## Abstract

Optical vortex arrays (OVAs) have successfully aroused substantial interest from researchers for their promising prospects ranging from classical to quantum physics. Previous reported OVAs still show a lack of controllable dimensions which may hamper their applications. Taking an isolated perfect optical vortex (POV) as an array element, whose diameter is independent of its topological charge (TC), this paper proposes combined phase-only holograms to produce sophisticated POV arrays. The contributed scheme enables dynamically controllable multi-ring, TC, eccentricity, size, and the number of optical vortices (OVs). Apart from traditional single ring POV element, we set up a *β*_{g} library to obtain optimized double ring POV element. With multiple selective degrees of freedom to be chosen, a series of POV arrays are generated which not only elucidate versatility of the method but also unravel analytical relationships between the set parameters and intensity patterns. More exotic structures are formed like the “Bear POV” to manifest the potential of this approach in tailoring customized structure beams. The experimental results show robust firmness with the theoretical simulations. As yet, these arrays make their public debut so far as we know, and will find miscellaneous applications especially in multi-microparticle trapping, large-capacity optical communications, novel pumping lasers and so on.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Owing to the twisted wavefront optical vortex (OV) possesses, substantial attention has been paid to its generation and applications. Significantly, Allen et al. related the screw-phase singularity to the term$\; \textrm{exp}({im\phi } )$, with *m* the topological charge and *ϕ* the azimuthal angle [1,2]. Meanwhile, each photon in an OV carries *m*ℏ orbital angular momenta (OAM) and the intensity profile resembles a “doughnut” consequently. In the wake of the maturity of OV creation [3–7], substantial applications in optical tweezers [8–10], large-capacity optical communications [11,12], among others [13,14], are spurred. However, the ring size of a conventional OV (say diameter or width or area) enlarges with larger topological charge [15,16], which poses challenges for fiber coupling, or special demands for large TCs but small ring diameters. The concept of perfect optical vortex (POV) proposed by Ostrovsky et al. addresses the aforementioned issue properly [17,18]. The so-called POV’s intensity pattern is impervious to its OAM thus is described as “perfect”. Subsequently, multitudinous modulations on POV arises [19–22]. Since a single OV contains limited information, optical vortex arrays (OVAs) are heuristically investigated [23–25]. A square POV array is firstly generated [26] based on a general grating-design framework proposed by Romero and Dickey [27,28] and then POV arrays with controllable diffraction orders and TCs were born [29]. What follows is a close-packed optical vortex lattice with expected arrangement generated by leveraging a series of logical operations to OV elements [30], which increases greatly the number of OVs in an array and produces prodigious structures. Recently, an elliptic OVA with designed orientation is likewise introduced via multicoordinate transformations [31]. The generated OVA looks remarkable except that this method only works for elliptic OV element. Another well-known method to generate OVAs is overlapping some particular optical beams like Laguerre-Gaussian beams [32], spherical waves [33], Bessel beams [34]. After superposing two POVs coaxially, a circular OVA is firstly demonstrated and the number of singularities and optimal overlap ratio as well as other properties of the OVA are investigated [35]. Further, an elliptic annular OVA by superposing two concentric elliptic POVs and an OVA along arbitrary curvilinear path by the superposition of two curve beams are proposed in succession [36,37]. But the sign of TC and relative location of the OV are difficult to modulate. Then an anomalous OVA is created via interference of two grafted POVs to meet the challenges [38]. In addition, OVAs are extended into three dimensions by two-dimensional (2D) Dammann grating accompanied with a Dammann zone plate [39], combination of 2D phase grating and added axial shifting modulation [40] and hybrid phase pattern in tight focusing region [41].

However, these existing OVAs may have their own constraints such as anomalous vortex distribution or lack of modulation aspects, fulfilling multi-degree of freedom control on OVA remains significant. Taking microparticles trapping for example, a ring trap combined with a point trap has proven better rotation performance of low-refractive-index particles [42], which can be achieved by double ring POV (i.e. two closely located rings) in some way [43]. And the larger TC is, the faster particle rotates. Then how to exactly achieve multiple particles’ rotation with diverse paths at different speeds concurrently? In another scenario of optical interconnect system constructed in [44], more complex OVAs will provide potential for higher transmission capacity. Moreover, controllable OVAs deserves to be researched because it will also pave the way for extensive applications in high-resolution imaging [45,46], quantum entanglement [47,48] and so on.

Herein, we present both theoretically and experimentally a protocol to produce more complicated perfect OVAs with multi-degree of freedom. The proposed scheme enables controllable multi-ring, TC, eccentricity, size, and the number of OVs. As for the multi-ring vortex beam, we select double ring POV as a special example. Although previous endeavors have introduced double ring POV in different ways, our work makes steps forward and shows its uniqueness and great potential in application scenarios. In particular, Liang et al. first generated double ring POV by the Fourier transform of azimuthally polarized Bessel beams [43] but it degrades with larger TC and it can not be utilized in vortex arrays. And lately, Yu et al. studied to control radial profiles of POV leveraging Dammann gratings [49] but the proposed theory is likewise more suitable for single POV rather than vortex array generation and it actually can not generate single ring POV because of the inherent sidelobe. Here we firstly analyze the formation mechanism of double ring POV in a quasi-analytical manner and focus on the optimization of it and sophisticated arrays with enhanced customization.

This work is organized as follows. In section 2, the basic principle is provided including POV element generation and optimization, array theory and hologram design. The designed phase-only hologram blends Bessel functions and cosine functions and detailed analytical derivations are given. We interfere the vortex arrays with Gaussian beams to ensure the existence of spiral phase and analyze them thoroughly from various aspects in section 3. The experimental results show firmness with the theoretical simulations. At last, we present some conclusions and remarks. As yet, these arrays make their public debut so far as we know and are envisaged to find miscellaneous applications.

## 2. Theory overview

An isolated POV with single bright ring can be described as the Dirac delta function embedded with spiral phase [17]. It is demonstrated that the Fourier transform of Bessel beam is this type of annular ring being independent of TC [18]. Such ideal Bessel beam can be expressed as:

where ${J_m}({\ast} )$ is the*m*-th order Bessel function of the first kind, $\alpha $ is the scaling parameter related to POV’s relative radius,

*m*is the TC and $({r,\theta } )$ are the polar coordinates of the incident plane perpendicular to the optical axis. After passing through a Fourier-transforming lens, a POV is formed as

With respect to double ring POV element, we employed the trigonometric function inspired by Amidror et al. [50]. They found that the spectrum of a circular cosine function represents an impulse ring which reads

*R*is the relative radius. Actually, the right hand of Eq. (3) represents an impulse ring with wake trails and those wake trails degenerate the main impulse ring. Similar results are fetched when it comes to the circular sine function [51]. However, when the spectrums of circular cosine and sine stack up, two impulse rings interfere at the Fourier plane and double bright ring may occur [52]. Based on the properties of trigonometric function, an additional angle $\beta $ is introduced into circular cosine function, i.e. $\cos ({2\pi Rr + \beta } )= \textrm{cos}\beta \cos ({2\pi Rr} )- \textrm{sin}\beta \textrm{sin}({2\pi Rr} )$:

Note that foregoing bright rings are not vortex beams because they don’t carry OAM indeed. When the spiral phase is added in circular cosine function of Eq. (4), which can be written as

its Fourier spectrum becomes obscure. In this case, one can also obtain the quasi-analytical expression of the far field pattern*R*=10 and $\beta = $3π/4 with TC

*m*=2, 5, 8 respectively and the first column of Fig. 1 depict the corresponding intensities of the far field. The ring profiles vary with TC irregularly and they are actually neither what we call “POV” nor double ring POV.

Our optimization process starts from the additional angle $\beta $. It can be modified to adjust the profile until it meets our requirements. Considering the ring profile contains a main ring and a relatively weak sidelobe, one can define $\eta $ as the ratio of energy density of the two bright rings

where ${w_o}$ and ${w_i}$ are energy densities of the outside and inside ring separately. It can be calculated as*m*=2, 5 and 8, those ${\beta _g}$ are (0.28$\pi $, 1.28$\pi $), (0.94$\pi $, 1.94$\pi $) and (0.66$\pi $, 1.66$\pi $) respectively. Substituting ${\beta _g}$ into Eq. (5) and recalculating the far field intensity of $E({r,\theta } )$, the inside and outside rings are both uniform with weak lobes neglectable, showing “perfectness”, as displayed in the third column of Fig. 1. In fact, based on the theory and simulation results, ${\beta _g}$ not only varies with TC

*m*, but alters with radius

*R*. Hence one can establish a ${\beta _g}$ library under certain

*m*and

*R*to map the far field intensity and input hologram, as elucidated in Table S1 of Supplement 1.

On the basis of generated single ring and double ring POV elements, eccentricity can be introduced as an additional degree of freedom to modulate the ring profile. Elliptic POV is discussed in very recent work [19,53,54] and it is generated by stretching Bessel beam into elliptic Bessel beam before Fourier transformation. Here we perform the similar stretch not only on Bessel beam to obtain elliptic POV, but also circular cosine (beam) embedded with spiral phase to demonstrate elliptic double ring POV:

*s*is the stretching factor that determines the eccentricity. More specifically, when $0 < s \le 1$, eccentricity $e = \sqrt {1 - {s^2}} $ and when $s > 1$, $e = \sqrt {{s^2} - 1} $ otherwise. It is surely that the same stretching is available on y-axis.

Aforementioned discussion focuses on one POV element, to obtain a 2D complicated POV array, we elaborate on the holographic grating motivated by the work [27,28,55]. In this case, the corresponding grating transmittance can be expressed as the superposition of Fourier series in a general sense:

*p*,

*q*) denotes the target diffraction order along

*x*-axis and

*y*-axis, $({{\gamma_x},\; {\gamma_y}} )$ are the corresponding spatial angular frequencies defined as reciprocals of the grating periods controlling the relative position of POV elements and ${b_{p,q}}$ denotes the Fourier coefficient. Most importantly, one can characterize diversiform ${b_{p,q}}$ to generate expected field distribution at the output plane. In this paper, we set $b = {J_m}({\alpha r} )\textrm{exp}({im\theta } )$ or $b = \cos ({2\pi Rr + {\beta_g}} )\exp ({im\theta } )$ for desired diffraction orders (

*p*,

*q*) and sum them up to constitute the grating. For example, in Fig. 2(a), to obtain a POV array with single and double ring elements, the grating transmittance satisfies$T({x,\; y} )= {J_{{m_1}}}({\alpha r} )\textrm{exp}({i{m_1}\theta } )\exp ({i{\gamma_x}x} )+ \cos ({2\pi Rr + {\beta_g}} )\exp ({i{m_2}\theta } )\exp ({ - i{\gamma_x}x} )$. Note that the obtained grating transmittance is complex-valued while existing display devices such as liquid crystal spatial light modulators (LC-SLMs) can only load either the phase or the amplitude components. Therefore the hologram needs to be converted into a phase-only one:

Consequently, a complex POV array with several single ring and double ring POV elements shows up at different diffraction orders (*p*, *q*) with selective *α, R, β, m, e*. Those degrees of freedom are combined together into a single holographic grating, which is never been done before so far as we know. It greatly expands the structure beams and also provides researchers enough room to fully utilize them in different scenarios.

## 3. Results and discussion

#### 3.1 Experimental layout

Figure 2(b) sketches the arrangement to generate this complex POV array for experimental demonstration. A 1617nm laser diode is employed as the source. The produced Gaussian beams are coupled into free space with a diameter of 3mm via a single mode fiber and a collimator, and then divided into two parts through a polarized beam splitter. One is p-polarizations going to be phase-only modulated by the LC-SLM (Holoeye, PLUTO-TELCO-013-C), the other is s-polarizations as the reference beams to measure the topological charge. The LC-SLM is placed at the front focal plane of the Fourier lens with focal length 200mm. As a result, the POV array appears at the rear focal plane and an infrared CCD camera (Xenics, Bobcat-320-star) is utilized to record the target intensity pattern. Besides, the first half wave plate works to adjust beam intensity and the second one is leveraged to transform the polarization state of reference beams from s-polarizations to p-polarizations which enables the interference.

#### 3.2 Topological charge

To begin with, a one-dimensional i.e. horizontal or vertical POV array is generated to demonstrate the proposed theory. In the case of POV array along *x*-axis, parameters *q* and ${\gamma _y}$ hold a fixed value of 0. Figures 3(a) and 3(c) illustrate the simulation and experimental results with four POV elements of different TC *m*. The present POV array consists of two double ring POV elements in the middle and two single ring POV elements outside. Besides, the TC distribution of the array is also evaluated. There have been plenty of methods to measure POV’s TC so far [56–58], here the approach of interference is employed [41]. An additional path is introduced as shown in the dotted box of Fig. 2(b), which enables POV array interfere with another reference Gaussian beams. In Figs. 3(b) and 3(d), the interference intensity pattern captured by CCD unravels TC, namely the bright fringe number denotes TC value 2, 4, 6, 8, separately.

The diameters of 9 elements in an array are measured, as shown in Fig. 4(c). Here the radius of a double ring POV is defined as the distance between its center and the midpoint of two bright rings given in Fig. 4(b). For concise comparison, we select the pixel numbers to represent their relative sizes. We capture the screenshot of the computer with resolution 1920×1080 to conduct data analysis. The calculated standard deviations are respectively 1.965 pixels (average diameter: 227.9 pixels) for single ring POVs and 3.0414 pixels (average diameter: 229.0 pixels) for double ring POVs which imply that POVs’ radii are independent of TC, showing quite “perfectness”.

#### 3.3 Size

In addition to the TC distribution, there are several controllable parameters that are worthwhile to conduct further exploration. To study the performance of $\alpha $ and *R*, another 1D array is generated, as displayed in Figs. 5(a) and 5(b). From Eqs. (2) & (3) one can tell $\alpha $ and *R* may dominate the radii of single ring POVs and double ring POVs separately. More concretely, those two radii are equal when $\alpha = 2\pi R$ is sufficed. Therefore the POV array composed of 4 elements can be generated by setting *R* as 4, 6 for two double ring POVs in the middle and$\; \alpha $ as $2\pi \cdot 2,\,\; 2\pi \cdot 8$ for the other two single ring POVs. All TCs of the four POVs in this array are fixed to 2. Note that smaller ring is supposed to look brighter while larger ring looks pale because bigger bright ring shares smaller energy density. Nevertheless, one can still adjust four elements’ weights to achieve homogeneous energy density apportion like Fig. 5. Visualization 1 is made for more vivid exhibition of POV arrays with different size elements, also showing dynamic control.

To quantify the POV’s size, each element’s area of the array is measured, as shown in Fig. 5(c). One can find that the POV’s area is quadratic to $\alpha /2\pi $ or *R*. The fitting curves based on the least square method depict the exact equation $Area = 1636 \cdot {({\alpha /2\pi } )^2}$ with adjusted R-square 0.9998 and $Area = 1577 \cdot {R^2}$ with adjusted R-square 0.9999. These equations coincide with above theory analysis saliently and also are of significance for further applications such as optical coupling because one can obtain POV arrays with desired sizes accurately. Here the reason why area is chosen rather than diameter as the evaluation index is that, area can reflect both circular POV’s and also elliptical POV’s size (see below), where diameter is incapable.

#### 3.4 Eccentricity

To extend above complex arrays, elliptical POV element is mooted to come on the scene. Eccentricity is an intriguing and meaningful degree of freedom for POV which was proposed very recently. Here not only elliptical POVs but also elliptical double ring POVs are generated in one array. For the elliptical POV array, as shown in Figs. 6(a) and 6(b), variable *s* are selected providing that the TC and size parameters are fixed as $m = 2$, $\alpha = 2\pi \cdot 4\; \textrm{and}\; R = 4$. The exact eccentricity of each POV element of experimental results is calculated. As expected, the degree of ellipse, i.e. eccentricity, enlarges with smaller *s*. And when *s*=1, an ellipse degenerates into a circle, which corresponds to normal POV or double ring POV. When *s*>1, the long shaft of an ellipse is along the horizontal direction rather than vertical direction like Fig. 6, and its eccentricity increases with *s*. The matching Visualization 2 demonstrates the lively changeable eccentricity of POV array. Figure 6(c) illustrates the relationship between *e* and *s*, and the correlation coefficient is evaluated as 0.9983 for single ring and 0.9976 for double ring, denoting high coincidence between the experimental and theoretical results.

On the other hand, as we squeeze *x*-axis to obtain elliptical array, we find the sizes of POV elements also change accordingly. The elliptic elements’ areas are calculated in Fig. 6(d) and one can find the parameter *s* not only generates ellipse but also results in size scaling for *s* times, compared to circle ones. In this regard, the fitting curves are $Area = 40850 \cdot s$ with adjusted R-square 0.9961 for single ring elliptic elements and $Area = 40730 \cdot s$ with adjusted R-square 0.9926 for double ring elliptic elements. This feature tells us size is actually dependent on$\; $ $\alpha (R )$ and *s* simultaneously, which also offers brute-force help for applications that demand peculiarly precise customized POV array such as novel pumping lasers and optical trapping.

#### 3.5 2D array

Next more versatile 2D POV arrays can be generated when *q* and ${\gamma _y}$ are not 0 any more. We start from normal square 2D array. Actually, one can select different parameters discussed in this paper to demonstrate flexible control capacity of this array theory. Specifically speaking, each single ring element of the array can be represented as $\left\{ {S\textrm{|}m\textrm{|}\frac{\alpha }{{2\pi }}\textrm{|}e} \right\}$ where *S* means single ring, *m* denotes TC, $\alpha /2\pi \; $defines its relative size and *e* is the eccentricity and similarly double ring element is $\{{D\textrm{|}m\textrm{|}R\textrm{|}e} \}$ where *D* means double ring and *R* defines its relative size. Setting five POVs as $\{{S\textrm{|}4\textrm{|}3\textrm{|}0} \}$,$\; \{{D\textrm{|}2\textrm{|}4\textrm{|}0.6} \}$,$\; \{{D\textrm{|}4\textrm{|}4\textrm{|}0} \}$*,*$\; \{{D\textrm{|}2\textrm{|}4\textrm{|}0.8} \}$ and$\; \{{S\textrm{|}4\textrm{|}5\textrm{|}0} \}$, experimental result in Fig. 7(b) manifests great accordance with simulation pattern. It’s worth mentioning that the spatial distance between neighboring elements depends on spatial angular frequencies $({{\gamma_x},\; {\gamma_y}} )$ and more detailed quantitative relationship is in [29].

Furthermore, one can generate more exotic arrays freely by selecting proper $\left\{ {S\textrm{|}m\textrm{|}\frac{\alpha }{{2\pi }}\textrm{|}e} \right\}$ and $\{{D\textrm{|}m\textrm{|}R\textrm{|}e} \}$. In this case, we obtain an intensity pattern named “Bear POV” as working example. Figure 8 illustrates the Bear POV with different facial expressions and Visualization 3 gives a more animated display of this “astonished bear” as well. The Bear POV opens up tremendous possibilities of more complex and novel POV arrays which only limited by one’s imagination. More importantly, this proves effectiveness of this method and indicates great potential applications like fabrication of micro-structured materials. Consider a specific optical communication protocol proposed by [44], OAM arrays with four elements are utilized for data transmission. Each element can be represented by $\{m \}$ while in our case each element is represented by $\left\{ {D(S )\textrm{|}m\textrm{|}R\left( {\frac{\alpha }{{2\pi }}} \right)\textrm{|}e} \right\}$, which means that data capacity is enlarged four times. Therefore the array is also highly promising in optical communication.

## 4. Conclusions

In summary, we propose both theoretically and experimentally an approach to generate complex POV array with unprecedentedly multiple selective degrees of freedom. The essential phase hologram contains controllable array information while its spectrum can effectively form desired array pattern. Among the array, the properties of each POV are well-controlled including multi-ring, TC, eccentricity, size and the number of OVs. In this regard, the method is elegant in tailoring customized structure beams with only one programable phase grating. To improve the element’s quality, we separately adopt Bessel function to produce conventional POV and set up a ${\beta _g}$ library for circular cosine function to generate optimized double ring POV. In addition, the topological charge is substantiated by interference field pattern and the calculated diameters confirm “perfectness”. The sizes of bright rings are dominated by $\alpha ,R$ and *s* provides another modulation dimension as eccentricity. And explicit analytical relations of these parameters are obtained. Based on the theory, a sequence of POV arrays are experimentally produced and especially the Bear POV is presented to show completely control on the array pattern. This work not only enriches the content of singular optics but also reveals high potential in myriads of applications such as multi-microparticle trapping, large-capacity optical communications, quantum secret sharing [59] and also high-resolution imaging.

Nonetheless, there still remains valuable issues to be further addressed. First of all, the phase-only grating loses encoded amplitude information which induces irrelevant diffraction orders and uneven energy distribution. Thus mode purity of the array has room for improvement. As for double ring POV element, simulation results indicate that the distance between two bright rings keeps constant with variable TC and size. Hence how to achieve controllable gap distance of double ring POV is still meaningful to explore.

## Funding

National Natural Science Foundation of China (11834001, 61905012); National Defense Basic Scientific Research Program of China (JCKY2020602C007); National Postdoctoral Program for Innovative Talents (BX20190036); China Postdoctoral Science Foundation (2019M650015); Beijing Institute of Technology Research Fund Program for Young Scholars.

## Disclosures

The authors declare no conflicts of interest.

## Supplemental document

See Supplement 1 for supporting content.

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