distribution into the orientation configuration of the nanobrick arrays according to the relation between amplitude modulation and orientation angle (see Fig. 1(c)). After the above process, the complete information of multiple flat images is recorded into the meta-hologram. Figure 2(d) shows the partial schematic view of the designed meta-hologram. According to the Fresnel diffraction integral formula [44], we numerically simulate the flat holographic images of preset planes by using MATLAB software and the results are shown in Fig. 2(e). We can see that the reconstructed holographic images agree quite well with our design.

 

Fig. 2. Design procedure of the amplitude-only meta-hologram for on-axis 3D display. (a) The computer generation of the 3D hologram. The optical field distributions that propagate from three image planes to the meta-hologram plane are respectively calculated by using Kirchhoff integral equation. Superposing the three optical field distributions to form the object light wave. (b) The normalized intensity distribution of the interference fringes formed by the object light wave and a plane reference light wave. (c) The normalized transmittance distribution of the meta-hologram after eliminating the zero-order light of the 3D hologram. (d) Schematic view of the amplitude-only meta-hologram. The transmittance profile of the CGH is translated into the orientation distribution of nanobricks by utilizing the relationship between amplitude and orientation. (e) The calculated results of the flat holographic images at preset planes.

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4. Experimental demonstration of the on-axis 3D meta-hologram

The designed meta-hologram is fabricated by the standard electron beam lithography (EBL), which has been demonstrated in many literatures [19,25,33]. The meta-hologram is composed of 1000 × 1000 orientation-variant nanostructures with identical dimensions, thus the area is 400 µm × 400 µm. Each orientation-variant nanostructure plays the role of a pixel of the hologram to generate the required continuous transmittance profile with normal incidence of LP light.

To investigate the performance of the fabricated meta-hologram, we use an optical microscope (Motic BA310MET-T) to obtain the reconstructed holographic images within the Fresnel range. As depicted in Fig. 3(a), we modify the optical setup inside the microscope, including employing a supercontinuum laser source (YSL SC-pro) in place of its own light source, inserting two linear polarizers (labeled as Polarizer and Analyzer) and employing a CMOS digital camera (Moticam X) to capture the optical microscopy images. The polarizer in front of the sample is used to obtain the desired x/y-LP light for illumination, and the second linear polarizer with orthogonal transmission axis is employed to ensure that only cross-polarized (y/x-LP) light is collected. As the objective lens has the finite depth of focus, the 3D holographic image cannot be directly captured, thus the fabricated sample is placed on a tunable object stage.

 

Fig. 3. Demonstration of the designed on-axis meta-hologram for 3D display. (a) Schematic of the optical setup for capturing the reconstructed holographic images at various observation planes. (b) The intensity distribution at the hologram plane (z = 0) experimentally observed by a 20× objective lens. The inset in the upper left corner is a scanning electron microscopy (SEM) image of the constituent nanostructures for top view. Scale bar: 1 µm. (c)-(e) Three letters of “W”, “H”, “U” are reconstructed at the preset image planes (z₁ = 700 µm, z₂ = 900 µm and z₃ = 1100 µm, respectively) and observed by a 100× objective lens. Solid and dashed lines in (b)-(e) represent the transmission axes of the bulk-optic polarizer and analyzer, respectively. (f) Evolution of images located at different distances z (ranging from 0 to 1100 µm) from the meta-hologram plane along the propagation direction. By tuning the position of the sample, three letters gradually appear. The operating wavelength is 633 nm.

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At first, we tune the position of object stage to make the hologram plane and focus plane overlapped. Intensity distribution at the surface of the meta-hologram in transmission is observed by a 20× (numerical aperture NA = 0.4) microscopic objective lens, as shown in Fig. 3(b). It can be seen that the brightness variations in the metasurface plane, which verifies the amplitude modulation capability of the metasurfaces. To clearly observe the reconstructed images in the center of the image planes, a 100× (NA = 0.8) microscope objective lens is employed to observe the images. By finely tuning the focus plane away from the hologram plane, the reconstructed images at different distances of 700 µm, 900 µm and 1100 µm relative to the meta-hologram plane appear successively, as shown in Figs. 3(c)–3(e). We can find that the measured results are coincident with the theoretical calculated ones shown in Fig. 2(e) and limited background noise exists in the observation, which is benefit from the extinction of the zero-order light in design processes. In addition, a dynamic on-axis evolution process of the captured images is exhibited in Fig. 3(f), which shows that clear holographic images only appear at the designed observation planes.

The designed meta-hologram for 3D display has a remarkable feature, that is, wavelength insensitivity. As proved in Eq. (3), the characteristic of continuous amplitude modulation originates from the orientation of the nanostructure, thus the contrast of the holographic image is independent of wavelength. To study the spectrum response of the meta-hologram, we irradiate the meta-hologram sample with different wavelengths ranging from 570 nm to 660 nm. Figure 4 shows the reconstructed images of three letters at four selected wavelengths: 570, 600, 630 and 660 nm, respectively. It is worth noting that the distance between the image plane and the hologram plane is inverse proportional to the wavelength according to the paraxial approximation. Thus, the positions of the images can be theoretically calculated according to the principle of geometrical optics [37,41]. In the experiment, as expected, the holographic images at four wavelengths are well reconstructed at each z-plane and they are almost exactly the same except the imaging distance is changed with respect to the operating wavelength. The experimentally measured positions of image planes coincide with their theoretical values in all wavelengths, which verifies the meta-hologram can work well over a wide wavelength range.

 

Fig. 4. Experimental verification of wavelength independence of meta-hologram. Reconstructed holographic images for different wavelengths of (a)-(c) 570 nm, (d)-(f) 600 nm, (g)-(i) 630 nm and (j)-(l) 660 nm. Their corresponding positions of image planes are marked at the bottom of the images for each case.

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5. Conclusions

In a summary, we apply conventional hologram design approach to design CGHs and develop an on-axis 3D display technique based on amplitude-only metasurface holograms. Merely by configuring the orientation angles of the nanostructured metasurface, the complex-amplitude information of a 3D object can be readily recorded into an amplitude-only meta-hologram. Benefitting from the capability of modulating amplitude continuously from negative to positive, the strong zero-order light that conventional holograms always suffer from could be directly eliminated. We experimentally demonstrate the on-axis meta-hologram by reconstructing multilayer flat holographic images at different planes within the Fresnel diffraction range. More interestingly, benefiting from the unique amplitude modulation mechanism governed by the nanostructure orientation, leading to the broadband response of the metasurface. The displayed effect can be further improved by increasing the size of the meta-hologram (that is, fabricating a metasurface with more nanostructures) to modulate the incident light with a larger area. Our research provides a simple but effective approach to develop ultracompact, on-axis and broadband 3D hologram, which can find its markets in 3D displays, optical information storage, virtual reality and so on.

Appendix: calculation process of hologram design

The calculation process of hologram including flow chart (see Fig. 5) and equations are described in detail as follow:

 

Fig. 5. The flow chart of the calculation process of hologram.

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Three layers containing three letters of “W”, “H” and “U” respectively are employed as the target flat images. The complex fields of three target images are described as U_i (x_i, y_i, z_i) = A_i (x_i, y_i, z_i) exp[jφ_i (x_i, y_i, z_i)] (i = 1, 2, 3), where x_i, y_i, z_i represent the coordinates in the image plane, A_i (x_i, y_i, z_i) and φ_i (x_i, y_i, z_i) denote the amplitude of the target image and the predefined random phase. Based on the Huygens-Fresnel diffraction theory, the corresponding complex field distribution propagating to the hologram plane (U_Hi (x_i, y_i, z_i)) can be computed by the Fresnel Kirchhoff integral:

(4)$${\textrm{U}_{\textrm{H}i}}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}) = \frac{\textrm{1}}{{j\lambda }}\int\!\!\!\int {{\textrm{U}_i}({x_\textrm{i}},{y_\textrm{i}},{z_i})} \frac{{\textrm{exp} (jkr)}}{r}dxdy\textrm{,}$$

where x₀, y₀, z₀ are the coordinates in the hologram plane, λ and k (k = 2π/λ) indicate the wavelength and wave vector, and $r = \sqrt {{{({x_\textrm{0}} - {x_i})}^\textrm{2}} + {{({y_\textrm{0}} - {y_i})}^\textrm{2}} + {{({z_\textrm{0}} - {z_i})}^\textrm{2}}}$ represents the distance between two points on the image plane and hologram plane. In our design, z₀ = 0 µm, z₁ = 700 µm, z₂ = 900 µm and z₃ = 1100 µm, respectively.

The complex field of the 3D object light wave can be expressed as

(5)$${\textrm{U}_\textrm{H}}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}) = \sum\limits_{i = \textrm{1}}^\textrm{3} {{\textrm{U}_{\textrm{H}i}}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})} = O({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})\textrm{exp} (j{\varphi _O}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})).$$

The interference fringes are formed by superimposing the object light wave and a plane reference (PR) light wave. The complex field of a plane reference (PR) light wave can be described as R(x₀, y₀, z₀) = A due to the on-axis design, thus the intensity distribution of the interference fringes can be written as

(6)$$\begin{aligned}\textrm{I}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}) &= ({U_\textrm{H}} + \textrm{R}) \times conj({U_\textrm{H}} + \textrm{R})\\ &= {A^\textrm{2}}\textrm{ + }{O^\textrm{2}}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})\textrm{ + 2}AO({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})\textrm{cos(}{\varphi _O}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})), \end{aligned}$$

where conj represents the conjugate operation. From Eq. (6) we can see that the first two non-interference terms form the zero-order light of the hologram, which is difficult to eliminate for conventional holograms. If we subtract the non-interference terms (the first and second terms) from the intensity distribution of the interference fringes, the zero-order light can be eliminated. Therefore, the transmittance distribution of the zero-order-free 3D meta-hologram can be written as

(7)$$t({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}) = \textrm{2}AO({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})\textrm{cos(}{\varphi _O}({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}})).$$

According to the relationship between the amplitude modulation and orientation angle (see Fig. 1(c)), we can readily obtain the orientation configuration of the nanobrick arrays:

(8)$$\theta ({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}) = \arcsin (t({x_\textrm{0}},{y_\textrm{0}},{z_\textrm{0}}))/\textrm{2}.$$

In experiment, the zero order light contains three parts: the undiffracted cross-polarized light of the hologram (that is, the non-interference terms in the intensity formula), the unconverted light due to fabrication imperfections, and the overfilling incident light [48,49]. The first part is eliminated in the design process of hologram, while the latter two occurring in fabrication and experiment processes are co-polarized and can be eliminated by using a pair of polarizer and analyzer with orthogonal polarization directions [50].

Funding

National Natural Science Foundation of China (11774273, 11904267, 91950110); National Postdoctoral Program for Innovative Talents (BX20180221); China Postdoctoral Science Foundation (2019M652688); Natural Science Foundation of Jiangsu Province (BK20190211); Open Fund of the Key Laboratory for Metallurgical Equipment and Control Technology of Ministry of Education in Wuhan University of Science and Technology (MECOF2020A01).

Disclosures

The authors declare no conflicts of interest.

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