Complex-amplitude hologram can provide a high-quality image that is free of twin image and zero-order noise. Multi-SLMs methods, single SLM with space-division modulations and super pixel methods were striving to meet the requirements of complex amplitude modulation. In this work, we propose an ultra-thin dielectric metasurface to improve the quality of holographic image by loading complex amplitude hologram. This metasurface controls the amplitude and phase simultaneously and independently in transmission mode. The multipole expansion method is applied to analyze the mechanism. The polarization conversion efficiency can be up to 99% and the diffraction efficiency is 71%. The simulations verify that the quality of holographic image could be greatly improved by a complex-amplitude metasurface.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Holography has numerous applications because of its capability in arbitrary wavefront modulation. In optical holography, there exists two steps. In the recording process, the phase and amplitude of an object light is encoded in the interference fringes as a hologram. In the reconstructing process, the object light can be reconstructed by illuminating the hologram with the reference light. Since the interference fringes recorded by the holographic film contains both amplitude and phase, it is able to reconstruct a high quality virtual three-dimensional (3D) scene. However, conventional optical holography requires interference, recording, as well as developing and fixing processes, which are cumbersome. In addition, a real 3D object is required to provide object light for interfering with reference light. Thus, the application of conventional optical holography is limited. With the availability of computer technology and the development of spatial light modulators (SLMs) technology, the wavefront recording and wavefront reconstruction processes can be replaced by numerical calculation. Holograms generated by numerical calculation are called computer-generated holograms (CGH). In computer-generated holography, the object light that is used to calculate the transmission function of a hologram can be described mathematically by a computer . Computer-generated holography eliminates the requirements of real objects for a hologram, making it possible to reconstruct all kinds of fictitious objects, greatly expanding the developments of holographic technology.
The SLM, a device consists of many repeated units, can modulates the amplitude or phase of the light wave independently in space and time at each unit. Typical SLMs such as liquid crystal on silicon (LCoS) devices and digital micro-mirror devices (DMD) were excellent candidates for controlling light fields. For DMD, the binary amplitude modulation can be realized by each pixel, a mirror that can switch between the states of ‘on’ and ‘off’. While for LCoS, the refractive index of liquid crystal in each pixel varies at different loading voltages, thereby giving an additional phase change of the reflected/transmitted light at the pixel. However, these two types of SLMs cannot control the complex amplitude of the light field directly. The complex amplitude hologram has to be converted into amplitude-only holograms or phase-only holograms before being encoded by these SLMs, resulting in the loss of information. Thereby, problems such as large speckle noise, DC noise and twin image occur in the holographic images . Different methods such as multi-SLMs , single SLM based on space-division modulation  and super pixel methods  were proposed to achieve complex amplitude modulation by available SLMs. Nevertheless, these methods are at the expense of complex alignment operation, low resolution, or large response time delay.
Metasurface, a kind of two-dimensional (2D) metamaterials with the thickness on an order of wavelength, have been widely studied to arbitrary manipulate the amplitude, phase and polarization of electromagnetic waves in recent years [6–8]. With sub-wavelength meta-atom, metasurface holograms could achieve sub-wavelength resolution, enlarged viewing angle, higher space-bandwidth product (SBP) and suppressed higher-order diffraction in the far field [9–11]. Several types of metasurface have been proposed to realize complex-amplitude modulation, from Terahertz to visible region [2,12,13]. Among them, the metasurface based on Huygens’ principle achieves high transmission with an ultra-thin (∼λ/6) nanodisk array . The phase is controlled by a block composed of multiple nanodisks. The metasurface sacrifices the high-resolution advantage. In another approach based on the geometric phase, an X-shaped meta-atom modulates the complex amplitude in a flexible way . Metasurface composed of this type of meta-atom preserves high resolution, with a polarization conversion efficiency of 49% 11. Dielectric nanofins improve the polarization conversion efficiency up to 90% while the height of the nanofins is approximately λ/2 .
In this work, a high efficiency and ultra-thin dielectric cross-shaped metasurface in the optical region is proposed. The advantages of Huygens’ metasurface and the geometric phase are combined. With the overlap of electric resonance and magnetic resonance, the transmission efficiencies along the two branches are close to unity. Meanwhile, the anisotropic meta-atom behaves as a half-wave plate, leading to the polarization conversion efficiency up to 99% under circularly polarized light. The amplitude is controlled by changing the width of one branch, while the phase is modulated by rotating the meta-atom based on geometric phase. Thus, arbitrary complex wavefront modulation can be flexibly modulated by this meta-atom with the use of a proposed search method. The thickness of meta-atom is less than λ/5. This thinner feature makes it possible to achieve mass manufacturing with low-cost and large-area by nanoimprinting technology. By utilizing this metasurface, the quality of holographic image is greatly improved by a complex amplitude hologram.
2. Principle of complex amplitude modulation
The rigorous formulation of Huygens’ principle reveals that to achieve purely forward propagating elementary waves, each individual elementary source should be described as an electrically small antenna that overlaps electric dipole (ED) with magnetic dipole (MD) resonances of the equal strength . In metasurfaces, under the equivalent surface boundary conditions, the collocated orthogonal electric and magnetic polarizable elements are applied to generate arbitrary field distributions for a given incident field.
For an isotropic antenna, like nanodisk, according to the coupling model theory, the transmission coefficient is 
If only the phase or the amplitude is manipulated, the isotropic Huygens’ meta-atoms have already met the need. However, to achieve complex amplitude modulation, more degrees of freedom are required. We can use the geometric phase principle to flexibly control the phase. As shown in Fig. 1(a), for a mirror symmetric meta-atom, the local orthogonal coordinate system is assigned as uov, the global coordinate system is xoy and the angle between u and x is θ, then according to Jones matrix, the scattering property of the meta-atom is 
In conventional Huygens’ metasurfaces, isotropic dielectric meta-atoms such as spheres and disks are commonly adopted. When the MD and ED are overlapping, these two dipole modes satisfy the first Kerker condition and constructively interfere with each other, leading to directional scattering. While for anisotropic dielectric meta-atom with broken rotational symmetry, new physics are introduced into the classical Mie scattering problem . The proposed structure is shown in Fig. 1(a), a silicon cross-shaped antenna is embedded in the PDMS (n = 1.406). Each branch has the same height H of 0.31 µm and possesses two sets of parameters, width W and length L. L1=0.7 µm and W1=0.35 µm for the longer branch and L2=0.4 µm, W2=0.2 µm for the shorter branch. The period of the metasurface is 1.2 µm.
The Cartesian multipole expansion technique is applied to analyze different multipole modes inside the anisotropic meta-atom . The first four terms, the ED, MD, electric quadrupole (EQ) and magnetic quadrupole (MQ) modes are considered in the multipole expansion. Figure 2 shows the multipole modes under the illumination of orthogonal polarized light. The FDTD simulations (solid black line) and the total scattering power (solid pink line marked with circles) are in good agreement with each other, indicating that higher-order multipoles are negligible. With light polarized along x-axis, the scattering spectrum shows a peak approximately 1.875 µm, as shown in Fig. 2(a). The peaks of the ED and MD resonant modes are close to each other and the MD mode dominates the scattering. The EQ and MQ resonant modes in this case contribute little to the scattering. While for light polarized along y-axis, two distinct peaks are presented at the positions of 1.789 µm and 1.825 µm in the scattering spectrum, which are caused by the MD mode and the ED mode separately, as shown in Fig. 2(b).
The insets in Fig. 2 present the electric field distributions of the ED resonance and the magnetic field distributions of the MD resonance. The upper inset in Fig. 2(a) shows that under x-polarized light, the electric field forms closed circle with strong circulating displacement currents in xz plane at the wavelength of 1.875 µm. It indicates that the corresponding induced magnetic response dominates the radiation. Under the illumination of y-polarized light, the peak at 1.789 µm, as shown in Fig. 2(b), is also caused by the induced magnetic response. While the peak at 1.825 µm is contributed by the electric response. As is shown in the upper inset of Fig. 2(b), the electric field points from one pole to another pole mimicking the radiation pattern of electric dipole in the far field. The lower insets in Figs. 2(a) and 2(b) depict the corresponding magnetic field distribution and electric field distribution in xy-plane under these two orthogonally polarized light at the wavelength of 1.825 µm and 1.875 µm, respectively.
The supported orthogonal multipole resonances are exquisitely constructed to fulfill the required surface polarizability tensor for a half-wave plate . Figure 3(a) shows the phase difference Δφ=φ1-φ2 and amplitudes t1, t2 of the transmission coefficients. The line in blue sky color indicates the position of 1.875 µm. At this wavelength, Δφ=π, and the amplitudes t1, t2 are nearly constant, both close to 1. As expected, with right circularly polarized (RCP) light travels through the meta-atom, the polarization conversion efficiency of the output light is great than 99% at the wavelength of 1.875 µm, as is shown in Fig. 3(b). The polarization conversion efficiency is defined as the ratio between Pl and the summation of Pl and Pr, where Pl and Pr are the intensities of the left circularly polarized (LCP) light and RCP light of the output light. In Fig. 3(a), Δφ is also equal to π at the wavelength of 1.825 µm. However, t1 is not equal to t2, leading to a low polarization conversion efficiency. The polarization conversion efficiency is greater than 80% in a wide range of 1.82-1.9µm.
Under the illumination of either x-polarized light or y-polarized light, the ED resonance peak is close to the MD resonance peak, leading to a high transmission coefficient. It can be deduced that by changing the geometric parameter W or L, the overlapped resonance peaks will separate gradually, leading to an amplitude change. Moreover, the phase can be manipulated by rotating the meta-atom based on geometric phase. Thus, the simultaneous and independent modulations of amplitude and phase can be achieved. As shown in Figs. 1(b) and 1(c), the amplitude and phase responses are investigated at the wavelength of 1.875 µm. The change in amplitude is monotonous from 0 to 1 with the change of W1, while the change in phase is almost linear when rotating the cross-shaped antenna.
3. Complex amplitude computer-generated hologram
The amplitude changes slightly with rotation angle θ at each fixed W1, making it hard to achieve arbitrary complex amplitude by directly finding the intersection of W and θ in Figs. 1(b) and 1(c). Here we propose a search algorithm with multiple back and forth to locate the corresponding modulation parameter (θ,W) for arbitrary complex amplitude Amexp(jφm). The flow chart of this algorithm is shown in Fig. 4.
- (a) Searching the amplitude value. We use A(0,W) as the initial amplitude to search for the target amplitude Am, namely, along vertical direction in Fig. 1(b). Assuming the corresponding W1 for the target amplitude Am is W.
- (b) Searching the phase value. By finding a suitable θ to meet the requirement of φ(θ,W)= φm in Fig. 1(c), namely, along the horizontal direction, the rotation angle θ can be determined.
- (c) Correcting the searched amplitude. By using coordinate (θ,W) in Fig. 1(b), the searched amplitude As can be obtained. However, since the amplitude is not uniform for a certain W1 when θ changes, the difference between As and Am may be greater than the allowable value δ. Noticing that the amplitude changes along the vertical direction in Fig. 1(b), we can correct the amplitude by changing W slightly with a value of Δw. Then we obtain the updated parameter (θ,W).
- (d) Correcting the searched phase. The updated phase φs=φ(θ,W) may not be equal to the target one φm. Based on geometric phase, we can then tune it by slightly rotating the meta-atom. Assuming the difference between φs and φm is Δφ, the corresponding rotating angle is half of Δφ. Hence, the final modulation parameters (Wf, θf) of each meta-atom is Wf=W,θf=θ-Δφ/2.
The reliability of our searching algorithm is verified by encoding a 300×300 complex hologram in the metasurface. The searched amplitude and phase by the proposed method are shown in Fig. 5. Comparisons of the searched values and calculated values along two arbitrary lines are depicted to illustrate the errors. As shown in Figs. 5(c) and 5(d), the searched amplitude and phase are consistent with the calculated values. The numbers of points deviating more than 20% are counted as 6.8% and 3.2% for amplitude and phase distributions with 90,000 pixels, respectively.
The complex CGH of the gate is obtained by using the angular spectrum algorithm without the use of random phase that are applied in conventional phase-only holograms [2,18]. For comparison, the conventional phase-only CGHs obtained by angular spectrum algorithm and GS algorithm are encoded in two metasurface, respectively. In these two algorithms, the random phase distribution is added to the amplitude information to simulate the diffusive effect of the object surface. The simulation is implemented by a commercial tool (Lumerical, FDTD solutions) based on the Finite-Difference Time-Domain (FDTD) method. Taking into account the limitations of the performance of our computer, a complex amplitude CGH with 160×160 pixels are encoded for simulation and only the near field distribution is recorded. The far-field distribution at the image plane, which is 77 µm away from the near-field monitor, is obtained by the angular spectrum propagation method. For the phase-only hologram generated by the GS algorithm, the holographic image is rebuilt by Fourier transform. Figure 6 shows the reconstruction images from these three metasurfaces, which are encoded with complex CGH, phase-only CGH by angular spectrum algorithm and phase-only CGH by GS algorithm, respectively. The image reconstructed from the complex-amplitude CGH is of higher quality than that reconstructed from the phase-only CGH. Most of the details of the object can be distinguished from Fig. 6(a), while Fig. 6(b) only displays the rough outline. Not only are the details lost, but also much noise is observed. Figure 6(c) shows higher quality than Fig. 6(b), but it exhibits more noise than Fig. 6(a), which can be evaluated by using MSE (mean-square error). In practice, with millions of pixels fabricated by mature processing technology, such as lithography and nanoimprinting, the holographic image could be very clear. For metasurface holography, the larger the number of pixels is, the higher the image resolution is, the larger the viewing angle is, and the longer the reconstruction distance is for 3D display. However, dues to the subwavelength pixel, it is not easy to manufacturing large-area small-error metasurface at relative low cost.
The polarization conversion efficiency of each meta-atom can be up to 99%. The fabrication fluctuation of W1, L1, W2 and L2 have little effect on polarization conversion efficiency while H have a greater effect on that. When W1 changes from 0.3 µm to 0.4 µm, the polarization conversion efficiency can still be higher than 87% at the wavelength of 1.875 µm, as shown in Fig. 7(a). However, in metasurface, each meta-atom is excited not only by the incident field, but also by the retarded fields from all other antennas. Therefore, it might not be appropriate to simply check the polarization conversion efficiency from an isolated antenna. Besides, for the amplitude modulation, the polarization conversion efficiency decreases. Therefore, we analyzed the polarization conversion efficiency in the near field. The efficiency around the pattern area can be higher than 90%.
The reconstruction from a hologram can be considered as the diffraction at the desired diffraction orders. The diffraction efficiency is defined as the power concentrated in the holographic image referenced to the total power transmitted by the hologram. The diffraction efficiency of the hologram is calculated about 71% in the proposed metasurface.
In conclusion, an ultra-thin metasurface that can realize complex amplitude modulation is proposed. The overlap of the electric resonance and magnetic resonance leads to a high conversion efficiency of up to 99% at the wavelength of 1.875 µm. The amplitude is controlled by changing the width of the long branch. The phase is manipulated by the rotation angle of the cross-shaped antenna. The monotonous distribution of amplitude and phase makes it easy to realize arbitrary amplitude and phase control by finding the intersection of amplitude and phase with the proposed searching algorithm. The complex amplitude CGH and the phase-only CGHs are encoded in the same type of metasurface. The reconstruction images exhibit that the quality of holographic image can be greatly improved by using complex amplitude CGH. This metasurface can be designed to operate in visible region and other specific band just by changing the geometric parameters, but with a decreased polarization conversion efficiency. It can also operate at each individual wavelength among a relatively wide wavelength range. The proposed metasurface also shows great potential in the applications of beam shaping, micro holographic display, 3D biological imaging, optical computing and any other fields that need to precisely modulate the complex amplitude.
National Natural Science Foundation of China (61775117); National Postdoctoral Program for Innovative Talents of China (BX201700313); China Postdoctoral Science Foundation (2018M630148).
We thank Xiaomeng Zhang at Tsinghua University for providing the code that used for multipole expansion. We thank the anonymous reviewers for the valuable suggestions.
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