A large field-of-view metasurface for complex-amplitude hologram breaking numerical aperture limitation

Yongheng Mu 1 , Mengyao Zheng 1 , Jiaran Qihttp://orcid.org/https://orcid.org/0000-0002-4086-5880 1 , Hongmei Li 1 , and Jinghui Qiu 1
  • 1 Department of Microwave Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, 150001, Harbin, China
Yongheng Mu
  • Department of Microwave Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, 150001, China
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, Mengyao Zheng
  • Department of Microwave Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, 150001, China
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, Jiaran QiORCID iD: https://orcid.org/0000-0002-4086-5880, Hongmei Li
  • Department of Microwave Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, 150001, China
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and Jinghui Qiu
  • Department of Microwave Engineering, School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, 150001, China
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Abstract

Owing to the potential to manipulate simultaneously amplitude and phase of electromagnetic wave, complex-amplitude holographic metasurfaces (CAHMs) can achieve improved image-reconstruction quality compared with amplitude-only and phase-only ones. However, prevailing design methods based on Huygens–Fresnel theory for CAHMs, e.g., Rayleigh–Sommerfeld diffraction theory (RSDT), restrict acquisition of high-precision reconstruction in a large field of view (FOV), especially in the small numerical aperture (NA) scenario. To this end, a CAHM consisting of Sine-shaped meta-atoms is proposed in a microwave region, enabled by a novel complex amplitude retrieval method, to realize large FOV holograms while breaking the large NA limitation. Calculations and full-wave simulations demonstrate that the proposed method can achieve superior-quality holograms, even for nonparaxial holograms in a relatively small NA scenario, thus improving FOV and aperture utilization efficiency of CAHMs. The reconstruction comparison of a complex multi-intensity field distribution between CAHM prototypes designed by our method and by RSDT further confirms this point. We also compare both theoretically and experimentally the CAHM by our method with the phase-only metasurface by weighted Gerchberg–Saxton algorithm. Superior-quality holograms with suppressed background noise and relieved deformation, promised by the extra amplitude manipulation freedom, is witnessed. Finally, due to its wavelength irrelevance, the proposed method is applicable to the entire spectrum, spanning from microwave to optics.

1 Introduction

Metasurfaces, a two-dimensional modality of metamaterials, have received extensive attention owing to their extraordinary capabilities to control amplitude, phase, and polarization of electromagnetic waves flexibly [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Among fascinating applications, metasurface hologram is receiving growing interests since fabrication of numerous subwavelength meta-atoms on a large aperture effectively eliminates undesired diffraction orders and thus provides unprecedented resolution for constructed images compared to traditional holograms based on spatial light modulators which may suffer from low resolution, small viewing angles, and undesired high-order diffractions [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. The marriage of computer-generated hologram and the metasurface platform further opens a bright avenue to construct multifunctional holographic system with exceptional image qualities. Recently, phase-only holographic metasurfaces (PHM), which engineer only the phase information in the hologram plane and assume the amplitude constant, have received most of the attention. Such a phenomenon mainly attributes to the availability of iterative optimization methods, among which Gerchberg–Saxton (GS) [38] is the most widely adopted approach for the propagation phase [39], [40], geometric phase [41], [42], [43], or Huygens’s metasurface design [44], [45], [46]. A dynamic weighting coefficient was further introduced to improve the robustness of the GS (GSW) [47], [48]. Advanced methods have been continuously proposed based on the GS to realize interesting applications, e.g., multiplexing [32], [49], [50], [51], spin or orbital angular momentum [32], [52], and nonlinearity [53], [54], [55]. However, due to the omission of amplitude information, PHM does not fully liberate the metasurface’s ability to regulate electromagnetic waves, and the realized hologram may suffer from noises in the preset area and image deformities, resulting in compromised imaging quality. Similar defects also exist in amplitude-only holographic metasurfaces (AHM) [56], [57], [58], [59], [60], [61]. Subsequently, with the emergence of the meta-atoms capable of simultaneously modulating the local phase and amplitude information, metasurfaces can be applied to perform the complex-amplitude holograms [35], [45], [62], [63], [64], [65], [66], [67]. Due to the additional degree of freedom in controlling the wave fronts, the complex-amplitude hologram metasurface (CAHM) is expected to exhibit superior imaging quality beyond PHM and AHM. However, such a perception has not yet been theoretically or experimentally demonstrated since the available synthesis methods for the CAHM fail to fulfill completely their potentials. The most widely used methods to synthesize CAHMs are based on the Huygens–Fresnel theory, e.g., Rayleigh–Sommerfeld diffraction theory (RSDT). It discretizes target fields as a limited number of point sources and calculates the diffraction fields of every point source by the RSDT, which are further superimposed in space to obtain amplitude and phase profiles in the hologram plane. Inevitably, the area ratio of the hologram plane to the imaging region in this case is required to be sufficiently large to collect dominating energies virtually radiated by the discretized point sources representing the target field, which corresponds to a large numerical aperture (NA) scenario. Otherwise, the truncation errors due to a moderate NA will severely deteriorate the imaging quality, especially for the target fields in the nonparaxial region. Thus, the prevailing RSDT-designed CAHMs tend to be large in dimensions, resulting in relatively bulky hologram systems, and nonparaxial image deformities are often encountered. Therefore, to achieve holographic images of higher quality in both paraxial and nonparaxial regions, it is of great demand to propose a novel methodology superior to the RSDT to improve the field of view (FOV) and break the large NA limitation for CAHMs. To this end, we propose a complex amplitude retrieval method, namely complex-weighted Gerchberg–Saxton (CGSW). The formulation of the proposed CGSW is first presented. Numerical experiments are then performed to demonstrate robustness and superior precision of the proposed CGSW regardless of the NA size and the imaging region (paraxial or nonparaxial). Furthermore, we propose a Sine-shaped meta-atom to physically realize the pixels in the hologram plane to construct the CAHM. Full-wave simulations and experiments are finally completed to verify the superiority of our CGSW and the corresponding CAHM, compared with the RSDT and the GSW.

2 Principle and formulation of CGSW and its subtle wave regulation capability

Figure 1 demonstrates the principle of complex-amplitude holograms (Figure 1a) and the proposed CGSW (Figure 1b). The marriage of our method with amplitude and phase engineering meta-atoms can accurately reconstruct complex fields. The CGSW method contains three types of iterations: phase iteration, amplitude iteration, and complex amplitude iteration, where the phase iteration and the amplitude iteration are categorized as the lower-level ones, while the complex amplitude iteration belongs to the top-level one. The hologram plane (represented by the purple bar in Figure 1b) is composed of P pixels, and the complex amplitude at every pixel is calculated by the CGSW method. The preset image (illustrated by the blue bar in Figure 1b) is discretized as Q discrete points, among which the amplitude distribution on the whole plane formed by the corresponding intensity of every point is defined as the preset value of the target field. After the initial amplitude and phase values of every pixel in the hologram plane are set as 1° and 0°, the first-round phase iteration (depicted by the yellow arrows in Figure 1b) is performed. The phase distribution on the hologram plane obtained from the last phase iteration is set as the initial phase value of the following first-round amplitude iteration process (portrayed by the purple arrow in Figure 1b), and this phase information remains during the amplitude iteration. In the first-round complex amplitude iteration, the phase information of the desired metasurface is obtained by the phase iteration, while the amplitude information is determined by the amplitude iteration. During every iteration, the calculated amplitude information in the imaging plane as shown in Figure 1b will be replaced by the preset amplitude distribution of the target fields to optimize the complex-amplitude information in the hologram plane. Meanwhile, in the hologram plane, a dynamic weighting factor defined subsequently is introduced for the same purpose. Therefore, by continuing the complex amplitude iterations, i.e., performing orderly and continuously the phase and the amplitude iterations, the nth complex amplitude distribution at the pth pixel in the hologram plane reads,
ApnejΦpn={|q=1Qejklp,qlp,qwqNAVqNA1|VqNA1||exp(jarg(q=1Qejklp,qlp,qwqnVqn1|Vqn1|))(Phaseiteration)|q=1Qejklp,qlp,qwqnVqn1|Vqn1||exp(jarg(q=1Qejklp,qlp,qwqNΦVqNΦ1|VqNΦ1|))(Amplitudeiteration)
where NA and NΦ represent the number of last amplitude and phase iteration, respectively, Q is the total number of the discrete points in the imaging plane, lp,q is the distance between the pth pixel and the qth discrete point, Vq denotes vector superposition of complex amplitude at the qth discrete point emitted from each pixel of the metasurface, and wq denotes the weighting factor and reads,
wqn=wqn1[Uqq=1Q|Vqn1||Vqn1|q=1QUq]g
where Uq is defined as the preset electric field amplitude at the qth discrete point, and g is the relaxation factor. What counts is by the innovatively introduced relaxation factor, the weighting factor can be adjusted flexibly according to the scale, which overcomes the dependence of the traditional GSW method on the phase initial values (Supplementary material S1). The index to evaluate the difference between the achievable electric field amplitudes in the imaging plane and the preset distribution is the sum-squared error (SSE), which reads
SSE=q=1Q(UqUmax|Vq||V|max)2q=1Q(UqUmax)2
where Umax and |V|max are the maximum values of Uq and Vq (q = 1 to Q), respectively. The SSE can be applied as a criterion for evaluating the convergence degree of the iteration, that is, when the SSE reaches the expected value and the variation trend of SSE with the number of iterations is stable, the entire iteration will be terminated. In addition, in lower-level iterations, the transition criterion from the phase iteration to the amplitude one and vice versa (depicted in gray around the lower middle part of Figure 1b) is fixed by
limn[q=1Q(UqUmax|Vqn||Vn|max)2q=1Q(UqUmax|Vqn1||Vn1|max)2q=1Q(UqUmax)2]0
Figure 1:
Figure 1:

Schematic diagram of the proposed complex-weighted Gerchberg–Saxton (CGSW) and complex-amplitude holograms. (a) The schematic representation of the proposed large field-of-view (FOV) complex-amplitude metasurface hologram. (b) The principle of the proposed CGSW. The input information includes the amplitude of each discrete point constituting the preset image and the initial value of the amplitude and phase in the hologram plane used as the initial value of the first phase iteration. The complex amplitude iteration can be decomposed into two processes of amplitude iteration and phase iteration, wherein the amplitude initial value or phase initial value used in the phase iteration or amplitude iteration is the final value obtained in the previous iterative process and remains unchanged during the iterative process.

Citation: Nanophotonics 9, 16; 10.1515/nanoph-2020-0448

To verify the proposed CGSW method, holographic imaging results of targeted “H”-, “I”-, and “T”-shaped field distributions by our CGSW with different NAs have been calculated and thus compared to those by the RSDT and by the GSW (shown in Figure 2a). It is noted that the preset amplitude profile of “H” is uniformly distributed, that of “I” piecewisely decreases from top to bottom, and that of “T” is uniform but the imaging position is in the nonparaxial region. All holographic metasurfaces in the theoretical calculation are composed of identical 34 × 34 pixels occupying an area of 408 × 408 mm2. The distance between the imaging and the hologram plane is 60, 200, and 300 mm, corresponding to NAs of 0.96, 0.71, and 0.56 at 10 GHz. The preset images remain for all three NA cases. It is noted that in microwave region, the physical dimension of available dielectric substrates is quite limited, and thus, the pixel number is here relatively small.

Figure 2:
Figure 2:

Comparison of image reconstruction capabilities of complex-weighted Gerchberg–Saxton (CGSW), Rayleigh–Sommerfeld diffraction theory (RSDT) and weighted Gerchberg–Saxton (GSW). (a) Under the condition that the NA is 0.96, 0.71 and 0.56, the three target images are reconstructed, respectively. The preset field distribution of the three target images is “H” with uniform amplitude distribution, “I” with gradient amplitude distribution with normalized intensity ratio 1:0.91:0.66:0.58, and “T” in the nonparaxial region, respectively. These three methods are used in turn to calculate the complex amplitude distribution or phase distribution in the hologram plane, and then, the reconstructed images in the imaging plane are obtained by Fresnel diffraction theory. (b) The schematic diagram of the preset field distribution of “I” as well as the field intensity distribution on the central axis of the reconstructed image obtained by three methods, respectively, when the NA is 0.56, where the black dotted line represents the ideal distribution of the preset field intensity. (c) The variation of SSE of CGSW (black curve) and GSW (blue curve) with the number of iterations when the NA is set to 0.71. The insert illustration shows the schematic diagram of the preset field distribution of ‘HIT1920’.

Citation: Nanophotonics 9, 16; 10.1515/nanoph-2020-0448

First, as shown in Figure 2a, the reconstruction result of “H” shows that the quality of the holographic image by the RSDT is limited by the NA. Moreover, the reconstructed “H”s acquired by the RSDT often suffer from the nonuniform field intensity distributions for all given NAs, difficult to eliminate by a denser discretization of the preset field (Supplementary material S2). On the other hand, the hologram performed by the GSW method breaks through the NA limitation, yet due to absence of the amplitude manipulation, the occurrence of stray energies outside the preset area in the holographic image seriously affects the sharpness and accuracy of the constructed image. In contrast, the CGSW method solves the problems inherent in the other two methods and performs high-quality holographic imaging even for a smaller NA, where cleaner, shaper, and more uniform images are achieved. It is noted that the reason why the line width in the holographic images becomes larger for decreasing NA is that the focal spot increases as the imaging distance becomes longer. Second, a more complex field distribution is selected as the target image to further reveal the high-precision characteristics of the CGSW. The preset amplitude distribution of “I” is divided into four parts with intensity ratio of 1:0.91:0.66:0.58, and the reconstructed image comparison indicates that only the hologram performed by the CGSW can distinguish the difference in intensity distribution as desired, allowing for more detailed and precise holographic constructions. Figure 2b shows that the distribution of the electric field intensity along the central axis of the imaging plane by three methods when NA = 0.56. The blue, green, and red solid lines represent the imaging results achieved at the target position by three methods, while the black dotted line represents the ideal distribution of the preset field intensity. It is illustrated that the CGSW provides the closest normalized intensity ratio to the ideal one. It is thus revealed that the greater flexibility of the CAHM in manipulating the wave front over the PHM and the AHM can be effectively liberated by the CGSW. Third, the comparison of the nonparaxial imaging capability is illustrated. The reconstructed nonparaxial “T”s show that the CGSW can restore the target image as well as in the paraxial cases, whereas the RSDT endures critical nonparaxial image deteriorations, more obvious deviating farther from the principle axis. This interesting yet a bit surprising phenomenon indicates that the CGSW can more efficiently utilize a metasurface aperture. To further illustrate this point, the comparison of the CGSW and the RSDT regarding their capability of generating a hologram larger than the area of the holographic metasurface will be demonstrated in section 3. Also, the serious noise interferences encountered by the GSW can be effectively inhibited. In general, it is demonstrated that our CGSW can reconstruct with supreme precision the preset fields even in the nonparaxial region, way outperforming the RSDT in boosting the aperture utilization efficiency and thus the miniaturization of holographic metasufaces.

It should finally be noted that the CGSW implements the optimization process for both the amplitude and the phase profiles of the hologram plane, making it possible for the first time to achieve a better complex-amplitude profile for CAHMs. Moreover, it will obviously improve the robustness and the convergence speed of the GSW, since there is an extra degree of freedom applied in optimization. Figure 2c clearly demonstrates this point. We perform a more comprehensive holographic imaging of “HIT1920” for a NA of 0.71 to show the superiority of the CGSW in holographic reconstruction of complex target fields, as detailed in Supplementary material S3. By comparing the SSE curves corresponding to two methods in the iteration process, the CGSW has more robust and quicker convergence and can achieve a much lower SSE. Its staircase convergence behavior indicates that the transitions from the phase iteration to the amplitude one (and vice versa) of the CGSW effectively advance the method in searching for a better solution, while the phase-only iterative process of the GSW appears rather chaotic. A closer observation on the SSE curve of the CGSW shown in Figure 2c reveals that the phase iterations is not stable as the amplitude ones, where similar SSE fluctuations as that of the GSW may be encountered. Such a phenomenon is attributed to the limited capability of the phase-only regulation approach, which again highlights the necessity of supplementing the extra amplitude regulation in the optimization process.

3 Ultrathin Sine-shaped meta-atom and its marriage with CGSW method

To numerically and experimentally demonstrate the validity of our CGSW and further implement physically the CAHM with a large FOV, a novel Sine-shaped meta-atom capable of simultaneously manipulating amplitude and phase of the transmitted wave front is proposed, whose operational mechanism is elucidated hereafter. The proposed Sine-shaped meta-atom will be applied to construct a series of proof-of-concept holographic metasurface prototypes, consisting of metal square lattices etched on a dielectric substrate. Among them, the metal square lattice has a size of 12 × 12 mm2 with a thickness of 0.035 mm, and a sinusoidal slit with a width of 1.2 mm exists in the center of the metal square lattices, as shown in Figure 3a. The thickness of the dielectric substrate is 4 mm with a relative permittivity of 2.85. The proposed Sine-shaped structure is designed to regulate the incoming electromagnetic wave with the wavelength of 30 mm. Thus, the arrangement period for both x and y directions of the meta-atoms is set to be 12 mm, roughly one third of the wavelength. It is noted that the Sine-shaped structure may potentially offer reduced mutual couplings since compared to other meta-atoms, its contour characteristics allow an effectively larger separation between adjacent meta-atoms. Such a meta-atom is also more feasible for fabrication and installation compared to the dielectric ones in the low frequencies. Similar to a birefringent crystal, the Sine-shaped meta-atom produces different refractive indices for two orthogonal electromagnetic waves, resulting in differences in transmittance and phase shift between two electric field vectors parallel to the fast and the slow axes, respectively, as illustrated in Figure 3b. Therefore, when the y-polarized electromagnetic wave having the angle θ with the symmetry axis of the Sine-shaped structure is normally incident on the meta-atom, the electric field vector of part of the incident wave is rotated by 2θ to the fast axis. After orthogonally decomposing the rotated electric field vector, the amplitude of the scattered wave with the x-polarized can be expressed as
Esca=σEin|cos(902θ)|
σ=12(AxcosΦxAycosΦy)2+(AxsinΦxAysinΦy)2
where 2σ denotes the cross-polarization conversion rate, and θ is the angle between the fast axis and the y-axis. Due to the onefold rotational symmetry and axis symmetry of the Sine-shaped structure, we can achieve the normalized transmission amplitude from 0 to 1 by varying the value of θ in the range of 0°∼45° (Figure 3e) or 90°∼135° (Figure 3g). In addition, the shape of the Sine-shaped slit is determined by the curve corresponding to the function y = Acos(0.86x), where x and y are the abscissa and ordinate values corresponding to the points on the curve in the local coordinate system with the center of the square lattices as the coordinate origin. The variation of A can lead to sinusoidal slits of different shapes to realize the transmission phase modulation under a linear excitation, governed by
Δφ=sinθcosθ|sinθcosθ|AxsinΦxAysinΦyAxcosΦxAycosΦy
Figure 3:
Figure 3:

Schematic diagram of meta-atoms. (a) Top view of the designed meta-atoms, where the red curve indicates the curve corresponding to the function y = Acos(0.86x), B = 1.2 mm, and E = 10 mm. Different values of A correspond to different element shapes. (b) The designed meta-atom produces different transmittance and phase shift between two electric field vectors respectively parallel to the fast axis and the slow axis. (c) When the y-polarized electromagnetic wave is normally incident, the amplitude and phase of the cross-polarized electromagnetic wave can be regulated by changing the rotation angle θ and A of the element. (d) The fabricated sample of the designed meta-atoms. The cross-polarization transmission amplitude and phase characteristics of different shaped elements when the variation range of θ is 0°∼45° (e, f) and 90°∼135° (g, h), respectively.

Citation: Nanophotonics 9, 16; 10.1515/nanoph-2020-0448

Therefore, we can achieve a phase shift covering 2π by varying A when θ varies in the range of 0°∼45° (Figure 3f) or 90°∼135° (Figure 3h), respectively. It is worth mentioning that as shown in Figure 3e–h when the value of A is changed to realize a phase shift covering 2π, the transmission amplitude of the scattered wave is almost unaffected. Moreover, the transmission phase is mainly determined by θ. Such a phenomenon greatly simplifies the meta-atom library construction. As a result, one can firstly achieve the desired transmission phase profile by varying A and thus fix the transmission amplitude by adjusting θ.

To further illustrate the meta-atom’s capacity to regulate the transmission amplitude and phase, and the advanced field reconstruction competence of the CGSW, we perform the full-wave simulation verification to reconstruct the “I”-shaped field distribution when NA = 0.56 shown in Figure 2a, for the synthesized CAHMs with Sine-shaped meta-atoms by RSDT and CGSW, respectively. Figure 4a and b shows that the simulation results of reconstructed fields are in good agreement with the theoretical ones. As aforementioned, only the reconstruction performed by the CGSW can distinguish the difference in intensity distribution as desired and achieve cleaner and shaper images. Furthermore, to verify that the CGSW can extend the FOV of CAHMs, where improved image reconstruction quality in a smaller NA scenario can be realized, we compare full-wave–simulated field reconstruction results of two CAHMs of the same aperture size and NA by RSDT and CGSW. The metasurface occupies an area of 408 × 408 mm2, whose outline is represented by the white dotted square in Figure 4c. The preset field consists of concentric square boxes, and the peripheral box covers an area of 612 × 612 mm2 with the line width of 12 mm, corresponding to 2.25 times the area of metasurfaces. The inner box has an area of 306 × 306 mm2 with the line width of 6 mm. The imaging plane is 300 mm away from metasurfaces, corresponding to a NA of 0.56 at 10 GHz. As shown in Figure 4c and d, the CAHM by RSDT can only reconstruct the whole inner box and lose particularly the information around four corners of the peripheral box, while that by CGSW can precisely reconstruct the whole target field. Results of PHM by GSW are shown in Supplementary material S3. The contrast of reconstruction results clearly shows the advantages of CGSW in extending the FOV of CAHMs by enhancing nonparaxial field reconstruction quality, even in the area larger than the metasurface aperture.

Figure 4:
Figure 4:

Simulation results of the image reconstruction by Rayleigh–Sommerfeld diffraction theory (RSDT) and complex weighted Gerchberg–Saxton (CGSW).

(a) and (b) Calculated and full-wave simulated reconstructed images of “I” by metasurfaces designed by RSDT and CGSW. (c) and (d) Calculated and simulated reconstructed images of the concentric square boxes by metasurfaces designed by RSDT and CGSW. The white dotted square represents the outline of metasurfaces and the insert illustration shows the preset image.

Citation: Nanophotonics 9, 16; 10.1515/nanoph-2020-0448

4 Experiment results and discussion

Finally, a series of proof-of-concept metasurface prototypes in microwave region are fabricated for experiments. An NA of 0.56 is selected to demonstrate the capability of the CAHM designed by CGSW to reconstruct hologram of the best quality even when the aperture is further miniaturized. As shown in Figure 5 and Supplementary material S4, totally three holographic metasurface prototypes are fabricated, i.e., the CAHM by CGSW, the CAHM by RSDT, and the PHM by GSW. The proposed holographic metasurface with four-level amplitude and twelve-level phase discretization consists of 34 × 34 Sine-shaped meta-atoms. The dimension of metasurfaces is 408 × 408 mm2 (larger dielectric substrates are not commercially available), and the distance between the imaging plane and the hologram plane is 300 mm, corresponding to the NA of 0.56 at 10 GHz. In the preset image, the preset images of “H,” “I,” “T,” ‘1,” “9,” “2,” and “0” are different with intensity ratio 1:2:1:2:1:2:1. Figure 5 demonstrates the theoretically (a–c), full-wave-simulated (d–f), and experimentally (g–i) reconstructed images by three holographic metasurfaces. The results show that the CGSW-designed CAHM overcomes the shortcomings of fragmentary image reconstruction of the RSDT-designed CAHM especially in the nonparaxial region. It also improves the issues of large amount of stray energy distributions outside the preset area, incomplete image details, and poor image quality in the reconstructed image by the GSW-designed PHM. Moreover, the fine details such as preset intensity ratio are witnessed to be far better reproduced than the other two cases. In addition, the CGSW displays a stable convergence rate, compared with the chaotic one of the GSW. Such a good robustness greatly extends the applicable range of the CGSW. Small differences between the theoretical calculation and the simulation results are possibly induced by the mutual coupling between the actual meta-atoms and the discretization of amplitude and phase. Finally, three holographic metasurface prototypes designed by the CGSW, the RSDT, and the GSW are fabricated by etching the patterned meta-atoms on copper-clad dielectric substrates. Gold sinking technology is performed to enhance the antioxidant properties of the metasurfaces. The experiments are performed in the microwave chamber, which can significantly reduce the influence of reflected waves and noise interference. In the experiment, a linear-polarized feed horn with the center frequency of 10 GHz is placed at the distance of 3 m from the metasurface to meet the requirements that the plane wave of y-polarized is normally incident on the holographic metasurface. In addition, a rectangular waveguide is used as the near-field test probe to scan and collect the electric field information of the x-polarized scattered wave in the target plane, as shown in Figure 5k. Experiment results (Figure 5g–i) are highly consistent with the simulation ones, and the peak signal-to-noise ratio (PSNR), defined as the ratio between peak intensity of every character in the experimentally reconstructed “HIT1920” to the standard deviation of the background noise, is adopted here to evaluate the imaging quality. The preset images of “H,” “I,” “T,” “1,” “9,” “2,” and “0” are different with intensity ratio 1:2:1:2:1:2:1, and the maximum, the minimum, and the averaged PSNRs are found to be 21.81 (Character “2”), 14.74 (Character “T”), and 18.7, which further validate the CGSW and the advantages in manipulating the wave front of the CGSW-designed CAHM. It should be noted that the CAHM prototype by the CGSW restores most precisely the preset image compared to other two metasurface prototypes, implying its superior immunity for fabrication errors. Such a phenomenon further confirms the robustness of the proposed CGSW and the advantageous imaging quality of the CAHM. Finally, it is noted that the field reconstruction quality can further be enhanced by increasing the meta-atom number comprising the metasurfaces.

Figure 5:
Figure 5:

Reconstruction of “HIT1920” by CGSW, RSDT, and GSW when NA = 0.56. Preset amplitudes of “H,” “I,” “T,” “1,” “9,” “2,” and “0” are different with intensity ratio 1:2:1:2:1:2:1. (a)–(c) Theoretically reconstructed images obtained by three methods. (d)–(f) Full-wave–simulated reconstructed images by the holographic metasurfaces designed by three methods. (g)–(i) Experimentally reconstructed images by the holographic metasurfaces designed by three methods. (j) Fabricated CGSW-designed CAHM. The inset is a zoomed-in view of the fabricated prototype. (k) The experimental environment for holographic imaging. The inserts are zoomed-in views of the linear-polarized feed horn and the near-field waveguide-probe.

Citation: Nanophotonics 9, 16; 10.1515/nanoph-2020-0448

5 Conclusion

In summary, complex-amplitude metasurfaces consisting of Sine-shaped meta-atoms are proposed in microwave region, enabled by a novel complex amplitude retrieval method, realizing a large FOV hologram while breaking the large NA limitation. Theoretical calculations are firstly conducted to preliminarily validate the proposed CGSW method. Full-wave simulations as well as proof-of-concept experiments for three holographic metasurfaces composed of novel Sine-shaped meta-atoms further verify the advantages and robustness of this method. It not only overcomes the shortcomings of the RSDT, a popular design method for the CAHM requiring the NA of the metasurface to be large enough but also solves the issues such as large stray energy distribution in the GSW, the most widely applied and advanced design method for the PHM. The proposed CGSW-designed CAHM can be utilized to reconstruct the most complete information of preset fields with the minimum requirement of the metasurface size, which may represent a step forward towards practical implementations of miniaturized large FOV and high-definition holographic system. It should be finally noted that the CGSW is a universal tool for the entire spectrum, spanning from low frequencies up to optical ones, while the proposed Sine-shaped meta-atom can also easily be reproduced in higher frequencies.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Grant nos. 61671178, 61301013, 61731007, and 62071152).

Author contributions: The manuscript was written through contributions of J. Qi, and Y. Mu. All authors have given approval to the final version of the manuscript. J. Qi conceived the idea of this manuscript. J. Qi, and Y. Mu together formulated and programed the CGSW, proposed the Sine-shaped meta-atom, fabricated the prototypes, and designed the experiments. Y. Mu and M. Zheng conducted the measurements. H. Li and J. Qiu provided valuable advice on the mauscript.

Research funding: This paper is supported by the National Natural Science Foundation of China (Grant nos. 61671178, 61301013, 61731007, and 62071152).

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Abbreviations

2D

two-dimensional

FOV

field-of-view

NA

numerical aperture

CGH

computer generated hologram

AHM

amplitude-only holographic metasurface

PHM

phase-only holographic metasurfaces

CAHM

complex-amplitude holographic metasurface

RSDT

Rayleigh–Sommerfeld diffraction theory

GS

Gerchberg–Saxton

GSW

weighted Gerchberg–Saxton

CGSW

complex weighted Gerchberg–Saxton.

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Footnotes

Supplementary Material

Dependence of the GSW on initial phase values; Influence of discrete point densities on imaging results of RSDT; Calculated and full-wave-simulated reconstructed images by the metasurface designed by GSW; Fabricated RSDT-designed and GSW-designed metasurfaces.

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0448).

If the inline PDF is not rendering correctly, you can download the PDF file here.

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    Schematic diagram of the proposed complex-weighted Gerchberg–Saxton (CGSW) and complex-amplitude holograms. (a) The schematic representation of the proposed large field-of-view (FOV) complex-amplitude metasurface hologram. (b) The principle of the proposed CGSW. The input information includes the amplitude of each discrete point constituting the preset image and the initial value of the amplitude and phase in the hologram plane used as the initial value of the first phase iteration. The complex amplitude iteration can be decomposed into two processes of amplitude iteration and phase iteration, wherein the amplitude initial value or phase initial value used in the phase iteration or amplitude iteration is the final value obtained in the previous iterative process and remains unchanged during the iterative process.

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    Comparison of image reconstruction capabilities of complex-weighted Gerchberg–Saxton (CGSW), Rayleigh–Sommerfeld diffraction theory (RSDT) and weighted Gerchberg–Saxton (GSW). (a) Under the condition that the NA is 0.96, 0.71 and 0.56, the three target images are reconstructed, respectively. The preset field distribution of the three target images is “H” with uniform amplitude distribution, “I” with gradient amplitude distribution with normalized intensity ratio 1:0.91:0.66:0.58, and “T” in the nonparaxial region, respectively. These three methods are used in turn to calculate the complex amplitude distribution or phase distribution in the hologram plane, and then, the reconstructed images in the imaging plane are obtained by Fresnel diffraction theory. (b) The schematic diagram of the preset field distribution of “I” as well as the field intensity distribution on the central axis of the reconstructed image obtained by three methods, respectively, when the NA is 0.56, where the black dotted line represents the ideal distribution of the preset field intensity. (c) The variation of SSE of CGSW (black curve) and GSW (blue curve) with the number of iterations when the NA is set to 0.71. The insert illustration shows the schematic diagram of the preset field distribution of ‘HIT1920’.

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    Schematic diagram of meta-atoms. (a) Top view of the designed meta-atoms, where the red curve indicates the curve corresponding to the function y = Acos(0.86x), B = 1.2 mm, and E = 10 mm. Different values of A correspond to different element shapes. (b) The designed meta-atom produces different transmittance and phase shift between two electric field vectors respectively parallel to the fast axis and the slow axis. (c) When the y-polarized electromagnetic wave is normally incident, the amplitude and phase of the cross-polarized electromagnetic wave can be regulated by changing the rotation angle θ and A of the element. (d) The fabricated sample of the designed meta-atoms. The cross-polarization transmission amplitude and phase characteristics of different shaped elements when the variation range of θ is 0°∼45° (e, f) and 90°∼135° (g, h), respectively.

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    Simulation results of the image reconstruction by Rayleigh–Sommerfeld diffraction theory (RSDT) and complex weighted Gerchberg–Saxton (CGSW).

    (a) and (b) Calculated and full-wave simulated reconstructed images of “I” by metasurfaces designed by RSDT and CGSW. (c) and (d) Calculated and simulated reconstructed images of the concentric square boxes by metasurfaces designed by RSDT and CGSW. The white dotted square represents the outline of metasurfaces and the insert illustration shows the preset image.

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    Reconstruction of “HIT1920” by CGSW, RSDT, and GSW when NA = 0.56. Preset amplitudes of “H,” “I,” “T,” “1,” “9,” “2,” and “0” are different with intensity ratio 1:2:1:2:1:2:1. (a)–(c) Theoretically reconstructed images obtained by three methods. (d)–(f) Full-wave–simulated reconstructed images by the holographic metasurfaces designed by three methods. (g)–(i) Experimentally reconstructed images by the holographic metasurfaces designed by three methods. (j) Fabricated CGSW-designed CAHM. The inset is a zoomed-in view of the fabricated prototype. (k) The experimental environment for holographic imaging. The inserts are zoomed-in views of the linear-polarized feed horn and the near-field waveguide-probe.