## Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal |

Optics Express, Vol. 20, Issue 23, pp. 25876-25883 (2012)

http://dx.doi.org/10.1364/OE.20.025876

Acrobat PDF (1396 KB)

### Abstract

We propose a method for controlling the local spin and orbital angular momentum (SAM and OAM) of a focused light beam in a uniaxial crystal by means of Pockels effect. For an input circularly polarized Bessel-Gaussian (BG) beam, both the local SAM and OAM of the output beam are circularly symmetric, their patterns and peak values vary with the applied electric field *E*_{0}. Let the output beam pass through a quarter-wave plate, the OAM keeps while the SAM varies. The local SAM density is nearly directly proportional to ± sin(2*φ*), where *φ* is the azimuthal angle and the signs are dependent on the radius and *E*_{0}.

© 2012 OSA

## 1. Introduction

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

2. H. He, M. E. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**(5), 826–829 (1995). [CrossRef] [PubMed]

4. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. **3**(2), 161–204 (2011) (and references therein). [CrossRef]

5. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. **91**(22), 227902 (2003). [CrossRef] [PubMed]

6. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science **292**(5518), 912–914 (2001). [CrossRef] [PubMed]

7. R. Dasgupta, S. Ahlawat, R. S. Verma, and P. K. Gupta, “Optical orientation and rotation of trapped red blood cells with Laguerre-Gaussian mode,” Opt. Express **19**(8), 7680–7688 (2011). [CrossRef] [PubMed]

8. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express **15**(23), 15214–15227 (2007). [CrossRef] [PubMed]

9. V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. **91**(9), 093602 (2003). [CrossRef] [PubMed]

10. K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B **21**, 1749–1757 (2004). [CrossRef]

11. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**(16), 163905 (2006). [CrossRef] [PubMed]

8. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express **15**(23), 15214–15227 (2007). [CrossRef] [PubMed]

12. Z. Bouchal, V. Kollarova, P. Zemanek, and T. Cizmar, “Orbital angular momentum of mixed vortex beams,” Proc. SPIE **6609**, 660907, 660907-8 (2007). [CrossRef]

13. I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express **19**(18), 16760–16771 (2011). [CrossRef] [PubMed]

14. S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express **14**(2), 535–541 (2006). [CrossRef] [PubMed]

15. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**(15), 6604–6612 (2006). [CrossRef] [PubMed]

16. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. **34**(7), 1021–1023 (2009). [CrossRef] [PubMed]

18. C. Loussert and E. Brasselet, “Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media,” Opt. Lett. **35**(1), 7–9 (2010). [CrossRef] [PubMed]

19. N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel light beams in uniaxial crystals,” Quantum Electron. **31**(1), 85–89 (2001). [CrossRef]

20. W. Zhu and W. She, “Electro-optically generating and controlling right- and left-handed circularly polarized multiring modes of light beams,” Opt. Lett. **37**(14), 2823–2825 (2012). [CrossRef] [PubMed]

21. A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(3), 036618 (2003). [CrossRef] [PubMed]

22. L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. **33**(7), 696–698 (2008). [CrossRef] [PubMed]

## 2. Theory

*γ*

_{33}= 1340 pm/V [23], and a dc electric field is applied along the same direction. Following [20

20. W. Zhu and W. She, “Electro-optically generating and controlling right- and left-handed circularly polarized multiring modes of light beams,” Opt. Lett. **37**(14), 2823–2825 (2012). [CrossRef] [PubMed]

*ε*and

_{o}*ε*being the permittivity of ordinary and extraordinary waves in the uniaxial crystal, respectively. It has been previously shown that, a circularly polarized Gaussian beam propagating along the optical axis of a uniaxial crystal has low conversion efficiency from SAM to OAM (lower than 50%) [24

_{e}24. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A **20**(1), 163–171 (2003). [CrossRef] [PubMed]

18. C. Loussert and E. Brasselet, “Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media,” Opt. Lett. **35**(1), 7–9 (2010). [CrossRef] [PubMed]

25. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-travelling paraxial beams,” J. Opt. Soc. Am. A **27**(3), 381–389 (2010). [CrossRef]

*m*-order right-handed circularly polarized paraxial BG beam, its field at the incident surface of SBN crystal (

*z*= 0) is

_{exp(−imφ)(X+iY)/2}, where

*β*and

*w*

_{0}are the transverse component of wave-number and beam waist, respectively

_{.}Following the method proposed in [25

25. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-travelling paraxial beams,” J. Opt. Soc. Am. A **27**(3), 381–389 (2010). [CrossRef]

*β*= 0 and

*m*= 0, Eqs. (2a) and (2b) reduce to that describing the propagation of Gaussian beam through a uniaxial crystal. They also describe the propagation of non-diffracting Bessel beam when

*w*

_{0}→∞.

26. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. **52**, 1045–1052 (2005). [CrossRef]

**r**and linear momentum density

**P**gives the angular momentum density. And the angular momentum density in the propagation direction is

*j*=

_{z}*l*+

_{z}*s*, with [8

_{z}8. R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express **15**(23), 15214–15227 (2007). [CrossRef] [PubMed]

*S*

_{3}is the Stokes parameter,

*E*and

_{x}*E*are the light field components along two orthogonal directions. By the relations

_{y}*E*, we get

_{y=i(E+−E−)/2}*l*= -

_{z}*ε*

_{0}/(2

*w*) [

*m*|

*A*

_{1}(

*r*,

*z*)|

^{2}+ (

*m*-2)|

*A*

_{2}(

*r*,

*z*)|

^{2}],

*s*=

_{z}*ε*

_{0}/(2

*w*) [|

*A*

_{1}(

*r*,

*z*)|

^{2}-|

*A*

_{2}(

*r*,

*z*)|

^{2}], where

*A*

_{1}(

*r*,

*z*) and

*A*

_{2}(

*r*,

*z*) are the azimuthal independent parts of

*E*

_{+}(

*r*,

*φ*,

*z*) and

*E*

_{-}(

*r*,

*φ*,

*z*) in Eqs. (2a) and (2b), respectively. The output local SAM and OAM densities rely on the local intensities of RHP and LHP components of the output beam. This can be easily understood by taking the output beam as a superposition of two circularly polarized waves, then performing an addition of their OAM but a subtraction between their SAM [8

**15**(23), 15214–15227 (2007). [CrossRef] [PubMed]

25. T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-travelling paraxial beams,” J. Opt. Soc. Am. A **27**(3), 381–389 (2010). [CrossRef]

*incident beam, the average SAM can be calculated and is with the formwhere*

_{m}*I*(

_{m}*x*) stands for the modified Bessel function of first kind. By employing the conservation law of angular momentum, we get the average OAM per photonSAM and OAM are not independent of each other, thus we can control SAM and OAM simultaneously. In the next section, we will use numerical method to investigate the influence of Pockels effect on the local (density) and global (average) properties of SAM and OAM in the output beam.

## 3. The electric control of local and global angular momentum of the output beam

*β*= 1 μm

^{−1}and

*w*

_{0}= 10 μm at the input plane (

*z*= 0), propagating along the optical axis of SBN crystal. The parameters of the beam are in paraxial regime [27

27. R. Borghi, M. Santarsiero, and M. A. Porras, “Nonparaxial Bessel-Gauss beams,” J. Opt. Soc. Am. A **18**(7), 1618–1626 (2001). [CrossRef] [PubMed]

28. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

*kw*

_{0}/

*β*only. The intensity distribution of BG beam cannot keep its initial profile and becomes single-ringed after a propagation distance

*z*>>D [28

28. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**(6), 491–495 (1987). [CrossRef]

*L*= 8 mm is much longer than D, thus the total intensity pattern of output light beam is a single ring when

*E*

_{0}= 0, which is evident in Fig. 1(b) . The pattern hardly changes when the applied electric field varies from −1.6883 to 3.55 kV/mm. The local SAM and OAM densities have circular symmetry. Figure 1(c) shows the dependences of normalized SAM (blue line) and OAM (red line) densities on radius for different

*E*

_{0}, from which one sees that, the densities can be tuned by the applied electric field. This is because the Pockels effect modulates the diffraction lengths of both ordinary and extraordinary beam components [24

24. A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A **20**(1), 163–171 (2003). [CrossRef] [PubMed]

*E*

_{0}[shown by Eqs. (2a) and (2b)]. As a result, the local SAM and OAM densities become electrically tunable. At

*E*

_{0}= −1.6883 kV/mm the OAM vanishes.

*E*

_{0}and reaches its maximum at

*E*

_{0}= 1 kV/mm. At the same time, the SAM density changes from positive to negative one. The reverse process occurs when

*E*

_{0}continues increasing. The distributions of OAM and SAM densities are interesting for some special values of

*E*

_{0.}For example, at

*E*

_{0}= 3.55 kV/mm, the OAM density is not longer single-ringed but double-ringed. There are two rings with opposite signs in the SAM pattern when

*E*

_{0}= −0.35 and 2.26 kV/mm. That is to say, the SAMs located at these two rings have different orientations. Therefore, two absorbing or birefringent particles trapped in these two different rings (off axis) may have opposite rotation directions. The sign of the inner ring is positive while that of the outer ring is negative at

*E*

_{0}= −0.35 kV/mm. The signs of these two rings reverse at

*E*

_{0}= 2.26 kV/mm. Form Fig. 1(b) and 1(c) one concludes that, the patterns of SAM and OAM densities are quite different with that of total intensity.

_{0}incident beam, the global SAM and OAM of the output beam can be calculated according to Eqs. (4) and (5). And their dependences on the applied electric field

*E*

_{0}are shown in Fig. 2 for

*w*

_{0}= 10 μm, where the blue and green lines represent the global SAM and OAM, respectively. As illustrated by Fig. 2, the conversion between SAM and OAM is electrically controllable. When

*E*

_{0}varies from −1.6883 to 0.87 kV/mm, SAM transfers to OAM gradually. The conversion efficiency of such a transformation defined as a ratio of the energy of LHP component to the total energy of output light beam, namely,

*η*= (1-

*S*)/2, has a maximum value more than 90% at

_{z}*E*

_{0}= 0.87 kV/mm. When further increasing

*E*

_{0}, OAM returns back to SAM. By comparing Fig. 1(c) and 2, one finds that, the applied electric field for obtaining the maximum value of global OAM (

*E*

_{0}= 0.87 kV/mm) is smaller than that corresponding to the maximum peak in OAM density pattern (

*E*

_{0}= 1 kV/mm). Although the ring in the OAM density pattern has a relatively smaller peak value at

*E*

_{0}= 0.87 kV/mm, it has a larger size, which leads to a maximum global OAM. One can further conclude that, light beam may have different local angular momentum density patterns even though its global angular momentum is identical. Besides, the local angular momentum changes in both magnitude and sign over the beam profile, while the global one describes the whole beam by a single value. Thus the signs of the local angular momentum density in some regions can be different with the global one.

*E*

_{0}but also on beam waist

*w*

_{0}. Figure 3 shows the efficiency as a function of

*E*

_{0}and

*w*

_{0}for

*L*= 8 mm and

*β*= 1 μm

^{−1}. One sees from Fig. 3 that, the maximum conversion efficiency increases with

*w*

_{0}, reaches 99.61% when

*w*

_{0}= 50μm. And the applied electric field has only a small change for different

*w*

_{0}at the maximum conversion efficiency.

## 4. The electric control of local and global angular momentum of the light beam after SBN and QWP

*x*-axis is placed near the output surface of SBN crystal, it will transfer RHP and LHP components to horizontal and vertical ones, respectively, and causes a phase difference of π/2 between the horizontal and vertical components [23]. Therefore, the light field after QWP becomes

*r*,

*z*) = |

*A*

_{2}(

*r*,

*z*)|/

*|A*

_{1}(

*r*,

*z*)| and

*δ*(

*r*,

*φ*,

*z*) = arg(

*A*

_{2}(

*r*,

*z*))-arg(

*A*

_{1}(

*r*,

*z*)) + 2

*φ*-π/2. After some simple calculations, we find that the total intensity and the OAM keep the same as those before QWP. The SAM, in contrast, shows new features. The global SAM vanishes whatever the applied electric field is. Of particular interesting, the local SAM density is nonzero, except for

*E*

_{0}= −1.6883 kV/mm. As shown by the Jones representation in Eq. (6), the phase difference between the horizontal and vertical components of light field depends on the azimuthal angle, while the parameter Ψ is independent of it. Therefore the parameter

*S*

_{3}= sin(2Ψ)sin

*δ*has 2-fold rotational symmetry around the axis of light beam, which means that the SAM density has lost its circular symmetry. The parameters Ψ and

*δ*rely on both radius

*r*and the applied electric field

*E*

_{0}, and the sign of sin

*δ*would change with

*r*and

*E*

_{0}. So, the SAM density will change its magnitude and sign against

*r*or

*E*

_{0}. Figure 4 shows the electric dependence of normalized SAM density for a RHP BG

_{0}incident beam, where the beam waist of the incident beam is chosen to be 10 μm in order to display the SAM density patterns in good quality. For

*E*

_{0}in range of −1.6883 and −0.35 kV/mm, the SAM density is closely proportional to -sin(2

*φ*), and has a ring shape pattern. The peak value of the ring increases with

*E*

_{0}and reaches its maximum at

*E*

_{0}= −0.35 kV/mm. Then it decreases when

*E*

_{0}further increases and vanish at

*E*

_{0}= 2.26 kV/mm. When

*E*

_{0}varies from −0.35 to 2.26 kV/mm, an outer ring emerges in the density pattern and grows gradually. And the SAM located at this ring is nearly directly proportional to sin(2

*φ*). The peak value of the outer ring reaches its maximum at

*E*

_{0}= 2.26 kV/mm, when the inner ring disappears and a new ring with a radius larger than the old outer ring begins to emerge. It is worth noting that the peak values of inner and outer rings are equal at

*E*

_{0}= 1 and 3.55 kV/mm. However, the signs of the SAM densities for these two cases are opposite. For higher-order BG incident beams, the patterns of SAM density have similar behavior, changing with

*E*

_{0}, if

*L*>>D.

## 5. Conclusion

*beam into a SBN crystal with an applied electric field*

_{m}*E*

_{0}along its optical axis. For a BG

_{0}beam incident onto a 8 mm long crystal, SAM converts to OAM with a very high conversion efficiency, for example 99.61% when

*w*

_{0}= 50μm. The patterns of SAM and OAM densities of the output light beam are ring-shaped and electrically controllable. When the output beam passes through a QWP, the OAM keeps the same, while the SAM shows new features. The local SAM density loses its circular symmetry and is proportional to ± sin(2

*φ*) with signs dependent on the radius and

*E*

_{0}.

## Acknowledgments

## References and links

1. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

2. | H. He, M. E. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

3. | S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express |

4. | A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. |

5. | A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. |

6. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science |

7. | R. Dasgupta, S. Ahlawat, R. S. Verma, and P. K. Gupta, “Optical orientation and rotation of trapped red blood cells with Laguerre-Gaussian mode,” Opt. Express |

8. | R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express |

9. | V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett. |

10. | K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B |

11. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

12. | Z. Bouchal, V. Kollarova, P. Zemanek, and T. Cizmar, “Orbital angular momentum of mixed vortex beams,” Proc. SPIE |

13. | I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express |

14. | S. H. Tao, X. C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express |

15. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

16. | E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Dynamics of optical spin-orbit coupling in uniaxial crystals,” Opt. Lett. |

17. | Y. A. Egorov, T. A. Fadeyeva, and A. V. Volyar, “The fine structure of singular beams in crystals: colours and polarization,” J. Opt. Soc. Am. A |

18. | C. Loussert and E. Brasselet, “Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media,” Opt. Lett. |

19. | N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel light beams in uniaxial crystals,” Quantum Electron. |

20. | W. Zhu and W. She, “Electro-optically generating and controlling right- and left-handed circularly polarized multiring modes of light beams,” Opt. Lett. |

21. | A. Ciattoni, G. Cincotti, and C. Palma, “Angular momentum dynamics of a paraxial beam in a uniaxial crystal,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

22. | L. Chen and W. She, “Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam,” Opt. Lett. |

23. | A. Yariv, |

24. | A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A |

25. | T. A. Fadeyeva and A. V. Volyar, “Extreme spin-orbit coupling in crystal-travelling paraxial beams,” J. Opt. Soc. Am. A |

26. | R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. |

27. | R. Borghi, M. Santarsiero, and M. A. Porras, “Nonparaxial Bessel-Gauss beams,” J. Opt. Soc. Am. A |

28. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

**OCIS Codes**

(230.2090) Optical devices : Electro-optical devices

(260.2160) Physical optics : Energy transfer

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 28, 2012

Revised Manuscript: October 2, 2012

Manuscript Accepted: October 5, 2012

Published: November 1, 2012

**Citation**

Wenguo Zhu and Weilong She, "Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal," Opt. Express **20**, 25876-25883 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-23-25876

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### References

- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992). [CrossRef] [PubMed]
- H. He, M. E. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett.75(5), 826–829 (1995). [CrossRef] [PubMed]
- S. H. Tao, X.-C. Yuan, J. Lin, X. Peng, and H. B. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express13(20), 7726–7731 (2005). [CrossRef] [PubMed]
- A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon.3(2), 161–204 (2011) (and references therein). [CrossRef]
- A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett.91(22), 227902 (2003). [CrossRef] [PubMed]
- L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science292(5518), 912–914 (2001). [CrossRef] [PubMed]
- R. Dasgupta, S. Ahlawat, R. S. Verma, and P. K. Gupta, “Optical orientation and rotation of trapped red blood cells with Laguerre-Gaussian mode,” Opt. Express19(8), 7680–7688 (2011). [CrossRef] [PubMed]
- R. Zambrini and S. M. Barnett, “Angular momentum of multimode and polarization patterns,” Opt. Express15(23), 15214–15227 (2007). [CrossRef] [PubMed]
- V. Garcés-Chávez, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multi-ringed light beam to an optically trapped particle,” Phys. Rev. Lett.91(9), 093602 (2003). [CrossRef] [PubMed]
- K. Volke-Sepulveda, S. Chávez-Cerda, V. Garcés-Chávez, and K. Dholakia, “Three-dimensional optical forces and transfer of orbital angular momentum from multiringed light beams to spherical microparticles,” J. Opt. Soc. Am. B21, 1749–1757 (2004). [CrossRef]
- L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006). [CrossRef] [PubMed]
- Z. Bouchal, V. Kollarova, P. Zemanek, and T. Cizmar, “Orbital angular momentum of mixed vortex beams,” Proc. SPIE6609, 660907, 660907-8 (2007). [CrossRef]
- I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express19(18), 16760–16771 (2011). [CrossRef] [PubMed]
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