Abstract

The propagation of Riemann–Silberstein (RS) vortices for Gaussian vortex beams with topological charges m=+1 through a lens is studied. It is shown that if there is an ideal lens, a RS vortex and a circular edge dislocation appear for Gaussian on-axis vortex beams, while only RS vortices take place for Gaussian off-axis vortex beams. In the presence of an astigmatic lens, there exist RS vortices but no edge dislocations for both Gaussian on-axis and off-axis beams. By varying the astigmatic coefficient, the off-axis parameter, and the propagation distance, the motion, creation, and annihilation of vortices may take place, and in the process, the total topological charge of RS vortices remains unchanged.

© 2012 Optical Society of America

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    [CrossRef]
  2. F. S. Roux, “Optical vortex density limitation,” J. Opt. Soc. Am. B 20, 1575–1580 (2003).
    [CrossRef]
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
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    [CrossRef]
  6. F. Wang, S. Zhu, and Y. Cai, “Experimental study of the focusing properties of a Gaussian Schell-model vortex beam,” Opt. Lett. 36, 3281–3283 (2011).
    [CrossRef]
  7. J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. A 409, 21–36 (1987).
    [CrossRef]
  8. J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. A 430, 413–421 (1990).
    [CrossRef]
  9. I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
    [CrossRef]
  10. G. Kaiser, “Helicity, polarization and Riemann–Silberstein vortices,” J. Opt. A 6, S243 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. G. Molina-Terriza, J. Recolons, and L. Torner, “The curious arithmetic of optical vortices,” Opt. Lett. 25, 1135–1137(2000).
    [CrossRef]
  15. S. A. Collins, “Lens system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177(1970).
    [CrossRef]
  16. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series, and Products, 7th ed. (Academic, 2007).
  17. I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
    [CrossRef]

2012

X. Lian, C. Deng, and B. Lü, “Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams,” Opt. Commun. 285, 497–502 (2012).
[CrossRef]

2011

2009

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11, 075410 (2009).
[CrossRef]

2004

G. Kaiser, “Helicity, polarization and Riemann–Silberstein vortices,” J. Opt. A 6, S243 (2004).
[CrossRef]

M. V. Berry, “Riemann–Silberstein vortices for paraxial waves,” J. Opt. A 6, S175 (2004).
[CrossRef]

2003

F. S. Roux, “Optical vortex density limitation,” J. Opt. Soc. Am. B 20, 1575–1580 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
[CrossRef]

2000

1997

1994

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

1993

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

1990

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. A 430, 413–421 (1990).
[CrossRef]

1987

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. A 409, 21–36 (1987).
[CrossRef]

1970

S. A. Collins, “Lens system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177(1970).
[CrossRef]

Alda, J.

Bernabeu, E.

Berry, M. V.

M. V. Berry, “Riemann–Silberstein vortices for paraxial waves,” J. Opt. A 6, S175 (2004).
[CrossRef]

Bialynicka-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
[CrossRef]

Cai, Y.

Collins, S. A.

S. A. Collins, “Lens system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177(1970).
[CrossRef]

Deng, C.

X. Lian, C. Deng, and B. Lü, “Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams,” Opt. Commun. 285, 497–502 (2012).
[CrossRef]

Freund, I.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series, and Products, 7th ed. (Academic, 2007).

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).

Hajnal, J. V.

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. A 430, 413–421 (1990).
[CrossRef]

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. A 409, 21–36 (1987).
[CrossRef]

Indebetouw, G.

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

Kaiser, G.

G. Kaiser, “Helicity, polarization and Riemann–Silberstein vortices,” J. Opt. A 6, S243 (2004).
[CrossRef]

Li, J.

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11, 075410 (2009).
[CrossRef]

Lian, X.

X. Lian, C. Deng, and B. Lü, “Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams,” Opt. Commun. 285, 497–502 (2012).
[CrossRef]

Lü, B.

X. Lian, C. Deng, and B. Lü, “Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams,” Opt. Commun. 285, 497–502 (2012).
[CrossRef]

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11, 075410 (2009).
[CrossRef]

Molina-Terriza, G.

Nye, J. F.

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. A 409, 21–36 (1987).
[CrossRef]

Recolons, J.

Roux, F. S.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series, and Products, 7th ed. (Academic, 2007).

Shvartsman, N.

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (North-Holland, 2001), pp. 219–276.

Torner, L.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (North-Holland, 2001), pp. 219–276.

Wang, F.

Zhu, S.

J. Mod. Opt.

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

J. Opt. A

J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11, 075410 (2009).
[CrossRef]

G. Kaiser, “Helicity, polarization and Riemann–Silberstein vortices,” J. Opt. A 6, S243 (2004).
[CrossRef]

M. V. Berry, “Riemann–Silberstein vortices for paraxial waves,” J. Opt. A 6, S175 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Alda and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737–2747 (1997).
[CrossRef]

S. A. Collins, “Lens system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177(1970).
[CrossRef]

J. Opt. Soc. Am. B

Nature

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).

Opt. Commun.

X. Lian, C. Deng, and B. Lü, “Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams,” Opt. Commun. 285, 497–502 (2012).
[CrossRef]

Opt. Lett.

Phys. Rev. A

I. Bialynicki-Birula and Z. Bialynicka-Birula, “Vortex lines of the electromagnetic field,” Phys. Rev. A 67, 062114 (2003).
[CrossRef]

I. Freund and N. Shvartsman, “Wave-field phase singularities: the sign principle,” Phys. Rev. A 50, 5164–5172 (1994).
[CrossRef]

Proc. R. Soc. A

J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. A 409, 21–36 (1987).
[CrossRef]

J. V. Hajnal, “Observation of singularities in the electric and magnetic fields of freely propagating microwaves,” Proc. R. Soc. A 430, 413–421 (1990).
[CrossRef]

Other

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” in Progress in Optics, Vol. XLII, E. Wolf, ed. (North-Holland, 2001), pp. 219–276.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals Series, and Products, 7th ed. (Academic, 2007).

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Figures (6)

Fig. 1.
Fig. 1.

Contour lines of phase of Ψ(x,y,z) at the geometrical focal plane for different values of the astigmatic coefficient, when an on-axis vortex Gaussian beam passes through an astigmatic lens. (a) C=0, (b) C=0.1×103m1, (c) C=0.25×103m1, and (d) C=0.297×103m1.

Fig. 2.
Fig. 2.

Contour lines of phase of Ψ(x,y,z) at the geometrical focal plane, when an off-axis vortex Gaussian beam passes through an astigmatic lens. (a) C=0, (b) C=0.035×103m1, (c) C=0.1×103m1, and (d) C=0.522×103m1.

Fig. 3.
Fig. 3.

Contour lines of phase of Ψ(x,y,z) at the geometrical focal plane for different values of the off-axis parameter. (a) a=0, (b) a=0.2mm, (c) a=0.265mm, and (d) a=0.625mm.

Fig. 4.
Fig. 4.

Contour lines of phase of Ψ(x,y,z) for different values of the propagation distance, when an on-axis Gaussian beam passes through an ideal lens. (a) z=0, (b) z=1000mm, (c) z=199.1mm, and (d) z=4000mm.

Fig. 5.
Fig. 5.

Contour lines of phase of Ψ(x,y,z) for different values of the propagation distance, when an off-axis vortex Gaussian beam passes through an ideal lens. (a) z=0mm, (b) z=200mm, (c) z=1000mm, and (d) z=4000mm.

Fig. 6.
Fig. 6.

Contour lines of phase of Ψ(x,y,z) for different values of the propagation distance, when an on-axis Gaussian vortex beam propagates through an astigmatic lens. (a) z=100mm, (b) z=142.9mm, (c) z=169.8mm, (d) z=200mm, (e) z=200.7mm, and (f) z=4000mm.

Equations (25)

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

Ex(x0,y0,0)=exp[x02+y02w02][(x0a)+isgn(m)y0]|m|,
Ey(x0,y0,0)=0,
Ex(x,y,z)=iλzexp[x2+y2w02]Ex(x0,y0,0)exp[ikC(x02y02)]exp{ik2z[(1zf)(x02+y02)2(xx0+yy0)]}dx0dy0,
Ey(x,y,0)=0,
+xnexp(ux2+2vx)dx=n!exp(v2u)πu(vu)nt=0|n|/21(n2t)!t!(u4v2)t,u>0,
Ex(x,y,z)=iπλzp1p2exp[ik2z(x2+y2)]exp(q124p1+q224p2)[q12p1+iq22p2+a],
Ey(x,y,0)=0,
p1=1w02ikC+ik2zik2f,p2=1w02+ikC+ik2zik2f,q1=ikxzandq2=ikyz.
ψ(x,y,z)=(E+icB)·(E*+icB*),
ψ(x,y,z)=12k2(2x22y2)(ExEx*)+ik22xy(ExEx*).
ψ(x,y,z)=gg*kz2exp(k2x22w02z2p1p1*+k2y22w02z2p2p2*)(A1+A+2iA3),
g=iπp02+k2C2λzp1p1*p2p2*,
A1=12|p1|2[hh*w02(k2x2w02z21)+ikx(p1*hp1h*)w02z+12],
A2=12|p2|2[hh*w02(k2y2w02z21)ky(p2h+p2h*)w02z+12],
A3=kw02|p1||p2|[kxyw02z2hh*+(x+iy)p1h2z(p2x+ip1*y)h*2z+C],
p0=1w02i(k2zk2f),p1=1w02i(kCk2z+k2f),p2=1w02+i(kC+k2zk2f),h=ikp1x2|p1|2z+kp2y2|p2|2z+a.
{Re[ψ(x,y,z)]=0Im[ψ(x,y,z)]=0,
{x0=0y0=0,x02+y02=w02.
x=y=0,
x2+y2=4w02z2pk2,
p=1w04+(k2zk2f)2.
rmin=2kw05f4f2+k2w041w04+(4f2+k2w042kw04fk2f)2.
{x0=0y0=0,{x0=ay0=0,{x=a+a2+4w022y0=0,{x=aa2+4w022y0=0.
{k3(x4y4)4w02z3pk2a(k2zk2f)x(x2y2)w02z2p+(a2w021)k(x2y2)z+k2a(x2y2)yw04z2p+2ax(k2zk2f)+2ayw02=0k3xy(x2+y2)4w02z3p+k2axy2w04z2p+ka2xyw02zk2ax2y(k2zk2f)w02z2p+kxyz+axw02+ay(k2zk2f)=0.
{x=0y=0,{x=0y=2afkw02,{x=0y=afk+fka2+4w02kw02,{x=0y=afkfka2+4w02kw02.

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