Abstract

We report what is to our knowledge the first all-optical detection of the frequency beats between Gaussian and Laguerre–Gaussian LG01 modes in their axial superposition, caused by the rotational Doppler effect. The relation between the observable off-axis optical vortex rotation and the rotational frequency shift of the Laguerre–Gaussian component is ascertained. The results can be used as a physical basis for recognition of Laguerre–Gaussian mode spectra along their orbital angular momenta.

© 2003 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
    [CrossRef]
  2. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  3. M. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, Huntington, N.Y., 1999).
  4. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  5. L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
    [CrossRef]
  6. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef]
  7. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef]
  8. G. Nienhuis, Opt. Commun. 132, 8 (1996).
    [CrossRef]
  9. A. Ya. Bekshaev, Opt. Spectrosc. 88, 993 (2000).
    [CrossRef]
  10. I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
    [CrossRef]
  11. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
    [CrossRef]
  12. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
    [CrossRef]
  13. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
    [CrossRef]

2002 (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

2000 (1)

A. Ya. Bekshaev, Opt. Spectrosc. 88, 993 (2000).
[CrossRef]

1998 (2)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

1996 (1)

G. Nienhuis, Opt. Commun. 132, 8 (1996).
[CrossRef]

1993 (1)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

1992 (1)

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

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

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

L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[CrossRef]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

Beijersbergen, M. W.

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

Bekshaev, A. Ya.

A. Ya. Bekshaev, Opt. Spectrosc. 88, 993 (2000).
[CrossRef]

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Dholakia, K.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Franke-Arnold, S.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Leach, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Nienhuis, G.

G. Nienhuis, Opt. Commun. 132, 8 (1996).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Padgett, M. J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[CrossRef]

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Slyusar, V. V.

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

Soskin, M. S.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

Spreeuw, R. J. C.

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

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

Vasnetsov, M. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

Woerdman, J. P.

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

Horizons of World Physics (1)

M. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, Huntington, N.Y., 1999).

JETP Lett. (1)

I. V. Basistiy, A. Ya. Bekshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002) Pis’ma Zh. Eksp. Teor. Fiz. 76, 566 (2002).
[CrossRef]

Opt. Commun. (2)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 103, 442 (1993).
[CrossRef]

G. Nienhuis, Opt. Commun. 132, 8 (1996).
[CrossRef]

Opt. Spectrosc. (1)

A. Ya. Bekshaev, Opt. Spectrosc. 88, 993 (2000).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. Lett. (4)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 81, 4828 (1998).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Other (2)

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1999), Vol. 39, p. 291.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Grating with linear modulation of the fringe contrast developed for LG01 mode creation. (b) Backreflection with the 90° prism rotating about the axis of the incident beam. The beam path for angle θ=0 (shaded area) is shown; the beam path for angle θ=180° is also shown.

Fig. 2
Fig. 2

Experimental observation of the off-axis OV rotation viewed in the focal plane. Left, output beam for selected angles θ from zero to 90°. Right, prism rotation in the same steps from zero to -90°.

Fig. 3
Fig. 3

Time dependence of the detector signal; double arrow, interval equal to the period of prism rotation (twice the period of the beats). The oscillogram was affected by laser output modulation (100 Hz); a median-smoothed curve is shown. The fundamental harmonic of the beats is shown by the lighter curve.

Equations (7)

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

E=EG exp-ρ2w2+ELGρw exp-ρ2w2+iφ+iδ,
Iρ,φ,tEG2+ρ2w2ELG2+2ρwEGELG×cosφ+Δωtexp-2ρ2w2.
Tρ,φ=T0+T1ρcos-Kρ cos φ+φ,
ET0+T1ρ2exp-iKρ cos φ+iφ+c.c.×exp-ρ2w2,
E1ρRexp-ρ2w2+iφ=x+iyRexp-ρ2w2,
E1x0+x+iyRexp-ρ2w2=x0+ρ expiφRexp-ρ2w2,
uρ,φ=p,lap,lρwlLpl2ρ2w2exp-ρ2w2+ilφ,

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