Abstract

We observed an optical cusp diffraction catastrophe with an initially smooth but elongated Gaussian beam with an aspect ratio of 2:1. Nonlinear and linear diffraction regimes account for the near-field elliptical annulus and the far-field spatially complex astroid.

© 1999 Optical Society of America

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References

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  1. A. E. Kaplan, Pis. Zh. Eksp. Teor. Fiz. 9, 58 (1969); [JETP Lett. 9, 33 (1969)]; G. A. Swartzlander, H. Yin, and A. E. Kaplan, J. Opt. Soc. Am. B 6, 1317 (1989).
    [CrossRef]
  2. S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukharukov, in Laser Handbook, F. T. Arecchi, ed. (North-Holland, Amsterdam, 1972).
  3. G. A. Swartzlander and A. E. Kaplan, J. Opt. Soc. Am. B 5, 765 (1988).
    [CrossRef]
  4. J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
    [CrossRef]
  5. R. Thom, Stabilitestructurelle et Morphogenes (Benjamin, Reading, Mass., 1972) [updated English translation, Structural Stability and Morphogenesis (Benjamin, Reading, Mass., 1975)].
  6. M. V. Berry and C. Upstill, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), p. 257.
    [CrossRef]
  7. J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
    [CrossRef]
  8. V. I. Arnol’d, Catastrophe Theory, 3rd ed. (Springer-Verlag, Berlin, 1992), Chap. 8.
    [CrossRef]
  9. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  10. L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).
  11. T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
    [CrossRef]

1997 (1)

L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).

1995 (1)

T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
[CrossRef]

1988 (1)

1977 (1)

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

1974 (1)

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

1969 (1)

A. E. Kaplan, Pis. Zh. Eksp. Teor. Fiz. 9, 58 (1969); [JETP Lett. 9, 33 (1969)]; G. A. Swartzlander, H. Yin, and A. E. Kaplan, J. Opt. Soc. Am. B 6, 1317 (1989).
[CrossRef]

1965 (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Ackermann, T.

T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukharukov, in Laser Handbook, F. T. Arecchi, ed. (North-Holland, Amsterdam, 1972).

Arnol’d, V. I.

V. I. Arnol’d, Catastrophe Theory, 3rd ed. (Springer-Verlag, Berlin, 1992), Chap. 8.
[CrossRef]

Berry, M. V.

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

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), p. 257.
[CrossRef]

Church, C.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Dunham, E.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Elliot, J. L.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

French, R. G.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Gierasch, P. J.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Gordon, J. P.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Kaplan, A. E.

G. A. Swartzlander and A. E. Kaplan, J. Opt. Soc. Am. B 5, 765 (1988).
[CrossRef]

A. E. Kaplan, Pis. Zh. Eksp. Teor. Fiz. 9, 58 (1969); [JETP Lett. 9, 33 (1969)]; G. A. Swartzlander, H. Yin, and A. E. Kaplan, J. Opt. Soc. Am. B 6, 1317 (1989).
[CrossRef]

Khizhnyak, A. I.

L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukharukov, in Laser Handbook, F. T. Arecchi, ed. (North-Holland, Amsterdam, 1972).

Kreminskaya, L. V.

L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).

Kriege, E.

T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
[CrossRef]

Lange, W.

T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
[CrossRef]

Leite, R. C.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Moore, R. S.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Nye, J. F.

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

Porto, S. P. S.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Sagan, C.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Soskin, M. S.

L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).

Sukharukov, A. P.

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukharukov, in Laser Handbook, F. T. Arecchi, ed. (North-Holland, Amsterdam, 1972).

Swartzlander, G. A.

Thom, R.

R. Thom, Stabilitestructurelle et Morphogenes (Benjamin, Reading, Mass., 1972) [updated English translation, Structural Stability and Morphogenesis (Benjamin, Reading, Mass., 1975)].

Upstill, C.

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), p. 257.
[CrossRef]

Veverka, J.

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

Whinnery, J. R.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

Astrophys. J. (1)

J. L. Elliot, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, and C. Sagan, Astrophys. J. 217, 661 (1977).
[CrossRef]

J. Appl. Phys. (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, J. Appl. Phys. 36, 3 (1965); J. R. Whinnery, D. T. Miller, and F. Dabby, IEEE J. Quantum Electron. QE-3, 382 (1968).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

L. V. Kreminskaya, M. S. Soskin, and A. I. Khizhnyak, Opt. Commun. 145, 337 (1997).

T. Ackermann, E. Kriege, and W. Lange, Opt. Commun. 115, 339 (1995); A. V. Il’yenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62, 465 (1996); A. V. Il’yenkov, L. V. Kreminskaya, M. S. Soskin, and M. V. Vasnetsov, Int. J. Nonlinear Opt. Phys. Mater. 6, 169 (1997).
[CrossRef]

Pis. Zh. Eksp. Teor. Fiz. (1)

A. E. Kaplan, Pis. Zh. Eksp. Teor. Fiz. 9, 58 (1969); [JETP Lett. 9, 33 (1969)]; G. A. Swartzlander, H. Yin, and A. E. Kaplan, J. Opt. Soc. Am. B 6, 1317 (1989).
[CrossRef]

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

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

Other (4)

V. I. Arnol’d, Catastrophe Theory, 3rd ed. (Springer-Verlag, Berlin, 1992), Chap. 8.
[CrossRef]

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukharukov, in Laser Handbook, F. T. Arecchi, ed. (North-Holland, Amsterdam, 1972).

R. Thom, Stabilitestructurelle et Morphogenes (Benjamin, Reading, Mass., 1972) [updated English translation, Structural Stability and Morphogenesis (Benjamin, Reading, Mass., 1975)].

M. V. Berry and C. Upstill, in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), p. 257.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup, showing a prism (Pr) that elongates a collimated Gaussian laser beam. A 200-mm focal-length lens (L1) focuses the beam in the center of a nonlinear cell (C) containing slightly absorbing liquid. The cell has a movable output window (W) and a maximum length Dmax=250 mm. An attenuator (AT1) and a lens (L3) are used to image the output face of the cell onto a CCD camera (CCD). The images are recorded with a frame grabber and a computer. The optical path comprising L2, AT2, and the beam splitter (BS) is the reference leg of an interferometer.

Fig. 2
Fig. 2

Nonlinear optical CDC. (a) Intensity profile and (b), (c) magnified interferograms at the cusps at the output face of the nonlinear cell (D=215 mm, P=156 mW). An astroid with an aspect ratio of 1/1.31 outlines the complex structure in the beam. Optical vortices with a topological charge of ±1 appear as (a) dark nodes and (b), (c) forks near two of the cusps.

Fig. 3
Fig. 3

Propagating beam, showing intensity profiles at the output face of a variable-length nonlinear cell P=156 mW. (a) Position of minimal beam area at D=103 mm. Aspect ratio, 1.31. (b) Onset of internal interference at D=122 mm. The inset depicts elliptical wave fronts and inward rays of equal length. (c) Nearly complete astroid formation at D=185 mm. The inset depicts rays of equal length that span the ellipse, forming an astroid-shaped evolute.

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