Optimized Catenary Metasurface for Detecting Spin and Orbital Angular Momentum via Momentum Transformation

Abstract

Theoretically, the topological charge l in the vortex can be any integer or fraction, thus the vortex carrying different topological charges can form an infinitely orthogonal orbital angular momentum state space, which can provide new dimensional resources for optical communication. However, high-capacity optical communication requires low delay, thus real-time detection of the OAM is significant for communication. Metasurfaces have the characteristics of low loss, ultra-thin, easy integration, and flexible phase controls, which provide a meaningful way to realize integrated OAM generation and detection. Here, an optimized streamlined metasurface (OSM) is presented, which can detect high-order vortex beams in a single, simple, and rapid manner by photon momentum transformation (PMT). Since different vortices are converted into focusing modes with distinct azimuthal coordinates on a transverse plane through PMT, a single measurement can determine OAMs in an ample mode space. In addition, the OSM can detect more and higher order OAMs compared with a discrete metasurface (DM) at the same size, due to its better wavefront sampling capabilities. With the merits of an ultra-compact device size, simple optical structure, and outstanding vortex recognition ability, our approach may underpin the development of integrated optics and quantum systems.

1. Introduction

There are two main types of angular momentum: spin angular momentum (SAM) [1,2] based on polarization vector rotation and orbital angular momentum (OAM) [3] based on wavefront phase structure rotation. In 1992, Allen et al. [4] precisely deduced that the orbital angular momentum in optical vortices comes from the phase factor of $\exp (il\varphi )$, where l is the topological charge, $\varphi$ is the azimuth angle, and each photon carries orbital angular momentum of $l\hslash$. Since then, OAM has drawn increasing attention in many realms, including super-resolution imaging [5], quantum information [6], optical communications [7,8,9], and signal processing [10,11,12,13], which is one of the important topics for SAM and OAM recognition. Moreover, different OAM modes can transmit the information as separate channels in a division multiplexing communication system, thus it is crucial to encode and decode different OAM modes separately. Therefore, detecting topological charges and identifying different OAM modes play a vital role in the application of vortex beams. Therefore, detecting topological charges and identifying different OAM modes play a crucial role in the application of the OAM beams.

Generally, vortex beams can be manipulated in a variety of ways, such as spiral phase plates [14,15], glued hollow axicons [16], birefringent elements [17,18], orbital–angular-momentum holography [19], microscopic ring resonators [20], spatiotemporal dynamics of plasmonic vortices [21], spatial light modulators (SLMs), ultrathin and planar metasurfaces [22,23], peritrophic multiplexing metasurfaces [24], and azimuthal-quadratic diffraction optical elements [25]. OAM identification is challenging due to the missing phase information for traditional intensity detectors. The early OAM detection was proposed based on interference with the mirror or reference wavefront [26,27], diffraction of specific apertures [28], and mutual projection of forked diffraction grating [6]. Nowadays, a variety of vortices have been successfully detected, such as single-photon vortices [27], supercontinuum femtosecond vortices, noninteger vortices [28], and entanglement vortices [4]. However, multiple repeated projection measurements and calculating the number of interference fringes are inefficient for detecting possible OAM modes, especially for higher-order OAM modes [29].

In addition, the traditional vortex optical recognition system is often complicated and bulky, and the recognition method of the OAM is usually complex, which brings certain limitations to the practical application. A more straightforward system will lead to more excellent stability from an engineering point of view. In recent years, a variety of functional devices with excellent properties have emerged, such as metastructures [30], nanoparticles [31], graphene [32], plasmas [33], and multi-functional structures [34], with the rapid development of new technologies. In addition, with the same secondary phase form of a radial converging lens, a more convenient design has been proposed, namely, angular lens [35], which focuses different OAM modes to corresponding azimuth positions in the focal plane at a constant intensity. It can significantly reduce the optical system’s complexity, improve the system’s stability, and increase the identification range of the vortex topological charge.

Based on the principle of symmetry transformations via quadratic phase [36], Guo et al. used a spin-decoupling metasurface to realize the sorting detection of topological charges of multiple incidents of left and right vortices simultaneously [37]. However, composed of discrete unit cells, the spin-decoupled metasurface suffered from the problem of insufficient phase sampling, which reduced the detection range of the OAM. Here, an optimized streamlined metasurface (OSM) [38] was adopted to improve detection capabilities and achieve the same size to increase the detection range of the OAM, especially for higher-order OAMs. In addition, compared with the l₀ = 40 conducted by the discrete metasurface (DM) at present, the detection range of the OAMs by the OSM in this paper was improved by about 6.5 times.

2. Theory and Methods

The OSM is an optimized dielectric nanostructure of catenary [39,40]. Compared with the DM, the OSM can achieve continuous wavefront sampling in 0–2π, with improved diffraction efficiency and continuous phase distribution in a wider spectral band [38]. Here, based on the photonic spin hall effect [41] and catenary optics [22], an OSM structure was designed. It can transform different vortices in a large mode space into focusing modes of different azimuth coordinates in the transverse plane by photon momentum transformation (PMT), thus, OAMs can be determined through a single-shot measurement. PMT is an optical transformation that transforms the received OAMs’ mode to the focusing modes on the focal plane with different angular positions [35,37], which can be expressed as:

$$M(r,\varphi )=g(r)\exp (l_0{\varphi }^2/2)$$

(1)

where:

$$\{\begin{array}{c}g(r)=1\\ g(r)=0\end{array}\begin{array}{c}\\ \end{array}\begin{array}{c}{r}_i\le r\le r_o\\ otherwise\end{array}$$

(2)

$$\Phi (\varphi )=\frac{l_0}{2}{\varphi }^2$$

(3)

where M is the phase mask, l₀ is the quadratic phase coefficient, $\varphi$ is the azimuthal coordinate defined as tan⁻¹ (y/x), and r_i and r_o represent the inner and outer radius of the ring phase mask. By flattening the polar coordinates into Cartesian coordinates, the azimuth-quadratic phase contour can be converted into a cylindrical quadratic phase contour. $\varphi$ can also be expressed as:

$$\varphi =\frac{s}{s_0}$$

(4)

where s₀ = πr₀ and r₀ = (r_i + r_o)/2, respectively, and s is the focusing position of the detection position from the origin position [37]. Then, after the vortex light with topological charge l acts on the OSM, the phase distribution of the outgoing light is as follows:

$$\Phi (\varphi )+l\varphi =\frac{l_0}{2}{(\frac{s}{s_0})}^2+l\frac{s}{s_0}=\frac{l_0}{2s_0^2}{(s+\frac{l{s}_0}{l_0})}^2-\frac{l^2}{2l_0}=\Phi ({\varphi }^{\prime })-\frac{l^2}{2l_0}$$

(5)

where $\varphi \prime$ denotes a new azimuthal coordinate. Neglecting the azimuth-independent term on the right-hand side of the equation, we find that there is only a mode-dependent rotation through the PMT:

$$s=-\frac{l{s}_0}{l_0}$$

(6)

According to Equation (4), the theoretical focus ${\varphi }_l$ can be expressed as:

$${\varphi }_l=-\frac{l}{l_0}, {\varphi }_l\in [-\pi , \pi ]$$

(7)

Here, ${\varphi }_l$ is the radian corresponding to the focus and the origin (the focus position when the topological charge is zero) of the vortex light with the topological charge l passing through the OSM. Then, the focusing spot is symmetric around the azimuth coordinate $\varphi =0$. As the topological charge of the vortex increases from zero to the positive and negative ends, the focusing mode will take the zero point as the origin and rotate to both ends with an azimuth angle spaced l/l₀ apart. By recording the azimuth coordinates of the focusing mode, the received OAM mode can be determined as follows.

$$l=-l_0{\varphi }_l, {\varphi }_l\in [-\pi , \pi ]$$

(8)

According to equation $f=\frac{k_0{r}_0^2}{l_0}$ [37], where k₀ is the wavenumber in free space, r₀ = (r_i + r_o)/2, and the focal length is independent of the topological charge of the incident OAM mode. Therefore, different OAM modes can be detected in the same transverse plane of the focal length.

According to Equation (4), the detection of vortex optical topological charge is correlated with l₀, and l₀ is positively correlated with the phase gradient that the designed metasurface can accommodate. As shown in Figure 1a, the 2π periodic phase distribution within the metasurface changes rapidly from right to left, reducing the structural space that accommodates 2π phase adjustment. To ensure the phase control ability of DM, it is necessary for the metasurface retention space to accommodate sufficient nanopillars for 2π phase adjustment, which limits the maximum l₀ and the maximum detectable OAMs. However, the single streamline line of the OSM has the 2π phase adjustment ability in both large and small Λ periods. When the structure period gradually decreases, the OSM can still accommodate more columns, effectively improving the distinguishing power of DM and significantly increasing the detection range and maximum detection mode of the OAMs. Then, the OSM is an equal width streamlined metasurface optimized according to the catenary formula. The catenary equation [22] is as follows:

$$y=\frac{\mathsf{\Lambda }}{\pi }\ln (|\sec (\pi x/\mathsf{\Lambda })|)$$

(9)

In full-model design, there is no difference between the OSM and DM in essence, and both fill the corresponding structure according to the phase distribution. However, DM requires the interaction of multiple nanocolumns to adjust the 2π phase, while the OSM requires only one structure for the 2π phase condition due to its unique design, which has significant advantages when the space range is limited. In addition, with the feasibility of manufacturing guaranteed, better phase adjustment ability can be obtained, and the streamline line is truncated at 0.75Λ (Λ is the catenary span). A vertical structure with width w₂ was added to the truncated region to increase the streamlined metasurface optimized for phase continuity and amplitude uniformity. Simultaneously, the w₂ structure was canceled when the distance between two adjacent catenary lines in the circumferential direction was less than 0.1 rad. This design maximizes the coefficient l₀ while ensuring adequate wavefront sampling and fabrication feasibility, making it have practical application potential.

3. Results

3.1. Analysis and Simulation

The wavefront sampling of DM phase modulation is insufficient, and the detection capability of the OAMs is low, especially for higher-order OAMs. Here, we give explanations from theoretical derivation. If the phase coverage range of a single rectangular nanocolumn structure is $\frac{2\pi }{n}$ and the period is $p$, the difference can be expressed as:

$$|\Phi (s2)-\Phi (s1)|=\frac{l_0}{2s_0^2}\left|s_2^2-s_1^2\right|=\frac{2\pi }{n},s_2-s_1>p$$

(10)

From Equation (10), it can be deduced that:

$$|s_1|<\frac{2\pi s_0^2}{n{l}_{0}p}-\frac{p}{2}$$

(11)

According to Formulas (4) and (11), it can be expressed as:

$$|{\varphi }_{\max }^{\prime }|<\frac{2\pi s_0}{n{l}_{0}p}-\frac{p}{2s_0}$$

(12)

When $\left|\varphi \right|>\left|{\varphi }_{\max }^{\prime }\right|$, the topology detection cannot be realized by the structure.

As shown in Figure 1a, the vortex passes through the metasurface with ideal phase distribution and form focusing spots at corresponding positions in the focal plane. The focus spots are distributed in a circular direction, and the course rotates clockwise. Then, it can be inferred that the whole optical system is relatively simple, and the differentiation of the different OAM modes is also relatively simple and convenient. The topological charge of incident light can be inferred only by comparing the location of the focusing light spot.

Here, the differences between the OSM and DM in sorting OAMs are further studied. The separation of LCP vortex light was calculated through the OSM and DM phase plates with l₀ = 200. Then, for a better comparison, a numerical simulation of the OSM and DM at l = −620 to l = 620, Δl = 10 was conducted here, and the simulation results were statistically sorted out to form Figure 1b,c. As can be seen in Figure 1b,c, the number of the OAM modes detected by the OSM is much larger than that detected by DM structure. Figure 1c indicates the simulation results are consistent with Equation (12), i.e., according to the square result, when ${\varphi }_l$ exceeds ±1.36 radians, OAMs detection fails using the DM structure. However, due to the advantages of its structure, the OAMs’ mode can be effectively detected by the OSM structure when DM detection fails, which significantly improves the detection range of the OAMs. Furthermore, in the same detection range, the OSM can also generate stronger focusing spots than DM, as shown in Figure 1d (normalization is performed here for the convenience of comparison). According to the above information, the OSM has higher efficiency than DM for PTM, which makes it have a larger mode resolution range for OAMs, and improves the practical application value in related fields.

3.2. Model Establishment and Simulation

Then, an l₀ = 200 OSM was designed to ensure the ease of production, and the left circular polarization (LCP) OAM was simulated and sequenced at wavelength 633 nm. Figure 2a illustrates the three-dimensional structure of the OSM. The Λ of each column of the catenary in the OSM decreases gradually from right to left. Considering the detection ability of the OSM and the feasibility of the device fabrication, the l₀ was set to 200, and the corresponding minimum Λ value of the metasurface was about 300 nm. The OSM was designed as a streamlined array with H = 600 nm and interval d₁ = 700 nm, where w₁ = 100 nm was the optimized streamlined catenary. To ensure the feasibility of manufacturing, the streamline was truncated by d₂ = 0.75Λ, and vertical bars of width w₂ = 100 nm were added to the truncated region to increase phase continuity and amplitude uniformity. Furthermore, a ring OSM with r_i = 50 μm and r_o = 80 μm was selected as an example of a proof-of-concept demonstration. The opening of the streamline catenary in the upper semicircle of the metasurface was outward, while the opening of the lower semicircle was inward. Then, the streamline catenary Λ gradually decreases with the circumference from right to left, and the Λ position reaches the minimum at the left radius of the inner circle. In addition, TiO₂ was selected as the constituent material of the OSM due to its high refractive index and low loss at visible frequency, and it realized this larger aspect ratio processing in metalenses at visible wavelengths [42], meta-optics [43], and spin-decoupled metasurfaces [37]. Then, Figure 2b shows the ordering of the OAMs by PTM. The designed OSM transforms vortex beams with different topological charges into circular focusing modes in the transverse focal plane by Pancharatnam–Berry phase regulation. Thus, OAMs can be determined through a single measurement, giving them simple and fast sorting capabilities that may find potential applications in quantum communication.

Then, FDTD simulation was used to simulate the designed OSM, and the simulation results are shown in Figure 3a (the LCP is filtered, and the RCP is retained to obtain a sharper focused image). Here, the FDTD simulation results of the different OAMs with topological charges between −600 and 550 through the PMT were shown, where the adjacent interval Δl was 50. According to the simulation results, it can be seen that all the focusing modes under the same focal length are located on the same circumference, and the spot position rotates clockwise as the topological charge increases from negative to positive. Moreover, when the topological charge is in the range of l = ±2.5l₀, the focus spot is double-arc cross-focusing, and the intensity of the focus spot is stronger. Then, when the topological charge changes in the range of ±2.5l₀ ~±3.14l₀, the focus gradually becomes a single arc, and the intensity gradually decreases. However, the position of the focal spot corresponding to the theoretical value does not change, thus it is still effective for OAM detection in this range.

To verify the accuracy of the OSM simulation, the corresponding relationship between simulation results and theoretical calculated values is given here, as shown in Figure 3b. Then, we can obtain that the azimuth is linearly related to the topological charge, and the difference between the azimuth coordinates of FDTD simulation and the theoretical calculated values is not more than 0.001 rad. For the coefficient l₀ = 200 of the OSM, the azimuth interval between the intensity peaks of the two superimposed adjacent modes is 1/l₀ = 0.005 rad. Therefore, the topological charge can be unambiguously predicted as follows:

$$l_{\mathrm{predict}}=-l_0{\varphi }_l$$

(13)

Theoretically, the designed OSM can detect the intensity modes of 1256 OAMs’ modes with topological charges ranging from −628 to 628, indicating an excellent OAM mode detection ability. In summary, the OSM has a wide mode resolution range for single isolated OAMs. In addition, the position of the focusing spot formed by the OAM through the OSM has a very high coincidence degree with the theory, which is conducive to reverse inferring the topological charge of incident vortex light from the position of the focal spot in a practical application and dramatically improves the detection accuracy.

3.3. Resolution of Superimposed OAMs

The resolution of multiple superimposed OAM modes was further investigated here. As shown in Figure 4a, the overlap area of two adjacent focusing spots decreases with the increase in Δl, their mutual interference gradually decreases, and their resolution gradually increases. When the topological charge interval is seven, the intensity curve of two-phase OAMs reduces to 38.2%, and the two focal spots can be distinguished clearly. Based on the interval Δl = 7, 180 different OAM modes ((200 × 2π)/7 + 1 = 180) can be detected simultaneously on the circle, and the large mode detection capability is demonstrated. In addition, decreasing the size of the focusing spot or increasing the radius of the focusing ring is expected to improve the angular resolution, such as adding a super-oscillatory phase on the original metasurface or designing a helical focusing mode.

To verify the validity of the OSM detection on the whole circumference, in this paper, FDTD was used to simulate the single topological charge, and then all the single simulation results were superimposed together to obtain all the focusing modes, as shown in Figure 4b. To receive a more uniform focusing light intensity, the theoretical focal length ($f=\frac{k_0{r}_0^2}{l_0}$) is increased by 50 μm as a new detection focal length. In addition, the intensity of the single OAM is normalized to compare two adjacent focusing spots more conveniently. As is shown in Figure 4c, for OAM with different topological charges, the azimuth Angle difference between adjacent normalized intensity peaks is about 0.05 rad, and adjacent peaks are separated obviously. Then, it can be concluded that 1256 topological charges can be detected by the OSM when measuring a single topological charge. Moreover, the OAMs of 180 different topological charges can be detected simultaneously in multiple topological detections. This improvement in the detection efficiency indicates that the OSM can detect large-mode OAMs efficiently and quickly, suggesting the application of the OSM on integrated optical systems that require high-capacity communications.

4. Discussion

In the present article, we first investigate the theoretical differences between the OSM and DM, showing that the OSM is easier to function in space-limited situations due to the unique properties of its individual microstructure when the full-model size is equal. Then, the streamlined structure has greater sampling efficiency than the discrete structure, thus the vortex light forms a more concentrated focus spot after passing through the metasurface, which is beneficial to improve the mode resolution ability. When DM arranges the modes of the OAMs, there is a large space in the left part of the focusing plane that cannot generate the focusing spot, reducing the maximum detection range and maximum detection value. The area in the left part of the focal plane where focal spots cannot be generated is related to the single unit structure in DM and the l₀ designed in the full model, as shown in Equation (12). When the l₀ of the full model is constant, the period p of the unit structure is smaller, and the ${\varphi }_{\max }^{\prime }$ is larger.

However, this problem does not exist in the OSM, which can effectively use the space in the focal plane and form the focused spot throughout the circumference, as shown in Figure 4b. The above advantages make the OSM significantly increase the detection range and maximum detection value of the OAMs’ modes and do not reduce the mode resolution when multiple superimposed OAMs. It cannot resolve two superimposed vortices with a mode interval smaller than Δl = 5, and it is consistent with the described spin-decoupled metasurface [37]. Still, its designed l₀ is only 40, and the OAMs’ modes resolution range is within ±100 topology charge, which is far below the maximum mode detection range in this paper. Due to the characteristics of the catenary metasurface, it can only regulate LCP (or RCP) at the same time. The OSM is designed for incident vortex for LCP and is not suitable for the vortex of RCP in this paper. In addition, there are some limitations on the resolution of adjacent OAM, and strict alignment is also required during detection.

Then, we demonstrated a resolving power of Δl = 1 when only a single isolated OAM mode is received at a time, there was no cross-talk between adjacent modes. However, when multiple overlapping vortices are received simultaneously, the intensity overlap between them leads to reduced resolution power. As shown in Figure 4a, FDTD simulation results show that they cannot resolve two superimposed vortices with a mode interval smaller than Δl = 5, which means that at most 180 superimposed vortices can be simultaneously detected. Similar mode overlaps and resolution limitations have been observed in the angular lens-based OAM sorting methods [35]. Since the perimeter of the ring in which the focus spot is located in the focal plane is fixed, as is the size of the focus spot, the resolution can be further improved by decreasing the focus spot (adding a super-oscillatory phase) or enlarging the perimeter of the ring (designing it into a spiral arrangement pattern). In addition, a modified azimuthal-quadratic phase mask can also effectively improve the mode resolution power, which is expressed as ${\varphi }_l^{\prime }=-\frac{n^{2}l}{l_0}$ [37] and indicates that the azimuth difference of adjacent OAM modes becomes n² times the original one. The detectable mode range is proportional to $l_0/n^2$, while the resolving power is inversely proportional to $l_0/n^2$. With $l_0/n^2$ decreasing, the mode resolving power for superimposed vortices increases, but it reduces the range of detectable modes. Therefore, the coefficients l₀ and n can be selected in practical applications according to the requirements of detection range and resolution.

For broadband PMTs and vortex sorting, we carried out simple simulation verifications for four wavelengths of 480, 532, 633, and 780 nm, although the focal length is wavelength dependent, and the focal depth is long enough that we can choose a transverse plane across all the focal planes. The results are not significantly different from the spin-decoupled metasurface [37], and it is possible to simultaneously detect wideband vortices covering the entire visible band, which greatly increases its potential application value. For example, there are two main ways to use vortex light in optical communication systems: one is to use vortex beams of the different OAM states as different carriers to transmit information (OAM division multiplexing); another method is to use different OAM states in the vortex as symbols of information transmission for data transmission (OAM coding). Therefore, the system that can easily and quickly identify the large capacity of the vortex with different OAM states has important potential application value for increasing the capacity of optical communication systems.

5. Conclusions

For the OSM structure, a single streamlined line can cover 2π phase control while multiple units are required in the DM structure. Therefore, in the limited space of the metasurface, the OSM can arrange more columns than the DM to achieve larger phase gradient regulation, thus significantly increasing the detection range and maximum detection value of the OAM. Moreover, the OSM also improves the problem of insufficient wavefront sampling of the DM, thus the detection resolution of the superimposed OAMs is improved. Then, when the detection range of a single OAM is greatly increased, a certain resolution of multiple superimposed OAMs can also be maintained. In addition, similar to the DM, the OSM can simultaneously detect wideband vortices covering the whole visible band. Then, in addition to being ultra thin and ultra light, the OSM also has an overall ultra-small size, making it highly integrated. At the same time, due to the large mode resolution ability of the OAMs, they can easily and quickly resolve high-capacity information, which makes it a potential application value in the medium application of optical quantum computers.

Then, we investigated the mode resolution of a single OAM and multiple superimposed OAMs and demonstrated its huge OAM mode resolution power. Furthermore, the mode detection method of the OAMs is shown. The whole optical system is straightforward, and the mode resolution method is also straightforward and fast. To predict the topological charge of the vortex, it is only necessary to determine the location of the focusing spot formed on the focal plane after the vortex light passes through the metasurface. This simple and effective system will have high stability in the practical application, thus it can adapt to various working environments and increase its potential application value.

It is worth mentioning that more structure arrays can be arranged on the circumference of the metasurface by adding the inner and outer radii of the full model, i.e., a more significant phase gradient can be controlled, and the scope of the OAMs’ mode sorting can be further expanded. On the contrary, if the system needs to be integrated with limited space, the inner and outer circle radius of the metasurface can be reduced to meet the requirements of the system. Of course, part of the range of the OAMs’ mode sorting will be lost. Moreover, if the demand for multiple superimposed OAMs’ mode resolution is high, increasing the mode resolution power can be achieved by appropriately decreasing the coefficient l₀. The scheme is simple, practical, convenient, and flexible and can be applied to many application scenarios, such as large-capacity OAM communication and highly integrated optical quantum systems.

In summary, we showed a comparison of the OAM discrimination efficiency between the OSM and DM, proving that the OSM has higher mode discrimination efficiency than DM. Then, we have demonstrated a single geometric TiO₂ OSM-based OAM mode discrimination method via PMT of the azimuthal-quadratic phase metasurface, where vortex beams are converted into focusing modes rotating clockwise on the screen as topological charges increase. Finally, we showed that the proposed approach could be extended to the ordering of superimposed OAMs with a proper mode interval. The results show that the OSM can significantly increase the mode detection range of the OAMs but does not reduce the ability to rank multiple superimposed OAMs’ modes. These results may have many important applications in momentum measurement of spin and angular momentum and detection of phase and polarization singularities.