Abstract

A method for generation of an array of optical vortices nested at off-axis positions in a pseudonondiffracting background beam is proposed. The free-space propagation of a single off-axis vortex is analyzed thoroughly. Particular attention is focused on evolution of the beam intensity profile, description of the trajectory of the original off-axis vortex, and nucleation of additional pairs of vortices. Conditions that provide propagation stability are also discussed. An experimental setup for generation of a pseudo-nondiffracting off-axis vortex beam and an array of off-axis vortices is proposed.

© 2004 Optical Society of America

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  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. J. F. Nye, Natural Focusing and Fine Structure of Light (IOP, Bristol, UK, 1999).
  3. V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  4. L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  5. L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (North-Holland, Amsterdam, 1999), pp. 291–372.
  6. L. Allen, S. M. Barnet, M. J. Padgett, Optical Angular Momentum (IOP, Publishing, Bristol, UK, 2003).
  7. M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
    [CrossRef] [PubMed]
  8. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  9. Z. Bouchal, J. Courtial, “Connection of singular and nondiffracting optics,” Pure Appl. Opt. 6, S1–S5 (2004).
  10. Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
    [CrossRef]
  11. I. D. Maleev, G. Schwartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
    [CrossRef]
  12. Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
    [CrossRef]
  13. Z. Bouchal, J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004).
    [CrossRef]
  14. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  15. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  16. U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
    [CrossRef]
  17. Z. Bouchal, R. Čelechovský, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic, are preparing a manuscript to be called “Mixed vortex states of light as information carriers.”

2004 (2)

Z. Bouchal, J. Courtial, “Connection of singular and nondiffracting optics,” Pure Appl. Opt. 6, S1–S5 (2004).

Z. Bouchal, J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004).
[CrossRef]

2003 (1)

2002 (3)

Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
[CrossRef]

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
[CrossRef]

1996 (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

1993 (1)

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992 (2)

V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1987 (1)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1974 (1)

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

Allen, L.

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (North-Holland, Amsterdam, 1999), pp. 291–372.

L. Allen, S. M. Barnet, M. J. Padgett, Optical Angular Momentum (IOP, Publishing, Bristol, UK, 2003).

Babiker, M.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (North-Holland, Amsterdam, 1999), pp. 291–372.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Barnet, S. M.

L. Allen, S. M. Barnet, M. J. Padgett, Optical Angular Momentum (IOP, Publishing, Bristol, UK, 2003).

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Berry, M. V.

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

Bouchal, Z.

Z. Bouchal, J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004).
[CrossRef]

Z. Bouchal, J. Courtial, “Connection of singular and nondiffracting optics,” Pure Appl. Opt. 6, S1–S5 (2004).

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

Z. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett. 27, 1376–1378 (2002).
[CrossRef]

Z. Bouchal, R. Čelechovský, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic, are preparing a manuscript to be called “Mixed vortex states of light as information carriers.”

Celechovský, R.

Z. Bouchal, R. Čelechovský, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic, are preparing a manuscript to be called “Mixed vortex states of light as information carriers.”

Courtial, J.

Z. Bouchal, J. Courtial, “Connection of singular and nondiffracting optics,” Pure Appl. Opt. 6, S1–S5 (2004).

Enger, J.

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Friese, M. E.

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Heckenberg, N. R.

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Kyvalský, J.

Z. Bouchal, J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004).
[CrossRef]

Maier, M.

U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
[CrossRef]

Maleev, I. D.

Nye, J. F.

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

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

Padgett, M. J.

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (North-Holland, Amsterdam, 1999), pp. 291–372.

L. Allen, S. M. Barnet, M. J. Padgett, Optical Angular Momentum (IOP, Publishing, Bristol, UK, 2003).

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schwartzlander, G.

Schwarz, U. T.

U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
[CrossRef]

Sogomonian, S.

U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
[CrossRef]

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Spreeuw, R. C. J.

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (4)

V. Y. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Z. Bouchal, J. Kyvalský, “Controllable 3D spatial localization of light fields synthesized by non-diffracting modes,” J. Mod. Opt. 51, 157–176 (2004).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

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

Opt. Commun. (3)

Z. Bouchal, “Resistance of nondiffracting vortex beam against amplitude and phase perturbations,” Opt. Commun. 210, 155–164 (2002).
[CrossRef]

U. T. Schwarz, S. Sogomonian, M. Maier, “Propagation dynamics of phase dislocations embedded in a Bessel light beam,” Opt. Commun. 208, 255–262 (2002).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

M. E. Friese, J. Enger, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Optical angular momentum transfer to trapped absorbing particles,” Phys. Rev. A 54, 1593–1596 (1996).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. C. J. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

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

Pure Appl. Opt. (1)

Z. Bouchal, J. Courtial, “Connection of singular and nondiffracting optics,” Pure Appl. Opt. 6, S1–S5 (2004).

Other (4)

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

L. Allen, M. J. Padgett, M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (North-Holland, Amsterdam, 1999), pp. 291–372.

L. Allen, S. M. Barnet, M. J. Padgett, Optical Angular Momentum (IOP, Publishing, Bristol, UK, 2003).

Z. Bouchal, R. Čelechovský, Palacký University, 17. listopadu 50, 772 07 Olomouc, Czech Republic, are preparing a manuscript to be called “Mixed vortex states of light as information carriers.”

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Figures (9)

Fig. 1
Fig. 1

Pseudo-nondiffracting array of 4×4 mutually incoherent vortices (a) at the initial plane z=0 and (b) at the plane z=L.

Fig. 2
Fig. 2

Graphic illustration of the positions of (a) the original vortex and (b) a nucleating pair of vortices. The positions of the vortices are given by intersections of the ring and the straight line given by Eqs. (29) and (30). The intersection denoted by the open circle in (a) is not accepted as a vortex point.

Fig. 3
Fig. 3

Positions of the original vortex at the transverse plane for typical propagation distances (solid circles). The vortex is placed at the x axis at the initial plane z=0 and then moves far from the x axis, reaches maximal distance, and returns to the x axis.

Fig. 4
Fig. 4

Change of the transverse coordinates of the original vortex as a function of propagation distance. At the plane z=0, the vortex coordinates are Δx and Δy=0.

Fig. 5
Fig. 5

Three-dimensional illustration of vortex trajectories. Open circles, original vortex; solid circles, pairs of nucleating vortices. The original vortex is shifted along the x axis at the initial plane.

Fig. 6
Fig. 6

Stable propagation of the original off-axis vortex (positions denoted by crosses). The figures illustrate the transverse intensity spot, the phase, and the interference of the vortex with the reference plane wave (a)–(c) at the plane z=0 and (d)–(f) at the plane z=L.

Fig. 7
Fig. 7

Same as Fig. 6 but for nonstable vortex propagation. The off-axis shift of the original vortex fulfills condition (45) required for nucleation of the additional pair of vortices.

Fig. 8
Fig. 8

Experimental implementation for generation of a pseudo-nondiffracting array of off-axis vortices.

Fig. 9
Fig. 9

Experimental realization of a single off-axis vortex (shift along x axis) for parameters m=1, α=15.5, λ=633 nm, w0=2 mm; intensity spots at propagation distances are (a) z=L/6 and (b) z=L/3.

Equations (52)

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u(r, t)=u002πA(ψ)exp[i(ωt-kr)]dψ,
A(ψ)=t(ψ)k=1Kl=1LT(Δxk, Δyl, ψ),
T(Δxk, Δyl, ψ)=Tk,lexp[iα(Δxkcos ψ+Δylsin ψ)],
u(r, t)=2πu0i-mk=1Kl=1LTk,lJm(αr¯k,l)×exp[i(mφ¯k,l+ωt-βz)],
r¯k,l=[(x-Δxk)2+(y-Δyl)2]1/2,
tan φ¯k,l=y-Δylx-Δxk,
u(r, φ, z)=K(r, z)exp[i(ωt-kz)]×02πA(ψ)exp[-iαrQ(z)cos(ψ-φ)]dψ,
K(r, z)=w0wexpi α2z2k-Fr2+α2z2k2+iΦ,
Q(z)=1-i 2zF(z)k,
F(z)=1w2+i k2Rg,
Φ(z)=arctanzq0cos θ,
Rg(z)=zcos θ+q02cos θz,
w(z)=w01+zq0cos θ21/2,
q0=πw02λ,
u(r, t)=U(r, t)k=1Kl=1LTk,lJm(αQr¯k,l)exp(imφ¯k,l),
U=im2πK(r, z)exp[i(ωt-kz)],
r¯k,l=[(x-Δxk/Q)2+(y-Δyl/Q)2]1/2,
tan φ¯k,l=(y-Δyl/Q)(x-Δxk/Q)-1.
Re(αQr¯)=qs,s=1, 2,,
Im(αQr¯)=0.
QRr¯R-QIr¯I=qs/α,
QIr¯R+QRr¯I=0.
RR=X2+Y2-QI|Q|22(Δx2+Δy2),
RI=2 QI|Q|2 (XΔx+YΔy),
X=x-Δx QR|Q|2,
Y=y-Δy QR|Q|2.
r¯R2=qsQRα|Q|22,
r¯I2=qsQIα|Q|22.
XΔx+YΔy=ys,s=1, 2,,
X2+Y2=ρs2,
ys=-qsα|Q|2QR,
ρs2=QI|Q|22(Δx2+Δy2)+qsα|Q|22(QR2-QI2).
φ¯R=12arctan(VI/VR),
φ¯I=-14ln(VR2+VI2),
VR=1-(ϕR2+ϕI2)ϕR2+(1+ϕI)2,
VI=2ϕRϕR2+(1+ϕI)2,
ϕR=XY+QI|Q|22ΔxΔyX2+QI|Q|22Δx2,
ϕI=QI|Q|2 (XΔy-YΔx)X2+QI|Q|22Δx2.
X1,2=±QI|Q|2 Δy,
Y1,2=±QI|Q|2 Δx.
x1=QR(z)|Q|2(z) Δx,
y1=-QI(z)|Q|2(z) Δx.
X1,2=1Δx (ys-ΔyY1,2),
Y1,2=2ysΔy±D2(Δx2+Δy2),
D=4[ys2Δy2-(Δx2+Δy2)(ys2-Δx2ρs2)].
ys2<(Δx2+Δy2)ρs2.
qs<α(Δx2+Δy2)1/2.
q2>α(Δx2+Δy2)1/2,
T=exp[-iR0(Kxcos ψ+Kysin ψ)+Kzzc],
Tk,l=exp-ikzcfCD,
Δxk=-fFD xk,
Δyl=-fFD yl,

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