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Abstract
Traditional OAM generation devices are bulky and can generally only create OAM with one specific topological charge. Although metasurface-based devices have overcome the volume limitations, no tunable metasurface-based OAM generators have been demonstrated to date. Here, a dynamically tunable multi-topological charge OAM generator based on an ultrathin integrable graphene metalens is demonstrated by simulation using the detour phase technique and spatial multiplexing. Different topological charges can be designed on different focal planes. Stretching the OAM graphene metalens allows the focal plane and the topological values to be changed dynamically. This design method paves an innovative route toward miniaturization and integrating OAM beam-type photonic devices for practical applications.
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1. Introduction
Optical beams that carry orbital angular momentum (OAM), which feature doughnut-shaped intensity distributions, have become a hot research topic since they were discovered by Allen et al. in 1992 [1]. It is well known that optical beams that carry OAM have helical phase fronts with an azimuthal phase term of exp(ilθ), where θ is the azimuthal angle and l is the topological charge. This topological charge l can have any value between −∞ and +∞, and its sign determines the handedness of the OAM beam. Importantly, OAM beams that carry different integer topological charges are inherently orthogonal to each other, providing a new degree of freedom for information encoding. The special characteristics of OAM beams have enabled and boosted developments in many research areas and technologies, including optical data storage [2], optical communications [3], quantum information processing [4,5], global weather pattern research [6], super-resolution imaging [7], trapping [8], and optical tweezers [9,10].
Continuing development of OAM-based technologies such as optical communications and optical tweezers means an urgent need for a dynamic tuning technique for OAM, which will require a dynamic OAM generator. Currently, most OAM generators are static, including Q-plates [11], cylindrical lens mode converters [12,13], fiber mode couplers [14], spiral phase plates [15,16], fork gratings [17], bound states in the continuum (BIC) based photonic crystal [18], and parts of metasurface OAM generators [19,20]. These methods and devices have all demonstrated the successful generation of OAM-carrying beams and even multi-OAM-carrying beams. However, once these devices have been fabricated, their OAM is fixed, which greatly limits their use in devices and systems that require a dynamic tuning capability. OAM generators based on spatial light modulators [21,22] are one possible solution, but their bulk limits their use amid the trend for integrated and miniaturized optics. Recently, researchers have developed OAM generators that can tune the OAM topological charges or OAM modes dynamically using BIC-based photonic crystal and metasurfaces [23–26]. However, these devices are limited by their design and fabrication technologies. Their tuning ability is confined to the two orthogonal polarizations, or their tuning wavelengths are in the microwave band. Additionally, there has been no demonstration of tunable metasurface OAM generators for operation in the communication and visible bands. Therefore, dynamically tunable, integratable, and multi-topological charge OAM generation devices that can be fabricated with low-cost and simple processes are highly desired to enable the miniaturization and integration of optical instruments.
Recently, ultrathin diffractive devices based on graphene and graphene derivatives [27,28] have been attracting extensive research attention. The mechanical robustness, outstanding thermal stability, and chemical and biological stability of the graphene family of materials allow for the broad application of these materials, not only in daily life but also in extreme environments [29]. In addition, ultrathin graphene-based devices can be fabricated using low-cost and scalable one-step laser writing techniques [30,31], and both phase and amplitude manipulation [32,33] of their wavefronts can be realized simultaneously. These properties make these graphene and graphene derivative-based devices stand out from devices composed of other two-dimensional materials that have been developed to support miniaturization and integration of photonic devices. Recently, we demonstrated a varifocal graphene metalens [27] that was designed using the detour phase technique; the graphene metalens was fabricated on a stretchable substrate, a focal length tuning range of more than 20% was achieved, and zoom imaging of different objects using this metalens was demonstrated. In this paper, we propose and provide a numerical demonstration of a tunable graphene OAM metalens with multiple focal planes that combines graphene metalenses and spiral phase plates into a single design. Although this method can be applied to other two-dimensional materials, the graphene and graphene family materials are chosen to simulate the tunable OAM metalens because of their excellent properties and can be stably stuck to PDMS to realize stretchable devices [27]. Different OAM topological charges can be generated in different focal planes using the graphene OAM metalens, as illustrated in Fig. 1(a). Furthermore, as shown in Fig. 1(b), transverse stretching of the metalens causes new focal planes with different topological charges to emerge because of the resulting enhancement of the second-order diffraction, which gives this graphene OAM metalens the ability to modulate the OAM topological charges dynamically. The proposed ultrathin graphene OAM metalens will likely find a wide range of applications, including miniaturized or highly compact optical manipulation and communications devices.
Fig. 1. (a) Graphene OAM metalens with three focal planes that correspond to three different topological charges. (b) Transversely stretching of the graphene OAM metalens enhances the second-order diffraction, leading to the emergence of new focal planes with different topological charges.
2. Design method and simulation results
Figure 2(a) shows the design method for the multifocal graphene OAM metalens. The method comprises three steps, and the first step is to design the graphene metalens. The graphene metalens consists of concentric reduced graphene oxide (rGO) rings, which can be fabricated using the direct laser writing method to reduce graphene oxide film. The radius of each rGO ring can be designed by both the detour phase [34] and Fresnel methods [35], as the Fresnel method offers a higher focusing efficiency, we use it to design the converging phase component of graphene metalens in this paper. The radius of each rGO ring can be expressed as ${r_n} = \sqrt {n\lambda f + {n^2}{\lambda ^2}/4} $, where n is a positive integer, λ is the incident beam wavelength, and f is the designed focal length of the graphene metalens. Here, we design three graphene metalenses with different focal lengths. The three focal lengths were designed to be f1 = 100 µm, f2 = 200 µm, and f3 = 300 µm to reduce the mutual interference among their focal spots. The second step involves OAM phase loading on the graphene metalens. To generate an OAM with a topological charge l, the wavefront phase distribution should satisfy the relation $\phi ({x,y} )= l \times \textrm{atan}({y/x} )$, where the Cartesian coordinates are represented by (x, y). We use the detour phase technique to apply the phase to the graphene metalens. The detour phase technique was introduced by Brown and Lohmann in 1966 [36] and has been used widely in holography, optical signal processing, and lens design applications [34,37,38]. When using the detour phase technique, wavefront modification can be realized through slight variations in the locations of the rGO rings. Therefore, to realize a phase distribution of $\phi ({x,y} )$, the location shift of each rGO ring can be expressed using $\varDelta d = \phi ({x,y} )\times \frac{\lambda }{{2\pi \textrm{sin}(atan\left( {\sqrt {{x^2} + {y^2}} /f} \right)}}$. OAM with arbitrary topological charge numbers can also be added to the graphene metalens; here, we select l = 1, 2, and 3, corresponding to f1, f2, and f3, respectively, as examples. The third step involves the spatial combination of these OAM graphene metalenses. The graphene metalens with the shortest focal length was set in the inner zone, and the metalenses with longer focal lengths were set in the outer zones. There are three boundaries, designated r1, r2, and r3, in the combined graphene OAM metalens, as shown in Fig. 2(a). The areas of these three zones should also be equal to obtain three foci with equal intensities. Therefore, the relationship r1: r2: r3=$\sqrt 1 :\; \sqrt 2 :\; \sqrt 3 $ should be satisfied. Furthermore, when the density of the rGO rings in the different zones was considered, a slight optimization was applied. Therefore, under the condition that the width of the rGO rings was 0.4 µm, three zones with r1 = 128 µm, r2 = 168 µm, and r3 = 200 µm were applied to the three-focus graphene OAM metalens. It is also important to note that OAM metalenses with additional foci can be designed readily based on the combination method used here. In real applications, the Gaussian beams are usually used as the incident beam. Fractional OAM states will be created if there is off-axis illumination [39,40]. Therefore, an enlarged Gaussian beam should be used to avoid this situation. In this simulation, as the graphene OAM metalens is on micrometer-scale, the centimeter scale Gaussian beam can be approximated a plane wave considering the small size of our graphene OAM metalens. The wavelength of this plane wave is 633 nm. The thickness of the graphene OAM metalens is 200 nm. Figure 2(b) shows the intensity distribution on the y-z plane, which was calculated based on the Rayleigh-Sommerfeld diffraction theory [41]. Three focused doughnut patterns with nearly equal intensities are shown clearly and discretely. The accuracy of the focal plane locations at 100 µm, 200 µm, and 300 µm, as we designed, verifies the design accuracy of the proposed method. The intensity and phase distributions in the x-y plane (indicated by the white dashed lines) are shown in Figs. 2(c1) and c2, 2(d1) and d2, and 2(e1) and e2, respectively. The phase distributions in these three x-y planes verify the clear topological charge numbers of l = 1, l = 2, and l = 3, as designed. The radii of the three focused doughnut shapes increase with the increase in the topological charges as expected. Therefore, a three-focus graphene OAM metalens is designed theoretically.
Fig. 2. (a) Design schematic of graphene OAM metalens with three focal planes produced by spatial multiplexing. (b) Normalized intensity distributions of the three doughnut shaped foci in the y-z plane. (c1), (c2) Normalized intensity and phase distributions, respectively, of the focus at f1 with topological charge l = 1. (d1), (d2) Normalized intensity and phase distributions, respectively, of the focus at f2 with topological charge l = 2. (e1), (e2) Normalized intensity and phase distributions, respectively, of the focus at f3 with topological charge l = 3.
Dynamic tuning of the focal length and the OAM topological charge number is challenging and remains a hot research topic in the miniaturization and integration of OAM generation devices. Lateral stretching is one possible solution that does not require longitudinal movement, thus allowing the thickness of the entire device to be minimized. Our previous work [27] demonstrated a varifocal graphene metalens that used a stretching mechanism. Here, we use a similar methodology to study and summarize the focal length changes using the stretch ratio (SR) of the three-focus graphene metalens without the OAM phase being loaded. The focal length is expected to increase parabolically with increasing SR, as shown in Fig. 3(a). Generally, our design is based on the first diffraction order, meaning the first diffraction order constitutes the main focal spot. There are also higher-order diffractions but with a negligible contribution to focal spots. The first diffraction order is the strongest when the SR is small (i.e., < 1.2). But as the SR increases further, the ring spacing is also enlarged, which strengthens the higher diffraction order and results in new focal spots. As a result, it is possible to tune the topological charges of this generated OAM. When SR > 1.3, the second diffraction order becomes non-negligible, which contributes to a new focal spot. The latter simulation results show a two-fold increase of the topological charge, which confirms to our assertion that dynamic tuning of the OAM topological charge through a stretching mechanism. In addition, as the SR increase, the relative power of the second order becomes larger than the first diffraction order. As shown in Fig. 3(a) and (c), a clear focus at f′3 appears when SR = 1.3, and then the foci f′2 and f'1 emerge when SR = 1.4 and 1.6, respectively. These new foci represent the second-order diffraction of the OAM graphene metalens with the double-phase modulation profile of the OAMs. Figure 3(b) shows the designed structures for the three-focus graphene OAM metalens in the cases where the SR is 1.1, 1.5, and 2. The corresponding normalized intensity distributions are shown in Fig. 3(c). Only three doughnut-shaped foci within the focal region occur when SR = 1.1 (first row, Fig. 3(c)). In comparison, when SR = 1.5, higher-order doughnut-shaped foci appear at f′3 and f′2 in the new focal regions. In this particular design, the focus of the first order at f1 overlaps slightly with the second order focus at f′3 (second row, Fig. 3(c)). Then, by stretching the metalens further to SR = 2, the focus at f′1 appears (third row, Fig. 3(c)). It is also worth mentioning that, because of the different sizes of the focused doughnut profiles, the covered or hidden doughnut f1 appears in the three-focus OAM graphene metalens, as shown in Fig. 3(c), when SR = 1.5 and 2.
Fig. 3. (a) Focal length change plot versus different SR values for the three-focus graphene metalens. (b) Structures of the three-focus graphene OAM metalens when SR = 1.1, 1.5, and 2. (c) Normalized intensity distributions in the y-z planes for the three-focus graphene OAM metalens when SR = 1.1, 1.5, and 2.
The intensity and phase distributions in the x-y planes of these doughnut-shaped foci at f1, f2, and f3 for SR = 1.1, 1.5, and 2 are shown in Fig. 4. The figure shows that the radii of the doughnut-shaped foci occurring at f2 and f3 (Figs. 4(b) and (c) increase with increasing SR because of the reduction in the effective numerical aperture (NA) of the metalens. In addition, the topological charges remain unchanged, as shown in the phase distributions. However, a different phenomenon can be observed in the doughnut-shaped focus at f1 (Fig. 4(a)). When SR = 1.5, the topological charge of l = 1 can be found at the center of the focal region. However, a topological charge of l = 6 can be found surrounding the central area that is caused by the interference from the higher diffraction order, which corresponds to the newly generated focus at f′3, as explained above. By changing the interference condition via further stretching of the lens to SR = 2, the topological charge is then restored to l = 1 because of the separation of the foci at f1 and f′3.
Fig. 4. Normalized intensity and phase distributions on the x-y planes of the focused doughnut shapes. (a) The focused doughnut shapes and phase distributions on the x-y plane of f1 when SR is 1.1, 1.5 and 2, respectively. (b) The focused doughnut shapes and phase distributions on the x-y plane of f2 when SR is 1.1, 1.5 and 2, respectively. (c) The focused doughnut shapes and phase distributions on the x-y plane of f3 when SR is 1.1, 1.5 and 2, respectively. The topological charges remain at 1, 2, and 3, respectively.
To show the topological charges of the new foci, we perform further simulations of the intensity and phase distributions on the x-y planes of f′1, f′2, and f′3, as shown in Fig. 5. The results show that the topological charges of the three second-order foci at f′1, f′2, and f′3 are l = 2, 4, and 6, respectively, which are twice the values of the corresponding topological charges at f1, f2, and f3, respectively. Given that the structure of the graphene metalens is a type of grating structure, we consider the grating equation Δrsinθm = mλ, where Δr is the ring space between two adjacent rings, θm is decided by the corresponding radius of ring and focal length (tanθm =r/f), and m is the diffraction order. For the original graphene metalens, m = 1, f1, f2, and f3 are the corresponding first-order diffraction. And when a spiral phase distribution of ϕ(x,y) is added into the graphene metalens, the location shift of each ring is Δd=ϕ(x,y)λ/(2πsinθ1). When m = 2, namely the second-order diffraction, the relationship sinθ2 = 2sinθ1 can be deduced according to the grating equation, and then, according to the location shift equation, ϕ(x,y) becomes to two times its value when m = 1, which means the OAM state in the second-order diffraction will double its original OAM topological charge number in the first-order diffraction. Therefore, the high-order diffraction m will lead to high-order OAM states. It is the same principle of a fork-grating to generation OAM beams with a higher topological charge on its higher diffraction order. The evident intensity and phase distributions in Fig. 5 demonstrate the doubled OAM states brought by the second-order diffraction. Theoretically, if we can stretch the graphene OAM metalens larger, it is possible to obtain clear doughnut-shaped foci with the topological charge three times of the original OAM states. In this way, the topological charge of the OAM can be tuned using different diffraction orders, which can only be achieved uniquely by using the diffractive metalenses. Therefore, we numerically demonstrate in-situ tuning of topological charges through lateral stretching of ultrathin metalenses in simulation.
Fig. 5. Normalized intensity and phase distributions on the x-y planes of the newly generated doughnut-shaped foci f′1 (a1 and a2), f′2 (b1 and b2), and f′3 (c1 and c2) when the SR is 2. The topological charges are 2, 4, and 6, respectively.
As the focusing efficiency (FE) is one of the most important parameters in generating high-order OAM states, we calculate the FE of all the doughnut-shaped foci. The relationship between each doughnut-shaped focal spot’s FE with SR is plotted in Fig. 6. The FE here is defined as the doughnut intensity in the focal plane (4 µm × 4 µm) divided by the intensity immediately behind the corresponding annular part of the multifocal graphene OAM metalens. As the SR increases, the FEs of all the three doughnut-shaped foci f1, f2, and f3 decrease; The FEs of the three new doughnut-shaped foci f′1, f′2, and f′3 increase with the increase of SR. In general, considering the diffraction efficiency follows the rule that the inner part of the metalens contributes more efficiently than the outer part to the focused beam. and as the outer areas increase faster than the inner areas, more absolute intensity comes in from the outer regions. There is a trade-off between FE and total intensity. As it is impossible that the FEs of high-order diffraction can continuously increase, there will be a critical value of SR for the highest FE of each diffraction order, which should be considered in the dynamic multifocal graphene OAM metalens design.
3. Conclusion
In conclusion, a multifocal graphene OAM metalens has been designed by using spatially multiplexed single-focus graphene OAM metalenses with different focal lengths and topological charges based on the detour phase technique. The multifocal metalens can achieve doughnut-shaped foci with highly uniform intensity distributions at different focal lengths. The dynamic tuning of the focal lengths and the topological charges via lateral stretching was also studied theoretically. At small SR values, the focal length of each lens can be tuned without changing its topological charges. In addition, second-order foci with higher topological charges can also be generated when the SR is large. Therefore, by simulation, we numerically demonstrate a stretch-tunable multifocal graphene OAM metalens that can simultaneously vary its focal lengths and topological charges. This work opens up new possibilities for the design of OAM metalenses to meet different requirements. Furthermore, the ultrathin and flat features of the proposed graphene OAM metalens can be integrated readily into optical/photonic devices to generate and detect OAM beams simultaneously; this feature will find a broad range of applications. In addition, the new design method can be applied generally to the design of different OAM photonic devices with size miniaturization and integration capabilities.
Funding
National Natural Science Foundation of China (62175162, 62205214); Natural Science Foundation of Guangdong Province (2016A030310130); Foundation of Guangdong Education Committee (2020KTSCX117); China Postdoctoral Science Foundation (2021M692173); Science, Technology and Innovation Commission of Shenzhen Municipality (KQTD20170330110444030, KQTD20180412181324255); Science and Technology Foundation of Shenzhen City (20200814100534001).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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