Abstract

Diffraction gratings that can be used to convert the Hermite–Laguerre–Gaussian (HLG) mode into the Gaussian mode were obtained; these modes are space modes of light beams. The HLG mode is inter mediate between the Hermite–Gaussian and the Laguerre–Gaussian modes. Generally, gratings produced from interfering two beams as holograms contain information of not only the phase modulation factor but also the amplitude modulation factor. Thus, they cannot be expected to provide high conversion efficiency. To produce high-efficiency gratings, the interference equation is divided into two factors. The gratings produced by considering only the phase factors act as phase converters.

© 2010 Optical Society of America

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    [CrossRef]

2006

2005

2004

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

2001

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

2000

A. T. O'Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35-45 (2000).
[CrossRef]

J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99-101 (2000).
[CrossRef]

1999

1995

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

1994

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

1993

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

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, 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, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Bandres, M. A.

Barnett, S. M.

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, 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, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Bentley, J. B.

Bezuhanov, K.

Campos, J.

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Cottrell, D. M.

Courtial, J.

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Davis, J. A.

Dennis, M. R.

Dreischuh, A.

Franke-Arnold, S.

Gibson, G.

Gilchrist, A.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Gutiérrez-Vega, J. C.

Haist, T.

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

Langford, N. K.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Mariyenko, I. G.

McNamara, D. E.

O'Brien, J. L.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

O'Neil, A. T.

A. T. O'Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35-45 (2000).
[CrossRef]

Padgett, M. J.

Pas'ko, V.

Paulus, G. G.

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Reicherter, M.

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Schätzel, M. G.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Strohaber, J.

Tiziani, H. J.

Uiterwaal, C. J. G. J.

van der Veen, H. E. L. O.

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

Vasnetsov, M.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Wagemann, E. U.

Walther, H.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, 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, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

J. Mod. Opt.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Nature

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Opt. Commun.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321-327 (1994).
[CrossRef]

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

A. T. O'Neil and J. Courtial, “Mode transformations in terms of the constituent Hermite-Gaussian or Laguerre-Gaussian modes and the variable-phase mode converter,” Opt. Commun. 181, 35-45 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8190 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Diffraction grating for the LG mode, phase distributions, and a simulated beam profile. The phase singularity is a point and is located at the center of the figure.

Fig. 2
Fig. 2

Normal diffraction grating. Because this grating represents the common phase plane, the phase relations can be visualized as a moiré pattern by superposing other diffraction gratings.

Fig. 3
Fig. 3

Diffraction grating drawn by using Eq. (5), phase distributions, and a simulated beam profile obtained by using a Gaussian mode. The phase singularity is shown as a line segment on the horizontal line.

Fig. 4
Fig. 4

Interference pattern between a HG mode and a plane wave. The interference pattern along the horizontal line at the center does not show a high contrast.

Fig. 5
Fig. 5

Diffraction grating for the HG mode, phase distributions, and a simulated beam profile obtained by using a Gaussian mode.

Fig. 6
Fig. 6

Arbitrary point on the Poincaré sphere represented in terms of the polar angle β and azimuthal angle α.

Fig. 7
Fig. 7

Diffraction grating for the HLG mode, phase distributions, and a simulated beam profile obtained by using a Gaussian mode. The grating is similar to the crushed shape of the grating for the LG mode.

Equations (13)

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LG q = E 0 r exp ( i q θ ) exp ( i k z z ) ,
u = exp ( i k x x i k z z ) ,
I = 1 + E 0 2 r 2 + 2 E 0 r cos ( k x x + q θ ) .
k x x + q θ = ( k x w i q log ( w ) ) Φ .
Φ = ( k x w i arccosh ( d w ) ) .
HG 10 = E 0 r 2 ( e i θ + e i θ ) · e i k z z .
| HG 10 + u | 2 = 2 E 0 2 r 2 cos 2 ( θ ) + 1 + E 0 r 2 ( e i θ + e i θ ) e i k x x + E 0 r 2 ( e i θ + e i θ ) * e i k x x
= 2 E 0 2 r 2 cos 2 ( θ ) + 1 + 2 2 E 0 r cos ( θ ) cos ( k x x ) .
E 0 r 2 ( e i θ + e i θ ) e i k x x + E 0 r 2 ( e i θ + e i θ ) * e i k x x = exp ( log ( E 0 r 2 ( e i θ + e i θ ) ) ) e i k x x + exp ( log ( E 0 r 2 ( e i θ + e i θ ) * ) ) e i k x x = exp ( R + i arg ( E 0 r 2 ( e i θ + e i θ ) ) ) e i k x x + exp ( R i arg ( E 0 r 2 ( e i θ + e i θ ) ) ) e i k x x = 2 e R cos ( arg ( E 0 r 2 ( e i θ + e i θ ) ) + k x x ) .
R = ( log ( E 0 r 2 ( e i θ + e i θ ) ) ) .
Φ = ( k x w i log ( E 0 r 2 ( e i θ + e i θ ) ) ) .
HLG = E 0 r ( cos ( β / 2 ) e i α / 2 LG 1 + sin ( β / 2 ) e i α / 2 LG 1 ) .
Φ = ( k x w i log ( E 0 r ( cos ( β / 2 ) e i α / 2 e i θ + sin ( β / 2 ) e i α / 2 e i θ ) ) ) .

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