## Deterministic approach to the generation of modified helical beams for optical manipulation

Optics Express, Vol. 13, Issue 10, pp. 3862-3867 (2005)

http://dx.doi.org/10.1364/OPEX.13.003862

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### Abstract

We present a deterministic method to generate modified helical beams which create optical vortices with desired dark core intensity patterns in the far-field. The experiments are implemented and verified by a spatial light modulator (SLM), which imprints a phase function onto the incident wavefront of a TEM00 laser mode to transform the incident beam into a modified helical beam. The phase function can be calculated once a specific dark core shape of an optical vortex is required. The modified helical beam is exploited in optical manipulation with verification of its orbital angular momentum experimentally.

© 2005 Optical Society of America

1. 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]

2. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. **21**, 827–829 (1996). [CrossRef] [PubMed]

3. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**, 217–223 (1995). [CrossRef]

4. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A **54**, 1593–1596 (1996). [CrossRef] [PubMed]

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

7. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. **29**, 1796–1798 (2004) [CrossRef] [PubMed]

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

*ilθ*), where the integer

*l*is the topological charge and

*θ*is the azimuth angle in the polar coordinates. The phase structure represented by the phase function is a |

*l*|-started helicoids and the sign of

*l*determines the spiral direction. The phase function of a modified helical beam can be written in a generalized form of exp[

*iφ*(

*θ*)], where

*φ*(

*θ*) represents the phase values and

*φ*(

*θ*)=

*lθ*in the case of conventional helical beam. According to Ref. [5

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

*a*

_{I}and

*b*

_{I}are constants depending on the beam’s radial amplitude profile,

*λ*is the wavelength of light and

*NA*is the numerical aperture of the focusing lens. Obviously,

*R*

_{I}(

*θ*) forms a close loop about the origin. Compared to the circumference of dark core of optical vortex, the maximum intensity loop

*R*

_{I}(

*θ*) is not easy to be visualized in terms of an intensity pattern. So the main purpose of this paper is to realize optical vortex with a desired dark core shape. Apparently the perimeter of the dark core is not exactly the loop

*R*

_{I}(

*θ*). For the sake of brevity, the perimeter of the dark core is referred to as the perimeter below. Since the experimental result in Ref. 5 indicated that the perimeter almost kept the same shape as the maximum intensity loop, it is reasonable to suggest that the perimeter of the dark core

*R*(

*θ*) is also dependent on the phase gradient

*d*

*φ*(

*θ*)

*dθ*linearly

*a*and

*b*are constants. For conventional helical beams, since the phase gradient is determined as a constant

*l*, the perimeter is formed in a circular shape. For modified helical beams, since the phase values can be modified to increase by the azimuth angle in a nonlinear fashion, it results in different local radii at different angles. The term of

*aλ*(

*b*×

*NA*) in Eq. (2) can be neglected during the calculation of the specific phase function because it does not affect the shape of the perimeter but only the size, whereas the size can be altered by choosing different focusing lens later.

*φ*(

*θ*) in a range between 0 and

*l*×2

*π*, where

*l*is the overall topological charge of the modified optical vortex, and in order to keep the overall topological charge as an integer

*l*, we assume

*R*(

*θ*) is determined in Eq. (4), then the phase function

*φ*(

*θ*) can be expressed as

*φ*(0)=0.

*n*vertices (

*P*

_{1},

*P*

_{2},…

*P*

_{n-1},

*P*

_{n}). For the sake of simplicity, vertex

*P*

_{1}is rotated onto the axis

*θ*=0.

*i*≤

*n*-1.

*b*. Consider the simplest polygon case, i.e. a triangle with

*l*=10. The amplitude of the modified helical beam is given as

*ψ*(

*r*,

*θ*)=exp(-

*r*

^{2}/

*iφ*(

*θ*)], where

*w*

_{0}is the beam waist. It is shown in Fig. 2 that the value of the constant b has a significant effort on the shape of the roundness of the reconstructed perimeter in the far field.

*b*decreases the shape of the perimeter tends to be a circle. Obviously an optimized value of

*b*is existed and it can be found out as an empirical formula

*b*=-0.3

*l*valid for

*l*≥10 in the simulation. It is also noted that Eq. (3) is only applied to helical beams with

*l*≫1. So Eq. (7) becomes

_{00}laser beam shining onto a reflective type SLM (Boulder Nonlinear Systems P512 with 512×512 pixels) was imposed by the designated phase function

*φ*(

*θ*). Then the modified helical beam passed through a focusing lens to generate the far-field intensity distribution as shown in Fig. 3 captured by a CCD beam profiler in the focal plane. A constant phase can be added to cancel out the Gouy phase shift effect in order to keep the patterns the same orientation with that of the designs.

*µm*diameter polystyrene spheres dispersed in water.

*l*, which indicates the existence of orbital angular momentum. The movie in Fig. 4 shows that the orbital angular momentum induced rotation of polystyrene spheres trapped in the bright rim of triangle shape optical vortex.

## Acknowledgments

## References and Links

1. | 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. |

2. | K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. |

3. | H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. |

4. | M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A |

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

6. | D. W. Zhang and X.-C. Yuan, “Optical doughnut for optical tweezers,” Opt. Lett. |

7. | W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. |

8. | N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(140.3300) Lasers and laser optics : Laser beam shaping

(140.7010) Lasers and laser optics : Laser trapping

(350.5030) Other areas of optics : Phase

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 24, 2005

Revised Manuscript: May 10, 2005

Published: May 16, 2005

**Citation**

J. Lin, X.-C. Yuan, S. H. Tao, X. Peng, and H. B. Niu, "Deterministic approach to the generation of modified helical beams for optical manipulation," Opt. Express **13**, 3862-3867 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3862

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### References

- 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]
- K. T. Gahagan and G. A. Swartzlander Jr., �??Optical vortex trapping of particles,�?? Opt. Lett. 21, 827-829 (1996). [CrossRef] [PubMed]
- H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, �??Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,�?? J. Mod. Opt. 42, 217-223 (1995). [CrossRef]
- M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, �??Optical angular-momentum transfer to trapped absorbing particles,�?? Phys. Rev. A 54, 1593-1596 (1996). [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, �??Modulated optical vortices,�?? Opt. Lett. 28, 872-874 (2003). [CrossRef] [PubMed]
- D. W. Zhang and X.-C. Yuan, �??Optical doughnut for optical tweezers,�?? Opt. Lett. 28, 740-742 (2003). [CrossRef] [PubMed]
- W. M. Lee, X.-C. Yuan, and W. C. Cheong, �??Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,�?? Opt. Lett. 29, 1796-1798 (2004). [CrossRef] [PubMed]
- N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, �??Laser beams with phase singularities,�?? Opt. Quantum Electron. 24, 951-962 (1992). [CrossRef]

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