Abstract

We present a novel family of paraxial optical beams having a confluent hypergeometric transverse profile, which we name hypergeometric Gauss modes of type-II (HyGG-II). These modes are eigenmodes of the photon orbital angular momentum and they have the lowest beam divergence in the waist of HyGG-II among all known finite power paraxial modes families. We propose to exploit this feature of HyGG-II modes for generating, after suitable focusing, a “light needle” having record properties in terms of size and aspect ratio, possibly useful for near-field optics applications.

© 2008 Optical Society of America

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  1. K. Dholakia, "Optics: Against the spread of the light," Nature 45, 413 (2008).
    [CrossRef]
  2. 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]
  3. G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
    [CrossRef]
  4. G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).
  5. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
    [CrossRef]
  6. J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  7. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  8. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  9. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, "Image processing with the radial Hilbert transform: theory and experiments," Opt. Lett. 25, 99-101 (2000).
    [CrossRef]
  10. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
    [CrossRef]
  11. B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
    [CrossRef]
  12. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindricalvector beams," Opt. Express 7, 77-87 (2000).
    [CrossRef]
  13. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  14. Q. Zhan, "Properties of circularly polarized vortex beams," Opt. Lett. 31, 867-869 (2006).
    [CrossRef] [PubMed]
  15. M. A. Bandres and J. C. Gutiérrez-Vega, "Circular beams," Opt. Lett. 33, 177-179 (2008).
    [PubMed]
  16. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
    [CrossRef] [PubMed]
  17. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian modes," Opt. Lett. 32, 3053-3055 (2007).
    [CrossRef] [PubMed]
  18. H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
    [CrossRef]

2008 (3)

K. Dholakia, "Optics: Against the spread of the light," Nature 45, 413 (2008).
[CrossRef]

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]

M. A. Bandres and J. C. Gutiérrez-Vega, "Circular beams," Opt. Lett. 33, 177-179 (2008).
[PubMed]

2007 (5)

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
[CrossRef]

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

2006 (2)

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Q. Zhan, "Properties of circularly polarized vortex beams," Opt. Lett. 31, 867-869 (2006).
[CrossRef] [PubMed]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

1987 (2)

J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

1959 (1)

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Bandres, M. A.

Brown, T. G.

Campos, J.

Chiu, D. T.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

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]

Chong, T.

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Cottrell, D. M.

Davis, J. A.

Dholakia, K.

K. Dholakia, "Optics: Against the spread of the light," Nature 45, 413 (2008).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Fong, C.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Gutiérrez-Vega, J. C.

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Jeffries, G. D. M.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Jr, J. J. M.

J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Karimi, E.

Khonina, S. N.

Kotlyar, V. V.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

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]

Marrucci, L.

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

McNamara, D. E.

Miao, X. S.

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
[CrossRef]

Piccirillo, B.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Richards, B.

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Santamato, E.

Shelby, J. P.

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

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]

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Skidanov, R. V.

Soifer, V. A.

Tan, W.

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
[CrossRef]

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]

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Wolf, E.

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

Youngworth, K. S.

Yuan, G.

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Zhan, Q.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

Zito, G.

Appl. Phys. Lett. (1)

H. Wang, L. Shi, G. Yuan, X. S. Miao,W. Tan, and T. Chong, "Subwavelength and super-resolution nondiffraction beam," Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

J. Mod. Opt. (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms" J. Mod. Opt. 42, 217-223 (1995).
[CrossRef]

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

Nano. Lett. (1)

G. D. M. Jeffries, J. S. Edgar, Y. Zhao, J. P. Shelby, C. Fong, and D. T. Chiu,"Using polarization-shaped optical vortex traps for single-cell nanosurgery," Nano. Lett.  7, 415-420 (2007).

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]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305-310 (2007).
[CrossRef]

Nature (1)

K. Dholakia, "Optics: Against the spread of the light," Nature 45, 413 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. Lett. (4)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, "Nondiffracting bulk-acoustic X waves in crystals," Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, "Spin-to-orbital angular momentum conversion in a strongly focused optical beam," Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

J. Durnin, J. J. M. Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Proc. R. Soc. London A (1)

B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system," Proc. R. Soc. London A 253, 358-379 (1959).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Variation of maximum intensity radius during the propagation for LG0,1, BG1, HyGG-II-1,1. (b) Scheme of the focusing system.

Fig. 2.
Fig. 2.

Intensity profile of the longitudinal and radial field components at the focal point of a high numerical aperture lens (a) and a system of high numerical aperture lens and binary phase mask (b). The black, gray and dashed lines are radial, longitudinal and total intensity, respectively.

Fig. 3.
Fig. 3.

Density plots of intensity distribution for (a) radial, (b) longitudinal components. (c) and (d) show the total intensity distribution for the HyGG-II-1,1 and the BG1 beams, respectively.

Equations (14)

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

( ρ , ρ + 1 ρ ρ + 1 ρ 2 ϕ , ϕ + 4 i ζ ) ψ ( ρ , ϕ ; ζ ) = 0 ,
HyGG-II pm u pm ( ρ , ϕ ; ζ ) = C pm ( 1 1 + i ζ ) p 2 + m + 1
× ρ m e ρ 2 ( 1 + i ζ ) e i m ϕ 1 F 1 ( p 2 , m + 1 ; ρ 2 ( 1 + i ζ ) ) ,
u pm ( ρ , ϕ , 0 ) = e ρ 2 + i m ϕ ρ m 1 F 1 ( p 2 , m + 1 ; ρ 2 ) .
u pm ( ρ , ϕ , 0 ) e ρ 2 1 + i ζ ρ p + m + ρ ( p + m + 2 ) Γ ( p 2 ) .
u m , m ( ρ , ϕ ; ζ ) = 1 2 ( 1 1 + i ζ ) 3 2 e i m ϕ ρ e ρ 2 2 ( 1 + i ζ )
× [ I m 1 2 ( ρ 2 2 ( 1 + i ζ ) ) I m + 1 2 ( ρ 2 2 ( 1 + i ξ ) ) ]
u 2 n , m ( ρ , ϕ ; ζ ) = 2 2 n + m + 1 π Γ ( 2 n + m + 1 ) Γ ( n + 1 ) ( 1 1 + i ζ ) n + m + 1
× e ρ 2 1 + i ζ ρ m e i m ϕ L n m ( ρ 2 1 + i ζ ) .
E ˜ ( r ˜ ) = i λ Ω a ˜ ( θ , φ ) e 2 π i ( z cos θ + ρ sin θ cos ( φ θ ) ) d Ω ,
E ρ ( r ˜ ) = f λ 0 α cos θ sin ( 2 θ ) l ( θ ) J 1 ( 2 π ρ sin θ ) e i ( 2 π z cos θ ) d θ
E z ( r ˜ ) = 2 i f λ 0 α cos θ sin 2 θ l ( θ ) J 0 ( 2 π ρ sin θ ) e i ( 2 π z cos θ ) d θ ,
l ( θ ) = e β 2 ( sin θ sin α ) 2 ( β sin θ sin α ) 1 F 1 ( 1 2 , 2 ; β 2 ( sin θ sin α ) 2 ) ,
η = 0 r 0 E z ( r , 0 ) 2 r dr 0 r 0 E ρ ( r , 0 ) 2 r dr + 0 r 0 E z ( r , 0 ) 2 r dr

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