## Reconfigurable beams with arbitrary polarization and shape distributions at a given plane |

Optics Express, Vol. 21, Issue 5, pp. 5432-5439 (2013)

http://dx.doi.org/10.1364/OE.21.005432

Acrobat PDF (6413 KB)

### Abstract

Methods for generating beams with arbitrary polarization based on the use of liquid crystal displays have recently attracted interest from a wide range of sources. In this paper we present a technique for generating beams with arbitrary polarization and shape distributions at a given plane using a Mach-Zehnder setup. The transverse components of the incident beam are processed independently by means of spatial light modulators placed in each path of the interferometer. The modulators display computer generated holograms designed to dynamically encode any amplitude value and polarization state for each point of the wavefront in a given plane. The steps required to design such beams are described in detail. Several beams performing different polarization and intensity landscapes have been experimentally implemented. The results obtained demonstrate the capability of the proposed technique to tailor the amplitude and polarization of the beam simultaneously.

© 2013 OSA

## 1. Introduction

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

2. S.C. Tidwell, G.H. Kim, and W.D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. **32**, 5222–5229 (1993) [CrossRef] [PubMed] .

5. A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. **31**, 1732–1734 (2006) [CrossRef] [PubMed] .

6. J.A. Davis, D.E. McNamara, D.M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. **39**, 1549–1554 (2000) [CrossRef] .

16. F. Kenny, D. Lara, O.G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express **20**, 14015–14029 (2012) [CrossRef] [PubMed] .

17. R. Tudela, E. Martin-Badosa, I. Labastida, S. Vallmitjana, and A. Carnicer, “Wavefront reconstruction by adding modulation capabilities of two liquid crystal devices,” Opt. Eng. **43**, 2650–2657 (2004) [CrossRef] .

18. V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. **28**, 1359–1361 (2003) [CrossRef] [PubMed] .

19. V. Arrizón, L. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. **44**, 1625–1634 (2005) [CrossRef] [PubMed] .

## 2. Optical setup

_{1}set at 45° with respect to the

*x*direction; this field (

**E**

*) is described by*

_{in}**E**

*=*

_{in}*E*

_{in}_{2}(

*x*,

*y*)

**e**

_{1}+

*E*

_{in}_{1}(

*x*,

*y*)

**e**

_{2}, where

**e**

_{1}and

**e**

_{2}are unit vectors in the

*x*and

*y*directions. Later,

**E**

*is split into two beams by means of a polarizing beam splitter, PBS*

_{in}_{1}. Reflected by mirror M

_{1}or M

_{2}, the split beams pass through different wave plates which rotate the oscillating plane and set the modulator in order to achieve the desired modulation curve. Afterwards, the light passes through modulator SLM

_{1}or SLM

_{2}which display complex transmittances

*h*

_{1}(

*x*,

*y*) and

*h*

_{2}(

*x*,

*y*), respectively. Each of the displays used is a translucent twisted nematic Holoeye HEO 0017 with a resolution of 1024 × 768 pixels and 32

*μm*of pixel pitch. Then, the light is recombined by means of the second polarizing beam splitter PBS

_{2}and fed into the on-axis reconstruction system consisting of a 4

*f*-Fourier lens system. Note that a spatial filter in the back focal plane of L

_{1}is needed to remove higher-order diffracted terms generated by the holograms

*h*

_{1}(

*x*,

*y*) and

*h*

_{2}(

*x*,

*y*). Finally, the resulting field is analyzed by means of P

_{2}and the final irradiance is recorded by the CCD camera. The output field (

**E**

*(*

_{out}*x*,

*y*)) at the camera plane is Where

*A*

_{1}(

*x*,

*y*) and

*A*

_{2}(

*x*,

*y*) are the amplitude distributions of

**E**

*(*

_{out}*x*,

*y*) in the

*x*and

*y*directions respectively and

*ϕ*

_{1}(

*x*,

*y*) and

*ϕ*

_{2}(

*x*,

*y*) are the corresponding accumulated phase shifts. The total phase delay between components

*x*and

*y*of

**E**

*(*

_{out}*x*,

*y*) is

*ϕ*(

*x*,

*y*) =

*ϕ*

_{2}(

*x*,

*y*) −

*ϕ*

_{1}(

*x*,

*y*). For convenience, we write

*A*

_{1}(

*x*,

*y*) =

*A*(

_{sh}*x*,

*y*)

*a*

_{1}(

*x*,

*y*) and

*A*

_{2}(

*x*,

*y*) =

*A*(

_{sh}*x*,

*y*)

*a*

_{2}(

*x*,

*y*) with

*A*(

_{sh}*x*,

*y*) is the beam shape distribution. In this way, the oscillation orientation distribution at each point (

*x*,

*y*) of the wavefront is

*θ*(

*x*,

*y*) = tan

^{−1}(

*a*

_{2}(

*x*,

*y*)/

*a*

_{1}(

*x*,

*y*)). For simplicity, the magnification introduced by the imaging 4

*f*system is not taken into account.

## 3. Codification procedure

19. V. Arrizón, L. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. **44**, 1625–1634 (2005) [CrossRef] [PubMed] .

_{1}(blue dots) and SLM

_{2}(red dots). Both curves have been determined using the method presented in [20

20. E. Martín-Badosa, A. Carnicer, I. Juvells, and S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. **8**, 764–772 (1997) [CrossRef] .

_{1},

*λ*/2 at 70° and

*λ*/4 at 145°; SLM

_{2},

*λ*/2 at 150° and

*λ*/4 at 35° (these angles are from the fast axis of the waveplate). From Fig. 2(a) it is apparent that: (i) the amplitude modulation is not constant and no phase-only modulation would be possible using this device; (ii) the phase values are limited to the range [0°, 240°]; and (iii) the displays perform in similar but not identical ways. Note that the phase origin of SLM

_{2}is shifted by fine tuning the optical path of the corresponding arm of the interferometer. The holographic algorithm takes advantage of the amplitude-phase coupling of these displays to achieve full-complex modulation with two SLMs. Let

*C*be the complex value to be coded at position (

_{nm}*n*,

*m*). If

*C*does not belong to the modulation curve

_{nm}*M*, it can be written as

*M*(see Fig. 2(a) for details). Selecting

*C*, as depicted in Fig. 2(b). Using this pixel arrangement, the optical Fourier transforms of

_{nm}*L*

_{1}. Finally, the desired distribution is reconstructed on-axis at the back focal plane of lens L

_{2}(CCD plane).

*C*that can be obtained as a combination of two points

_{nm}_{1}or SLM

_{2}, respectively. Consequently, if the phase origin of SLM

_{2}is shifted 60° with respect to the other display, the system can access any amplitude value and phase difference

*ϕ*(

*x*,

*y*) between the two components within the circle of transmittance

*T*= 0.4. Although the relative phase delay,

*ϕ*(

*x*,

*y*), can be achieved from many pairs

*ϕ*

_{1}(

*x*,

*y*) and

*ϕ*

_{2}(

*x*,

*y*), it is necessary to emphasize that this pair of phase distributions has to be smooth and without phase jumps across the beam.

## 4. Results

*θ*(

*x*,

*y*) = 4tan

^{−1}(

*y/x*). In these two cases, the shape

*A*(

_{sh}*x*,

*y*) remains constant. However, the illumination is not uniform due to the expanded Gaussian incident beam. The third case, (Fig. 4(c)), is a Laguerre-Gauss 10 mode (Eq. 2a), where the inner part of the beam is radially polarized and the external ring is azimuthally polarized. The last case considered, (Fig. 4(d)), is a doughnut-shaped beam following Eq. 2b; in this case the oscillation orientation and the phase delay are

*θ*(

*x*,

*y*) = tan

^{−1}(

*y/x*) and

*ϕ*(

*x*,

*y*) = ±2tan

^{−1}(

*y/x*) respectively. The + sign stands for right-handed polarization states whereas the − sign is used for left-handed cases.

*w*

_{0}is the beam waist radius and

_{2}. Rows 2 to 5 display the results obtained for each beam when the analyzer P

_{2}is set at 0°, 45°, 90° and 135°, respectively.

*T*= 0.4, see Fig. 2).

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}) (SP) has been carried out [21]. The beam used to perform this measure is a non-circular radially polarized Gaussian beam as shown in Fig. 6(a). In practice, these parameters are easily obtained according to the following relations where

*I*(

*α*,

*β*) stands for the recorded intensity when the analyzer P2 is set at an angle

*α*with respect to the

*x*direction;

*β*is the retardation between the

*x*and

*y*directions. Related to the SP, the degree of polarization

*P*(DP) is

*S*

_{0}(Fig. 6(b)) represents the total intensity.

*S*

_{1}(Fig. 6(c)) displays red and blue pixels in those areas where the dominant polarization is in the

*x*or

*y*directions respectively. Note that the polarization direction changes smoothly, according to a radially polarized pattern. A similar interpretation is possible for

*S*

_{2}(Fig. 6(d)) but for

*α*= 45° and

*α*= 135°.

*S*

_{3}(Fig. 6(e)), that compares the amount of right and left handed circular polarization of the field, is almost zero everywhere. Finally, the DP is shown in Fig. 6(f): notice that

*P*= 1 for the points of the beam and is nearly zero outside.

## 5. Conclusions

*π*and present amplitude-phase coupling, a cell-oriented computer generated hologram algorithm has to be used to encode the information; in this way, full complex modulation can be achieved. Different beams with arbitrary polarization and shape distributions have successfully been obtained experimentally thereby demonstrating the feasibility of the proposed technique.

## Acknowledgment

## References

1. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics |

2. | S.C. Tidwell, G.H. Kim, and W.D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. |

3. | M.A.A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. |

4. | K.C. Toussaint Jr, S.P. Park, J.E. Jureller, and N.F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. |

5. | A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. |

6. | J.A. Davis, D.E. McNamara, D.M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. |

7. | R.L. Eriksen, P.C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. |

8. | C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

9. | X.L. Wang, J. 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. |

10. | H. Chen, Z. Zheng, B.F. Zhang, J. Ding, and H.T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. |

11. | H.T. Wang, X.L. Wang, Y. Li, J. Chen, C.S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express |

12. | H. Chen, J. Hao, B.F. Zhang, J. Xu, J. Ding, and H.T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. |

13. | I. Moreno, C. Iemmi, J. Campos, and M.J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Lett. |

14. | S. Tripathi and K.C. Toussaint, “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Lett. |

15. | S. Liu, P. Li, T. Peng, and J. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid sagnac interferometer,” Opt. Express |

16. | F. Kenny, D. Lara, O.G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express |

17. | R. Tudela, E. Martin-Badosa, I. Labastida, S. Vallmitjana, and A. Carnicer, “Wavefront reconstruction by adding modulation capabilities of two liquid crystal devices,” Opt. Eng. |

18. | V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett. |

19. | V. Arrizón, L. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. |

20. | E. Martín-Badosa, A. Carnicer, I. Juvells, and S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol. |

21. | M. Born and E. Wolf, |

**OCIS Codes**

(090.1760) Holography : Computer holography

(260.5430) Physical optics : Polarization

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 17, 2013

Revised Manuscript: February 15, 2013

Manuscript Accepted: February 15, 2013

Published: February 26, 2013

**Citation**

David Maluenda, Ignasi Juvells, Rosario Martínez-Herrero, and Artur Carnicer, "Reconfigurable beams with arbitrary polarization and shape distributions at a given plane," Opt. Express **21**, 5432-5439 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-5432

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### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics1, 1–57 (2009). [CrossRef]
- S.C. Tidwell, G.H. Kim, and W.D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt.32, 5222–5229 (1993). [CrossRef] [PubMed]
- M.A.A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett.27, 1929–1931 (2002). [CrossRef]
- K.C. Toussaint, S.P. Park, J.E. Jureller, and N.F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett.30, 2846–2848 (2005). [CrossRef] [PubMed]
- A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett.31, 1732–1734 (2006). [CrossRef] [PubMed]
- J.A. Davis, D.E. McNamara, D.M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt.39, 1549–1554 (2000). [CrossRef]
- R.L. Eriksen, P.C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun.187, 325–336 (2001). [CrossRef]
- 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]
- X.L. Wang, J. 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, 3549–3551 (2007). [CrossRef] [PubMed]
- H. Chen, Z. Zheng, B.F. Zhang, J. Ding, and H.T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett.35, 2825–2827 (2010). [CrossRef] [PubMed]
- H.T. Wang, X.L. Wang, Y. Li, J. Chen, C.S. Guo, and J. Ding, “A new type of vector fields with hybrid states of polarization,” Opt. Express18, 10786–10795 (2010). [CrossRef] [PubMed]
- H. Chen, J. Hao, B.F. Zhang, J. Xu, J. Ding, and H.T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett.36, 3179–3181 (2011). [CrossRef] [PubMed]
- I. Moreno, C. Iemmi, J. Campos, and M.J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Lett.19, 4583–4594 (2011).
- S. Tripathi and K.C. Toussaint, “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Lett.20, 10788–10795 (2012).
- S. Liu, P. Li, T. Peng, and J. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid sagnac interferometer,” Opt. Express20, 21715–21721 (2012). [CrossRef] [PubMed]
- F. Kenny, D. Lara, O.G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express20, 14015–14029 (2012). [CrossRef] [PubMed]
- R. Tudela, E. Martin-Badosa, I. Labastida, S. Vallmitjana, and A. Carnicer, “Wavefront reconstruction by adding modulation capabilities of two liquid crystal devices,” Opt. Eng.43, 2650–2657 (2004). [CrossRef]
- V. Arrizón, “Complex modulation with a twisted-nematic liquid-crystal spatial light modulator: double-pixel approach,” Opt. Lett.28, 1359–1361 (2003). [CrossRef] [PubMed]
- V. Arrizón, L. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt.44, 1625–1634 (2005). [CrossRef] [PubMed]
- E. Martín-Badosa, A. Carnicer, I. Juvells, and S. Vallmitjana, “Complex modulation characterization of liquid crystal devices by interferometric data correlation,” Meas. Sci. Technol.8, 764–772 (1997). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999).

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