Abstract

We demonstrate that a Laguerre-Gauss spectrum of plane waves distribution optimize the variance of the spectrum-bandwidth product. In the space domain, the axial Ez (TM01) and the azimuthal Eϕ (TE01) have also a Laguerre-Gauss profile that describes correctly some experimental published and calculated results in the focal plane.

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References

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  1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
    [CrossRef] [PubMed]
  2. D. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001).
    [CrossRef] [PubMed]
  3. H. Dehez, M. Piché, and Y. De Koninck, “Enhanced resolution in two-photon imaging using a TM01 laser beam at a dielectric interface,” Opt. Lett.34(23), 3601–3603 (2009).
    [CrossRef] [PubMed]
  4. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
    [CrossRef]
  5. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express10(7), 324–331 (2002).
    [PubMed]
  6. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
    [CrossRef] [PubMed]
  7. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  8. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A253(1274), 358–379 (1959).
    [CrossRef]
  9. B. Jia, X. Gan, and M. Gu, “Direct measurement of a radially polarized focused evanescent feield facilitated by a single LCD,” Opt. Express13(18), 6821–6827 (2005).
    [CrossRef]
  10. B. Hao and J. Leger, “Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam,” Opt. Express15(6), 3550–3556 (2007).
    [CrossRef] [PubMed]
  11. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, Sixth Edition (Academic Press, 2000) Chap. 6.5.
  12. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), chap. 17.6.
  13. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A24(6), 1793–1798 (2007).
    [CrossRef] [PubMed]
  14. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A15(10), 2705–2711 (1998).
    [CrossRef]
  15. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.
  16. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett.31(6), 820–822 (2006).
    [CrossRef] [PubMed]
  17. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000).
    [CrossRef]

2009 (1)

2007 (2)

2006 (1)

2005 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

D. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9(10), 490–497 (2001).
[CrossRef] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

2000 (3)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000).
[CrossRef]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
[CrossRef] [PubMed]

1998 (1)

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, Ser. A253(1274), 358–379 (1959).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Biss, D.

Brown, T. G.

De Koninck, Y.

Dehez, H.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Gan, X.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Gu, M.

Hao, B.

Jia, B.

Kozawa, Y.

Leger, J.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Nesterov, A. V.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000).
[CrossRef]

Niziev, V. G.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

Piché, M.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[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. London, Ser. A253(1274), 358–379 (1959).
[CrossRef]

Sato, S.

Tovar, A. A.

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, Ser. A253(1274), 358–379 (1959).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000).
[CrossRef] [PubMed]

Zhan, Q.

J. Opt. Soc. Am. A (2)

J. Phys. D Appl. Phys. (1)

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000).
[CrossRef]

Opt. Commun. (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179(1-6), 1–7 (2000).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes Probed by Single Molecules,” Phys. Rev. Lett.86(23), 5251–5254 (2001).
[CrossRef] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper Focus for a Radially Polarized Light Beam,” Phys. Rev. Lett.91(23), 233901 (2003).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. 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, Ser. A253(1274), 358–379 (1959).
[CrossRef]

Other (3)

I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products, Sixth Edition (Academic Press, 2000) Chap. 6.5.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), chap. 17.6.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publication Inc., New York, 1964), 22.12.7.

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Figures (2)

Fig. 1
Fig. 1

Normalized Ez from Eq. (18) (a) and normalized Eϕ fields from Eq. (19) (b).

Fig. 2
Fig. 2

a: Normalized Ez from Eq. (23). b: Normalized Eϕ fields from Eq. (23). For both graphs, a = −2/3 and f 2 = 0.113.

Equations (25)

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E z or H z =[ 0 A(τ) e ikz 1 τ 2 J o (krτ)τdτ ]
TM{ E r =i 0 A(τ) 1 τ 2 e ikz 1 τ 2 J 1 (krτ)dτ H φ = i η 0 A(τ) e ikz 1 τ 2 J 1 (krτ)dτ
TE{ H r =i 0 A(τ) 1 τ 2 e ikz 1 τ 2 J 1 (krτ)dτ E ϕ =iη 0 A(τ) e ikz 1 τ 2 J 1 (krτ)dτ
σ ^ 0 2 = k 2 0 A 2 τ 2 dτ τ 0 A 2 dτ τ
σ 0 2 = 0 | E ϕ | r 2 (rdr) 0 | E ϕ | 2 (rdr)
0 J n (ar) J n (br)rdr = 1 a δ(ab)
k 2 σ 0 2 = 0 ( dA dτ ) 2 dτ τ 0 A 2 dτ τ
[ 0 [ ( dA dτ ) 2 dτ τ + Λ 0 0 A 2 dτ τ + Λ 1 0 A 2 τ 2 dτ τ ] ]=0
d dτ ( 1 τ dA dτ ) Λ 0 A τ Λ 1 Aτ=0
A m (τ)= A 0 τ 2 e τ 2 2 f 2 L m (1) ( τ 2 f 2 )withm=0,1,2,... Λ 0 f 2 =4(m+1), Λ 1 = 1 f 4
σ ^ 0 2 = k 2 0 τ 2 A 2 τdτ 0 A 2 τdτ =2 f 2 k 2
0 e τ 2 f 2 L m (1) ( τ 2 f 2 ) L n (1) ( τ 2 f 2 ) τ 3 dτ=0 nm = f 4 2 (n+1) n=m
x L m (1) (x)=(m+1)[ L m (x) L m+1 (x)]
E z = c 0 (1) m (m+1)exp[ (kfr) 2 2 ][ L m ( (kfr) 2 )+ L m+1 ( (kfr) 2 )]
E ϕ = c 1 (1) m exp[ (kfr) 2 2 ](kfr) L m (1) ( (kfr) 2 )
E z (r,0) E 0 =exp[ (kfr) 2 2 ]( 1 (kfr) 2 2 )
E ϕ (r,0) E 0 = 2 (kfr)·exp[ 1 2 ]·exp[ (kfr) 2 2 ]
E z (r,0) E 0 =exp[ r 2 2 σ 0 2 ][ 1 r 2 2 σ 0 2 ]
E ϕ (r,0) E 0 = 2 e 1 2 ( r σ 0 )exp[ r 2 2 σ 0 2 ]
SpotSiz e FWHM = 0.0263 λ 2 f 2 1.038 σ 0 2
R=2 f 2
E z (r,0) E 0 =exp[ r 2 2 σ 0 2 ][1+ 6a1 24a ( r σ 0 ) 2 a 24a ( r σ 0 ) 4 ]
E ϕ (r,0) E 0 =( r σ 0 )exp[ r 2 2 σ 0 ][1+ a 12a ( r σ 0 ) 2 ]
exp[ 1 2 f 2 (1 τ 2 )ikz 1 τ 2 ]= n=0 ( 1 2 f 2 ) n 2 n! H n ( kfz 2 ) ( 1 τ 2 ) n 2
0 π/2 J μ (xsint) sin μ+1 (t) cos 2υ+1 dt= 2 υ Γ(1+υ) x υ+1 J μ+υ+1 (x)

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