Abstract

We show how the effects of azimuthal optical aberrations on singular light beams can result in an intensity modulation in the beam waist or focal point spread function (PSF) that is directly proportional to the amplitude of the applied phase aberration. The resulting distortions are enough to significantly degrade the utility of the singular beams even in well corrected optical systems. However we show that pattern of these intensity modulations is related to the azimuthal order of the applied aberration and we suggest how this can be used to measure those aberrations. We demonstrate a closed loop system using a liquid crystal spatial light modulator as a programmable diffractive optical element to both generate the beam and correct for the sensed aberrations based on feed back from a CCD detected intensity image of the focal point spread function.

© 2006 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  2. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  3. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pasko, S. M. Barnett, and S. Franke-Arnold, "Free-space information transfer using light beams carrying orbital angular momentum," Opt. Express 12, 5448-5456 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-22-5448
    [CrossRef] [PubMed]
  4. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef] [PubMed]
  5. J. E. Curtis and D. G. Grier "Structure of Optical Vortices," Phys. Rev. Lett. 90, 133901 (2004).
    [CrossRef]
  6. F. K. Fatemi and M. Bashkansky, "Cold atom guidance using a binary spatial light modulator," Opt. Express 14, 1368-1375 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1368
    [CrossRef] [PubMed]
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    [CrossRef]
  8. K.I. Willig, S.O. Rizzoli, V. Westphal, R. Jahn, and S.W. Hell, "STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis," Nature (London) 440, 935-939 (2006).
    [CrossRef]
  9. C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 88, 153901 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. M. A. A. Neil, M. J. Booth, and T. Wilson, "New modal wave-front sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000).
    [CrossRef]
  15. M. A. A. Neil, M. J. Booth, and T. Wilson, "Dynamic wave-front generation for the characterization and testing of optical systems," Opt. Lett. 23, 1849- 1851(1998).
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    [CrossRef] [PubMed]

2006 (1)

2005 (1)

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 88, 153901 (2005).
[CrossRef]

2004 (2)

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

2001 (1)

T. A. Klar, E. Engel, and S. Hell, "Breaking Abbes diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes," Phys. Rev. E 64, 066613 (2001).
[CrossRef]

2000 (2)

M. A. A. Neil, T. Wilson and R. Juskaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

M. A. A. Neil, M. J. Booth, and T. Wilson, "New modal wave-front sensor: a theoretical analysis," J. Opt. Soc. Am. A 17, 1098-1107 (2000).
[CrossRef]

1998 (1)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

1990 (1)

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Barnett, S. M.

Bashkansky, M.

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Booth, M. J.

Courtial, J.

Curtis, J. E.

J. E. Curtis and D. G. Grier "Structure of Optical Vortices," Phys. Rev. Lett. 90, 133901 (2004).
[CrossRef]

Engel, E.

T. A. Klar, E. Engel, and S. Hell, "Breaking Abbes diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes," Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Fatemi, F. K.

Franke-Arnold, S.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Gibson, G.

Grier, D. G.

J. E. Curtis and D. G. Grier "Structure of Optical Vortices," Phys. Rev. Lett. 90, 133901 (2004).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Hell, S.

T. A. Klar, E. Engel, and S. Hell, "Breaking Abbes diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes," Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Ju, R.

M. A. A. Neil, T. Wilson and R. Juskaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

Klar, T. A.

T. A. Klar, E. Engel, and S. Hell, "Breaking Abbes diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes," Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Mahajan, V. N.

Neil, M. A. A.

Padgett, M. J.

Pasko, V.

Paterson, C.

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 88, 153901 (2005).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Tamm, C.

Vanderveen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

Vasnetsov, M.

Weiss, C. O.

Wilson, T.

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Microsc. (1)

M. A. A. Neil, T. Wilson and R. Juskaitis, "A wavefront generator for complex pupil function synthesis and point spread function engineering," J. Microsc. 197, 219-223 (2000).
[CrossRef] [PubMed]

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

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

Opt. Commun. (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular-momentum, " Opt. Commun. 96, 123-132 (1993)
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, and R. J. C. Spreeuw and J. P. Woerdman, "Orbital angular momentum of light and transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. E (1)

T. A. Klar, E. Engel, and S. Hell, "Breaking Abbes diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes," Phys. Rev. E 64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett. (4)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, "Measuring the orbital angular momentum of a single photon," Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. E. Curtis and D. G. Grier "Structure of Optical Vortices," Phys. Rev. Lett. 90, 133901 (2004).
[CrossRef]

C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 88, 153901 (2005).
[CrossRef]

Other (2)

K.I. Willig, S.O. Rizzoli, V. Westphal, R. Jahn, and S.W. Hell, "STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis," Nature (London) 440, 935-939 (2006).
[CrossRef]

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed. Cambridge, England: Cambridge University Press, 1999).
[PubMed]

Supplementary Material (2)

» Media 1: GIF (83 KB)     
» Media 2: GIF (290 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

Intensity patterns in the Fourier plane for a normal beam (l = 0) and singular beams (l = 1,2,3) aberrated with -0.447Zi (left column) and +0.447Zi (right column) of Z i=5,6,7,8,9,10,11,14,and 15 (from top to bottom).

Fig. 2.
Fig. 2.

Gray scale images representing the magnitude of the sensitivity matrix, |Sij | for (a) as singular 1 = 1 beam and (b) a non-singular 1 = 0 beam. The displayed gray scale is normalised such that the maximum value, S 55 with l = 1, is set to white. (c) The segmented sensor detector pattern.

Fig. 3.
Fig. 3.

(a) Sensitivity, Δ i , as a function of normalised sensor radius, ρ/ρo . (b) Simulated and (c) experimental sensor signal, Δ i , vs aberration, ai , for Z 5,6,7,8,9,10,14,15.

Fig. 4.
Fig. 4.

Experimental set up for closed-loop aberration correction.

Fig. 5.
Fig. 5.

(a)Experimental (top) and simulated (bottom) intensity PSFs for the l = 1 singular beam aberrated with 0.447X Z i=5,6,7,8,9,10,14,15 (left to right). (b) (92 KB) l = 1 PSFs (left) and corresponding normal l = 0 PSFs (right) as a sequence (t=0→4) during the correction procedure for a well corrected beam (t=0), aberrated by a glass slide. (c) (320 KB) Sequence (t=0→8) of l = 1 (top left), l = 0 (top right) PSFs and the corresponding holograms (bottom) during the correction of a heavily aberrated beam. (d) Peak intensity of the l = 0 PSFs as a Strehl ratio vs iteration, for the images shown in (b) and (c).

Tables (1)

Tables Icon

Table 1. Geometry and performance parameters for the proposed wave-front sensor, calculated by numerical simulation

Equations (14)

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U ( ρ , ξ ) = [ P ( r , θ ) ] = [ e jlθ e j a i Z i ( r , θ ) ]
[ f ( r , θ ) ] = 0 2 π 0 1 f ( r , θ ) e jρr cos ( θ ξ ) rdrdθ
I ( ρ , ξ ) = U ( ρ , ξ ) U * ( ρ , ξ )
s i = d I da i | a i = 0 = 2 [ U ( ρ , ξ ) d U * ( ρ , ξ ) d a i ] | a i = 0
s i = 2 [ [ e jlθ ] ( [ j Z i ( r , θ ) e jlθ ] ) * ]
[ R n ( r ) e jmθ ] = 2 π j m e jmξ 0 1 R n ( r ) J m ( ρr ) rdr
s i = ( 1 ) m 2 F l [ ( 1 ) l G m l G m + l ] sin ( ) m even
s i = ( 1 ) m + 1 2 F l [ ( 1 ) l G m l + G m + l ] cos ( ) m odd
s i = ( 1 ) m 2 F l [ ( 1 ) l G m l + G m + l ] cos ( ) m even
s i = ( 1 ) m + 1 2 F l [ ( 1 ) l G m l + G m + l ] sin ( ) m odd
F k = 2 π 0 1 J k ( ρr ) rdr
G k = π 0 1 R n ( r ) J k ( ρr ) rdr
Δ i = δ n + δ n δ n
S i j = d Δ i d a j | a j = 0

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