Abstract

Elliptical polarization can appear in only monochromatic optical fields. In polychromatic vector fields the polarization is a Lissajous figure, but in only commensurate fields do the figures have well-defined shapes; in other fields the shapes are undefined. Nonetheless, I show that a given paraxial polychromatic vector field has a coherency ellipse field associated with it that contains polarization singularities and stationary points that are surrogates for the corresponding critical points of the parent optical field.

© 2003 Optical Society of America

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References

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  1. M. Born and E. W. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959).
  2. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  3. M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).
  4. E. Wolf, Opt. Lett. 28, 5 (2003). (The correct starting page in Ref. 7 of our Ref. is page 884.)
    [CrossRef] [PubMed]
  5. A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
    [CrossRef]
  6. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, England, 1999).
  7. D. A. Kessler and I. Freund, Opt. Lett. 28, 111 (2003).
    [CrossRef] [PubMed]
  8. I. Freund, Opt. Lett. 27, 1640 (2002).
    [CrossRef]
  9. A. I. Konukhov and L. A. Melnikov, J. Opt. B 3, S139 (2001).
    [CrossRef]
  10. I. Freund, Opt. Lett. 26, 1996 (2001).
    [CrossRef]
  11. I. Freund, M. S. Soskin, and A. I. Mokhun, Opt. Commun. 208, 223 (2002).
    [CrossRef]
  12. E. Wolf, Nuovo Cimento XII, 884 (1954).
    [CrossRef]

2003

2002

I. Freund, Opt. Lett. 27, 1640 (2002).
[CrossRef]

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

2001

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

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

1999

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

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

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, England, 1959).

1954

E. Wolf, Nuovo Cimento XII, 884 (1954).
[CrossRef]

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, England, 1959).

Dogariu, A.

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

Freund, I.

Kessler, D. A.

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. I.

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

Nye, J. F.

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

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

Popescu, G.

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[CrossRef]

Soskin, M. S.

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

Staliunas, K.

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

Vasnetsov, M.

M. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

Wolf, E.

Wolf, E. W.

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

J. Opt. B

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

Nuovo Cimento

E. Wolf, Nuovo Cimento XII, 884 (1954).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Phys. Rev. Lett.

A. Dogariu and G. Popescu, Phys. Rev. Lett. 89, 243902 (2003).
[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. Vasnetsov and K. Staliunas, Optical Vortices (Nova Science, Commack, N.Y., 1999).

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

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

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Figures (2)

Fig. 1
Fig. 1

Multimode He–Ne laser polarization figures. For illustrative purposes three longitudinal modes centered on the gain profile are assumed to oscillate with amplitudes 1:2:1 at frequencies ν equal to the mode number multiplied by the cavity mode spacing Δν, so that all frequencies are commensurate. Shown are figures traced by the electric vector E over one Lissajous period T=1/Δν. Typically, T25 ns, and ρ=Δν/ν10-6. For ρ1 the basic shapes of the figures no longer depend on this parameter, but the spacing between the lines of the figure does, and here ρ was adjusted to avoid overly crowded lines. (a) ρ=0.05. A linearly polarized laser with all modes initially oscillating in phase passes through a quarter-wave plate that is effectively achromatic over the gain profile. Although all modes become right circular, because of their small frequency difference, they become out of phase after half a Lissajous period, and, as shown below the figure (left to right), E spirals inward to zero. Over the remaining half-period, E spirals outward (right to left). (b) ρ=0.01. Linearly polarized laser light is multiply scattered through a 1.5-mm-thick sample with a transport mean free path of 10 µm. The scattered light is depolarized, and the relative mode amplitudes and phases in the two orthogonal field components Ex and Ey, and therefore the resulting Lissajous polarization figures, differ at different points in the scattered field. Shown here is a typical example. In practice, of course, the modes are pulled slightly toward the center of the gain curve, say by δν, the mode frequencies are no longer strictly commensurate, the lines of the figures in both (a) and (b) never retrace themselves, and instead they tend to fill in the figures over a time of order 1/δν.

Fig. 2
Fig. 2

Spectral perturbation of an ellipse field. (a) Unperturbed unit amplitude ellipse field at frequency ν containing an IC=+1/2 C point at its center. The ellipses surrounding the C point rotate by π around the point. (b) Lissajous field produced by adding an ellipse field to (a) at 2ν with amplitude 1/3 that contains an IC=-1 C point at its center.8 Shown also are the gray lines that mark the ellipse symmetry axes in (a). There is no obvious relationship between these lines and the corresponding Lissajous figures. Although here the central Lissajous figure has threefold rotational symmetry, in general, Lissajous figures, including Lissajous singularities, are asymmetric. (c) Φ12 of the ellipse field in (a). This phase field has a charge q=+1 vortex at its center corresponding to the C point in (a).9,10 (d) Φ12 of the Lissajous field in (b). This field has a charge q=+1 vortex at its center corresponding to a (perhaps not readily apparent) Lissajous singularity in (b). In both (a) and (b) the phase is coded -π to π, black to white.

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