Abstract

We show the dynamics of the evolution of screw phase dislocations in the wave fronts of Gaussian beams with nested multiple-charged vortices that propagate in quadratic nonlinear crystals under conditions for seeded second-harmonic generation. The number of existing vortices is shown to depend on the input light and material conditions, including the topological charge, the width and intensity of the pump and seed signals, and the propagation length inside the crystal. A closed-form model, which holds for arbitrary topological charges of the pump and seed inputs under conditions of negligible depletion of the pump beam, is developed to predict the number of vortices that exist at any instant in the propagation. It is discovered that different combinations of input charges yield a fascinating variety of vortex patterns.

© 2000 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. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
    [CrossRef] [PubMed]
  3. M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10(9), 35–38 (1997).
  4. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  5. B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14, 3045–3053 (1997).
    [CrossRef]
  6. D. Rozas, C. T. Law, and G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  7. 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]
  8. 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]
  9. T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
    [CrossRef]
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    [CrossRef]
  11. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
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    [CrossRef]
  14. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [CrossRef]
  15. D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
    [CrossRef]
  16. D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 58, 7903–7907 (1998).
    [CrossRef]
  17. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
    [CrossRef]
  18. L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
    [CrossRef]
  19. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  20. A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
    [CrossRef]
  21. G. Molina-Terriza, L. Torner, and D. V. Petrov, “Vortex streets in walking parametric wave mixing,” Opt. Lett. 24, 899–901 (1999).
    [CrossRef]
  22. P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
    [CrossRef]
  23. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
    [CrossRef]

1999 (4)

D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
[CrossRef]

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

G. Molina-Terriza, L. Torner, and D. V. Petrov, “Vortex streets in walking parametric wave mixing,” Opt. Lett. 24, 899–901 (1999).
[CrossRef]

1998 (4)

L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 58, 7903–7907 (1998).
[CrossRef]

1997 (5)

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10(9), 35–38 (1997).

B. Luther-Davies, J. Christou, V. Tikhonenko, and Y. S. Kivshar, “Optical vortex solitons: experiment versus theory,” J. Opt. Soc. Am. B 14, 3045–3053 (1997).
[CrossRef]

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

1996 (1)

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

1995 (1)

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

1993 (3)

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]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992 (1)

1987 (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

1974 (1)

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

Ackemann, T.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10(9), 35–38 (1997).

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]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Beijersbergen, M. W.

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]

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]

Berzanskis, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

Chinaglia, W.

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Christou, J.

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Freund, I.

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

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]

Ilyenkov, A. V.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Khiznyak, A. I.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Kivshar, Y. S.

Kreminskaya, L. V.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Kriege, E.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

Lange, W.

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

Law, C. T.

Luther-Davies, B.

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Matijosius, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

McDuff, R.

Minardi, S.

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, L. Torner, and D. V. Petrov, “Vortex streets in walking parametric wave mixing,” Opt. Lett. 24, 899–901 (1999).
[CrossRef]

D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
[CrossRef]

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]

Padgett, M.

M. Padgett and L. Allen, “Optical tweezers and spanners,” Phys. World 10(9), 35–38 (1997).

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Parametric down-conversion for light beams possessing orbital angular momentum,” Phys. Rev. A 59, 3950–3952 (1999).
[CrossRef]

Petrov, D. V.

D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
[CrossRef]

G. Molina-Terriza, L. Torner, and D. V. Petrov, “Vortex streets in walking parametric wave mixing,” Opt. Lett. 24, 899–901 (1999).
[CrossRef]

L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 58, 7903–7907 (1998).
[CrossRef]

Piskarskas, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

Rozas, D.

Sapone, S.

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

Smilgevicius, V.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

Smith, C. P.

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Soto-Crespo, J. M.

L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

Stabinis, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

Swartzlander, G. A.

Tikhonenko, V.

Torner, L.

D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
[CrossRef]

G. Molina-Terriza, L. Torner, and D. V. Petrov, “Vortex streets in walking parametric wave mixing,” Opt. Lett. 24, 899–901 (1999).
[CrossRef]

L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

D. V. Petrov and L. Torner, “Observation of topological charge pair nucleation in parametric wave mixing,” Phys. Rev. E 58, 7903–7907 (1998).
[CrossRef]

Torres, J. P.

L. Torner, J. P. Torres, J. M. Soto-Crespo, and D. V. Petrov, “From topological charge information to sets of solitons in quadratic nonlinear media,” Opt. Quantum Electron. 30, 809–827 (1998).
[CrossRef]

Trapani, P. D.

P. D. Trapani, A. Berzanskis, S. Minardi, S. Sapone, and W. Chinaglia, “Observation of optical vortices and J0 Bessel-like beams in quantum-noise parametric amplification,” Phys. Rev. Lett. 81, 5133–5136 (1998).
[CrossRef]

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. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

White, A. G.

Woerdman, J. P.

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]

Yamane, T.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Appl. Phys. B: Lasers Opt. (1)

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3: Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

J. Mod. Opt. (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

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

Nature (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330, 769–771 (1987).
[CrossRef] [PubMed]

Opt. Commun. (7)

I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–380 (1998).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[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]

T. Ackemann, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. 115, 339–346 (1995).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Conversion of topological charge of optical vortices in a parametric frequency converter,” Opt. Commun. 140, 273–276 (1997).
[CrossRef]

D. V. Petrov, G. Molina-Terriza, and L. Torner, “Vortex evolution in parametric wave mixing,” Opt. Commun. 162, 357–366 (1999).
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[CrossRef]

Phys. Rev. A (2)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
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[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Radial profiles of two coaxial singular beams with different amplitudes, widths, and charges of the existing on-axis vortices. The arrows mark the radial locations where vortices appear when the two beams are superposed. This plot was made with beams with the shape given by Eq. (5), with amplitudes A0, widths w1, and charges m1 given by 1.8, 1, and 3 and 0.9, 2, and -1, respectively.

Fig. 2
Fig. 2

Combined beam obtained by the superposition of the two beams shown in Fig. 1. (a) Amplitude and (b) interference pattern obtained by superposition of the total beam with a reference plane wave tilted slightly relative to the propagation axis. Dashed circles, the radial positions where vortices appear.

Fig. 3
Fig. 3

Number of single-charge vortices present in the wave front of the SH beam predicted by the model of negligible depletion of the pump fundamental frequency beam. Combination of input topological charges of the input pump and seed beam: [m1, m2]=[3, -1]. Width aspect ratio between pump and seed beams: (a) κ=1, (b) κ=0.5. Solid curves, exact phase matching (βv=0); dotted curves, βv=10. Open circles in (a) indicate the locations of the simulations shown in Fig. 8 below. m and n in the labels [m; n] stand for the number of single-charge vortices present and for their total topological charges, respectively. See text for details.

Fig. 4
Fig. 4

Analogous to Fig. 3 but for the combination of input topological charges [m1, m2]=[2,2].

Fig. 5
Fig. 5

Analogous to Fig. 3 but for the combination of input topological charges [m1, m2]=[1,3].

Fig. 6
Fig. 6

Analogous to Fig. 3 but for the combination of input topological charges [m1, m2]=[2,5].

Fig. 7
Fig. 7

Analogous to Fig. 3 but for the combination of input topological charges [m1, m2]=[2, -4]. Open circles in (a) indicate the locations of the simulations shown in Fig. 9 below.

Fig. 8
Fig. 8

Numerically obtained interferograms for the SH beam at various propagation distances inside the nonlinear crystal. Open circles, positive vortices such as those of the pump FF beam; filled circles, negative vortices. Input conditions: FF pump peak amplitude A0=0.1; SH seed peak amplitude A0s=0.01; width of both beams, w1=w2=2, hence κ=1. Input charges: m1=3, m2=-1; exact phase matching (β=0). Propagation distances: (a) ξ=0.5, (b) ξ=1.5, (c) ξ=2, (d) ξ=3.

Fig. 9
Fig. 9

Analogous to Fig. 8 but with the following differences. Input conditions: FF pump peak amplitude A0=0.1; SH seed peak amplitude A0s=0.002; width of both beams w1=w2=2, hence κ=1. Input charges: m1=2, m2=-4; exact phase-matching (βv=0). Propagation distances: (a) ξ=0.1, (b) ξ=0.4, (c) ξ=2, (d) ξ=4.

Equations (23)

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ia1ξ-α122a1+a1*a2 exp(-iβξ)=0,
ia2ξ-α222a2+a12 exp(iβξ)=0,
a2(x, y, ξ)=- dx - dya2(x, y, ξ=0)×K(x-x, y-y, ξ)+i- dx - dy 0ξ dξa12(x, y, ξ)×exp(iβξ)K(x-x, y-y, ξ-ξ).
K(u, v, w)=i2α2πwexpu2+v2i2α2w.
a1(x, y, ξ=0)=A0rw1|m1| exp-r2w12exp(im1ϕ),
a2(x, y, ξ=0)=A0srw2|m2| exp-r2w22exp(im2ϕ),
a1(x, y, ξ)=A0rw1|m1|1(1+iξv)|m1|+1×exp-r2w12(1+iξv)exp(im1ϕ),
a2(x, y, ξ)=A0srw2|m2|1(1+iξs)|m2|+1×exp-r2w22(1+iξs)exp(im2ϕ)+iA02rw12|m1|f(ξv; βv)(1+iξv)2|m1|+1×exp-2r2w12(1+iξv)exp(i2m1ϕ),
f(ξv; βv)=0ξvexp(iβvζ)1+iζdζ,
f(ξv; βv)=-i exp(-βv)ln(1+iξv)+k=1 βvk(1+iξv)k-1kk!.
F(r, ϕ)=A(r)exp(imAϕ)+B(r)exp(imBϕ),
mr,0=12πr,0 Φ·dl.
mr,0=mA|A(r,0)|>|B(r,0)|mB|A(r,0)|<|B(r,0)|.
|A2p(r, ξ)|=|A2s(r, ξ)|,
Φ2p(r, ξ)=Φ2s(r, ξ)±π,
ρ2|m1|=ρ|m2|A0sz0vA02C(ξv; κ, m1, m2)|f(ξv; βv)|exp[α(ξv; κ)ρ2],
ρ=rw1,κ=w2w1
α(ξv; κ)=21+ξv2-1κ2+ξv2/4κ2,
C(ξv; κ, m1, m2)=1κ|m2|(1+ξv2)(2|m1|+1)/2(1+ξv2/4κ4)(|m2|+1)/2.
A0sA02z0v>Δmα(ξv; κ)Δm|f(ξv; βv)|C(ξv; κ, m1, m2)exp(Δm),
A0sA02z0v<Δmα(ξv; κ)Δm|f(ξv; βv)|C(ξv; κ, m1, m2)exp(Δm).
A0sA02z0v=Δmα(ξv; κ)Δm|f(ξv; βv)|C(ξv; κ, m1, m2)exp(Δm).
A0sA02z0v>|f(ξv; βv)|C(ξv; κ, m1, m2).

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