## Abstract

In this paper, we propose a type of rectilinear lattices of polarization vortices, each spot in which has mutually independent, and controllable spatial polarization distributions. The lattices are generated by two holograms under special design. In the experiment, the holograms are encoded on two spatial light modulators, and the results fit very well with theory. Our scheme makes it possible to generate multiple polarization vortices with various polarization distributions simultaneously, for instance, radially and azimuthally polarized beams, and can be used in the domains as polarization-based data transmission system, optical manufacture, polarization detection and so on.

© 2016 Optical Society of America

## 1. Introduction

Light beams with isotropic polarizations, such as linearly, elliptically, and circularly polarized beams, have been widely studied and applied in lots of fields. Polarization vortices [1], also known as vector beams, which can be described by the higher-order Poincaré sphere [2], changes the understanding of polarization considerably, and has contributed to an improvement of optical systems. For instance, particle trapping [3], optical communications [4], laser material processing [5], surface plasma excitation [6], nonlinear optics [7] and so on. In contrast to homogeneously polarized beams, polarization vortices have anisotropic spatial polarization distributions, and can be written as [8]:

*p*the polarization order,

*φ*the azimuthal angle, and

*A*(

*r*) the amplitude distribution.

*φ*is the initial orientation of the field vector for

_{0}*φ*= 0, which depends the state of polarization of a polarization vortex. For instance

*p*= 1 and

*φ*= 0 or π/2 characterizes radial or azimuthal polarization, respect. Particularly

_{0}*p*= 0 represents linearly polarized beams, which can be regarded as zero-order polarization vortices.

Polarization vortices can be generated through inserting mode-selection elements [9–11] like q-plates [11] in the resonator. Another welcomed approach called transforming other beams as optical vortices into polarization vortices outside the resonator is widely used. Optical vortices are a kind of structure beams have helical wavefronts and carry orbital angular momentum (OAM), which can be easily realized by computer-generated holograms [12,13]. Two optical vortices with opposite spin angular momentum (SAM) and OAM can be combined together, resulting in polarization vortices, as expressed by Eq. (2) [14]:

In Eq. (2), |*R _{p}*〉and |

*L*〉represent the orthonormal circular polarized optical vortex beams with topological charge -

_{p}*p*and

*p*carrying SAM of -

*ħ*and

*ħ*, OAM of -

*pħ*and

*pħ*, and can be written as:

Based on Eqs. (2)-(4), schemes as Sagnac interferometer [15], Twymann-Green interferometer [16], Wollaston prism [17], spatial light modulation [5,18] were used to generate vortex-beams. However, to the best of our knowledge, only one-vortex beams were generated by these systems. In some cases, beams with different polarization distributions are needed simultaneously. For instance, in laser manufacturing, radially polarized beams are used for laser cutting [19] and azimuthal polarized beams for punching [20]. In addition, in polarization vortices based optical communications, generating beams with different spatial polarization is also of importance [5].

In this paper, motivated by the gratings proposed by Romero and Dickey [21], and based on Eq. (2), we demonstrate a way to generate multiple polarization vortices in the form of rectilinear lattices, each spot in which has mutually independent, and controllable spatial polarization distributions. The basic principle of generating the polarization vortices lattices is analyzed firstly. Then, a setup consists of two phase-only liquid-crystal spatial light modulators (SLM) encoded by special-designed holograms, is built to do the experiment. Finally, polarization vortices with different polarization states, including radial and azimuthal polarized beams, are generated simultaneously. The states of polarization in different diffraction orders are measured with a rotating polarizer, which fit well with the theory.

## 2. Designing holograms

If an additional phase *σ* is introduced into |*L _{p}*〉, then Eq. (2) can be written as:

*C*= exp(

*iσ)*. Equation (5) indicates the introduction of

*σ*contributes to the changing of initial polarization, which means setting different

*σ*when generating optical vortices results in the polarization’s variation of polarization vortices.

Based on Eq. (5), for the sake of generating rectilinear lattices of polarization vortices with different polarization states, a feasible way is designing two special holograms. One is to create multiple circular polarized optical vortices with their initial phase varying with diffraction orders, while the other has same functionalities but identical initial phase. Meanwhile, the SAM and OAM carried by the two optical vortices arrays are opposite. By building a proper optical system, the two optical vortices arrays could be combined together between corresponding diffraction orders. And thus rectilinear lattices of polarization vortices can be obtained.

Therefore, the key is how to design the two holograms. Thanks to Ref [21], we can design a hologram that produces a selected number of diffraction orders with selective intensity and phase control. The hologram’s phase distribution *θ*(*x*) can be Fourier expanded as:

*m*the diffraction order,

*T*the grating constant.

*c*is the Fourier coefficients and can be written as:

_{m}*c*are the complex amplitude of each diffraction order generated by the phase grating indeed and can be expressed as:

_{m}*p*denotes the topological charge,

*φ*is the azimuthal angle. |

*c*| and

_{m}*σ*are the amplitude and initial phase, respect. Equation (8) indicates that the desired target orders

*m*, amplitude |

*c*|, initial phase

_{m}*σ*, as well as the topological charge

*p*can be defined. Therefore, optical vortices array with spots of arbitrary initial phases and topological charge can be obtained by setting parameters in Eq. (8) appropriately, which paves the way for the rectilinear lattices of polarization vortices.

## 3. Experimental setup

The experimental setup sketched in Fig. 1 realizes the combination of two optical vortices arrays with orthogonal circular polarizations. The fundamental Gaussian mode with the wavelength of 1550 nm is generated by a laser diode (LD), and emerged from a collimator (Col.). A 45° arranged polarizer (P1) is placed behind the collimator, leading to the 45° linear polarized Gaussian beams with equal weight of horizontal and vertical polarization components. Then, the orthogonal linear polarized beams are incident into SLM1 (Holoeye, PLUTO-TELCO-013-C, nominal resolution 1920 × 1080 pixels, pixel pitch 8.0 μm). Since the performance of the polarization control [22] of SLMs, only the horizontal linear polarization component can be modulated, while the vertical linear component unaffected. After passing through a 45° arranged half wave plate (HWP), the horizontal and vertical polarization components will be reciprocal interchanged. And thus the initial vertical polarization component could be modulated by SLM2. To image the SLM1’s encoding hologram on SLM2 with unit magnification, a 4-*f* system is utilized, with SLM1 and SLM2 located at the object focus of lens L1 and the image focus of lens L2 separately, as shown in Fig. 1. A 45° arranged quarter wave plate (QWP) is placed behind SLM2, to transform the two combined orthogonal linearly polarized beams into left and right circularly polarized helical beams. An infrared CCD camera with the spectral range of 900~1700 nm is placed at the focal plane of L3, to observe the far field diffraction patterns. A rotating polarizer P2 is arranged in front of CCD, to analyze the polarization distribution.

## 4. Results and discussions

In this section, three cases are carried out as the examples. Let’s begin with the rectilinear lattices of two spots. Based on the designing procedure mentioned in section 2, we design two holograms to generate polarization vortices at ± 1st diffraction order, where beams at −1st diffraction order are radially polarized, while at + 1st diffraction order are azimuthally polarized. The parameters’ setting of designing the holograms are shown as follows. For the -1st diffraction order, *σ*SLM1*-1=0, σSLM2-1=0, pSLM1-1=1, pSLM2-1=1. For the +1st diffraction order, σSLM1+1=0, σSLM2+1=π, pSLM1+1=1, pSLM2+1=1. One can find that the choosing of the topological charge of the two holograms are the same. The reason is the using of reflective SLM in the setup, which will introduce an additional reflection during the beams’ propagation. One reflection will contribute to the opposite topological charge. The initial horizontal polarized part modulated by SLM1 is reflected by two SLMs twice totally, while the initial vertical polarized part modulated by SLM2 is reflected by SLM2 once. Hence, only by setting the same topological charge when designing the holograms can Eq. (5) be satisfied. In this case, according to Eq. (5), the polarization vortex in the −1st diffraction order is |ψ_{1,0}〉and in + 1st diffraction order is |ψ_{1, π/2}〉. The holograms that encoded on SLM1 and SLM2 are displayed in Figs. 2(a) and 2(b). The experimental and simulated results are sketched in Figs. 2(c) and 2(d).*

*Rectilinear lattice with four states, where | ψ_{-2,0}〉, |ψ_{-1,0}〉, |ψ_{1, π/2}〉and |ψ_{2, π/2}〉are in the −2nd, −1st, 1st and 2nd diffraction orders, is also proposed. To accomplish these states, both of the two holograms have the same topological charge distribution as pSLM1-2=pSLM2-2=-2, pSLM1-1= pSLM2-1=-1, pSLM1+1= pSLM2+1=1, and pSLM1+2= pSLM2+2=2. And the relative phases are forced to σSLM1-2=σSLM1-1=σSLM1+1=σSLM1+2=σSLM2-2=σSLM2-1=0, σSLM2+1=σSLM2+2=π. Based on the constraints above, the two holograms are computed as Figs. 3(a) and 3(b). The experimental and simulated rectilinear lattices with four states are displayed in Figs. 3(c) and 3(d).*

*The previous two cases have common characteristics that the existent diffraction orders are in symmetrical distribution. Here we show that rectilinear lattice with asymmetrical states distribution can also be generated. Take the lattices with | ψ_{1,0}〉, |ψ_{2,0}〉, |ψ_{3,0}〉in +1st, +2nd and +3rd diffraction orders as example, the relative phase σSLM1+1=σSLM1+2=σSLM1+3=σSLM2+1=σSLM2+2=σSLM2+3=0 and the topological charge pSLM1+1= pSLM2+1=1, pSLM1+2= pSLM2+2=2, pSLM1+3= pSLM2+3=3. Then the holograms can be obtained as shown in Figs. 4(a) and 4(b). And the experimental and simulated results are displayed in Figs. 4(c) and 4(d).*

*We have shown the generation of rectilinear lattices of polarization vortices with mutually independent, and controllable spatial polarization distributions. The experimental patterns fit well with theory. However, it can be found that some of the patterns obtained by CCD shown in Figs. 2-4 have a bright spot in the 0th diffraction order. The reasons can be known as two aspects as the SLM we use and the holograms’ property. On the one hand, some of the holograms have a large phase leap in the center (for instance, Figs. 3(a) and 3(b)), which the SLMs’ resolution can’t meet, leading to the unmodulated rays in 0th order. Therefore, it’s better to arrange no polarization vortex in the 0th order. On the other hand, the holograms designed by the methods given by Ref [21]. are not very perfect as the diffraction efficiency can’t reach 100% in theory, which means some of rays will be project to undesirable orders. This phenomenon is obvious especially for the asymmetrical states distribution, just like the patterns in Fig. 4(d).*

*One thing should be noticed that in our scheme, multiple polarization vortices are generated as diffracted beams with different orders, which may introduce aberration or astigmatism for higher-order diffraction orders. However, it will have little influence on the practice application, for we can generate two-dimensional lattices to avoid high order diffraction orders.*

*5. Conclusions*

*5. Conclusions*

*In brief, we have reported a kind of rectilinear lattices of polarization vortices with various spatial polarization distributions, which make it available to generate multiple polarization vortices simultaneously. The presented experimental results demonstrate the ability of the proposed method to generate arbitrary rectilinear lattices of polarization vortices, which makes sense for such domains as optical trapping, polarization detection, optical manufacture and so on.*

*Funding*

*Funding*

*National Basic Research Program of China (973 Program) (2014CB340002, 2014CB340004).*

*References and links*

*References and links*

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