Abstract

We propose and experimentally implement a method for the generation of a wide class of partially spatially coherent vortex beams whose cross-spectral density has a separable functional form in polar coordinates. We study phase singularities of the spectral degree of coherence of the new beams.

© 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. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
    [CrossRef]
  3. L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 34, p. 291.
  4. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
    [CrossRef]
  5. S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001).
    [CrossRef]
  6. G. S. Agarwal, J. Opt. Soc. Am. A 16, 2619 (1999).
    [CrossRef]
  7. As is customary, by singular beams we mean beams with phase singularities.
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 4.3.2.
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.
  10. We define ηs to be equal to unity for nonnegative values of its argument and to zero otherwise.
  11. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
    [CrossRef]
  12. I. Gradstein and I. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, New York, 1980).
  13. The power of the source Pn is equal to Pn≡λnm∫d2ρψnmρ,02.
  14. Th. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
    [CrossRef]
  15. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.9.
    [CrossRef]

2001

1999

G. S. Agarwal, J. Opt. Soc. Am. A 16, 2619 (1999).
[CrossRef]

1998

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

1992

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
[CrossRef]

1974

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

1804

Th. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, J. Opt. Soc. Am. A 16, 2619 (1999).
[CrossRef]

Allen, L.

L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 34, p. 291.

Babikier, M.

L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 34, p. 291.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
[CrossRef]

Berry, M. V.

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

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.9.
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

Gradstein, I.

I. Gradstein and I. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, New York, 1980).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 4.3.2.
[CrossRef]

Nye, J. F.

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

Padgett, M. J.

L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 34, p. 291.

Ponomarenko, S. A.

Ryzhik, I.

I. Gradstein and I. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, New York, 1980).

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

Soskin, M. S.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

Vasnetsov, M. V.

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 4.3.2.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.9.
[CrossRef]

Young, Th.

Th. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[CrossRef]

J. Mod. Opt.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, J. Mod. Opt. 45, 539 (1998).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, J. Mod. Opt. 39, 985 (1992).
[CrossRef]

J. Opt. Soc. Am. A

S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001).
[CrossRef]

G. S. Agarwal, J. Opt. Soc. Am. A 16, 2619 (1999).
[CrossRef]

Philos. Trans. R. Soc. London

Th. Young, Philos. Trans. R. Soc. London 94, 1 (1804).
[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. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 219.
[CrossRef]

L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 34, p. 291.

I. Gradstein and I. Ryzhik, Tables of Integrals, Series and Products, 5th ed. (Academic, New York, 1980).

The power of the source Pn is equal to Pn≡λnm∫d2ρψnmρ,02.

As is customary, by singular beams we mean beams with phase singularities.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 4.3.2.
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 16.

We define ηs to be equal to unity for nonnegative values of its argument and to zero otherwise.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999), Sec. 8.9.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Properties of partially coherent singular beams composed of LG01 and LG11 modes: (a) contours of constant phase of the spectral degree of coherence at a pair of points with coordinates ρ,ϕ,z and 0.1 wz,0,z in the plane z=1.5zd, (b) modulus of the spectral degree of coherence μ at the same pair of points as a function of the dimensionless radial variable ρ/wz. The ratio of the powers of the two modes is taken to be the same as in the experiment, P1/P0=0.45.

Fig. 2
Fig. 2

Experimental arrangement for generating and testing partially coherent singular beams with a separable phase: L, laser; BSs, beam splitters, Ms, mirrors; CSH-01 and CSH-11, computer-synthesized holograms producing LG01 and LG11 modes, respectively; SSs, spatial selectors of diffraction orders; OS, opaque screen; OP, observation plane. Inset: Young interference fringes behind the opaque strip illuminated by (a) vortex-free LG00 mode and (b) LG01 mode.

Fig. 3
Fig. 3

Experimental results: interference pattern behind the opaque strip illuminated by (a) LG01 mode, (b) LG11 mode, and (c) incoherent superposition of the LG01 and LG11 modes. The distance between the strip and the observation plane is 470 mm; the width of strip d is 1 mm, and the width of the beam in the plane of the strip, wz1.62 mm.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Wρ,ρ,z,ω=n=0Nλnmψnm*ρ,z,ωψnmρ,z,ω,
ψnmρ,z,ωw0wzρwzmLnm2ρ2wz2×exp-iΦρ,zexp-ρ2/wz2.
Φρ,z=mϕ-kz+Q arctanzzd-kρ22Rz,
μρ,ρ,z,ωWρ,ρ,z,ωIρ,z,ωIρ,z,ω.
μρi,ρ0,z,ω=0.
Ψρ,z=ϕ-ϕ0-kρ2-ρ022Rz+πiηρ-ρiz,

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