Abstract

Polarization singularities and elliptic stationary points (collectively, elliptic critical points) were measured experimentally via the complex Stokes field S1+iS2, where S1 and S2 are Stokes parameters. This new, easily implemented method yielded detailed, high-resolution experimental data for all elliptic critical points. These data confirm with high precision the elliptic-field topological sign rule, loop rules, and Stokes singularity relations introduced recently.

© 2003 Optical Society of America

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References

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  1. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, UK, 1999).
  2. J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 447 (1987).
    [CrossRef]
  3. J. V. Hajnal, IEE Proc. H 134, 350 (1987).
  4. J. V. Hajnal, Proc. R. Soc. London Ser. A 430, 413 (1990).
    [CrossRef]
  5. O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
    [CrossRef]
  6. O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
    [CrossRef]
  7. A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
    [CrossRef]
  8. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  9. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
  10. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, 1959).
  11. I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  12. I. Freund, Phys. Rev. E 52, 2348 (1995).
    [CrossRef]
  13. I. Freund, Waves Random Media 8, 119 (1998).
  14. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, Opt. Lett. 27, 545 (2002).
    [CrossRef]

2002 (5)

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, Opt. Lett. 27, 545 (2002).
[CrossRef]

2001 (2)

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

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

1999 (1)

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

1998 (1)

I. Freund, Waves Random Media 8, 119 (1998).

1995 (1)

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

1990 (1)

J. V. Hajnal, Proc. R. Soc. London Ser. A 430, 413 (1990).
[CrossRef]

1987 (2)

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 447 (1987).
[CrossRef]

J. V. Hajnal, IEE Proc. H 134, 350 (1987).

1959 (1)

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

Angelsky, O.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

Angelsky, O. V.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, Opt. Lett. 27, 545 (2002).
[CrossRef]

Born, M.

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

Freund, I.

Hajnal, J. V.

J. V. Hajnal, Proc. R. Soc. London Ser. A 430, 413 (1990).
[CrossRef]

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 447 (1987).
[CrossRef]

J. V. Hajnal, IEE Proc. H 134, 350 (1987).

Konukhov, A. I.

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

Melnikov, L. A.

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

Mokhun, A.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

Mokhun, A. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, Opt. Lett. 27, 545 (2002).
[CrossRef]

A. I. Mokhun, M. S. Soskin, and I. Freund, Opt. Lett. 27, 995 (2002).
[CrossRef]

Mokhun, I.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

Mokhun, I. I.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, Opt. Lett. 27, 545 (2002).
[CrossRef]

Nye, J. F.

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

Soskin, M.

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

Soskin, M. S.

Wolf, E. W.

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

IEE Proc. H (1)

J. V. Hajnal, IEE Proc. H 134, 350 (1987).

J. Opt. B (1)

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

Opt. Commun. (2)

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, Opt. Commun. 207, 57 (2002).
[CrossRef]

I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (2)

O. V. Angelsky, A. I. Mokhun, I. I. Mokhun, and M. Soskin, Phys. Rev. E 56, 036602 (2002).
[CrossRef]

I. Freund, Phys. Rev. E 52, 2348 (1995).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

J. V. Hajnal, Proc. R. Soc. London Ser. A 430, 413 (1990).
[CrossRef]

J. V. Hajnal, Proc. R. Soc. London Ser. A 414, 447 (1987).
[CrossRef]

Waves Random Media (1)

I. Freund, Waves Random Media 8, 119 (1998).

Other (2)

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

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

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Figures (5)

Fig. 1
Fig. 1

Experimental apparatus. Lens L1 focuses the linearly polarized He–Ne laser into the 80µm-core multi-mode fiber. The outgoing elliptically polarized speckle field (degree of polarization,10 0.41) is collimated by lens L2 and, when required (I5 and I6), first projected through a λ/4 retardation plate (accuracy, λ/100). The beam is then projected through an analyzing polarizer (Pol) onto the face of camera CCD, whose output is fed to computer PC. The fiber used in this study was chosen because it produced an ellipse field rich in polarization singularities and their elliptic stationary points.

Fig. 2
Fig. 2

Sign rules for C points. C points (Stokes S12 vortices) with negative (positive) charges are shown by filled circles with white rims (open circles with black rims). (a) Sign rule for C points on zero crossings Z1 of S1 (thicker curves) and Z2 of S2 (thinner curves). Left-, S3<0 (right-, S3>0) handed regions are shaded light gray (white). The boundaries between regions of opposite handedness are L lines. As required by the sign rule, C point signs alternate along both Z1 and Z2; the net charge on all closed zero crossings is zero. L lines do not affect the sign rule. (b) Sign rule for C points terminating the X-shaped special contour lines (bifurcation lines) that pass through an azimuthal saddle point (filled circles). Shown is the phase of S12 coded -π to +π, black to white. As required by the sign rule, C points terminating opposite ends of a bifurcation line have the same sign, whereas the four C points surrounding an open saddle point alternate in sign. Collision and annihilation of C points of opposite sign cause adjacent bifurcation line segments to form closed bifurcation loops and lead to the loop rules illustrated in Fig. 3.

Fig. 3
Fig. 3

Loop rules for C points and azimuthal stationary points. Shown is the phase of S12 coded -π to +π, black to white. Negative (positive) C points are shown by filled circles with white rims (open circles with black rims); azimuthal saddle points, by filled circles; and azimuthal maxima (minima), by filled closed contours shaded light gray (dark gray). (a) Loop rule for one-loop saddles. Three azimuthal extrema (top to bottom: maximum, minimum, maximum) run down the left-hand side of the figure. Each extremum is enclosed within the bifurcation loop of a one-loop saddle that has the free ends of its bifurcation lines terminated by C points. The nature of each of the three extrema (maximum or minimum) agrees with the following loop rule: Starting at the negative terminating C point and moving along the bifurcation line, if one traverses the bifurcation loop in the positive, counterclockwise direction (negative, clockwise direction) the enclosed extremum must be a maximum (minimum). (b) Loop rule for an inner, two-loop figure-eight saddle (to the right of the center line) enclosed within the bifurcation loop of an outer, one-loop saddle. The loop rule for a figure eight requires that the extrema in both loops be of the same type, two maxima or two minima, whereas the loop rule for a one-loop saddle discussed for (a) requires that, for the configuration shown here, these extrema both be minima. This conclusion follows because, starting at the negative terminating C point (in the upper left-hand corner) and moving along the bifurcation line, the bifurcation loop that encloses the figure-eight saddle is traversed in the negative, clockwise direction.

Fig. 4
Fig. 4

Stokes singularity relations. Vortices of Stokes fields Sab=Sa+iSb (a,b,c=1,2,3) with positive (negative) charge qab are shown by open (filled) circles. (a) Zero crossings. +Sa- identifies zero crossing Za of Sa and shows the sign σa of Sa on both sides of the contour. Sab vortices are located at the intersections of Za with Zb. Starting at the highest point on Za and proceeding clockwise around the contour, in the discussion below, qab and σc of the first (second) Sab vortex encountered are labeled qab (qab) and σc (σc), respectively. (b) S12 (C points) and S31 vortices on a closed Z1 that contains an internal S23 (Ex) vortex. The vortex charges satisfy 2σ1q23=σ3q12+σ3q12=σ2q31+σ2q31, where σ1=+1, q23=-1, σ3=-1, q12=+1, σ3=+1, q12=-1, σ2=-1, q31=+1, σ2=+1, and q31=-1. (c) S12 (C points) and S23 (Ex and Ey) vortices on a closed Z2 that contains an internal S31 vortex. Here the vortex charges satisfy 2σ2q31=σ3q12+σ3q12=σ1q23+σ1q23, where σ2=+1, q31=-1, σ3=+1, q12=-1, σ3=-1, q12=+1, σ1=+1, q23=-1, σ1=-1, and q23=+1. (d) S23 (Ex and Ey) and S31 vortices on a closed Z3 (L line) that contains an internal S12 vortex (C point). Vortex charges satisfy 2σ3q12=σ2q31+σ2q31=σ1q23+σ1q23, where σ3=-1, q12=+1, σ2=-1, q31=+1, σ2=+1, q31=-1, σ1=-1, q23=+1, σ1=+1, and q23=-1.

Fig. 5
Fig. 5

Component field vortex reactions. Coordinate rotation β increases sequentially by 100 mrad for each curve, 1 to 6. An invariant L line is shown by thick curves; negative (positive) vortices, by filled (open) circles; and regions of S1<0 (S1>0) are shaded light gray (white). At a, a positive/negative Ex (S1<0) vortex pair nucleates, whereas vortex collision and annihilation occur at b. A nearby, negative Ey (S1>0) vortex (bottom) is a spectator that is unaffected by these reactions.

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