## Optimal annulus structures of optical vortices

Optics Express, Vol. 12, Issue 19, pp. 4625-4634 (2004)

http://dx.doi.org/10.1364/OPEX.12.004625

Acrobat PDF (1945 KB)

### Abstract

An idea of an optimal annulus structure phase mask with helical wavefront is suggested; the resulting helical mode can be focused into a very clear optical vortex ring with the best contrast. Dependences of the optimal annulus width and the radius of the optical vortex ring on topological charge are found. The desired multi-optical vortices as the promising dynamic multi-optical tweezers are realized by extending our idea to the multi-annulus structure. Such multi-optical vortex rings allow carrying the same or different angular momentum flux in magnitude and direction. The idea offers flexibility and more dimensions for designing and producing the complicated optical vortices. For the Gaussian beam illumination, the optimal spot size, which ensures the high energy/power efficiency for generating the best contrast principal ring, is also found.

© 2004 Optical Society of America

## 1. Introduction

1. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef] [PubMed]

*iℓθ*), a plane wavefront can be converted into a helical mode, where ℓ is the topological charge and

*θ*is the azimuthal angle; each photon in a helical mode can carry an orbital angular momentum ℓħ [2–5

2. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995). [CrossRef]

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]

1. D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef] [PubMed]

14. M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. **78**, 547–549 (2001). [CrossRef]

15. A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science **296**, 1841–1844 (2002). [CrossRef] [PubMed]

16. A. Terray, J. Oakey, and D. W. M Marr, “Fabrication of linear colloidal structures for microfluidic applications,” Appl. Phys. Lett. **81**, 1555–1557 (2002). [CrossRef]

17. G. A. Swartzlander Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. **69**, 2503–2506 (1992). [CrossRef] [PubMed]

18. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**, 123–132 (1993). [CrossRef]

4. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

5. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. **88**, 053601 (2002). [CrossRef]

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

13. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**, 169–175 (2002). [CrossRef]

11. J. E. Curtis and D. G. Grier, “Structure of optical vortices” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

## 2. Theory

*θ*-2

*π*int(ℓ

*θ*/2

*π*) and generated by a dynamic PSLM, where the function int(

*x*) gives the integer part of

*x*. The field in the focal plane of a lens with a focal length

*f*can be obtained by implementing the Fourier transform to the field on the plane of the PSLM, that is,

*r*,

*θ*) and (

*ρ*,

*ϕ*) are the polar coordinate systems in the front and back focal plane, respectively,

*J*

_{ℓ}(

*x*) is the ℓ -th order Bessel function of the first kind, Γ() is the Gamma function,

_{1}

*F*

_{2}[ ] is the generalized hypergeometric function,

*R*

_{0}is the outer radius of the optical vortex phase mask,

*λ*is the wavelength of the illuminating beam and the dimensionless parameter

*p*=

*πR*

_{0}ρ/(

*λf*). Fig. 1(a) shows a typical example of a simple spoke-like structure phase mask encoded by ℓ = 40 [6

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]

*R*

_{0}and a changeable inner radius of

*R*

_{1}ranging from 0 to

*R*

_{0}. The field intensity distribution in the focal plane along the radial direction can be calculated from Eq. (1), provided that the lower limit of the integral is replaced by

*R*

_{1}instead of zero. For the case of

*R*

_{0}= 256 pixels and ℓ = 40 , the calculated results are depicted in Figs. 2(a), (b), (c) and (d) at

*R*

_{1}= 0, 125, 200 and 230 pixels, respectively. The parameters used in our theoretical analyses, for example, the wavelength

*λ*= 632.8 nm and the focal length

*f*= 730 mm, are the same as those in the experiments below. It is evident that the too small or too large

*R*

_{1}gives rise to the inferior contrast of the inner principal ring, owing to the appearance of the relative strong outer subsidiary rings as background. Consequently, we believe that there should exist an optimal inner radius

*R*

_{1}(in other words, an optimal annulus width), at which value the subsidiary rings are significantly suppressed and the inner principal ring exhibits the best contrast.

*R*=

*R*

_{0}-

*R*

_{1}at five different values of ℓ = 10, 20, 30, 40 and 50. Fig. 3 gives the calculated results. As shown in Fig. 3(a), the peak intensity of the principal ring at first increases monotonously as Δ

*R*expanding and then arrives at its maximum saturation value once Δ

*R*is over a certain critical value. In contrast, with increasing Δ

*R*, the peak intensity of the first subsidiary ring at first rapidly grows up to its sub-maximum, then drops down to a minimum (near zero) at a characteristic annulus width (which is marked as Δ

*R*) and finally arrives at its maximum again, as shown in Fig. 3(b). Δ

_{opt}*R*is called the optimal annulus width, originating from the two dominant factors: (i) the first subsidiary ring is completely subdued and the other subsidiary rings have the extremely low intensity simultaneously, making the principal ring exhibits the best contrast and (ii) the principal ring also has the very high intensity being at least about 90% of the maximum value. The criterion for finding this optimal annulus width is that the intensity of the first subsidiary ring has the lowest level. Based on this criterion and further numerical fitting, a simple scaling relation of Δ

_{opt}*R*. with

_{opt}*R*

_{0}and ℓ is found that

*R*increases linearly with the increase of the outer radius

_{opt}*R*

_{0}of the annulus phase mask while decreases nonlinearly with the topological charge ℓ increasing; that is to say, the large

*R*

_{0}requires the large Δ

*R*, while the large ℓ gives rise to the small Δ

_{opt}*R*. In addition, we also reveal the dependence of the radius

_{opt}*ρ*

_{P}of the principal ring of optical vortex on the annulus width Δ

*R*, which exhibits the analogous behavior as the peak intensity of the principal ring. When Δ

*R*> Δ

*R*,

_{opt}*ρ*

_{P}becomes to be independent of Δ

*R*. In fact, this is a very important nature and the third reason choosing the above criterion of the optimal annulus width, and allows to create a simple relation of

*ρ*

_{P}with

*R*

_{0}and ℓ, that is,

*ρ*

_{P}versus ℓ, like in the traditional spoke-like structure [6

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]

*ρ*

_{P}can be controlled by choosing

*R*

_{0}and ℓ. The direction (clockwise or anti-clockwise) of the orbital angular momentum carried on the optical vortex can be also arbitrarily selected, by changing the sign of ℓ.

*N*narrow annuluses with the same width of Δ

*r*; Eq. (1) can be rewritten as

*u*

_{0}(

*r*,

*ω*)) is the amplitude of the illuminating Gaussian beam

*R*is the average radius of the annulus phase mask with the optimal annulus width Δ

_{m}*R*, i.e.,

_{opt}*R*=

_{m}*R*

_{0}-Δ

*R*/2 or =

_{opt}*R*

_{1}+Δ

*R*/2. We give a simulated test regarding the dependence of the intensity of the principal ring on the spot size of Gaussian beam, where an optimal annulus phase mask (

_{opt}*R*

_{0}= 256 pixels, ℓ = 40 and Δ

*R*= 50 pixels) is designed by Eq. (2). The solid line in Fig. 4 shows our simulated result. We can see that when the spot size of Gaussian beam is near 330 pixels, the intensity of the principal ring in the focal plane arrives at its maximum, which is in good agreement with the value predicted by Eq. (6).

_{opt}## 3. Experiment

*λ*= 632.8 nm was used as light source and the designed phase mask was displayed on a twisted nematic liquid crystal display (LCD) (with 1024×768 pixels, each pixel with 18

*μm*×18

*μm*) and a lens of focal length

*f*= 730 mm was used. Phase-only operation for the LCD can be achieved by use of a combination of two polarizers and two quarter-wavelength plates [19

19. A. Marquez, C. Lemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. **40**, 2558–2564 (2001). [CrossRef]

*R*

_{0}= 256 pixels, ℓ = 40 and Δ

*R*= Δ

*R*= 50 pixels. The circles in Fig. 4 show the measured results, which agree with the theory. In all the experiments below we adjust the spot size of the input Gaussian beam to satisfy the optimal condition of Eq. (6). The optimal annuls phase mask (as shown in Fig. 5(a)) was displayed on the LCD. Fig. 5(b) and (c) are the simulated intensity and phase distribution of the optical vortex ring in the focal plane, respectively. The center spot in Fig.5 (b) is formed by the light coming from the center section of the annular phase mask, because we did not block it in this simulation. The simulated result indicates that the radius of the principal ring produced in the focal plane is well in agreement with the value predicted by Eq. (3). The thick dashed circle in Fig. 5(c) gives the corresponding position of the principal ring in the focal plane, which indicates the helical wavefront of the vortex ring. Fig. 5(d) shows the observed optical vortex ring in real experiments, which is consistent with the simulated result shown in Fig. 5(b). Compared with Fig. 1(b) of the conventional spokelike structure phase mask with the same

_{opt}*R*

_{0}and ℓ, it can be seen that the subsidiary rings was successfully suppressed and the principal ring becomes more clear.

*R*

_{0}of phase mask, the wavelength

*λ*, the focal length

*f*, the desired radii

*ρ*

_{P}(ℓ

_{1}) and

*ρ*

_{P}(ℓ

_{2}) of two principal rings of the bi-optical vortices.

*R*

_{0}is usually determined by the size of PSLM used in experiments.

*ρ*

_{P}(ℓ

_{1}) and

*ρ*

_{P}(ℓ

_{2}) depend upon the experimental aims. (ii) Calculating the topological charge ℓ

_{1}of the outer annulus of the phase mask according to Eq. (3) and its inner radius

*R*

_{1}=

*R*

_{0}-Δ

*R*(ℓ

_{1}) by Eq. (2). (iii) Calculating the topological charge ℓ

_{2}of the inner annulus using Eq. (3) and its inner radius

*R*

_{2}=

*R*

_{1}-Δ

*R*(ℓ

_{2}) by Eq. (2), but

*R*

_{0}and

*ρ*

_{P}(ℓ

_{1}) should be replaced by

*R*

_{1}and

*ρ*

_{P}(ℓ

_{2}), respectively.

*R*

_{0}=256 pixels,

*ρ*

_{p}(ℓ

_{1}) = 750

*μ*m and

*ρ*

_{P}(ℓ

_{2}) = 550

*μ*m. We estimate ℓ

_{1}≈41,

*R*

_{1}-207 pixels, ℓ

_{2}≈ 20 and

*R*

_{2}-149 pixels, respectively. Figure 6(b) indicates the observed pattern of bi-ring optical vortex produced in the focal plane. In this bi- ring structure situation, although the spot size of the incident Gaussion beam cannot be directly determined by Eq. (6), the calculated values offer a useful instruction for experiments. In our experiment the spot size is approximately equal to the optimal spot size calculated only according to the parameters of the outer annulus, which makes a smaller difference between the intensities of the two vortex rings. Because the two principal rings are from two different annulus structures of the phase mask, their orbital angular momentums carried could be set to the same or opposite directions, depending on the relative signs of the topological charges ℓ

_{1}and ℓ

_{2}of the two annulus. Figure 6(a) and (b) are the case in which the orbital angular momentums carried by the two principal rings have the opposite directions. Figure 6(c) and (d) are the other designed example, the difference from the former is that the latter has

*ρ*

_{P}(ℓ

_{1}) -

*ρ*

_{P}(ℓ

_{2}) = 750

*μ*m and the same sign of

*ℓ*

_{1}= 41 and ℓ

_{2}= 33. In this situation, the two vortex rings produced by this bi-annulus structure are overlapped each other in the focal plane. The interference between these overlapped principal rings produces a new optical vortex distribution. It is clear that more complicated structure phase mask could be created with the present method.

## 4. Discussion

*π*at the phase step points of phase mask. For the convenience of description, we define the phase deviation Δ

*ϕ*=

*ϕ*- 2

*π*, where

*ϕ*is the maximum phase at the phase step points. For an ideal programmable PSLM,

*ϕ*is exactly equal to 2

*π*, that is, Δ

*ϕ*= 0 . In most practical situations, however, the phase deviation Δ

*ϕ*is often either smaller or larger than 0, which results in the periodic modulation of the reconstructed optical vortices.

*ϕ*= -0.5

*π*when these phase masks are displayed onto the LCD described above. Figures 7(a), (b), (c) and (d) show the observed images of the optical vortex rings produced in the focal plane. Compared with Fig. 1(b), Fig. 5(d), Fig. 6(b) and Fig. 6(d), the corresponding optical vortices shown in Fig.7 exhibit very serious intensity modulations, which are obvious because of the introduction of the large phase deviation. To further verify this idea, we study the effect of the phase deviation Δ

*ϕ*on the depth of the azimuthal modulation in detail; as an example, the theoretical result is plotted in Fig. 8 for the optimal annalus phase mask with

*R*

_{0}= 256 pixels, ℓ = 40 and Δ

*R*(40) = 50 pixels. We find that the depth of the azimuthal modulation increases nearly with exacerbating the phase deviation δ

_{opt}*ϕ*. This fact could be useful in adjusting the modulation depth for studying Brownnian transport in modulated potentials.

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. G. Grier, “A revolution in optical manipulation,” Nature |

2. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

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

4. | L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. |

5. | A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. |

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. | N. B. Simpson, L. Allen, and M. J. Padgett, “Optical tweezers and optical spanners with Laguerre-Gaussian modes,” J. Mod. Opt. |

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

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

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

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

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

13. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

14. | M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, “Optically driven micromachine elements,” Appl. Phys. Lett. |

15. | A. Terray, J. Oakey, and D. W. M. Marr, “Microfluidic control using colloidal devices,” Science |

16. | A. Terray, J. Oakey, and D. W. M Marr, “Fabrication of linear colloidal structures for microfluidic applications,” Appl. Phys. Lett. |

17. | G. A. Swartzlander Jr. and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. |

18. | M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

19. | A. Marquez, C. Lemmi, I. Moreno, J. Davis, J. Campos, and M. Yzuel, “Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,” Opt. Eng. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(090.1760) Holography : Computer holography

(140.7010) Lasers and laser optics : Laser trapping

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 25, 2004

Revised Manuscript: September 14, 2004

Published: September 20, 2004

**Citation**

Cheng-Shan Guo, Xuan Liu, Jing-Liang He, and Hui-Tian Wang, "Optimal annulus structures of optical vortices," Opt. Express **12**, 4625-4634 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4625

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### References

- D. G. Grier, �??A revolution in optical manipulation,�?? Nature, 424, 810-816 (2003) [CrossRef] [PubMed]
- M. J. Padgett and L. Allen, �??The Poynting vector in Laguerre-Gaussian laser modes,�?? Opt. Commun. 121, 36�??40 (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]
- L. Allen, M. J. Padgett, and M. Babiker, �??The orbital angular momentum of light,�?? Prog. Opt. 39, 291�??372 (1999) [CrossRef]
- A. T. O�??Neil, I. MacVicar, L. Allen, and M. J. Padgett, �??Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,�?? Phys. Rev. Lett. 88, 053601 (2002). [CrossRef]
- 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]
- N. B. Simpson, L. Allen, and M. J. Padgett, �??Optical tweezers and optical spanners with Laguerre-Gaussian modes,�?? 43, 2485�??2491 (1996) [CrossRef]
- K. T. Gahagan and G. A. Swartzlander Jr., �??Optical vortex trapping of particles,�?? Opt. Lett. 21, 827�??829 (1996) [CrossRef] [PubMed]
- K. T. Gahagan and G. A. Swartzlander, �??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]
- 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]
- J. E. Curtis and D. G. Grier, �??Structure of optical vortices�?? Phys. Rev. Lett. 90, 133901 (2003) [CrossRef] [PubMed]
- J. E. Curtis and D. G. Grier, �??Modulated optical vortices,�?? Opt. Lett. 28, 872-874 (2003) [CrossRef] [PubMed]
- J. E. Curtis, B. A. Koss, and D. G. Grier, �??Dynamic holographic optical tweezers,�?? Opt. Commun. 207, 169�??175 (2002) [CrossRef]
- M. E. J. Friese, H. Rubinsztein-Dunlop, J. Gold, P. Hagberg, and D. Hanstorp, �??Optically driven micromachine elements,�?? Appl. Phys. Lett. 78, 547�??549 (2001) [CrossRef]
- Terray, A., Oakey, J. & Marr, D. W. M. �??Microfluidic control using colloidal devices,�?? Science 296, 1841�?? 1844 (2002) [CrossRef] [PubMed]
- Terray, A., Oakey, J. & Marr, D. W. M, �??Fabrication of linear colloidal structures for microfluidic applications,�?? Appl. Phys. Lett. 81, 1555�??1557 (2002) [CrossRef]
- G. A. Swartzlander, Jr. and C. T. Law, �??Optical vortex solitons observed in Kerr nonlinear media,�?? Phys. Rev. Lett. 69, 2503-2506 (1992) [CrossRef] [PubMed]
- M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, �??Astigmatic laser mode converters and transfer of orbital angular momentum,�?? Opt. Commun. 96, 123-132 (1993) [CrossRef]
- A. Marquez, C. Lemmi, I. Moreno, J. Davis, J. Campos and M. Yzuel, �??Quantitative prediction of the modulation behavior of twisted nematic liquid crystal displays based on a simple physical model,�?? Opt. Eng. 40, 2558-2564 (2001) [CrossRef]

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