Abstract

We analytically and experimentally study the Fraunhofer diffraction of an optical vortex beam possessing noninteger values of the azimuthal index. We show that the Fraunhofer diffraction of this beam presents the birth of a vortex at α=n+ε, where n is an integer number and ε is a small fraction. We discuss this behavior on the basis of the born vortex movement from a position of low intensity to high intensity when α is increased of an integer number in fractional steps of ε.

© 2012 Optical Society of America

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
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  3. J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004).
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  4. W. M. Lee, X. C. Yuan, and K. Dholakia, Opt. Commun. 239, 129 (2004).
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  5. S. H. Tao and X. C. Yuan, J. Opt. Soc. Am. A 21, 1192 (2004).
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  6. H. Garcia-Gracia and J. C. Gutierrez-Vega, J. Opt. Soc. Am. A 26, 794 (2009).
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  7. J. B. Gotte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, Opt. Express 16, 993 (2008).
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  8. P. H. Jones, M. Rashid, M. Makita, and O. M. Marago, Opt. Lett. 34, 2560 (2009).
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  9. A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, Opt. Express 19, 5760 (2011).
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  10. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
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  11. J. P. Kirk and A. L. Jones, J. Opt. Soc. Am. 61, 1023 (1971).
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  12. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).
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  14. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Opt. Express 15, 9378 (2007).
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2011 (2)

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

2009 (2)

2008 (1)

2007 (1)

2004 (4)

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004).
[CrossRef]

W. M. Lee, X. C. Yuan, and K. Dholakia, Opt. Commun. 239, 129 (2004).
[CrossRef]

S. H. Tao and X. C. Yuan, J. Opt. Soc. Am. A 21, 1192 (2004).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1971 (1)

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Barnett, S. M.

Baumgartl, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Berry, M. V.

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

Chavez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Dholakia, K.

Flossmann, F.

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Franke-Arnold, S.

Garcia-Gracia, H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

Gotte, J. B.

Gutierrez-Vega, J. C.

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Jones, A. L.

Jones, P. H.

Kartashov, Y. V.

Kirk, J. P.

Kumar, A.

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004).
[CrossRef]

Lee, W. M.

W. M. Lee, X. C. Yuan, and K. Dholakia, Opt. Commun. 239, 129 (2004).
[CrossRef]

Makita, M.

Marago, O. M.

Mourka, A.

O’Holleran, K.

Padgett, M. J.

Preece, D.

Rashid, M.

Shanor, C.

Singh, R. P.

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Tao, S. H.

Torner, L.

Vaity, P.

Vysloukh, V. A.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Wright, E. M.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004).
[CrossRef]

Yuan, X. C.

W. M. Lee, X. C. Yuan, and K. Dholakia, Opt. Commun. 239, 129 (2004).
[CrossRef]

S. H. Tao and X. C. Yuan, J. Opt. Soc. Am. A 21, 1192 (2004).
[CrossRef]

J. Opt. A (1)

M. V. Berry, J. Opt. A 6, 259 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, New J. Phys. 6, 71 (2004).
[CrossRef]

Opt. Commun. (1)

W. M. Lee, X. C. Yuan, and K. Dholakia, Opt. Commun. 239, 129 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chavez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup. A triangular aperture was placed over each vortex and the diffracted light was acquired by a CCD camera.

Fig. 2.
Fig. 2.

Theoretical result at Fraunhofer plane for amplitude (left) and phase (right) using α=3.3.

Fig. 3.
Fig. 3.

Total vortex strength S of the field at Fraunhofer plane as a function of α computed from Eq. (6).

Fig. 4.
Fig. 4.

Experimental measurements of intensity profile of a LG beam for fractional value of α varying from 3 to 4 with steps of 0.1 at Fraunhofer plane.

Fig. 5.
Fig. 5.

Experimental measurements of the intensity profile of a LG beam for values of α=3.3, 4.7 at Fraunhofer plane and the diffraction pattern by a triangle aperture showing the value of TC of α=1 and α=1 for fractional TC of α=3.3 and α=4.7, respectively.

Equations (6)

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Ei(ρ,ϕ)=Aραexp(ρ2w02)exp(iαϕ),
E(k,θk)02π0+Ei(ρ,ϕ)eikρcos(θkϕ)ρdρdϕ.
exp(iαϕ)=exp(iπϕ)sin(πα)π+exp(inϕ)αn.
E(k,θk)kexp(π2k2w022)+(i)|n|exp(inθk)BIn(k)αn,
BIn(k)=I(|n|1)/2(π2k2w022)I(|n|+1)/2(π2k2w022),
Sα=limk12π02πdθkθkargE(k,θk)=limk12π02πdθkRe[(i)θkE(k,θk)E(k,θk)].

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