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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  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

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

1995

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

1977

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

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

1969

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

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.

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.

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

Opt. Commun.

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.

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

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

Other

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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