Abstract

The transition of the vortex structure of fractional elegant Laguerre–Gaussian beams is discussed in detail as the angular mode index of the beam is continuously varied between integer values. Under this kind of variation, the vortices can be classified into five groups. Contrary to the behavior of the vortices of the nondiffracting Bessel beams of fractional order, the nodal lines of the vortices in the case of the fractional eLG beams exhibit intricate shapes.

© 2013 Optical Society of America

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References

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  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  2. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
    [CrossRef]
  3. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
    [CrossRef]
  4. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [CrossRef]
  5. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre–Gaussian modes,” Opt. Express 15, 6300–6313 (2007).
    [CrossRef]
  6. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
    [CrossRef]
  7. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
    [CrossRef]
  8. C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16, 11411–11422 (2008).
    [CrossRef]
  9. S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
    [CrossRef]
  10. I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
    [CrossRef]
  11. T. Fadeyeva, C. Alexeyev, A. Rubass, and A. Volyar, “Vector erf-Gaussian beams: fractional optical vortices and asymmetric TE and TM modes,” Opt. Lett. 37, 1397–1399 (2012).
    [CrossRef]
  12. J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
    [CrossRef]
  13. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
    [CrossRef]

2012 (1)

2009 (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

2008 (3)

2007 (1)

2005 (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

2004 (3)

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

2003 (1)

2002 (1)

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Alexeyev, C.

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

Bekshaev, A. Ya.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[CrossRef]

Burnham, D.

Courtial, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Fadeyeva, T.

Flossmann, F.

Franke-Arnold, S.

Götte, J. B.

Gutiérrez-Vega, J. C.

Leach, J.

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

López-Mariscal, C.

C. López-Mariscal, D. Burnham, D. Rudd, D. McGloin, and J. C. Gutiérrez-Vega, “Phase dynamics of continuous topological upconversion in vortex beams,” Opt. Express 16, 11411–11422 (2008).
[CrossRef]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[CrossRef]

McGloin, D.

O’Holleran, K.

Orlov, S.

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Pas’ko, V. A.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

Preece, D.

Regelskis, K.

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Rubass, A.

Rudd, D.

Slyusar, V. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

Smilgevicius, V.

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Soskin, M. S.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Stabinis, A.

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Vasnetsov, M. V.

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Volyar, A.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

J. Opt. A (3)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[CrossRef]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[CrossRef]

I. V. Basistiy, V. A. Pas’ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A 6, S166 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (2)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Opt. Commun. (1)

S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Prog. Opt. (2)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Transverse amplitude and phase distributions of the fractional beam Uη,λ(r) at the plane z=0 for the angular indices λ={2,2.25,2.5,2.75,3} and the radial indices (a), (b) η=1; (c), (d) η=2. Transverse dimensions ±4w0.

Fig. 2.
Fig. 2.

Phase map for beams with η=0 in 0.1λ2.9 at plane z=0. Red lines are equally spaced contour phase lines. Black and white dots are vortices with topological negative and positive charge, respectively; green dots are saddle points. Transverse dimensions ±4w0.

Fig. 3.
Fig. 3.

Migration of vortices of the beam U1,λ(x,y,0) as the angular order increases from 0 to 3. Red and black lines represent positive and negative vortices, respectively. Dotted circles are the rings of phase dislocation when λ becomes integer. Lines with dotted ends indicate that vortices come from or go to infinity. Transverse dimensions ±4w0.

Fig. 4.
Fig. 4.

Unfolding and refolding of higher-order axial vortices during the transition between consecutive integer angular mode numbers. Numbers represent the order λ. Transverse dimensions ±0.8w0.

Fig. 5.
Fig. 5.

Vertical displacement of the vortices along the y axis as the angular order increases for the beams with radial indices η=0,1, and 2. Red and black lines represent positive and negative helicity of the vortices, respectively.

Fig. 6.
Fig. 6.

Behavior of the phase map of the fractional beam U1,λ(r) for λ=1.3 and 1.5 within the range 0z/zR1. Red and blue lines correspond to zero contour lines ReU=0 and ImU=0, respectively. White and black dots represent positive and negative vortices. Transverse dimensions ±3w0.

Fig. 7.
Fig. 7.

Evolution of vortex nodal lines of the eLG beam with indices η=1 and λ=1.5 in the waist region |z|zR. Red arrows show the helicity of the vortex lines at the plane z=0.

Equations (6)

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

U0,0(r)=2π1μw0exp(r2μw02),μμ(z)=1+izzR,
A±w02(x±iy)=w02exp(±iθ)(r±i1rθ),
Uη,λ(r)=cη,λ(A+)η+λ(A)ηU0,0,
Uη,λ(r)=(i)λl=(1)lsinc(lλ)Uη,l(r),
Uη,l(r)=i2η+|l|Γ(η+|l|+1)Γ(|l|+1)(2μw0)2η+|l|R|l|U0,0(r)F11(η,|l|+1;R2)exp(ilθ),
Rrμw0,ηη+λ|l|2.

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