Abstract

We consider the formation of a small focal spot, which has been reported in [J. Opt. Soc. Am. A 24, 1793 (2007) [CrossRef]  ], by separating a higher-order radially polarized Laguerre–Gaussian beam into two parts: a Bessel-like multi-ring part and an annular part. The latter forms a dominant small spot of longitudinal electric field component near the focus, while the former mainly contributes to a weak annular pattern of radial component.

© 2012 Optical Society of America

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References

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2012 (1)

2011 (2)

2010 (3)

2008 (2)

G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567–4581 (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]

2007 (2)

2006 (1)

2004 (1)

2003 (2)

C. C. Sun and C. K. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28, 99–101 (2003).
[CrossRef]

M. Alfaro, J. J. Moreno-Balcázar, and M. L. Rezola, “Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II,” J. Approx. Theory 122, 79–96 (2003).
[CrossRef]

2001 (2)

M. A. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre–Gauss and Bessel–Gauss beams,” J. Opt. Soc. Am. A 18, 177–184 (2001).
[CrossRef]

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

2000 (1)

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

1998 (1)

1997 (1)

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

1995 (1)

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[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. A 253, 358–379 (1959).
[CrossRef]

1872 (1)

J. W. Strutt, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).

Alfaro, M.

M. Alfaro, J. J. Moreno-Balcázar, and M. L. Rezola, “Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II,” J. Approx. Theory 122, 79–96 (2003).
[CrossRef]

April, A.

Bajer, J.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

Borghi, R.

Bouchal, Z.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

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]

Choudhury, A.

Courjon, D.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[CrossRef]

Dehez, H.

Dorn, R.

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Everler, M.

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Flores-Pérez, A.

Glöckl, O.

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Grosjean, T.

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[CrossRef]

Hashimoto, N.

Helseth, L. E.

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

Hernández-Hernández, J.

Hibi, T.

Horák, R.

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

Horanai, H.

Huang, W.

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[CrossRef]

Ito, A.

Jáuregui, R.

Kitamura, K.

Kozawa, Y.

Kurihara, M.

Lerman, G. M.

Leuchs, G.

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Levy, U.

Liu, C. K.

Lü, B.

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[CrossRef]

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]

Moreno-Balcázar, J. J.

M. Alfaro, J. J. Moreno-Balcázar, and M. L. Rezola, “Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II,” J. Approx. Theory 122, 79–96 (2003).
[CrossRef]

Nemoto, T.

Noda, S.

Piché, M.

Porras, M. A.

Quabis, S.

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Rezola, M. L.

M. Alfaro, J. J. Moreno-Balcázar, and M. L. Rezola, “Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II,” J. Approx. Theory 122, 79–96 (2003).
[CrossRef]

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. A 253, 358–379 (1959).
[CrossRef]

Sakai, K.

Santarsiero, M.

Sato, A.

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]

Sheppard, C. J. R.

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]

Strutt, J. W.

J. W. Strutt, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).

Sun, C. C.

Tovar, A. A.

Volke-Sepúlveda, K.

Vyas, S.

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]

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. A 253, 358–379 (1959).
[CrossRef]

Yokoyama, H.

Zhang, B.

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[CrossRef]

Appl. Opt. (1)

J. Approx. Theory (1)

M. Alfaro, J. J. Moreno-Balcázar, and M. L. Rezola, “Laguerre–Sobolev orthogonal polynomials: asymptotics for coherent pairs of type II,” J. Approx. Theory 122, 79–96 (2003).
[CrossRef]

J. Opt. Soc. Am. A (6)

Mon. Not. R. Astron. Soc. (1)

J. W. Strutt, “On the diffraction of object-glasses,” Mon. Not. R. Astron. Soc. 33, 59–63 (1872).

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. (5)

S. Quabis, R. Dorn, M. Everler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161–172 (2001).
[CrossRef]

T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272, 314–319 (2007).
[CrossRef]

R. Horák, Z. Bouchal, and J. Bajer, “Nondiffracting stationary electromagnetic field,” Opt. Commun. 133, 315–327 (1997).
[CrossRef]

B. Lü, W. Huang, and B. Zhang, “Fraunhofer diffraction of a Bessel beam focused by an aperture lens,” Opt. Commun. 119, 6–12 (1995).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Proc. R. Soc. 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. A 253, 358–379 (1959).
[CrossRef]

Other (1)

A. Erdélyi, ed., Tables of Integral Transforms (McGraw-Hill, 1954), Vol. 1.

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Figures (4)

Fig. 1.
Fig. 1.

Intensity profile of a vector LG5,1 beam (solid curve) with the result of the fitting to the squared Bessel function (J1) (dashed curve).

Fig. 2.
Fig. 2.

Calculated intensity distributions at the focal plane for vector LG5,1 beams with different focusing conditions. The intensity distributions at the pupil plane are shown in leftmost column. The pupil of the focusing lens is depicted by a dashed circle in each figure. The intensity distributions at the focal plane are normalized to the maximum value of the total pattern for the outermost ring of a vector LG5,1 beam. The dimension at the focal plane is 10λ×10λ.

Fig. 3.
Fig. 3.

Intensity profile at the focal plane of the longitudinal component for the focusing of a vector LG5,1 beam. The dashed curve is the result of the fitting to the squared Bessel function (J0).

Fig. 4.
Fig. 4.

Intensity profiles at the focal plane generated by (a) the whole, (b) the inner five rings, and (c) the outermost ring of a vector LG5,1 beam. The total (black solid curve), the radial (blue dotted curve), and the longitudinal (red dashed curve) components are represented in each figure.

Equations (5)

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Ep,m(s)(r,ϕ,z)Yp,m(r,z)exp(imϕ),
Yp,m(r,z)=ω0ω(z)[2rω(z)]|m|Lp|m|[2r2ω2(z)]exp[ikz+i(2p+|m|+1)ψ(z)]exp{r2[1ω2(z)ik2R(z)]},
Ep,m(v)(r,ϕ,z)Yp,m(r,z){sin(nϕ)ρ+cos(nϕ)φcos(nϕ)ρ±sin(nϕ)φ},
Lpm(x)exp(x2)(px)m/2Jm(2px).
Yp,m(r,ϕ,z)ω0ω(z)p|m|/2J|m|[22prω(z)]exp[ikzi(2p+|m|+1)ψ(z)]exp[ikr22R(z)].

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