Abstract

We study the evolution of phase singularities and coherence singularities in a Laguerre–Gauss beam that is rendered partially coherent by letting it pass through a spatial light modulator. The original beam has an on-axis minumum of intensity—a phase singularity—that transforms into a maximum of the far-field intensity. In contrast, although the original beam has no coherence singularities, such singularities are found to develop as the beam propagates. This disappearance of one kind of singularity and the gradual appearance of another is illustrated with numerical examples.

© 2009 Optical Society of America

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References

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    [CrossRef]
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2008

2006

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428-435 (2006).
[CrossRef]

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733-5745 (2006).
[CrossRef] [PubMed]

2005

2004

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097-2102 (2004).
[CrossRef]

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]

G. A. Swartzlander, Jr., and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

2003

2002

2001

S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150-156 (2001).
[CrossRef]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

1999

1998

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Angelsky, O. V.

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

Bogatyryova, G. V.

Browne, S. L.

Dai, M.

Dayton, D. C.

Dennis, M. R.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

Fel'de, C. V.

Fischer, D. G.

Freund, I.

Gahagan, K. T.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 45, E.Wolf, ed. (North Holland, 2003), pp. 119-203.
[CrossRef]

Gbur, G.

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 2000). See Chap. 8.3.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Kudryashov, A. V.

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Maleev, I. D.

I. D. Maleev and G. A. Swartzlander, Jr., “Propagation of spatial correlation vortices,” J. Opt. Soc. Am. B 25, 915-922 (2008).
[CrossRef]

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]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambrige, 1995).

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]

Mokhun, A. I.

Mokhun, I. I.

Nye, J. F.

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

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]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Polyanskii, P. V.

Ponomarenko, S. A.

Sandven, S. P.

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Schmit, J.

G. A. Swartzlander, Jr., and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

Schoonover, R. W.

T. D. Visser and R. W. Schoonover, “A cascade of singular field patterns in Young's interference experiment,” Opt. Commun. 281, 1-6 (2008).
[CrossRef]

R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14, 5733-5745 (2006).
[CrossRef] [PubMed]

Schouten, H. F.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Soskin, M. S.

Swartzlander, G. A.

I. D. Maleev and G. A. Swartzlander, Jr., “Propagation of spatial correlation vortices,” J. Opt. Soc. Am. B 25, 915-922 (2008).
[CrossRef]

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]

G. A. Swartzlander, Jr., and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

K. T. Gahagan and G. A. Swartzlander, Jr., “Simultaneous trapping of low-index and high-index microparticles observed with an optical-vortex trap,” J. Opt. Soc. Am. B 16, 533-537 (1999).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Tyson, R. K.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219-275.
[CrossRef]

Visser, T. D.

Wang, Z.

Wolf, E.

Yin, J.

Z. Wang, M. Dai, and J. Yin, “Atomic (or molecular) guiding using a blue-detuned doughnut mode in a hollow metallic waveguide,” Opt. Express 13, 8406-8423 (2005).
[CrossRef] [PubMed]

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 45, E.Wolf, ed. (North Holland, 2003), pp. 119-203.
[CrossRef]

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 45, E.Wolf, ed. (North Holland, 2003), pp. 119-203.
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428-435 (2006).
[CrossRef]

T. D. Visser and R. W. Schoonover, “A cascade of singular field patterns in Young's interference experiment,” Opt. Commun. 281, 1-6 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

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]

G. A. Swartzlander, Jr., and J. Schmit, “Temporal correlation vortices and topological dispersion,” Phys. Rev. Lett. 93, 093901 (2004).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London, Ser. A 457, 141-155 (2001).
[CrossRef]

Other

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 2000). See Chap. 8.3.

A. E. Siegman, Lasers (University Science Books, 1986).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambrige, 1995).

J. Yin, W. Gao, and Y. Zhu, “Generation of dark hollow beams and their applications,” in Progress in Optics, Vol. 45, E.Wolf, ed. (North Holland, 2003), pp. 119-203.
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, 1999).

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier, Amsterdam, 2001), pp. 219-275.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Scaled on-axis spectral density z 2 S ( ρ = 0 , z , ω ) , calculated from Eq. (20), normalized by the radiant intensity in the forward direction J [ s = ( 0 , 0 , 1 ) , ω ] . In this example σ S = 15 λ and σ μ = 4 λ .

Fig. 3
Fig. 3

Normalized spectral density S ( ρ , z , ω ) , calculated from Eq. (17), in several cross sections of the beam. In this example σ S = 15 λ and σ μ = 4 λ .

Fig. 4
Fig. 4

Evolution of the modulus of the spectral degree of coherence, as calculated from Eq. (17), along the two directions of observation at which a coherence singularity occurs in the far field. In this example σ S = 15 λ and σ μ = 4 λ .

Fig. 5
Fig. 5

(a) Position of two far-zone observation points (dots) on a circle centered around the z axis. (b) The spectral degree of coherence of the field at the two points as a function of the angle ϕ for three values of θ. In this example σ S = 15 λ and σ μ = 4 λ .

Equations (24)

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U ( inc ) ( ρ , ω ) = A exp ( i ϕ ) ρ exp ( ρ 2 4 σ S 2 ) ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = U ( 0 ) * ( ρ 1 , ω ) U ( 0 ) ( ρ 2 , ω ) ,
μ ( 0 ) ( ρ 1 , ρ 2 , ω ) = W ( 0 ) ( ρ 1 , ρ 2 , ω ) S ( 0 ) ( ρ 1 , ω ) S ( 0 ) ( ρ 2 , ω ) ,
S ( 0 ) ( ρ , ω ) = W ( 0 ) ( ρ , ρ , ω ) = A 2 ρ 2 exp ( ρ 2 2 σ S 2 ) ,
μ ( 0 ) ( ρ 1 , ρ 2 , ω ) = μ ( 0 ) ( ρ 2 ρ 1 , ω ) = exp [ ( ρ 2 ρ 1 ) 2 2 σ μ 2 ] ,
W ( 0 ) ( ρ 1 , ρ 2 , ω ) = A 2 ρ 1 ρ 2 exp [ ( ρ 1 2 + ρ 2 2 ) 4 σ S 2 ] exp [ ( ρ 2 ρ 1 ) 2 2 σ μ 2 ] .
J ( s , ω ) = ( 2 π k ) 2 S ̃ ( 0 ) ( 0 , ω ) μ ̃ ( 0 ) ( k s , ω ) cos 2 θ ,
μ ( ) ( r 1 s 1 , r 2 s 2 , ω ) = S ̃ ( 0 ) [ k ( s 2 s 1 ) , ω ] exp [ i k ( r 2 r 1 ) ] S ̃ ( 0 ) ( 0 , ω ) ,
S ̃ ( 0 ) ( f , ω ) = 1 ( 2 π ) 2 S ( 0 ) ( ρ , ω ) e i f ρ d 2 ρ ,
μ ̃ ( 0 ) ( f , ω ) = 1 ( 2 π ) 2 μ ( 0 ) ( ρ , ω ) e i f ρ d 2 ρ .
S ̃ ( 0 ) ( f , ω ) = ( 2 f 2 σ S 2 ) σ S 4 A 2 exp ( f 2 σ S 2 2 ) 2 π ,
μ ̃ ( 0 ) ( f , ω ) = σ μ 2 exp ( f 2 σ μ 2 2 ) 2 π .
J ( s , ω ) = 2 k 2 σ S 4 σ μ 2 A 2 cos 2 θ exp ( k 2 σ μ 2 sin 2 θ 2 ) ,
μ ( ) ( r s 1 , r s 2 , ω ) = [ 1 2 k 2 σ S 2 sin 2 θ ] exp ( 2 k 2 σ S 2 sin 2 θ ) .
sin θ CS = ( 2 k 2 σ S 2 ) 1 2 .
k 2 σ μ 2 2 1 .
W ( ρ 1 , ρ 2 , z , ω ) = ( z = 0 ) W ( 0 ) ( ρ 1 , ρ 2 , ω ) G * ( ρ 1 , ρ 1 , z , ω ) G ( ρ 2 , ρ 2 , z , ω ) d 2 ρ 1 d 2 ρ 2 ,
G ( ρ , ρ , z , ω ) = i k 2 π z exp ( i k z ) exp [ i k ( ρ ρ ) 2 2 z ] .
S ( ρ = 0 , z , ω ) = W ( ρ 1 = 0 , ρ 2 = 0 , z , ω )
= ( k A z ) 2 0 0 ρ 1 2 ρ 2 2 exp ( ρ 1 2 2 σ + 2 ) exp ( ρ 2 2 2 σ 2 ) × I 0 ( ρ 1 ρ 2 σ μ 2 ) d ρ 1 d ρ 2 ,
1 2 σ ± 2 = 1 4 σ S 2 + 1 2 σ μ 2 ± i k 2 z .
s 1 = ( sin θ , 0 ) ,
s 2 = ( sin θ cos ϕ , sin θ sin ϕ ) ,
μ ( ) ( r s 1 , r s 2 , ω ) = [ 1 2 k 2 σ S 2 sin 2 θ sin 2 ( ϕ 2 ) ] × exp [ 2 k 2 σ S 2 sin 2 θ sin 2 ( ϕ 2 ) ] .

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