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Abstract
The operation of near-field and far-field can be employed to display holographic and nanoprinting images, which significantly improves the information density. Previous studies have proposed some approaches to display the images independently or simultaneously, but cannot satisfy these two characteristics in a single structure under the same incident light. Here, a single layer multifunctional metasurface is proposed to display a nanoprinting image and a holographic image independently and simultaneously. By tailoring the dimensions of each nanobricks and adopting different orientation angle, the amplitude and phase can be artificially designed. Moreover, enabled by the simulated annealing algorithm, we take the impact of both amplitude and phase of each nanobrick into consideration, which eliminates the unnecessary influence of amplitude on holographic image. Compared with previous work, our metasurfaces markedly improve the quality of holographic image with simple structures while not affecting the nanoprinting image. To be exact, it breaks the coupling between the near-field and far-field, achieving independent and simultaneous control of both fields. Our proposed metasurfaces carry characteristics of simple manufacture, little crosstalk, and great compactness, which provides novel applications for image displays, optical storage and information technology.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Near-field and far-field are important concepts in wireless communication, antenna design and information multiplexing [1–3], which are employed to analyze the response of signals in different regions. The near-field is the region with a distance less than one wavelength from the device while the far-field is the region where the propagation distance of light is much larger than one wavelength [4]. According to Fresnel diffraction theory, there are inherent interrelation and restriction within near-and far-field [5–8], leading to the far-field inevitably influenced by the amplitude control in near-field pattern. Therefore, the arbitrarily regulation of far-field and near-field is beset with difficulties. With the increase of information density, breaking this coupling relationship to independently and simultaneously control near-field and far-field have become urgent needs in information technologies.
With unprecedented capability of manipulating electromagnetic fields, metasurfaces composed of two-dimensional (2D) nano arrays have paved a new path for independent and simultaneous control of near-field and far-field [9–22]. In terms of image display, near-field and far-field usually display nanoprinting image and holographic image respectively [23–29]. To date, lots of approaches have been proposed to encode both holography and nanoprinting into a single metasurface, which can display images in both fields [30–39]. According to different physical properties, these proposed approaches can be divided into three main aspects: orientation degeneracy, structurally birefringence and interference of double-nanoblock cell (DNC). For oriented degeneracy, benefiting from the combination of propagation phase and amplitude based on Malus laws, holographic image can be obtained in far field under circularly polarized(CP) light and nanoprinting image can be obtained under a linearly polarized (LP) beam passing through a polarizer respectively [40]. For structurally birefringence, the phase and amplitude are independently designed by the modulation of birefringence and orientation angle of nanostructure, so as to generate nanoprinting and holographic images [41]. For DNC, the nanoprinting intensity is continuously controlled by adjusting the rotation angle difference between two atoms while holographic image is reconstructed by orientation of one atom [42].
In summary, previous studies have constructed specific light field distribution in near-field and far-field with different artifices. However, most of the solutions unavoidably accompany with crosstalk between the two fields. Besides, the solutions achieving complete decoupling for both fields require two different polarization source. It is difficult to independently display images in two fields under a light source with the same polarization state, which increases the complexity of application.
Here, we proposed a dual-channel metasurfaces to independently and simultaneously control far-field and near-field with high quality under CP light. As for near-field, crystalline silicon nanobricks with different geometric sizes are utilized to control amplitude. As for far-field, we design the orientation angles of the nanobricks to modulate the phase obtained by Simulated Annealing algorithm(SAA). Normally, the holographic image is determined by phase while the amplitude in near-field needs to be as consistent as possible to guarantee it does not affect the holographic image. Here, the near-field amplitude is used as an input item to optimize phase of the hologram. Based on the target amplitude distribution, the proposed method optimizes a better phase distribution with the help of geometric phase modulation and SAA, which significantly decreases the constraint between far-field and nearfield. Hence, the holographic image in far-field is greatly improved and the amplitude in near-field also can be precisely modulated. This approach provides a simple and crosstalk free mode in independent and simultaneous controlling near-field and far-field, which can be applied in high information density or multichannel imaging.
2. Design and method
As shown in Fig. 1(a), with the capability of controlling amplitude and phase, the proposed metasurfaces can independently and simultaneously display a grayscale nanoprinting image and a holographic image in near-field and far-field respectively under CP light.
Fig. 1. a) Schematic diagram of decoupling far-field and near-field. When circularly polarized (CP) light is incident, the nanoprinting image and holographic image will appear in the near-field and far-field respectively. b) Schematic diagram of a metasurface unit-cell. The blue part represents the Si nano-bricks, and the white part is the SiO2 substrate. c) The simulated transmittance and the phase of cross polarized light with different orientation angle under CP light. The blue curve and the red curve represents transmittance and cross-polarized light phase respectively.
The gray level of nanoprinting image is determined by amplitude in near-field. Due to intrinsic loss in visible range, crystalline silicon is employed to be the material of nanostructures. The amplitude of crystalline silicon nanostructure is strongly dependent on the dimensions. Each unit cell is composed of a silicon nanobrick and a planar substrate (SiO2), as shown in Fig. 1(b). By adjusting the dimensions of nanobricks, the resonance effect can be controlled to create different transmittance. Therefore, if a grayscale nanoprinting need to be encoded in metasurfaces, each meta-atom needs to be designed to match the corresponding gray level.
Next, we consider the phase modulation for recording holographic image. In the principle of the geometric phase, when the LCP light is incident on the metasurfaces, the complex-amplitude of output light can be written as [43]
We use FDTD Solutions (Lumerical Inc., Vancouver, BC, Canada) to scan the length and width of the silicon nanobricks and obtain the transmittance and the corresponding propagation phase. For boundary conditions, the x-axis and y-axis adopt periodic boundaries, and the z-axis adopts perfectly matched layer (PML). According to grating formula, the large cell sizes will cause high diffraction orders of the metasurfaces. However, small cell sizes will shorten the distance between each of nanobricks, leading to near-field coupling. We carefully choose the period of each cell to be 400 nm and the identical height is 600 nm. Then we scan the length of the nanobricks from 150 nm to 300 nm at a wavelength of 582 nm, and scan the width from 50 nm to 200 nm to obtain the corresponding propagation phase and transmittance.
To eliminate the interference of co-polarized part of light and improve diffraction efficiency, we screened the nanobricks with the highest polarization conversion rate (PCR) at each gray level. For the purpose of analysis, Fig. 2(a) shows the transmittance and phase delay between x axes and y axes in each transmittance we selected, which is tuned to the half-wave-plate condition. The transmittance covers 10 gray levels, so as to better restore the gray level of the image. The proportion of co-polarized component and cross-polarization component in each gray level under CP light are shown in Fig. 2(b). Since the co-polarized component is very small, its interference to holographic image can be approximately ignored. The details of geometric parameters, transmittances and propagation phases are listed in Fig. 2(c). By matching the gray level of the near-field image with the transmittance of nanobricks, the near-field image can be encoded into metasurfaces.
Fig. 2. Each gray level parameters. (a)Transmittance and phase delay between x axes and y axes obtained by scanning the length and width of the nanobricks in each transmittance. Δφ represents the phase delay between x axes and y axes. (b)Proportion of light power in Co-polarized light and Cross-polarized light. (c) Geometric parameters and the corresponding transmittances and propagation phases of the selected nanobricks.
With the aim of reconstructing the desired holographic image, computer-generated holograms (CGH) were carefully designed. For the hologram, the phase φ(x, y) is given by Gerchberg–Saxton(GS)algorithm and Simulated Annealing algorithm (SAA). SAA is a random optimization algorithm which can obtain the global optimal solution by probability jumping out of the local optimal solution. As most common phase optimization algorithm, GS works well when optimizing phase of hologram with consistent amplitude. However, due to the uneven amplitude in near-field, the phase obtained by GS can bring blur of far-field holograms. Unlike GS, SAA optimizes different phase distributions for different near-field amplitudes, which significantly reduce the blur in far-field. By probably accept new solution, the algorithm can jump out of the local optimal solution and gradually approach the global optimal solution. The details of the SAA optimization can be summarized as the following steps. First, adding random phase perturbations at random points based on the phase distribution φ(x, y) obtained by GS algorithm and calculate the diffraction results by combining amplitude distribution U(x, y) and phase distribution φ(x, y). Taking the phase obtained by GS algorithm as the initial solution of SAA can reduce the optimization time. Then judge whether the diffraction results are better than the previous solution. If it is better than the previous solution, it can be replaced as new solution. Otherwise, it has a specific probability to jump out to be new solution. Lastly, the probability of jumping out of the local optimal solution decreases with the increasing of iteration number. Repeat the above process until convergence.
Conventional SAA requires to compute new diffraction results for many times when new solutions are obtained. Here, we proposed an improved SAA to optimize the phase profile. The amplitude of the near-field is used as the input parameter to optimize the phase and the phase distribution obtained by GS algorithm is used as the initial solution.
Rayleigh Sommerfeld diffraction (RS) is used in the case of large angle exit. The formula of diffraction can be given by [27]
From the Eq. (3), the complex amplitude of far-field is obtained by the sum of the response of each point in near-field, which will take lots of time to compute the CGH. To obtain the new results efficiently, the new diffraction result is calculated by the difference of response of a random point. For an instance, when a random phase perturbation eiθ is added at the point (x’, y’), the new results of diffraction can be calculated by
Fig. 3. (a)Objective function to evaluate the superiority of the solution. “RS” is the Rayleigh Sommerfeld diffraction formula for large angle outgoing light. “Corrcoef” is the correlation degree between the holographic image and the target image. (b)Flowchart of the SAA for simultaneously display of near-field and far-field.
Next, we use geometric phase to encode the optimized phase obtained by SAA. The desired phase can be expressed as the difference between target phase φ2 and propagation phase φ1. Figure 4 shows the whole flowchart of detailed design. The first step is to scan the cell structure to select the appropriate cell dimensions to correspond to different gray values. Second, input the target image to match the gray value with the selected structure. Then obtain the required phase distribution φ2 through the SAA, and finally modulate the orientation angle to compensate the phase φ3 for the difference between the target phase φ2 and the propagation phase φ1.
3. Results
To verify our design, we employ a metasurface composed of 80×80 nanobricks and the overall size of metasurface is 32×32um. LCP light(582nm) is used to be incident light to illuminate the metasurface and set the distance of holographic image 30um. Figure 5 shows the simulated results for light intensity in near-field and far-field. Channel 1 is the nanoprinting image and channel 2 is the holographic image.
Fig. 5. The simulated results of near-field and far-field images. The first row shows nanoprinting and holographic images in sample 1 and the second row shows nanoprinting and holographic images in sample 2.
We choose the two badges as the nanoprinting images in sample 1 and sample 2, as shown in Fig. 5. The target holographic images in far-field are “A” and “B” respectively. As the results shown, the nanoprinting image and holographic image can be generated respectively in near-field and far-field when LCP light is incident. The simulated results indicate that the images in both fields is totally clear, which agrees well with our design.
Holographic diffraction efficiency can be defined as the ratio of holographic image light intensity to incident light intensity. At the design wavelength of 582 nm, the efficiency of holographic image is 24.86% in sample 1, and the efficiency of holographic image in sample 2 is 24.31%. For metalens with focal spot, based on definition, the numerical aperture (NA) can be expressed as
Where f is the focal length and d is the length of metalens. For two-dimensional hologram, f can be defined as the distance from the hologram plane to the metasurface plane. According to Eq. (5), the NA in our design is 0.4751.After that, we investigate the wideband characteristics of the proposed metasurfaces in the range of 562nm to 602nm. Due to the different required phase based on diffraction theory, the variation of wavelength will lead to the sharp decline of image quality. Therefore, holographic images do not have wideband characteristics. However, the near-field image is less affected by the disturbance of wavelength. Figure 6(a) shows the nanoprinting image in sample1 and sample2 when the wavelength ranges from 562nm to 602nm. It can be seen that at the range of 562 to 602nm, the quality of nanoprinting image still remains high though the transmittance is wavelength dependent. To further explore the performance of the nanoprinting, we adjusted the polarization of the light source and explore the imaging quality under different polarized light. As shown in Fig. 6(b), the image profile under different polarized light has some differences with target image but it still possible to be legible.
Fig. 6. (a)Simulation results of nanoprinting images in near-field with LCP light ranging from 562 nm to 602 nm in steps of 10 nm. (b) The nanoprinting image under Right circularly polarized light(RCP), X-Linearly polarized light(XLP) and Y-Linearly polarized(YLP) light.
4. Discussion
The proposed metasurfaces shows the ability of controlling the far and near-fields under a same LCP light. First, the phase distribution obtained by GS algorithm ignores the effect of amplitude in near-field while the holographic image is controlled by complex amplitude. In our design, benefiting from SAA optimization, the phase distribution can be designed correctly in the case of uneven amplitude, which significantly improves the clarity of holographic image. In the numerical calculation, the correlation between the diffraction results and the target image is 0.84 by GS, while the correlation can reach 0.98 in the case of “GS + SAA”. Figure 7(a) shows the comparison between the holographic images obtained by GS and by “GS + SAA” in the case of the same nanoprinting image. Obviously, holographic image obtained by “GS + SAA” works better in far-field. In addition, our improved SAA greatly decreased the computing time when calculating diffraction results. Other optimization algorithms such as particle swarm optimization algorithm or genetic algorithm require to do more changes to the previous solution to obtain suitable new solution and need to recalculate the diffraction result with Eq. (3) while the improved SAA only requires to calculate the response of a single point to obtain new solution. Therefore, the improved SAA we design shows great advantages compared with other optimization algorithms.
Fig. 7. (a) Holographic image at z = 30um obtained by GS algorithm and holographic image obtained by “GS + SAA”. (b) Average transmittance value from 562 nm to 602 nm at each gray level and desirable value. The blue curve is the average transmittance from 562 nm to 602 nm. The red curve is the desirable value in each gray level.
Secondly, in the near-field, we take transmittance as the screening standard of gray scale while the previous work used half-wave-plate with different amplitude responses. Though we still tune each nanobrick to be half-wave-plate, actually we use transmittance as the grayscale standard, which better restores images without polarization filter.
At last, the nanoprinting image has broadband characteristic, which improves the robustness of the designed metasurfaces. We analyze the wideband characteristics in each gray level and find that the average transmittance value from 562 nm to 602 nm is near the desirable value, as shown in Fig. 7(b). Actually there are no violent resonance in these nanobricks and the transmittance is steady near 582 nm. Therefore the near-field image can work well over a broad wavelength range.
Despite these advantages, the demonstrated metasurfaces have some flaws. Due to the inherent loss of silicon material at visible wavelength, the diffraction efficiency of the far-field is low. In truth, it can be attributed to the transmittance of cross polarized light that is always lower than the total transmittance in near-field. As a result, the metasurfaces with high gray level nanoprinting image in near-field can increase the efficiency of holographic image. Besides, it can be seen that the holographic image has the background noise caused by co-polarization component. This is because the phase of co-polarization component is not controllable under LCP light. By increasing PCR or set a polarization filter to select RCP component, the image quality can be significantly improved. In addition, NA is an upper limit to the spatial frequencies in a metasurface hologram. Due to the sampling theorem, a higher NA metasurface encodes a wider range of spatial frequencies and the metasurface behaves like a conventional grating, which results in the loss of high frequency components. In other words, there are still rooms for further improvement of image quality by increasing NA.
5. Conclusion
In summary, we combine the modulation of the transmittance and the phase to demonstrate a metasurface with independent control of near-field and far-field under a same LCP light. With the help of the angular insensitivity of the CP light transmittance, the orientation of each nanobricks can be artificially designed to modulate the phase. Then we consider amplitude into the optimization of phase and calculate the hologram with SAA. As a result, the displayed holographic image exhibits a high-quality result compared with the previous work obtained by GS algorithm. The proposed metasurfaces provide a new functionality that independently controls the near field and the far field with a simple structure under the same incidence of CP light, which significantly reduces the complexity of design. With the superior characteristic mentioned above, our proposed metasurfaces can be employed to achieve nano-scale information encryption, information storage, multi-channel display, etc.
Funding
National Natural Science Foundation of China (61774062, 61875057, 11674109, 11674107); Natural Science Foundation of Guangdong Province (2021A151501035); Guangzhou Municipal Science and Technology Project (2019050001).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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