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Generation of arbitrarily structured optical vortex arrays based on the epicycle model

Yuping Tai, Haihao Fan, Xin Ma, Wenjun Wei, Hao Zhang, Miaomiao Tang, and Xinzhong Li

Open Access Open Access

Abstract

Optical vortex arrays (OVAs) are complex light fields with versatile structures that have been widely studied in large-capacity optical communications, optical tweezers, and optical measurements. However, generating OVAs with arbitrary structures without explicit analytical expressions remains a challenge. To address this issue, we propose an alternative scheme for customizing OVAs with arbitrary structures using an epicycle model and vortex localization techniques. This method can accurately generate an OVA with an arbitrary structure by pre-designing the positions of each vortex. The influence of the number and coordinates of the locating points on customized OVAs is discussed. Finally, the structures of the OVA and each vortex are individually shaped into specifically formed fractal shapes by combining cross-phase techniques. This unique OVA will open up novel potential applications, such as the complex manipulation of multiparticle systems and optical communication based on optical angular momentum.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

An optical vortex array (OVA) is a structured-light field with abundant modes [1–6]. Unlike a single optical vortex (OV), OVAs are the result of manipulating two degrees of freedom: the optical angular momentum (OAM) and spatial position. It consists of multiple OVs with identical or different OAM states at distinct spatial positions. OVAs provide greater degrees of freedom and more information, facilitating tremendous applications in large-capacity optical communication [7–10], and multiple-particle manipulations [11–16].

Creating well-performing OVAs is important for these applications. In recent years, various methods have been widely used to generate OVAs, including spatial light modulators [17], Dammann gratings [18–20], and microstructural materials [21,22] and mode superposition [23,24]. Among these techniques, researchers have achieved modulation with multiple parameters, except for OAM in OVAs, such as the number of OVs, topological charge (TC), ellipticity, size, and amplitude [1,25–27]. Spatial position, another crucial degree of freedom in OVAs, has also received extensive attention. Researchers have developed different OVAs with versatile structures, such as triangle [28], close-packed [29], orientation-selective elliptic [30], “bear” [31], and Gibbs–Wulff [32] OVAs. These studies have greatly facilitated the development of OVAs and their applications. However, owing to the inflexibility of spatial position modulation in these methods, the OVA modes that can be generated are restricted.

To develop functional OVAs, Li et al. recently produced cycloidal OVAs in which the unit OVs were arranged along cycloids using cycloidal equations and phase-shifting techniques [33]. The cycloidal OVAs provide more abundant spatial modes. However, this method generates only cycloidal OVAs because of its dependence on the cycloid equation. Traditional methods of generating OVAs are effective only in specific scenarios where the structures of the OVA do not have analytical expressions. Universal methods to realize the free customization of OVAs with arbitrary structures, independent of analytical expressions, have not yet been developed. In addition, the beam shaping method via the epicycle model in astrophysics is an effective method for generating a single OV with customized trajectories and without analytical expressions [34].

To address this demand, inspired by the concept of an epicycle model and combines the and vortex localization techniques, we propose a universal method for generating OVAs (characterized by the OVs’ location distributions with arbitrary structures regardless of with and without explicit analytical expressions). Specifically, this method exploits the epicyclic model to design arbitrary curves and combines OV positioning techniques to achieve an accurate distribution of OVs along arbitrary trajectories. Various customized OVAs are demonstrated through numerical simulations and experiments. Moreover, the influence of the number and coordinates of the locating points on customized OVAs is discussed. Furthermore, the shapes of OVAs and unit OVs are simultaneously controlled using cross-phase techniques.

2. Theory

For simplicity, we chose a perfect optical vortex (POV) as the unit OV to construct the novel OVA. A POV is an OV whose diameter is independent of its topological charge. The POV is usually obtained using the Fourier transform of the Bessel beam [35]. The complex amplitude of the Bessel beam is expressed as:

$$E({r,\varphi } )= {J_m}({{k_r}r} )\exp ({\textrm{i}m\varphi } )\exp ({{{ - {r^2}} / {\omega_0^2}}} ), $$
where ω0 represents the waist of the Gaussian beam, which limits the size of the Bessel beam. Jm is the m-th order Bessel function of the first type. A convex lens with focal length f is employed to perform the Fourier transformation of the Bessel beam as follows:
$$E({\rho ,\theta } )= \frac{{{k_0}}}{{\textrm{i}2\mathrm{\pi }f}}\int_0^\infty {\int_0^{2\mathrm{\pi }} {E({r,\varphi } )} } \exp \left( { - \frac{{\textrm{i}{k_0}}}{f}\rho r\cos (\theta - \varphi )} \right)r\textrm{d}r\textrm{d}\varphi. $$

Then, the OVA is obtained by introducing a matrix of phase shifts in the original plane according to the Fourier transform shift theorem:

$$\begin{array}{c} E({\rho ,\theta } )= \frac{{{k_0}}}{{\textrm{i}2\mathrm{\pi }f}}\int_0^\infty {\int_0^{2\mathrm{\pi }} {\sum\limits_{n = 1}^N {E(r,\varphi )} } } \exp [{\textrm{i}{k_0}(n^{\prime} - 1)\alpha r} ]\\ \exp \{{\textrm{i}2\mathrm{\pi }[{{S_{n,x}}\rho r\cos (\theta - \varphi ) + {S_{n,}}_y\rho r\sin (\theta - \varphi )} ]} \}r\textrm{d}r\textrm{d}\varphi , \end{array}$$
where N is the number of OVs in OVA, n’ and α are the refractive index and cone angle of an axicon, respectively. (Sn,x, Sn,y) represents the position matrix of the n-th OV in the OVA, which determines the positions of the OVs.

In contrast to previous studies, herein, (Sn,x, Sn,y) is determined via the epicycle model, which is a mathematical model employed in astrophysics to predict the trajectory of planets. The epicycle model can be used to decompose the expression of an arbitrary curve into a series of sine and cosine functions, which are expressed as the sum of the Fourier series [34,36],

$$F(t )= \sum\limits_{k = {{ - N} / 2}}^{{N / 2}} {{c_k}} {e^{\textrm{i}2\mathrm{\pi }kt/N}}, $$
and
$${c_k} = \frac{1}{N}\sum\limits_{k ={-} {N / 2}}^{{N / 2}} {{f_{}}(n)} {e^{ - \textrm{i}\omega n}}, $$
where N and k denote the number of epicycles and k-th epicycle, respectively. ck determines the radius and initial rotation angle of each epicycle. f (n) = xn + iyn, n is the ordinal number of each key point, (xn, yn) represent the coordinates of the desired distribution of the OVs, and ω = 2πk/N is the rotation rate, respectively. For convenience, the coordinates of each control point are expressed as complex numbers. Therefore, f(n) is not a set of consecutive real numbers but a set of coordinate values for two-dimensional (2D) vectors.

Mathematically, if (xn, yn) on an arbitrary curve is a function of variable t, the curve can be expressed as F(t) = [x(t), y(t)]. Therefore, according to Euler’s formula, the position matrix (Sn,x, Sn,y) can be written as:

$$\left\{ {\begin{array}{{c}} {{S_{n,x}} = \sum\limits_{k = {{ - N} / 2}}^{{N / 2}} {[{\textrm{Re} ({{c_k}} )\cos (\omega_{tn}) - {\mathop{\rm Im}\nolimits} ({{c_k}} )\sin (\omega_{tn})} ]} }\\ {{S_{n,y}} = \sum\limits_{k = {{ - N} / 2}}^{{N / 2}} {[{\textrm{Re} ({{c_k}} )\sin (\omega_{tn}) + {\mathop{\rm Im}\nolimits} ({{c_k}} )\cos (\omega_{tn})} ]} } \end{array}} \right.,$$
where Re(·) and Im(·) represent the real and imaginary parts of the complex number, respectively. Note that the epicycle model here is used to locate the position of each OV on the OVA with arbitrary curvilinear trajectories, which is essentially different from the individual vortex beams with arbitrary curvilinear structures produced in Ref. [34].

Figure 1 illustrates the process of customizing an OVA with arbitrary structures. The predesigned structure is H-shaped, as shown in Fig. 1(a). Because an arbitrary geometric curve can be considered a collection of points, a specific number of locating points marked by green dots are considered on the H-shaped curve. The coordinates of the location points (xn, yn) were then obtained by establishing their Cartesian coordinates, where n represents the n-th location point. In this case, n = 44. The radius of each epicycle was determined by substituting the coordinates of the 44 key points into Eq. (5), and the rotation rate of each epicycle satisfies ω = 2πk/N. The constructed epicycle model, shown in Fig. 1(b), contains 44 epicycles corresponding to the 44 locations. For clarity, only three epicycles have been shown, where the center of each circle is on the circumference of the previous circle. The trajectory of the point is on the last circle when each epicycle rotates at different speeds, as shown in the inset in the lower-left illustration, that is, H. The numerical expression for the H-shaped curve is obtained by substituting the calculation result of Eq. (5) into Eq. (6). Finally, customized OVAs with the same structure as shown in Fig. 1(a) were generated via vortex positioning techniques, as shown in Fig. 1(c), where the number of OVs is M = N = n = 44 and the positions of the OVs in the OVAs correspond to the positions of the input locating points. Figure 1(d) shows the flow chart for customizing the OVAs using the proposed epicycle model.

 figure: Fig. 1.

Fig. 1. Process of customizing OVA with arbitrary structures using epicycle models. (a) Predesigned structure of OVA. (b) Epicycle model. (c) OVA with H shape. (d) Flow chart for customizing OVA with arbitrary structures.

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3. Results and discussion

The experimental setup used the same equipment as that in our previous study [30]. To verify the effectiveness of the proposed method, five OVAs with different letter structures of “OPTICS” were generated. Figures 2(a1)−(f1) and (a2)−(f2) illustrate the simulated and experimental intensity patterns of the OVAs arranged along different letters, where the white lines represent the predesigned structures. Herein, the number of OVs in the OVA of the six structures was set as M = 16, 30, 24, 24, 19, and 28. The number of OVs in an OVA with an arbitrary structure can be adjusted easily by changing the number of location points n in Eq. (1). The intensity patterns show that the OVs in OVAs were distributed along the contours extracted from “OPTICS,” where the OVs were distributed evenly [Figs. 2(a)−(d)] or unevenly [Figs. 2(e)−(f)] by adjusting the coordinates of the locating points. However, the experimental results agreed well with the simulated results.

 figure: Fig. 2.

Fig. 2. OVAs with different structures to illustrate “OPTICS”. (a1)−(f1) Simulated intensity patterns. (a2)−(f2) Experimental intensity patterns. (a3)−(f3) Experimentally retrieved phase patterns. (a4)−(f4) Interferograms between OVAs and a plane beam. The insets are the magnified images of the encircled patterns, 2.5×. For simplicity, the TC of each OV in the OVAs is set as a unit, m = 1; similarly, hereinafter.

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To verify the existence of OVs in the OVA, interferograms were obtained between the OVAs and a plane beam, as shown in Figs. 2(a4)−(f4), where the number and direction of forks determine the magnitude and sign of the TC for m = 1. The experimental phase patterns were recovered by the Fourier and inverse Fourier transformation of the interferograms [37], as shown in Figs. 2(a3)−(f3), where each OV position is marked by a dashed circle. The phase of each OV increases from 0 to 2 along a circle, which also determines the TC for m = 1. The structures shown in Figs. 2(a)−(d) are described by explicit analytical expressions, whereas no analytical expressions exist for the structures shown in Figs. 2(e) and (f). Therefore, OVAs with arbitrary structures can be easily customized according to the application requirements. OVAs with more complex distributions are easily generated using the proposed method, regardless of whether the structures have analytical expressions. An advantage of this method is that it can be constructed from actual complex objects to form more versatile and functional structures.

In addition, the epicycle model has several adjustable parameters that precisely regulate the spatial modes of OVAs. Specifically, the number of location points described in Eq. (5), and the location point coordinates (xn, yn) are the two key parameters for adjusting the OVA structures. The specific structure was constructed using a convex structure superposed on a rectangle, as shown in Figs. 3(a1)–(d1). To verify the performance of the proposed method, we generated four OVAs by adjusting the number of location points n, as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. OVAs with different number of OVs. (a1)−(d1) Positions of the OVs. (a2)−(d2) Simulated intensity patterns. (a3)−(d3) Experimental intensity patterns. (a4)−(d4) Experimental retrieved phase patterns.

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As shown in the second and third rows of Fig. 3, the numbers of OVs in the OVAs are M = 14, 25, 25, and 28. OVAs maintained the same structure, but the number and distribution of OVs were individually modulated. The number of OVs in Fig. 3(b) is the same as that in Fig. 3(c); however, their distributions were modulated differently. Compared with Fig. 3(d1), when the three location points on the left were removed from the preset structure [Fig. 3(b1)], the three consecutive OVs on the left were removed correspondingly, as shown in Figs. 3(b2) and (b3). In contrast, three OVs were removed, and the remaining OVs were redistributed evenly to obtain Figs. 3(c2) and (c3). By adjusting the number and position of the locating points, OVAs with different position distributions and OV numbers were generated, whereas the structure of the OVA remained unchanged. This property is important for constructing a functional OVA, where the density and gap can be easily modulated to realize specific particle trapping and the formation of micro- or nano-materials.

Furthermore, it is necessary to accurately control and modulate the structure of OVA in versatile application scenarios. To verify the capacity for accurate modulation, the OVA shape was modulated in real time by adjusting the coordinates of the location points (xn, yn). In this case, taking the hexagon as an example, we regulated the position of a single OV encircled by the white dashed rectangle in Fig. 4.

 figure: Fig. 4.

Fig. 4. Accurate control of the structure of OVAs. (a1)−(d1) Simulated intensity patterns. (a2)−(d2) Experimental intensity patterns. (a3)−(d3) Experimental retrieved phase patterns. (e) Displacement of the specific locating point vs. the displacement of the specific OV. (f) Structure correlation within the OVAs removing the specific moving OV.

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The specific OV moved upwards with an increase in the longitudinal coordinates of the corresponding location point. Here, we set the position of vertex OV in the left column of Fig. 4 as the starting point. Subsequently, the specific OV was controlled to move along the vertical direction, where the displacements were d = 0.4, 0.8, and 1.2, as shown in Figs. 4(b)−(d). Both the intensity patterns and retrieved phase patterns demonstrated this process well. Additional data on the displacement of a specific OV were measured and compared with the change in its coordinate values, as shown in Fig. 4(e). According to the calculations, the OV moves linearly with an increase in the ordinate of the location point [fitting coefficient R2 = 0.9974]. To verify the change in the other OVA parts, the structural correlation of the intensity was calculated, and the results are shown in Fig. 4(f), where the structure deducted the specific moving OV. A high correlation between both the structure and OV distribution was revealed, with a fitting coefficient of 0.9545 (average value). All results prove that the method can precisely control the position of arbitrary OVs in an OVA, whose structure remains unchanged.

Furthermore, the OVs in the OVA were simultaneously modulated into polygonal structures using cross-phase techniques [38]. The cross phases possess the form shown in Fig. 5(a2), which can be written as

$$\psi (x,y) = \exp[{\textrm{i}u({x^p}\cos\phi - {y^q}\sin\phi )({x^p}\sin \phi + {y^q}\cos \phi )} ].$$

 figure: Fig. 5.

Fig. 5. Simultaneous mixed modulation of the structures of OVA and OV. (a1)−(a3) Generation of phase pattern shaped by cross phase in the source plane. (b1)−(b4) Experimental intensity patterns of OVAs with different OV shapes arranged along an ellipse. (c1)−(c4) Simulated phase patterns corresponding to (b1)−(b4). (d1)−(d4) Experimental intensity patterns of OVAs with the same shapes of individual OV and OVA. (e1)−(e4) Simulated phase patterns corresponding to (d1)−(d4). Insets are the 2× magnified images of the encircled OVs.

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This phase has the simplest form of exp(iuxpyq) when ϕ = 0. The parameter u controls the conversion rate, and the azimuthal factor ϕ characterizes the rotation angle of the converted beams in the Fourier plane. The exponential indices p and q are positive integers. The sum of p and q denotes the number of OV edges. The polygonal OV can be obtained using the Fourier transform of the initial field E′(x, y), which is the product of the Bessel vortex beam in Eq. (1) and the cross phase expressed as E′(x, y) = E(x, y)ψ(x, y). Figures 5(a1)−(a3) show the phase generation of the initial pattern E′(x, y). Figure 5(b) shows the experimental intensity patterns of OVAs with different shapes of OVs, including elliptical, triangular, square, and pentagonal shapes, whereas the overall structure of the OVA remained unchanged in an elliptical shape. Different unit OV shapes were realized by setting different combinations of (p, q). Similarly, the existence of an OV was verified via a 2π phase jump at the position of the OV in the phase patterns, which were retrieved using interferograms, as illustrated in Fig. 5(c).

Based on these techniques, we generated fractal OVAs in which each OV possessed the same structure as the OVA. As shown in Figs. 5(d1)−(d4), the generated OVAs have elliptical, triangular, quadrilateral, and pentagonal fractal shapes, respectively. The shapes and arrangements of the OVs are ellipses, triangles, squares, and pentagons. Figures 5(d) and (e) show the intensity patterns and the corresponding retrieved phase patterns. Moreover, the shaping modulation of the unit OVs can be easily controlled by adjusting the conversion rate, u.

The proposed method can be used to customize OVAs along an arbitrary trajectory as desired and form specific functional structures, such as chiral shapes, optical gears, and spiral structures, for various applications. The remarkable advantages are that the proposed OVAs can be multidimensionally and accurately modulated, including their overall shape, OV number and distribution, and the shape, position, and order of each OV. Note that the OVs will have the similar radius and larger OAM with the larger TC. To exploit the potential applications, the proposed OVA should be expanded to 3D space, a vectorial OVA should be developed, and specific structures should be explored in other coordinates. Furthermore, the ensemble parameters should be designed to characterize all the properties of the OVAs. The OAM distribution, energy flow, and optical force field should be further studied.

4. Conclusions

In summary, we proposed an entirely new method for generating OVAs with arbitrary structures by combining an epicyclic model with vortex localization techniques. This method allows for the precise control of customized, arbitrarily structured OVAs in a 2D space. By adjusting the number and positions of locating points and other spatial degrees of freedom, we controlled the number of OVs in the OVA, whereas the overall structure of the OVA remained unchanged. Each OV in OVA was reshaped using cross-phase techniques, such that the OVA formed a fractal shape. This study provides a universal method to generate OVAs with arbitrary structures. Specifically, the method allows structure mapping from objects, thereby facilitating potential applications for the complex manipulation of multiparticle systems and optical-sensitive material formation.

Funding

Natural Science Foundation of Henan Province (232300421019, 222300420042); National Natural Science Foundation of China (12274116, 12174089); Key Scientific Research Projects of Institutions of Higher Learning of the Henan Province Education Department (21zx002); State Key Laboratory of Transient Optics and Photonics (SKLST202216).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Process of customizing OVA with arbitrary structures using epicycle models. (a) Predesigned structure of OVA. (b) Epicycle model. (c) OVA with H shape. (d) Flow chart for customizing OVA with arbitrary structures.

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Fig. 2.
Fig. 2. OVAs with different structures to illustrate “OPTICS”. (a1)−(f1) Simulated intensity patterns. (a2)−(f2) Experimental intensity patterns. (a3)−(f3) Experimentally retrieved phase patterns. (a4)−(f4) Interferograms between OVAs and a plane beam. The insets are the magnified images of the encircled patterns, 2.5×. For simplicity, the TC of each OV in the OVAs is set as a unit, m = 1; similarly, hereinafter.

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Fig. 3.
Fig. 3. OVAs with different number of OVs. (a1)−(d1) Positions of the OVs. (a2)−(d2) Simulated intensity patterns. (a3)−(d3) Experimental intensity patterns. (a4)−(d4) Experimental retrieved phase patterns.

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Fig. 4.
Fig. 4. Accurate control of the structure of OVAs. (a1)−(d1) Simulated intensity patterns. (a2)−(d2) Experimental intensity patterns. (a3)−(d3) Experimental retrieved phase patterns. (e) Displacement of the specific locating point vs. the displacement of the specific OV. (f) Structure correlation within the OVAs removing the specific moving OV.

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Fig. 5.
Fig. 5. Simultaneous mixed modulation of the structures of OVA and OV. (a1)−(a3) Generation of phase pattern shaped by cross phase in the source plane. (b1)−(b4) Experimental intensity patterns of OVAs with different OV shapes arranged along an ellipse. (c1)−(c4) Simulated phase patterns corresponding to (b1)−(b4). (d1)−(d4) Experimental intensity patterns of OVAs with the same shapes of individual OV and OVA. (e1)−(e4) Simulated phase patterns corresponding to (d1)−(d4). Insets are the 2× magnified images of the encircled OVs.

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Equations (7)

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E ( r , φ ) = J m ( k r r ) exp ⁡ ( i m φ ) exp ⁡ ( − r 2 / ω 0 2 ) ,
E ( ρ , θ ) = k 0 i 2 π f ∫ 0 ∞ ∫ 0 2 π E ( r , φ ) exp ⁡ ( − i k 0 f ρ r cos ⁡ ( θ − φ ) ) r d r d φ .
E ( ρ , θ ) = k 0 i 2 π f ∫ 0 ∞ ∫ 0 2 π ∑ n = 1 N E ( r , φ ) exp ⁡ [ i k 0 ( n ′ − 1 ) α r ] exp ⁡ { i 2 π [ S n , x ρ r cos ⁡ ( θ − φ ) + S n , y ρ r sin ⁡ ( θ − φ ) ] } r d r d φ ,
F ( t ) = ∑ k = − N / 2 N / 2 c k e i 2 π k t / N ,
c k = 1 N ∑ k = − N / 2 N / 2 f ( n ) e − i ω n ,
{ S n , x = ∑ k = − N / 2 N / 2 [ Re ( c k ) cos ⁡ ( ω t n ) − Im ( c k ) sin ⁡ ( ω t n ) ] S n , y = ∑ k = − N / 2 N / 2 [ Re ( c k ) sin ⁡ ( ω t n ) + Im ( c k ) cos ⁡ ( ω t n ) ] ,
ψ ( x , y ) = exp ⁡ [ i u ( x p cos ⁡ ϕ − y q sin ⁡ ϕ ) ( x p sin ⁡ ϕ + y q cos ⁡ ϕ ) ] .
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