Abstract

Critical foliations are special two-dimensional slices (planes of observation) of a three-dimensional optical field in which an infinitesimally small change in the angle of observation produces major qualitative differences in the observed field structure. They are common, but previously unrecognized, features of optical fields that contain vortices. An experimentally realizable, on-axis example of such a foliation is described for a paraxial Gaussian laser beam.

© 2001 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. M. Berry, Proc. SPIE 3487, 1 (1998).
    [CrossRef]
  3. I. Freund, Opt. Commun. 181, 19 (2000).
    [CrossRef]
  4. S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley, Reading, Mass., 1994), pp. 174–180.
  5. I. Freund and N. Shvarstman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef] [PubMed]
  6. I. Freund, Opt. Commun. 159, 99 (1999).
    [CrossRef]
  7. I. Freund, Opt. Commun. 163, 230 (1999).
    [CrossRef]
  8. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).
  9. N. R. Heckenberg, R. McDuff, C. P. Smith, and G. A. White, Opt. Lett. 17, 221 (1992).
    [CrossRef]
  10. G. A. Swartzland and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
    [CrossRef]

2000 (1)

I. Freund, Opt. Commun. 181, 19 (2000).
[CrossRef]

1999 (2)

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

I. Freund, Opt. Commun. 163, 230 (1999).
[CrossRef]

1998 (1)

M. Berry, Proc. SPIE 3487, 1 (1998).
[CrossRef]

1994 (1)

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

1992 (2)

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

1974 (1)

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

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Berry, M.

M. Berry, Proc. SPIE 3487, 1 (1998).
[CrossRef]

Berry, M. V.

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

Freund, I.

I. Freund, Opt. Commun. 181, 19 (2000).
[CrossRef]

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

I. Freund, Opt. Commun. 163, 230 (1999).
[CrossRef]

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

Heckenberg, N. R.

Law, C. T.

G. A. Swartzland and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef]

McDuff, R.

Nye, J. F.

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

Shvarstman, N.

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

Smith, C. P.

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Strogatz, S. H.

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

Swartzland, G. A.

G. A. Swartzland and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[CrossRef]

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

White, G. A.

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, JETP Lett. 52, 429 (1990).

Opt. Commun. (3)

I. Freund, Opt. Commun. 159, 99 (1999).
[CrossRef]

I. Freund, Opt. Commun. 163, 230 (1999).
[CrossRef]

I. Freund, Opt. Commun. 181, 19 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

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

Phys. Rev. Lett. (1)

G. A. Swartzland and C. T. Law, Phys. Rev. Lett. 69, 2503 (1992).
[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]

Proc. SPIE (1)

M. Berry, Proc. SPIE 3487, 1 (1998).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Trajectories and bundles. In (a) black (white) arrowheads correspond to positive (negative) charges as seen by the observer; in (b)–(d), to positive (negative) index critical points. Vortex (stationary point) trajectories are shown by thick (thin) curves. Stationary point trajectories that lie above (below) the plane of the vortex trajectory are shown by solid (dashed) curves. (a) Vortex charge trajectory. The planes of a foliation are labeled 1–3. Filled (open) circles represent positive (negative) vortices as seen by the observer in this foliation. (b)–(d) Index trajectories: (b), (c) phase bundles, (d) intensity bundle.

Fig. 2
Fig. 2

Critical foliation: (a) vortex trajectory, (b) trajectory projected onto the xZ plane. The vertical inflection point (here located at the origin) is the hallmark of a critical foliation. Lines labeled 1–5 are analogs of planes 1–5 in Fig.  3.

Fig. 3
Fig. 3

Intensity index trajectories for a subcritical foliation. In plane 1 a single zero-amplitude minimum corresponding to a positive vortex (thick solid curve), a single saddle point (dotted curve), and a single nonzero minimum (thin solid curve) are present. On crossing the upper turning point (junction) located in plane 2, the nonzero minimum transforms into a positive–negative vortex dipole and a second saddle point; in plane 3, three vortices and two saddle points are present. At the lower turning point in plane 4, the negative vortex created in plane 2 collides with the vortex and the saddle point that were initially present in plane 1, yielding a new nonzero amplitude intensity minimum. Finally, in plane 5, as in plane 1, a zero-amplitude minimum (positive vortex), a saddle point, and a nonzero minimum are present. Note that the index, which equals +1, is conserved in all the planes of the foliation.

Fig. 4
Fig. 4

Intensity contour maps. Zero-amplitude minima (vortices) are shown by filled ovals; saddle points, by open circles. The scale of the horizontal x axis (vertical y axis) is -0.0667w0 to +0.0667w0 (-0.002w0 to +0.002w0). (a), (a′) Z=-0.002Z0; (b), (b′) Z=0; (c), (c′) Z=+0.0002Z0. (a), (b), (c) Leaves of critical foliation corresponding to planes 1, 3, and 5, respectively, of Fig.  2(b). Only a single vortex is seen, as expected. (a′)–(c′) Subcritical foliation. The stationary points seen here and their Z-dependent reactions are in complete accord with Fig.  3 for the corresponding planes 1, 3, and 5.

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