Abstract

Optical vortices are always created or annihilated as vortex dipoles—pairs with opposite topological charge. Here a quantity, consisting of the transverse first and second derivatives of the optical field, is derived with which one can distinguish between vortex dipole annihilation and creation events. Analytical and numerical examples are provided as a demonstration of the method.

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
    [CrossRef]
  3. M. V. Berry, Proc. SPIE 3487, 1 (1998).
  4. M. V. Berry, J. Phys. A 11, 27 (1978).
    [CrossRef]
  5. N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).
  6. I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
    [CrossRef]
  7. I. Freund, J. Opt. Soc. Am. A 11, 1644 (1994).
    [CrossRef]
  8. N. Shvartsman and I. Freund, Phys. Rev. Lett. 72, 1008 (1994).
    [CrossRef]
  9. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef]
  10. N. Shvartsman and I. Freund, Opt. Commun. 117, 228 (1995).
    [CrossRef]
  11. I. Freund and M. Wilkinson, J. Opt. Soc. Am. A 15, 2892 (1998).
    [CrossRef]
  12. M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
    [CrossRef]
  13. K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
    [CrossRef]
  14. T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
    [CrossRef]
  15. P. Senthilkumaran and F. Wyrowski, J. Mod. Opt. 49, 1831 (2002).
    [CrossRef]
  16. M. Chen and F. S. Roux, J. Opt. Soc. Am. A 27, 2138 (2010).
    [CrossRef]
  17. I. Freund and D. Kessler, Opt. Commun. 187, 71 (2001).
    [CrossRef]
  18. F. S. Roux, J. Opt. Soc. Am. B 21, 655 (2004).
    [CrossRef]
  19. F. S. Roux, J. Opt. Soc. Am. B 21, 664 (2004).
    [CrossRef]
  20. F. S. Roux, Opt. Commun. 266, 433 (2006).
    [CrossRef]
  21. M. V. Berry, J. Opt. A 6, 675 (2004).
    [CrossRef]

2010 (1)

2006 (1)

F. S. Roux, Opt. Commun. 266, 433 (2006).
[CrossRef]

2004 (3)

2002 (1)

P. Senthilkumaran and F. Wyrowski, J. Mod. Opt. 49, 1831 (2002).
[CrossRef]

2001 (1)

I. Freund and D. Kessler, Opt. Commun. 187, 71 (2001).
[CrossRef]

2000 (1)

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

1998 (3)

I. Freund and M. Wilkinson, J. Opt. Soc. Am. A 15, 2892 (1998).
[CrossRef]

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

M. V. Berry, Proc. SPIE 3487, 1 (1998).

1995 (2)

N. Shvartsman and I. Freund, Opt. Commun. 117, 228 (1995).
[CrossRef]

K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
[CrossRef]

1994 (3)

I. Freund, J. Opt. Soc. Am. A 11, 1644 (1994).
[CrossRef]

N. Shvartsman and I. Freund, Phys. Rev. Lett. 72, 1008 (1994).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

1993 (1)

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

1989 (1)

P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

1981 (1)

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

1978 (1)

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

1974 (1)

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

Aoki, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Baranova, N. B.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

Berry, M. V.

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

M. V. Berry, Proc. SPIE 3487, 1 (1998).

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

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

Berzanskis, A.

K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
[CrossRef]

Chen, M.

Cullet, P.

P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Dennis, M. R.

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Freund, I.

I. Freund and D. Kessler, Opt. Commun. 187, 71 (2001).
[CrossRef]

I. Freund and M. Wilkinson, J. Opt. Soc. Am. A 15, 2892 (1998).
[CrossRef]

N. Shvartsman and I. Freund, Opt. Commun. 117, 228 (1995).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

I. Freund, J. Opt. Soc. Am. A 11, 1644 (1994).
[CrossRef]

N. Shvartsman and I. Freund, Phys. Rev. Lett. 72, 1008 (1994).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Gil, L.

P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Jarustis, V.

K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
[CrossRef]

Kessler, D.

I. Freund and D. Kessler, Opt. Commun. 187, 71 (2001).
[CrossRef]

Komiyama, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Mamaev, A. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

Miyamoto, Y.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Nye, J. F.

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

Ozawa, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Pilipetskii, N.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

Rocca, F.

P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Roux, F. S.

Senthilkumaran, P.

P. Senthilkumaran and F. Wyrowski, J. Mod. Opt. 49, 1831 (2002).
[CrossRef]

Shkunov, V. V.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

Shvartsman, N.

N. Shvartsman and I. Freund, Opt. Commun. 117, 228 (1995).
[CrossRef]

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

N. Shvartsman and I. Freund, Phys. Rev. Lett. 72, 1008 (1994).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

Sotomaru, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Staliunas, K.

K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
[CrossRef]

Takeda, M.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Wilkinson, M.

Wyrowski, F.

P. Senthilkumaran and F. Wyrowski, J. Mod. Opt. 49, 1831 (2002).
[CrossRef]

Zel’dovich, B. Y.

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

J. Mod. Opt. (1)

P. Senthilkumaran and F. Wyrowski, J. Mod. Opt. 49, 1831 (2002).
[CrossRef]

J. Opt. A (1)

M. V. Berry, J. Opt. A 6, 675 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

J. Phys. A (1)

M. V. Berry, J. Phys. A 11, 27 (1978).
[CrossRef]

Opt. Commun. (6)

P. Cullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

I. Freund, N. Shvartsman, and V. Freilikher, Opt. Commun. 101, 247 (1993).
[CrossRef]

N. Shvartsman and I. Freund, Opt. Commun. 117, 228 (1995).
[CrossRef]

K. Staliunas, A. Berzanskis, and V. Jarustis, Opt. Commun. 120, 23 (1995).
[CrossRef]

I. Freund and D. Kessler, Opt. Commun. 187, 71 (2001).
[CrossRef]

F. S. Roux, Opt. Commun. 266, 433 (2006).
[CrossRef]

Opt. Rev. (1)

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, and M. Takeda, Opt. Rev. 5, 374 (1998).
[CrossRef]

Phys. Rev. A (1)

I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

N. Shvartsman and I. Freund, Phys. Rev. Lett. 72, 1008 (1994).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

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

M. V. Berry and M. R. Dennis, Proc. R. Soc. London Ser. A 456, 2059 (2000).
[CrossRef]

Proc. SPIE (1)

M. V. Berry, Proc. SPIE 3487, 1 (1998).

Sov. J. Exp. Th. Phys. Lett. (1)

N. B. Baranova, B. Y. Zel’dovich, A. V. Mamaev, N. Pilipetskiĭ, and V. V. Shkunov, Sov. J. Exp. Th. Phys. Lett. 33, 195 (1981).

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Figures (2)

Fig. 1.
Fig. 1.

Comparison of the phase functions of polynomial Gaussian beams that contain vortices dipoles (a) prior to annihilation and (b) after pair creation. The diagrams beneath the phase functions show the respective beams propagating upward, with the locations of the observation planes and the trajectories of the vortex dipoles as indicated.

Fig. 2.
Fig. 2.

Sequence of four color-coded phase functions of a polynomial Gaussian beam is shown. Regions with positive (negative) vitality are shown in red (turquoise). The four images in the sequence denote consecutive slices of the phase of the beam along the propagation direction, separated by a tenth of a Rayleigh range (δz=zR/10). The phase function in (a) represents a case just prior to the creation of a vortex pair. In (b), the vortex pair has just been created in the positive (red) region. The vortices crossed the boundary in (c), moving in a negative (turquoise) region. In (d), we see the phase function just after the vortices annihilated in the negative (turquoise) region.

Equations (21)

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g(x,y,z)=P(x,y,z)(1iz)3exp(x2+y21iz),
Pan(x,y,z)=x2+y2+35(1+2i)(1iz)y+11+12i1060(1iz)(9+48i53iz)
Pcr(x,y,z)=x2+y235(42i)(1iz)yiz(1iz).
Ω=a(x,y,z)×b(x,y,z)=i2g(x,y,z)×g*(x,y,z),
T=Ω|Ω|=a(x,y,z)×b(x,y,z)|a(x,y,z)×b(x,y,z)|.
Ωz=axbyaybx=i2(gxgy*gygx*)=0,
T(t)=x(t)x^+y(t)y^+z(t)z^,
a(t)=tT(t)=tx(t)x^+ty(t)y^+tz(t)z^.
g(x,y,z)=gxx+gyy+12gxxx2+gxyxy+12gyyy2igxx+gyykz.
az=2k[(A1H0A2H5)Ωz+(H3H1H4H2)A0](4k2Ωz2+A0)2,
H0=axxaxy+ayyaxy+bxxbxy+byybxy,
H1=ayaxx+ayayy+bybyy+bybxx,
H2=bxbxx+axaxx+bxbyy+axayy,
H3=aybxxaxbxy+bxaxybyaxx,
H4=aybxyaxbyy+bxayybyaxy,
H5=axx2+bxx2ayy2byy2,
A0=H12+H22,
A1=H12H22,
A2=H1H2,
az=2k(H3H1H4H2)(H12+H22).
annihilation eventaz<0,creation eventaz>0.

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