Abstract

In a 4Pi focusing system radially polarized laser beams can be focused to a spherical focal spot. For many applications, e.g., for moving trapped particles or for scanning a specimen, one would like to change the position of focal spot along the optical axis without moving lenses or laser beams. We demonstrate how this can be achieved by modulating the phase of the input field at the pupil plane of the lens. The required phase modulation function is determined by spherical wave expansion of the plane wave factors in the Richards–Wolf integral.

© 2011 OSA

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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. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  3. Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
    [CrossRef]
  4. P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
    [CrossRef]
  5. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
    [CrossRef]
  6. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
    [CrossRef] [PubMed]
  7. S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007).
    [CrossRef]
  8. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
    [CrossRef] [PubMed]
  9. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
    [CrossRef] [PubMed]
  10. M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
    [CrossRef] [PubMed]
  11. 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(8), 501–505 (2008).
    [CrossRef]
  12. K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
    [CrossRef] [PubMed]
  13. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
    [CrossRef] [PubMed]
  14. W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
    [CrossRef] [PubMed]
  15. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959).
    [CrossRef]
  16. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, 1941).

2010

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

2009

M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
[CrossRef] [PubMed]

W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
[CrossRef] [PubMed]

2008

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(8), 501–505 (2008).
[CrossRef]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
[CrossRef] [PubMed]

2007

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007).
[CrossRef]

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

2006

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
[CrossRef]

2004

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
[CrossRef]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
[CrossRef] [PubMed]

2003

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

2000

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

1959

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Bokor, N.

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
[CrossRef] [PubMed]

Brown, T. G.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Chen, W.

W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
[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(8), 501–505 (2008).
[CrossRef]

Davidson, N.

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
[CrossRef] [PubMed]

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]

Fortin, P.-L.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

Hayashi, T.

M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
[CrossRef] [PubMed]

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
[CrossRef]

Heckenberg, N. R.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
[CrossRef] [PubMed]

Huang, K.

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

Kang, X. L.

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
[CrossRef]

Kawauchi, H.

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

Kozawa, Y.

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

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]

Li, Y. P.

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[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(8), 501–505 (2008).
[CrossRef]

Michihata, M.

M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
[CrossRef] [PubMed]

Nieminen, T. A.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
[CrossRef] [PubMed]

Piché, M.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

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]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Rubinsztein-Dunlop, H.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
[CrossRef] [PubMed]

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
[CrossRef]

Sato, S.

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

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(8), 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(8), 501–505 (2008).
[CrossRef]

Shi, P.

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

Takaya, Y.

M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
[CrossRef] [PubMed]

Varin, C.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[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(8), 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. Roy. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Yan, S.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007).
[CrossRef]

Yao, B.

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007).
[CrossRef]

Yonezawa, K.

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Zhan, Q.

W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
[CrossRef] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

Zhang, X.

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

Appl. Opt.

M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48(32), 6143–6151 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004).
[CrossRef]

J. Phys. At. Mol. Opt. Phys.

P.-L. Fortin, M. Piché, and C. Varin, “Direct-field electron acceleration with ultrafast radially polarized laser beams: scaling laws and optimization,” J. Phys. At. Mol. Opt. Phys. 43(2), 025401 (2010).
[CrossRef]

Nat. Photonics

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(8), 501–505 (2008).
[CrossRef]

Opt. Express

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004).
[CrossRef] [PubMed]

Opt. Lett.

H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, “Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam,” Opt. Lett. 32(13), 1839–1841 (2007).
[CrossRef] [PubMed]

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33(2), 122–124 (2008).
[CrossRef] [PubMed]

K. Huang, P. Shi, X. L. Kang, X. Zhang, and Y. P. Li, “Design of DOE for generating a needle of a strong longitudinally polarized field,” Opt. Lett. 35(7), 965–967 (2010).
[CrossRef] [PubMed]

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4pi focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004).
[CrossRef] [PubMed]

W. Chen and Q. Zhan, “Creating a spherical focal spot with spatially modulated radial polarization in 4Pi microscopy,” Opt. Lett. 34(16), 2444–2446 (2009).
[CrossRef] [PubMed]

Phys. Rev. A

S. Yan and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76(5), 053836 (2007).
[CrossRef]

Y. I. Salamin, “Electron acceleration from rest in vacuum by an axicon Gaussian laser beam,” Phys. Rev. A 73(4), 043402 (2006).
[CrossRef]

Phys. Rev. Lett.

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. Roy. Soc. A

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253(1274), 358–379 (1959).
[CrossRef]

Other

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, New York, 1941).

Supplementary Material (1)

» Media 1: MPG (395 KB)     

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Figures (4)

Fig. 1
Fig. 1

A 4Pi focusing system consists of two confocal high-NA objective lenses illuminated by two counter-propagating radially polarized beams. Their fields at the pupil planes of the lens are denoted by l left (θ) with 0°≤θ≤90°and l right (θ) with 90°≤θ≤180°or, equivalently, l(θ) with 0°≤θ≤180° for the total field.

Fig. 2
Fig. 2

Intensity (a) and phase (b) of the input field l(θ) for generating a spherical intensity spot centered at a point z 0 = 1.5λ. The range of θ is from 0° to 90° for l left (θ) and from 180° to 90° for l right (θ). The insets show the gray scale values of the phase.

Fig. 3
Fig. 3

Line scans of the normalized total intensity of the electric fields of a spherical spot centered at z = 1.5λ. The solid line refers to the axial intensity distribution and the dashed line corresponds to the transversal intensity, overlaid on the axial line scan.

Fig. 4
Fig. 4

(Media 1) Evolution of the intensity of the spherical focal spot in the vicinity of the focus as the parameter z 0 changes continuously from −2λ to 2λ. (a)-(e) are successive frames extracted from the movie at t = 0s, 1s, 2s, 3s and 4s (or z 0 = −2λ, −1λ, 0λ, 1λ and 2λ), respectively.

Equations (11)

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

E ρ ( ρ , z ) = 0 π l ( θ ) X ( θ ) sin 2 θ J 1 ( k ρ sin θ ) e i h ( θ ) z d θ ,
E z ( ρ , z ) = 2 i 0 π l ( θ ) X ( θ ) sin 2 θ J 0 ( k ρ sin θ ) e i h ( θ ) z d θ .
J 0 ( k ρ sin θ ) e i h ( θ ) z = n = 0 i n ( 2 n + 1 ) P n ( cos θ ) P n ( cos α ) j n ( k R ) ,
E z n = 0 A n j n ( k R ) P n ( cos α ) ,
A n = i n ( 2 n + 1 ) 0 π l ( θ ) P n ( cos θ ) sin 2 θ d θ .
j 0 ( k | R u 3 z 0 | ) = n = 0 ( 2 n + 1 ) P n ( cos α ) j n ( k z 0 ) j n ( k R ) .
A n = ( 1 ) n ( 2 n + 1 ) j n ( k z 0 ) ,
i n 2 A n = 2 n + 1 2 1 1 ( 1 x 2 ) 1 / 2 g ( x ) P n ( x ) d x .
g ( x ) = ( 1 x 2 ) 1 / 2 k = 0 i n 2 A n P n ( x )
l ( θ ) = A ( θ ) l 0 ( θ )
A ( θ ) = n = 0 i n 2 A n P n ( cos θ ) ,

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