Abstract

We propose and generate a new radial-variant vector field (RV-VF) with a distribution of states of polarization described by the square of the radius and exploit its focusing property. Theoretically, we present the analytical expressions for the three-dimensional electric field of the vector field focused by a thin lens under the nonparaxial and paraxial approximations based on the vectorial Rayleigh–Sommerfeld formulas. Numerical simulations indicate that this focused field exhibits bifocusing spots along the optical axis. The underlying mechanism for generating the bifocusing property is analyzed in detail. We give the analytical formula for the interval between two foci. Experimentally, we generate the RV-VFs with alterable topological charge and demonstrate that the interval between two foci is controllable by tuning the radial topological charge. This particular focal field has specific applications for biparticle trapping, manipulating, alignment, transportation, and accelerating along the optical axis.

© 2014 Optical Society of America

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References

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  1. G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).
  2. X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
    [CrossRef]
  3. C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
    [CrossRef]
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    [CrossRef]
  7. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
    [CrossRef]
  8. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
    [CrossRef]
  9. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34, 2444–2446 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (3)

2011 (3)

2010 (2)

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, “Nonparaxial propagation of a Gaussian optical vortex with initial radial polarization,” J. Opt. Soc. Am. A 27, 372–380 (2010).
[CrossRef]

2009 (1)

2008 (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

2006 (2)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006).
[CrossRef]

2005 (2)

2000 (1)

Arlt, J.

Bautista, G.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Biener, G.

Chen, H.

Chen, J.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Chen, W.

W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34, 2444–2446 (2009).
[CrossRef]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

Chen, Z.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Chremmos, I.

Christodoulides, D. N.

Cui, Y.

Ding, J. P.

H. Chen, J. Hao, B. F. Zhang, J. Xu, J. P. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Efremidis, N. K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), Sect. 5.

Gu, B.

Guo, C. S.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Hao, J.

Hasman, E.

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Huttunen, M. J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Kauranen, M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Kleiner, V.

Kontio, J. M.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Kotlyar, V. V.

Kovalev, A. A.

Kozawa, Y.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Li, P.

Li, Y. N.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Li, Y. P.

Liu, S.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

Mäkitalo, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Niv, A.

Padgett, M. J.

Peng, T.

Prakash, J.

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Sato, S.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Simonen, J.

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Wang, H. T.

H. Chen, J. Hao, B. F. Zhang, J. Xu, J. P. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011).
[CrossRef]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Wang, X. L.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Xu, J.

Zhan, Q.

Zhang, B. F.

Zhang, P.

Zhang, Y.

Zhao, J.

Zhao, Y.

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

Nano Lett. (1)

G. Bautista, M. J. Huttunen, J. Mäkitalo, J. M. Kontio, J. Simonen, and M. Kauranen, “Second-harmonic generation imaging of metal nano-objects with cylindrical vector beams,” Nano Lett. 12, 3207–3212 (2012).

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[CrossRef]

Opt. Commun. (1)

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (7)

Phys. Rev. Lett. (2)

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, “Revealing local field structure of focused ultrafast pulses,” Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Other (2)

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1966).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005), Sect. 5.

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Figures (6)

Fig. 1.
Fig. 1.

Nonparaxial intensity distributions of a RV-VF in the xz plane when n=1, φ0=0, r0=1.9mm, f=2mm, and y=0. Dotted and dashed lines in (d) are positions of the true foci and the lens’ geometrical focus, respectively.

Fig. 2.
Fig. 2.

Interval between bifoci Δzf verse n for the nonparaxial focused vector fields when r0=1.9mm and f=2mm. The circles are the simulated results while the solid line is obtained by Eq. (9).

Fig. 3.
Fig. 3.

Experimental arrangement for measuring the bifoci of the focused vector fields. The dashed-line box shows the generating unit of the RV-VFs.

Fig. 4.
Fig. 4.

(a)–(c) Measured and (d)–(f) simulated intensity patterns with a horizontal polarizer of the RV-VFs with φ0=0 for n=0.5, 1.0, and 1.5.

Fig. 5.
Fig. 5.

Paraxial intensity distributions of a RV-VF when n=1, φ0=0, r0=1.9mm, and f=150mm. (a) Simulated longitudinal intensity pattern of the vector field through the focus. (b)–(d) Simulated and (e)–(g) measured transverse intensity patterns of the vector field at the positions of the true foci and the lens’ geometrical focus.

Fig. 6.
Fig. 6.

Value of Δzf versus n for the paraxial focused vector fields when f=150mm and r0=1.9mm. The circles are the measured data while the solid line is obtained by Eq. (9). The error bar is ±15%.

Equations (16)

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E⃗(r,ϕ,0)=Er(r,ϕ,0)e^r+Eϕ(r,ϕ,0)e^ϕ,
Er(r,ϕ,0)=A(r)P(r)cos(2nπr2r02ϕ+φ0),
Eϕ(r,ϕ,0)=A(r)P(r)sin(2nπr2r02ϕ+φ0).
Er(ρ,θ,z)=12π[Er(r,ϕ,0)cosΦEϕ(r,ϕ,0)sinΦ]Uzrdrdϕ,
Eϕ(ρ,θ,z)=12π[Er(r,ϕ,0)sinΦ+Eϕ(r,ϕ,0)cosΦ]Uzrdrdϕ,
Ez(ρ,θ,z)=12π{Er(r,ϕ,0)[rρcosΦ]+Eϕ(r,ϕ,0)ρsinΦ}1RURrdrdϕ,
Rz2+ρ2+r22z2+ρ2ρrcos(ϕθ)z2+ρ2.
Er(ρ,θ,z)=ikzE0ξ2eikξ0r0cos(2nπr02r2+Ψ)×exp(ikr22ξikr22f)J0(βr)rdr,
Eϕ(ρ,θ,z)=ikzE0ξ2eikξ0r0sin(2nπr02r2+Ψ)×exp(ikr22ξikr22f)J0(βr)rdr,
Ez(ρ,θ,z)=ikE0ξ2eikξ0r0[ρJ0(βr)+irJ1(βr)]×cos(2nπr02r2+Ψ)exp(ikr22ξikr22f)rdr,
Er(ρ,θ,z)=ikzE0ξ2eikξl=1Λ(l)[Ω1(l)+Ω2(l)],
Eϕ(ρ,θ,z)=kzE0ξ2eikξl=1Λ(l)[Ω1(l)Ω2(l)],
Ez(ρ,θ,z)=ikE0ξ2eikξl=1Λ(l){ρ[Ω1(l)+Ω2(l)]+iβ2l[Ω1(l+1)+Ω2(l+1)]},
t(r)exp[i(k2f±2nπr02)r2].
f=f(1±2nfλr02)1f(12nfλr02).
Δzf=4nf2λr02.

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