## Generation and dynamics of optical beams with polarization singularities |

Optics Express, Vol. 21, Issue 7, pp. 8815-8820 (2013)

http://dx.doi.org/10.1364/OE.21.008815

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### Abstract

We present a convenient method to generate vector beams of light having polarization singularities on their axis, via partial spin-to-orbital angular momentum conversion in a suitably patterned liquid crystal cell. The resulting polarization patterns exhibit a C-point on the beam axis and an L-line loop around it, and may have different geometrical structures such as “lemon”, “star”, and “spiral”. Our generation method allows us to control the radius of L-line loop around the central C-point. Moreover, we investigate the free-air propagation of these fields across a Rayleigh range.

© 2013 OSA

## 1. Introduction

1. J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt. **53**, 293–363 (2009). [CrossRef]

2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt. **53**, 293–363 (2009). [CrossRef]

1. J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

3. M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” *Prog. Opt.*42, 219–276 (2001). [CrossRef]

*topological charge*, given by the phase change in units of 2

*π*along a closed path surrounding the singularity [3

3. M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” *Prog. Opt.*42, 219–276 (2001). [CrossRef]

4. S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. **2**, 299–313 (2008). [CrossRef]

5. J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A **387**, 105–132 (1983). [CrossRef]

11. M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. **28**, 1475–1477 (2003). [CrossRef] [PubMed]

*polarization*” state of the field. The polarization can be also parametrized by introducing the three

*reduced Stokes parameters*,

*s*

_{1},

*s*

_{2}and

*s*

_{3}, or the polar and azimuthal angles on the

*Poincaré sphere*. The vector-field singularities form points or lines in the transverse plane where one of the polarization parameters is undefined. Examples are the so-called

*C-points*and

*L-lines*, where the orientation and the handedness of the polarization ellipse are undefined, respectively [6

6. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A **389**, 279–290 (1983). [CrossRef]

7. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**, 201–221 (2002). [CrossRef]

12. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincare beams,” Opt. Express **18**, 10777–10785 (2010). [CrossRef] [PubMed]

13. 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]

*polarization-singular beams*” (PSB). The PSB may have different transverse polarization patterns, characterized by a different integer or half-integer topological index

*η*as defined by the variation of the polarization ellipse orientation for a closed path around a C-point, such as “star”, “monstar”, “lemon”, “spiral” (with a central “source/sink” singularity), “hyperbolic” (with a central “saddle” singularity), etc. Some examples are shown in Fig. 1. Optical PSB possessing high values of positive and negative topological charge form so-called “polarization flowers” and “hyperbolic webs”, respectively [8

8. I. Freund, “Polarization flowers,” Opt. Commun. **199**, 47–63 (2001). [CrossRef]

14. P. Kurzynowski, W. A. WoŸniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express **20**, 26755–26765 (2012). [CrossRef] [PubMed]

15. T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt. **51**, C224–C230 (2012). [CrossRef] [PubMed]

16. 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]

17. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. **51**, C1–C6 (2012). [CrossRef] [PubMed]

## 2. Polarization singular beams

*ρ*,

*ϕ*,

*z*are cylindrical coordinates,

*δ*is a tuning parameter, |

*L*〉 and |

*R*〉 stand for left- and right-circular polarizations, respectively,

*α*is a phase, and LG

*denotes a Laguerre-Gauss mode of radial index*

_{p,m}*p*and OAM index

*m*, with a given waist location (

*z*= 0) and radius

*w*

_{0}.

*m*≠ 0 have a doughnut shape, with vanishing intensity at the center. The PSB resulting from Eq. (1) then have a polarization pattern, shown in Fig. 1, characterized by the following four elements: (i) at the center (that is the beam axis), the polarization is left-circular (C-point), since the last term in Eq. (1) vanishes here; (ii) as the radius increases, the second term contributes and changes the circular polarizations into left-hand elliptical; (iii) at a certain radius

*ρ*=

*ρ*

_{0}(

*δ*), where the two waves in Eq. (1) have equal amplitude, the polarization becomes exactly linear (L-lines); (iv) at larger radii, that is for

*ρ*>

*ρ*

_{0}, the second term in Eq. (1) prevails and the polarization state changes into right-handed elliptical. It is worth noting that, in this case, the polarization pattern has a topological index

*η*=

*m*/2. The initial angle at azimuthal location

*ϕ*= 0 of the polarization ellipse major axis is fixed by the relative phase

*α*in Eq. (1). Of course, the |

*L*〉 and |

*R*〉 polarizations in Eq. (1) can also be swapped, leading to an inverted polarization ellipticitiy everywhere.

16. 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]

18. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. **13**, 064001 (2011). [CrossRef]

20. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express **19**, 4085–4090 (2011). [CrossRef] [PubMed]

*δ*, the q-plate converts a varying fraction of the spin angular momentum change suffered by the light passing through the device into OAM with

*m*= ±2

*q*, where

*q*is the q-plate topological charge and the sign is determined by the input circular polarization handedness. The conversion is full when the q-plate optical phase retardation is

*δ*=

*π*(or any odd multiple of

*π*). For other values of

*δ*, the q-plate is said to be untuned and the spin-to-orbital conversion is only partial. In the untuned q-plate a fraction of the incoming beam passes through unchanged. The field at the exit plane of an untuned q-plate for a Gaussian left-circular polarized input beam is given by the following expression: where

21. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. **32**, 3053–3055 (2007). [CrossRef] [PubMed]

*q*, thickness

*d*and average refractive index

*n̄*for a Gaussian input (see Ref. [22

22. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett. **34**, 1225–1227 (2009). [CrossRef] [PubMed]

*α*in the second term depends on the initial angle (on the

*x*axis) of the q-plate optical-axis pattern [16

16. 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]

18. L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. **13**, 064001 (2011). [CrossRef]

*iqϕ*), can be expanded in a series of LG modes [21

21. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. **32**, 3053–3055 (2007). [CrossRef] [PubMed]

*m*= 2

*q*. The optical retardation

*δ*of the q-plate, that is the q-plate tuning, can be adjusted continuously by applying an appropriate voltage [20

20. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express **19**, 4085–4090 (2011). [CrossRef] [PubMed]

*q*= −1/2, +1/2, and +1, generating correspondingly equal indices of the polarization patterns.

## 3. Experiment

*H*), vertical (

*V*), anti-diagonal (

*A*), diagonal (

*D*),

*L*and

*R*polarization states, respectively. The polarization pattern was thus entirely reconstructed, as shown in form of little ellipses in Fig. 2 and subsequent figures. In order to reduce noise and experimental errors, the average intensity was recorded in a grid of 20×20 pixel square area [17

17. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. **51**, C1–C6 (2012). [CrossRef] [PubMed]

*δ*with the applied voltage. The radius where the L-line forms in the polarization pattern depends on the ratio between the absolute value of the two coefficients in Eq. (2) and, hence, on the voltage applied to the q-plate. In particular, for optical retardation

*δ*= 0 and

*δ*=

*π*the beam profile is Gaussian fully left-circular and doughnut-shaped fully right-circular, respectively. In the intermediate case, the pattern is a superposition and a circular L-line appears in the polarization pattern, whose radius

*ρ*

_{0}depends on

*δ*. From Eq. (1) one finds

*δ*=

*π*/4, and the corresponding measured L-lines radii. As shown in the figure, also the intensity profile changes from a Gaussian (left-circular) to a doughnut (right-circular) shape. In Fig. 3 (c) and (g), the optical retardations are

*π*/2 and 3

*π*/2, respectively; the

*π*difference in the relative phases results into a 180° rotation of the polarization patterns. The topological index of the polarization pattern, however, depends only on the

*q*-value of the q-plate, and in the case of Fig. 3 we have

*η*=

*q*= 1/2.

*η*= −1/2, +1/2 and +1 topologies under free-air propagation for fixed parameter

*δ*=

*π*/2. To this purpose, we moved the CCD camera along the propagation axis around the imaging-lens focal plane. We recorded the beam polarization patterns at six different planes in the range −

*z*≤

_{R}*z*≤

*z*, where

_{R}*z*is the lens Rayleigh parameter and

_{R}*z*= 0 corresponds to the beam waist location. The experimental results are shown in Fig. 4. As it can be seen, the polarization pattern evolves during propagation. Indeed LG

_{00}and LG

_{0,}

*modes of Eq. (1) have different*

_{m}*z*dependences of their Gouy phases; the relative Gouy phase between them is given by

*ψ*= |

*m*|arctan(

*z/z*

_{0}), which in the explored region varies in the range |

*m*|[−

*π*/4,

*π*/4]. In the case

*m*= ±1 (i.e.

*η*= 1/2) the phase evolution leads to a rigid rotation of the whole polarization structure by an angle equal to

*ψ*[as shown in Fig. 4 (a) and (b)], while when

*m*= 2 (i.e.

*η*= 1), this dephasing leads to a more complex pattern dynamics, in which the polarization structure changes from radial to spiral and then to azimuthal (Fig. 4 (c)).

## 4. Conclusion

23. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. **106**, 123901 (2011). [CrossRef] [PubMed]

24. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. **3**, 989 (2012). [CrossRef] [PubMed]

## Acknowledgments

*Framework Programme for Research of the European Commission, under grant number 255914 - PHOR-BITECH.*

^{th}## References and links

1. | J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A |

2. | M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt. |

3. | M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” |

4. | S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. |

5. | J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A |

6. | J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A |

7. | M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. |

8. | I. Freund, “Polarization flowers,” Opt. Commun. |

9. | O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. |

10. | M. V. Berry, M. R. Dennis, and R. L. Lee Jr., “Polarization singularities in the clear sky,” New J. Phys. |

11. | M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett. |

12. | A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincare beams,” Opt. Express |

13. | 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. |

14. | P. Kurzynowski, W. A. WoŸniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express |

15. | T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt. |

16. | L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. |

17. | F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. |

18. | L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. |

19. | E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. |

20. | S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express |

21. | E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. |

22. | E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett. |

23. | C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett. |

24. | A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun. |

**OCIS Codes**

(260.5430) Physical optics : Polarization

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 13, 2012

Revised Manuscript: December 21, 2012

Manuscript Accepted: December 21, 2012

Published: April 3, 2013

**Citation**

Filippo Cardano, Ebrahim Karimi, Lorenzo Marrucci, Corrado de Lisio, and Enrico Santamato, "Generation and dynamics of optical beams with polarization singularities," Opt. Express **21**, 8815-8820 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8815

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### References

- J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A336, 165–190 (1974). [CrossRef]
- M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical Vortices and Polarization Singularities,” Prog. Opt.53, 293–363 (2009). [CrossRef]
- M. S. Soskin and M. V. Vasnetsov, (Ed. E. Wolf), “Singular optics,” Prog. Opt.42, 219–276 (2001). [CrossRef]
- S. Franke-Arnold, L. Allen, and M. J. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev.2, 299–313 (2008). [CrossRef]
- J. F. Nye, “Polarization effects in the diffraction of electromagnetic waves: the role of disclinations,” Proc. Roy. Soc. Lond. A387, 105–132 (1983). [CrossRef]
- J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. Roy. Soc. Lond. A389, 279–290 (1983). [CrossRef]
- M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213, 201–221 (2002). [CrossRef]
- I. Freund, “Polarization flowers,” Opt. Commun.199, 47–63 (2001). [CrossRef]
- O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun.207, 57–65 (2002). [CrossRef]
- M. V. Berry, M. R. Dennis, and R. L. Lee, “Polarization singularities in the clear sky,” New J. Phys.6, 162 (2004). [CrossRef]
- M. S. Soskin, V. Denisenko, and I. Freund, “Optical polarization singularities and elliptic stationary points,” Opt. Lett.28, 1475–1477 (2003). [CrossRef] [PubMed]
- A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincare beams,” Opt. Express18, 10777–10785 (2010). [CrossRef] [PubMed]
- 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]
- P. Kurzynowski, W. A. WoŸniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express20, 26755–26765 (2012). [CrossRef] [PubMed]
- T. A. Fadeyeva, C. N. Alexeyev, P. M. Anischenko, and A. V. Volyar, “Engineering of the space-variant linear polarization of vortex-beams in biaxially induced crystals,” Appl. Opt.51, C224–C230 (2012). [CrossRef] [PubMed]
- 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]
- F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt.51, C1–C6 (2012). [CrossRef] [PubMed]
- L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt.13, 064001 (2011). [CrossRef]
- E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94, 231124 (2009). [CrossRef]
- S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express19, 4085–4090 (2011). [CrossRef] [PubMed]
- E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett.32, 3053–3055 (2007). [CrossRef] [PubMed]
- E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett.34, 1225–1227 (2009). [CrossRef] [PubMed]
- C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing Local Field Structure of Focused Ultrashort Pulses,” Phys. Rev. Lett.106, 123901 (2011). [CrossRef] [PubMed]
- A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, and P. Maddalena, “Light-induced spiral mass transport in azo-polymer films under vortex-beam illumination,” Nat. Commun.3, 989 (2012). [CrossRef] [PubMed]

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