Abstract

A theoretical model is proposed to describe a novel vortex beam named anomalous vortex (AV) beam. Analytical propagation formula for the proposed beam passing through a paraxial ABCD optical system is derived, and the propagation properties of such beam in free space are studied numerically. It is interesting to find that an AV beam will eventually become an elegant Laguerre–Gaussian beam in the far field (or in the focal plane) in free space. Furthermore, we report experimental generation of an AV beam and measure its propagation properties. Our experimental results are consistent with the theoretical predictions.

© 2013 Optical Society of America

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References

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2013 (1)

2012 (2)

Y. Yang, M. Mazilu, and K. Dholakia, Opt. Lett. 37, 4949 (2012).
[CrossRef]

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

2011 (2)

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

C. Zhao and Y. Cai, Opt. Lett. 36, 2251 (2011).
[CrossRef]

2010 (1)

M. Uchida and A. Tonomura, Nature 464, 737 (2010).
[CrossRef]

2008 (1)

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

2004 (1)

2003 (1)

2001 (1)

1998 (1)

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

1992 (1)

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

1973 (1)

1970 (1)

Ahmed, N.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Allen, L.

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

April, A.

Bandres, M. A.

Beijersbergen, M. W.

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

Borghi, R.

Cai, Y.

Cizmar, T.

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

Collins, S. A.

Dholakia, K.

Y. Yang, M. Mazilu, and K. Dholakia, Opt. Lett. 37, 4949 (2012).
[CrossRef]

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

Dolinar, S.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Fazal, I.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Gutiérrez-Vega, J. C.

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

Huang, H.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Lin, Q.

Lu, X.

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Mazilu, M.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Nasalski, W.

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Porras, M. A.

Ren, Y.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

Siegman, A. E.

Spreeuw, R. J. C.

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

Tonomura, A.

M. Uchida and A. Tonomura, Nature 464, 737 (2010).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Tur, M.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Uchida, M.

M. Uchida and A. Tonomura, Nature 464, 737 (2010).
[CrossRef]

Wang, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Willner, A.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Woerdman, J. P.

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

Yan, Y.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Yang, J.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Yang, Y.

Yue, Y.

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Zhao, C.

J. Mod. Opt. (1)

S. Saghafi and C. J. R. Sheppard, J. Mod. Opt. 45, 1999 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Nat. Photonics (2)

K. Dholakia and T. Cizmar, Nat. Photonics 5, 335 (2011).
[CrossRef]

J. Wang, J. Yang, I. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. Willner, Nat. Photonics 6, 488 (2012).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Nature (1)

M. Uchida and A. Tonomura, Nature 464, 737 (2010).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (1)

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

Other (2)

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

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Figures (4)

Fig. 1.
Fig. 1.

Intensity distribution (contour graph) of an AV beam for different values of m and n with w0=1mm.

Fig. 2.
Fig. 2.

Normalized 3D intensity distribution (top), the corresponding contour graph (middle), and the phase distribution (bottom) of an AV beam at different propagation distances in free space (a), (d), (g) z=0; (b), (e), (h) z=zR; and (c), (f), (i) z=20zR.

Fig. 3.
Fig. 3.

Experimental setup for generating an AV beam and measuring its focused intensity. DPSSL, diode-pumped solid-state laser; NDF, neutral density filter; L1,L2, L3, thin lenses; SLM, spatial light modulator; CA, circular aperture; SPP, spiral phase plate; BPA, beam profile analyzer; PC1, PC2, personal computer.

Fig. 4.
Fig. 4.

Theoretical and experimental results of the intensity distribution of the generated AV beam in the source plane [(a-1) to (c-1) and (a-2) to (c-2)] and focal plane [(a-3) to (c-3) and (a-4) to (c-4)] for different values of m and n.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

En,m(ρ0,θ0,0)=E0(ρ0w0)2n+|m|exp(ρ02w02)exp(imθ0),
En,m(ρ0,θ0,0)=E0n!2ns=0n(1)s(ns)Ls(2ρ02w02)×exp(ρ02w02)(ρ0w0)|m|exp(imθ0).
En,m(ρ,θ,z)=iλBexp(ikz)02π0En,m(ρ0,θ0,0)exp{ik2B×[Aρ022ρρ0cos(θθ0)+Dρ2]}ρ0dρ0dθ0,
En,m(ρ,θ,z)=im+1πn!λBexp(ik2BDρ2ikz)g(n+|m|+1)(kρ2B)|m|×exp(k2ρ24gB2)Ln|m|(k2ρ24gB2)exp(imθ),
g=1w02+ikA2B.
02πexp[inθ1+ikbrcos(θ1θ2)]dθ1=2πexp[in(π/2θ2)]Jn(kbr),
0exp(ax2)Jv(2bx)x2n+v+1dx=n!2bvanv1exp(b2/a)Lnv(b2/a).
(ABCD)=(1z01).
En,m(ρ,θ,z)=im+1πn!λzexp(ikz)w02(n+|m|+1)(kρ2z)|m|×exp(w02k2ρ24z2)Ln|m|(w02k2ρ24z2)exp(imθ)=im+1πn!λzexp(ikz)w02n+|m|+2exp(imθ)×(ρw)|m|exp(ρ2w2)Ln|m|(ρ2w2),
En,m(0)(ρ0,θ0,0)=E0(ρ0w0)2n+|m|exp(ρ02w02).

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