## Abstract

Complex vector light fields, classically entangled in polarization and phase, have become ubiquitous in a wide variety of research fields. This has triggered the demonstration of a wide variety of generation techniques. Of particular relevance are those based on computer-controlled devices due to their great flexibility. In particular, spatial light modulators have demonstrated their high capabilities to generate any vector beam, with various spatial profiles and polarization distributions. Here, we put forward a novel technique that exploits the superposition principle in optics to enable the simultaneous generation of many vector beams using a single digital hologram. As proof-of-principle, we demonstrated the simultaneous generation of sixteen vector vortex beams with various polarization distributions and spatial shapes on a single SLM, each with their own spatial shape and polarization distribution.

© 2017 Optical Society of America

## 1. Introduction

The ability to tailor the spatial properties of light has significantly changed the landscape of photonic-based applications. This is the case of complex light fields classically entangled in polarization and phase, a topic of late due to their various applications. In particular, classically entangled fields with cylindrical symmetry, commonly known as Cylindrical Vector Beams (CVB), are nowadays routinely used in fields such as laser material processing, optical tweezers, high-resolution microscopy, optical metrology, and classical and quantum communication, among many others [1–5]. The generation of CVB has been achieved internal or external to laser cavities, using geometrical phase elements [6–9] or optical interferometers [10, 11]. In the latter, two vortex beams with opposite topological charge and orthogonal polarizations are recombined interferometrically to generate CVB with various polarization distributions and spatial shapes. Remarkably, interferometric techniques have been fueled by the advent of Spatial Light Modulators (SLM), which have provided with one of the most flexible and versatile methods to shape light [12–14]. Importantly, SLMs also enable the simultaneous generation of multiple scalar beams (multiplexing), a feature that has found applications in optical tweezers and optical communications [15–19].

Most common interferometric methods, aimed at the generation of CVB, rely on splitting the amplitude of a scalar beam into two new beams to manipulate their polarizations, topological charge and amplitude independently. Their topological charge can be manipulated via SLMs or spiral phase plates while their polarizations can be controlled with half- and quarter-wave plates. CVB are then generated from a coaxial superposition of these beams carrying opposite topological charges and orthogonal circular polarization. Manipulation of the spatial shape and polarization of each beam as well as phase delays between them allows to switch from one vector beam to another. This approach, in general, can only produce one vector beam at a time.

Here we put forward a novel generation technique that enables the rapid generation of any CVB and allows for the simultaneous generation of multiple vector beams (vector-beam multiplexing) using a single hologram. Our proposed method is also interferometric in nature but differs from the rest as it relies on dividing the wavefront of the initial beam and not its amplitude. This principle allow us to split the original beam into many that can be later recombined in pairs to generate multiple vector beams simultaneously. The phase, amplitude and shape of each beam can then be digitally manipulated in an independent way to generate any vector beam without the need to manipulate any external optical element. That is, any combination of multiple vector beam can be generated by simply changing the digital hologram displayed on the SLM. As proof-of-principle, we demonstrated the simultaneous generation of sixteen CVB, each with different spatial shapes and polarization distribution. Even though we restricted to the generation of vector beams with cylindrical symmetry, our digital method enables the generation of vector beams with arbitrary polarization distributions and spatial shape. Moreover, even though we only generated close to 20 CVB, it has being previously demonstrated that SLMs allows to multiplex close to 200 scalar modes [20], which will lead to the generation of one hundred CVB.

## 2. Experiment

#### 2.1. Experimental setup

To experimentally demonstrate the simultaneous generation of multiple CVB, we used a horizontally polarized laser (*λ* = 532 nm Verdi G Coherent) and a reflective SLM (Holoeye Pluto) to generate multiplexed scalar beams. An schematic representation of the implemented setup is shown in Fig. 1. On the SLM, a set of holograms were multiplexed to generate two sets of scalar fields, each with a unique carrier frequency (grating). The frequency of each grating is carefully selected to separate the multiplexed holograms into two groups, one traveling along path **A** and another along path **B**. To accomplish this, the holograms are multiplexed such that the grating frequencies within each group are relatively small compared to the grating of the two different sets. The beams propagating along path **A** are kept with the same horizontal polarization while the ones traveling along path **B** are rotated to vertical polarization by means of a Half-wave plate (HWP) orientated at 45°. Both sets of beams are recombined into a single set, using a Polarizing Beam Splitter (PBS). To simplify the overlapping of all beams, a judicious choice of the gratings is encoded so that the alignment of one ensures the alignment of the rest. A Quarter-wave Plate (QWP) at 45° transform their states of polarization to left- and right-handed, respectively, to generate in this way multiple CVBs, each with its own polarization state or spatial shape. The resulting CVBs were observed with a CCD camera and their state of polarization analyzed with a linear polarizer placed before the CCD. A further analysis of their quality was performed using the vector quality factor, a measure that assigns the value 1 to pure vector beams and 0 to scalar beams. In this proof-of-principle experiment we restricted ourselves to the generation of vector beams with cylindrical symmetry (CVB) but any other vector beam can be generated by simply modifying the displayed hologram.

#### 2.2. Generation principle

In general, CVB can be generated as linear combinations of optical vortices of opposite topological charge *ℓ* (with *ℓ* ∈ **Z**) and orthogonal circular polarization as [21–23],

*ϕ*is the azimuthal angle of the cylindrical coordinates. As mentioned before, in order to generate multiple CVB simultaneously, two independent sets of scalar beams traveling along different paths are first generated. This can be easily achieved on the SLM by multiplexing each beam’s hologram with different spatial gratings into a single hologram, as illustrated in Fig. 2(a). The spatial shape, amplitude and phase of each beam can be manipulated accordingly via the digital hologram. The purpose of sending two sets of beams along different paths is to rotate the polarization of each group to orthogonal circular polarization ${\widehat{e}}_{L}$ and, ${\widehat{e}}_{R}$ as illustrated in Fig. 2(b). Coaxial superposition of these two sets of beams enables the generation of the desired CVB. Manipulation of the spatial shape and phase difference between both beams, via the digital hologram, makes possible to generate CVB with various polarization states as conceptualyy illustrated in Fig. 2, were six CVB are shown. It should be stressed that the gratings shown in this figure should only be taken as illustration of our concept. Analogously the intensity profiles and CVB shown only illustrates the concept, as the intensity, topological charge and phase shift of the beam within each group has to be properly selected.

To multiplex various CVB spatially separated in the far field, each must be encoded with a unique carrier frequency. This can be achieved by adding a linear phase grating to the hologram of each beam. Linear gratings in combination with spatial filters are commonly used to isolate the first diffraction order from undesired zero and higher diffraction orders. The transfer function of a linear phase grating is given by [20],

*U*and

*V*are the spatial coordinates of the generated beam in the far field, achieved with a Fourier lens of focal length

*f*.

*U*and

*V*are related to the grating frequencies (

*u*,

*v*) as

*U*=

*uλf*and

*V*=

*vλf*. Here,

*λ*is the wavelength of the input light beam. The final expression for the multiplexed hologram that we display on the SLM to generate a number

*M*of CVB takes the final form,

*π*. The first sum represents the scalar modes traveling along path

**A**whereas the second one, those traveling along path

**B**.

*h*and

_{j}*h*represent the holograms required to generate each scalar beam. In order to generate each CVB, the coordinates (

_{k}*U*,

_{j}*V*) of each beam generated by the hologram

_{j}*h*is carefully selected to match the coordinates (

_{j}*U*,

_{k}*V*) of the beam generated by the hologram

_{k}*h*.

_{k}## 3. Results

#### 3.1. Multiplexing of vector beams on the high order Poincaré sphere

The Higher Order Poincaré Sphere (HOPS) is a very useful geometrical interpretation of CVB, according to which any vector state can be represented as a point (*α*, *ϕ*) on a sphere [8, 21, 22]. In this representation, left and right circularly polarized states are positioned on the poles, CVB are located along the equator and states with elliptical polarizations occupies the rest of the sphere. A mathematical representation can be derived from Eq. (1) by choosing, ${\mathrm{\Psi}}_{R}^{-\ell}=\mathrm{cos}(\phi /2)$, ${\mathrm{\Psi}}_{L}^{\ell}=\mathrm{sin}(\phi /2)$ and −*α*_{1} = *α*_{2} = *α*/2 as,

*α*∈ [0, 2

*π*] and

*φ*∈ [0,

*π*]. In this representation, each

*ℓ*value gives rise to a unique HOPS. The digital method presented here enables the generation of any CVB on the full Poincaré sphere. Moreover, it allows for the generation of CVB corresponding to different HOPS. For example, by simply varying the phase offset e

*, we can generate any vector mode along the equator. In addition, we can vary the weighting terms, related to*

^{iα}*φ*, on either path of the interferometer by changing the phase modulation depth of the encoded hologram to move from a full vector mode to a full scalar mode on either pole of the HOPS, as previously demonstrated in [24, 25]. Figure 3 illustrates the simultaneous generation of twelve modes, defining two different HOPS, one associated to

*ℓ*= +1 [Fig. 3(e)] and the other to

*ℓ*= −1 [Fig. 3(f)]. Multiplexing of these twelve HOPB, required the multiplexing of twenty four holograms, to generate twelve beams traveling along path

**A**and twelve along path

**B**. Table 1 contains specific values

*α*,

*ϕ*of each generated mode. This table also shows the phase difference Δ

*φ*between ${\widehat{e}}_{R}$ and ${\widehat{e}}_{L}$, a value that is encoded in the SLM to generate the desired CVBs. Here we also show the frequency values (

*u*,

*v*) used for each encoded hologram. The values (

*u*,

_{A}*v*) correspond to the beams encoded along path

_{A}**A**, where as the values (

*u*,

_{B}*u*) to those traveling along path

_{B}**B**.

To test the robustness of our proposed method, we measured the quality of the multiplexed modes using the Vector Quality Factor (VQF) [26, 27]. The VQF measures the purity of a vector beam, assigning a value 0 to scalar modes and the value 1 to vector modes. The methodology to determine the VQF is further explained in the appendix. From our generated modes, we randomly selected two of them and measured their VQF before and after multiplexing. We deliberately selected a scalar and and a vector mode (mode labeled as 8 and mode labeled as 3). For the scalar case, we obtained a value *VQF _{b}* = 0.01 before multiplexing and a value

*VQF*= 0.01 after multiplexing. For the vector beam we obtained

_{a}*VQF*= 0.98 and

_{b}*VQF*= 0.96, respectively. This strongly indicates that multiplexing of twelve CVB does not affect their purity.

_{a}#### 3.2. Multiplexing of Bell states

As another example to highlight the capabilities of our technique, we multiplexed sixteen Bell states, eight of which were generated with *ℓ* = ±1 and eight with *ℓ* = ±2. Bell states are of great relevance in optical communication and quantum computing [5, 28, 29], mathematically they can be represented as,

Figure 4 shows the intensity profile of the 16 Bell states generated by properly adjusting the hologram’s topological charge and phase. To generate the *TM* modes, two holograms with opposite topological charges −*ℓ* and +*ℓ* were multiplexed on the hologram, with frequency grating values similar to the previous case. A similar procedure is applied to generate the mode *TE*, but in this case, a *π* phase offset between both beams is digitally encoded on the hologram. To generate the *HE ^{o}* and

*HE*modes, we simply interchange the

^{e}*ℓ*values in the hologram. The intensity profile after an analyzer orientated at

*ϕ*= 0°, 45°,

*ϕ*= 90° and 135°, inserted before the CCD, are shown in Figs. 4(b), 4(c), 4(d) and 4(e) respectively. In this case, the VQF values before and after multiplexing for a randomly selected mode are

*VQF*= 0.99 and

_{b}*VQF*= 0.98, respectively. Showing that the purity of the generated CVB remains practically constant.

_{a}#### 3.3. Multiplexing of vector Bessel beams

To show that our technique can be applied to CVB with other spatial shapes, we multiplexed sixteen Vector Bessel Beams (VBB). While previous methods have used SLMs to generate single VBB [30, 31], here we show the multiplexing of sixteen VBB simultaneously. To generate these modes, we can substitute the amplitude terms ${\mathrm{\Psi}}_{R}^{-\ell}$ and ${\mathrm{\Psi}}_{L}^{\ell}$ of Eq. (1) by *J*_{−}_{ℓ}*(k _{t}ρ*), respectively. Here,

*J*(

_{ℓ}*x*) is the Bessel functions of the first kind and

*k*is the transverse component of the wave vector. Bessel beams of any order can be easily generated by encoding a digital axicon on an SLM [20]. To generate multiple VBB simultaneously, we proceed in an analogous way as before by first generating two sets of scalar Bessel beams,

_{t}*J*

_{±}

*). Experimentally, we can not generate a pure Bessel beam, as it will carry infinite energy. Instead, we generate a Bessel-Gaussian beam, by multiplying the term*

_{ℓ}(k_{t}ρ*J*

_{−}

*(*

_{ℓ}*k*) by a Gaussian envelope $\mathrm{exp}(-{\rho}^{2}/{\omega}_{0}^{2})$, where

_{t}ρ*ω*

_{0}is the beam width. Figure 5(a) shows a set of sixteen multiplexed vector Bessel modes with

*ℓ*= ±1. The corresponding intensity profile for the analyzer placed at

*φ*= 0°,

*φ*= 45°.

*φ*= 90° and

*φ*= 135° are shown in 5(b), 5(c), 5(d) and 5(e), respectively. Moreover, as it is well-known, the intensity profile of a Bessel beam forms a delta ring in the far field. This provides with a straight forward way to confirm that in fact, we are generating Bessel beams, as illustrated in Fig. 5(f).

## 4. Discussion and conclusion

In this article we have presented a novel method to generate and multiplex vector modes using a single spatial light modulator. Our method is based on the multiplexing concept implemented with a single hologram displayed on an SLM. By multiplexing several holograms, it is possible to split an input light beam into many more, each with their own spatial shape and phase. To multiplex a given set of vector beams, various holograms are combined into a single one, each with a unique carrier frequency. The multiplexed holograms are split into two groups to generate two sets of beams traveling along two spatially separated paths where each set is endowed with left and right polarizations, respectively. In this scheme, the beams within each group are spatially separated to avoid any overlapping. Both sets of beams are then coaxially superimposed to generate the desired set of CVB. The digital nature of this method allows full control of each beam’s amplitude weighting, spatial shape and phase, which allows the generation of arbitrary vector beams with various spatial shapes and polarization distribution. Here we only generated cylindrical vector beams but our technique can be applied to any other geometry. Although we only demonstrated the generation of sixteen CVB, this technique allows for the generation of many more. Moreover, we believe that this generation technique also allows for the precise digital sorting of any vector beam in a similar way as [32], which is of great relevance in optical communications. Notice that previous works have reported the generation of rectilinear latices of multiple vector beams, using two SLMs [33, 34] or by splitting the SLMs’ screen into two independent screens [32]. Our generation scheme is different as it only requires a single hologram to generate multiple vector beams. A great advantage in using a single hologram is that it enables full digital control of all degrees of freedom of vector beams, including their spatial coordinates in the observation plane. Finally, it should also be considered that any beam shaping method based on a multiplexing approach suffers from an intensity decrease after the addition of each hologram, which is proportional to the number of multiplexed beams. Hence, the maximum number of beams that can be multiplexed on a single hologram depends among other things on the specific device (see [20] for more details).

## Appendix

## Vector quality factor

In order to validate our generation method, we performed a quality measure of the CVB generated in this way. A tool that has become quite popular for this purpose is the Vector Quality Factor VQF [27]. The VQF takes advantage of the similitude between quantum entanglement and non-separability of phase and polarization in vector beams. It is defined using quantum mechanics tools as,

where*C*is the concurrance [35].

The VQF ranges from 0 for scalar beams to 1 for vector beams. The parameter, *s* defined by,

*σ*〉 represents the expectation values of the Pauli operators for

_{i}*i*= {1, 2, 3}. These are obtained by a set of 12 normalized, on-axis intensity measurements, six identical measurements for two different basis states [27, 36]. For our experiment we chose the circular polarisation basis and project these into the orbital angular momentum (OAM) basis. For this, the two circular polarization (left and right) of the vector beam, are projected into a set of six holograms with topological charge

*ℓ*, −

*ℓ*and four superposition states given by exp(i

*ℓϕ*) + exp(i

*γ*) exp(−i

*ℓϕ*) with

*γ*= {0,

*π*/2

*ℓ*,

*πℓ*, 3

*π*/2

*ℓ*}, as illustrated in Fig. 6

The expectation values 〈*σ _{i}*〉 are calculated from the twelve intensity measurements as,

To determine the VQF experimentally, we measure the on-axis intensity values *I _{mn}* with

*m*,

*n*∈ {1, 2, 3}. A multiplexing approach allowed us to measure six intensity values simultaneously corresponding to one of the two polarizations. To change from one polarization basis to another, we used a QWP (

*λ*/4) set to ±45° in combination with a polarisation sensitive spatial light modulator SLM

_{2}, and OAM projections by a phase pattern on SLM

_{2}.

Figures 6 show typical intensity distribution obtained experimentally for pure vector beams 6(a) and a scalar beams 6(c). Figure6(b) and 6(d) shows the intensity values, properly normalized and arranged in the form of Table I, from which the VQF can be easily computed.

## Funding

National Research Foundation (NRF); the Claude Leon Foundation; CONACyT;CSIR-DST;

## References and links

**1. **Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics **1**, 1–57 (2009). [CrossRef]

**2. **M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photon. Rev. **7**, 839–854 (2013). [CrossRef]

**3. **H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. **19**, 013001 (2017). [CrossRef]

**4. **B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, “Characterizing quantum channels with non-separable states of classical light,” Nat. Phys. **13**, 397–402 (2017). [CrossRef]

**5. **G. Milione, M. P. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, and R. R. Alfano *et al.*, “4× 20 gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de) multiplexer,” Opt. Lett. **40**, 1980–1983 (2015). [CrossRef] [PubMed]

**6. **A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial bessel beams obtained by use of quantized pancharatnam–berry phase optical elements,” Opt. Lett. **29**, 238–240 (2004). [CrossRef] [PubMed]

**7. **L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. **96**, 163905 (2006). [CrossRef] [PubMed]

**8. **D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photon. **10**, 327–332 (2016). [CrossRef]

**9. **Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher-order poincarŐ sphere with metasurface,” Appl. Phys. Lett. **104**, 191110 (2014). [CrossRef]

**10. **S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. **29**, 2234–2239 (1990). [CrossRef] [PubMed]

**11. **S. Liu, P. Li, T. Peng, and J. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express **20**, 21715 (2012). [CrossRef] [PubMed]

**12. **A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photonics **8**, 200–227 (2016). [CrossRef]

**13. **C. Rosales-Guzmán and A. Forbes, *How to Shape Light with Spatial Light Modulators* (SPIE, 2017). [CrossRef]

**14. **C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**, 78 (2007). [CrossRef]

**15. **T. Čižmár, L. C. D. Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B **43**, 102001 (2010). [CrossRef]

**16. **J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photon. **6**, 488–496 (2012). [CrossRef]

**17. **N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science **340**, 1545–1548 (2013). [CrossRef] [PubMed]

**18. **A. Trichili, C. Rosales-Guzmán, A. Dudley, B. Ndagano, A. B. Salem, M. Zghal, and A. Forbes, “Optical communication beyond orbital angular momentum,” Sci. Rep. **6**, 27674 (2016). [CrossRef] [PubMed]

**19. **P. Li, B. Wang, and X. Zhang, “High-dimensional encoding based on classical nonseparability,” Opt. Express **24**, 15143 (2016). [CrossRef] [PubMed]

**20. **C. Rosales-Guzmán, N. Bhebhe, N. Mahonisi, and A. Forbes, “Multiplexing 200 modes on a single digital hologram,” arXiv:1706.06129v1 (2017).

**21. **G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. **107**, 053601 (2011). [CrossRef]

**22. **E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré beam patterns produced by nonseparable superpositions of laguerre-gauss and polarization modes of light,” Appl. Opt. **51**, 2925–2934 (2012). [CrossRef] [PubMed]

**23. **S. Chen, X. Zhou, Y. Liu, X. Ling, H. Luo, and S. Wen, “Generation of arbitrary cylindrical vector beams on the higher order poincaré sphere,” Opt. Lett. **39**, 5274–5276 (2014). [CrossRef]

**24. **I. Moreno, J. A. Davis, T. M. Hernandez, D. M. Cottrell, and D. Sand, “Complete polarization control of light from a liquid crystal spatial light modulator,” Opt. Express **20**, 364–376 (2012). [CrossRef] [PubMed]

**25. **Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express **23**, 17701–17710 (2015). [CrossRef] [PubMed]

**26. **M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A **92**, 023833 (2015). [CrossRef]

**27. **B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. **41**, 3407 (2016). [CrossRef] [PubMed]

**28. **M. A. Cox, C. Rosales-Guzmán, M. P. J. Lavery, D. J. Versfeld, and A. Forbes, “On the resilience of scalar and vector vortex modes in turbulence,” Opt. Express **24**, 18105–18113 (2016). [CrossRef] [PubMed]

**29. **G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. **40**, 4887 (2015). [CrossRef] [PubMed]

**30. **J. A. Davis, I. Moreno, K. Badham, M. M. Sánchez-López, and D. M. Cottrell, “Nondiffracting vector beams where the charge and the polarization state vary with propagation distance,” Opt. Lett. **41**, 2270–2273 (2016). [CrossRef] [PubMed]

**31. **A. Dudley, Y. Li, T. Mhlanga, M. Escuti, and A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. **38**, 3429–3432 (2013). [CrossRef] [PubMed]

**32. **I. Moreno, J. A. Davis, K. Badham, M. M. Sánchez-López, J. E. Holland, and D. M. Cottrell, “Vector beam polarization state spectrum analyzer,” Sci. Rep. **7**, 2216 (2017). [CrossRef] [PubMed]

**33. **S. Fu, S. Zhang, T. Wang, and C. Gao, “Rectilinear lattices of polarization vortices with various spatial polarization distributions,” Opt. Express **24**, 18486–18491 (2016). [CrossRef] [PubMed]

**34. **S. Fu, C. Gao, T. Wang, S. Zhang, and Y. Zhai, “Simultaneous generation of multiple perfect polarization vortices with selective spatial states in various diffraction orders,” Opt. Lett. **41**, 5454–5457 (2016). [CrossRef] [PubMed]

**35. **W. Wootters, “Entanglement of formation and concurrence,” Quantum Information and Computation **1**, 27–44 (2001).

**36. **M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A **92**, 023833 (2015). [CrossRef]