Abstract

An optical-vortex filament is characterized by a dark core of vanishing size and fluidlike propagation dynamics in the near-field region. This type of phase singularity does not naturally occur as an eigenmode of a cylindrically symmetric system, but it can be easily formed by computer-generated holography. The size of the core is an important attribute affecting vortex–vortex interactions within a laser beam. Here we demonstrate a means to minimize the core size, and we experimentally show that a beam-to-core size ratio exceeding 175 may be readily achieved.

© 1998 Optical Society of America

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [Crossref]
  2. M. V. Berry, Singularities in Waves and Rays, Physics of Defects, Les Houches Sessions XXXV, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North Holland, Amsterdam, 1981), pp. 453–543.
  3. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].
  4. M. S. El Naschie, ed., “Special issue on nonlinear optical structures, patterns, chaos,” Chaos Solitons Fractals4(8/9) (1994).
  5. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [Crossref]
  6. M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
    [Crossref] [PubMed]
  7. G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [Crossref] [PubMed]
  8. G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
    [Crossref]
  9. A. W. Snyder, L. Poladian, and D. J. Mitchell, “Stable black self-guided beams of circular symmetry in a bulk Kerr medium,” Opt. Lett. 17, 789–791 (1992).
    [Crossref] [PubMed]
  10. F. S. Roux, “Dynamical behavior of optical vortices,” J. Opt. Soc. Am. B 12, 1215–1221 (1995).
    [Crossref]
  11. G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.
  12. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [Crossref]
  13. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
    [Crossref]
  14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref] [PubMed]
  15. V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].
  16. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [Crossref] [PubMed]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 67.
  18. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), p. 860.
  19. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 320.
  20. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [Crossref] [PubMed]
  21. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [Crossref]
  22. W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
    [Crossref]

1997 (2)

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

1992 (5)

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[Crossref] [PubMed]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

A. W. Snyder, L. Poladian, and D. J. Mitchell, “Stable black self-guided beams of circular symmetry in a bulk Kerr medium,” Opt. Lett. 17, 789–791 (1992).
[Crossref] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

1991 (1)

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
[Crossref]

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

1989 (1)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

1981 (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Baranova, N. B.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

M. V. Berry, Singularities in Waves and Rays, Physics of Defects, Les Houches Sessions XXXV, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North Holland, Amsterdam, 1981), pp. 453–543.

Brambilla, M.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Cattaneo, M.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Coates, A. B.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

D’Angelo, E. J.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Firth, W. J.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

Gahagan, K. T.

Gambogi, W. J.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
[Crossref]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 320.

Gerstadt, W. A.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
[Crossref]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 67.

Green, C.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Heckenberg, N. R.

Kent, A. J.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), p. 860.

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), p. 860.

Law, C. T.

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[Crossref] [PubMed]

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

Lugiato, L. A.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Mackara, S. R.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
[Crossref]

Mamayev, A. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

McDonald, G. S.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

McDuff, R.

Mitchell, D. J.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[Crossref]

Oppo, G.-L.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Pilipetskii, N. F.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

Pirovano, R.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Poladian, L.

Pratti, F.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

Roux, F. S.

Rozas, D.

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

Sacks, Z. S.

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

Shkukov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

Smith, C. P.

Snyder, A. W.

Soskin, M. S.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Swartzlander, G. A.

D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[Crossref] [PubMed]

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[Crossref] [PubMed]

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

Syed, K. S.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

Tredicce, J. R.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Vasnetsov, M. V.

V. Yu. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” Pis’ma Zh. Eksp. Teor. Fiz. 52, 1037–1039 (1990) [ JETP Lett. 52, 429–431 (1990)].

Webber, A. M.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, and A. M. Webber, “Holographic transmission elements using improved photopolymer films,” Proc. SPIE 1555, 256–266 (1991).
[Crossref]

Weiss, C. O.

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

White, A. G.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Zel’dovich, B. Ya.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

Zhang, X.

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[Crossref]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Optical vortices in beam propagation through a self-defocusing medium,” Opt. Commun. 94, 469–476 (1992).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

M. Brambilla, M. Cattaneo, L. A. Lugiato, R. Pirovano, F. Pratti, A. J. Kent, G.-L. Oppo, A. B. Coates, C. O. Weiss, C. Green, E. J. D’Angelo, and J. R. Tredicce, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

G. A. Swartzlander and C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
[Crossref] [PubMed]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

Pis’ma Zh. Eks. Teor. Fiz. (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamayev, N. F. Pilipetskii, and V. V. Shkukov, “Dislocations of the wavefront of a speckle-inhomogeneous field (theory and experiment),” Pis’ma Zh. Eks. Teor. Fiz. 33, 206–210 (1981) [JETP Lett. 33, 195–199 (1981)].

Pis’ma Zh. Eksp. Teor. Fiz. (1)

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[Crossref]

Proc. SPIE (1)

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M. V. Berry, Singularities in Waves and Rays, Physics of Defects, Les Houches Sessions XXXV, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North Holland, Amsterdam, 1981), pp. 453–543.

M. S. El Naschie, ed., “Special issue on nonlinear optical structures, patterns, chaos,” Chaos Solitons Fractals4(8/9) (1994).

G. A. Swartzlander, Z. S. Sacks, X. Zhang, D. Rozas, and C. T. Law, “Formation and propagation of optical vortices,” in Digest of the International Quantum Electronics Conference, 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), p. 31.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 67.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1961), p. 860.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 320.

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Figures (8)

Fig. 1
Fig. 1

Transverse profiles of a beam containing a vortex of topological charge m=1. (a) Intensity profile showing a dark core of diameter, 2wv, on a Gaussian background field of radial size w0. (b) Phase profile where white and black correspond to a vortex phase of zero and -2π, respectively.

Fig. 2
Fig. 2

Radial intensity distribution (arbitrary units) of a propagating beam initially containing an ideal point vortex on a Gaussian background field of size w0, showing rapid oscillations in the near-field regime and a smooth beam for z/z0>1, where z0 is the diffraction length.

Fig. 3
Fig. 3

Numerical integration of Eq. (13), showing the radial intensity distributions in the focal plane for an initial point vortex (m=1) on Gaussian and pillbox beams, with characteristic sizes w0 and R=w0, respectively. For comparison the nonvortex beams (m=0) are also shown. The radial coordinate is scaled by the diffraction-limited spot size, w0=λf/πw0, and the amplitude is scaled by the optical gain factor, w0/w0.

Fig. 4
Fig. 4

(a) Interferogram of a single point vortex of charge m=1. The vortex core is located at the fork of the equiphase lines. Far from the core, the lines are separated by the grating period, Λ. (b) One-bit gray-scale rendering of (a), showing grating lines of width Λ/2. The interference fringes are composed of line segments, resulting in η=ΛNp (η=4 in this example) distinct phase domains whose boundaries radiate from the core, where Np is the resolution of the laser printer (typically measured in dots per inch).

Fig. 5
Fig. 5

(a) Finite phase resolution in the CGH produces η arms radiating from the vortex core in the focal plane of a lens. An aperture of diameter da may be used to spatially filter the holographic image. (b) Large apertures produce smaller cores and more distorted background fields compared with small apertures (c). The effect of aperture size on the beam-to-core size ratio, βHWHM, is plotted in (d).

Fig. 6
Fig. 6

Schematic of the optical system for converting the CGH into a thick phase hologram. A beam from a Spectra-Physics frequency-stabilized argon-ion laser (Ar) is directed through a glass wedge beam splitter (BS). The object beam is transmitted through the computer-generated hologram (CGH), and the first-order diffracted beam is allowed through a pinhole (P) in a spatial filter assembly. The object and reference beams are balanced with the aid of an attenuator (Atn) and made to interfere at equal angles, θd, with respect to the normal of the holographic recording film attached to a glass window (HRF). A helium–neon laser (HeNe) and an optical power meter (OPM) are used to monitor the efficiency of the hologram in real time.

Fig. 7
Fig. 7

Schematic of the optical system to examine the vortex recorded onto HRF. The vortex beam is imaged to a CCD array (C) by means of a lens (L) with focal length f=50.8 mm and imaging distance di=137 mm. Both C and L are mounted on a translation stage (TS), which can move along the optical axis, allowing us to monitor the transverse intensity profile at different propagation distances, z. The camera C is connected to a Macintosh IIfx computer with a frame grabber (not shown).

Fig. 8
Fig. 8

HWHM size of a vortex plotted versus the propagation distance, z. The discrete points are experimental results. The curve was obtained from numerical simulations, assuming a large Gaussian beam (w0=12.8 mmwV) with a vortex of the amplitude profile A(r)=tanh(r/wV), with initial wV=165 μm (i.e., wHWHM=145 μm).

Equations (31)

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u(ρ, ϕ, z)=Am(ρ, z)exp(-imϕ)exp[iΦm(ρ, z)],
u(x, y, z)=(iλz)-1-- dx0dy0u0(x0, y0)×exp{i(k/2z)[(x-x0)2+(y-y0)2]},
u(ρ, ϕ, z)=(iλz)-1 exp[i(k/2z)ρ2]×002π rdrdθh(r)u(r, θ)×exp[-i(k/z)rρ cos(θ-ϕ)],
Jm(a)=(2π)-102π dθ exp[-im(θ-π/2)]×exp(-ia cos θ)
u(ρ, ϕ, z)=(2π/iλz)exp[-im(ϕ+π/2)]exp[ikρ2/2z]×0 rdrh(r)Am(r)Jm(ρkr/z).
Hm{f(r)}=0 rdrf(r)Jm(ξr).
Am(r, z=0)=(r/L)m exp(-r2/w02),
Hm{f(r)}=0 drrm+1 exp(-pr2)Jm(ξr)=[ξm/(2p)m+1]exp(-ξ2/4p),
Am(ρ, z)=[w0/w(z)][ρw0/Lw(z)]m exp[-ρ2/w2(z)],
Φm(ρ, z)=-(m+1)tan-1(z/z0)+kρ2/2R(z),
|u(ρ, ϕ, z)/u0(z)|2=(2e/m)m[ρ/w(z)]2m ×exp[-2ρ2/w2(z)].
Hm{f(r)}=0r exp(-pr2)Jm(ξr)dr=(π1/2ξ/8p3/2)×exp(-γ)][I(m-1)/2(γ)-I(m+1)/2(γ)],
γ=ξ2/8p=(1/2)[ρ/w(z)]2(1+iz0/z),
ξ/8p3/2=(w0z0/4)(ρ/z)(1+(z0/z)2)-3/4 ×exp{(3i/2)[π/2-tan-1(z/z0)]}.
Am(ρ, z)=π2 ρw0 z0z5/4z0R(z)3/4 exp-12 ρ2w2(z)×[I(m-1)/2(γ)-I(m+1)/2(γ)],
Φm(ρ, z)=-m π2+π4+ρw02 z0z 1-w022w2(z)-32 tan-1zz0,
uf(ρf, ϕf)=(iλf )-1002π u0(r0, θ0)r0 ×exp[-i(kr0ρf/f )cos(θ0-ϕf)]dr0dθ0=(k/f )exp[-i(m+1)π/2]×exp(imϕf)0 Am(r0)Jm(kr0ρf/f )r0dr0exp[-i(m+1)π/2]exp(imϕf)Am(ρf),
Am(ρf)=π1/22 z0f2 ρfw0 exp(-γf)[I(m-1)/2(γf)-I(m+1)/2(γf)],
A1(ρf)=-(πR/2ρf)[J1(ρf/ρ0)H0(ρf/ρ0)-J0(ρf/ρ0)H1(ρf/ρ0)],
Eobj=C1 exp(-iθ),
Eref=C0 exp(i2πx/Λ),
Iz=0(x, θ)=|Eobj+Eref|z=02=2C02[1+cos(2πx/Λ+θ)].
f(x, θ)=m=-Cm exp(imθ)exp(i2πmx/Λ)
ψm=arcsin(mλ/Λ)arcsin(mλNp/η),
η=ΛNp
β=w0/wV2w0Np/η1,
da,min=cbλf/w0,
da,max=f tan ψ1fλ/Λ.
Q=4πdθd/Λ2πλd/n(Λ)2.
ηB=sin2(πdΔn/λ cos θd),
(xc(i), yc(i))=S(x, y)I(x, y, z(i))dxdy/SI(x, y, z(i))dxdy,

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