Abstract

A laser beam with phase singularities is an interesting object to study in optics and may have important applications in guiding atoms and molecules. We explore the characteristics of a singularity in a nondiffracting Bessel beam experimentally by use of a programmable spatial light modulator with 64-level phase holograms. The diffraction efficiency with 64-level phase holograms is greatly improved in comparison with that obtained with a binary grating. The experiments show that the size and deflection angle of the beam can be controlled in real time. The observations are in agreement with scalar diffraction theory.

© 2003 Optical Society of America

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References

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  1. Y. Song, D. Milam, and W. T. Hill III, Opt. Lett. 24, 1805 (1999).
    [CrossRef]
  2. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  3. A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
    [CrossRef] [PubMed]
  4. I. Khoo and S. Wu, Optics and Nonlinear Optics of Liquid Crystals (Utopia, Singapore, 1993), pp. 1–6, 269–286.
    [CrossRef]
  5. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
    [CrossRef]
  6. N. R. Heckenberg, Opt. Lett. 17, 221 (1992).
    [CrossRef]

1999 (1)

1997 (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

1993 (1)

I. Khoo and S. Wu, Optics and Nonlinear Optics of Liquid Crystals (Utopia, Singapore, 1993), pp. 1–6, 269–286.
[CrossRef]

1992 (1)

1989 (1)

A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
[CrossRef] [PubMed]

1987 (1)

Durnin, J.

Friberg, A. T.

A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
[CrossRef] [PubMed]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

N. R. Heckenberg, Opt. Lett. 17, 221 (1992).
[CrossRef]

Hill III, W. T.

Khoo, I.

I. Khoo and S. Wu, Optics and Nonlinear Optics of Liquid Crystals (Utopia, Singapore, 1993), pp. 1–6, 269–286.
[CrossRef]

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Milam, D.

Song, Y.

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Turunen, J.

A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
[CrossRef] [PubMed]

Vasara, A.

A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
[CrossRef] [PubMed]

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Wu, S.

I. Khoo and S. Wu, Optics and Nonlinear Optics of Liquid Crystals (Utopia, Singapore, 1993), pp. 1–6, 269–286.
[CrossRef]

J. Opt. Soc. Am. A (1)

A. Vasara, J. Turunen, and A. T. Friberg, J. Opt. Soc. Am. A 6, 1748 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Phys. Rev. A (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, Phys. Rev. A 56, 4064 (1997).
[CrossRef]

Other (1)

I. Khoo and S. Wu, Optics and Nonlinear Optics of Liquid Crystals (Utopia, Singapore, 1993), pp. 1–6, 269–286.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. The reference beam is reflected off the SLM and then off three mirrors (M1–M3) to increase the propagation length. It then passes through at least one neutral-density filter (NDF) before being recorded by a Pulnix TM-72EX CCD camera.

Fig. 2
Fig. 2

Phase masks used to produce two different sizes of the dark core of a Bessel beam with a Bessel function of the first kind of order n=2. The first column consists of the actual phase masks that are generated on the computer and placed on the SLM. The parameter is ρ0=1.4 mm for the top; notice the larger amount of deviation from the horizontal of the fork at the center of the mask due to the smaller magnitude of ρ0. This results in a smaller hole. The parameter for the bottom is ρ0=3.2 mm, resulting in a smaller deflection from the horizontal for the singularity of the grating, causing a larger hole. The middle column is a two-dimensional picture of the Bessel beam generated from the gratings to their left, where the intensity is represented by brightness. The column on the far right is a three-dimensional image of the hollow beam, where both brightness and height represent intensity.

Fig. 3
Fig. 3

Comparison between two nonlinear fits of the Bessel beam (J7) profiles. The top profile was generated from a 64-level phase grating, whereas the bottom profile was generated with a binary phase grating. The ratio of the power in the first ring obtained from the 64-level grating compared with the binary grating is 12.4. The experimental profiles are denoted by the thick curve, whereas the Bessel function fit is denoted by the thin curve, and the amplitudes of both profiles are in similar units. The scale of the x axis from 0 to 500 is equal to the length of the CCD on the camera used to take the profile. The same neutral-density filter was used for both profiles to eliminate the saturation on the CCD camera.

Fig. 4
Fig. 4

By adjusting the bias voltage across the liquid crystals, we can control the phase shift produced by the grating. On the left, the bias is not optimized, and we get a significant amount of light in the zeroth-order diffraction beam. This picture was taken at a bias voltage of 1.47 V. On the right, the bias voltage is optimized at 1.88 V, and almost all of the power is in the first-order diffracted beam, whereas only a small amount remains in the zeroth order.

Equations (2)

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Ex,y,z0,t=expiβz-ωt×02πAϕexpiαx cos ϕ+y sin ϕdϕ,
Ψρ,ϕ=nϕ+2πρ/ρ0.

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