Abstract

The dynamical behavior of optical vortices during propagation of an electromagnetic wave is investigated. The propagation of these waves is computed with rigorous scalar diffraction theory. It was observed that a single optical vortex propagates perpendicularly to the wave front. Two optical vortices with the same charges (isopolar vortex pairs) are observed to gyrate around each other during propagation. Two optical vortices with opposite charges (bipolar vortex pairs) are observed to drift laterally away from the general direction of propagation. In cases in which interference between waves is allowed to occur or in which wave fronts are converging to focal points, bipolar vortex pairs are sporadically created and annihilated. An elementary model for the propagation of vortex pairs is proposed to explain the gyration and drift phenomena.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
    [CrossRef] [PubMed]
  3. F. S. Roux, "Diffractive optical implementation of rotation transform performed by using phase singularities," Appl. Opt. 32, 3715–3719 (1993).
    [CrossRef] [PubMed]
  4. F. S. Roux, "Branch-point diffractive optics," J. Opt. Soc. Am. A 11, 2236–2243 (1994).
    [CrossRef]
  5. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computergenerated holograms," Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  6. G. A. Swartzlander, Jr., and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503–2506 (1992).
    [CrossRef] [PubMed]
  7. E. Kreyszig, Advanced Engineering Mathematics, 5th ed. (Wiley, New York, 1983), Chap. 13, p. 621.
  8. F. Wyrowski and O. Bryngdahl, "Iterative Fourier-transform algorithm applied to computer holography," J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, p. 488.
  10. A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973);also in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Vol. MS51 of SPIE Milestone Series (Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1992), pp. 422–439.
    [CrossRef]

1994 (1)

1993 (1)

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, "Generation of optical phase singularities by computergenerated holograms," Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

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

1991 (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

1988 (1)

1974 (1)

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

1973 (1)

A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973);also in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Vol. MS51 of SPIE Milestone Series (Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1992), pp. 422–439.
[CrossRef]

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[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]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, p. 488.

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Bryngdahl, O.

Devaney, A. J.

A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973);also in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Vol. MS51 of SPIE Milestone Series (Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1992), pp. 422–439.
[CrossRef]

Heckenberg, N. R.

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics, 5th ed. (Wiley, New York, 1983), Chap. 13, p. 621.

Law, C. T.

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

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

McDuff, R.

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]

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Roux, F. S.

Sherman, G. C.

A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973);also in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Vol. MS51 of SPIE Milestone Series (Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1992), pp. 422–439.
[CrossRef]

Smith, C. P.

Swartzlander, G. A.

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

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

White, A. G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, p. 488.

Wyrowski, F.

Appl. Opt. (1)

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

Opt. Lett. (1)

Phys. Rev. A (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, and C. O. Weiss, "Transverse laser patterns. I. Phase singularity crystals," Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

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

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

SIAM Rev. (1)

A. J. Devaney and G. C. Sherman, "Plane-wave representations for scalar wave fields," SIAM Rev. 15, 765–786 (1973);also in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Vol. MS51 of SPIE Milestone Series (Society of Photo-Optical Instrument Engineers, Bellingham, Wash., 1992), pp. 422–439.
[CrossRef]

Other (2)

E. Kreyszig, Advanced Engineering Mathematics, 5th ed. (Wiley, New York, 1983), Chap. 13, p. 621.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8, p. 488.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Diagrammatical representation of the propagation of a finite input function with a limited spatial bandwidth.

Fig. 2
Fig. 2

Central portion of the phase of a complex-valued function with one optical vortex.

Fig. 3
Fig. 3

Predicted and observed results obtained from the propagation of a complex-valued function with one optical vortex, showing the locations of the optical vortex.

Fig. 4
Fig. 4

Central portion of the phase of a complex-valued function with two similarly charged optical vortices.

Fig. 5
Fig. 5

Predicted and observed results obtained from the propagation of a complex-valued function with two similarly charged optical vortices, showing the orientation angles of the separation vector, the separation distance, and the drift locations of the optical vortices.

Fig. 6
Fig. 6

Central portion of the phase of a complex-valued function with two optical vortices with opposite charges.

Fig. 7
Fig. 7

Predicted and observed results obtained from the propagation of a complex-valued function with two optical vortices with opposite charges, showing the orientation angles of the separation vector, the separation distance, and the drift locations of the optical vortices.

Equations (25)

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

F ( a , b ) = - - f ( x , y ) exp [ j 2 π ( a x + b y ) ] d x d y ,
g ( x , y , z 0 ) = - - F ( a , b ) × exp [ - j 2 π ( a x + b y + c z 0 ) ] d a d b ,
c = ( 1 λ 2 - a 2 - b 2 ) 1 / 2 .
F ( a , b ) = F ( a , b ) exp [ - j 2 π z 0 ( 1 λ 2 - a 2 - b 2 ) 1 / 2 ] ,
c Ψ ( x , y ) · d s ^ = n 2 π ,
f ( x , y , z ) = A ( x , y ) exp [ j ( q arg ( x , y ) - k z ) ] ,
P = - [ q arg ( x , y ) - k z ] = k e ^ z - q ( - y e ^ x + x e ^ y ρ 2 ) ,
δ r = δ r k P ¯ = δ z e ^ z + δ z q ( - y e ^ x + x e ^ y k ρ 2 ) .
δ r = δ z e ^ z + δ z q ( ρ × e ^ z k ρ 2 ) .
d r ( x , y ) d z = e ^ z + q ( ρ × e ^ z k ρ 2 ) .
ρ = r - s
d s ( x , y ) d z = e ^ z - q ( ρ × e ^ z k ρ 2 ) .
d ρ d z = d r d z - d s d z = 2 ( ρ × e ^ z k ρ 2 ) .
k ρ 2 ( k · ) ρ = 2 ( ρ × k ) ,
ρ = ρ 0 [ sin ( 2 z k ρ 2 ) e ^ x + cos ( 2 z k ρ 2 ) e ^ y ] ,
Period = π k ρ 2 .
m = 0.5 ( r + s ) .
d m d z = 0.5 ( d r d z + d s d z ) = e ^ z .
( k · ) m = k .
d ρ d z = d r d z - d s d z = 0 ,
( k · ) ρ = 0.
d m d z = 0.5 ( d r d z + d s d z ) = e ^ z + ( ρ × e ^ z k ρ 2 ) ,
( k · ) m = k + ( ρ × k k ρ 2 ) .
m = z e ^ z + z k ρ e ^ x .
Drift rate = d m d z - e ^ z = 1 k ρ e ^ x .

Metrics