## Interference from multiple trapped colloids in an optical vortex beam

Optics Express, Vol. 14, Issue 16, pp. 7436-7446 (2006)

http://dx.doi.org/10.1364/OE.14.007436

Acrobat PDF (1925 KB)

### Abstract

Laguerre-Gaussian (LG) beams are important in optical micromanipulation. We show that optically trapped microparticles within a monochromatic LG beam may lead to the formation of unique intensity patterns in the far field due to multiple interference of the forward scattered light from each particle. Trapped colloids create far field interference that exhibits distinct spiral wave patterns that are directly correlated to the helicity of the LG beam. Using two trapped particles, we demonstrate the first microscopic version of a Young’s slits type experiment and detect the azimuthal phase variation around the LG beam circumference. This novel technique may be implemented to study the relative phase and spatial coherence of two points in trapping light fields with arbitrary wavefronts.

© 2006 Optical Society of America

## 1. Introduction

1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A **336**, 165–90 (1974). [CrossRef]

2. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. **198**, 21–27 (2001) [CrossRef]

1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A **336**, 165–90 (1974). [CrossRef]

3. S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Görlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, “Observation of vortex phase singularities in Bose-Einstein Condensates,” Phys. Rev. Lett. **87**, 080402 (2001) [CrossRef] [PubMed]

4. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D.N. Christodoulides, and J.W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature **440**, 1166–1169 (2006) [CrossRef] [PubMed]

5. 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**, 8185–8189 (1992) [CrossRef] [PubMed]

6. H. He, M. E. J. 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**, 826–829 (1995). [CrossRef] [PubMed]

7. D. Rozas, C. T. Law, and G.A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B **14**, 3054–3065 (1997) [CrossRef]

8. G. A. Swartzlander Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. **93**, 093901 (2004). [CrossRef] [PubMed]

9. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. **92**, 143905 (2004) [CrossRef] [PubMed]

10. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger ,“Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001). [CrossRef] [PubMed]

11. A Jesacher, S Furhapter, S Bernet, and M. Ritsch-Marte ,“Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. **94**, 233902 (2005) [CrossRef] [PubMed]

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

13. W. Wang, S.G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. **94**, 103902 (2005) [CrossRef] [PubMed]

6. H. He, M. E. J. 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**, 826–829 (1995). [CrossRef] [PubMed]

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

14. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt Lett **22**, 52–54 (1997) [CrossRef] [PubMed]

20. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986) [CrossRef] [PubMed]

6. H. He, M. E. J. 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**, 826–829 (1995). [CrossRef] [PubMed]

14. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt Lett **22**, 52–54 (1997) [CrossRef] [PubMed]

18. 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**, 093602 (2003), [CrossRef] [PubMed]

19. K Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass Opt. **4**, S82–S89 (2002) [CrossRef]

14. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt Lett **22**, 52–54 (1997) [CrossRef] [PubMed]

18. 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**, 093602 (2003), [CrossRef] [PubMed]

19. K Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass Opt. **4**, S82–S89 (2002) [CrossRef]

## 2. Numerical simulation

*et al*. [21

21. J. -P. Chevaillier, J. Fabre, and P. Hamelin, “Forward scattered light intensities by a sphere located anywhere in a Gaussian beam,” Appl Opt **25**, 1222–1225 (1986) [CrossRef] [PubMed]

*p*and

*l*:

*p*+1 denotes the number of radial nodes whereas

*l*denotes the number of cycles of azimuthal 2π phase shift around the mode circumference. Single ringed LG beams (

*p*=0) are considered in this work. A key characteristic of an LG beam or optical vortex field is its unique helical phase structure, denoted in the mode description by

*e*where ϕ is the azimuthal phase variation around the circumference of the optical field and

^{ilϕ}*l*denotes the number of cycles of azimuthal 2π phase shift around the beam [5

5. 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**, 8185–8189 (1992) [CrossRef] [PubMed]

**75**, 826–829 (1995). [CrossRef] [PubMed]

22. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. **14**, 1874–1889 (1975) [CrossRef] [PubMed]

*E*(

*x*

_{0},

*y*

_{0},0) propagated towards the far-field at incremental z values, ∆z is given as

*k*,

_{x}*k*are the transverse wave vectors, the longitudinal wave vector

_{y}*E*(

*x*

_{0},

*y*

_{0},0) with an analytical expression for an LG beam containing diffracting disks, we are able to numerically simulate the propagation of the LG beams diffracted by microspheres towards the far-field in free space. By using a split-step Fourier method based on Eq. (1), we numerically propagate the LG beam containing microspheres in its annular intensity ring towards the far-field (Fraunhofer plane) with increments of z values, given as ∆z. Since the microspheres are well described in the Mie Regime and the scattering angle is small, we consider mainly diffracted rays from the microspheres in the simulation. The scattering geometry is illustrated in Fig. 1. The result of the numerical simulation of nine microspheres scattering the trapping light is presented in Fig. 1(B) to Fig. 1(D).

## 3. Experimental setup

18. 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**, 093602 (2003), [CrossRef] [PubMed]

_{00}beam at wavelength of 1070 nm laser beam (10W, CW, Yb-fiber laser (IPG Corporation)). The LG beams generated for the experiments described are of radial index

*p*=0, azimuthal index

*l*=1 and

*l*=3. The LG beam is expanded to fill up the back aperture of the plano-convex lens, L1, and hence achieve a focused spot with diameter of around 15μm. The role of the lens (L2) is to relay the scattered light that is collected by a 60X microscope objective (OB, NA = 0.85), onto the observation plane (CCD camera). By shifting the position of the relay len, we can image the optically trapped particles in the LG beam. We can also observe the different propagation plane towards the far-field (Fraunhofer diffraction patterns) using the relay lens.

_{2}0 solution to reduce heating effects and placed in a cylindrical sample chamber of diameter 1cm and depth 100μm. The size of the microsphere is chosen to approximately match the width of the annular intensity profile of the LG beam. The trapping light field is almost total covered by the microspheres. The inset of Fig. 2 shows a LG beam (

*l*=3,

*p*=0) trapping nine silica colloids of 6.84μm around the beam circumference.

19. K Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass Opt. **4**, S82–S89 (2002) [CrossRef]

## 4. Results

### 4.1 Spiral wave pattern

*l*=3,

*p*=0) beam diffracting from a fully filled microsphere are traced at z

_{R}/2, where z

_{R}is the Rayleigh range.

### 4.2 Two dimensional optical vortices light field

2. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. **198**, 21–27 (2001) [CrossRef]

23. K. O′Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express **14**, 3039–3044 (2006) [CrossRef]

24. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. **7**, 55 (2005). [CrossRef]

*l*=3,

*p*=0) beam. The interference of the forward scattering of the trapped colloidal crystals generates interesting interferometric patterns. Figure 5(A) to Fig. 5(F) show that the far-field intensity patterns consist of a large array of patterns with regions of high and low intensity, ranging from simple interference fringes to complex intensity landscapes. The shows data from numerical simulation (intensity (i), phase (ii)) and experimental (intensity, (iii)) data side by side.

*l*=3,

*p*=0) beam diffracted by 3 or more colloidal particles trapped within annular intensity pattern contains a large amount of unity and opposite charge vortex pairs. This is seen in Fig. 5 (B) (ii) to Fig. 5(F) (ii). Therefore, from the simulation and experimental results, we show that the diffracting light fields from the microspheres are very much like numerous individual spherical beams that interfere as plane waves in the far-field. Thus by simply controlling the trapped position of the colloids in a LG beam, a series of complex two dimensional optical vortex landscapes can be generated, without the need to employ any complex mathematical algorithm or any complex multiple beam interferometer setups.

### 4.3 Young slits experiment with optically trapped particles

*l*=1,

*p*=0) (with a single helical phase ramp from 0 to 2π around the beam circumference) and observe the far field interference pattern in each case. The optical trapping system permits us to place the two objects at will at any azimuthal position within the LG beam (

*l*=1,

*p*=0) beam profile, as shown before in Fig. 3.

25. U. Eichmann, J.C. Bergquist, J.J Bollinger, J.M Gilligan, W.M. Itano, D.J Wineland, and M.G Raizen, ”Young′s interference experiment with light scattered from two atoms,” Phys Rev Lett **70**, 2359–2362 (1993). [CrossRef] [PubMed]

*l*=1,

*p*=0) (Fig. 7(B)), where each of the beam has been aperture by two 6.84μm silica spheres, we are able to detect the effect of the displacement and separation of the intensity fringes solely due to the azimuthal phase variation. Since the intensity of the superposition of two fields in a Young’s double slit is directly proportional to

*α*is the optical phase difference of the two apertures (which are the 6.84μm silica spheres) and Δ

*ϕ*is the additional azimuthal phase difference between the two colloids.

*l*=1,

*p*=0). This yields a 2π linear azimuthal phase ramp around the beam circumference. The small separation between the colloids, due to the smaller annular intensity profile of the LG beam (

*l*=1,

*p*=0), means we are able to obtain a relatively similar pitch of interference fringes compared to interference fringes from the Gaussian beam. The experimental and numerically evaluated interference fringes are shown in Fig. 7. We see that the interference fringes are shifted by π, compared to a Gaussian beam, when an LG beam propagates through the two colloids. Though there is a strong central intensity due to the propagating Gaussian and LG beam in each instance, it is clear that from the diffracting interference fringes; an obvious π phase increment is observed as expected (emphasized using the dashed red line). We also notice a slight tilt in the interference fringes, which we attribute to the finite sampling area of the beam due the finite size of each colloid. The shift in the fringe correlates to the relative phase difference sampled by the two colloids is π as shown in Fig. 6. Experimentally, we have found that the average fringe visibility to be 0.67. Thus illustrates a high fringe visibility/contrast and stability of the interference fringes.

26. J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt Lett **28**, 872–874 (2003). [CrossRef] [PubMed]

**91**, 093602 (2003), [CrossRef] [PubMed]

27. C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express **14**, 4183–4188 (2006) [CrossRef] [PubMed]

28. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. **31**, 649–651 (2006) [CrossRef] [PubMed]

29. P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express **13**, 6657–6666 (2005) [CrossRef] [PubMed]

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A |

2. | J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. |

3. | S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Görlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, “Observation of vortex phase singularities in Bose-Einstein Condensates,” Phys. Rev. Lett. |

4. | B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D.N. Christodoulides, and J.W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature |

5. | 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 |

6. | H. He, M. E. J. 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. |

7. | D. Rozas, C. T. Law, and G.A. Swartzlander Jr., “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B |

8. | G. A. Swartzlander Jr. and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. |

9. | D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. |

10. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger ,“Entanglement of the orbital angular momentum states of photons,” Nature |

11. | A Jesacher, S Furhapter, S Bernet, and M. Ritsch-Marte ,“Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. |

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

13. | W. Wang, S.G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. |

14. | N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt Lett |

15. | K.T. Gahagan and G.A. Swartzlander Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B. |

16. | J. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett |

17. | K. Ladavac and D. G. Grier, “Microoptomechanical pump assembled and driven by holographic optical vortex arrays,” Opt Express |

18. | 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. |

19. | K Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B: Quantum Semiclass Opt. |

20. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

21. | J. -P. Chevaillier, J. Fabre, and P. Hamelin, “Forward scattered light intensities by a sphere located anywhere in a Gaussian beam,” Appl Opt |

22. | E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. |

23. | K. O′Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express |

24. | J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. |

25. | U. Eichmann, J.C. Bergquist, J.J Bollinger, J.M Gilligan, W.M. Itano, D.J Wineland, and M.G Raizen, ”Young′s interference experiment with light scattered from two atoms,” Phys Rev Lett |

26. | J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt Lett |

27. | C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express |

28. | J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. |

29. | P. Fischer, C. T. A. Brown, J. E. Morris, C. López-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, “White light propagation invariant beams,” Opt. Express |

30. | J.M.R. Fournier, M. M. Burns, and J..A. Golovchenko, “Writing Diffractive Structures by Optical Trapping,” Proc SPIE -Int Soc Opt Eng |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(050.5080) Diffraction and gratings : Phase shift

(260.3160) Physical optics : Interference

**ToC Category:**

Trapping

**History**

Original Manuscript: June 13, 2006

Revised Manuscript: July 24, 2006

Manuscript Accepted: July 24, 2006

Published: August 7, 2006

**Virtual Issues**

Vol. 1, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

W. M. Lee, V. Garcés-Chávez, and K. Dholakia, "Interference from multiple trapped colloids in an optical vortex beam," Opt. Express **14**, 7436-7446 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7436

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### References

- J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. London Ser. A 336,165-90 (1974). [CrossRef]
- J. Masajada and B. Dubik, "Optical vortex generation by three plane wave interference," Opt. Commun. 198, 21-27 (2001) [CrossRef]
- S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Görlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard, and W. Ketterle, "Observation of vortex phase singularities in Bose-Einstein Condensates," Phys. Rev. Lett. 87,080402 (2001) [CrossRef] [PubMed]
- B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D.N. Christodoulides and J.W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006) [CrossRef] [PubMed]
- 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, 8185-8189 (1992) [CrossRef] [PubMed]
- H. He, M. E. J. 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, 826-829 (1995). [CrossRef] [PubMed]
- D. Rozas, C.T. Law, G.A. Swartzlander, Jr., "Propagation dynamics of optical vortices," J. Opt. Soc. Am. B 14, 3054-3065 (1997) [CrossRef]
- G. A. Swartzlander, Jr. and J. Schmit, "Temporal correlation vortices and topological dispersion," Phys. Rev. Lett. 93, 093901(2004). [CrossRef] [PubMed]
- D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr, "Spatial correlation singularity of a vortex field," Phys. Rev. Lett. 92, 143905 (2004) [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001). [CrossRef] [PubMed]
- A Jesacher, S Furhapter, S Bernet, M. Ritsch-Marte, "Shadow effects in spiral phase contrast microscopy," Phys. Rev. Lett. 94, 233902 (2005) [CrossRef] [PubMed]
- L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant and K. Dholakia, "Controlled rotation of optically trapped microscopic particles," Science 292, 912-914 (2001). [CrossRef] [PubMed]
- W. Wang, S.G. Hanson, Y. Miyamoto and M. Takeda, "Experimental investigation of local properties and statistics of optical vortices in random wave fields," Phys. Rev. Lett. 94, 103902 (2005) [CrossRef] [PubMed]
- N. B. Simpson, K. Dholakia, L. Allen and M. J. Padgett, "Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner," Opt Lett 22,52-54 (1997) [CrossRef] [PubMed]
- K.T. Gahagan and G.A. Swartzlander, Jr, "Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap," J. Opt. Soc. Am. B. 16, 533-537 (1999) [CrossRef]
- J. Curtis and D. G. Grier, "Structure of optical vortices," Phys. Rev. Lett 90, 133901 (2003) [CrossRef] [PubMed]
- K. Ladavac and D. G. Grier, "Microoptomechanical pump assembled and driven by holographic optical vortex arrays," Opt Express 12, 1144-1149 (2004). [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, 093602 (2003), [CrossRef] [PubMed]
- K Volke-Sepulveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt and K. Dholakia, "Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum Semiclass Opt. 4, S82-S89 (2002) [CrossRef]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986) [CrossRef] [PubMed]
- J. -P. Chevaillier, J. Fabre, and P. Hamelin, "Forward scattered light intensities by a sphere located anywhere in a Gaussian beam," Appl Opt 25,1222-1225 (1986) [CrossRef] [PubMed]
- E. A. Sziklas and A. E. Siegman, "Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method," Appl. Opt. 14, 1874-1889 (1975) [CrossRef] [PubMed]
- K. O'Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four, and five plane waves," Opt. Express 14, 3039-3044 (2006) [CrossRef]
- J. Leach, M. R. Dennis, J. Courtial and M. J. Padgett, "Vortex knots in light," New J. Phys. 7, 55 (2005). [CrossRef]
- U. Eichmann, J.C. Bergquist, J.J Bollinger, J.M Gilligan, W.M. Itano, D.J Wineland, M.G Raizen, "Young's interference experiment with light scattered from two atoms," Phys Rev Lett 70, 2359-2362 (1993). [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, "Modulated optical vortices," Opt Lett 28, 872-874 (2003). [CrossRef] [PubMed]
- C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholakia, "Orbital angular momentum transfer in helical Mathieu beams," Opt. Express 14, 4183-4188 (2006) [CrossRef] [PubMed]
- J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006) [CrossRef] [PubMed]
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