Abstract

Higher-order extrema with topological indices greater than unity are discussed. Explicit constructions are given for their wave functions, and simple geometric rules are presented for analysis of their topological indices. Experimental means for verifying the theory with use of Gaussian laser beams are considered, unusual properties of optical vortices contructed from this new type of critical point are described, and applications to topologically based optical arithmetic are suggested.

© 2000 Optical Society of America

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  1. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. P. Szwaykowski, J. Ojeda-Casteneda, “Nondiffracting beams and self imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
    [CrossRef]
  4. R. Piestun, J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [CrossRef]
  5. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
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  6. W. D. Montgomery, “Algebraic formulation of diffraction applied to self imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968).
    [CrossRef]
  7. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  8. A. W. Lohman, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  9. M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
    [CrossRef]
  10. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  11. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.
  12. V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).
  13. G. A. Swartzlander, C. T. Law, “Optical vortex solitons observed in Kerr nonlinear media,” Phys. Rev. Lett. 69, 2503–2506 (1992).
    [CrossRef] [PubMed]
  14. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  15. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  16. I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
    [CrossRef]
  17. M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
    [CrossRef] [PubMed]
  18. E. Abromochkin, V. Volostnikov, “Spiral-type beams; optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
    [CrossRef]
  19. D. Rozas, Z. S. Sacks, G. A. Swartzlander, “Experimental observation of fluid-like motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
    [CrossRef]
  20. M. V. Berry, C. Upstill, “Catastrophe optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.
  21. M. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan, J. L. Safko, eds. (World Scientific, London, 1994), pp. 55–65.
  22. G. P. Karman, M. W. Beijersbergen, A. van Duiji, D. Bouwmeester, J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A 15, 884–899 (1998).
    [CrossRef]
  23. M. V. Berry, “Wave dislocation reactions in non-paraxial beams,” J. Mod. Opt. 45, 1845–1858 (1998).
    [CrossRef]
  24. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1988), Chap. 4, pp. 73–83.
  25. I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
    [CrossRef]
  26. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6, pp. 174–180.
  27. V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973), Chap. 5, pp. 254–268.
  28. B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry—Methods and Applications (Springer-Verlag, Berlin, 1984,1986,1990), Part II. The geometry and topology of manifolds (1986); Part III. Introduction to homology theory (1990).
  29. J. Milnor, Morse Theory (Princeton U. Press, Princeton, N.J., 1963).
  30. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  31. I. Freund, “Critical point explosions in two-dimensional wave fields,” Opt. Commun. 159, 99–117 (1999).
    [CrossRef]
  32. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 369, No. 3.616.8.
  33. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. Applied Mathematics Series No. 55 (U.S. GPO, Washington, D.C., 1964).
  34. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [CrossRef]
  35. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  36. G. H. Kim, J. H. Jeon, K. H. Ko, H. J. Moon, J. H. Lee, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. 36, 8614–8621 (1997).
    [CrossRef]
  37. B. Y. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 3.
  38. A. M. Deykoon, M. S. Soskin, G. A. Swartzlander, “Nonlinear optical catastrophe from a smooth initial beam,” Opt. Lett. 24, 1224–1226 (1999).
    [CrossRef]

1999 (4)

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

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

A. M. Deykoon, M. S. Soskin, G. A. Swartzlander, “Nonlinear optical catastrophe from a smooth initial beam,” Opt. Lett. 24, 1224–1226 (1999).
[CrossRef]

1998 (3)

1997 (3)

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

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

G. H. Kim, J. H. Jeon, K. H. Ko, H. J. Moon, J. H. Lee, “Optical vortices produced with a nonspiral phase plate,” Appl. Opt. 36, 8614–8621 (1997).
[CrossRef]

1996 (1)

E. Abromochkin, V. Volostnikov, “Spiral-type beams; optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

1994 (2)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

1993 (2)

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

I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

1992 (2)

1991 (1)

P. Szwaykowski, J. Ojeda-Casteneda, “Nondiffracting beams and self imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

1988 (1)

A. W. Lohman, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

1987 (2)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1974 (1)

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

1973 (1)

1968 (1)

1967 (1)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. Applied Mathematics Series No. 55 (U.S. GPO, Washington, D.C., 1964).

Abromochkin, E.

E. Abromochkin, V. Volostnikov, “Spiral-type beams; optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Ordinary Differential Equations (MIT Press, Cambridge, Mass., 1973), Chap. 5, pp. 254–268.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Beijersbergen, M. W.

G. P. Karman, M. W. Beijersbergen, A. van Duiji, D. Bouwmeester, J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A 15, 884–899 (1998).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Berry, M.

M. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan, J. L. Safko, eds. (World Scientific, London, 1994), pp. 55–65.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.

Berry, M. V.

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

M. V. Berry, “Wave dislocation reactions in non-paraxial beams,” J. Mod. Opt. 45, 1845–1858 (1998).
[CrossRef]

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

M. V. Berry, C. Upstill, “Catastrophe optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.

Bodenschatz, E.

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

Bouwmeester, D.

Brambilla, M.

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Bryngdahl, O.

Catteneo, M.

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Deykoon, A. M.

Dubrovin, B. A.

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry—Methods and Applications (Springer-Verlag, Berlin, 1984,1986,1990), Part II. The geometry and topology of manifolds (1986); Part III. Introduction to homology theory (1990).

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

Fomenko, A. T.

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry—Methods and Applications (Springer-Verlag, Berlin, 1984,1986,1990), Part II. The geometry and topology of manifolds (1986); Part III. Introduction to homology theory (1990).

Freilikher, V.

I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Freund, I.

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

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, 1988), Chap. 4, pp. 73–83.

Gorshkov, V. N.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 369, No. 3.616.8.

Heckenberg, N. R.

Indebetouw, G.

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

Jeon, J. H.

Karman, G. P.

Kim, G. H.

Ko, K. H.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Law, C. T.

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

Lee, J. H.

Lohman, A. W.

A. W. Lohman, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Lugiato, L. A.

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

McDuff, R.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

Milnor, J.

J. Milnor, Morse Theory (Princeton U. Press, Princeton, N.J., 1963).

Montgomery, W. D.

Moon, H. J.

Novikov, S. P.

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry—Methods and Applications (Springer-Verlag, Berlin, 1984,1986,1990), Part II. The geometry and topology of manifolds (1986); Part III. Introduction to homology theory (1990).

Nye, J. F.

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

Ojeda-Casteneda, J.

P. Szwaykowski, J. Ojeda-Casteneda, “Nondiffracting beams and self imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Piestun, R.

Pilipetsky, N. F.

B. Y. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 3.

Pirovano, R.

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

Rozas, D.

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

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 369, No. 3.616.8.

Sacks, Z. S.

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

Shamir, J.

Shkunov, V. V.

B. Y. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 3.

Shvarstman, N.

I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Smith, C. P.

Soskin, M. S.

A. M. Deykoon, M. S. Soskin, G. A. Swartzlander, “Nonlinear optical catastrophe from a smooth initial beam,” Opt. Lett. 24, 1224–1226 (1999).
[CrossRef]

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

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Natl. Bur. Stand. Applied Mathematics Series No. 55 (U.S. GPO, Washington, D.C., 1964).

Strogatz, S. H.

S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), Chap. 6, pp. 174–180.

Swartzlander, G. A.

A. M. Deykoon, M. S. Soskin, G. A. Swartzlander, “Nonlinear optical catastrophe from a smooth initial beam,” Opt. Lett. 24, 1224–1226 (1999).
[CrossRef]

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

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

Szwaykowski, P.

P. Szwaykowski, J. Ojeda-Casteneda, “Nondiffracting beams and self imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.

van Duiji, A.

Vasnetsov, M. V.

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

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Volostnikov, V.

E. Abromochkin, V. Volostnikov, “Spiral-type beams; optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

White, A. G.

Woerdman, J. P.

G. P. Karman, M. W. Beijersbergen, A. van Duiji, D. Bouwmeester, J. P. Woerdman, “Airy pattern reorganization and subwavelength structure in a focus,” J. Opt. Soc. Am. A 15, 884–899 (1998).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Zel’dovich, B. Y.

B. Y. Zel’dovich, N. F. Pilipetsky, V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985), Chap. 3.

Appl. Opt. (1)

J. Mod. Opt. (3)

M. V. Berry, “Wave dislocation reactions in non-paraxial beams,” J. Mod. Opt. 45, 1845–1858 (1998).
[CrossRef]

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

M. V. Berry, E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).
[CrossRef]

J. Opt. Soc. Am. (3)

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

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Opt. Commun. (6)

E. Abromochkin, V. Volostnikov, “Spiral-type beams; optical and quantum aspects,” Opt. Commun. 125, 302–323 (1996).
[CrossRef]

P. Szwaykowski, J. Ojeda-Casteneda, “Nondiffracting beams and self imaging phenomenon,” Opt. Commun. 83, 1–4 (1991).
[CrossRef]

I. Freund, N. Shvarstman, V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[CrossRef]

I. Freund, “Saddle point wave fields,” Opt. Commun. 163, 230–242 (1999).
[CrossRef]

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

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, J. P. Woerdman, “Helical wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

A. W. Lohman, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Phys. Rev. A (2)

M. Brambilla, M. Catteneo, L. A. Lugiato, R. Pirovano, F. Prati, “Dynamical transverse laser patterns. I. Theory,” Phys. Rev. A 49, 1427–1451 (1994).
[CrossRef] [PubMed]

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

Phys. Rev. Lett. (3)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

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

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

Other (12)

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–549.

M. V. Berry, C. Upstill, “Catastrophe optics,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.

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Figures (9)

Fig. 1
Fig. 1

Second-order saddle. (a) Contour map with superimposed gradient vectors (arrows). The equation of this saddle is f(x, y)=y(y-3x)(y+3x). Passing through the saddle point at the origin are three bifurcation lines (thick lines) that divide the plane into six sectors with alternating signs for f, which, for example, is positive within the upper vertical sector. These lines are the three roots of f(x, y)=0: y=0 (x axis) and y=±3x. Vectors on bifurcation lines point from negative to positive sectors, vectors pointing away from the saddle point are on lines of steepest ascent (f>0), and vectors pointing toward the saddle point are on lines of steepest descent (f<0). The vectors rotate in the negative (clockwise) direction by 4π in one complete counterclockwise circuit around the saddle point, so the Poincare–H́opf index of this saddle is τ=-2. (b) Three-dimensional (3D) view of the saddle as a surface above the xy plane. The saddle point is at the intersection of the bifurcation lines shown as the three thick lines.

Fig. 2
Fig. 2

First-order maximum described by the fourth-order polynomial f(x, y)=-(x4+y4-ax2y2), with a=5/3. (a) Contour map with superimposed gradient vectors (arrows). Although the direction of rotation of the vectors oscillates near the corners of the figure, their net rotation is +2π and the index is +1. These corners sharpen and the accompanying oscillations of the directions of the gradient vectors become more pronounced as a increases. For a>2 the function describes a third-order saddle with τ=-3. (b) 3D view of the maximum in (a). As a increases, the corners remain nearly flat (constant height) out to increasing distances before finally dropping. For a>2 they rise instead of fall, and the figure acquires four directions of steepest ascent in addition to the four directions of steepest descent already present.

Fig. 3
Fig. 3

On the surface of a sphere, a first-order saddle (τ=-1) at the north pole (N) induces a third-order extremum (τ=+3) at the south pole (S), thereby conserving the Euler characteristic of the sphere (χ=+2). (a) Coordinate systems. (b) Northern and (c) southern polar caps after removal from the sphere before being flattened into planes. (d), (e) Directions of steepest ascent and descent are shown in (d) for the saddle at the north pole and in (e) for the extremum at the south pole. Note the discontinuity at the origin in (e) that is characteristic of higher-order extrema. This figure is discussed in more detail in Subsection 3.A.

Fig. 4
Fig. 4

Fourth-order extremum. This critical point is described by f(r, θ)=(2-r)3 cos(3θ), r<2. (a) Contour map with superimposed gradient vectors (arrows). The three bifurcation lines [cos(3θ)=0] are shown as thick lines. The remaining contours form “petals” that asymptote to these lines as r0. Along directions of steepest ascent (descent), the vectors point away from (toward) the extremum. Since the vectors rotate through +8π along a path encircling the extremum, the index is +4. (b) 3D view of the extremum in (a). Note that there is a singularity at the origin and that directions of steepest descent (ascent) are positive (negative). This figure may be compared with the second-order saddle in Fig. 1.

Fig. 5
Fig. 5

DBA sequences (Subsection 2.C) for (a) a higher-order extremum, (b) a regular saddle point, and (c) an irregular saddle. As one recedes from the origin, the members of a sequence come together for an extremum [(a)] but separate for saddles [(b), (c)]. Note the discontinuities at the origin in (a) and (c). (d), (e) 3D views of the irregular functions in (a) and (c): (d) the extremum in (a) and (e) the saddle in (c). Note the seemingly minor differences between these two surfaces. This figure is discussed in more detail in Subsection 3.B.

Fig. 6
Fig. 6

(a) Intensity (thick line) normalized to unit peak height of the approximation in Eq. (16) to a tenth-order extremum (n=9) embedded in a Gaussian laser beam (thin line) with waist parameter w0=5 mm for a propagation distance Z=100 mm. The parameters used in Eq. (16) are listed in the text in Subsection 4.B. (b) Contour map of extremum in (a) at beam waist. Note the 18 petals characteristic of an extremum of tenth-order. (c) Contour map of Gauss–Laguerre mode with n=9. This map contains an eighth-order saddle at the origin surrounded by a ring of nine maxima alternating with nine minima.

Fig. 7
Fig. 7

Third-order (n=3) vortex V(r, θ)=g(r)×exp(3iθ)exp(-r2/w02) in the presence of a background field Δ(1+i), where g(r) is the envelope function in Eq. (16) with =0.3. (a) Star of zero crossings (ZC’s) for the real (ZR, thick lines) and imaginary (ZI, thin lines) parts of the wave function for Δ=0 [Eq. (16), m=δ=0, w0=1]. (b) Breakup of saddle-type vortex into three first-order vortices located at the intersections of ZR and ZI for Δ=0.4 [Eq. (16), m=3, δ=-3, w0=1]. (c) Extrema-type vortex for Δ=0.4 [Eq. (16), m=0, δ=0, w0=1]. Note the petallike ZC’s and the three new first-order vortices. (d) Breakup of Gauss–Laguerre vortex into six vortices (2n) for Δ=0.4 [Eq. (16), m=3, δ=-3, w0=0.3]. The field of view in all figures is 0r1.

Fig. 8
Fig. 8

Relative stability of extrema-type and saddle-type vortices: coherent superposition of laser beam 1 containing an embedded third-order positive (n=+3) vortex at (x, y)=(+0.5, 0) with laser beam 2 containing an n=-3 vortex at (-0.5, 0). (a) Amplitude dependence g1(x, 0) of beam 1 and g2(x, 0) of beam 2 for extrema-type vortices [Eq. (16), =0.015, m=3, δ=0.3]. The curve labeled Gauss is the Gaussian profile with waist parameter w0=1 common to both beams. (b) Real (thick lines) and imaginary (thin lines) ZC’s of the composite beam in (a). Note that both extrema-type vortices remain intact, i.e., they locally maintain the six armed zero-crossing star characteristic of a third-order vortex [Fig. 7(a)]. (c) As in (a) but for saddle-type vortices [Eq. (16), =0.015, m=3, δ=-3]. (d) ZC’s of the vortices in (c). Here both vortices have been completely fragmented by their mutual perturbation. This figure together with Fig. 7 illustrates the exceptional stability of extrema-type vortices as compared with (common) saddle-type vortices.

Fig. 9
Fig. 9

Topological arithmetic using extrema-type (ee-) vortices. A vortex addendum field Va*(qa) with topological charge -qa is coherently added to a vortex base field Vb(qb) with charge qb. The resulting intensity U=|Va*(qa)+Vb(qb)|2 has |qa+qb| lobes (petals) with contrast ratio C=Umax/Umin. The base and addendum ee-vortex wave functions are each of the form V(r; q)=g(r; q)×[cos(qθ)+i sin(qθ)]exp(-r2), where g(r; q) is the envelope function in Eq. (16) with m=q, =0.01, and δ=0.3. For this form the contrast C=[g(r; qa)+g(r; qb)]2/[g(r; qa)-g(r; qb)]2 can be very large. (a)–(d) Vortex addition for qb=4 and qa=0, 1, 2, 3. (e)–(h) Vortex subtraction for qb=4 and qa=-1, -2, -3, -4.

Equations (19)

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ΔӨ±=2π(1±n).
τs(n)=-(n-1).
τe(n)=n+1.
fe(m)=(c-r)m-1 cos[(m-1)θ],r<c.
f(r, θ)=gn(r)hn(θ),
Өn=arctantan(θ)-ρn tan(nθ)1+ρn tan(θ)tan(nθ),
ρn=ngn(r)rgn(r).
τn=12π02πdθӨnθr,
Өnθr=1-nρn1+(ρn2-1)sin2(nθ),
τn=1-n sgn(ρn).
22 f=0.
fn(r, θ)=gn(r)cos(nθ-θ0),
r22gnr2+rgnr-n2gn=0.
gn(r)=c+rn+c-r-n,
Gn(r, θ; Z=0)=gn(r)cos(nθ)×2π1w0exp(-r2/w02),
Gn(r, θ; Z)=Γn(r; Z)cos(nθ),
Γn(r; Z)=-22πin-1w0λZexp(-ikZ)exp[-ikr2/(2Z)]×0r dr gn(r)exp(-r2/w02)×exp[-ikr2/(2Z)]Jn(krr/Z)
fn(r, θ)=rmm+δ+r(m+δ)cos(nθ)
V(r, θ; k)=R(r)cos[ρ(θ; m)]+iI(r)cos[ι(θ; n)],

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