Abstract

Laguerre–Gaussian beams have become increasingly important in areas like optical manipulation and quantum communications. A good understanding of the accuracy and sensitivity of their displacement measurements are required for these applications. In this Letter, we present a method to analyze and compare the performance of these measurements, and an estimation technique is proposed to improve their measurement accuracies. The effectiveness of our methods is verified by simulation studies.

© 2011 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdma, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. D. Grier, Nature 424, 810 (2003).
    [CrossRef] [PubMed]
  3. M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
    [CrossRef] [PubMed]
  4. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
    [CrossRef]
  5. J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, Opt. Express 16, 10117 (2008).
    [CrossRef] [PubMed]
  6. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
    [CrossRef]
  7. G. M. Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  8. F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
    [CrossRef] [PubMed]
  9. M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
    [CrossRef]
  10. Y. D. Liu, C. Gao, X. Qi, and H. Weber, Opt. Express 16, 7091 (2008).
    [CrossRef] [PubMed]
  11. M. Merano, N. Hermosa, and J. P. Woerdman, Phys. Rev. A 82, 023817 (2010).
    [CrossRef]
  12. V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
    [CrossRef]
  13. J. Lin, X. C. Yuan, M. Chen, and J. C. Dainty, J. Opt. Soc. Am. A 27, 2337 (2010).
    [CrossRef]
  14. M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
    [CrossRef]
  15. A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
    [CrossRef]
  16. G. Meyer and N. M. Amer, Appl. Phys. Lett. 53, 1045 (1988).
    [CrossRef]
  17. J. Ma, H. Sun, S. Wang, and D. Yan, Appl. Opt. 28, 2123 (1989).
    [CrossRef] [PubMed]
  18. S. Cui and Y. C. Soh, Appl. Phys. Lett. 96, 081102 (2010).
    [CrossRef]
  19. S. Cui and Y. C. Soh, IEEE Trans. Electron Devices 57, 2310 (2010).
    [CrossRef]
  20. S. Cui and Y. C. Soh, Phys. Rev. A 82, 062510 (2010).
    [CrossRef]
  21. N. Hermosa, A. Aiello, and J. P. Woerdman, Opt. Lett. 36, 409 (2011).
    [CrossRef] [PubMed]
  22. C. Sheppard, Opt. Express 17, 3690 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
  24. A. Aiello and J. P. Woerdman, Opt. Lett. 36, 543 (2011).
    [CrossRef] [PubMed]

2011 (2)

2010 (6)

S. Cui and Y. C. Soh, Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

S. Cui and Y. C. Soh, IEEE Trans. Electron Devices 57, 2310 (2010).
[CrossRef]

S. Cui and Y. C. Soh, Phys. Rev. A 82, 062510 (2010).
[CrossRef]

J. Lin, X. C. Yuan, M. Chen, and J. C. Dainty, J. Opt. Soc. Am. A 27, 2337 (2010).
[CrossRef]

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

2009 (2)

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

C. Sheppard, Opt. Express 17, 3690 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

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

2006 (1)

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

2005 (1)

M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
[CrossRef]

2003 (1)

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

2000 (1)

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

1997 (2)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

1996 (1)

A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
[CrossRef]

1994 (1)

M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
[CrossRef] [PubMed]

1992 (1)

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

1989 (1)

1988 (1)

G. Meyer and N. M. Amer, Appl. Phys. Lett. 53, 1045 (1988).
[CrossRef]

Aiello, A.

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
[CrossRef] [PubMed]

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

Amer, N. M.

G. Meyer and N. M. Amer, Appl. Phys. Lett. 53, 1045 (1988).
[CrossRef]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

Arlt, J.

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

Babiker, M.

M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
[CrossRef] [PubMed]

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

Barsuglia, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Beijersbergen, M. W.

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

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

Buy, C.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Carruthers, A. E.

Chen, M.

Cizmar, T.

Cojoc, D.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

Cui, S.

S. Cui and Y. C. Soh, Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

S. Cui and Y. C. Soh, IEEE Trans. Electron Devices 57, 2310 (2010).
[CrossRef]

S. Cui and Y. C. Soh, Phys. Rev. A 82, 062510 (2010).
[CrossRef]

Dainty, J. C.

Dholakia, K.

J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, Opt. Express 16, 10117 (2008).
[CrossRef] [PubMed]

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

Ferrari, E.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Fischer, P.

Gao, C.

Garbin, V.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Granata, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Grier, D.

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

Hermosa, N.

N. Hermosa, A. Aiello, and J. P. Woerdman, Opt. Lett. 36, 409 (2011).
[CrossRef] [PubMed]

M. Merano, N. Hermosa, and J. P. Woerdman, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Kostamovaara, J. T.

A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Lin, J.

Liu, Y. D.

Ma, J.

Mäkynen, A. J.

A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
[CrossRef]

Mazilu, M.

Merano, M.

M. Merano, N. Hermosa, and J. P. Woerdman, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

Meyer, G.

G. Meyer and N. M. Amer, Appl. Phys. Lett. 53, 1045 (1988).
[CrossRef]

Morris, J. E.

Myllylä, R. A.

A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
[CrossRef]

Padgett, M. J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

Pas’ko, V. A.

M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
[CrossRef]

Petrov, D.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Power, W. L.

M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
[CrossRef] [PubMed]

Qi, X.

Reece, P. J.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Sheppard, C.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Soh, Y. C.

S. Cui and Y. C. Soh, IEEE Trans. Electron Devices 57, 2310 (2010).
[CrossRef]

S. Cui and Y. C. Soh, Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

S. Cui and Y. C. Soh, Phys. Rev. A 82, 062510 (2010).
[CrossRef]

Soskin, M. S.

M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
[CrossRef]

Spreeuw, R. J. C.

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

Sun, H.

Tamburini, F.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

Terriza, G. M.

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

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

Torner, L.

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

Torres, J. P.

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

Umbriaco, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

Vasnetsov, M. V.

M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
[CrossRef]

Versluis, M.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Volpe, G.

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Wang, S.

Ward, R.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Weber, H.

Woerdma, J. P.

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

Woerdman, J. P.

Yan, D.

Yuan, X. C.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

S. Cui and Y. C. Soh, Appl. Phys. Lett. 96, 081102 (2010).
[CrossRef]

G. Meyer and N. M. Amer, Appl. Phys. Lett. 53, 1045 (1988).
[CrossRef]

IEEE Trans. Electron Devices (1)

S. Cui and Y. C. Soh, IEEE Trans. Electron Devices 57, 2310 (2010).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

A. J. Mäkynen, J. T. Kostamovaara, and R. A. Myllylä, IEEE Trans. Instrum. Meas. 45, 324 (1996).
[CrossRef]

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

Nat. Phys. (1)

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

Nature (1)

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

New J. Phys. (2)

M. V. Vasnetsov, V. A. Pas’ko, and M. S. Soskin, New J. Phys. 7, 46 (2005).
[CrossRef]

V. Garbin, G. Volpe, E. Ferrari, M. Versluis, D. Cojoc, and D. Petrov, New J. Phys. 11, 013046 (2009).
[CrossRef]

Opt. Commun. (1)

J. Arlt and K. Dholakia, Opt. Commun. 177, 297 (2000).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (4)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[CrossRef]

M. Merano, N. Hermosa, and J. P. Woerdman, Phys. Rev. A 82, 023817 (2010).
[CrossRef]

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

S. Cui and Y. C. Soh, Phys. Rev. A 82, 062510 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, Phys. Rev. Lett. 97, 163903 (2006).
[CrossRef] [PubMed]

M. Babiker, W. L. Power, and L. Allen, Phys. Rev. Lett. 73, 1239 (1994).
[CrossRef] [PubMed]

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, Phys. Rev. Lett. 78, 4713 (1997).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

LG 1 1 beam ( K = 1 and w z = 1 mm ) projected on the QD. The color map for the beam intensity is shown on the right.

Fig. 2
Fig. 2

(a)  D ( r ) of a LG 2 2 beam, a LG 0 2 beam, and a LG 2 0 beam ( K = 1 and w z = 1 mm ). (b) Their position estimates calculated by conventional formulas.

Fig. 3
Fig. 3

(a) Estimation errors when the incident beam is a LG 0 2 beam ( K = 1 and w z = 1 mm ). (b) Estimation errors when the incident beam is a LG 2 0 beam ( K = 1 and w z = 1 mm ).

Tables (3)

Tables Icon

Table 1 Measurement Sensitivities k x for Different LG Beams

Tables Icon

Table 2 Linearity Index LH x 3 for Different LG Beams

Tables Icon

Table 3 Optimized Value for a in Our Proposed Formulas [Eqs. (3, 4)]

Equations (8)

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

X = I A + I C I B I D I A + I B + I C + I D , Y = I A + I B I C I D I A + I B + I C + I D ,
D p l ( r ) = K 2 p ! ( p + | l | ) ! w z 2 exp [ 2 r 2 w z 2 ] ( 2 r 2 w z 2 ) | l | ( L p | l | ( 2 r 2 w z 2 ) ) 2 ,
X = I A + I C I B I D I A + I B + I C + I D + a ( I A + I C I B I D I A + I B + I C + I D ) 3 ,
Y = I A + I B I C I D I A + I B + I C + I D + a ( I A + I B I C I D I A + I B + I C + I D ) 3 .
d f ( x 0 ) d x 0 | x 0 = 0 = 2 0 D ( r ) d r , d 2 f ( x 0 ) d x 0 2 | x 0 = 0 = 0 ,
d 3 f ( x 0 ) d x 0 3 | x 0 = 0 = 0 [ d 2 D ( r ) d r 2 + 2 d D ( r ) r d r ] d r .
k x = 2 0 D ( r ) d r 0 D ( r ) π r d r , LH x 3 = 0 [ d 2 D ( r ) d r 2 + 2 d D ( r ) r d r ] d r 2 0 D ( r ) d r .
a = 0 [ d 2 D ( r ) d r 2 + 2 d D ( r ) r d r ] d r ( 0 D ( r ) π r d r ) 2 24 ( 0 D ( r ) d r ) 3 .

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