## Abstract

We proposed an approach for creating three-dimensional (3D) multifocal perfect vortices arrays by using a high numerical aperture objective. The position, orbital angular momentum states, number and diameter of the perfect vortices can be freely modulated by a special designed hybrid phase plate (HPP). HPP could be calculated by 3D phase shifting expression which is derived from Fourier transform theory of the Debye diffraction integral. Furthermore, we developed a novel pixel checkerboard method for adding phase information into the HPP. The segmentation of HPP is related to vortex quality and intensity uniformity. This method could fully use each pixel to modulate the light, since the spatial light modulator has to be used. Small size lattices could generate high quality and uniform intensity vortex arrays in tight focusing region, which may have potential applications in coupling, optical coding and decoding.

© 2016 Optical Society of America

## 1. Introduction

Vortex beams carrying orbital angular momentum (OAM) are attracting more and more attention due to their wide applications such as fluid flow vorticity measurement [1], underwater optical communication [2], ultra-small rotations [3–5] and measurement of the rotational Doppler frequency shift [6]. As the OAM beam provides a new photonic degree of freedom for optical communications, researchers are attracted toward their applications on free space [7–9] and fiber communication systems [10–13]. However, in these types of construction, the intensity profile and beam divergence of the these conventional vortex beams are strongly limited as their ring diameter depends on their order or topological charges [14]. Thus, two constituent optical vortices cannot closely coincide with each other. This property may create problems when coupling multiple OAM beams into an air-core fiber with fixed annular index profile [15,16]. In recent years, the concept of perfect vortex whose annular intensity profile is independent of topological charge was proposed to overcome this limitation [17,18] and the generated perfect vortex could be used to enhance surface plasmon excitation [19] and trap particles [20]. More recently, the generation of the perfect vortex is research hotspot of optical vortex application via the Fourier transforming optical system and a liquid-crystal spatial light modulator (SLM) [21], digital micromirror device [22] and two-dimensional (2D) encoding continuous-phase gratings [23]. Perfect optical vortex with continuous controlled ring radius by adjusting the separation between the lens and axicon is implemented [24]. However, the perfect vortices are obtained at a fixed propagation distance, and the position and topological charge of each perfect vortex in the array cannot be modulated. Another, with the development of OAM optical fiber communication, the specially designed air-core fibers will play an important role in future communication network [25–27]. The multiplexing OAM communication will greatly improve the channel capacity. Although the generation of the perfect vortices could be achieved in 2D surface by using a 2D continuous-phase gratings [23], the diameter of the generated optical vortex is much larger than a typical air-core fiber which is used to propagate OAM beam [26,28]. Therefore, in tight focusing system, the generated perfect optical vortices have smaller diameter (less than 10μm) and the focused perfect vortices have potential application on coupling multiple OAM modes into a specific air-core fiber (diameter of ring-core smaller than 8μm), since the perfect vortices have tunable diameter.

The central area of 2D continuous-phase gratings is divided less than one pixel and the phase information is inaccurate, although they could be used to generate multifocal arrays [29–34]. Three-dimensional (3D) optical vortices arrays can also be generated by using 2D continuous-phase gratings, but these arrays contain fixed position and topological charge [35]. In order to overcome these problems, in this paper, we attempt to generate focused perfect optical vortices arrays in 3D focal region, which not only have 3D tunable position and topological charges, but also possess stable diameter of the perfect vortices. Thus, we proposed a new method to calculate a hybrid phase pattern at the back aperture plane of an aberration-free high numerical aperture (NA) objective for generating the multifocal perfect vortices arrays. In this tight focusing system, 3D phase shift expression is developed by using Debye diffraction integral and a new phase segmentation method called Pixel Checkerboard Method (PCBM) with fully using of SLM pixels is demonstrated. There are two independent parts for generating perfect vortices arrays. The first part is the combination of spiral phase and axicon phase for single perfect vortex in tight focusing system. The second part contains the number and 3D positional information of the perfect vortices which are added by PCBM. In section 2, we introduced the theory and principle of generating single perfect vortex in tight focusing system. We also discussed the relation between perfect vortex diameter and the axicon parameter, which can be used to control the diameter of perfect vortex in tight focusing system. In section 3, we demonstrated the PCBM which is used to code the hybrid phase to generate a 3D perfect vortices array. For characterizing property of PCBM, we also discussed the impact of SLM pixels distribution on generating high quality 3D perfect vortices, which have potential applications on optical storage, parallel optical trapping and manipulation, fast material laser processing and high-resolution multi-dimensional optical imaging, especially they could be used to couple multiplexing OAM modes into fiber communication network.

## 2. Single perfect vortex in tight focusing system

In tight focusing system, based on the Debye diffraction integral, the electric field in the neighborhood of the focusing spot can be expressed as a Fourier transform [36]:

*U*(

*θ*,

*ϕ*) = exp(i

*φ*) is the pure-phase modulation function on the back aperture plane of an aberration-free high NA objective (NA = 1.32), that the incident light obeys the Abbe’s sine condition. The wave number

*k*could be given by

*k*= 2π/λ and λ is the vacuum wavelength.

*k*,

_{x}*k*,

_{y}*k*represent the wave vector in

_{z}*x*,

*y*,

*z*directions which can be expressed as

*k*= -

_{x}*k*cos

*ϕ*sin

*θ*,

*k*= -

_{y}*k*sin

*ϕ*sin

*θ*and

*k*=

_{z}*k*cos

*θ*. Then the transmitted field ${\overrightarrow{E}}_{t}$ could be expressed as,

*t*(

_{p}*θ*,

*ϕ*) and

*t*(

_{s}*θ*,

*ϕ*) are the transmission coefficients for p- and s-polarization, respectively. ${\overrightarrow{e}}_{p}$ and ${\overrightarrow{e}}_{\text{s}}$ are the unit vectors for p- and s-polarization. Upon transmission, the unit vector ${\overrightarrow{e}}_{p}$ is deflected by

*θ*and becomes ${\overrightarrow{e}}_{\text{r}}$. As shown in Fig. 1,

*α*is the maximum angle of the aperture and

*θ*= arcsin(

*r/R*× NA/

*n*) is the deflection angle, in which

_{t}*R*is the maximum radius of the aperture.

*n*= 1.518 is the medium index behind high NA objective.

_{t}*r*and

*ϕ*are the normalized polar coordinates in back aperture plane.

In the focusing region, the electric field of the each focused spot could be expressed as a Fourier transform by weighted field ${\overrightarrow{E}}_{t}$. The blue line in Fig. 1 shows the light without phase modulation and the incident light focuses on the original focusing spot. The red line in Fig. 1 represents the modulated transmitted field by a hybrid phase plate (HPP) and the local position of the focusing spot can be controlled by the hybrid phase in the focusing region. Therefore, in tight focusing system, we can use a special designed HPP, which is set at the back aperture plane of high NA objective, to modulate the electric field in the neighborhood of the focus.

To start with, we introduced the HPP which is used to generate single perfect vortex in focusing region. The HPP contains two parts and the equation could be expressed as,

in which two parts of the hybrid phase are vortex phase exp(*ilϕ*) and axicon phase exp(

*iηr*), respectively. In Eq. (3), parameter

*l*represents the topological charge of the optical vortex and

*η*represents the axicon parameter which could be used to control the diameter of the perfect vortex in the focusing region.

Using Eq. (3), HPPs can be calculated as shown in Figs. 2(a)-2(c) for generating single perfect vortex with topological charges *l* = 1, 10, 20 and the axicon parameter *η* = 11.2, 9.5, 5.8, respectively. The number of spiral branches in each spatial phase pattern equals to the topological charge of optical vortex. The corresponding intensity distribution of the three perfect vortices in the focal plane are shown in Figs. 2(d)-2(f), respectively. Three perfect vortices have same diameter 7.03μm which are independent from the topological charges. Here the diameter of perfect vortices in tight focusing system is defined as the diameter of the highest intensity circular ring on the perfect vortices. These same size perfect vortices show the diameter control capability in focal plane by adjusting the axicon parameter. But, from the normalized intensity patterns, we can see that high order perfect vortex has larger annular width. The high order perfect vortex has larger vortex diameter and in the same focal plane the modulated same diameter shows larger annular width. According to the simulation results, the intensity quality and diameter uniformity of the focused perfect vortices are excellent in tight focusing system. In order to identify the topological charges of the optical vortices, the interference patterns between the designed perfect vortices and a spherical beams are given in Figs. 2(g)-2(i). By counting the bright fringes number, the topological charge can be identified.

For modulating the diameter of perfect vortex, we investigated the relationship between the diameter and axicon parameter *η*. And we also investigated the intensity quality of the perfect vortices. Firstly, as the axicon parameter could control the diameter of perfect vortex, we can change the *η* to generate perfect vortices with same diameter and different topological charges. Since the diameter is defined as the highest intensity circular ring on the perfect vortices, the cross section intensity distribution of perfect vortices should be given. And for ensuring the uniformity of different vortices and calculating the diameters of perfect vortices, different topological charges perfect vortices (*l* = 1, 5, 10, 15 and 20) were calculated and the intensity distribution is shown in Fig. 3(a). Although the perfect vortices have same diameter, the circular intensity rings have a tendency to broaden the ring width with increasing of the topological charges. High order optical vortex (large topological charge) has larger vortex diameter, so in the tight focusing system, high order optical vortex has strong diffraction to generate small vortex diameter by using axicon phase. However, all the maximum intensity rings still have the same diameter, which is significant to couple the multiple OAM beams into a fiber with fixed annular core (diameter more than 7μm) [26]. Secondly, we calculated the relationship between the axicon parameter *η* and the diameter of perfect vortex. For coupling OAM beam into air-core fiber, small diameter ring 7.03μm was chosen as the diameter of the focused perfect vortices. As shown in Fig. 3(b), for maintaining the same diameter of optical vortices with different topological charges, the axicon parameter *η* declines with increasing of topological charges. Therefore, in tight focusing system, the topological charges and diameter of the optical vortices could be modulated by an axicon for the application on different devices or communication fibers.

## 3. 3D perfect vortices arrays

As shown in Fig. 2, the diameter of perfect vortices could be controlled in tight focusing system by using the HPP. As a coupling device, tight focusing system could also couple multiple vortices into air-core fiber or multicore fiber. Therefore, 3D multifocal perfect vortices with controllable position and topological charges are proposed and investigated. In tight focusing system, with the shift theorem of Fourier transform, the focal spot shifted in 3D space could be expressed as,

*∆x*,

*∆y*and

*∆z*are the relative displacement components from the original focusing spot as shown in Fig. 1. Then the phase information of HPP which is used to generate 3D perfect vortices arrays in focusing region can be rewritten as,

*∆x*,

*∆y*and

*∆z*), axicon parameter

*η*for controlling diameter, and topological charge

*l*. According to Eq. (5), we can calculate a HPP for generating 3D multiple perfect vortices arrays in focusing region. For maximizing the utilization of HPP phase information, we present a novel method called Pixel Checkerboard Method based on SLM pixel blocks. So the size of HPP is designed to be 1000 × 1000 in pixels and pure phase HPP could be loaded by a pure phase SLM such as: HoloEye, LETO, with 1920 × 1080 pixels of pitch 6.4 μm and calibrated for a 2π phase shift at 532 nm.

The segmentation of HPP was investigated for improving the quality of the perfect vortices in the tight focusing system. The whole back aperture plane is considered as a square checkerboard with length of 1000 pixels, which also could be considered as part of SLM. The phase information is added into lattices as shown in Fig. 4(a), in which there are four color lattices and each lattice contains 8 × 8 pixels for generating four perfect vortices in focal plane. The yellow lattices have been filled with the hybrid phase which is used to generate perfect vortex with the topological charge *l* = 2 and the perfect vortex is located at (10, 0) μm in the focal plane. For creating perfect vortices array in original focal plane (*∆z* = 0) according to Eq. (5), the red, blue, and green lattices are calculated with the topological charges *l* = 4, 6, 8 and the perfect vortices coordinates are (*∆x*, *∆y*) = {(0, 10), (−10, 0), (0, −10)} μm, respectively. These four colors lattices as a basic cell fills up the whole plane for generating the HPP. The calculated HPP is shown in Fig. 4(d), and the corresponding intensity distribution of perfect vortices is shown in Fig. 4(g), in which four perfect vortices could be seen.

However, in the focal plane we can see broken rings which are the interference with undesired rings caused by bigger lattices (8 × 8 pixels). Then we set smaller lattice (5 × 5 pixels) as shown in Fig. 4(b) for improving the phase distribution uniformity in the HPP. The calculated HPP is shown in Fig. 4(e) and the corresponding perfect vortices array are shown in Fig. 4(h). From the intensity distribution we can clearly find the quality of perfect vortices is improved. That is the lattice size could influence the vortex quality. As the SLM can modulate phase in pixel level, the lattices can be set as one pixel. Figure 4(c) shows the schematic pattern of HPP with smallest lattices (single pixel). The corresponding HPP and intensity distribution of the four perfect vortices could be seen in Figs. 4(f) and 4(i). Comparing Figs. 4(h) and 4(i), the quality of this array is not obviously improved again. The reason is that the lattices with 5 × 5 pixels are enough to generate four uniform perfect vortices array in the focal plane. It also means that the single pixel lattices could be used to generate more perfect vortices. Using PCBM we can control the number and position of perfect vortices in 2D focal plane.

To investigate the intensity uniformity caused by lattice size, we calculated the intensity of each perfect vortex. The intensity of vortex is defined as the sum of each pixel’s gray value (the image is a 256 order gray image) of the optical vortex. From Figs. 4(g) and 4(h), we could see that with changing of lattice size the generated perfect vortices have different intensity. Big lattice size generates visually inhomogeneous vortices. So the uniform intensity is investigated for improving the vortex quality. As shown in Fig. 5(a), with decreasing of lattice size, the vortices intensity increase. When the lattices are 5 × 5 pixels, all the perfect vortices have almost identical intensity which is irrelevant to the topological charge. Moreover, we also compared the intensity difference of perfect vortices in the same array. The uniformity is shown in Fig. 5(b). The cost function of the uniformity of the four vortices array is defined as *ξ* = 2*I _{min}*/(

*I*+

_{max}*I*), where

_{min}*I*and

_{max}*I*represent the maximum and minimum intensity of vortices. The uniformity of this array reaches 99% when the length of lattice is 6 × 6 pixels from Fig. 5(b). So if there are more foci in the focusing region, we could shorten the lattice length for guarantee the perfect vortex quality.

_{min}According to the discussion above, we could use PCBM to control the number and position of perfect vortices. By combining axicon which could be used to control topological charges and diameter of perfect vortices, the 3D perfect vortex arrays in tight focusing region with controllable number, position, topological charges and vortex diameter could be realized. Based on Eq. (5) we can calculate a HPP for generating more than one perfect vortices in focusing region. Using PCBM with single pixel segmentation, we created a HPP as shown in Fig. 6(a), for generating 9 perfect vortices in the focusing region. Along the beam propagation, we designed two focal plane in focusing region, first (original) focal plane (*∆z* = 0) and the second focal plane (*∆z* = 80μm). The first focal plane contains 4 perfect vortices with the topological charges *l* = 6, −7, 8 and −9 and the corresponding intensity shows in Fig. 6(b). The diameter and center coordinates of perfect vortices are 7.03μm and (*∆x*, *∆y*, *∆z*) = {($5\sqrt{2}$, $5\sqrt{2}$, 0) ($-5\sqrt{2}$, $5\sqrt{2}$, 0) ($-5\sqrt{2}$, $-5\sqrt{2}$, 0) ($5\sqrt{2}$,$-5\sqrt{2}$, 0)} μm, respectively. In the second focal plane, there are five perfect vortices with topological charges *l* = 1, −2, 3, −4 and 5 as shown in Fig. 6(c). The uniform vortex diameter is 7.03μm and the center coordinates are (*∆x*, *∆y*, *∆z*) = {(0, 0, 80) (10, 0, 80) (0, 10, 80) (−10, 0, 80) (0, −10, 80)} μm, respectively. From Figs. 6(b) and 6(c) we can see perfect vortices decline in quality. Since in the focusing region there are 9 perfect vortices and two focal planes, two reasons cause the loss of quality. First, 9 lattices are used to generate 9 perfect vortices and in the HPP the pixel number for generating one perfect vortex decreases. Second, the two focal plane result in crosstalk each other. Although the divergence angle (maximum angle of the aperture, α, as shown in Fig. 1) is more than 60 degree in tight focusing system, the interference still affect the quality of perfect vortices. Therefore, increasing the distance of two focal planes can achieve high quality perfect vortices arrays in the focal region. The simulation results show that the distance should be more than 80μm. In Fig. 6(d), we can clearly see two perfect vortex arrays in the focusing region. From the simulation results we can see that all the vortices have same diameter, it is difficult to identify the topological charges. So the simulated interference patterns are given for counting the bright fringes as topological charges. From Figs. 6(e) and 6(f), we can see the positive vortices are clockwise and negative vortices are anticlockwise.

## 4. Conclusions

In summary, we have proposed a method to create 3D perfect vortices arrays with uniform intensity. 3D phase shifting expression and PCBM are used to calculate HPP for modulating 3D position, topological charges, diameter and number of perfect vortices in tight focusing region. The HPP based on pixel segmentation can generate high quality and uniform intensity perfect vortices arrays. The method PCBM could fully use each pixel of SLM to modulate the incident light. Because of the flexibility of the position, topological charges, number and diameter of perfect vortices, the generated vortices have potential applications on optical storage, parallel optical trapping and manipulation, fast material laser processing and high-resolution multi-dimensional optical imaging, especially they also could be used to couple multiplexing OAM modes into fiber communication network.

## Funding

National Natural Science Foundation of China (NSFC) (11304064, 11304065, 51506035); National Key Technology R&D Program (2014BAA02B03); Special Scientific Research Found of Public Service Industry of Quality Inspection Project (201410030-03).

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