## Abstract

As a spatial structured light field, the optical vortex (OV) has attracted extensive attention in recent years. In practice, the OV lattice (OVL) is an optimal candidate for applications of orbital angular momentum (OAM)-based optical communications, microparticle manipulation, and micro/nanofabrication. However, traditional methods for producing OVLs meet a significant challenge: the OVL structures cannot be adjusted freely and form a close-packed arrangement, simultaneously. To overcome these difficulties, we propose an alternative scheme to produce close-packed OVLs (CPOVLs) with controllable structures. By borrowing the concept of the close-packed lattice from solid-state physics, CPOVLs with versatile structures are produced by using logical operations of expanding OV primitive cells combined with the technique of phase mask generation. Then, the existence of OAM states in the CPOVLs is verified. Furthermore, the energy flow and OAM distribution of the CPOVLs are visualized and analyzed. From a light field physics viewpoint, this work increases the adjustment dimensions and extends the fundamental understanding of the OVL, which will introduce novel applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a structured light field, optical vortices possess a helical phase of exp(j*lϕ*) and an orbital angular momentum (OAM) of *l*ħ, where *ϕ* is the azimuthal coordinate, *l* is the topological charge (TC), and ħ is the reduced Planck’s constant [1]. In past decades, optical vortices have attracted extensive attention in numerous research fields, such as micromanipulation [2], optical imaging [3], quantum information [4–6], optical communications [7–11], and micro/nano-fabrication [12,13]. For these applications, an optical vortex lattice (OVL) is always involved, because it contains multiple optical vortices that provide additional information and flexibility [7,14–16]. Thus, the generation, detection, and verification of OVLs are of great importance in the related research.

In general, there are three typical methods for producing an OVL. Among these, Dammann grating, which can uniformly redistribute the intensity on different diffraction orders, is well-known as a good candidate [13,14,17,18]. Owing to the convenience of the modulation of each OV, this method is a popular technique for obtaining OVLs. However, the OVL generated via this method can only have a square structure, and the position of each OV is restricted by the diffraction rule. To realize an OVL with versatile structures, another popular method was proposed, involving the superposition of two or more special optical beams. For instance, different circular OVLs were formed via the superposition of Laguerre–Gaussian beams [19], Bessel beams [20], and perfect optical vortex beams [21]; rectangular OVLs were generated via Ince-Gaussian orthogonal-mode superposition [22,23]; arbitrary curvilinear arranged OVLs were reported via interfering two 2D curve laser beams [24]; and a random distribution of OVs was obtained via the superposition of three [25] and more [26–28] plane waves. Nevertheless, the distribution of OVs in these OVLs is nonuniform in the observed plane, and the TC of each OV is always a unit, i.e., TC = ± 1, which is restricted for the cases of applications. Third, an OVL was produced by designing specific microstructure materials using the topological defects in a nematic liquid crystal mesophase [29,30], chromophore nanolattices [31], holographic polymer-dispersed liquid crystal films [32], and self-assembled defects in smectic liquid crystals [33]. By exploiting the microfabrication technique, this method can yield smaller OVLs than the aforementioned methods. However, it is limited by the material structures, which result in complexity and difficulty in modulation once the materials are fabricated.

On the other hand, for some advanced applications, producing a close-packed OVL (CPOVL) with a controllable structure is a fundamental physical issue. For optical communications based on OAM, a trend few-mode multi-core fiber needs a hexagonal OVL [34,35]. For the fabrication of micro-functional chiral materials [12,13] induced by light illumination, a CPOVL with a controllable and versatile structure will provide flexibility and improve the microstructure density. For vast microparticles or particle cluster manipulation [36,37], a CPOVL will provide more adjustment dimensions and flexibility. Therefore, developing a novel method for the production of CPOVLs with controllable structures is a significant scientific task.

Herein, we propose an alternative scheme for producing a CPOVL with a controllable structure. Each optical vortex serves as a lattice element, and a CPOVL with the desired structure is obtained by a series of logical operations involving the expansion of OV primitive cells. By using the technique of the phase mask generation, CPOVLs with a controllable structure are experimentally realized, including square, rhombic, hexagonal, and triangular OVLs. Then, the existence of the spiral phase is verified, and the OAM states are determined via the phase-shift method. Moreover, the expansion of the OVLs and their intensity profiles are analyzed. Further, the properties of the energy flow and OAM distribution of the OVLs are visualized and analyzed under the close-packed arrangement condition. This work provides a new perspective for producing OVLs and will introduce fantastic applications.

## 2. Close-packed arrangement method

To produce close-packed arrangement OVLs with controllable structures, the challenge is accurately determining the position of each lattice element and simultaneously ensuring the compact arrangement. To solve this problem, we borrow the crystal lattice theory and close-packed arrangement concept from solid-state physics, considering each individual optical vortex element as a Bravais net. As demonstrated in Fig. 1, there are two typical close-packed arrangements: the square close-packed arrangement shown as Route I and the most close-packed arrangement shown as Routes II and III. For simplicity, each individual OV in the CPOVL has the same radius and ring width; i.e., the radius of each OV is independent of its TC.

For the square close-packed arrangement of Route I, the primitive cell (PC1) is constructed by the basic vectors ** a** and

**, and their angle is**

*b**θ*, which forms lattice coordinates, as shown in Fig. 1(a). By expanding PC1, larger lattices are easily obtained, such as 3 × 3 and 5 × 5 lattices, as shown in Figs. 1(b) and 1(c), respectively.

The relationship between the lattice coordinates (** a**,

**) and the Cartesian coordinates (**

*b**x*,

*y*) satisfies the following expression:

*d*is the length of the basis vectors. Here,

*d*= |

**| = |**

*a***|.**

*b**L*(.) is the matrix representing the coordinates of the lattice sites, which is obtained by permutations with repetition of the integers. The integer numbers are equal to those of the rows of the square lattice; for instance, the lattice coordinates of Fig. 1(b) are constructed by the integers 0, 1, and 2. The lattice coordinates and corresponding matrix

*L*are shown in Fig. 1(b). Similarly, the lattice coordinates of the 5 × 5 square close-packed arrangement are constructed by the integers in the range of 0-4. The Cartesian coordinates are deduced using the formula

*S*=

_{sq}*L*×

*T*, which is suitable for the square close-packed arrangement scheme.

For the most close-packed arrangement, two kinds of rhombic primitive cells are constructed, as shown in Fig. 1(d), which are represented as PC2 and PC3, respectively. Larger rhombic OVLs can be generated by expanding these. Controllable versatile structured lattices are obtained via logical operations of the rhombic lattices, such as a hexagonal OV lattice (Route II) and a triangular OV lattice (Route III). To produce the hexagonal lattices, as shown in Figs. 1(e) and 1(f), two rhombic lattices conduct the “AND” operation via superposition at different positions. Although the hexagonal lattice nets have different lattice coordinates, they have the same Cartesian coordinates after transformation by Eq. (1). By the same operation of two rhombic lattices, triangular lattices are obtained, as shown in Figs. 1(g) and 1(h). Consequently, the Cartesian coordinates of the hexagonal and triangular lattices satisfy the same expression: *S _{he}* =

*S*= (

_{tr}*L*×

*T*

_{1})∩(

*L*×

*T*

_{2}). In a different case, the “AND” operation is conducted with different spatial positions of these two rhombic lattices, i.e.,

*L*

_{1}(

*L*

_{2}) has different values. For encoding simplicity, the coordinate origins and lattice coordinates

*L*are selected to be different for hexagonal and triangular lattices. In general, by some logic-operation (including “AND”, “OR”, and “NOT”) combination of rhombic OV lattices, researchers can obtain a CPOVL with any desired structure.

Thus far, we have located the center position of each individual OV of the OVL. The following work aims to assign the desired OAM to each OV in the OVL. However, it is difficult to achieve this goal, as the radius of the OV bright ring increases with its TC. Fortunately, the perfect optical vortex (POV) [38], in which the radius of the POV ring is independent of its TC, is an optimal candidate to overcome this issue. This property of the POV ensures that the lattice element has a consistent radius and the desired OAM, simultaneously. Usually, the POV is produced via a Fourier transform of an optical vortex through an axicon [39]. After the generation of a POV as a lattice element, an OVL is obtained via repetition and displacement using the position matrix *S* (*S _{sq}*,

*S*,

_{he}*S*) mentioned above at the Fourier plane. This process is easily conducted by inducing a phase shift at the object plane, according to the shift theorem of the Fourier transform [40,41].

_{tr}To realize a CPOVL with a controllable structure, a phase mask of *Lattice* is designed via the superposition of a series of phase masks with different phase-shift amounts at the object plane according to the principle of independence of light propagation:

*ρ*,

*φ*) are the polar coordinates at the object plane,

*k*is the wavenumber,

*n*is the refractive index of an axicon,

*α*is the cone angle of the axicon,

*N*is the total number of the OVL, and

*S*

_{n}_{,1}and

*S*

_{n}_{,2}are the elements of the OVL position matrix

*S*, respectively.

For convenience, the phase of the axicon is embedded in the phase mask, as indicated by the first term of Eq. (2). The second and third terms indicate the spiral phase and phase-shift factor, respectively.

In this case, the theoretical expression of the CPOVL is deduced via combining the expanding single perfect optical vortex [39] and the shift theorem of the Fourier transform [40,41]:

*u*,

*v*) are the Cartesian coordinates at the Fourier plane,

*ω*= 2

*f*/

*kω*

_{g}is the Gaussian beam waist at the Fourier plane, and

*ω*

_{g}is the beam waist of the incident Gaussian beam at the object plane,

*f*is the focus length of the lens used in Fourier transform. In this case, the ring radius and width of each optical vortex are represented by

*r*and 2

_{n}*ω*, respectively.

## 3. Experimental setup

The generation process of the phase mask is illustrated in Fig. 2(a). The phase term of *Lattice* obtained by Eq. (2) multiplies the phase term of a blazed grating, and consequently the phase mask is produced and written into a reflective liquid-crystal spatial light modulator (SLM). A schematic of the experimental setup is shown in Fig. 2(b). A green solid-state Nd:YAG laser with 50-mW power is used as the light source. After passing through a pinhole filter (PF) and a convex lens (L1), the laser beam changes from a Gaussian profile to flat-top profile, approximately truncated by a circular aperture. Then, reflected by a beam splitter, the beam illuminates the SLM plane. The SLM is located between two polarizers of P1 and P2 to eliminate the parasitic light [38]. Finally, with another convex lens (L2, focus length of 200 mm), the OVL field is recorded using a charge-coupled device camera at the Fourier plane of the SLM.

In principle, the generation process of the OVL uses the technique of Fresnel holography. Thus, the diffraction reproduction contains the 0th and ± 1st diffraction orders. The + 1st diffraction order is the desired OVL, and the −1st diffraction order is its conjugate image, which indicates that they have opposite OAM states. In the experiment, the 0th and −1st diffraction orders are blocked by an aperture to avoid degrading the imaging quality of the OVL.

## 4. Results and discussion

By combining the close-packed arrangement method with the phase mask, four typical kinds of CPOVLs are produced experimentally: square, rhombus, hexagonal, and triangular structures, as shown in Fig. 3. The OAM states are represented by Dirac symbols, and the value corresponds to the TC of each lattice element. Here, the reduced Planck’s constant, ħ, is omitted for simplicity. Distinctly, this method is effective for producing OVLs with a close-packed arrangement and controllable structures. By adjusting the length of the basic vector, *d*, the interval between two adjacent lattice elements is freely controlled. The different columns of Fig. 3 demonstrate the separation of OVL elements by changing the parameter *d*. As shown in the schematic of the generation principle in Fig. 1, for the close-packed arrangement condition, it is theoretically deduced that the relationship between the length of the basic vector and the radius of the lattice element should satisfy *d* = 2*R*. Interference phenomena appear as two adjacent lattice elements intersect. At the condition of *d* = 2*R*, interference slightly affects the adjacent border of each lattice element, as shown in column b3. Consequently, crosstalk always appears at the connection of two adjacent OV elements because of the parasitic light coherence and non-ideal POV lattice element.

Intuitively, the crosstalk is eliminated if the length of the basic vector *d* is sufficiently greater than 2*R*. However, if the length of the basic vector is too large, the close-packed arrangement condition is destroyed. A threshold exists to ensure the close-packed arrangement condition and exact distinction simultaneously. To determine this threshold, we illustrate comparatively the simulation results of a hexagonal OVL, as shown in Figs. 3(c) and (d), respectively. From Fig. 3(c), we find that the resolution of two adjacent lattice elements is affected by the magnitude, sign, and odevity of their OAM states. There is a simple criterion that the valley in double peaks falls below the value of the 1/e peak. For OAM-based optical communications, the crosstalk must be eliminated to ensure the fidelity of the reconstructed signal. However, for microparticle manipulation and micro/nanofabrication, partial superposition of lattice elements may increase the flexibility and the potential applications. For example, when we pull out a particle cluster, the OVL should be designed in a process from superposition to separation.

After OVL generation, it is necessary to verify the existence and determine the magnitude of the OAM. To achieve this, one method exploits the interference between a spherical wave and the OVL [17]. However, this method needs an additional interference optical path, which increases the complexity of the optical system. Moreover, the magnitude of the TC cannot be determined by counting the number of spiral fringes, owing to the non-symmetry. Here, phase-shift techniques are applied to the OVL and its conjugate lattices, and the resulting interferograms of 3 × 3 square and rhombus OVLs are shown in Fig. 4. The superposition is represented by the Dirac symbols, into which two integer numbers are written. The first and second integers in the Dirac symbol represent the OVL element and the conjugate OVL elements, respectively.

0th-order bright spots always exist in the center of each pattern. The conjugate lattice has opposite OAM states to the OVL. For single-lattice element superposition, the number of spiral fringes is 8 and 10, as shown in insets of Figs. 4(a1) and 4(b1), respectively, which is twice the number of TCs of OAM. This result is consistent with the previous works [41–43]. When four and nine lattice elements are superposed, the number of spiral fringes is equal to the absolute value of the difference between the TCs of the superposed elements. Consequently, the existence of the OAMs of the OVL is verified, and the OAMs are measured using the interference of the ± 1st diffraction orders.

Thus far, an OVL with a close-packed arrangement was produced effectively using the logical operation of a primitive cell, combined with the phase mask generation technique. OVLs with versatile square, triangular, and hexagonal structures are generated, and their OAM states are verified. These large OVLs can easily be expanded in size, as shown in Fig. 5. During the expansion process, the OAM reserves the original states, as shown in second column in Fig. 5. The OAM states on the outer sides are assigned random integers within [-10 10]. Here, the matrix of the permutations with the repetition of *L*(.) is constructed using theintegers from 0 to 5. The profiles demonstrate the intensity fluctuation present the adjacent connection on the same structures with the OVLs. The numbers ① and ② are used to determine the position of the profiles, which rotates at a solid angle for clear visualization. Careful observation of the profile curves reveals that most of the connections have a double-peak structure. For single-peak structures, by comparing the FWHM and the designed ring width, discrimination can be achieved via quantitative analysis. Owing to the effect of parasitic light, the background exhibits little fluctuation. Nevertheless, as indicated by the overall pattern, the OVLs are of high quality.

The energy flow and OAM are two basic physical parameters. For a sparsely arranged OVL, these parameter distributions of the lattice element are same as an individual optical vortex [44,45]. For a close-packed arrangement, there is an interesting question: how does the energy flow or OAM between two adjacent lattice elements in the OVL interact? In response to this issue, Fig. 6 shows a numerical simulation of the energy flow and OAM distributions in the observed plane, which correspond to the experimental results in the first column of Fig. 5.

The energy flow is shown in the upper panel of Fig. 6. The direction and length of the red arrow represent the direction and magnitude of the energy flow at this point, respectively. The energy flow is mainly concentrated on the bright rings, and its magnitude is proportional to the magnitude of the TC. The direction of the energy flow is the same as the spiral direction of each lattice element. The energy flow is approximately null, while the intensity has a high magnitude, as shown in Fig. 6(a1). This is because the pattern only demonstrates the distribution of the energy flow in the transverse plane of the free-propagation space and is related to not only the amplitude but also the phase distribution [44,45]. If there is no spiral phase, there is no energy flow in the transverse plane. That is, the energy flow only has a longitudinal component; the transverse component is zero.

At a connection, the energy can flow from one lattice element to an adjacent one because they have the same flow direction at the connection for opposite OAM states, as indicated by A and C in Fig. 6. Conversely, energy hardly flows from one lattice element to another for two adjacent OAM states with the same sign, because they have the opposite flow direction at the connection, as indicated by B and D in Fig. 6. Using this property, we can transfer microparticles from one circle to another if two lattice elements are close enough.

The lower panel of Fig. 6 presents the numerical simulation results for OAM distributions, corresponding to the first column of Fig. 5. In these patterns, the symbol “⊙” indicates the location of the selected helical axis and the direction of the OAM, which is outward vertically from the paper surface. In accordance with the definition of OAM, the magnitude of the OAM is determined by the location of the selected symmetry axis in the calculation. As a result, only one lattice element has a uniform OAM distribution on the ring of its lattice site, where the locations superpose between the selected symmetry axis and the center of the lattice element. Even so, significant results are obtained by plotting the symmetry axes in the transverse plane connecting centers of lattice elements with the screw axis. These in-plane symmetry axes are marked as I, II, III, and IV. The OAM circle of the lattice element is divided into two semicircles by one of the symmetry axes. The OAM on the two semicircles is symmetrically distributed with respect to the dividing axis. Along each semicircle, the magnitude of the OAM uniformly increases or decreases, which indirectly indicates that the OAM of each lattice element preserves its structure during the close-packed arrangement. Owing to the OAM interaction, there are singular points at the connections between lattice elements.

For a specific spatially structured OVL, the OAM state of each lattice element is easily assigned according to the desire of the researchers. Using this property, we can obtain OVLs with a structure more complex than the four aforementioned structures. Figure 7(a) demonstrates a fractional hexagonal OVL with modulated gaps, where the TC of each lattice element is 1.5. The gaps of the outer six OVs rotate anticlockwise by a step of π/3, and after a clockwise circle, the gap returns to the origin. This property allows us to release the microparticles at the different gap positions [46].

Many kinds of spatially structured OVLs are obtained. By conducting different logical operations of the expanding primitive cells, OVLs with several interesting structures are produced experimentally, including a hollow rhombus-shaped lattice [Fig. 7(b)], an “Olympic rings”-shaped lattice [Fig. 7(c)], a honeycomb-shaped lattice [Fig. 7(d)], and a hollow hexagram-shaped lattice [Fig. 7(e)]. These specific structures introduce new potential applications, such as self-assembled chiral superstructures induced by light illumination, the fabrication of honeycomb microstructured materials, microparticle cluster manipulation, and the fabrication of other specific microstructured materials. The structures of the CPOVL and its potential applications are merely limited by our imagination.

The proposed method is effective for producing a CPOVL with controllable and versatile structures. Each lattice element can be accurately located at the position required for the structure. The radius and width of the lattice element can be adjusted individually by changing the cone angle of the axicon and the waist of the incident Gaussian beam, which increases the flexibility for controlling the structure of the optical field. However, several issues should be further studied to optimize this method. For instance, the diffraction efficiency and mode purity should be calculated and improved. Furthermore, for different applications, the experimental optical path should be adjusted by employing some specific optical elements. For instance, a high numerical aperture (N. A.) objective is needed to replace the convex lens of L2 when conducting microparticle manipulation.

## 5. Conclusions

An alternative scheme for producing a CPOVL with controllable structures is proposed. Borrowing the concept of a close-packed arrangement in a crystal lattice, the structures of the OVL are precisely constructed by conducting logical operations of expanding OV primitive cells. By exploiting the phase mask generation technique, OVLs with different structures are generated experimentally. The OAM state of each lattice element is tailored freely and verified by the interference between the OVL and its conjugate lattice. The energy can flow from a lattice element to the adjacent one, and the OAM distribution interacts in the connection area. The proposed method is effective for producing CPOVLs with controllable structures. Our results are an important step towards extending the fundamental understanding of the generation of OVLs.

## Funding

National Natural Science Foundation of China (Grant nos. 61775052, 11704098, 11525418, 91750201).

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