Abstract

Coherent optical Lissajous states are easily created by nonlinear processes such as second-harmonic generation (SHG). Singular properties of such states are discussed and illustrated theoretically with non-phase-matched SHG of an ellipse field containing a C point.

© 2003 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. J. F. Nye, in Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), p. 545.
  3. J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
    [CrossRef]
  4. J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
    [CrossRef]
  5. J. F. Nye, Natural Focusing and the Fine Structure of Light (Institute of Physics, Bristol, England, 1999).
  6. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 2001), Vol. 42, Chap. 4.
  7. G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef]
  8. G. Popescu and A. Dogariu, Phys. Rev. Lett. 88, 183902 (2002).
    [CrossRef]
  9. I. Freund, Opt. Lett. 27, 1640 (2002).
    [CrossRef]
  10. G. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chaps. 1–2.
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec. 1.4.2.

2002

G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

G. Popescu and A. Dogariu, Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef]

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

1987

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

1983

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

1974

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[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. Wolf, Principles of Optics (Pergamon, Oxford, England, 1959), Sec. 1.4.2.

Boyd, G.

G. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chaps. 1–2.

Dogariu, A.

G. Popescu and A. Dogariu, Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef]

Freund, I.

Gbur, G.

G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

Hajnal, J. V.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

Nye, J. F.

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

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

J. F. Nye, in Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), p. 545.

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

Popescu, G.

G. Popescu and A. Dogariu, Phys. Rev. Lett. 88, 183902 (2002).
[CrossRef]

Soskin, M. S.

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

Vasnetsov, M. V.

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

Viser, T. D.

G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

Wolf, E.

G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

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

Opt. Lett.

Phys. Rev. Lett.

G. Gbur, T. D. Viser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef]

G. Popescu and A. Dogariu, Phys. Rev. Lett. 88, 183902 (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]

J. F. Nye, Proc. R. Soc. London Ser. A 389, 279 (1983).
[CrossRef]

J. F. Nye and J. V. Hajnal, Proc. R. Soc. London Ser. A 409, 21 (1987).
[CrossRef]

Other

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

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

J. F. Nye, in Physics of Defects, R. Balian, M. Kleman, and J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), p. 545.

G. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chaps. 1–2.

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

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Figures (3)

Fig. 1
Fig. 1

Handedness of a Lissajous state. The origin is shown by a black dot; electric vector E, by black arrows; regions of positive (negative) rotation of E are red (blue), dots of these colors show the position of the endpoint of E at equally spaced intervals of time, and the colored arrows show the direction of motion of this endpoint. (a) Lissajous trefoil generated by use of Eqs. (4) with n=2 (SHG), N1=3, and N2=2, so that mz=+. However, since E rotates in the negative, clockwise direction at every point in the optical cycle, the handedness h must be assigned as h=-1. (b) General asymmetric Lissajous state. Although in this particular example E spends equal time rotating clockwise and counterclockwise (there are equal numbers of red and blue dots), the negative area that is subtended clockwise exceeds the positive, counterclockwise area, and from Eqs. (5) the net handedness of this state is h=-1. The green line shows the principal axis that defines azimuthal orientational angle α.

Fig. 2
Fig. 2

(a) Positive index input C point, (b) output Lissajous state. In (a) and (b) figures with h=0 are shown in gray and lie on a circle of radius r=1 that separates RH and LH regions. Other color conventions are the same as those used in Fig. 1. All figures are plotted on a square grid with spacing Δx, Δy=1/2, with the origin at the center of the figure. The purple squares are positive, counterclockwise paths. Azimuthal singularities enclosed by a path have a positive (negative) index if the axes (green lines) of the figures on the path rotate in the same (opposite) direction as the path.

Fig. 3
Fig. 3

Map of azimuthal angle α coded 0 to π black to white, including contours. Positive (negative) singularities with index IC=+1/2 IC=-1/2 are shown by a black circle with a white border (white circle with a black border) and correspond to the Lissajous singularities in Fig. 2(b).

Equations (8)

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fx=-2Cd22exey, fy=Cd22ey2-ex2,
E=ReAe+f,
ex=1+x+iy, ey=i1-x+iy.
Ex=N11/2 cost+nNn1/2cosnt,
Ey=N11/2 sint-nNn1/2sinnt.
h=sign-z·E×dE/dt=
signS3ω+2S32ω,
Mxx-Myy=Mxy=0.

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