Abstract

The critical points of generic paraxial ellipse fields consist of singular points of circular polarization, called C-points, and azimuthal stationary points, i.e., maxima, minima, and saddle points. We define these stationary points here and review their properties. The sign rule for ellipse fields requires that the sign of the singularity indices IC=±1/2 of the C-points on non-self-intersecting lines of constant azimuthal ellipse orientation (modulo π/2), i.e., a-lines, alternate along the line. We verify this rule experimentally, using a newly developed interferometric technique to measure C-points and a-lines in an elliptically polarized random optical field.

© 2002 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. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
    [CrossRef]
  3. M. V. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, New York, 1999).
  4. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4.
  5. J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP Publishing, Bristol, UK, 1999).
  6. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  7. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
  8. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.
  9. I. Freund and N. Shvartsman, Phys. Rev. A 50, 5164 (1994).
    [CrossRef] [PubMed]
  10. N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).
  11. O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
    [CrossRef]

2002

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[CrossRef]

2001

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

1999

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP Publishing, Bristol, UK, 1999).

M. V. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, New York, 1999).

1998

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

1994

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

1981

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

1974

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

1959

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

Angelsky, O. V.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[CrossRef]

Baranova, N. B.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Berry, M. V.

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

Born, M.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

Freund, I.

I. Freund, Opt. Lett. 26, 1996 (2001).
[CrossRef]

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

Kivshar, Y. S.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Konukhov, A. I.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Luther-Davies, B.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Mamaev, A. V.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Melnikov, L. A.

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

Mokhun, A. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[CrossRef]

Nye, J. F.

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP Publishing, Bristol, UK, 1999).

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

Pilipetskii, N.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Shkukov, V. V.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Shvartsman, N.

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

Soskin, M. S.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4.

Wolf, E. W.

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

Zel'dovich, B. Y.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

J. Opt. B

A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
[CrossRef]

JETP Lett.

N. B. Baranova, B. Y. Zel'dovich, A. V. Mamaev, N. Pilipetskii, and V. V. Shkukov, JETP Lett. 33, 195 (1981).

Opt. Lett.

Phys. Rep.

Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. A

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

Phys. Rev. E

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, Phys. Rev. E 65, 036603(5) (2002).
[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

M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Sec. 1.4.2.

M. V. Vasnetsov and K. Staliunas, eds., Optical Vortices (Nova Science, New York, 1999).

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, Chap. 4.

J. F. Nye, Natural Focusing and the Fine Structure of Light (IOP Publishing, Bristol, UK, 1999).

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Figures (5)

Fig. 1
Fig. 1

Generic C-points (shaded circles). The components of the field model used here are Ex=3+V± and Ey=i3-V±, where V±=x±iy. (a) V+, IC=+1/2; (b) V-,IC=-1/2. The ellipses surrounding a positive IC (negative IC) generic C-point rotate in the same counterclockwise (clockwise) direction as a counterclockwise path that encircles the point. Note that on any line passing through the C-point the ellipses on opposite sides of the point have perpendicular orientations. Generic L-lines may be either open or closed: An example of a closed L-line is shown in Fig. 3 below.

Fig. 2
Fig. 2

Experimental apparatus: Beam splitters BS1,2 and mirrors M1,2 form a Mach–Zehnder interferometer. Lenses L1,2 and pinhole PH expand and spatially filter the reference beam. Lens L3 focuses the linearly polarized He–Ne laser beam onto multiple-scattering sample S, and lens L4 projects an enlarged image of the scattered ellipse field onto charge-coupled device camera CCD, whose output is fed to computer PC. Quarter-wave plates λ/41,2 and analyzer A manipulate the polarization of the sample and reference beams to produce interferograms with forks whose positions locate C-points, L-lines, and a-lines. Detailed operation of the interferometer is described in Ref. 11. Shown on the PC monitor is a two-fork interferogram of a C-point dipole measured in real time.

Fig. 3
Fig. 3

C-points, a-lines, and the sign rule. (a) C-points A–C in a random ellipse field are shown by large circles labeled with the signs of their singularity index IC=±1/2. a-lines a1,2, corresponding to ellipse contours with constant azimuthal orientation α1,2 (modulo π/2), are shown by heavy dark curves, and an L-line is shown by the thick, lighter, closed curve. Measurement points on a-lines and on the L-line are shown by small open circles. As may be seen, the signs of the C-points alternate along the a-lines, in accord with the requirements of the sign rule. This requirement remains unchanged even if the a-line crosses an L-line. (b) Ellipses surrounding the three C-points in (a). The ellipses rotate in the expected direction and have perpendicular orientations on opposite sides of the C-point, as expected (Fig. 1). There are many a-lines, each of which satisfies the sign rule, that pass through every C-point. The a-lines shown were randomly selected for measurements by arbitrary orientation of wave plate λ/42 in Fig. 2. Further details of how such measurements are performed may be found in Ref. 11.

Fig. 4
Fig. 4

Azimuthal stationary points (shaded ellipses). (a) α-minimum. Ex=1, Ey=expiπ/42/3+5x2+5y2. As may be seen, azimuthal angle α increases along all directions away from the central stationary point. (b) α-maximum. Ex=1+8x2+y2,Ey=expiπ/42.5-5x2-7y2. Here α decreases along all directions away from the central point. (c) α-saddle point. Ex=0.5+1.5y2, Ey=expiπ/40.5+3x2-1.25y2. α increases (decreases) along directions of steepest ascent (steepest descent) corresponding to the horizontal x axis (vertical y axis) and maintains the orientation of the central saddle point along the ±45° diagonals that correspond to bifurcation lines. Generic C-points and α-extrema are characterized by Poincaré–Hopf index η=+1; α-saddles, by η=-1. IC and η must be simultaneously conserved in every singularity reaction, which is a strong constraint with many important implications.

Fig. 5
Fig. 5

C-point quadrupole. (a) Contour map of Stokes phase field φ12 [Eq. (1)]. (b) Corresponding ellipse field. Each C-point (circles) is labeled by the sign of its singularity index, IC=±1/2. Ex=1+Qx,y, Ey=i+Qx,y, Q=x-b+iy-bx+b+iy+bx-b-iy+bx+b-iy-b, b=0.6. As required by the sign rule, C-points of the same sign terminate a bifurcation line (thick lines), whereas C-points with opposite signs alternate about the saddle point, shown by the small filled circle in (a) and the filled ellipse in (b).

Equations (3)

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S12x,y=S1x,y+iS2x,y=A12 expiφ12,
S1=Ex2-Ey2,  S2=2 ReEx*Ey.
φ12x,y=2αx,y,

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