Abstract

We show that the intensity distribution of an optical vortex contains information of its order. Specifically, the number of dark rings in the Fourier transform of the intensity is found to be equal to the order of the vortex. Based on this property and the orthogonality of Laguerre polynomials, we demonstrate the feasibility of an experimental technique for determining the order of optical vortices. It shows the beauty of going to complementary spaces, which has been employed earlier also to find the information not available in other domains.

© 2011 Optical Society of America

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References

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    [CrossRef]

2011 (3)

2010 (1)

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

2009 (3)

2008 (1)

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

2007 (1)

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

2006 (2)

E. Fraczek, W. Fraczek, and J. Masajada, Optik 117, 423(2006).
[CrossRef]

H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 999 (2006).
[CrossRef] [PubMed]

2003 (2)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901(2003).
[CrossRef] [PubMed]

1998 (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

1996 (1)

G. S. Agarwal and R. P. Singh, Phys. Lett. A 217, 215 (1996).
[CrossRef]

1995 (1)

D. E. James, H. C. Kandpal, and E. Wolf, Astrophys. J. 445, 406 (1995).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and R. P. Singh, Phys. Lett. A 217, 215 (1996).
[CrossRef]

Alfano, R. R.

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Anderson, M. E.

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Baida, L.

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, Opt. Laser Technol. 43, 1233 (2011).
[CrossRef]

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Cottrell, D. M.

Curtis, J. E.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901(2003).
[CrossRef] [PubMed]

Davis, J. A.

de Araujo, L. E. E.

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Fraczek, E.

E. Fraczek, W. Fraczek, and J. Masajada, Optik 117, 423(2006).
[CrossRef]

Fraczek, W.

E. Fraczek, W. Fraczek, and J. Masajada, Optik 117, 423(2006).
[CrossRef]

Ghai, D. P.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 47, 123 (2009).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ruzhik, Table of Integrals, Series and Products (Academic, 2007).

Grier, D. G.

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901(2003).
[CrossRef] [PubMed]

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Guo, C.-S.

Hua, T.

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, Opt. Laser Technol. 43, 1233 (2011).
[CrossRef]

James, D. E.

D. E. James, H. C. Kandpal, and E. Wolf, Astrophys. J. 445, 406 (1995).
[CrossRef]

Jixiong, P.

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, Opt. Laser Technol. 43, 1233 (2011).
[CrossRef]

Kandpal, H. C.

D. E. James, H. C. Kandpal, and E. Wolf, Astrophys. J. 445, 406 (1995).
[CrossRef]

Krishna, Y.

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

Kumar, A.

A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, Opt. Lett. 36, 1161 (2011).
[CrossRef] [PubMed]

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

Lu, L.-L.

Masajada, J.

E. Fraczek, W. Fraczek, and J. Masajada, Optik 117, 423(2006).
[CrossRef]

Melvin, B.

Mitry, M. J.

Molina-Terriza, G.

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

Moreno, I.

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Pascoguin, L.

Prabhakar, S.

Ruzhik, I. M.

I. S. Gradshteyn and I. M. Ruzhik, Table of Integrals, Series and Products (Academic, 2007).

Senthilkumaran, P.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 47, 123 (2009).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

Singh, R. P.

A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, Opt. Lett. 36, 1161 (2011).
[CrossRef] [PubMed]

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

G. S. Agarwal and R. P. Singh, Phys. Lett. A 217, 215 (1996).
[CrossRef]

Sirohi, R. S.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 47, 123 (2009).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

Sztul, H. I.

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]

Vaity, P.

A. Kumar, S. Prabhakar, P. Vaity, and R. P. Singh, Opt. Lett. 36, 1161 (2011).
[CrossRef] [PubMed]

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

Vyas, S.

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

Wang, H.-T.

Wolf, E.

D. E. James, H. C. Kandpal, and E. Wolf, Astrophys. J. 445, 406 (1995).
[CrossRef]

Yongxin, L.

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, Opt. Laser Technol. 43, 1233 (2011).
[CrossRef]

Astrophys. J. (1)

D. E. James, H. C. Kandpal, and E. Wolf, Astrophys. J. 445, 406 (1995).
[CrossRef]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Nat. Phys. (1)

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

Nature (1)

D. G. Grier, Nature 424, 810 (2003).
[CrossRef] [PubMed]

Opt. Laser Technol. (1)

L. Yongxin, T. Hua, P. Jixiong, and L. Baida, Opt. Laser Technol. 43, 1233 (2011).
[CrossRef]

Opt. Lasers Eng. (3)

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, Opt. Lasers Eng. 48, 276 (2010).
[CrossRef]

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 47, 123 (2009).
[CrossRef]

D. P. Ghai, S. Vyas, P. Senthilkumaran, and R. S. Sirohi, Opt. Lasers Eng. 46, 419 (2008).
[CrossRef]

Opt. Lett. (5)

Optik (1)

E. Fraczek, W. Fraczek, and J. Masajada, Optik 117, 423(2006).
[CrossRef]

Phys. Lett. A (1)

G. S. Agarwal and R. P. Singh, Phys. Lett. A 217, 215 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

J. E. Curtis and D. G. Grier, Phys. Rev. Lett. 90, 133901(2003).
[CrossRef] [PubMed]

Other (2)

I. S. Gradshteyn and I. M. Ruzhik, Table of Integrals, Series and Products (Academic, 2007).

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Experimental setup to generate optical vortices and to find their order.

Fig. 2
Fig. 2

Intensity distribution of optical vortices of orders m = 1 to 4 (from left to right): theoretical (top) and experimental (bottom).

Fig. 3
Fig. 3

Distribution of G m ( ω 1 , ω 2 ) computed from the in tensity distributions of Fig. 2: theoretical (top) and experimental (bottom).

Fig. 4
Fig. 4

Normalized orthogonal integrals C m n for optical vortices of orders m = 1 to 4. Inset shows C m n and the intensity record of vortex for m = 10 .

Equations (13)

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

E m ( x , y ) = ( x + i y ) m exp [ ( x 2 + y 2 ) / σ 2 ] ,
I m ( x , y ) = ( x 2 + y 2 ) m exp [ 2 ( x 2 + y 2 ) / σ 2 ] .
F m ( ω 1 , ω 2 ) = ( x 2 + y 2 ) m × exp [ 2 ( x 2 + y 2 ) / σ 2 i ( ω 1 x + ω 2 y ) ] d x d y ,
F m ( ω 1 , ω 2 ) = n = 0 m ( m n ) I n ( ω 1 ) I m n ( ω 2 ) ,
I n ( ω 1 ) = x 2 n exp ( 2 x 2 / σ 2 i ω 1 x ) d x .
0 x 2 n exp ( β 2 x 2 ) cos a x d x = ( 1 ) n π ( 2 β ) 2 n + 1 exp ( a 2 4 β 2 ) H 2 n ( a 2 β ) ,
H 2 n ( x ) = ( 1 ) n 2 2 n n ! L n 1 / 2 ( x 2 ) ,
I n ( ω 1 ) = π σ 2 n + 1 2 n + 1 / 2 n ! exp ( ω 1 2 σ 2 8 ) L n 1 / 2 ( ω 1 2 σ 2 8 ) ,
n = 0 m L n α ( x ) L m n β ( y ) = L m α + β + 1 ( x + y ) ,
F m ( ω 1 , ω 2 ) = π σ 2 m + 2 2 m + 1 m ! exp ( ζ ) L m ( ζ ) ,
0 exp ( ζ ) L m ( ζ ) L n ( ζ ) d ζ = δ m , n
C m n = 0 F m ( ζ ) L n ( ζ ) d ζ
G m ( ω 1 , ω 2 ) = log [ 1 + | F m ( ω 1 , ω 2 ) | ] .

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