Abstract

We report a new method to create high purity longitudinally polarized field with extremely long depth of focus in the focal volume of a high numerical aperture (NA) objective lens. Through reversing the radiated field from an electric dipole array situated near the focus of the high-NA lens, the required incident field distribution in the pupil plane for the creation of an ultra-long optical needle field can be found. Numerical examples demonstrate that an optical needle field with a depth of focus up to 8λ is obtainable. Throughout the depth of focus, this engineered focal field maintains a diffraction limited transverse spot size (<0.43λ) with high longitudinal polarization purity. From the calculated pupil plane distribution, a simplified discrete complex pupil filter can be designed and significant improvements over the previously reported complex filters are clearly demonstrated.

© 2010 OSA

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  1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon 1(1), 1–57 (2009).
    [CrossRef]
  2. L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
    [CrossRef] [PubMed]
  3. R. D. Romea and W. D. Kimura, “Modeling of inverse Ĉerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
    [CrossRef] [PubMed]
  4. Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
    [CrossRef]
  5. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000).
    [CrossRef] [PubMed]
  6. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
    [PubMed]
  7. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
    [CrossRef]
  8. H. Kang, B. H. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18(10), 10813–10821 (2010).
    [CrossRef] [PubMed]
  9. I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
    [CrossRef]
  10. A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
    [CrossRef] [PubMed]
  11. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
    [CrossRef]
  12. H. F. Wang, L. P. 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]
  13. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008).
    [CrossRef] [PubMed]
  14. 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]
  15. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
    [CrossRef] [PubMed]
  16. S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. Soc. Am. A 10, 125103 (2008).
  17. 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]
  18. S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
    [CrossRef]
  19. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
    [CrossRef]
  20. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
    [CrossRef] [PubMed]
  21. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15558–15570 (2009).
    [CrossRef] [PubMed]
  22. Y. S. Xu, J. Singh, C. J. R. Sheppard, and N. G. Chen, “Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter,” Opt. Express 15(10), 6409–6413 (2007).
    [CrossRef] [PubMed]
  23. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  24. A. Balanis, Antenna Theory: Analysis and Design, 3rd Edition, (Wiley-Interscience, 2005).
  25. M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006).
    [CrossRef] [PubMed]
  26. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007).
    [CrossRef] [PubMed]

2010

2009

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15558–15570 (2009).
[CrossRef] [PubMed]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon 1(1), 1–57 (2009).
[CrossRef]

2008

S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. Soc. Am. A 10, 125103 (2008).

H. F. Wang, L. P. 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]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008).
[CrossRef] [PubMed]

2007

2006

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[CrossRef] [PubMed]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006).
[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]

2002

2001

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

1998

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

1995

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

1994

S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
[CrossRef]

1990

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
[CrossRef] [PubMed]

R. D. Romea and W. D. Kimura, “Modeling of inverse Ĉerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[CrossRef] [PubMed]

Abouraddy, A. F.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[CrossRef] [PubMed]

Banzer, P.

Beversluis, M. R.

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006).
[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]

Bouchal, Z.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

Brown, T. G.

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. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Chen, N. G.

Chen, W.

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
[CrossRef]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

Chong, C. T.

H. F. Wang, L. P. 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]

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
[CrossRef] [PubMed]

Cižmár, T.

Dholakia, K.

Ding, J.

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]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Gu, M.

Guo, C. S.

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
[CrossRef] [PubMed]

Huang, K.

Iglesias, I.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[CrossRef]

Jia, B. H.

Kang, H.

Kang, X. L.

Karimi, E.

Kimura, W. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Ĉerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[CrossRef] [PubMed]

Kitamura, K.

Leger, J. R.

Leuchs, G.

Li, Y. P.

Lukyanchuk, B.

H. F. Wang, L. P. 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]

Marrucci, L.

Ni, W. J.

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Noda, S.

Novotny, L.

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006).
[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]

Olivik, M.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

Peschel, U.

Piccirillo, B.

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
[CrossRef] [PubMed]

Quabis, S.

Romea, R. D.

R. D. Romea and W. D. Kimura, “Modeling of inverse Ĉerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[CrossRef] [PubMed]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Sakai, K.

Santamato, E.

Sheppard, C.

H. F. Wang, L. P. 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]

Sheppard, C. J. R.

Shi, L. P.

H. F. Wang, L. P. 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.

Shimoda, K.

S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
[CrossRef]

Singh, J.

Stranick, S. J.

Sugihara, R.

S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
[CrossRef]

Takeuchi, S.

S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
[CrossRef]

Toussaint, K. C.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[CrossRef] [PubMed]

Vohnsen, B.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[CrossRef]

Wang, H. F.

H. F. Wang, L. P. 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]

Wang, H. T.

Wang, X. L.

Xu, Y. S.

Yang, S.

S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. Soc. Am. A 10, 125103 (2008).

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. Express 7(2), 77–87 (2000).
[CrossRef] [PubMed]

Zhan, Q.

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
[CrossRef]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon 1(1), 1–57 (2009).
[CrossRef]

S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. Soc. Am. A 10, 125103 (2008).

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[PubMed]

Zhang, X.

Adv. Opt. Photon

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon 1(1), 1–57 (2009).
[CrossRef]

J. Mod. Opt.

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995).
[CrossRef]

J. Opt.

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010).
[CrossRef]

J. Opt. Soc. Am. A

S. Yang and Q. Zhan, “Third-harmonic generation microscopy with tightly focused radial polarization,” J. Opt. Soc. Am. A 10, 125103 (2008).

J. Phys. Soc. Jpn.

S. Takeuchi, R. Sugihara, and K. Shimoda, “Electron acceleration by longitudinal electric field of a Gaussian laser beam,” J. Phys. Soc. Jpn. 63(3), 1186–1193 (1994).
[CrossRef]

Nat. Photonics

H. F. Wang, L. P. 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]

Nature

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998).
[CrossRef]

Opt. Commun.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265(2), 411–417 (2006).
[CrossRef]

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun. 271(1), 40–47 (2007).
[CrossRef]

Opt. Express

H. Kang, B. H. Jia, and M. Gu, “Polarization characterization in the focal volume of high numerical aperture objectives,” Opt. Express 18(10), 10813–10821 (2010).
[CrossRef] [PubMed]

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 and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10(7), 324–331 (2002).
[PubMed]

P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18(10), 10905–10923 (2010).
[CrossRef] [PubMed]

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17(18), 15558–15570 (2009).
[CrossRef] [PubMed]

Y. S. Xu, J. Singh, C. J. R. Sheppard, and N. G. Chen, “Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter,” Opt. Express 15(10), 6409–6413 (2007).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Improved focusing with hypergeometric-gaussian type-II optical modes,” Opt. Express 16(25), 21069–21075 (2008).
[CrossRef] [PubMed]

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14(7), 2650–2656 (2006).
[CrossRef] [PubMed]

K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

L. Cicchitelli, H. Hora, and R. Postle, “Longitudinal field components for laser beams in vacuum,” Phys. Rev. A 41(7), 3727–3732 (1990).
[CrossRef] [PubMed]

Phys. Rev. D Part. Fields

R. D. Romea and W. D. Kimura, “Modeling of inverse Ĉerenkov laser acceleration with axicon laser-beam focusing,” Phys. Rev. D Part. Fields 42(5), 1807–1818 (1990).
[CrossRef] [PubMed]

Phys. Rev. Lett.

A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006).
[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]

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]

Other

A. Balanis, Antenna Theory: Analysis and Design, 3rd Edition, (Wiley-Interscience, 2005).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the method of pupil plane field synthesis in order to achieve specific focal field characteristics through the reversing of the radiation pattern of a dipole array. (a) System layout with dipole array oscillating along optical axis in the focal volume and the coordinates used in the calculation; and (b) Far-field geometry and conceptual diagram of the dipole array with 2N dipole elements.

Fig. 2
Fig. 2

Total intensity distribution in the transverse rz plane and axial intensity distribution in the focal volume for the obtained optical needle field with extended DOF for (a) (c) N = 2; and (b) (d) N = 3. The corresponding required incident field in the pupil plane P i are illustrated for (e) N = 2 and (f) N = 3.

Fig. 3
Fig. 3

Structure and parameters of discrete complex pupil filter originating from Fig. 2(e).

Fig. 4
Fig. 4

Focal intensity distribution using the discrete complex pupil filter. (a) Total intensity distribution in the transverse rz plane; (b) Axial total intensity distribution; (c) Comparison of beam quality η in the main DOF for different filter designs; and (d) Transverse intensity distribution when z = 0,1λ, 2λ for the discrete complex pupil filter.

Tables (1)

Tables Icon

Table 1 Parameters of dipole array and obtained intensity distributions (NA = 0.95)

Equations (4)

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

E 0 ( θ ) = E D A ( θ ) a θ = C sin θ A F N a θ ,
A F N = n = 1 N A n [ e j ( k d n cos θ + β n ) / 2 + e j ( k d n cos θ + β n ) / 2 ] ,
E i ( ρ i , θ ) = E D A ( θ ) ( cos φ x i + sin φ y i ) / cos θ ,
E ( r , ψ , z ) = i λ 0 θ max 0 2 π E D A ( θ ) ( cos θ cos φ i + cos θ sin φ j + sin θ k ) exp [ i k r sin θ cos ( φ ψ ) i k z cos θ ] sin θ d θ d φ ,

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