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Abstract
Computer-generated holography typically generates terahertz (THz) holographic images with a pixel size larger than wavelength. We propose a multi-foci metalens model to reconstruct THz holographic images with subwavelength resolution. The designed devices are realized based on dielectric metasurfaces consisting of silicon micropillars with spatially variant orientations. By exploiting quasi-continuous profile of focal points as the pixels of a holographic image, a metalens can reconstruct a high-resolution target image on its focal plane. The effects of size and pitch of each sub-diffraction focal point on imaging quality and pixel resolution are discussed. The intensity distribution at each focal point indicates that the reconstructed images have subwavelength resolution. In comparison with conventional hologram designs, this design method can be used to reconstruct THz holographic images with subwavelength resolution, which have potential applications in THz communication, information security and anti-counterfeiting.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) holograms have great potential applications in many research fields, including THz communications, information storage, and image encryption [1]. Although spatial light modulators can be used to develop THz holograms [2,3], these THz spatial modulation devices are still insufficient due to the weak response of natural materials in the THz range [4]. Dielectric lenses can achieve holograms through phase accumulation, but they are not suitable for THz on-chip integration due to the large volume [5]. Metasurfaces, two-dimensional counterparts of metamaterials, can be artificially designed to respond to the THz wavelength range at subwavelength scale, providing a new platform for the development of THz metasurface devices [6]. Benefiting from unique properties of metasurfaces, many unusual THz devices have been developed, such as holograms [1,4], beam deflectors [7,8], metalenses [9,10], and vortex beam generators [11,12] in the THz range.
So far, computer-generated holography (CGH) is a widely used method of digitally generating holographic images [13,14]. Most metasurface holograms in the THz range have been designed based on the CGH method [1,15–19]. Various THz holograms have been developed for different applications such as wavelength multiplexing [15,16], polarization multiplexing [1,17], dynamic thermal control [18], and image transforming operations [19]. Encoding and decoding of THz holograms have also been studied [20], which is of significance for information security and anti-counterfeiting. Compared with only phase-modulated metasurface holograms [4,21], the simultaneous implementation of amplitude modulation and phase modulation can achieve higher resolution [22]. However, the resolution in the abovementioned THz holographic images is not at the subwavelength scale. To further develop THz techniques for more practical applications, how to reconstruct holographic images with a subwavelength resolution is urgently needed.
Recently, the unprecedented capability of metasurfaces has led to the development of THz metalenses with unusual functionalities, including spin-selective metalenses [23,24], multi-foci metalenses with polarization rotation functionality [9,10], asymmetric metalenses [25], and active metalenses [26]. Inspired by these works and the fact that a lens can achieve a focal point with a subwavelength size, we build a metalens model to tackle the resolution challenge in the conventional THz metasurface hologram designs. Unlike a conventional lens, which cannot reconstruct a holographic image, the THz devices we design based on the metalens model can reconstruct THz holographic images on their focal planes. Benefiting from our unique design method, each focal point corresponds to a pixel of the holographic image, leading to the subwavelength resolution. By exploiting quasi-continuous profiles of focal points, several THz holographic patterns with subwavelength level pixel resolution are reconstructed. The resolution and quality of THz holographic images on the focal plane are analyzed. Finally, THz holographic patterns on the different z-planes in the transmission direction are presented. Our method has potential applications in many fields such as THz communication, information security and anti-counterfeiting.
2. Principles and methods
According to Fermat’s principle, the phase profile required by a typical lens is given as
Initially, a multi-foci metalens can be realized by designing each phase profile for one focal point in different regions [23]. Since multiple phase profiles can be integrated into a single phase profile, sub-regional design is no longer required [27,28]. This suggests that the phase profile of a metalens can generate more focal points. The combined phase profile for a multi-foci lens can be described by
where ${\varphi _j}(x,y) ={-} 2\pi {f_0}\left( {\sqrt {{f^2} + {{(x - {x_j})}^2} + {{(y - {y_j})}^2}} - {f_{\textrm{D}j}}} \right)/c$. (xj, yj) are the coordinates of the jth focal point on the focal plane z = f, and ${f_{\textrm{D}j}} = \sqrt {{f^2} + x_j^2 + y_j^2}$, where j = 1 to n (positive integer number).Each pixel in the target image corresponds to a focal point, whose coordinates are (xj, yj), which are included in Eq. (2). A holographic image is reconstructed on the focal plane after an incident THz beam passes through the metalens.
Here, we utilize a dielectric geometry metasurface to achieve the phase profile of the metalens. The dielectric metalens consists of silicon micro-pillars with spatially variant orientations. Each micro-pillar in the metasurface can be considered as a half-wave plate, which can generate a Pancharatnam-Berry (PB) phase of ${\pm} 2\phi (x,y)$, where $\phi (x,y)$ is the angle between the long side of micro-pillar and x-axis, and the sign of the phase is determined by the helicity of the incident beam. Therefore, the unit structure orientation profile needs to satisfy $\phi (x,y) = \Phi (x,y)/2$ to achieve the desired function of the metalens to generate a holographic image. In addition, due to the use of the PB phase, each ${\varphi _j}(x,y)$ in Eq. (2) can only converge for left-handed circularly polarized (LCP) THz wave but diverge for right-handed circularly polarized (RCP) wave. Thus, a metalens designed according to Eq. (2) can only reconstruct the holographic image consisting of multiple focal points under the illumination of LCP wave [29].
The schematic of the metalens structure is shown in Fig. 1(a). The converted RCP can be reconstructed into a holographic pattern under the illumination of LCP wave. The height, length, width of each silicon pillar and the period of one unit cell are optimized to be h1 = 500 µm, l = 55 µm, w = 20 µm and p = 70 µm. The finite element method is used to simulate one unit cell of the whole structure under the periodic condition to calculate the polarization conversion efficiency of circularly polarized wave. A silicon substrate with the height h2 = 500 µm is also simulated, and the permittivity of silicon is ɛSi = 11.9. The calculated circularly co-polarized and cross-polarized transmittances are given in Fig. 1(b). At the frequency of 0.95 THz, the circularly cross-polarized transmittance can be as high as 98.8%, while the co-polarized transmittance is less than 0.1%. The circular polarization conversion efficiency tCross / (tCross + tCo) is more than 99.9%, meaning that each unit cell can be considered as an excellent half-wave plate at the operating frequency of 0.95 THz. Thereby, in the following metalens design and simulation, the operating frequency f0 is chosen to be 0.95 THz (corresponding to λ0 = 315.79 µm). At this frequency, when the LCP plane wave shines on the metalens at normal incidence, the majority of the LCP wave is converted into RCP with an additional phase $\Phi (x,y)$, and the incident wave can be focused at each focal point on the focal plane to form a holographic image. The Fresnel-Kirchhoff diffraction integral method is used to evaluate the far-field reconstructed patterns, in terms of the intensity distributions on the focal plane.
Fig. 1. (a) Schematic of the dielectric metalens consisting of silicon pillars with spatially variant orientations to generate the required PB phase. (b) Calculated co-polarized (blue line) and cross-polarized (red line) transmittances.
3. Results and discussion
Since the multiple focal points generated by a metalens are used as the pixels of the holographic images, the size and diffraction situation of a focal spot represent the resolution and quality of an individual pixel in a holographic image, which are related to the focal length f and clear aperture D of the metalens. The clear aperture of metalens is D = pN, where N is the phase pixel number along the diameter of the metalens. Figure 2(a) presents the states of a single focal point generated by the metalenses with different focal lengths f and different phase pixel numbers N. It can be seen in Fig. 2(a) that when the phase pixel number remains the same, the size of the focal spot can increase with the focal length of the metalens. In contrast, for a fixed focal length, the size of the focal spot can decrease as the phase pixel number of the metalens increases, meaning that clear aperture is increased. Such dependence on the focal length and clear aperture is compatible with the classical formula of the lens focal spot diameter d0 = 2λ0f / D. At this time, the focal spot contains 86.5% of the focusing energy. When the focal length is a constant, the larger clear aperture can make the diffraction more severe, resulting in more concentric ring fringes on the focal plane. In order to make the pixel points have a more uniform intensity distribution, the center intensity is chosen to drop to 1/e2 (13.5%) as the focal spot radius instead of the focusing energy containing 86.5%, which makes the focal spot diameter d slightly smaller than d0. Figure 2(b) shows the relationship between the normalized size d/λ0 and the focal length f with different phase pixel numbers N. To explain the relationship between d/λ0 and N within the appropriate range, the focal lengths of f = 5 mm and 10 mm are typically taken in Fig. 2(c). In the case where f is constant, the focal spot diameter d is inversely proportional to the phase pixel number N. As the phase pixel number N continues to increase, the d approaches to the limit of 0.63λ0. Taking the focal points as the pixels of a holographic image, the size of the focal spot determines the pixel resolution. Meanwhile, the quality of the holographic image should be taken into consideration. The cost of highest resolution is that the diffraction becomes more serious, which affects the quality of the holographic image. A trade-off between the resolution and the quality of the holographic image is necessary. Consequently, two key factors of the focal length f = 15 mm and the phase pixel number N = 500 are used to obtain the focal point at the wavelength level without excessive diffraction.
Fig. 2. (a) Normalized intensity distributions at the focal plane of the metalenses with different combinations of focal lengths f and phase pixel numbers N. (b) Relationship between the normalized size of the single focal point and the focal length with the phase pixel numbers N = 100 (black line), N = 300 (red line), N = 500 (blue line), and N = 1000 (green line), respectively. (c) Relationship between the normalized size of the single focal point and the phase pixel number for metalenses with focal lengths f = 5 mm (blue line) and f = 10 mm (green line).
Next, to achieve a better holographic image quality, the distance between the centers of two focal points plays an important role. Figure 3(a) presents the focusing states of two focal points on the x axis generated by the metalenses with different center distances Δ between the two focal points, and Fig. 3(b) presents the corresponding normalized intensity distributions along the line that passes through the two focal points (y = 0 mm). When the center distance Δ is greater than λ0, the two focal points can be clearly distinguished. But the intensity in a single pixel is not uniform, with obvious cutoff between two pixels. When the center distance Δ = λ0, the ratio of the maximum value to the minimum value of the intensity is 1: 0.738, which is close to the resolvable intensity ratio of energy receiver 1: 0.735. However, the pixels are square while the focal points are circular. Taking the two focal points used as the pixels, at the two corners where the two pixels are connected, the intensity is weaker with the sense of discontinuity. When the center distance Δ = 0.95λ0, the continuity between the two focal points treated as square pixels has been significantly improved. As the center distance Δ continues to decrease, the two focal points are no longer distinguished due to the severe overlap. The continuous and non-overlapping arrangement of pixels with uniform intensity is a prerequisite for the formation of holographic images with better quality. Therefore, in the following metalens designs for generating holographic images, the center distance of adjacent focal points is chosen as Δ = 0.95λ0. It can also reconstruct holographic images with a subwavelength pixel resolution. In addition, when the two focal points along the x direction are relatively close, there are two intensity points along the y axis, which is due to the coherent superposition of the respective diffraction fringes of the two focal points. However, the intensity of the two points is much weaker than that of the two focal points along the x direction, which is acceptable for reconstructing holographic images.
Fig. 3. (a) Normalized intensity distributions of two focal points along the x axis generated by the metalenses with different center distances Δ between the two focal points. (b) Corresponding normalized intensity distributions along the line passes through the centers of two focal points (y = 0 mm).
After the coordinates of all the pixels in a holographic image are determined, with the selected f0 = 0.95 THz, f = 15 mm, N = 500, and Δ = 0.95λ0, the phase profile of the metalens that generates the holographic image can be obtained based on Eq. (2). In Figs. 4(a)–4(f), various THz holographic images are displayed on the focal plane of the metalenses after LCP wave passes through the metalenses, including a regular triangle with a side length of 10 mm, a square with a side length of 10 mm, a regular hexagon with a side length of 6 mm, a ring with a radius of 6 mm, a regular pentagram with a circumscribed circle radius ro of 6 mm (the inscribed circle radius ri = (1 + tan218°)ro / (3–tan218°)), and a word “THz”. It can be seen from Fig. 4 that by exploiting quasi-continuous profile of focal points as the pixels of holographic image, the metalenses can reconstruct the target image on its focal plane and work well. These holographic patterns are realized by designing the phase profiles of the metalenses instead of optimization algorithms and using the focal points as the pixels of holographic images. The pixel resolution of the THz holograms can reach subwavelength level, in contrast to the previous THz metasurface holograms based on optimization algorithms [1,15–19].
Fig. 4. Normalized intensity distributions on the focal plane of the metalenses to present the THz holographic patterns, including (a) a regular triangle, (b) a square, (c) a regular hexagon, (d) a ring, (e) a regular pentagram, and (f) the word “THz”.
To evaluate the holographic image consisting of focal-point pixels, metalenses with the number of focal points reduced by half are designed, corresponding to each metalens in Fig. 4 respectively. In Figs. 5(a)–5(f), the same patterns as in Figs. 4(a)–4(f) can be observed on the focal plane, but only half of the focal-point pixels are used, showing the formation of dot plots. By selecting the focal-point positions at intervals to achieve the phase profiles, these focal points can be clearly seen on the same focal plane of the metalenses, which can explain that the patterns in Fig. 4 are composed of focal-point pixels. This design method based on metalens model can reconstruct high-resolution THz holographic images, which renders this technology very attractive for security and anti-counterfeiting when high-resolution THz holographic images are needed.
Fig. 5. Normalized intensity distributions on the focal plane of the metalenses to present the THz holographic dot images with only half number of the focal-point pixels in Fig. 4. (a) a regular triangle, (b) a square, (c) a regular hexagon, (d) a ring, (e) a regular pentagram, and (f) the word “THz”.
The effects of the phase pixel number N and the focal length f on the quality of holographic images are further discussed, based on the selected N = 500 and f = 15 mm. Take the ring holographic pattern in Fig. 4(d) as an example, the normalized intensity distributions on the focal plane f = 15 mm of metalenses with different N are presented (Fig. 6(a)). The corresponding radial normalized intensity distributions are shown in Fig. 6(b), in which the red dashed line represents the normalized intensity of 0.135. It can be seen from Fig. 6 that the resolution can increase as the phase pixel number, but the undesirable diffraction tends to be serious as the phase pixel number increases. Thereby, the resolution can be improved but with the serious side effects of the surrounding diffraction, which is detrimental to the quality of the holographic image. It can get the same conclusion as the previous analysis for a single focal point.
Fig. 6. (a) Normalized intensity distributions to present the effect of the phase pixel number N on the THz holographic ring pattern generated by metalenses. (b) Corresponding normalized radial intensity distributions of the THz holographic ring pattern. The red dashed line represents the normalized intensity of 0.135.
The normalized intensity distributions on the different focal planes of metalenses with the same phase pixel number N = 500 are presented (Fig. 7(a)). The corresponding radial normalized intensity distributions are shown in Fig. 7(b). It can be seen that for the metalenses with the same clear aperture, the higher resolution of the holographic image can be obtained as the focal length decreases. The focal length cannot be infinitely reduced. It is unable to complete the convergence effect under a small focal length, as shown in Fig. 7(a) with f = 5 mm. When the focal length becomes large, the intensity of all diffraction fringes is less than 13.5% of the central intensity, as shown in Fig. 7(b) with f = 25 mm, which can reduce the resolution of the holographic image. Therefore, the use of metalenses to achieve holograms requires a trade-off between the holographic resolution and the holographic quality.
Fig. 7. Normalized intensity distributions to present the effect of the focal length f on the THz holographic ring pattern generated by metalenses. (b) Corresponding normalized radial intensity distributions of the THz holographic ring pattern. The red dashed line represents the normalized intensity of 0.135.
Finally, the imaging region where the holographic patterns are generated by metalenses is discussed. Here, we take the word pattern “THz” in Fig. 4(f) as an example. Figure 8(a) presents the normalized intensity distributions in the transmission direction on the planes of y = –2 mm, y = 0 mm and y = 2 mm, respectively. The planes correspond to the lower, middle, and upper boundaries of the word pattern “THz”, respectively. It can be seen from Fig. 8(a) that the upper, middle and lower boundaries of the word pattern “THz” are clear and the focal points on the focal plane z = 15 mm has a small focal depth. As shown in Fig. 8(b), the normalized intensity distributions on different z-planes are given, and the holographic word pattern “THz” is the clearest on the focal plane z = 15 mm. At a plane 0.5 mm away from the focal plane, the pattern information can be observable but severely blurred, especially at z = 14.5 mm. When it is 1 mm away from the focal plane, the pattern information cannot be distinguished no matter before and after the focal plane, which is related to the metalens model.
Fig. 8. Normalized intensity distributions of the THz wave on different planes of the metalens that can generates the holographic pattern “THz”. (a) Normalized intensity distributions on different x-z planes, including y = –2 mm, y = 0 mm, and y = 2 mm, respectively. (b) Normalized intensity distributions on x-y planes of z = 14 mm, z = 14.5 mm, z = 15 mm, z = 15.5 mm, and z = 16 mm, respectively.
4. Conclusion
A metalens model is proposed to design THz holograms that can reconstruct subwavelength resolution holographic images. Unlike typical metasurface hologram design methods, the quasi-continuous focal points generated by a metalens are used as the pixels of holographic images, which feature subwavelength resolution. The image quality and pixel size of the simulated holographic images render such design method very attractive for realizing higher resolution THz holographic images, which are of importance in THz communication, information security and anti-counterfeiting.
Funding
Program for New Century Excellent Talents in University (NCET-12-0844); Natural Science Foundation of Hebei Province (A2019502044); Fundamental Research Funds for the Central Universities (2019MS118).
Disclosures
The authors declare no conflicts of interest.
Data availability
The datasets presented in this paper are available from the corresponding authors upon reasonable request.
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