Abstract

Spatial correlation vortex dipoles may form in the four-dimensional mutual coherence function when a partially coherent light source contains an optical vortex. Analytical and numerical investigations are made in near- and far-field regimes.

© 2004 Optical Society of America

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References

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  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. G. A. Swartzlander, Jr., “Optical vortex filaments,” in Optical Vortices, M. Vasnetsov and K. Staliunas, eds., Vol. 228 of Horizons in World Physics (Nova Science, Huntington, N.Y., 1999).
  3. A. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
  5. Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
    [CrossRef]
  6. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
    [CrossRef]
  7. G. A. Swartzlander, Jr., “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
    [CrossRef]
  8. D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
    [CrossRef] [PubMed]
  9. H. Gross, “Numerical propagation of partially coherent laser beams through optical systems,” Opt. Laser Technol. 29, 257–260 (1997).
    [CrossRef]
  10. H. Schouten, G. Gbur, T. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [CrossRef] [PubMed]
  11. G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  12. G. Bogatyryova, C. Fel’de, P. Polyanskii, S. Ponomarenko, M. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
    [CrossRef] [PubMed]
  13. I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
    [CrossRef]
  14. D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
    [CrossRef] [PubMed]
  15. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
    [CrossRef]
  16. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical votex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
    [CrossRef]
  17. G. A. Swartzlander, Jr., “Optical vortex solitons,” in Spatial Solitons, S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001), p. 299.
  18. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C ++ (Cambridge U. Press, Cambridge, 2002), p. 501.
  19. The value of χ may diverge as the intensity vanishes. To prevent this problem we truncate the value of χ when the intensity falls below 2% of the maximum value.

2004

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

2003

2002

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[CrossRef]

D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

2001

1998

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical votex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
[CrossRef]

1997

H. Gross, “Numerical propagation of partially coherent laser beams through optical systems,” Opt. Laser Technol. 29, 257–260 (1997).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

1974

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Bogatyryova, G.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Bouchal, Z.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[CrossRef]

Fel’de, C.

Gbur, G.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Gross, H.

H. Gross, “Numerical propagation of partially coherent laser beams through optical systems,” Opt. Laser Technol. 29, 257–260 (1997).
[CrossRef]

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
[CrossRef]

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Palacios, D.

D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

Perina, J.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[CrossRef]

Polyanskii, P.

Ponomarenko, S.

Rozas, D.

D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical votex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Sacks, Z. S.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical votex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Schouten, H.

Soskin, M.

Swartzlander Jr., G. A.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

I. D. Maleev and G. A. Swartzlander, Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
[CrossRef]

D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

G. A. Swartzlander, Jr., “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
[CrossRef]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, Jr., “Holographic formation of optical votex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
[CrossRef]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Visser, T.

Wolf, E.

J. Mod. Opt.

Z. Bouchal and J. Perina, “Non-diffracting beams with controlled spatial coherence,” J. Mod. Opt. 49, 1673–1689 (2002).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

Opt. Laser Technol.

H. Gross, “Numerical propagation of partially coherent laser beams through optical systems,” Opt. Laser Technol. 29, 257–260 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. Palacios, D. Rozas, and G. A. Swartzlander, Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. 88, 103902 (2002).
[CrossRef] [PubMed]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, Jr., “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[CrossRef] [PubMed]

D. Rozas, Z. S. Sacks, and G. A. Swartzlander, Jr., “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[CrossRef]

Proc. R. Soc. London, Ser. A

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other

G. A. Swartzlander, Jr., “Optical vortex filaments,” in Optical Vortices, M. Vasnetsov and K. Staliunas, eds., Vol. 228 of Horizons in World Physics (Nova Science, Huntington, N.Y., 1999).

A. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

G. A. Swartzlander, Jr., “Optical vortex solitons,” in Spatial Solitons, S. Trillo and W. Torruellas, eds. (Springer-Verlag, Berlin, 2001), p. 299.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C ++ (Cambridge U. Press, Cambridge, 2002), p. 501.

The value of χ may diverge as the intensity vanishes. To prevent this problem we truncate the value of χ when the intensity falls below 2% of the maximum value.

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Figures (6)

Fig. 1
Fig. 1

Initial (a) intensity and (b) phase profiles of an electromagnetic field vortex of topological charge m=1. Vector s shows the displacement of the vortex core from the centroid of the beam envelope. Position vectors r and r, respectively, make angles ϕ and ϕ with the x axis and are related: r=r+s. When m=1 the vortex phase is equivalent to angle ϕ.

Fig. 2
Fig. 2

Notation used to describe the mutual coherence function in the (a) initial and (b) far-field planes. Unprimed (primed) variables are measured from the beam envelope (vortex) center. FT, Fourier transform.

Fig. 3
Fig. 3

Numerically calculated far-field distributions of intensity I(k) and normalized cross correlation χ(k) for an initial on-axis vortex (s=0) of topological charge m=1. Intensity profiles (a)–(c) depict an increasingly distinct vortex core as the relative coherence length, σc=lc/w0, increases from low (σc=1) to medium (σc=2) to high (σc=5) coherence. The corresponding magnitudes (d)–(f) and phases (g)–(i) of χ(k) demonstrate the formation of a ring dislocation of diameter 2kV that persists for arbitrary values of σc. When s=0, χ(k) is real and positive (negative) inside (outside) the dislocation ring. Line plots of values along the kx and ky axes are overlaid on the corresponding images as an aid to the eye.

Fig. 4
Fig. 4

Numerically calculated far-field distributions of intensity I(k) and normalized cross correlation χ(k) for an initial off-axis vortex (sx0, sy=0) of topological charge m=1 and medium coherence (σc=2). Intensity profiles depict a diffuse core along the -ky axis for (a) sx/w0=0.2 and (b) sx/w0=0.4. The magnitudes (c) and (d) and the phases (e) and (f) of χ(k) reveal a pair of oppositely charged spatial correlation vortices separated by 2kV and that have opposite topological charges.

Fig. 5
Fig. 5

Same as Figs. 4(a), 4(c), and 4(e) except for the low- (σc=1) and high- (σc=4) coherence cases.

Fig. 6
Fig. 6

Plots of radial distance between spatial correlation vortices, kV, as a function of initial electromagnetic vortex displacement s for three values of initial relative coherence length σc=lc/w0. Values along both axes are normalized by the initial beam size, w0. Numerically calculated values are shown as data marks, and second-order polynomial fits are included to aid the eye.

Equations (11)

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E(r, ϕ, t)=A(r)g(r)exp(imϕ)exp(iωt)exp(iβ),
Γ(r1, r2)=E(r1, t)E*(r2, t),
Γ(r1, r2)=C(r1-r2)A(r1)A(r2)g(r1)g(r2)×exp[im(ϕ1-ϕ2)],
I(r)=Γ(r, r)=A2(|r-s|)g2(r),
χ(r)=Γ(r, -r)/[I(r)I(-r)]1/2=C(2r)exp(imΦ),
Γ(k1, k2)=(1/λzd)2Γ(r1, r2)exp[-i(k1·r1-k2·r2)]dr1dr2,
C(r1-r2)=exp(-|r1-r2|2/lC2),
Tx=exp(-(x12+x22)/rbs2+2x1x2/lC2)×exp(ik2xx2-ik1xx1),
Ty=exp(-(y12+y22)/rbs2+2y1y2/lC2)×exp(ik2yy2-ik1yy1),
Γm=1(k1, k2)=(E0/w0λzd)2x1x2Txdx1dx2×Tydy1dy2+x1Txdx1dx2iy2Tydy1dy2+x2Txdx1dx2iy1Tydy1dy2+Txdx1dx2y1y2Tydy1dy2.
Γm=1(k1, k2)=(E0/λzdw0)2[x1x2-s(x2+x1)]Txdx1dx2Tydy1dy2+s2Txdx1dx2Tydy1dy2+Txdx1dx2[y1y2+is(y2-y1)]Tydy1dy2+x2Txdx1dx2iy1Tydy1dy2-x1Txdx1dx2iy2Tydy1dy2.

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