## Abstract

A flexible approach is presented to generate vector beams with arbitrary polarization and complex amplitude by means of two cascaded transmissive liquid crystal spatial light modulators (LCSLMs). The configuration of the cascaded LCSLM system and its modulation characteristic are introduced. Theoretical analysis and experimental demonstration prove that the system in combination with a double-pass computer-generated hologram and a black-and-white pattern can generate vector beams with arbitrary polarization and complex amplitude by respectively controlling the complex amplitudes of two orthogonal polarization components of the beams. Using this system, we successfully generate radially polarized vector beams with helical phase distributions and vector Bessel beams with inhomogeneous amplitude distributions in experiments.

© 2014 Optical Society of America

## 1. Introduction

In recent years, optical vector beams with spatially variant polarization states have attracted considerable attention for their potential applications in high-resolution imaging, plasmonic focusing, nanoparticle manipulation, laser machining, remote sensing, and singular optics [1]. = Many methods for generation of optical vector beams have been proposed [2–24], which can be classified into static and dynamic methods. Of the static methods, polarization-selective computer-generated holograms etched on birefringent substrates [2], liquid crystal gel [3], uniaxial crystal [4], subwavelength metal grating [5–7], rotating Glan polarizing prism [8] have been used to convert a homogeneous polarization into inhomogeneous one. On the other hand, various types of interferometers such as Mach-Zehnder interferometer [9], diffractive optical element interferometer [10], Sagnac interferometer [11], and pentaprism interferometer [12] also have been used to combine two orthogonal polarization modes into different polarization distributions. Although these methods contribute to the generation of vector beams and some of them possess diffraction efficiency theoretically approaching 100%, they have a common disadvantage that they are not versatile and dynamic with the designed configurations and optical elements.

In order to overcome the limitation of the above static approaches, programmable spatial light modulators (SLMs) have been widely used for generating arbitrary vector beams in recent years [13–24]. In [13, 14], a wavelength-independent tunable liquid crystal (LC) polarization rotator using two cascaded nematic LC retarders was reported. In [15, 16], nematic liquid-crystal SLMs (LCSLMs) were used as a controlled wave plate to control the polarization. In [17–20], transmissive twist-nematic LCSLMs (TN-LCSLMs) were adopted to control the amplitude and phase of two orthogonal polarization components for the purpose. Furthermore, Maurer *et al*. [21], Davis *et al*. [22], and Moreno *et al*. [23] proposed their methods to control the orthogonal polarization components using a nematic LCSLM or a parallel-aligned LCSLM. Clegg *et al.* [24] also proposed a method for control of the arbitrary polarization using a ferroelectric LCSLM.

In this paper we present a method for generating vector beams with arbitrary polarization and complex amplitude. In this method, two transmissive TN-LCSLMs are adopted and cascaded through a telecentric imaging assembly, because a single TN-LCSLM can’t control the amplitude and phase independently and simultaneously at each pixel. We display a specially designed double-pass computer generated hologram (DPCGH) and a black-and-white pattern (BWP) on two TN-LCSLMs respectively to implement a double-pass, common path interference. This configuration has advantage that the orthogonal polarization components share the same optical path and elements so that the output is less affected by the external disturbances [20]. It can generate vector beams with arbitrary polarization and complex amplitude distributions simply by changing the parameters of the DPCGH. The theoretical analysis and experimental investigation for this approach will be introduced in Section 2 and Section 3.

## 2. Principle and analysis

#### 2.1 Experimental setup with the cascaded LCSLM system

Figure 1 shows the principle architecture of the experimental setup used in our method. An expanded and collimated laser beam is incident on a cascaded LCSLM (C-LCSLM) system composed of two identical transmissive TN-LCSLMs (SLM-1 and SLM-2) and a telecentric imaging assembly. The telecentric imaging assembly consists of two identical telecentric lenses, which can restrict the change of magnification with the help of a common diaphragm between the two telecentric lenses. SLM-1 is placed in the front object plane of the telecentric imaging assembly and SLM-2 is in the conjugate image plane. Because the telecentric imaging assembly can form an image equivalent to the object in size, SLM-1 and SLM-2 can be aligned for pixel-to-pixel superimposition and finally implement a C-LCSLM system with more freedom and control parameters for generation of a desired output beam.

In this paper, we mainly try to apply the system for generation of arbitrary vector beams. For the purpose, a linear polarizer P with its polarizer axis parallel to the input LC director of SLM-2 is inserted in front of SLM-2. Under this condition, SLM-2 can be used as a programmable polarization rotator. According to the wave-guiding effect of the twist-nematic LCSLM [25], if we display a binary gray pattern with only white and black gray levels onto SLM-2, the pixels with black gray level (corresponding to a zero control voltage) will rotate the linear polarization beam passed through the polarizer P 90 degrees, while the linear polarization direction of the beam will be left unchanged through the pixels with white gray level (corresponding to a high control voltage). Such two orthogonal polarization encoding channels can be gotten through SLM-2, one including the pixels with white gray level and the other with zero gray level. If we further design a DPCGH and display it on SLM-1 to respectively control the complex amplitudes of the beams propagating to the pixels on SLM-2, an arbitrary vector beam can be generated by recombination of two orthogonal polarization components.

#### 2.2 Modulation characteristic of the C-LCSLM system

The Jones matrix of the TN-LCSLM can be generally described as [25, 26]

where*A*

_{g},

*B*

_{g},

*C*

_{g}, and

*D*

_{g}are Jones parameters of the TN-LCSLM. These parameters usually are functions of the exerted voltage (often corresponding to the gray level

*g*of the picture displayed on the TN-LCSLM), which can be calibrated by experiments.

Generally speaking, if a beam with a Jones vector of **E _{i}** is incident on the SLM-1, the Jones vector of the output beam from the SLM-2 of the C-LCSLM system can be written as

**E**is an x-linear polarization beam with its amplitude is

_{i}*E*. Then Eq. (2) can be further expressed as

_{ix}_{g,1}, A

_{g,2}, and C

_{g,2}are Jones parameters of SLM-1 and SLM-2, respectively. From Eq. (3) we can see that the output of the C-LCSLM can be controlled by the values of

*A*

_{g,1},

*A*

_{g,2}, and

*C*

_{g,2}, where

*A*

_{g,1}can be determined by the DPCGH displayed on SLM-1 and

*A*

_{g,2},

*C*

_{g,2}will be decided by the gray levels of the picture displayed on SLM-2. According to the wave-guiding effect of 90 degree TN-LCSLM mentioned above, we find that

*A*

_{g,2}in Eq. (3) will be equal to zero for the pixels controlled by black gray level. In contrast,

*C*

_{g,2}keeps to be zero for the pixels with white gray level. If the picture displayed on SLM-2 is designed into a black-and-white interleaved pattern with black gray level in even lines and white gray level in odd lines, Eq. (3) can be further expressed as

*comb*() is the sampling function,

*d*is the period of the interleaved pattern. From Eq. (4) it can be seen that, if the pixels of SLM-1 are also divided into two groups (one corresponding to even lines and another corresponding to odd lines) and controlled by a specially designed DPCGH, we can control the complex amplitudes of two orthogonal polarization components independently to generate the desired output vector beams.

#### 2.3 Design of the DPCGH

It is well known that a vector beam with arbitrary polarization and complex amplitude can be divided into two orthogonal polarizations, which can be expressed as

*u*and

*v*are the complex amplitude distributions of the two components; $tx$, $ty$ and ${\varphi}_{x}$,${\varphi}_{y}$are their normalized amplitudes and phases, respectively. The polarization state distribution of the vector beam will be determined by the phase difference between the two components.

As mentioned above, we should display a DPCGH on SLM-1 to control the complex amplitudes of the orthogonal polarization components of the output vector beam, respectively. The scheme of designing the DPCGH is shown in Fig. 2.

Firstly we design two computer generated holograms (CGHs) of CGH-*u* and CGH-*v* as shown in Figs. 2(a) and 2(b). CGH-*u* is encoded for the complex amplitude of the component *u*, and CGH-*v* is encoded for the complex amplitude of the component *v*, which can be expressed as

*f*is the carrier frequency of the reference beam. Then, we design a couple of complementary sampling masks of MASK-1 and MASK-2 as shown in Figs. 2(e) and 2(f), which are used to sample CGH-

_{0}*u*and CGH-

*v*respectively. CGH-

*u*and CGH-

*v*are sampled into CGH-1 and CGH-2 as shown in Figs. 2(g) and 2(h). The final DPCGH as shown in Fig. 2(i) is combined by the two interleaved CGHs; its odd lines and even lines are coming from the corresponding lines of CGH-1 and CGH-2 respectively. The transmission function of such a DPCGH can be expressed asWhen we display the designed DPCGH on SLM-1, the parameter

*A*

_{g,1}in Eq. (4) will be proportional to the gray level of the DPCGH. Furthermore, MASK-1 just satisfies the need of the BWP displayed on SLM-2. Finally the output beam

**E**of the C-LCSLM system can be written as

_{o}#### 2.4 Spatial filtering and inverse Fourier transform

The 4-*f* spatial filtering system can extract the diffraction order we wanted and reconstruct the desired vector beam through an inverse Fourier transform. The Fourier transform of the orthogonal polarization components can be expressed as

*n = 0*. The extracted order can be written as

According to the above analysis, we can summarize our method for generation of a vector beam based on the C-LCSLM system as follow:

- 1) Design a DPCGH according to Eq. (8). In this step, for avoiding the influence of the relative phase difference, we need to multiply $ty$ by a conjugate additional phase term of $\mathrm{exp}(i\Delta \beta )$.
- 2) Respectively display the DPCGH and the BWP on SLM-1 and SLM-2 to realize the desired modulation of the two orthogonal polarization components.
- 3) Extract the diffraction order of the beam we wanted through a spatial filter and reconstruct the final output vector beam after an inverse Fourier transform.

## 3. Experiments and discussions

In our experiment, a horizontal polarization He-Ne laser with $\lambda =632.8nm$ is expanded, collimated and directed onto the C-LCSLM system, two transmission TN-LCSLMs with $1024\times 768$ pixels and pixel size of 14μm are adopted. The relative phase difference ($\Delta \beta $) between the two orthogonal polarization components introduced by SLM-2 is about $0.49\pi $. The object distance (or the image distance) of the telecentric imaging assembly is 150mm (from the object or image plane to the first optical surface of the telecentric lens). For matching the size of the CCD, we use two Fourier lenses with the focal length of 400mm and 180mm respectively.

A typical type of vector beams is the cylindrically symmetric polarization beams, which can be expressed as

*m*. Figures 3(e)–3(g) show the concentric doughnut extinction intensity distributions. The concentric doughnuts will emit or gather as we rotate the polarization analyzer. The number of concentric doughnut is proportional to

*n*. Figure 3(h) shows the situation with

*m*= 2 and

*n*= 1; the spiral extinction intensity distribution is also rotate as we rotate the polarization analyzer.

Of course, we can add a phase distribution to the vector beams as shown in Eq. (5). Here we take $tx=\mathrm{cos}\theta $, $ty=\mathrm{sin}\theta \cdot \mathrm{exp}(i\Delta \beta )$, and${\varphi}_{x}={\varphi}_{y}=5\theta $. In this condition, a radially polarized vector beam with a helical phase wavefront can be generated. When we put a polarization analyzer in front of the CCD sensor, the intensity distribution shown in Fig. 4(e) is similar to the results of radially polarized beams shown in Fig. 3(b). The central dark spot originates from the singularity in both polarization and phase. The size of central dark spot shown in Fig. 3(b) is less than that in Fig. 4(e). This may be caused by the helical phase existed in the latter situation. For figuring out the phase structure of the vector beams, we can superimpose the vector beams with a plane wave. The fork in the interference patterns shown in Figs. 4(b) and 4(f) indicate the topological charge of the helical phase. We can also separate the polarization singularity and the phase singularity of the vector beams by controlling the phase parameter and polarization parameter of the vector beams. Figures 4(c), 4(d), 4(g), and 4(h) show the intensity distributions and the interference patterns of vector beams with separated polarization singularity and phase singularity.

The C-LCSLM system also can be used to produce vector beams with inhomogeneous amplitude distributions. For example, if we substitute $tx={J}_{0}(r)\cdot \mathrm{cos}\theta $, $ty={J}_{0}(r)\cdot \mathrm{sin}\theta \cdot \mathrm{exp}(i\Delta \beta )$, and ${\varphi}_{x}={\varphi}_{y}=0$ into Eqs. (6) and (7), a so-called vector Bessel beam can be generated. This vector beam has the cylindrically symmetric zero-order Bessel intensity distribution with a polarization singularity in the center. Figures 5(a)–5(d) show the intensity distributions of the vector Bessel beams with different analyzer orientations. After further diffracting a distance of about 100mm, the corresponding intensity distributions are shown in Figs. 5(e)–5(h). Figures 5(i)–5(l) show the intensity profiles along the analyzer orientations on the two record planes. The blue solid lines and red dashed lines correspond to the two cases. It can be seen from Fig. 5 that the vector Bessel beam is also a non-diffraction beam.

## 4. Conclusion

In summary, we present a flexible configuration to generate vector beams with arbitrary polarization and complex amplitude by a C-LCSLM system composed by two TN-LCSLMs and a telecentric imaging assembly. In this configuration, an encoded DPCGH is displayed on the first TN-LCSLM of SLM-1 and a BWP is displayed on another one of SLM-2. The theoretical analysis and experimental demonstration have proved that the C-LCSLM system in combination with the DPCGH and the BWP can generate the vector beams with arbitrary polarization and complex amplitude by respectively controlling the complex amplitudes of two orthogonal polarization components of the beams. As examples, we successfully generate radially polarized vector beams with helical phase distributions and vector Bessel beams with inhomogeneous amplitude distributions. We hope this method could provide us a convenient way to study the properties of other complex vector beams in experiments.

## Acknowledgments

This work is supported by the NSFC (Grant No. 11074152 and No. 10934003) and the Research Foundation for the Doctoral Program of Higher Education (Grant No. 20113704110002), as well as a Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J11LA52).

## References and links

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

**2. **F. Xu, J. E. Ford, and Y. Fainman, “Polarization-selective computer-generated holograms: design, fabrication, and applications,” Appl. Opt. **34**(2), 256–266 (1995). [CrossRef] [PubMed]

**3. **H. W. Ren, Y. H. Lin, and S. T. Wu, “Linear to axial or radial polarization conversion using a liquid crystal gel,” Appl. Phys. Lett. **89**(5), 051114 (2006). [CrossRef]

**4. **A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express **18**(10), 10848–10863 (2010). [CrossRef] [PubMed]

**5. **Z. Bomzon, V. Kleiner, and E. Hasman, “Computer-generated space-variant polarization elements with subwavelength metal stripes,” Opt. Lett. **26**(1), 33–35 (2001). [CrossRef] [PubMed]

**6. **W. B. Chen, W. Han, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Generating cylindrical vector beams with subwavelength concentric metallic gratings fabricated on optical fibers,” J. Opt. **13**(1), 015003 (2011). [CrossRef]

**7. **W. J. Cai, A. R. Libertun, and R. Piestun, “Polarization selective computer-generated holograms realized in glass by femtosecond laser induced nanogratings,” Opt. Express **14**(9), 3785–3791 (2006). [CrossRef] [PubMed]

**8. **Q. Hu, Z. H. Tan, X. Y. Weng, H. M. Guo, Y. Wang, and S. L. Zhuang, “Design of cylindrical vector beams based on the rotating Glan polarizing prism,” Opt. Express **21**(6), 7343–7353 (2013). [CrossRef] [PubMed]

**9. **S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. **29**(15), 2234–2239 (1990). [CrossRef] [PubMed]

**10. **K. C. Toussaint Jr, S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. **30**(21), 2846–2848 (2005). [CrossRef] [PubMed]

**11. **V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. **45**(33), 8393–8399 (2006). [CrossRef] [PubMed]

**12. **C. Y. Han, R. S. Chang, and H. F. Chen, “Solid-state interferometry of a pentaprism for generating cylindrical vector beam,” Opt. Rev. **20**(2), 189–192 (2013). [CrossRef]

**13. **O. Aharon and I. Abdulhalim, “Liquid crystal wavelength-independent continuous polarization rotator,” Opt. Eng. **49**(3), 034002 (2010). [CrossRef]

**14. **A. Safrani and I. Abdulhalim, “Liquid-crystal polarization rotator and a tunable polarizer,” Opt. Lett. **34**(12), 1801–1803 (2009). [CrossRef] [PubMed]

**15. **M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express **18**(1), 212–217 (2010). [CrossRef] [PubMed]

**16. **S. Tripathi and K. C. Toussaint Jr., “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Express **20**(10), 10788–10795 (2012). [CrossRef] [PubMed]

**17. **D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express **21**(5), 5432–5439 (2013). [CrossRef] [PubMed]

**18. **X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. **32**(24), 3549–3551 (2007). [CrossRef] [PubMed]

**19. **H. Chen, J. J. Hao, B. F. Zhang, J. Xu, J. P. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. **36**(16), 3179–3181 (2011). [CrossRef] [PubMed]

**20. **S. Liu, P. Li, T. Peng, and J. L. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express **20**(19), 21715–21721 (2012). [CrossRef] [PubMed]

**21. **C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. **9**(3), 78 (2007). [CrossRef]

**22. **J. A. Davis, G. H. Evans, and I. Moreno, “Polarization-multiplexed diffractive optical elements with liquid-crystal displays,” Appl. Opt. **44**(19), 4049–4052 (2005). [CrossRef] [PubMed]

**23. **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**(1), 364–376 (2012). [CrossRef] [PubMed]

**24. **J. H. Clegg and M. A. A. Neil, “Double pass, common path method for arbitrary polarization control using a ferroelectric liquid crystal spatial light modulator,” Opt. Lett. **38**(7), 1043–1045 (2013). [CrossRef] [PubMed]

**25. **P. Yeh and C. Gu, *Optics of Liquid Crystal Displays* (John Wiley, 1999), Chap. 4.

**26. **I. Moreno, P. Velásquez, C. R. Fernández-Pousa, M. M. Sánchez-López, and F. Mateos, “Jones matrix method for predicting and optimizing the optical modulation properties of a liquid-crystal display,” J. Appl. Phys. **94**(6), 3697–3702 (2003). [CrossRef]